MOLECULAR MGTION iN
CONDENSED PHASES. r
NUCLEAR MAGNETIC RELAXATION 7

AND RAMAN LINESHAPE STUDIES OF "

SEVERAL SMALL MOLECULES

Eééagefiaiiea fer the 533333 33 3 3 3-

E‘Eififiéé‘éfafl SHE WWEQSEW
3.3%} ALLEN WEéGHT
i 9 7 4

 

 

This is to certify that the

thesis entitled
MOLECULAR MOTION IN CONDENSED PHASES.
NUCLEAR MAGNETIC RELAXATION AND RAMAN LINESHAPE

STUDIES OF SEVERAL SMALL MOLECULES
presented by

David Allen Wright

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Chemistry

 

7Wfl/Q5; ; L 077 L30

Major profess

Date MW ‘2’) 36374

0-7639

 

     
 

3' 1
BINDING IY ‘7
I

 

v t. Y INC. A
“2555? moms m
4MAM'ELHII ':

 

 

 

 

it}. I..! ... : ir I 11.43.1114! 1:13‘41 - ‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: ‘0
U a‘ ".‘ a
E: .:P '1‘; Jib

tQ-ln‘q ‘ i,
I .-
nu-l'fi '3 t“
’4'...

vr1

?\ ‘_
"'3'. ‘3 | “ES

3 3' PAT}
h ." Vet tE
I
I‘-
u“ 9‘ ‘
....‘:"u:le
.|
0 ...
6 VI; l-“a F"
S h 5‘- "I.
1?
i . ~91;.:
... ‘.~
H’
. ‘ -
u' .3).- 51"1‘:
I‘ .
rt”... “
.|'d.:F

ABSTRACT

MOLECULAR MOTION IN CONDENSED PHASES.
NUCLEAR MAGNETIC RELAXATION AND RAMAN LINESHAPE
STUDIES OF SEVERAL SMALL MOLECULES

By
David Allen Wright

Raman lineshape measurements were used to augment NMR relaxation
data in order to investigate molecular motion in liquids and solids.
The Raman experiments were done only at room temperature while pulsed
NMR measurements of T1, T2 and the self-diffusion coefficient 05
were carried out over as wide a range of temperature as possible.

A general method has been developed by which two-pulse experi-
ments may be largely automated by interfacing a minicomputer to an
NMR pulse spectrometer. The necessary pulses and time delays are
supplied by the computer and the interface converts these to thelevels
necessary for the rf oscillator and power amplifier of the spectrom-
eter. Values of the relaxation times (T1, T2, Tlp) are extracted
from the collected data by data analysis subroutines which perform a
least-squares fit of the data to an exponential decay. Deviations
of the data points from the least-squares slope are displayed to permit
an immediate visual check for experimental errors. The advantages
of the simple pulse sequences are retained while their disadvantages.

particularly inefficient data collection for very long or very short

 

‘chOII.' ...et. ‘
C

i.

.I
..vnl

"5’ (‘1' 'é: 3'-

.--

 

I...-.-' ‘9.

, V

I ’ '
_%,..‘..'o.u:'a "'
.... I 0 Q». \v

4" I
‘l ..r .:' .

.... .o OIO.‘ '

. ~ 0-60.;
‘. 19v .
.

d. .a it. v

 

u u .
‘4‘- »p": ;.Ia.
‘0‘ I. u U. -
I .
0-,. I.

t . , ' _
M:
". \O y

“ a . "
V :0. ... I,“
" .‘p. ... ‘
O'o." ‘Opa
. . -( "
.'

 

‘0
" I...
’9': 1 ‘A
_ .: Eta” 4:
. 5
.... -.'<,:F‘
c i“ .‘
.\-,;.!‘
0" ...“ new .‘
V v
.A'h. ‘
T“ “ E “F...‘
‘V‘VI
l“
......
5 . .9 P s
'5 J L'-

q. t F
'1. ‘ .
v’: 4"! ‘

~. " u.
' 1n .1“: ;‘
I‘U.,‘
3.5:.
" jet ..
u( \ I
- waC‘cd.
\U
I l
\‘n' a
"ml-fin;
‘5‘
,‘- I
‘1 V
P
' «I. .L 1-,...
V.-
3""
"z.
‘-:"‘n
a: x .

David Allen Wright

relaxation times, are greatly reduced. The lower limit of relaxation
times which may be measured by this method is about 100 usec.

A simpler interface was constructed which allowed Raman spectra
to be digitized and punched onto IBM cards for analysis on the CDC
6500 computer. A FORTRAN computer program was written which calculated
the reorientational broadening of Raman A1 lines, from which the rota—
tional correlation time was calculated.

Deuterium quadrupole coupling constants have been obtained for
two symmetric-top molecules in the liquid phase by combining NMR relaxa-
tion data with Raman line-shape analysis of bands of 51 symmetry.
The Raman lines have been corrected for vibrational and instrumental
broadening by comparing the polarized and depolarized components of a
single line. More than one A1 line has been studied for each molecule.
The previously uncertain deuterium quadrupole coupling constant for
CDBr3 has been determined in this work to be l7015 kHz and deuterium
coupling constants in CDX3 molecules are discussed. The procedure
described here provides a different method for obtaining nuclear

quadrupole coupling constants in the liquid phase. The temperature

2

dependence of the D spin-lattice relaxation rate was used to analyze

published 13C-relaxation data for CHBr3, and limits were placed on
the anisotropy of rotational motion, which at 20° C were l.5<D||/Dlsl.0.
Motion in liquid CDBr3 was compared to motion in the much more inten-
sively studied liquid CDCl3 system.

Spin-lattice and spin-spin relaxation rates at 56 MHz were mea-
sured in CF3CCl3 from l4l° K to 432° K. More limited measurements
were made of T] at 15.87 MHz and the self-diffusion coefficient in

the liquid phase. A phase transition was observed in the solid at

.....‘zutv -e T‘;

r .'.'.'.I1'.' I!
"O 5 . I I

‘ nu.

. . n.-~,' " ‘
'3' o ,l
etc It I C '

 

r.

’0
fie. ... ‘ l O a.
' ‘. ' A’OO
' ' UI‘ _
O
.l
I' .. Q .al.
. *r ..
' lag I.
‘vn,
. ...I
.
DI p
1' \fi'lr:‘-
'iv¢..‘

I523 3f .. .

|
3“». ‘

D a P
..T‘ 'vd':.-i.&r

3.
:; N‘:o.’, ‘.
. w..."
8.. .
‘4. Te N5 e
|| 4.3 ‘
i
k.
Ik'l-r
~ Q
.... . .‘c'
"bl‘
~.._‘
1' :7";th
‘M'ul‘. -r
IRA".
’d

4' ‘

...“:‘A' \ ‘
1‘:‘ a; ei;_
\n'
'J'~I:‘¢;_

“'vu

12‘-

'3‘...
i
.3'?
5-. '3'
I \
v
M...
h n
’\ F.

 

David Allen Wright

l47° K from a discontinuity in the T] data.

By means of linewidth, T2, and variable-field T1 measurements,
spin-lattice relaxation in plastic crystalline CF3CC13 was found to
have contributions from intramolecular dipole-dipole interactions,
translational diffusion, and spin-rotation. The activation energies
for these processes were determined to be 1.8 kcal/mole, 12.9 kcal/
mole, and «~1.8 kcal/mole, respectively.

At 56 MHz, spin-rotation was found to be more important in the
solid than translational diffusion. The value of the translational
diffusion coefficient at the melting point was determined to be 2.1

x 10'8

or 1.3-10'8, depending on whether the crystal structure is fcc
or bcc.
The separation of the liquid phase 19F relaxation in CF3CC13
was made on the basis of self-diffusion measurements and a single room
temperatureRaman measurement coupled with the temperature dependence
of the rotational correlation time determined from the solid phase
data. The rotational motion was discussed in terms of Gordon's extended
diffusion model, and it was found that the Hubbard relation for isotropic
reorientation predicted angular momentum correlation times which were
in approximate agreement with the diffusion models, indicating that
(Hf-diagonal elements of the spin-rotation tensor are small in CF3CCl3.
Spin-lattice relaxation rates were measured from the melting point
to the critical point for CF3Br and CFZBrz; also the self-diffusion
coefficient was measured in liquid CF3Br from room temperature to the
mElting point. The separation of Rl,tota1 into the contributions

fronlvarious mechanisms was discussed but was not quantitatively suc-

Cessibl due to difficulties in obtaining satisfactory values for the

3;. '12 .se :‘

e... . I
In: a ot'l "6

A

, a
f " ‘wu
n t. w 0 - .

«1 0 A0
'. :‘2‘, f’.
u

\
In. '0‘. .0

 

 

 

»yo.r0)01 qr A".
I I*.' '- .v

.:r‘--r 0-9-0
.I

U IU' -

v; .
: ‘Eit‘. 19'

31116 relaxat-

‘ I ‘ v
A. ‘1 rte.:‘lF

 

David Allen Wright

intermolecular dipole-dipole relaxation rate.

Spin-lattice and spin-spin relaxation rates were measured over
a limited temperature range for CF2C1CC13, CFCTZCF612, CF3CF3, and
CF31. The use of the difference RZ-R1 to obtain the scalar coupling
constant JFX’ where X is the other halogen, was discussed, but it was
not, in general, possible to obtain reliable values. These substi-
tuted ethanes were found to behave very similarly to CF3CC13 in that
Spin-rotation dominated the liquid range, with translational diffusion
only a minor contribution to the total spin-lattice relaxation rate
(R1) both in the liquid and solid, at 56 MHz. Also the presence of a
minimum in R1 in the solid indicated the probable presence of a spin-
rotational relaxation mechanism.

"Effective" spin-rotation interaction constants were calculated
from the relaxation rate at the critical point, and were compared with
chemical shielding derived values. The agreement was found generally
to be good, indicating that off-diagonal elements of the spin-rotation

tensor were small for these compounds and that the motion was roughly

isotropic.

MOLECULAR MOTION IN CONDENSED PHASES.
NUCLEAR MAGNETIC RELAXATION AND RAMAN LINESHAPE
STUDIES OF SEVERAL SMALL MOLECULES

By

David Allen Wright

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirement
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1974

To my Parents

ii

 

 

 

I. :fll'...’
l '0‘ u-v l

.: :p‘ ‘I' O

a... us 1‘ IV. ..

‘i‘no'n...l ..
Av. Idod'V‘ .5
l

O
.P.. ‘. .A-‘.'
p
H. ' '9' ha

2" a

N...

.3" Ir
5 J . n

. U .

 

. “
o"-
O
[-1, 'a
a 5 -
~'.
.9
' . v P.-
. ,.'c “6".
D
‘30
”u,

ACKNOWLEDGMENTS

The author would like to express his appreciation to Professor
Rogers for allowing him the freedom to pursue a research problem which
at times strayed from the area of magnetic resonance.

In addition, thanks are due to Dr. Scott Blackwell and Professor
G. E. Leroi for their theoretical and experimental assistance in Raman
spectroscopy, to Professor R. I. Cukier for helpful discussions on a
number of topics in this Dissertation, and to the Departmental staff,
particularly Mr. Martin Rabb and Mr. Wayne Burkhart, without whose
skills much of the experimental equipment would have been much more
difficult to construct.

Finally, the author wishes to acknowledge the financial support
of the Department of Chemistry throughout his years as a Graduate

student.

iii

 

an

. .
l‘. i‘:‘
A.
E
I
...
Q
-
a... D.‘ :
\- 1
‘
.-K
E: {0"
c‘..-
9
... ' P..
aux. . .

I"!

r11

 

TABLE OF CONTENTS

Page
LIST OF TABLES ........................
LIST OF FIGURES .......................
INTRODUCTION ......................... l
THEORETICAL ......................... 4
I. Correlation Functions in Spectroscopy ....... 4
A. Calculation of Correlation Functions
from Molecular Models ............. 9
B. The Correlation Function for Anisotropic
Rotational Diffusion ............. 13
II. The Theory of NMR Relaxation ............ 18
A. Spin-Lattice Relaxation ............ 18

8. Density Matrix Theory of NMR Relaxation. . . . 21

III. The Theory of Reorientational Broadening of

Raman Bands .................... 24

HISTORICAL .......................... 38

I. NMR Relaxation Mechanisms ............. 38

A. Dipole-Dipole Relaxation ........... 38

B. Scalar—Coupled Relaxation ........... 41

C. Quadrupolar Relaxation ............. 42

D. Spin-Rotational Relaxation .......... 43

II. Rotational Motion in Liquids ............ 47
A. Modifications to the Bloembergen, Purcell

and Pound-Debye Treatment ........... 47

Anisotropic Rotational Diffusion ....... 49

Non-Diffusional Models ............ 53

The Extended Diffusion Model ......... 54

Experimental Tests of Extended Diffusion . . . 59
Investigations of Extended Diffusion by
Visible and Infrared Spectroscopy ....... 62

mmonw
.0000

iv

 

 

 

I I. I II
a It . o a Q C V O I O ' ... t. a I; ‘ l' .I’ .!L III-4 Ill
0 . lulI ‘- PbL HUN \ a I d -
(c .3 u u a) p\.
.0. u or.» a a o a o a sac I o u e as u u I a I -: .Au
. nu.- » u . I! and PL. HJ .35 F “K lip any C hi. hi. 2. Ah. nuu FL. .» tr.
a v u.- o V I v I S u i ..
o .‘H‘. o I O I I I O a
NW» N u a. o v t I. H H H NJ 1" !\i I '1‘
a . . o u a o 9 o \ ~ O A
. o . u
. - l-l'bk N

Chapter Page
III. Plastic Crystals .................. 64
A. Phase Transitions in the Solid ........ 64
B. Spectroscopic Studies of Phase Transitions . . 65
C. The Pople-Karasz Theory of Melting ............ .68
EXPERIMENTAL
I. The NMR Spectrometer ................. 73
A. Introduction ................. 73
B. The Pulse Spectrometer ............ 78
C. The Computer Interface ............ 81
D. The Probe ................... 90
E. Field Gradients for Diffusion Measurements . . 94
F. Temperature Control and Measurement ...... 97
II. Measurement of Relaxation Times .......... 100
A. The Computer Program ............. 100
B. Performance of the Computer-Controlled
NMR System .................. 107
C. Adjustments for Various Pulse Sequences. . . . 108
III. The External Lock ................. 111
IV. The Raman Spectrometer ............... 118
V. Sample Preparation ................. 123
A. NMR Samples .................. 123
8. Raman Samples ................. 123
C. Preparation of Materials ........... 126
RESULTS ........................... 127
1. Measurements of Spin-Lattice Relaxation Rates . . . 128
11. Measurements of Spin-Spin Relaxation Rates ..... 139
III. Self-Diffusion Coefficients ............ 145
DISCUSSION ........................... 155

'.o
...:9

.-..
. o
I'i'
.. .. H
00"
win '0
w
I
F

B

I

Us

9

I

V.

A

U.

P

p

be
u, . ‘
05.. "2'3:
V
cl. ..-.E;r

m

Ih (:3

 

 

Chapter Page
1. Raman and NMR Relaxation Studies of

CDC13 and CDBr3 .................. 155
A. Introduction ................. 155

8. Analysis of Raman Spectra of CDBr3 and
CDCl3 ..................... 159
c. Effect of 20 on Diffusional Motion ...... 165
D. Deuteron Quadrupole Coupling Constants . . . . 166
E. The Rotational Diffusion Tensor in CDBr3 . . . 168
II. NMR and Raman Studies of CF3CC13 .......... 179
A. Determination of the Second Moment ...... 181
B. Translational Diffusion in the Solid ..... 182
C. Translational Diffusion in the Liquid ..... 190
D. Rotational Motion. . . ............ 195

E. Demonstration of Isotropic Motion From
Raman Data .................. 200
111. Relaxation of ‘9F in craer and crzsrz ....... 201
IV. Other Systems ................... 212
V. Spin-Spin Relaxation ................ 218
A. Anamalous T2 Behavior ............. 218
VI. Spin-Rotation Tensors ............... 220
SUMMARY ........................... 229
REFERENCES .......................... 233
APPENDIX A ....................... 247
A. RELAX2 ....................... 247
1 . Comands and Constants ............ 247
2. Acquiring Data ................ 248
Multiple Pulse Experiments .......... 248
Two Pulse Experiments ............. 248
3. Display Routines ............... 249
Extracting Data from Pulse Trains ....... 250

vi

Chapter Page
4. The Discrete Data Set ............. 252
Fitting to an Exponential Decay ........ 252
5. Other Features ................ 254
6. Listing .................... 259
B. Other Nicolet Programs ............... 300
1. Pulsed Field Gradient Timing and Data
Collection .................. 300
2. Triplet T1 Timing and Data Collection ..... 301
3. Calculation of the Second Moment of an
Absorption Line ................ 308
4. Raman Correlation Function Division ...... 3]]
C. RMANFIT ...................... 313
1. Operation ................... 313
2. Suggestions for Improvements ......... 315
3. Listing .................... 317

vii

C.

“.3

v .
r":"_:e:
0' b '

3:932":

11‘2“?!

. . ’
"‘3'.“
. ... 'vd
" 1 .'* a

 

TABLE

10

11

12

13

LIST OF TABLES

The relationships between the Cartesian and

spherical components of a 3x3 tensor .........

Experimental measurements of anisotropic

motion in liquids ..................
Experimental tests of extended diffusion .......

Activation energies for nuclear relaxation

processes in plastic crystalline solids ..............

The output from the "pulse booster" unit as
determined by the input from the computer ......
A comparison of experimentally determined

temperatures with accepted literature values .....

Measured values of temperature gradients across
the sample area of the probe for temperatures below

300° K ........................
Spin-lattice relaxation rates in CF3CC13 .......

Spin-lattice relaxation rates in CF3CC13 at
15.87 MHz ......................

Spin-lattice relaxation rates of 2D in CDBr3 .....
Spin-lattice relaxation rates of 19F in CF3Br . . . .
Spin-lattice relaxation rates of 19F in CFzBrz. . . .

19

Spin-lattice relaxation rates of F in CF2C1CC13 . .

viii

Page

31

52

6O

69

89

101

102

128

130
131
132
134

135

'."-J30 C

 

K.

‘p

‘I

‘
‘9'
tool. )
."
0'

I
t.... z..'
It!

0
‘ .00-
\"'.. a

I
on on.
I.\ '

I.

:IO'.(IQI'
cl 1"

'u' a;
\ I. P
I! p

. ,
IO n
\ '.( ‘I-
I. ‘ ..l

o - _
A-

\ ”.(FI'
I. '

 

 

Table

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

Spin-lattice relaxation rates of 19F in CFC12CFC12. .

19

Spin-lattice relaxation rates of F in CF31 .....

Spin-lattice relaxation rates of 19F in CF3CF3. . . .

Spin-spin relaxation rates of 19F in solid CF3CC13. .
Spin-spin relaxation rates of 19F in liquid CF3CC13 .
Spin-spin relaxation rates of 19F in CFzBr2 .....

19

Spin-spin relaxation rates of F in CF3CF3 .....

Spin-spin relaxation rates of 19F in CF2C1CC13. . . .

Determination of the field gradient and measurement

of the self-diffusion coefficient in CF3CC13 .....

Determination of the field gradient and measurement

of the self-diffusion coefficient in CF3Br ......
Raman linewidth measurements .............

Raman correlation times reported for CDC13 and

CDBr3 ........................

NMR and Raman results and deuteron quadrupole

coupling constants in CDC13 and CDBr3 ........

The relations between coupling constants, force

constants, and structure for sp3 hybridized carbon. .

The temperature dependence of 13C spin-lattice

relaxation in CDBr3 .................

Physical properties of CDCl3 and CDBr3 ........

ix

Page

136
137
138
139
140
141
142

143

145

146

148

163

164

167

169

178

N

0" I0."'

' Q
D-U' no.
0' .':|
o.” 0"

a o'-
c . ...
4‘ .5 ..
5 a

 

Table

30

31

32
33

34

35

36

37

A1
A2

Page
The contribution of translational diffusion to
spin-lattice relaxation in solid CF3CC13 and the
derived values of the mean jump time, Td ........ 189

The contribution of translational diffusion to
spin-lattice relaxation in liquid of CF3CC13 calculated

from the self-diffusion coefficient or the macroscopic

viscosity ....................... 192
Physical properties of CF3CC13 ............. 193
Smoothed relaxation rates in CF3CC13 .......... 194

The contribution of translational diffusion to
the total relaxation rate in CF3Br, calculated

from the reported density and viscosity ........ 205

The contribution of translational diffusion to
the total relaxation rate in CFZBrz, calculated
from the reported density and viscosity ........ 206

Spin-rotation constants for 19F in various

molecules as determined from molecular beam (MB),

chemical shielding (a),or NMR relaxation (NMR)

data .......................... 224

19

Effective F spin-rotation constants obtained

from the spin-lattice relaxation rate at the

critical point with the assumption of 13 = 1.0 ..... 227
Legal commands for RELAXZ ............... 256
Useful addresses ................... 258

 

Figure

LIST OF FIGURES

Page
The relationship between re and TJ for the
perturbed free rotor model and the Debye

Raman 90° scattering geometry ............ 27
The effect of axial ratio n and the angle

90 upon the ratio Teff/Tc‘ Teff is the effec-

tive correlation time obtained by assuming the mole-
cule as (a) a small prolate spheroid, (b) a small

oblate spheroid, and Tc is the correlation time

derived by the Debye-Bloembergen theory, assuming

the molecule as a small sphere ............ 51
The temperature dependence of the second moment

of the resonance line for a typical plastic

crystalline solid, hexamethylethane. Also shown

are the second moments calculated for different

types of motions .................. 67
Plot of reduced transition temperature (T:) and

reduced melting temperature (T;) versus r for

molecular crystals. Solid points are T; data,

open circles T: data ................. 72
Block diagram of computer-interfaced pulsed

NMR spectrometer ................... 76
Timing diagram (x axis to scale). (a) Timing

of rf pulses. (b) internal timing in the

xi

 

 

_.

(A)

4
'\9

I“
l

 

Figure

10

11

12

13

Page
interface. The levels are 0 Vand +5 V.
(c) Timing of the interface output. Positive
voltages are «:2 V, negative voltages -15 V ...... 77
The arrangement of crossed diodes and
tuned cables for the purpose of isolating
the transmitter from the receiver .......... 80

Modification of the tuned spectrometer to

allow nuclei to be observed in the frequency

range 4-30 MHz. Crossed diodes are denoted

by S, low-noise amplifiers are denoted by

[> ......................... 82
Interface circuit diagram. All npm transistors

are T1598; all pnp are 2N3645. A11 resistors

are 1/4 w 7.5 kn except where noted; all

capacitors are 10,000 pfd 500 VDC mica except

where noted ..................... 83
Details of the switching circuit for T1/T1p

control ....................... 85
The circuit diagram of the switch for trans-

ferring control of the spectrometer from the

pulse programmer to the computer. The other

wiring details of all IC's are identical with

those shown on the top IC .............. 87
The pulse booster circuit and the circuit

for generating a third pulse. A11 resistors

are'lkn, 1/4 W, except as noted ........... 88

xii

 

to

'l

I
0.. . p
' I. ‘
V a
9' pl ’
p O D
,,

p. I.
i U
:9. .D

c .
M Mr
b 51

0' u
.IA ‘ _
F o
‘u v u
9. . '
'é DU

 

Figure

14

15

16

17

18

19

20

21

22
23

24

Page
The logical circuit for generating the
third pulse ..................... 89
Circuit diagram of the probe. . . .......... 91
Photograph of the probe showing the place-
ment of the Dewar, the field gradient coils
and the external lock ................ 92
The circuit for providing a variable-length
field-gradient pulse by means of altering
the current to the z-shim coils ........... 96
Details of the coils and coil form for
generating a pulsed field gradient .......... 98
Generalized flow diagram of the software
for timing and measurement control .......... 104
Basic timing for sequences involving more
than two pulses. (a) the pulsed field-
gradient sequence, and (b) generalized multiple-
pulse sequence .................... 106
Correctly adjusted phasing as shown by a
train of 90° pulses ................. 110
The probe circuit for the external lock ....... 113
The arrangement of modulation coils and
transmitter/receiver coil in the external
lock ......................... 114
The circuit for coupling the transmitter
and receiver in the external lock .......... 116

xiii

 

 

... ADO
. . 0'
. .. ..-
.] 0"

”'1‘; I

 

Figure

25

26

27

28

29

30

31

32

A block diagram of the external lock

circuit .....................

The optical train for the Raman scattering

experiment ....................

A block diagram of the apparatus for digitally

recording Raman lineshapes ............

The design of 5 mm NMR sample tubes for

relaxation time measurements up to the critical

temperature ...................

A semi-logarithmic plot of magnetization
versus time in the 180°-T-90° T1 measuring

sequence, showing the typical linearity and

degree of scatter observed ............

A typical KINFIT plot of experimental and
theoretical values of magnetization versus
time in the two-pulse spin echo experiment,

from which the diffusion coefficient is

calculated ....................

Polarized spectrum (——-), depolarized
spectrum(- - ~), and calculated convolu-
tion of the polarized component with a

Lorentzian reorientational spectrum

(- - -), for the 222 cm'1 line of CDBr3 .....

Polarized spectrum (———9, depolarized spectrum
(- - -), and calculated convolution of the

polarized component with a Lorentzian

xiv

Page

. .117

. .119

. .121

. .124

. .144

. .147

. .149

1.:

 

 

Of the

Figure

32

33

34

35

36

37

reorientational spectrum (- - -), for the

521 cm'] line of CDBr3 ..............

Polarized spectrum (—-—), depolarized
spectrum (° - -), and calculated convolution
of the polarized component with a Lorentzian

reorientational spectrum (- - -), for the

2250 cm" line of CDBr3 .............

Polarized spectrum (-——), depolarized spectrum
(- - o), and calculated convolution of the
polarized component with a Lorentzian re-

orientational spectrum (- - -), for the 650

cm"1 line of CDC13 ................

Polarized spectrum (-——), depolarized spectrum
(- - -), and calculated convolution of the
polarized component with a Lorentzian re-

orientational spectrum (- - -), for the 2256

cm"1 line of CDCl3 ................

Polarized spectrum (-——), depolarized spectrum
(- - -), and calculated convolution of the
polarized component with a Lorentzian re-

orientational spectrum (- - -), for the 714

-1
cm line of CF3CC13 ...............

Correlation functions calculated from CDBr3
Raman lines; (a) uncorrected low frequency side

offthe 521 cm'] line, (b) low frequency side

XV

Page

. .150

. .151

. .152

. .153

. .154

.o‘

u
o
.‘.

it-
.—_4

 

A: 9"
u ' '
upg’;'

 

Figure

38

39

4o

41

42

Page

of the 521 cm'] line corrected for the
presence of a hot band, and (c) low fre-
quency side of the 222 cm'1 line ........... 160

Spin-lattice relaxation rates as a function of

13

temperature for 2D in CDBr3 (0), and C in

13CHBr3 (|:]). Also shown are the calculated

13C relaxa-

dipole-dipole contribution to the
tion rate (- - -), and [R1 tota1(]3c) - R1 dd
(”ell“) ...................... 172

The temperature dependence of the contribution

to the 13C relaxation rate in ‘3

CHBr3 from

scalar coupling to 79Br, assuming isotropic
reorientation .................... 175
The temperature dependence of the contribution

to the 13C relaxation rate in 13

CHBr3 from scalar
coupling to bromine predicted by (a) isotropic re-
orientation with JCBr = 91 Hz (-—-), and (b) aniso-
tropic reorientation with Ea,sc = 1.59 kcal/mole,

JCBr = 55 Hz and D'llpl.= 1.49 at 20° C (- - -) . . .177
Spin-lattice relaxation of 19F in liquid and

solid CF3CC13 .................... 180
The temperature dependence of spin-spin relaxa-

tion and the temperature and frequency

dependence of spin-lattice relaxation

in solid CF3CC13. . ................. 183

xvi

O
.9:

 

N I

. :1?-
D

h .
“it"

 

 

Figure

43

44

45

46

47

48
49

Page

The temperature dependence of the width of

the 19F resonance line in solid CF3CC13 ....... 187
The temperature dependence of the mean

jump time for translational diffusion in

solid CF3CC13 .................... 188
A comparison of the experimental relations

between t; and 13 with the predictions of the

extended diffusion model ............... 199
The temperature dependence of spin-lattice

relaxation in CF3Br, and the calculated

contribution from translational diffusion ...... 202
The temperature dependence of spin-lattice

and spin-spin relaxation in CFzBrz, showing

a separation into different contributions based

on the calculated contribution from translational
diffusion and equal but opposite temperature

dependence of R1,dd and R1,sr‘ The dipole-

dipole relaxation rate predicted from dielectric
relaxation data is given by (O) ........... 203
Choice of axes for the CFzBr2 molecule ........ 208
The temperature dependence of the self-

diffusion coefficient (05) measured by the

two-pulse, steady gradient, spin-echo method,

compared with the temperature dependence of DS

as derived from the macroscopic viscosity and density

for CF3Br ..................... .211

xvii

Figure

50

51

52

53

54

55

A1
A2

The temperature dependence of the spin-
1attice spin-relaxation rate of 19F in liquid

and solid CF2C1CC13 and the spin-spin relaxa-

tion rate in the liquid .............

The temperature dependence of the spin-

lattice relaxationiate of 19F in solid

and liquid CFClZCFC12 ..............

The temperature dependence of the spin-

1attice and spin-spin relaxation rate of

19
F in CF3CF3 ..................

The temperature dependence of the spin-
lattice relaxation rate of 19F in liquid

CF31 .......................

The temperature dependence of the spin-spin

relaxation rate of 19F in solid CFZClCC13 . . . .

The temperature dependence of the difference

RZ'Rl for various fluorohalocarbons, (0)

Core map of RELAX2 ................
Flow chart for the operation of RELAX2 ......

xviii

Page

. .213

. .214

. .215

. .216

. .217

. .219

. .255
. .257

 

'-

CE? :3; r

' . (I I .
.OI.DT',' ..
.. .01. I 2 ‘
Q.
, . O
‘9‘. . U p ’
.n- 0 . ' '

-‘v'I Oln'( .- .
I. 0" I 0 CU -

:LW '3 z" .
no t j C. -
O

‘ a
U

“39 n o
.2... 5.1,,

 

‘o-u o .
.: I’ If, F‘j.
“our “led-"pr

'5' a .

T: .9: Ryan-l -. .
9 ." ..

5" g . a . .
s o Ii. (try 'E-

2:131:15 a r":

Lifted 39

>1,

-r'

313179“.

. ‘90

 

~I
" - i 1'
ll ‘
WW
‘1. ‘
I:':...Q
"° I0 ;
I“? ,'
M
l._I 3;.-.
’b-I

 

INTRODUCTION

Nuclear magnetic relaxation studies have been extremely useful
in understanding the details of molecular motion in fluids. This
is especially true for nuclei which have large spin-rotational
contributions to the relaxation rate, as information about the cor-
relation of angular momentum is important in terms of testing a
molecular model but difficult to obtain by other methods. To date,
studies which rigorously test a model of molecular reorientation
have been performed solely on molecules of rather special nature.
Since the predictions of the extended diffusion model have been
borne out very well for those molecules, it would be of interest to
attempt such a rigorous study on molecules of a more common nature.
To this end, 19F spin-lattice relaxation measurements have been per-
formed on molecules of C3v symmetry containing -CF3 groups (CF3CC13,
CF3CF3,CF3C1, CF3Br, and CF31) to varying degrees of thoroughness.

In addition, less symmetric molecules containing other -CF2 and -CF-
groups were investigated to aid in understanding the results in the
-CF3 containing molecules.

A major problem in NMR relaxation studies is that there are often
too many unknowns for precise values of the correlation times to be
determined. To alleviate this difficulty, the NMR data may be aug-
mented with Raman line-shape data, which can reduce by one the number
of NMR unknowns by supplying the rotational correlation time. Although
variable-temperature Raman data would be most useful, even a single

room-temperature measurement would be helpful. In this investigation,

.. ‘l' -
.' g ‘
'0‘: I 10“ U-

.A. .‘.”."A.I _
\ I
it .

3:35 ‘rcr. t’
’ 331° , and 1'
3’3 ifa 535

q
'5 ”tango.
Q ‘.:~' ~‘

 

a single Raman measurement in conjunction with NMR temperature-dependent
relaxation data.has been used to separate the spin-rotational contribu-
tion to the relaxation rate from the dipole-dipole contribution in
CF3CC13, and also to determine the coupling constants (JFCl in CF3CC13
and the 2D quadrupole coupling constant in CDC13 and CDBr3) which are
not observable in the high-resolution spectrum due to the rapid
relaxation of the quadrupolar nucleus.

Any experimental measurement which is sensitive to the details
of molecular motion may be divided into two parts; the determination
of the microscopic parameter(s) of interest (correlation times, in
this case) from the measured macroscopic property (namely, the relaxa-
tion time), and the interpretation of the microscopic parameter(s)
in terms of a specific model for molecular motion.

The Theoretical section of this dissertation will consider only
the latter part of this problem by discussing the role of time cor-
relation functions in spectroscopy. The result of this section will
be to show how a correlation function is calculated from a given
model of molecular motion. This approach is taken in order that it
be clear how this correlation function may then be applied to explain
motional effects in many seemingly diverse experiments. The two
experiments which will be discussed explicitly in terms of this cor-
relation function are NMR relaxation through molecular motion and
the broadening of Raman lines. I

The Historical section will address itself to current methods
of extracting the microscopic parameters of interest from the ex-

perimental measurements. The experiments most pertinent to the

u
I
-

 

 

present work will be those which transcend the boundaries of the
NMR experiment for a better understanding of the motions responsible
for spin-lattice relaxation.

Accurate and rapid measurements of relaxation times are generally
more conveniently performed when a computer handles a major portion
of the experimental details. A considerable section of this work
is devoted to describing a computer-controlled two-pulse NMR spectrom-
eter which, within its design specifications, is extremely versatile

and was used to acquire nearly all of the NMR data discussed herein.

v o: 00.: .A'...

on 0 U "
a
..o' ...-o ‘-
0 l .
I. .I ' '
U
.
”"O‘a- a O a
’ D.

‘“ 0-‘I. I

‘.., o. O.
'

'o
‘ l
o, ‘
I“? 1F?
4,. :F-"
' "13 m.
It}; . _
I... Are a.
V
‘5 _
,‘ c I-.:P
‘,
r. “Fl 3!.
h
1"“ h
's " ,-
': ":’,Hn "
4" V Ir
U I.
’2". i
5' \ M
I. ‘
' "'3 HE»
‘ t
‘ I
p. ‘
F ‘ .
‘ HESS}.
'-
I.
. o“,_n
‘ .‘
'5

' .I'I
.- o-
"

‘-.

 

THEORETICAL

1. Correlation Functions in Spectroscopy

Although the early development of NMR relaxation theory was in
terms of time correlation functions, in learning this theory one
rather rapidly discards correlation functions for correlation times
and consequently misses much of the significance of the correlation
function approach. Recently many diverse non-equilibrium phenomena
have been discussed in terms of time correlation functions. There
exist a large number of excellent reviews on the use of time correla-
tion functionSI'S; the best introductory articles in this field are
those of Zwanzig1 and Gordonz. Of particular concern to this investi-
gation is the time correlation function describing the rotational
Inotion of molecules in fluids, which must be governed by the aniso-
tropic part of the intermolecular potential. While classical methods
of investigating fluids, such as measurements of viscosity or heat
capacity, are insensitive to angle-dependent intermolecular forces,
spectroscopic methods such as NMR relaxation times and Raman and IR
lineshapes are affected by these types of forces. The time correla-
tion function approach provides that the broadening of spectral lines
land the change in NMR relaxation times with temperature (and other
effects) be treated on a COlllllOll basis.

The classical definition of a time correlation function for two

dynamical properties A and B may be stated as

CAB = <A(O)-B(t)> (l)

it“
'8
O I-
I.
‘
S.-
0‘

—
a“

.8

...... '0‘; "

.o-o'

"'9'.

 

-5°".P=" o.
a ... '1. 'S
U
.' .
.f' " ‘h; o ,.
I ..~ U

6. .... .
‘nt "1.4
n"“
a, ‘
|.
6":E ‘fi
Burr>
o
’u. “. .
*3” l:
Il'i , n
w} e:.;
‘
....
" ‘._.~‘.|"
I ' A ‘
"'I

..3 -.‘
c ': JE y] a~
.
$33..
. .‘ Eta: fr-
.l. :;1

 

where the brackets indicate an equilibrium ensemble average. when A
and B are different properties CAB is termed a cross-correlation
function, when they are identical CAA is termed an auto-correlation
function.

A correlation function tells concisely how a given dynamical
property at time t correlates with its value at time t=0. One could
calculate the time correlation functions by this method: Imagine
that it is possible to follow the motion of a single molecule and
the dynamical property p(t) is the orientation of the molecular dipole
moment. One measures its orientation at time t=0 and at time t; the
correlation is the projection of p(t) on p(O). To get an average
value for the function one repeats this experiment a large number
of times choosing various reference times t=0, and averages over the
starting times. If the usual assumption is made that the system is
ergodic, that is, ensemble averages are the same as time averages,
then this procedure will give a time correlation as defined by Equa-
tion (1).

A time correlation function can be calculated no matter what
type of motion is occurring, however, usually the motion is random
and the knowledge of its time dependence is in the form of a condi-
tional probability function P(y2,y]|t2,t]) which gives the probability
(If observing the dynamical variable with value yz at time t2 if it
had a value y] at time t]. The time correlation function may then
be calculated from P as follows: If y is a random function of time
and f(y) is a function of y then the average value of f at time t1
will be found to be

 

‘0
mg.

c—J

:5" a. "
‘Iblod~b 5'
J}.

I“
. .
I. '

Hid.
._= I:
‘5
N.
3...? ‘ _
( HI) -

 

fl—Tt] =fP<y1.t1)f(y1)dy1 (2)
and the average value at time t2 will similarly be
T‘iftz =fv<y2.t2)f(y2>dy2. (3)

If we form the product f(t1)-f(t2) and then perform the averaging,

the result is clearly a time correlation function,

 

5(t2-t1) = f(t])-f(t2)
‘j/T’U].t])P(.Y2.t2)f(y])f(y2)dy1dy2. (4)

The product P(y],t])-P(y2,t2) is a joint probability function and is
related to the conditional probability function P by the obvious

relation
P(y1.t])-P(y2.t2) = P(y1.t1)P(y1.y2;t].tz). (5)

therefore
Gog-t.) =ffp(y].t1my,.y2;t].tzuwfozwyldyz. (6)

Some comnon assumptions made in the theory of random processes6

are that the random functions are Gaussian and are stationary; that
is, they are invariant to a shift of the time axis, and the process is a

Markoff process. In a Markoff process the function P does not depend on

‘n

:3.
w“: ‘5
' of!

'__4

 

times prior to t]. If,and only if;these assumptions hold, then the
time correlation functions are exponential7.

Time correlation functions appear in spectroscopy in two ways:
in the Heisenberg description of the frequency spectrum as the
Fourier transform of the appropriate time correlation function and
also in an equation for the transition rate. The difference between
these two appearances is that a spectrum is the complete Fourier
transform of the time correlation function (the “spectral density")
limited only by the resolution and sensitivity of the instrument,
whereas a transition rate constant measures only a single frequency
component of the spectral density. Obviously then.a spectrum gives
much more information than a transition rate (relaxation time) about
its time correlation function. While in principle the frequency
dependence of transition rates would provide additional information,
in practice the frequencies at which they are measured are orders
of magnitude too low for any frequency dependence to be observed.

The Heisenberg expression for the shape of an absorption band
may be obtained2 from the corresponding Schrodinger expression,

where for example the infrared absorption is given by

and the various terms have their usual meaning. In the Schrodinger
picture the spectrum is viewed as the set of transitions between the
time-dependent states Ii> and |f>.

To obtain the Heisenberg expression from this we introduce the

Fourier expansion of the 6-function

1

 

o p 0
v :-"D<f ':.
' ,‘g ’4' : '

v

w

:‘b" Ir .....r-
. ‘ ')v' .

‘0
o n
‘ 2:959»; ..
5'.
r
\“A .

 

6(a)) = %J eiwtdt (8)

and, expressing the dipole moment operator in the interaction repre-

sentation.
p(t) = eiJCt/hfie-iJCt/M; (9)
thus,
3 -lwt A
1(a)) =71?! dte <e-n(0)e-a(t)>. (10)

and for an isotropic system

I(w) = giff dte'iwt<fl(0)°fl(t)>- (N)

The Heisenberg picture views the spectrum as the Fourier transform
of the time correlation function for the dipole moment operator of
the absorbing molecules.

An equivalent transformation can be done to the Schrddinger

expression for the transition rate5
t 2
_ l -i t
“m -i-n-g]<fldc1(t)li>e “f1 dt . (12)
O ,

 

where Hf+i is the average over some time interval t and mfi = (Bf-£1)”.

If we let

 

‘3
"(.i
o.
‘-r
‘I
I
II

"* 0- pc-
.' :1 ’fi‘
| w’v. ' g.

"3r as ‘rc

'-

y' “t“:r-t...
‘ r

 

 

 

f" lea-'4‘...
" 9H
' "Ii 5'.

I <‘~‘
' ‘DII
,2“, .
"' w, {PM
tn
;' . ‘ .
‘ Q‘.. a P
. |'
3 s
a". ’
on I}. I’ll;
\

.d a”!
.123
“ “Ere.
‘2‘”.
a: ”it

Kfi(t) = <f|JC](t)|i>,

Equation (l2) becomes5

("fl-i) = if] d16fi(r)e'1wfiT. (13)

This same type of expression was obtained by Callen and Nelton8

in an extension of the Nyquist relation for electrical circuits and

is known as the fluctuation-dissipation theorem since it relates the
power dissipated by the system, I(w), to the function <u(O)-u(t)>
which describes the way spontaneous fluctuations return to equilibrium.
The advantages of the Heisenberg picture have been thoroughly discussed
by Gordonz’g.

A. Calculation of Correlation Functions from Molecular Models

Two simple models of rotational motion exist and the time cor-
relation functions calculated from these models may be considered
as limiting cases for possible time correlation functions of real
liquids. The parameters which emerge from any of these models which
are pertinent to NMR relaxation theory are the correlation time for
angular orientation, 16’ and the correlation time for angular
momentum, TJ.

An excellent physical picture which explains the relationship

10 and bears

between TJ and T9 has been given by Green and Powles
repeating here. Consider a molecule undergoing rotational diffu-

sion in a liquid. Since the rotational step size is very small,

.00 ~r ‘.
I! {:1 '

«up a. 9":3
2.. I. - "

I

'0-
I"'
y. d..-

.s
I \
D‘-

1-r;!';""f ‘
.v u “I V' "

:‘a' mi;

- ton-alt...
v l\~, l u-
I l‘ 000‘! .05-

 

 

“ I

.‘t‘ girls: ..

335:.12 is

:fih-n | .

u
:0. .. . n

N no.~ :y a

2:3 .11, ..

"- «Lit-‘-

 

\
”I
.. ..A
' A
'=.r3] - ‘
w H n,
U
lt.‘
I'M?
will ‘

 

10

a large number of diffusive steps will be required before an arbitrary
vector with orientation 90 reaches orientation 90 + 69, where 69 2 l
radian. Consequently many diffusive steps will be necessary before
the orientation becomes uncorrelated. But since the molecular
angular momentum undoubtedly changes with each diffusive step, TJ
may be associated with the time between collisions and TJ << 16.
Now suppose the collisions occur less frequently and the step sizes
begin to increase. The angular momentum correlation time increases
while, since the path traversed by the orientation vector is shorter,
16 decreases. This trend continues until the step size is such that
the orientation is uncorrelated after a single step, hence Ta = 16.
The first model is the perturbed free-rotor model11 in which
the molecule is depicted as undergoing essentially free rotation,
governed by the molecular inertia tensor, but is occasionally
interrupted by a collision. The resulting time correlation function
may be calculated from Equation (l). For example, for a linear
molecule Gn(t) = Pn(coswt), where n indicates which spherical harmonic
the dynamical variable transforms as.and m is the rotational fre-
quency. Averaging over an ensemble of such molecules is accomplished

by integrating over a Boltzmann distribution, so

<Pn(c05wt)> = f Pn(cosmt)wexp(- %w2)dw. (14)

O
This integral has been obtained numerically by Gordon9

problem has been examined more generallylz. The relationship be-

; also, the

tween T9 and Id for this model is T6 = TJ/(2J+]). One would not

expect this description of rotational motion to be too good for

... a .910;
u I i...”

 

o
s '3- '
‘11 I i.

7.51.:0-1.

I

I
. '0“..‘. 5’ '

... .I.. U

77:.o:oc 1' R-

." J-

..-,..a ...
. ‘ l
. . I I
' '
M
~ .
'.| :IA“ .
' curl. 'u-
.
I

...“ .‘:'-:'°
' I... .'
so. "
.E 3::Mqlo .
up ‘
"“.'qu‘,
"was. pr“:
‘ 55‘
. I . I
'J g
'- _\ P
' O .46 r:.:
in

V .
D...\' i.

"" ~41. '6 el

 

ll

the liquid state, except perhaps as the critical point is approached.

A more liquid-like theory is the rotational diffusion theory
originally formulated by Debye13 to explain the anamolous dispersion
of radiofrequency waves in liquids. In this model the molecular
reorientation occurs through a large number of angular steps of
vanishingly small size. The molecule is treated as a sphere embedded
in a viscous fluid where the retarding force is given by Stokes'

3

Law,f = 8na n,where a is the molecular radius and n is the macro-

scopic viscosity. By analogy
-§% P(no,n;t) = -DV2P(O°,Q;t), (l5)

in the isotropic case with no external forces: here P(Q,t) is the
conditional probability function for the coordinates and time interval

t, o is the rotational diffusion coefficient, and v2

is the Laplacian
operator. He examine the motion of a vector over the surface of a
sphere by subjecting the Laplacian to the condition-§% = O. The co-
ordinates a now become the polar angles 9 and o.

The solution can be written in terms of the eigenfunctions of

V2 (the spherical harmonics),
vzv'gm) = -£(2.+l)Y'2$2, (l6)
as

(17)

P(oo,n;t) = Z \(‘f'ma)\!€(n)e"5”~t
2.,m

A
. |
.... ‘ =
.l t:

d.
...;4,;...:
a. v. 0 .6

I
'9" d an:
I' ' ' b.-
II
c
o
: :' Ir 9,
‘I 0" I .
b.’
*‘v' e ‘.
:.‘ | ..
' [ ..
l'.$. ...”
" \\— r
" u .
‘e
o o
. ,,-7”
-'fl..
E I..."
U 5‘

—ln
ll,

 

12

where E = NRA-l )0. Then, from Equation (6)

Gt-to = 952 Y'"* Y'“ 'Efitf f d o. l8
( ) ff(0>§‘,(o,)£(o)e (no) (man < >

If the dynamical variable can be written in terms of a spherical
harmonic Pn(cosO) then, from the orthonormal properties of Pn’ a
single term in the sum in Equation (18) will be picked out, the
integral will become trivial, and the time correlation function may
be expressed in terms of a single exponential with time constant
£(£+l)D = T.

The difficulties with the Debye treatment are:

(l) The particle is treated as a sphere.

(2) Rotation is described as a sphere turning in a viscous
fluid, i.e., a "stick" boundary condition. (However,
recently friction coefficients have been calculated14
for a "slip" boundary condition, namely, zero friction
for rotation about a symmetry axis.)

(3) The short time behavior of the time correlation function
is not correct. The requirement that a classical time
correlation function be symmetric with respect to time

inversion implies5 that

d_"§.

n = O for n odd, (l9)
dt. t:

0

which is clearly at variance with the non-zero deriva-
tives of an exponential function.
The failure of the Debye model is in not considering inertial

effects, as was done in the perturbed free-rotor model, since even

 

 

»' a“ f
.: i-. :5 (3" .‘

.r'l;':“ ";A. 3r.

- fl .-
"‘v 3 Pa':‘ "r f
L. l 5' c '5' I
.- I.

   
  
    

23.51::er :rctn :

5‘5'9’31. the '

 

:.I.-‘ 1 ..-
I :3"? 31...?

.1 “ l
"he ‘6‘: “ TIC'nT

13

models which allow large-step reorientation result in exponential

time correlation functions‘s’m.

The relation between T6 and TJ for these two simple models is
shown in Figure l. In terms of the time between collisions (the)
the Debye model applies when tbc << 16 and the perturbed free rotor
model holds when tbc >> 16. In a later section a model will be
considered which bridges the intermediate region between these two
cases by considering the rotational time correlation function to
be a function both of time and of the time between collisions

(equivalently, the length of a diffusive step).

B. The Correlation Function for Anisotropic Rotational Diffusion

The well-known equation for isotropic translational diffusion

5%- P('F,t) = -ovzp(?,t), (20)

which describes the probability P that a molecule will be at location
7' at time t, may be derived by a conservation of mass argument or,
more significantly, by a random walk mechanism originally due to
Einstein in which the particle completely loses the memory of the
previous step”. In his monograph on polar molecules, Debye13
considered the analogous case of isotropic rotational diffusion and
obtained an exponential time correlation function as discussed in

the previous section. This treatment was later generalized by Perrin18
to describe the rotational diffusion of an ellipsoid and then re-

derived in terms of the eigenfunctions and eigenvalues of a

 

 

 

14
l l I
TO. +— ”T
1”
\ ’/
I
\\ ’,’
1' \\ 1’
0 \ 1”
\ I
1.0 - \ I” "‘
\\ I”
‘\ ,’
\ 1’
\ ,I'
\ X
\ ,I’
,X
I” \\
0.1 _ \ ..4
I”’ \
1 x J \ L
Ol L0 l0
13'
Figure l. The relationship between T and fa for the perturbed free

rotor model and the Debye model.

 

{frgtfi‘TtEI r"
:73? “I V? Z
r‘.:‘:-' we I)??? 3
_.. as: -.r:'ess'
“ad's Ker.

:iufls 'ESJ‘nt .

51.5538 111':
E7: 2: 1193119 '
2'35 :e'ssr also <

l~ we:

'Ir;)

0
r.

 

15

quantum-nechanical rigid rotor by Favro‘g. The correlation functions

appropriate for NMR of a particle undergoing anisotropic rotational

diffusion have been calculated by Woessner20 using the notation of

Perrin, and Huntressfl has recently reviewed the relevant sections

of Favro's paper.

Favro's result for a completely random process i521
3— 9(9 t) = -E-5oE P(Q t) (21)
at O 9 9

where If is the dimensionless angular momentum operator. In a sym-
metric top molecule the coordinate system which diagonalizes the

inertia tensor also diagonalizes the diffusion tensor, so Equation

(21) becomes

3
bgf P(Qat) = -§ 011E12P(Qst)° (22)

In order to calculate a time correlation function from Equation
(22) it is convenient to introduce the Wigner rotation matrices22
DiMm) which are a complete orthonormal set spanning the space of
Euler angles 52 = o.B,v. Consequently,the probability function P in

terms of a linear combination of rotation matrices becomes

. .. J
P(O.t) = Z Z agmm oifnm). (23)
J=0 K,M=-J

with the expansion coefficient

.J J ZJ l
0K,M(t) = an(n,t)DK’M(n)[—t§-] - (24)

81!

 

on
V \
o 1
' I

we '."€ 1':

 

1

 

 

16

Since all the time dependence is now in the expansion coefficient

c1: M(t), we place Equation (23) in (22) to obtain

a: J m J
E Z ongm) aa—L— "(t “2014?: 2 «K “moi" Km) (25)
J=O K,M=-J i=l J= O M,K=-

From the quantumnmechanical rigid-rotor problem we know that the 0
matrices are eigenfunctions of the total angular momentum operator

A

L , and of one component, which we may choose to be {2;

C2 DJ %M(9) = n WJ(J+1)D M(:2) L 20K, M(a) = -nMoK M(s2). (26)

Rearranging the first summation on the right-hand side of Equation

(25) we have

Zuni?

i:

3
A2 A2
1 oil. + (Dll- i)Lz, (27)

Thus, substituting Equation (26) and (27) into Equation (25),
multiplying through by D: “(9), and employing the orthogonal properties

of the rotation matrices we obtain

3 J _ _ _ 2 J

5E41K’M(t) - [919(J+1) + (Dll pl)" JQK’M(t)- (28)
This equation is readily integrated; we then have

0:,M(t) = 0:,M(o)exP{-[D.LJ(J+1)+(D| l- .L)M2]t}. (29)

17
The coefficient at zero time is found from the initial condition
P(0,0) = 6(90-9), (30)

which also allows us to replace the function P in Equations (22)
and (23) with a conditional probability function 6(9 ,Q;t). From
Equations(23) and (30) we have

m .J
6(90-9) = 2:2 agmmmgfnm) (31>

J=0 M,K=-J

which, through the orthogonality properties of the D matrices, requires

that

J J
“Km‘o’ ‘ ”mm. 7 ' (32)

Also, from Equations (23), (29), and (32) we have

emo ,-o ,t) = 2 23:2— 2“ DK,M(“.)°::M(“)°'t/Tm' (33)
J= 0 K, M=- J?

where l/Tm = DlJ(J+l) + (DH-Di) M2.

To calculate an actual time correlation function such as
<DJ: M(Q ) D: M(£2)> we substitute Equation (33) into Equation (6),
replace p(Oo) with l/Bn2 , which is equivalent to an initial isotropic

distribution, and obtain

r

. .. :
:Y'Jfi m ...";

 

 

‘Jr as; gi.

b- .‘ ‘ j
.1. ‘ ‘3' [;*;'.

wh'

only

i.» nq‘l. ‘. ‘
I '2 [3" '38.. .

:giet": razor ii.

..I

t;

:0» OJ
.uu3;.t

‘t
(m. (—

UK‘E'.

 

to: . .
" “‘ nwar of

“ *3 K‘ecular

Q
\

I l.
\-

:,(
“.mo 3“:‘
....we I;

l

LUE) With :
Hr M N $1

”‘95:! in the
fir .

r!

“3‘"
(" eq’u'étio

18
on J
<DJ* ”M“: )0: M(Sl)>- = —2- 13:20 dQZ ZJ—BJT
J= 0 K,M=-
x Dime.wmmvi’fmn.Wimme'mm- <34)

Clearly, for any given value of J, only a certain number of terms will
be non-zero from the orthogonality relations. Thus if J=2, which is
the result for Raman and NMR correlation functions, only five terms
will be picked out from the infinite series in (34), and for an
asymmetric rotor five correlation times will be necessary to describe

the motion. Equation (34) is simplified to

 

<1)": "(:20 mi, "(ab = 2 231+] e‘t/T'". (35)

where the number of terms in the sum of exponentials is seen to depend

upon the molecular symmetry.
11. The Theory of NMR Relaxation

A. Spin-Lattice Relaxation

Nuclei with spin I f 0 possess a magnetic moment and when a
collection of N such nuclei is placed in a magnetic field Ho the
degeneracy in the 21+l states of different spin quantum number is
removed. The populations of the various states are given by the

Boltzmann equation

 

refs Elf: stir

 

i a; -
...-.C
"m an ur

ill a

:7
"‘I

...':.'at‘;0n bee.

‘v

19

Pm = exp{-Em/kT}
= exP{-M mHo/kT}, (36)

where m is the spin quantum number. Thus, the induced bulk magne-

tism becomes

I
Z Ium
Mo = NY” 9}1———— . (37)

23 Pm

m=-I

For temperatures higher than a few millidegrees Kelvin, exp(-Em/kT)
3 l + MYmHo/kT, so

hH
{mi—ken?

O Y” 2] + {m

 

_ 2 2 I(I+l)
‘ NY ” 3kT ° (38)
In addition to absorbing energy from the static (Zeeman) field
in going from an unmagnetized state to a state described by Equation
(38), a sample of nuclear magnets may interact with the magnetic
component of electromagnetic radiation through resonant absorption.

The separation between states (Am=tl) is

AE = yflHo. (39)

 

3.1:" n. ‘.
nasce-

:fie rescrant

-.£ :rstant T

13
‘o.. {...
'M == wows,

 

7?:3 2 . p
. 1. me CT.“

l
o

 

20

Expressing the energy difference in terms of frequency, AE = Hm,

we obtain

w = YHO, (40)
which is the basic equation relating the strength of the static field
and the frequency of the resonant radiation.

The approach of the magnetization to equilibrium, whether from
zero to "o as the Zeeman field is turned on, or the return to "0
after the resonant radiation is turned off, may be characterized by
a time constant T], the spin-lattice relaxation time. This can be
shown as follows.

Consider a system S characterized by eigenstates |a> and popula-
tions Pa. The differential equation describing the time dependence

of P is
a
dPa
1? = 25: "GBPB'?BaPa ’ (4] )

where was is a transition rate of the form of Equation (12). Since

"as = "Ba.the steady-state solution (dPa/dt = 0) of (41) is Pa = P

For the purposes of this simple presentation we omit the ad hgg_

3°

argument for making the steady-state populations follow the Boltzmann
distribution, which is usually introduced at this point. The net
magnetism can be expressed in terms of the time-dependent popula-

tions,

Mztt) = <Mz(t)> = um amnpmm. (42)

p

. 0.. Oé"
rztm -£ w E

1:11; ElithtS

 

 

 

8731.2,

   

ME?" = 2‘. 9““:

M33 arising fr-j

 

“"5 9 the const-

-
.

a “V

“g‘

I! .‘P ,
" \13‘Jdtl0n

1’!

Me by Pozula

M" This res
“”15 as the

.,-'i ir-diagcr‘a] c

x.
w. _
Gar-r?
‘ -

Cf “UCleg

 

2]

and taking the time derivative,

sz(t) de(t)

dt =Nflv§m dt . (43)

 

 

Combining Equations (4l) and (43), it is a simple matter to show that

when m = l/2,

sz(t) 1
T— = - 1'? "2(t)’ (44)

where T] = 2". However when m ¥ l/2 it is not apparent that the sum
of terms arising from B>l in Equation (4l) will be expressible as

a single time constant. An additional problem is that the master
equation (Equation (4l)) is applicable only when the system may be
described by populations, that is, when the density matrix is
diagonal. This restraint is commonly violated in NMR relaxation
experiments as the spin system following a 90° pulse is characterized
by a non-diagonal density matrix. Consequently a more rigorous

treatment of nuclear relaxation is desirable.

8. Density Matrix Theory of NMR Relaxation

Relaxation theory has been developed through two formalisms;

the Bloch-Hangsness-Redfield density matrix approac and the

26.27
27

linear response theory of Kubo and Tomita Although the former

is the more commonly used, it has been argued that the linear
response approach is entirely equivalent. and in addition is the more

intuitively acceptable theory, as the only approximations made are

 

...-a—‘

I
c1 ‘
r.

NRC is the I
“or ‘nctisn of
(on .‘, u'.
' 3‘ duke.

it the intere:

dt

 

22

to the behavior of macroscopic variables.

The semiclassical density matrix theory of relaxation will be
outlined here. In this approach the spin system is described quantum
mechanically but the lattice is treated classically. This approxima-
tion means that terms in the interaction Hamiltonian which pertain
to the lattice will be treated as random functions of time, describable
as a correlation function.

The equation of motion of the density matrix 0 for the system
S is

(@551 = -[aco + J61(t).o]. (as)

where HJCo is the Zeeman Hamiltonian and hJC1(t) is a stationary
random function of time describing the interaction of the spin system
with the lattice.

In the interaction representation, Equation (45) becomes

ire—34.2 = {ac-Rom. (46)

The density matrix, correct to second order, can then be formally

solved as

o*(t) = o*(0)-ifdt'[JC‘;(t'),o*(0)]

t t'
- f at] dt"[JC‘f(t').[ch(t").o*(0)]] (47)

 

l
‘4 ’ '
....“ ' :.'
0. IL.
I.

§ , .
2'2 frteractz-
3.0._ t
"M will!“ In:

7’6 2y

 

23

and the time derivative of Equation (47) is

t
W t = -i[ac7(t).a*(o>1- I dt'[JC?(t).[JC’{(t').o*(0)]]- (48)

Making a number of approximations based on the shortness of the cor-

28

relation time , and introducing the variable 1 = t-t', Equation (48)

becomes

(D

1%? = - dr<toc§(t).[ac‘{(t-r).o*(t):u>. (49)
o
The interaction Hamiltonian, which is time dependent because
of the molecular motion, can be related to its value at different

times by
ac;(t) = U'](I)JC;(t-I)U(T). (50)

where U is the unitary operator describing the molecular motion

which is effective in changing JCI. Thus, Equation (49) becomes

@5131... ./ dt<[Jc:(t),[U'1(t-1)JC;(T)U(t-'t).o*(t)]]>- (51)

Equation (5l) is the fundamental equation from which the explicit
expressions for the relaxation time may be obtained. The procedure
one follows is to choose the appropriate interaction Hamiltonian,
express the time dependence in the evolution operator U, and calculate

the expectation value <Iz> from the relation

- it'pr
"W; MEN...
' .-

 

 

M‘M Lie

- > 3mm:

I <

A...“ .
‘1. .-
I.l N .ln

9 Eque-

24
<Iz> = Tr(oIz). (52)

In many cases a Bloch equation (Equation (44)) is obtained, showing

that the relaxation is indeed expressible as an exponential decay.

III. The Theory of Reorientational Broadening

The orientational broadening of Raman lines has been discussed
by a number of workersz’3’29'32; here we review the results which
are important for this investigation.

Let 3 be the molecular polarizability tensor, which we expand

in a Taylor series in the normal coordinates q(t),
am = 3°(t) + 23"(t)q"(t). (53)
v

where the summation runs over all the normal coordinates and

3%) = 3319- ; (54)
aqv(t) qvgo

higher term are neglected.
We employ the Heisenberg picture discussed in Section I to state
that the scattered light intensity is proportional to the Fourier

transform of the polarizability correlation function
_ l -iwt = =
1(w) - in' dte <a(t) a(0)>. (55)

SWbStituting Equation (53) into Equation (55) and labeling each tensor

'o
_~ur' "M
til"! Sb!"-

1"-

.
\‘i '
L

.
_ _

9': E nave Tait

‘ 0

up ‘ p‘ ' I
...‘ 3'? 31511511;

.::'":,a:‘on fur:

attic: invoi «fir:

 

 

25

with a superscript capital letter indicating the molecule to which

it refers, we have

Hut) 2117 jdte'iwt {2;<ZA(t)§B(O)>}

g—"fite'iwt {2<3°A(t)303(0)>
B

+

g;<3VA<t)§VB(o)><qV"<t)qV3(0)>}. (56)
V

Here we have made the usual assumption that the vibrations and rota-
tions are statistically independent. This may be rewritten in terms
of correlation functions involving a single particle and correlation

functions involving two particles

at) = <§A(t)3"(0)> + Z<EA(t)§B(0) . (57)
31A

Expanding both terms in Equation (57), just as in Equation (56),
with the condition <qVA(t) qVB(o)> = 6AB<qVA(t) qVA(0)>, i.e., that
vibrations on different molecules are uncorrelated, the remaining

terms are

«p(t) = <3°A(t)3°"(0)> + Zémmammb
BfA

+ z<EVA(t)§VA(0)><q"A(t)q"A(0)>. (58)
V

whhflican be identified as the correlation functions responsible

 

ofi‘f-Efi‘k
0i '
1. 23:9..121.

r ‘ - .
l've die' (-

. . 'vn‘
...;Otx '.4. n '
a... - \IU,‘

TI"? 362.315

(D

Q'Ei :1

'tr (3

3 xx
‘ J3 Elation

”“9 tie +5,

"39 cor

.31 ITS-M. be idei‘.

 

"We Using
31“.,

MMMEQ llhh

I. 3"

“'92 (90°

' Mt “ic‘ .
r tar}

Zdtj

”M be Eithe

I, T" F
Many po

26

for single-particle Rayleigh, cooperative Rayleigh, and Raman scatter-
ing, respectively.

Since Raman scattering occurs displaced by a frequency wv from
the Rayleigh line we may experimentally suppress the first two terms
in Equation (58); we drop the superscripts and the time correlation

function becomes

W) = Z<3<t>3(0)><q(t)q(0)>. (59)
v
He separate the polarizability tensor into a trace and a trace-
less part,

E = E - atrT’ (60)

_ 1_ = . . _
where “tr - 3 (axx + ayy + “22) and l is the unit matrix. Conse

quently, Equation (59) may be written
¢(t) = Z {<B(t)8(0)><q(t)q(0)> + <q(t)q(0)>}. (61)
V

since the time correlation function of a time independent quantity
(ctr) must be identically unity.

Before using Equation (55) and (6l) to determine the equation
fin~scattered light intensity we must consider the scattering geometry
hiFigure 2 (90° scattering). There are four possible arrangements
fln-the polarization of the incident and detected light, each of
whfifllcan be either vertically polarized (with reference to Figure 2)

or horizontally polarized; stating the incident polarization first,

 

27

/"
Y / ’ giM/fi/a’ZD—

J V beam

incident beam

 

Figure 2. Raman 90° scattering geometry.

, . v' ‘3?
.11! :"a 15: "5
.. .

w. “an r'
:2. “natal -'.

”ll‘.;p;filh;‘;.i
I b.

of! I lliJ

2?.2, f}? I." “F‘

 

'3'*':v'q~'.€l‘
”M ‘ 5:4; 11:,
d

’26 Rf}. .
““ a Sin:

”tr L.

.. Uftms’ for

ti“

.5“)

u
A
('1)
YD"

 

28

these arrangements are referred to as VV, VH, HV, and HH. Thus with
VH geometry the incident light is vertically polarized and the
detected light is horizontally polarized. In a scattering experi-
ment with one of these four types of polarization one replaces the
general equation relating the incident and scattered light through
the polarizability tensor with a scalar equation which is, for

example, for VH geometry,

e2 8 - ex = 32x. (62)
The polarizability tensor
Bxx Bxy sz
B = Byx 31y 8All (63)
B B B

is in laboratory frame Cartesian coordinates and the VH geometry
"picks out" a single component. The time correlation function

then becomes, for VH geometry,
¢(t) = <e1-§(t)-€jei-§(0)~€j><q(t)q(0)> + <q(t)q(0)>
= <e,j(t)e,j(0)><q(t)q(0)> + <q(t)q(0)>. (64)

Similarly, the other geometries pick out other single components

 

 

 

'M' If the fade

‘3'.“ "
'5 “Cress.

2:3,”: .

29

vv + 822 VH +,Bzx

HV-+ Byz HH'+ syx (65)

and appropriate time correlation functions can be written.

He now wish to express these tensor components, and consequently
each time correlation function, in the molecule-fixed coordinate
system. He transform an arbitrary Cartesian tensor A in the labora-
tory frame (LE) to the molecular frame (HF) through a unitary trans-

formation using the (as yet unspecified) rotation operator R giving
KLF(h) = R“(h)-E"F.R(n), (as)

where Q = a,B,y, is the set of Euler angles specifying the orienta-
tion. If the molecule is rotating.then Q is time-dependent and this
time dependence will be contained in the rotation operator. If the
tensor is expressed in the spherical basis.rather than the Cartesian
basis.then the rotation operators are the familiar Higner rotation

matrices,
u“.ALF(a).u = [u".R"(n)-u1 . u.A"F.u" - [u“-R(n).u1, (67a)
or
sLF(o) = 0(9) - SMF - o“(o), (67b)

where S is the spherical tensor derived from the Cartesian tensor A

 

 

.' IE 17955

l; o'ncr fr’T‘
. "1: .:‘.-l, V

"tees: am;

2,3187 agar-e

 

k‘ ‘
~ I. ‘I‘ lr“
NEDttER’.

'-‘ a
'1 W ‘
~ 3 :1: g o

' ”1(sz -
Za":at".m gem
3 Zir'tTatfi 3n .

”‘9': 3313‘?” f

1;"
st hie SCa‘+
*L

30

and 0(9) is the rotation matrix. The relationships between the
components of a 3 x 3 Cartesian tensor and a second-rank spherical
tensor are given in Table I for convenience.

If we represent the time correlation function (Bij(t)3k2(o)>
by Bijk

of interest among the Bl possible combinations:

2 then,from relation (65),there are four correlation functions

VV'+ B VH‘+ B

2222 ZXZX

HV'+ B HH'+ B

yzyz yxyx' (68)

By symmetry arguments32 it is possible to reduce these Bl combinations
to two independent time correlation functions and, in the process,

show that B = B = Bxyxy; consequently. the three "crossed"*

xzxz zyzy

polarization geometries, VH, HV, and HH all are measuring the same
time correlation function. Substituting Equation (64) into Equation

(55), we obtain for crossed polarization

Ivn<wi = dte““t<sxz(t)ex,(0)><q(t)q(0)>. (69)

where the second term in Equation (64) is clearly zero since off-
diagonal elements of the unit matrix are zero.

Similarly, the polarized spectrum can be written

 

*
Shme the scattered polarization vector is rotated 90° with respect

to the incident vector.

 

zz:

 

—4
u

 

 

31

Table l. The relationships between the Cartesian and spherical

Components of a 3x3 tensor.

 

 

XX

yy

22

Xy

ZX

2y

1
§{Txx M Tyy + Tzz]
‘[T —T 1:17
'2 xx yy xy

$[sz t isz]

2 l
jEITzz ’ 2(Txx M Tyy)]

1
TI '5‘: To + (T+2 + T_2)/2

1
TI 7? To - (1+2 + T_2)/2

TI + TO

2_
J6
(T+2 ‘ T-2)/2i

(T+l ' T-l)/2

(T+] ' T_])/21

 

 

 

 

tare ::aef'."i:‘
3225(5).
“, iti‘r:
° * we. 3 '-

 

 

32

mm = bljdte'inBzzumzzW)><Cl(t)q(0)>

+ bzfdte““’t<q(t)q(o)>. (70)

where the coefficients b] and b2 have been shown31 to be 4/3 and l,

respectively.

If we designate the depolarized and polarized components, respec-
tively, of the Raman line as

Ianis(m) = Ivu<w> = ate““‘<sxz(t)exz(o)><q(t>q(0)>. (71a)

and

1150‘“) = Iwm) - g Imam)

=fdte'1M’M<q(t)q(0)> (71b)

then, clearly, these components may be determined by measuring the VV

and VH lineshapes. Since the trace of a tensor is unchanged by rota-

tion of the coordinate system,the time correlation function <atr(t)-

“tr(0)> = l and rotational motion contributes nothing to the iso-

tropic lineshape 1150(w). 0n the other hand the anisotropic component

cfi’the line is broadened by the presence of the time-dependent cor-
relation function <sz(t)3xz(0)>. The next step is to rewrite this
time correlation function in terms of the rotation matrices in order

to use the results of Section IB.

To use the rotation matrices we first must convert to a spherical

hi.
n

(2‘5 2'2 SUE'SC'
"din. t or
"' ‘08 I‘YS'U

:ze t'ec.'er

 

a‘éi‘tt‘iig 172'

M“ = fit) a
"M”? the foi

:19 'n3

. If!

“If“ a
'5

 

I
(0‘2)!
"mi

33

basis set

Bx: H(s -BE§). (72)

where the superscripts remind us that we're still in the laboratory

coordinate system. He then employ the rotation matrices to change

to the molecular frame

2
B§F= IE3 D‘f)(a)ef. (73)
M=- 2
Substituting (72) into (73),
&=-% jggto‘zi (n)- ofz’n(n)Js: (74>

M=-2

and the correlation function is

<3§§<t>e§§<0>> = 2'" 3&32 <[o‘f’* (a )- 052’ .(n')1

x [0(2) (n) - 0‘2) M(n)1e"f*e:F>. (75)

where 9‘ = 9(t) and n = 9(0). He reduce the number of terms by

employing the following relationship, valid for an equilibrium

ensembl e3 .

o<2>(n )o(§)(n)> = <o§2’(n )o‘§’(n)>anjsm (76)

 

 

. ‘.'. «'6
midi»?-

 

 

and are left with

2 *
(sz(t)8xz(0)> 3 %’&E:2<[D£€& (9')D§f&(9)

+ 05?};(9')D($2M(9)JIBGFI2>. (77)

The solution to a time correlation function of rotation matrices for
the case of rotational diffusion was seen in Section 18 to be a sum

of 2J+l exponentials independent of the value of K (Equation (35)).

Employing this result, we simplify Equation (77)

J..

2
<szit)exz(0)> = Z IaflFI2 2.1+ exp{-J(J+i)ol+nz(oll-oim. (78>
M=-2

We note that Equation (77) is a model-independent result and that the
diffusion model merely gives the specific form (decaying eXponentials)
for the time-dependent functions. From Equation (78) we have that
the rotational component of the depolarized Raman line is, in general,
a superposition of five Lorentiian lines. However the symmetry of
a given Raman vibrational mode will allow us to set many of the co-

30

efficients lsnFl2 equal to zero , as we will show for the case of

C3v symmetry.
For this symmetry the polarizability tensor may be separated

33

for the A1 and doubly degenerate E vibrations as follows:

a1 0 O
BX:,M = o ai o q1(t) (79)
o 0 b1

 

 

 

35

and
"F Cj 0 0 NF 0 'Cj 'dJ
BE’j] g 0 'Cj dj Qj](t) BE’jz = ’Cj 0 0 qj2(t). (80)

Referring to Table l, it is clear that for A] modes |8:F|2 in Equa-

tion (78) is, for different M values,

MF 2 _ MF 2 _ MF 2
|3¢2| z o, IB.]I = o, leol y o (8la)
and for E modes
.3232 f o. IBZ‘fIZ f o. lean? s 0. (81b)

Consequently,for A1 vibrations Equation (78) may be simplified to

<exz(t)sxz(0)>M1 =1} Is” 2 (wr- ,

0 (82a)

which shows that A] lines are broadened only by rotations perpendicular
to the symmetry axis ("tumbling") and are unaffected by motion parallel

to the symmetry axis ("spinning“). 0n the other hand, lines of E
symmetry are affected by both since

-(tD -D )t -(4D -ZD )t
<sz(t)8xz(0)>E=§r[lB?FI2€-' i H +|e§F|2e H i 1. mm

In order to test a model such as the diffusion model one needs

u>extract the rotational time correlation function from the

V i )
.... -.
[)3
Y I \|
AP.S\.I .
I

we <1), <R> at

“"3” functicr

 

“-35%. There

it 3‘9 cor

 

 

 

 

36

experimental spectra (Equation (7l)). The problem is slightly more
complicated than Equation (7l) indicates since all real spectra will
be additionally broadened by a slit function. Thus, Equation (71)

is modified to account for instrumental effects

Iisoho) if dwexpfiwt) <V(0)V(t)><S(O)S(t)>

= ‘3[<V><S>] (83a)

Ianisw) = fdwexp(iwt)<V(0)(t)><R(O)R(t)><S(O)S(t)>
= ’3 [<V><R><S>] , (83b)

where <V>, <R> and <S> are the vibrational, rotational and slit cor-
relation functions, respectively, and ‘3’ indicates a Fourier trans-
formation. There are two alternatives for obtaining <R>, the convolu-

tion method of Bartoli and Litowitz3o, and the Fourier inversion
method.

From the convolution theorem we may write

3’[<V><R><S>] =3[<V><S>]* 3[<R>] , (84a)

or

Ianis(m) = Iiso(w) *Ior(w)’ (84b)

Where the * indicates a convolution and Ior(w) is the Fourier

 

6..., 3f the r

l ._
l'j'fitlve .'

 

 

37

transform of the rotational correlation function.

Alternatively, we may write

-l
T‘M<V><R><S>]} =3 {Ianiswn =
3"]{3[<V><S>]} s‘Hlisomn

 

<R>. (85)

Clearly the Fourier inversion method (Equation (85)) is superior when

the functional form of <R> (or Ior(“)) is complex.

3‘ Ié It‘tten ir

 

 

HISTORICAL

I. NMR Relaxation Mechanisms

As was discussed in Section I, the perturbation expression for

probability per unit time of an induced transition
w = 2" |<k|V|m>|26{E -E -u } (86)
km '7? k m m

can be written in the Heisenberg representation as

no

wkm = E12- f d1<V(t)V(0)>e'i(m'k)T, (87)

where V are the interactions and are merely the various terms in the
Hamiltonian of the system which are made time dependent by molecular
motions.

Since the total relaxation rate is, in general, a sum of many
terms of the form of Equation (87), the problem of interpreting the
relaxation time comes down to the problem of deciding which terms in
the Hamiltonian of the system give important contributions to
Rl,total' It is possible to study Rl,total and obtain information
about each correlation function of the type defined by Equation
(87).

A. Dipole-Dipole Relaxation

The first investigation into relaxation effects in NMR was done

very shortly after the first nuclear resonance signal was observed

38

.1 I'.‘ v,
.: . uvfiS‘ ‘ ’

:;-0- P. ‘Hr
:u'vd IvU V

....'¢ Q'Oa'§ ...-
. a
«n . I I. d 5

‘ F
:‘iziar be r‘

 

f.

I"

er»
JEn, PU!“

 

39

in l945. Bloembergen, Purcell and Pound published34 in 1947 their
measurements of proton relaxation times in water and a variety of
hydrocarbon liquids, plus a few relaxation times for nuclei other
than protons. They showed that the observed relaxation rates could
be accounted for by two effects: inter- and intramolecular dipole-
dipole interactions. A dipole-dipole interaction between two spins

I and S can be written28

HJCI 3 Z F(q)A(Q) (88)
q

where the F(Q) are random functions of the relative positions of the
two spins and the A(Q) are functions of the spin variables. Con-
sequently, the time dependence of ”ac produced by molecular motion
is solely contained in the F(Q), which are proportional to the Y;

spherical harmonics
F‘o) = (l-3cosze)rig, F(]) = (sinecosee’i¢)rig,

F(2) = (sinzee'21¢)r;g (89)

and F(Q) = F(‘Q)*. With the assumption of isotropic random motion

the correlation of the F(Q)'s becomes
<F(q)(0)F(q)*(t)> = qu.G(t). (90)

Bloembergen, Purcell and Pound chose as a form for 6(1)

 

‘5
'3‘
8

I'\
M

=3 emu:—

:0...

fr .

15%;)
”‘W on the

Term] f

$739) and C

 

40
6(1) = <|F(Q)|2>e'|t|/Tc (91)
and so obtained the spectral density J(w) as

J(w) if dtG(t)e-im
(92)

Since at the resonant frequencies used mzt2 << 1, this expression

reduces to
J(])(w) = fg-rggrc, J(2)(w) --%§’r;§Tc (93)
(after evaluating the averages) and, consequently,

2 -
RI’dd = 2y4u I(I+l)rlgrc . (94)

Bloembergen, Purcell and Pound treated the intermolecular contribu-
tion by considering the correlation time for that process to depend

inversely on the diffusion coefficient

Tt ‘ 125' (95)
(the numerical factor is l2 rather than 6 because both spins are

diffusing) and calculated

4 2

, 31% P N
Rl,trans a ' (96)

.40.: A'F- '-

' Q
a
. .. nirv' '

 

«Ho‘d"-.. .
1...! 11c o
I ’

Etta visc:

Y. ;. .1
,1, 1. 5O '
. quali' :.

 

 

 

41

13

They evaluated both of these correlation times by Debye's application

01’ hydrodynamic theory appropriate to spheres in a viscous fluid:

3 3
Tc'%¥—’ Tt=%’ (97)

Applying this theory to water, Bloembergen, Purcell and Pound got
rather good agreement because the junp mechanism is appropriate for
such a strongly hydrogen-bonded system. In comparing the relaxation
time to the viscosity for a nunber of hydrocarbons the results were
only in qualitative agreement, undoubtedly because they neglected
the effects of dissolved oxygen in shortening the relaxation times.
This error seems rather ironic, as Bloembergen, Purcell and Pound
did study the effect on R1 of adding paramagnetic impurities, which

4

increase R1 trans since 7 in Equation (96) is then replaced with

Y2<u2>. where <u2> is the effective magnetic moment of the paramag-

netic impurity and is about l03 larger than 72.

L Scalar-Coupled Relaxation

For aqueous solutions of salts of lilnM and (.ld+++ it was found35
that TZ/T] was less than unity and. dependent on the proton frequency.
This was explained as due to the presence of a scalar coupling of
the form M 1-5 between the proton and the metal ion. The contribu-

tions to T.l and T2 from this term were shown to be

 

2 1’
“(ml-ms) TS

 

 

:e': Strength 15‘

33;": i" ( I .
J Z ]

eaten:

.a, j P ‘ .
23:11.) sral:

 

“by I: w‘ ‘
”fl, \' 4))“ r5 I

:‘ic.‘ln' " 3 . 4'
' uf'rl :cbzi‘lng ‘

z'f, 510. as SnCl

 

it": T.| even at '

 

 

' a
,_.J
-, «2mm: ar RE‘

   
 
 

:' . ~
«~WTQQR, P)
“M in 8%“. i

‘7 F“) '3 uteri an
‘3 he“
waxear quadr-

i‘v ~
~=fluctuatil

 

42

2 1.’
A S
R = _ 5 5+] + , (98b)
2,Sc 3 ( ) {1+(wl_ws)21_52 TS}

where 15 is the relaxation time of the electron spin. Clearly, at

a field strength low enough so that (ml-ms)z‘rz << l, T1 = T2 but
otherwise T2 < T]. The scalar coupling relaxation mechanism is also
Sometimes found to be important when S is a nuclear win”. In this
case the much smaller coupling constant is balanced by a much longer
nuclear (S spin) relaxation time. In molecules for which nuclear
Spl n-spin coupling constants are large and magnetic dipoles are
small, such as SnCl4, scalar coupling may give an important contribu-

tion to T1 even at high fields.

C. Quadrupolar Relaxation

Bloembergen, Purcell and Pound also compared relaxation rates
of H and D in 50% heavy water and realized that the faster relaxa-
tion of deuterium (although YD < y") was due to an interaction of
the nuclear quadrupole moment (present for all nuclei with I<l/2)

with a fluctuating electric field. The Hamiltonian is

me, =3 - Wt). (99)
where 6 is the quadrupolar coupling tensor and 7 is the electric
field-gradient tensor. The time dependence of the interaction arises
since 5 is quantized in the laboratory frame by the external mag-
netic field while l=l is fixed in the molecular frame, which is under-
going rapid reorientational motions. The V tensor is exactly of the

form of the F tensor in Equation (89), where now the 6 in Equation (89)

 

 

9
{142159 angle
we) wear. F

a o ' '
5272:3120

“m 3

 

‘Z'é’ir‘lt .5 << I

3‘19: that an ir;

“-75 expressed

 

43

refers to the angle between the Zeeman field and the synlnetry axis
01" the V tensor. Proceeding through the same arguments as for the

case of intramolecular dipole-dipole relaxation, we obtain

21+3 Q2 We a 2v 2
R = 1 + ) ( (100)
"q 2“51 (21-(1) TC

in the limit wt << I.

L $91 n-Rotati onal Relaxation

34 38-42

In gases, Bloembergen, Purcell and Pound and others
realized that an important relaxation mechanism would be the inter-
action between the nuclear spin and the overall rotation of the
molecule expressed as

ch = T . c - 3(t), (101)
where 3 indicates the rotational quantum state and E is called the
spin-rotation tensor. This interaction is made time dependent by
collisions which alter J. The magnetic field can be thought to arise
as the rotating electron cloud produces a field opposing the Zeeman
field. In the gas phase the time between collisions is directly

related to the rotational correlation time 16 which governs the

di pole-di pol e interacti on1 1

T
J
“re ' m, ("’2’

 

5233396 1'" Sect

L322] : L

@5312” )5 tr

 

3:3: gin-rater
:3 he second 1
3"2': at nucleus
3234 gass.

has first t'
are: in liqu‘ds,
ficmsistent witr
[351531060 ar

Fm hat fluori re

 

Ts‘imiecu‘aes 5
3533316419016 1“
. Earlier mez
TB in liquid HF)
:‘Ja‘ied on the b?-
‘cfne in an HF m
::‘ar::iin such as
'«I act on the fi
fiatien through
lastudy of thc
.: Mutter of y

{I

a ”.1
“fixation tip

44

as discussed in Section I, thus the total rate can be expressed a534

R = 272[%-H'2J(J+l)+3H'2 i J*‘ + 11c. (103)

l,total
The first term is the spin-rotation contribution and for hydrogen
gas, the spin-rotation constant H' was found to have the value 27
gauss. The second term is the intramolecular contribution with H"
the field at nucleus l originating from nucleus 2 and for H2 has the
Value 34 gauss.

It was first thought43

that the spin-rotational relaxation was
Quenched in liquids, since relaxation rates in the early experiments
were consistent with dipole-di pole interactions. However, measure-

44'45 on oxygen-free fluorocarbons

ments by Gutowsky and coworkers
showed that fluorine Tl's were consistently shorter than the proton
Tj's in molecules such as CHFClz, where the correlation time 16 for
the dipole-dipole interaction should surely be the same for the two
nuclei. Earlier measurements of hydrogen and fluorine relaxation
times in liquid HF47 also gave 11F < 1]” but this inequality was
explained on the basis of unequal correlation times for hydrogen and
fluorine in an HF molecule. However, in the case of CHFCl2 no easy
explanation such as this was possible. The one known mechanism which
wouliiact on the fluorine nuclei more strongly than on the protons,

4a 44

relaxation through chemical shift anisotropy 9 was PM“ Wt

from a study of the field dependence of T".
A number of years after the initial observation of shorter

9
F relaxation times. it was proposed46 that a different statistical

L: as PESDORS‘
31mm. This he
.13 :~:‘.e‘ned unje'
W's-"ea tart.
‘rzmei Dreamed .
s-s-g;-;,, m
:a 1‘: the mien.
‘1": arc-tier, ...-3
7:51.11? any)“ .‘
75 ‘nich descri
333: ES 3035 the a

‘3 S"ii-Tar than t

V‘”

 

 

3‘: int :0 “HR is
‘53. 49
mm em?

~13!)

45

process was responsible for the extra contribution to the 19F
relaxation. This new approach was necessary because the relaxation
times obtained under the assumption of rotational Brownian diffusion
gave incorrect temperature dependence to the 19F Tl's in CHFClZ.

The model proposed to account for the fluorine relaxation was a
transient-Jump model, where the mechanism became operative only during
the time the molecule was making a rotational jump from one orienta-
tion to another, with an interactive strength proportional to the
molecular angular velocity. The angular velocity correlation func-
tions, which describe the time dependence of the angular velocity,
decay as does the angular position correlation function in a time

much shorter than the Larmor frequency, hence the measurable quantity
important to NMR is the angular momentum correlation time.

Hubbard49 examined this mechanism by calculating the angular
velocity correlation function for a spherical molecule undergoing
rotational Brownian diffusion and making the assumption that the
angular velocity obeyed an equation analogous to the Langevin equa-
tion, which is often invoked to describe translational Brownian dif-
fusion. Hith the condition that either TJ << T9’ or TJ >> ‘9’ he
was able to obtain a fundamental relationship between these two cor-

relation times,
n=1 (104)
a.) BET’

and an expression for the spin-rotational relaxation rate

_ ZIkT 2
Rl,sr ' "2 eff TJ ’ (105)

 

 

es. is the more"

. a ‘ O.
#37:: interac.ic

 

 

5513;131:295 a:
.67» 7515. usf
“1' ‘5 COmared

 

7’:‘1:reasing ra
2‘ :JE'iChl'ng' Of
2: “"5“ K.) as

Siifi-mtati One

existent possit

be hOS
‘zs
reorient

.3. .
v 1“

46

where I is the moment of inertia and ceff is the effective spin-

rotation interaction constant (in radians/sec)

2 _ l 2 2

(l06)
Inserting values appropriate to the liquid state it is apparent that
TJ << Te. Thus, using a quite different model (rotational Brownian
motion as compared to a transient-jump model), Hubbard was able to
obtain the correct temperature dependence for spin-rotational relaxa-
tion (increasing rate with increasing temperature) and also explain
the 'quenching' of the spin-rotation interaction in the liquid
(IJ = I/6kT 16) as opposed to the gas phase (Ta = ‘9 (2J+l)).
Spin-rotational relaxation in the solid state was not considered

an important possibility until the observations of Blinc and Lahajner50

of the ‘9

F relaxation in the liquid and solid phases of SFG, TeF6,
and SeF6. These molecules all have plastic crystalline phases and
show a frequency-independent relaxation rate which decreases as the
temperature decreases, both in the liquid and plastic crystalline
phases. Since, in the case of SF5, spin-rotation had been shown51
to be the dominant relaxation mechanism in the liquid phase, the
continuation of this relaxation curve into the solid phase led Blinc
and Lahajner to suggest that spin-rotation is also important in the
solid. Spin-rotation was later recognized to be important in the

‘9F in MoF6 and wF652, ‘H in NH 153'54.

4
57, and 31F in phosphine58.

plastic crystalline phase for
3‘? in white phosphorusss’ss, ‘H in "(14cm4
The reorientational motion giving rise to spin-rotational relaxa-

tion in the solid phase may be described either as small-step

 

Trig"; 30:8

:3“? '15 an ”Of:

Q

“3332!, ‘0' MS

—9.

‘1 3e ill-35 ’0’

t:
te-ZIES"

(5'5: is the "EV
:‘2':;:=:s expressi c.
33:33. (135))-

e"'e:resented t7”:
isefiine white :
if": an eight-fol
‘m‘fzai calculati

ear orientatior

Ite‘ .
..ave been re“.

 

 

 

47

diffusion53, as in the Hubbard model49 for liquids, or by a tran-
sient-jump model“'52 in which molecular libration about an equilibrium
position in an n-fold well is interrupted by random torque impulses
which rotate the molecule to another potential well. At the Larmor
frequency, for plastic phases, the contribution to TJ comes mainly

from the jumps from well to well52

55

and the spin-rotation relaxation

rate becomes

R g ZIkT

l,sr ‘;2"C§ff¢2TJ ’ (‘07)

where o is the mean jump angle. This may be compared with the
analogous expression for the case of isotropic rotational diffusion
(Equation (105)). Boden and Foiiand55 found that Equation (107)
well represented the spin-rotational relaxation rate in plastic
crystalline white phosphorus (P4) when ¢ = l20°. For most plastic
solids an eight-fold (cubic) potential is more appropriate, and
numerical calculations of the full time-correlation functions for

angular orientation and angular velocity for the transient-jump

model have been reportedsg.

II. Rotational Motion in Liquids

A. Modifications to the Bloembergen, Purcell and Pound-Debye Treat-

ment

There were two early attempts to correct the failure of the

60

Debye theory for nonviscous fluids. Gierer and Hirtz took into

account the discontinuous nature of the liquid in a simple quasi-steady

._.. 0;“
gnu-3““

C‘. ‘
I I '-

.'- fl
.2 'é 5126's a ='-

$55 is the ra

$3.... For 5

t": ‘25 U5 the ‘

‘zszia 433 (a S:

9:11)“

based
. 62

..es theory
taxation time wh

vent pair. App

 

. a) be combo

Iii-E‘s '
'HTEdlCIS th=

\ l
\
i 1'

see
I“)? Spheri r
2's»

wen

 

48

state calculation of the torques acting on a rotating molecule by

its neighbors and obtained a friction constant
a = 8ma3[6(%) + (1 +941“ . (108)

where a is the radius of the molecule and b is the radius of the

neighbor. For b = a,

g = m. (109)

which is l/6 the value predicted by the Stokes-Einstein relation.

For b/a 2100 (a spherical macromolecule in solution)
a = 8wna3(o.970). (110)

Hil16] based her calculation of dielectric relaxation times on

Andrade's62

theory of viscosity and obtained an expression for the
relaxation time which depended on the mutual viscosity of a solute-

solvent pair. Applying this theory to pure polar liquids she obtained

Tc 3 3 3;{5' na3 , (111)

which may be compared with Debye's Tc = 4nna3/ki. Thus, the Hill
model predicts that Tc would be smaller by a factor of 3(3-/§)/B
z l/5 than the Debye Tc’ Experimentally, dielectric relaxation times
fbr nearly spherical molecules (for which one would expect the Debye

treatment, or a modification, to work well) were found63 that were

 

 

~i'.‘ I. P.
..P M ‘P
'- ‘.. ...". ~-

 

"e first exte

 

rarest of rate?
smeared mg!
series of ani:
73353113, d,

5.23:: for a r‘l :‘

3‘5 “Skits Shown

 

the: We tlf'e
33.32% '\
° (33.) as:

‘n' :04
' I

- ".7!" functii OHS
«atria Maxat
'95 carreiati

fit‘gn tllTEs _ F

DI 1/

:fiiFn ‘
den: Pu

3.

“4:54

49

as small as eight one-thousandths of the predicted value.

B. Anisotropic Rotational Diffusion

The first extension of the Bloembergen, Purcell and Pound-Debye
treatment of rotational motion was given in three independent papers

20,64,65

which appeared roughly simultaneously. These authors considered

the problem of anisotropic rotational diffusion, basing their work

1118 and Favro‘g, of the diffusion

on the solution, due to Perri
equation for a rigid ellipsoid, which was discussed in Section IB.
These results showed that rotational diffusion of an asymmetric top
required five time correlation functions for t = 2 spherical harmonics
(Equation (35)) appropriate to magnetic relaxation or three time cor-
relation functions for the 2 = l spherical harmonics appropriate to
dielectric relaxation. The NMR relaxation time then depends not on

a single correlation time but on a complex function of all the cor—

relation times. For a symmetric top this function 1521
T =(3cosze-l)2 + 3sin26cosze + 35in4e (112)

where B is the angle between the vector interaction in the relaxa-
tion mechanism and the principal axis of the diffusion tensor, and
B1 is the perpendicular component of the diffusion tensor. In order
to show the relation between the effective correlation time Tee =

ff
f(e,K) (K s DIIIBL) and the correlation time obtained from the

Bloembergen, Purcell and Pound-Debye treatment (we), Shimizd66

computed the ratio Teeff/Tc as a function of axial ratio for prolate

 

 

"2.1 It] N ob'
" fear that it
-.e“s:°.r3$‘7c mi
:33: reiaxeti cn
{term tines f:
52's:";.isning :

safely true fcr

7E "E ixdtlon TitE

 

3'51 EC nuclei w‘

£33537“ mechan‘
“$934195, inves
with“ rate fc
f‘rt ; - n .

" ‘ ln Eduatlo

Hazel chosen.
mic-vial diffus‘
Toe sym-etri c-t:
‘32”

“y be cai
' /_
|‘WEY‘E I l

' “'-"ess discu

50

(Figure 3a) and oblate (Figure 3b) symnetric tops. From the figure

it is clear that for many molecules, i.e., those with axial ratio

54, anisotropic motion does not make a large difference in the cal-
culated relaxation times. Consequently, absolute calculations of
relaxation times for a given nucleus in a molecule have little hope

of distinguishing between anisotropic and isotropic motion. This is
especially true for spin-l/Z nuclei, where several mechanisms,
including relaxation through paramagnetic impurities, compete for

the relaxation rate. Recognizing this problem, many investigators
turned to nuclei with spin I > l/Z, with the understanding that the
quadrupolar mechanism would dominate the relaxation rate. A number

of molecules, investigatedm-m by measuring the temperature-dependent
relaxation rate for two quadrupolar nuclei in the same molecule.for
which a in Equation (llZ) is different, showed different activation
energies for the two nuclei. This fact shows that, independent of

the model chosen, the motion in these molecules is anisotropic. If
rotational diffusion is used to describe the molecular motions in
these symmetric-top molecules then the two components of the diffusion
tensor may be calculated. Table 2 compares the ratio DIIIQI.W1th
Il/Ill (where I is the moment of inertia) for these and other molecules.
As Huntress discusse52] in his review of anisotropic molecular re-
orientation, the diffusion equation is equivalent to the Schrodinger
equation for rotation with the diffusion tensor D replacing the
moment-of—inertia tensor MZ/ZI. Since the relative magnitudes of
the components of the inertia tensor determine the spinning and
tumbling rotational velocities in the gas phase, by analogy the

inverse components of the diffusion tensor determine the rate of

Figure 3.

51

 

 

 

 

 

 

The effect of axial ratio n and the angle 60 upon the ratio
reff/tc: Teff is the effective correlation time obtained by
assuming the molecule as (a) small prolate spheroid, (b) a
small oblate spheroid; Tc is the correlation time derived by
the Debye-Bloembergen theory, assuming the molecule as a
small sphere.

52

Table 2. Experimental measurements of anisotropic motion in liquids.

 

 

 

Molecule (3?). D /D I /I E E Ref.
|| .L .1 ll 91 all
5030" 25 8.89 10.11 1.7 30.1 0.8 30.2 71
10.35 2.00 0.73 74
v0c13 30 1.4 0.529 1.9730.05 2.3030.05 73
cc13cu 25 2.1 0.917 2.5830.14 1.9 30.3 73
8013 0 .75 0.500 1.2830.02 1.5530.14 73
00013 20 1.9 0.530 1.5 30.10 0.7 30.1 72
003000 -30 14 10.1 1.7 30.2 0.5 30.1 75
P013a 25 2.38 1.78 1.7 1.7 75
P8r3a 25 3.15 1.89 2.3 1.7 75
017013a 25 3.32 1.35 2.3 ---- 75
c505 b 57 4.5 0.500 2.7 30.3 0.3 77
0585"“ -1 2.1 0.498 3.1 30.3 1.0 30.3 78
d3-s-Triazine l52 2.3 0.498 3 4 :l 0.3 3] 79
0031 25 4515‘ 20.32 1.9a 0.8 30.1 80
20 2233d 2.1b 0.4 30.1 81

 

 

aPerpendicular motion estimated by microviscosity theory.
bPerpendicular motion determined by Raman lineshape analysis.

cPartially deuterated molecules were studied.
d

Uncertainty due to estimate of deuteron quadrupole coupling constant.

 

 

efffifm 35C;
5:119 Emmi
s33. '10 the COT
s:i“.sion liri
i'es'ar irteratt
z't'taiiy th‘s
{Estate we r
2135.1113th ax
i" dated to the
371232.33: intent

33:73 notion
she to guest
i'tne spinning hi
73?:etic or die

:15 CH4 0" CU

|
’1:-

n-D‘Effusiond

.SOImpic 8rd
'="e;axation r:
..iifsng] difoS‘

3763.1
e Such as f

 

W a
‘3 d descri;

 

h |

£9244
.ai "Diem-(I
”Nunez-8

mtion are

53

reorientation about the different molecular axes in the liquid phase.
Since the components of the diffusion tensor can be shown to be
related to the components of the inertia tensor2] as D1 ”kT/Ii in

the diffusion limit, then in liquids with sufficiently weak inter-
molecular interactions we would expect to find Dll/DuaIl/Ill. Ex-
perimentally this is not the case, as all molecules in Table 2 except
BCl3 rotate more rapidly about their symmetry axis than perpendicular
to the symmetry axis. The fact that the diffusion tensor is not at
all related to the inverse inertia tensor shows the importance of
anisotropic intermolecular forces in the liquid.

Rapid motion about the symmetry axis, as in CD3CECD or CD3CN
leads one to question the validity of small-step rotational diffusion
for the spinning motion. Alternatively, one may consider the problem
in magnetic or dielectric relaxation with much more symmetric molecules,

such as CH4 or CCl3Br.

C. Non-Diffusional Models

Isotropic Brownian diffusion gives correlation times (and there-
fore relaxation rates) which are much too large. Clearly, if isotropic
rotational diffusion is not a valid description of the motion in a
molecule such as CH4, then rotational diffusion will not likely be
valid as a description of the spinning motion in a CH3X molecule.

The calculation of correlation times for isotropic rotation in
spheroidal molecules was the subject of a series of papers by Steele

82-84

and coworkers They observed that the forces governing transla-

tional motion are quite different from those restricting free rotation

"IE 3:254: “A
sattztst‘P b‘et'e
.3333, and I?”
:2 .‘r‘l:t‘? . .

:2 .2 be well 1'
.7 ayerine'itai
”Beaten, F’W‘Ce
:"t-iSte-ele’s re
27:.Tat‘en based
‘:.7:' as detemin
I'eufar interact
extant tine cor
5") non-Farkovia
ismsiaae. Th
Patina] molecul
$2 resent) y, win
Tif'ate should i
‘37 of rotation,
33*“ otservatior
‘53th of d P

-:. kid-lived PC

‘
p.
4.

 

Inc
'- P'siide

lei.
'3’ Vary Sma‘

54

in the liquid, and one should not in general expect to find any
relationship between the viscosity, which depends on the translational
friction, and relaxation times, which usually depend on the rota-
tional friction. The calculated time correlation functions were
found to be well represented by a Gaussian function and comparisons84
with experimental data showed better agreement than the Debye-
Bloembergen, Purcell and Pound model had shown. Atkins85 has ob-
tained Steele's result as the classical limit to a quantum-mechanical
calculation based on a model wherein the molecules rotate in the
liquid as determined by their free-rotor Hamiltonian. The inter-
molecular interactions enter the model as a Gaussian damping of the
resultant time correlation functions. This theory was modified to
allow non-Markovian process in order to analyze the relaxation data

86

for ammonia The effects of spin-rotational relaxation in these

spheroidal molecules was not considered for this inertial model until

quite recently, when it was shown87

that the spin-rotational relaxa-
tion rate should increase as Tl/z. The importance of an inertia
model of rotation, at least for small molecules, is evident from

88‘ 92 of rotational fine structure in the far

recent observations
ir spectrum of a number of molecules dissolved in SF6, which indicate

that long-lived rotational states are present in these systems.

D. The Extended Diffusion Model

Gordon's93

model of rotational motion of molecules was an attempt
to provide a smooth transition between strictly diffusive motion,

that is, very small step size, and motion characterized by large

 

:3? $185- TIE
3:2: 2! “my
2:5 3" art? in”
".1223 The ff)
213‘ 3‘15" is "
113372“ the reOFiC”
:33: the miec
:L-ezentiy in t?“-
E-ajs exaonent‘? a)
fetish functions
:zie'iiertial Inc

Tie urinary v
e- tie between
$73 momentum c
353:3 generaliz
1'3; although the
is studied in t
fatness-y as It a
3'33: features a

“tresui

t5 for tr.

 

55

angular steps. The problem of large step reorientation has been
treated by Ivanov16 and a general theory of reorientation through
steps of arbitrary size by Cukier and Lakatos-Lindenberg‘5 and
Cukier94. The fundamental difference between these approaches and
that of Gordon is that Ivanov, and Cukier gt_al,, employ a model
wherein the reorientations occur instantaneously, whereas in Gordon's
approach the molecule rotates as governed by its inertia tensor.
Consequently in the "jump" models the correlation functions are
always exponential while in the extended diffusion model the time cor-
relation functions are Gaussian for an initial portion of time due
to the inertial motion of the molecules between collisions.

The primary variable in the extended diffusion model is the
mean time between collisions, which may be identified with the
angular momentum correlation time. The extended diffusion model
has been generalized to include sphericalgs'98 and symmetric”’100
tops; although the symmetric-top treatment is required for the mole-
cules studied in this investigation, in this section we will consider
the theory as it applies to linear molecules, where all of the im-
portant features are present but little of the mathematical complexity.
The results for the symmetric-top case will then simply be stated
as necessary.

The correlation function is calculated by following the micro-
scopic time evolution of a single molecule and then averaging over
a Boltzmann distribution of velocities. The motion is pictured as
periods of free rotation, during which the molecule behaves classically,

interrupted by collisions which instantaneously randomize the angular

momentum vector without affecting the orientation of the molecule.

 

 

Zr 33 H'dlff':
21",; VECtOr TS
region and tr:
71‘s ”£3me

:2.- far a rolec

 

r‘a'restsr (lines
First diffusiil

so
§

"I t] the m]
rate-tion of J;

I... On ll!

“3) is, 1

 

56

In the M-diffusion model only the orientation of the angular
momentum vector is randomized; in the J-diffusion model both the
orientation and the magnitude are randomized. Clearly the M-diffusion
linfit is inappropriate for real systems, as the angular momentum
vector for a molecule must change through collisions, but the con-
sideration of M- and J-diffusion limits allow the strength of the
intermolecular forces to vary.

93

Following Gordon , we now calculate the correlation function

of a vector (linear molecule) undergoing extended diffusion. During

the first diffusive step the molecule is rotating with velocity “8

so
U(O) - U(t) = COSw1t. (ll3)
At time t1 the molecule undergoes a collision which changes the
orientation of J; at the end of this diffusive step the projection
of U on U(O) is, from spherical geometry,
U(O) - U(t) - cos mltlcosw2(t-t])

- cosasinwltlsinw2(t-t]) , (114)

where cxis the angle between J] and J2. Since the collision is

postulated to randomize this angle,

<cosa> = 0, (ll5)

 

#21 2‘8 “8'3. QC

:2 ‘32 first c"
V’E’IE) IVE HMS.

0f tI

(,Tf".1

”\JIV-
53’ heralize

.I ,
C “no!
' 'riq

.-

11))“1 ‘

 

57
where the average is taken over many molecules. Therefore
<U(0)U(i.'.)>2 = coszt-lcosw2(t-t1). (116)

Since the first diffusive step may be terminated at any time in the
interval t we must properly average this projection over different

values of t]
t
<U(0)U(t)>2 = t"1 [dtlcoswlqcoswzu-tl). (117)

We can generalize this result to an arbitrary number of steps; for

molecules in the (n+l)th diffusive step

_ n!
<U(0)U(t)>n+1 - fifdtncoswn+](t-tn)f dtn_1coswn(tn-tn_])
O 0

t3 2
x...x f dt2c05w3(t3-t2)f dt1c05u32(t2-t1)c05w1t]. (118)
O O

The correlation function will be a sum of contributions from molecules
in the first, second,...,nth diffusive steps. If we represent the
distribution of number of steps by a Poisson distribution. since the

total time t is not long compared to the time between collisions,

f(n) = 3‘; (5'24". (119)

th

then the contribution from molecules in the n step is weighted

by this probability that n steps will occur, and therefore

 

- L I

ignohfion .’

€110): _
l
'~“ the Nil“

.2. In mm c

 

 

58

no. t
<U(0)U(t)>=e"t/Tz't'nfdtncOSmth-tn)

n=0 °

tn

*1

t3
dtn_-ICOSwn(tn-tn_] )X. o .XI dt3COSw3(t3-t2)

n
xfK dtzcoswz(tz-t1)c05m1t1 . (120)
O

For the M-diffusion limit wh+l = “n

time correlation function is Equation (l20) averaged over a Boltzmann

distribution of rotational velocities wexp(-'%w2)

= =w2=w~l, and the

co

<U(0)U(t)>m ijf.dmexp(-«%w2)<U(O)U(t)>. (lZl)

In the J-diffusion limit the rotational velocities are not equal
and each must be separately averaged over a Boltzmann thermal distribu-

tion. In terms of the free-molecule dipole correlation function Fo(t),

oo

F0(t) = <coswt> = f cos(wt)wexp(- %w2)dm, (122)

the J-diffusion correlation function is

no t
<U(o)uu:)>J = e't/‘Z 1‘"[ f dtnF0(t-tn)
n=0 °

tn t3
X f dtn-1Fo(tn'tn_1)xooon dt2F0(t3‘t2)
0

t2
x f dt1F0(t2-t])Fo(t1)]. (123)

 

 

 

 

«
CT

'3‘; .I
-..“.3: {hairy b

 

- ‘Ecrientatjona
‘1 "cm-d1

ffusion

I
, II
M .

n

"5 M, the;
'09“ .
J“: in SpIte ‘
{Emtamy TE]

fixation Pale

59

Both the M— and J- correlation functions were evaluated numer-
ically in Gordon's paper 93.

The M- diffusion limit has been examined101 for collision
distributions other than Poisson. The primary change in these

95 99 molecules are considered

formulas when spherical or symmetric-top
is the replacement of the various coswt terms with the 0(9) rotation
matrices.

A number of other papers on the subject of extended diffusion
have appeared which have treated Gordon's M- and J- diffusion
models‘oz']04 or a similar modellos. Also a series of papers has
been published, based on the projection operator treatment of linear

1106.107

response theory by Mar , which attempts a more general approach

to reorientational relaxation by including both extended diffusion

and non-diffusional reorientation as limiting cases‘os'lll.

E. Experimental Tests of Extended Diffusion

As yet, there have been few tests of the extended diffusion
model, in spite of countless relaxation studies in which the data
are probably reliable. This is so because in the usual case the
relaxation rate has contributions from several mechanisms which can-
not be separated. Since one wishes to know TO and tJ,the best type
of system to study would be a linear or spherical-top molecule with
one type of nucleus relaxing through spin-rotation and a second type
through the quadrupolar mechanism. Table 3 lists those molecules
which have been investigated by NMR or NMR-Raman methods as a fluid
phase in order to test the applicability of the extended-diffusion

 

 

 

 

 

 

 

 

N. p a .. u - - c J? i o r -n .
.bvz ncovm‘a—UEOU h?@L.¢U.m:OU U «.0 watt! 3.0m: UNLOQ‘COE Q~DUQ~QE
>£CLGC0¢C< atzpt> 6.15%... mUOCUan §W~UEZ
. 05!! - H§-.§ L b ‘w finaifiv-ufii,‘ 1.: L :6 with In: 9 Q li.‘ hi..-§\ nih‘h.‘ 1., ..i i It ti‘ REP! \

 

60

 

 

 

 

N G F .
om_ q o: 0 am A—umm. p Aaa_v c Pom" um_ a c_u
a: F . n
aNF a o: o P ow a_m an
mNF Pavbcacc ---- «:29 4a ~c.F» cmmpp eucm
amp Pawuaacm ---- «:20 4a Nh._» cmmF. acme”
em a ---- «:20 4a N».Fc cmaFF epucm
---- as: N ._ a
RN, um=OFaeo=a Do 4a b » aamo~ Foam
---- N . p a
oNP a u an Apummv c Ala—v p .ummum_ you
m~_ a ---- ago an .sam._p do. ecu
¢~_-_~. ---- mzu bu o._h :— emu
covuoucmpgoms
cup asan-maaap ---- as _h um, um
mp_.mFF umaopaeoca ---- use an NF.F» zm_m= an:
mFF-mFF unsopmsoca ---- use A“ Nb.Ph Fumm.= .0:
b
Np_ a ---- 0 4a u_h UMP mmu
.eum mcoymapucou «umcmvwmcou u we magma vow: omsouwcoz upsumpoz
aaoguompg< amapm> m.asoh muogpmz wmpuzz
.covmammpu umucmuxa mo mummy Foacwspcwaxm .m mpnah

 

.fih .Aah.

 

 

 

 

 

 

 

.551 fiCnuvmnaPUr-OU «...vmewvvaCU U .50 ”aw-Lam Univ“
\fi..CL&mef-< ficmwuapfl) .QEQF “nation—w: UmwawnNWWon‘OQz Wu‘unvmvhoz

 

 

 

 

 

 

 

61

.Lm. mmu.F

Pm soc»

.~.o u my u my bacon Featurcu on» on can

u

a camcmamm op mgumcmcpm v—opm cmeomN acmcwtwpc :. vocammme mm: —pu

.cowumpmc vgmnaax «mm m:_m= anew copamxmpmc mzz acct tucwmano m? «:2
u

can .ucmepcqum Emma cmpzumpos a a? vocamuoe mw

.muuu mcpupavzm paupemgu an» east nm>pcmu m“

a
.Aacpoq

—mo.uvco as» uaoao cowmac assumcwaeou mguv c.3Fm mmcovumo .uc.on Pmuwuvcu as» a» acroa mcwu_me on» sect
mace; u*=UVF «crucmngm .Aucvoa mapppon ms» op acwoq wavy—we mg» soc; appmsmav mucus vwaaPF pmwucmqudga

 

 

 

 

a a _. a m D
2: «2 a o: “a E ._. cm “.2 n. a u
_m, a «H». __ap no an ape VN».Au V.h _u .d am_uu
__ u ._. mm mp mm mp
H a
p
.ymm mcovm:_ucou umcmuvmcou 0 mo mmcnm can: umcouwcoz «Paompoz
xaoguOmwc< m=Fm> ~.aemh muogum: Popuzz

 

 

.ua==.p=oo - m aFDah

 

 

z‘s:.'.ar srznetr

2': act inert:

T3715; my D»:

255.5 is off-a;

g

I“. Cfmxlca‘afl

'5‘ o

:‘3- "
... 9“

~oamo‘.

(T‘

'4'

i‘ié‘ed.

l. ..
I.E'.‘H‘Fnh

w

 

62
model37’112'133. Table 3 shows that a variety of techniques have
been used to investigate the temperature dependence of re and TJ and
also that J-diffusion has been remarkably successful for these small
molecules, the majority of which are spherical tops (or were analyzed
as such). In all of these cases the spin -l/2 nucleus is on the
molecular symmetry axis; so if anisotropic motion was not considered,
or is not important for reason of symmetry, as in CCl4 or C52, Equa—
tion (l05) may be used to obtain TJ. Naturally, if the spin -l/2
nucleus is off-axis or if the motion is anisotropic, the analysis is
more complicated. In only one case131 (CCl3F) was extended diffusion
applied to a molecule of lower symmetry and the effects of anisotropy

considered.

F. Investigations of Extended Diffusion bkaisible and Infrared
Spectroscopy

Classical forms of spectroscopy (Raman, Rayleigh, and IR) can
be considered to test the extended diffusion model directly since
the full rotational correlation function is, in principle, measured;
from this 16 and TJ may be extracted. These experiments are less
satisfactory in that the two correlation functions are not determined
independently. At the present time NMR relaxation experiments are
the only independent measure of TJ, although other methods have been
discussed‘34’135. The difficulty with spectroscopic methods of
determining time correlation functions is shown by Equation (83),
since generally it is not possible to simply Fourier invert the

bandshape. In addition, Raman scattering experiments suffered from

 

 

299’“,

D n‘
.341”

":7 3353558j ir

:::ér‘:e'ta] 1y.

'It "I ‘ .'
.v: ‘E‘f‘fha‘mr

 

 

aired in data
a: tie zemth n:

. 3991 shown 3

U“
i‘.

5E5 ‘ except ‘
Ami-r 0f meaS~
:: the IR tech”):
:Ptzeiure 1'5 to "

'::er of

Rayieli
Em“
em of ti

-:;i€‘;;h effect.

iii-IR] 58’

.
a“ .
9,5”

ments repc

 

63

an ill-defined scattering geometry before the introduction of the
laser as a light source.

Raman2’9’31’93’136 and infrared2’9’93’137'137 band shapes have
been discussed in terms of molecular motions, also the relationship
between Raman and nuclear spin relaxation has been examined and it
has been pointed out that Raman bandshapes can be used to predict

nuclear relaxation times‘40.

That this has not been exploited is
perhaps a result of the ease of measuring nuclear relaxation times
experimentally. Recently the experimental techniques for isolating
the reorientational broadening component of Raman lines have been
explored in detail with a number of systems studied125’141'157,

and the zeroth moment of the Raman rotational correlation function
has been shown to agree satisfactorily with the NMR correlation

timesl4l 155_

except in the case of chloroform at high pressures
A number of measurements of the rotational time correlation function
by the IR technique have been reported152’153’157'166; the usual

procedure is to neglect other broadening mechanisms‘58’159.
dl49-15l,167-l7l

A smaller
number of Rayleigh experiments have appeare , due to
the problem of the uncertain contribution of the intermolecular
Rayleigh effect. In addition, there have been a small number of

158,172

Joint NMR-IR , NMR-Raman77’81’125’155, and NMR-Rayleigh174’175

experiments reported.

'0; p:
'1’; TranS‘ '. .3

:a'r‘retric net?
fret-e! anorely i
22.? is a scl‘id-
2'21 snarges to cv-
'-'-S"'=.v spserical or

5:13;], originail:

' o
‘Floe

“tion signals 3
1' :is model it is
asztential contain"

357 (T

I” an alt:

:ewiers are greatev

Tzn in

my?“ 0f transi‘

a“? i. the number

l'ysm s

E ,

mi: c"Mal str

$3,533. ‘ I
9 0f C3v SY-"i

6d tre

Wet!"

 

gig media
its-cules of th‘;

i the rotation

64

III. Plastic Crystals

A. Phase Transitions in the Solid

A solid-solid phase transition is a not uncommon phenomenon;
McCullough176 states that of 300 organic compounds studied in detail
by calorimetric methods, more than one-third exhibited some sort
of thermal anomaly in the solid. The most obvious cause for such
behavior is a solid-solid phase transition wherein crystal struc-
ture A changes to crystal structure B. When the molecules are
roughly spherical or possess an axis of high symmetry another pos-

sibility, originally suggested by Pauling‘77

, is that the phase
transition signals the onset of molecular rotation in the solid.

In this model it is helpful to think of a molecule as rotating in

a potential containing n wells, with the barrier between wells less

l178, the

than kT. In an alternative model, proposed by Franke
barriers are greater than kT and a jump from one preferred orienta-
tion to another occurs through thermal fluctuations. The observed

entropies of transition for many molecules which have plastic phases

agree reasonably well179

with entropies calculated from AS = R in n
where n, the number of preferred orientations, is determined by

the crystal symmetry and the molecular symmetry. For example, in

a cubic crystal structure there are eight potential wells, and a
molecule of C3v symmetry will have eight distinguishable orientations
for a predicted transition entropy of 4.13 e.u. Crystals made up

of molecules of this type should have quite unusual physical properties

due to the rotational freedom; quite early it was recognized that

2:; 553213 were the
ratifies 0‘ f..5'
:2131 the for”?!
gaiastic) 53"“: i
manslaticnal a'
15361335335] 2.1:
zea'riaiy gained
255:": will only
it"ZTm, so LSm = ‘3
fast solid will b
":3? since a plas:
be :naracteri sti c
“? vapor pressures
fistic solids oft-3

’he early meas.

.se'l-‘Ddynaml C a

five?

ic data are or:.

he ' '
.‘e indication

n n.
— — — —

Ir? *
.nermcdynami c t

 

65

many solids where the molecules were of spherical shape had unusually

low entropies of fusion‘ao.

That this should be so may be seen quite
readily by the following argument. The entropy of fusion for a normal
(non-plastic) solid consists of the entropy gained as the molecules
gain translational and orientational freedom upon melting. For a
rigid ellipsoidal molecule this is 3R/Tm. However, molecules which
have already gained orientational freedom at a phase transition in
the solid will only gain the translational freedom at the melting
point Tm, so Asm = l.5R/Tm. Furthermore, the melting point of a
plastic solid will be higher than the melting point of a non-plastic
isomer since a plastic crystal process more degrees of freedom.
Other characteristic properties of plastic solids are relatively
high vapor pressures in the solid, small liquid temperature range
(plastic solids often sublime) and a soft, waxy solid state.

The early measurements on the nature of plastic solids were

all thermodynamic, and as Darmon and Brot18]

point out, thermo-
dynamic data are only a measure of the disorder of the system and
give no indication of the dynamics of this disorder. Consequently,
from thermodynamic data alone one is unable to distinguish between
a high-barrier, thermally-activated jump model and a low-barrier

continuous-rotation model.

B. Spectroscopic Studies of Phase Transitions

In order to understand the dynamics of the thermodynamic dis-
order in plastic solids it is necessary to study more than the

equilibrium properties. Dielectric relaxation studies182 have

1712:1715 “7:5"
mean 15 ““3;
:3 me CfeiECT-HC
w: 3;; liquid 5
1'12th requires
7.x1armlecules
:zi‘ec. Since the:
:e'e:tric relaxati:
father hand, do '
we complete 1"
Ethis method. “I
j |
at; both continue-I
22o; the temeratu I
:‘astic crystal; th-j
ottiscontinuitiel
53"!) a Phase trarl
AIthe change in
Therm for the

eeoerce of the ll

lire“

(8/in2)mi
his .
lift; '

, ” before, and B

 

66

shown that the rotational motion contributing to the dielectric
polarization is unaffected upon freezing a plastic solid13’184;
both the dielectric constant and the activation energy change little
between the liquid and the solid183tlg4. However since dielectric
relaxation requires the presence of a reorientating electric dipole,
non-polar molecules or rotations about the dipolar axis cannot be
studied. Since these are precisely the cases of special interest,
dielectric relaxation has proven of limited use. NMR studies, on

the other hand, do not require a permanent dipole moment, and most of
the more complete investigations into motion in plastic solids have
used this method. NMR studies of plastic solids have been performed
using both continuous wave (cw) and pulse techniques. Figure 4

shows the temperature-dependent second moment M2 for a characteristic
plastic crystal; the linewidth behaves similarly. In the cw experi-
ment discontinuities in the second moment, or in the linewidth,
signal a phase transition. The nature of the motion may be deduced

185

from the change in the value of the second moment and the activa-

tion energy for the process may be determined from the temperature-

 

 

dependence of the linewidth through the semi-emperical relation34
2 2
rd . C '3 . (124)
ade

where a = (B/2n2)'],-yis the magnetogyric ratio of the resonant spin,
GR is the line width in the transition region, and C is the line
width before, and B after, the narrowing. In Figure 4 the second moment

is seen to drop at Tt from a value consistent with a rigid lattice

Ir
a

.ll

:mflD/va

‘.

L

F 2 .lefibi C20 .-v nun

.I 5
AV
4:

 

l
! so”

' The te

Eire 4

a

PESOn
h

exam:

5

 

 

Calcui
R

C
\

efer

 

67

32.29....

20-

J
L!"
V

 

SECOND MOMENT (GAussfi
‘8
j 5‘

 

m
I
u
-4
'0
_,_-__-_-_--.z
.0

effusion 1 .

260
TEMPERATURE PK)

 

Figure 4. The temperature dependence of the second moment of the
resonance line for a typical plastic crystalline solid
hexamethylethane. Also shown are the second moments
calculated for different types of motions. From
Reference l97.

 

zi’iJe in agr
'7’.

\- =633d m
9
so": transu .

' .0 ‘

2—2 h '1
#115953 of T] ’
..zn'agg fraJ-f) C
is‘v'e iii” and

35.
gummy

‘ ’1: P2379415- P‘
\—

or; detennim
fixation (Ema
we? diffusion .
l

Izvs‘elline phase‘
‘22: for isotropil
'ta,dd- The P8]:
tiers)

ation and Ti

3‘32 theory of n

68

to a value in agreement with quasi-free isotropic rotation. At Tsd
the second moment undergoes a smooth decline to zero, an indication
of rapid translational diffusion. In pulse experiments discontinu-
ities in T] indicate a phase transition and strong temperature
dependence of T],T2 and T1p (the spin-lattice relaxation time in the
rotating frame) can be used to determine the mean time between dif-
fusive jumps and the activation energy for diffusion using the theory

l86,lB7 l88.

due to Torrey , as corrected by Torrey and Resing

C. The Pople-Karasz Theory of Melting,

NMR determinations of activation energies for dipole-dipole
relaxation (Ea dd), spin-rotational relaxation (E and transla-

tional diffusion (E

a,sr)’
a trans) for compounds which exhibit a plastic

50-58,]89—206

crystalline phase , are summarized in Table 4. Recall

that for isotropic rotational diffusion one predicts that Ea sr .
- Ea dd’ The relationship between the activation energies for
translation and rotation may be studied on the basis of the Pople-

Karasz theory of meltin9207'209.

In this theory the melting and
transition temperatures are shown to depend on the parameter v,

which is a measure of the energy barriers for molecular rotation
relative to those for a jump to an interstitial site. Following
Smith191’2°%.we associate these barriers with the activation energies
for molecular reorientation (Ea,dd) and molecular diffusion (Ea,trans)’
respectively; thus

_ Ea dd
varzr—L— (125)

a,trans .

9:

a:
on. an

.tcz

p.~

 

inll

 

 

 

~.c...v_ Ac gooey env.o --- mam
E: 1}... -
o I‘ killII
— N. oé
o6- 5m man 2m 2: we
acuLa.-U Luann Uni-am 6mg. Eh uh D~DUU~CZ

69

 

 

 

 

as, am.m Pm._ --- emN --- scapuacaoazoz
am. m.pp o.Ps. --- cum --- a=a_»econzoz
Nm m.m ~.~F N.~a~ ~.m- h.¢m~ was
~m ~.m m.~F ~.mom o.oa~ o.mm~ mac:
amp op ~.P
map ¢.ook.m o_.ooo~.~ --- mmw ~.P- «Am=0v_m_mmfim=ov
mm, o.m_ ~.~
map N“ m_ m.F
km, a. m.ooo.ap ..oa~.~ o.cmm as" m.~m. «AmzuvuomAmzuv
ma. NH mp N.oow.~ --- use m.o.~ PuwamzuvuumFo
mm. m.p «._H_.mp ~.oam.m o.~ma “.mcm F.om~ e.ua ucpa>.a
km _._- ..coo~.o --- a omv eopuezz
amp m.ep m.~ Nae mac omm togasao
no, ~.e o.oo o._ mmN NmN ca. eAmzuvo
Nap o.~ m.mm m-m com). oem aow acaucasau<
_m_ m.oom.¢~ o..o~.m~ mums. mam .mq aAzowzovu
om, ~.¢ ¢.o«m.¢F Ah zo_v N._
am, F.~ ~.oo_.¢_ AF gm.gv om¢.o --- can we, ~_amu
map ..NP o.¢
om.mm o.m- ~.m mmm Rpm ma. N;
a a a /
.tam aa+o_xma meat» am am am ca am «no 5p oh a_=ua_oz
U],

mmmgq a=__puumxgu u'uma_q as» =_ mmwmgocm zowpa>vu

o< .¢ azoh

 

 

 

 

 

 

 

NC“ N.@ m N CCP th . :,\\.v.v
_o~.nmp o «.8 men mmm um. Nfiumamtuyu
pow. no p m m . .8 own new «42 Banamtucu

.bmwz atflkualfl Lflnflfl henna.“ ”Db Eh. UL. U~30m-£

€5.3-xv0

 

70

.ucpon mcvupme mg» an acmwupmmmou copmamevvuwpwm mg» $0 osp~> mghn

.acwoa Gav—won u ah "Heron mcwppos . sh “omega mapppaumxgu ovumnpa on» o» :c*uwm=acu any to «cauacuaeuu . aha

 

 

 

 

 

ooN F.o hm.o --- ¢.-~ m~_ Ho
ecu ~.o mo._ ~.oo~ ~.ea_ _~— can
now a." mm.o m.mw. .m_ mo. .ua
mou ¢.o e.~ o.pv ¢.__~ o.~mp ~.o~F mug
mm 9.? m.¢ m.o.;. m.o --- mm, mm . no;
¢c~.mo~ FF.oomm.p mo.ooma.o ~.mm_ .oo o.om maz
~o~ . ~.o N.“ can a“. --- «Amzuv.m
.c~.mmp m ms. ”an mmN “mp ~_0~Am=uvu
_om.mm_ m m.=e «mm mom mm. _umfimzuvg
a a a 1/

.moa am+opxma meme» um cm am we mm mm» Eh up m_=uw_o:
//

.Ufl=:9#=OU 1 ¢ w—Qflh

In Figure 5 we

imperatures, l
’ 3 C
To
aids the boil"I
U

ones are from Re‘

'8‘. Iith the pred‘f

 

7]

In Figure 5 we plot r versus reduced melting (Ta) and transition

(TE) temperatures, where
T; = D.72(Tm/Tb) T; -- 0.72(Tt/Tb) (125)

and Tb is the boiling point at one atmosphere pressure. The smooth

curves are from Reference 191, Figure 6, and give qualitative agree-

ment with the predictions of the Pople-Karasz theoryzos.

 

 

 

 

 

72

 

     

 

 

3 1 1 1 1 1 1 1 1
.7 <1) om ~
6 “ 0(9) ‘i
5- 'm' '1
°.. (3:9 \ g_§? ' _S? _+
0(5)

 

 

Figure 5. Plot of reduced transition temperature (Tt) and reduced
melting temperature (T* ) versus r for molecular crystals.
Solid points are hm da a, open circles Tt data.

 

:, I-‘t'tductioo

through SEVE'
22212 saving in
toes in liquids 3’.
Itus the advanta-
rose detection) a
it 4] honogenei t3
arxr‘rents. Howe
:5'ection becomesl
"ea induction decl
Excise ratio. Fl
-“?s difficulty bl
ris advantage is
Miller uncertaj

We elaborate

'u
lfi‘flrp
l“ i;

”mains 1

r. -
5.!“-

I
F01- VEl‘y sho r'
i
i

gent lWilli re,—

 

EXPERIMENTAL

I. The NMR Spectrometer

A. Introduction

 

Although several multiple-pulse sequences have been suggested
for time saving in the measurement of long spin lattice relaxation
times in liquids 37’210'2”, the 130°-r-90° method is still useful.
It has the advantages that only two pulses and one rf channel (hence,
diode detection) are needed, and the requirements on pulse lengths
and H1 homogeneity are not as stringent as in the multiple-pulse
experiments. However, if T] is much longer than one second, data
collection becomes tedious, particularly if the addition of several
free induction decays (FID) is necessary for a satisfactory signal-

213’214 alleviate

to-noise ratio. Progressive saturation methods
this difficulty by not waiting for the return to equilibrium but
this advantage is offset by the smaller value of Mo-Mm, which leads
to larger uncertainties in the calculated relaxation times, and by
the more elaborate calculations necessary.

For very short relaxation times (<l0"'2 sec) the lBO°-I-90°
sequence remains the preferred method. In addition to the less
stringent requirements on the spectrometer, the assumption that
T] >> nTs, where Ts is the length of pulse(s) necessary to sample
the magnetization, becomes valid only for n small whereas the pur-
pose of the multiple-pulse experiment is to sample the decay of the

magnetization many times in the span of a few relaxation times.

Thus, even pulse sequences which average out errors in phasing and

73

(13222211137 31-
22' ill-3m] of :|
222122131“ cannot
l
mu equilib"
l

i) interfaciF-i
f“:'encies in tl
'st'rents have r»
‘2 partially-rel: ,
m; arounts o.‘
2161210) a type
excited below b...

resent approach 5

i1—

‘21 stadyi ng sing‘
seztroteters and
kasurerent c
kw .
-- convenient?
he lethod for 7
1‘ he shape of 1
“l the case of
livid

ea USEful 1

3951‘
mi exchange

3:.
it scalar coup

Nils (( T]I).J
r .

ASHsht dig
it

“W mahner.

 

74

H1 homogeneity37

are not adequate for short relaxation times. However,
manual control of the lBD°-t-90° experiment is now inefficient since
the operator cannot collect the data at the rate allowed by the rapid
return to equilibrium.

By interfacing a computer to the pulse spectrometer these in-
efficiencies in the two-pulse experiment may be minimized. Commercial
instruments have recently become available which include a capability
for partially-relaxed Fourier transform (PRFT) experiments with

varying amounts of data reduction 215’216-

These systems presum-
ably employ a type of computer-controlled timing similar to that
described below but no details appear to have been published. The
present approach should provide a useful and less expensive alternative
for studying single-spin systems and is applicable to a variety of
spectrometers and minicomputers.

Measurement of T1‘) (T1 in the rotating frame) may also be per-
fbrmed conveniently by the method described here. Since the only
pulse method for T1p measurement requires many repetitions to trace
out the shape of the relaxation curve, the same inefficiencies apply
as in the case of the lBD’-t-90° T1 experiment. Such measurements

provide a useful method for studying ultra-slow motions in solid5217.

2‘8 and spin-spin relaxation time5219. Also,

chemical exchange rates
where scalar coupling exists between a resonant nucleus I and a spin
S (TIS << T11), J15, the scalar coupling constant, and T15 can be
determined220 from T11.

A slight disadvantage to this simple scheme for relaxation
time measurements is that T2 cannot be determined in a straight-

forward manner. The two-pulse experiment for measuring T2

 

I.
ooh)-

If H .9
v.0 0'

my '1':
. 0“» DI -
o o-

r. r:- e:
P U

I v

'D ..
...... ;
l b ..

v V. I .
‘3" P
‘I‘L-ii bra
"l," .r

3.- 2

Tc..-
\h

I‘m

“a." 0‘

75

(90°-t-l80°-t) is affected by diffusion221 and may not give an

222

exponential decay; the Meiboom-Gill modification to the Carr-

Purcell method 822‘l

removes this effect and produces an exponential
return to equilibrium but requires more than two pulses. Therefore,
either the computerfcontrolled two-pulse experiment may be performed
and the data analyzed as discussed in Section II to determine the
self-diffusion coefficient, or the timing of the Carr-Purcell spin-
echo train may be arranged externally and the data collected by the
computer and analyzed on-line. The program and procedure for analyz-
ing multiple-pulse T2 experiments is fast and efficient but entirely
routine and is discussed in Appendix A.

Computer-controlled data collection may in one sense be con-
sidered a convenience, although in repetitive experiments, like the
180L90°T1 sequence, the convenience can be substantial. The true
advantage to the procedure described herein lies in the interactive
relationship between the experimenter and the experiment. The
ability to evaluate an experiment immediately as it is performed
is an advantage which cannot be stressed too strongly.

The components of a modified NMR Specialties MP-lOOO spectrom-

216 and a home-built interface

eter, with a Nicolet 1082 computer
substituted for the pulse programmer, are shown in Figure 6. The
pulse and timing sequences for both T1 and Tlp measurement are shown
in Figure 7(a). In addition to providing the necessary pulses and
time delays, the computer provides on-line data analysis, giving

an immediate numerical value for the relaxation time and a visual
check of the fit to the expected log (Hz-Mm) _v_s_. time relationship.

'This visual check is useful in tracing the cause of a large standard

 

76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

;{ PROBE PREAMPIIFIER
Powen “Ll—U— '
AMPUFIER RECEIVER
+11—‘ 1
[ 1.1—- ,
“14"*4“' PHASE
as UNIT INTERFACE ...____,
f oenxmoa
e L LIL
oscntoscoee ~ N'COLET ourreaemw
l082 AMPLIFIER
oowwuna
TELETYPE l

 

 

Figure 6. Block diagram of computer-interfaced pulsed NMR spectrometer.

 

 

 

 

10

L.)

1,1,]

PHASE:

PHASE I

391119 7.

77

3.

   

 

COMPUTER
PULSE 2

---/

T

 

2
E

-::::::::5"<

   
   
  

 

FF2 . .
[c] a . a s
: : I I
1 l i :
T1[T2] : ' ' 3 '
HA
lWiAEHEIDET. ' .
5 i 1 i :
TIP 3 '
Phi
PHASE DET.
Figure 7. Timing diagram (x axis to scale). (a) Timing of rf pulses.

(b) Internal timing in the interface. The levels are 0 V
and +5 V. (c) Timing of the interface output. Positive
voltages are'mz V,negative voltages -l5 V.

 

“vhf; $9131
:1 ,, v
'"I’C:Eret
rat a .
1:131:36 of b

- in
..;:un.: a S‘:

r-‘E

:e' :1e coarse
We oniy no
sesftite detect
2:22:32 descri
ass in chair-e
a: sion in Fig;
21“; poise f1
:cllecti on

21:21 l1 and '

1‘oll.

The spectr
Steiaities HP-

(Sii is Shown 1’

air

“i2 qua
5359 control,

we DhiSE-Sen
aerate at 56. 4
it 14‘“ Nil ar.
me”lent to u
is he Wluenc

£13Weed sa

78
deviation since, for systems in which T1 3 T2, it has been shown223
that inhomogeneties in H1 produce a non-exponential relaxation. A
poor choice of baseline will also produce systematic deviations,
indicating a significant drift from optimum operating conditions
over the course of the experiment.

The only modification to a spectrometer employing phase-
sensitive detection required to perform Tlp experiments by the
procedure described here is a variable attenuation for the locking
pulse in channel B. The timing for the two experiments is identical,
as shown in Figure 7 (a), with pulse 2 serving to terminate the
locking pulse for T1p measurements. The procedure for more efficient
data collection and reduction in two-pulse experiments applies directly

to both T1 and T1p experiments.

8. The Pulse Spectrometer

The spectrometer used in this work was a highly modified NMR
Specialties MP-lDDD. A block diagram of the components which were
used is shown in Figure 6. The rf unit originally consisted of
a 15.00 MHz quartz crystal, two doubler stages, the delay lines for
phase control, an rf amplifier consisting of two nuvistor tubes,
and a phase-sensitive detector. This unit was first modified to
operate at 56.4 MHz by exchanging the crystal for one which oscillated
at 14.11 MHz and retuning all of the stages, but it proved more
convenient to use a frequency synthesizer operating at 56.4 MHz
as the frequency source and bypassing the two doubler stages. This

also allowed samples to be brought to resonance by changing the rf

 

"I

 

221:) other ’.

he never as;
n at second 5:
2,212: as use
17' 2211.‘i-goratior
in load.

The lobe was
3:72:1‘01 between
WEI-idle (if;
lowed
£1.52? taxioum
“he ”Wm
32 vhltages
”these one
111.210“. C1

3515 Of diOdes

51iéhuati0n hi

on ":5ng m1
£39335 35 an
33%an
12hesent a )
it truism rte

hot-device when

a?” ”7"? of
Me preamp
5W1- 11

)r

WWW was

 

79

frequency rather than changing the magnetic field.

The power amplifier was a two-stage, high-power unit but for this
work the second stage of amplification was not necessary. The one
stage that was used contained two 3E29 tubes operating in a push-
pull configuration and delivering an output of 350 V p-p across a
50 ohm load.

The probe was of the single-coil design described in Section ID.
Isolation between the transmitter and receiver was achieved with

quarter-wave (A/4) cables and crossed diodesZM’245

shown in Figure
8. Both the transmitter-probe and probe-receiver distances were
A/Z for maximum transmitted voltage. The useful properties of diodes
in this application is their non-linear voltage-current curve. For
small voltages (<0.5 V) a set of crossed diodes will present a high
impedence while for large voltages (>0.5 V) they will behave as
conductors. Consequently, when an rf pulse is being delivered both
sets of diodes conduct; the pulses arrive at the probe with little
attenuation but set D2 prevents the input voltage to the preamplifier
from rising much above 0.5 V. The effective short circuit at DZ
appears as an open circuit at Dl since the two sets of diodes are a
quarter-wavelength apart. After the pulse is over both sets Dl and
DZ present a high impedence to the nuclear signal, thus isolating
the transmitter from the receiver circuit. In order to lower the
impedence when the diodes are conducting, each set was made up of
eight pairs of diodes.

The preamplifier and receiver were standard components, tuned
to 56.4 MHz. The recovery time of the receiver system following

an rf pulse was 7 microseconds, consequently the total dead-time

 

 

 

 

 

 

 

 

 

RECL

 

are

 

 

Figure 8. The arrangement of crossed diodes and tuned cables for
the purpose of isolating the transmitter from the
receiver.

‘uzw‘!

 

2159790139
The t'oinS

:1 1'12 “:3 952:5
3252225 and a
2'32 oer soot;
stains three
mtg SEZu€"£E
iris investi
i: ‘2‘.?‘.d:.ci 119
2' rich were 1
fit it) be c1
Sets, are dis
In order
35". Quenc)
PEG 5-30 He;

he) enlist
rd

:1.“

L - This
5‘. .
"rlhhe cha

iihahced

81

for a 90° pulse was ll usec.

The timing for various pulse sequences was originally supplied
by the NMRS pulse programmer which is capable of generating two pulse
sequences and multiple pulse sequences with up to four different
pulses per subcycle (for example, the triplet T] sequencezw’212
contains three pulses per subcycle, and the basic Waugh dipolar nar-‘

226’227 contains four pulses per subcycle). As a part

rowing sequence
of this investigation, a major improvement was made to the spectrometer
in introducing the option of computer-controlled timing, the advantages
of which were discussed in Section IA. The different pulse sequences
which may be computer-controlled, and their respective computer pro-
grams, are discussed in Section II.

In order to measure the spin-lattice relaxation times for deuterium
at a frequency of 9.2l MHz, this system was modified to operate in the
range 5-30 MHz with the addition of an rf mixing network and broadband
power amplifier, probe and preamplifier as described by Traficante gt
.31228. This modification, shown in Figure 9, affects the discussion of

the probe characteristics only; the rest of the basic spectrometer is

unchanged.

C. The? Interface
In order to perform the function of the pulse programner, the
interface must provide pulses of varying lengths from the computer-

229 and must shift the logic levels from

generated trigger pulses
the +5 V and D V used by the computer to those required by the
spectrometer. The circuit diagram is shown in Figure 10 with

the sections responsible for the different outputs identified.

The interface is switched from TI to T1p mode by a single control

 

 

fl“).g
WW ..31

    

 

 

‘5

1 I
3
1

(‘

43...... .ZH
~ 1.w.....ir 7.2.114..-

 

 

in...“ - .

 

82

asm wave:
. xn* an umpocmv was mcmpmp_ a
.Z x 32.3 age 33.6 339.0 .sz omua mmcmcoawuwnwwmrcuu .m 955
-ww»comno on w» vopuzc zappm ow copoEocuumam woes» «cu a P
5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

(a

 

 

 

 

 

 

guzwumakutimzép
”:2x=<
$232.23:sz <3 5.02: mz 823
02(393 _ .
n - 1“ «yawUmu
mus—u. 1V “ a . .Zm:( M ~12 on
uo<z_ “ a M
w u ,
. _ «nzuwmx_
or u m mm<za
aun=u _ m i
z b. s. — _
_ _
_ .
. .
_ _
_ .
. n " nac—
. _
_ _ $50.2
“ . d u i
n a ”
mug >6. _ a m 1)-
£1 «4 u , «Emmi. . mutumpz.
. _ x _ , _
_ .
u u
_ 1 $28 . .525.
_ . .sag .
. 3.5.. _ :2 9E
.(c . .
:xxn.zm " _
_ M
u 2 r
s m
.

#-

83

PP< .maomzm dam gen .Pm swamp» dam mtoumcmcat» an: FF<

.umuo: mews: pamuxm wows uo>
com neg ooo.op men mcouwumamu ppm "capo: econ: ugmuxw as m.n .3 e\F one mcopmwmmg

.Ewcmmpu uwsocwu mummsmucu

.op dt=m_u

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e :3... n gszcammoz
1L
*QO (.215
may. . .:2
3. F”? n 5%.. m
.n
1111111 111111111111141 111111- 11111i.v 05:00
0 as. . .&8_._.s8_. " . s m;\:
. v o o. : h... . _
b C-ha‘ ‘— . _
N. a“: v2 n1 _
. u
u )“9 '- 0. . Q. 0 . v. u —
15.14 gm» 2... x. .6. and u
_ 111111111111111111111111111111111111111
*189 h! h. .. ._. . .
n t i
. a. .1 G 1 . . n . .18»
._OuUQuQD. . . ._o a
82¢ a... E... g 5 El.

.n. x. i. .
_
.

 

 

 

 

 

 

 

 

1x“. llllllllllllllllll L

 

 

 

‘

u . xnx
" an a: g? I
_ nan

_ _ gas xs x.
_ u 25: .
u _ 25v an ” . su
. M _

.—

.." I. 'I - -1 l.-. to.
. .
i 7". |

 

 

 

‘etfggj‘ng to a
3:111: gates w‘
hiring for th
25'.l node the l
2's :acsisting
2:.15frxthes
2212.22 the 01.
so: the mltivit
mmMmr
size A to clear
ales iron the 1
hit-1e spectrone
“this ilipflo;
hi‘h level 1) '
hhiiut of th
":irtuit B.

The length
.1; 5059) is var
2”) Mention“
”hate-rs Hill 111
h"".£hi11g to U) E
211:.

he NMR Spe

1:. .
‘JSC‘.

ilator an
1:!
"m 0R hetwe

t may be p
535

.ldSed On at

3h211'

84

line leading to a front panel SPDT switch. This line controls two
quad NAND gates which are connected to function as SPDT switches.
The wiring for the TIF) section is given fully in Figure ll. In
the T] mode the input pulses from the computer gate identical cir-
cuits consisting basically of a monostable multivibrator M]. The
outputs from these monostables are then level-shifted and combined
to produce the outputs required by the spectrometer. In the T1‘)
mode the multivibrator circuit B (MlB) is replaced as shown in Figure
l0, where two monostables (M3 and M4) produce a pulse 6 usec after
pulse A to clear the flipflop FF2 (l and 2 are used here for the
pulses from the computer while pulse A and pulse B [Figure 7(a)]
are the spectrometer pulses). Input pulse 2 is used as the clock
fbr this flipflop, and since the J and K inputs are unconnected
(logic level 1) the outputs change state after the clock pulse.
The output of this flipflop (Q of FF2)replaces the output of M18
in circuit B.

The length of each pulse in the T] mode (and of pulse A in the
Tip mode) is varied manually from l-lDD usec in two steps by three-
turn potentiometers in the timing circuits of the two primary multi-
vibrators "IA and "18' The use of a calibrated dial facilitates
switching to the T10 mode, as the A pulse must be changed from l80°
to 90°.

The NMR Specialties MP-lDOD requires -l5 V pulses to gate the
rf oscillator and power amplifier, with the output to the PA a
logical 0R between A and B pulses. The phase detector in this
circuit may be protected from feed-through by a diode gate which

is biased on at all times except during, and 2 usec after, the PA

85

o
.Pocucou Fh\~p cow uwsucwu m:_;ou_zm wzu mo m_wmpmo .Fp oczmwm

Dunomvnn_

 

 

 

 

 

-:;pdse. l
Vesisshowr
we detector 1
:1 172‘? mm:
e", this sect
re: to green:
rerintoe 1

11 order 1
2e :o‘se prcgy
291212). 1
2hneme'
5‘53? lhogram

The exper
:3 the Slect
333" Wises t
3i"- ‘he mlti
StaSE11: adds
3mmh-c<
“he gain 1

“$0 ShO'

his? for 3.9

uses (IOP).
ii 500 nanos
sec allotted
Ill) and IOPZ

Mi 0R op.

86

output pulse. The internal timing and the timing of the output
muses is shown in Figures 7(b) and (c); the ON voltage for the
phase detector diode gate is 2 V and the 2 usec delay is produced
by multivibrators MZA and M28. Currently, this protection is not
used; this section of circuitry, with the l kn output resistor re-
noved to produce +5 V pulses, is used as the input to the pulse
driver in the multinuclear system (Figure 9).

In order to facilitate switching control from the computer to
the pulse programmer an additional board was added to the interface
(Figure l2). This is simply a series of NAND gates wired as switches
to change the input to the power amplifier from the computer to the
pulse programmer with a single front panel switch.

The experimental setup placed the computer about 20 feet away
from the spectrometer and it was found that attenuation of the com_
puter pulses through the approximately 40 feet of cable was so great
that the multivibrators would not trigger. Consequently an amplifier
stage was added to the computer output (Figure l3) which consisted
of a common-collector transistor amplifier for each pulse with a
voltage gain of approximately unity but a large current gain.

Also shown in Figure l3 is the circuitry to generate the trigger
pulse for 3-pulse sequences such as the pulsed-field-gradient ex-
periment23o. The Nicolet 1080 series computer has two input/output
pulses (IOP), readily available for control of an experiment, which
are 500 nanoseconds long and occur at different times during the 4
usec allotted to the associated commands. The commands which generate
IDPl and IOP2 (4l02 and 4l04, respectively) may be combined in a

logical 0R operation to produce a command (4106) which causes both

Ti . "..I‘ 1" Al‘ FEE

4‘

 

mos-r 111

{HID—
1:50—
silo-—

unmo—

programmer in

87

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ch A°—-—-!
scope synco—m
: 1:
a —-41
I

. 1: "———° PAoui

' l "-——--—-O I out
: A out

' 1—-——-o scope sync.

F—Q +5 V

: 13

: ' . programmer
1
Linearioce
interface in __
ch AB c '
ch A c
ch 8 or s u .___‘ B s
ync.

3 fl)

Figure l2.

 

 

 

 

 

 

jLAtpu.

The circuit diagram of the switch for transferring control
of the spectrometer from the pulse programmer to the
computer. The other wiring details of all IC's are
identical with those shown on the top IC.

 

 

i

 

 

632. mm 33.3 .3 a} .5. F
9; 23.0.59. :< .325 2.23 a 9.3985... com 3.39.5 «5 23 «.585 ..333 3:5 2:. .2 2:3...

 

 

m1 .
1_ H1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n
o l.—
OOK 3.:
«-
¢
'— vflllL V .
o = n
TL _ . a. _
A h
n V t .filulllll‘o
o 00: w. n .E: J
1 =
v. 1

 

 

 

 

 

 

!‘I

 

:mnfim

3% that it C

mummn'

'fiiTMo

mei

 

 

p
u.t
ut,‘°'

in.

Cam:

89

IOPl and IOPZ to appear. The two multivibrators in Figure 13 delay
IOPl so that it occurs simultaneously with IOP2. Now there are four

possible output states, shown in Table 5.

Table 5. The output from the "pulse booster" unit as determined by

the input from the computer

 

 

 

 

 

Computer Resulting pulse
Command Out from computer Out to spectrometer
IOPl IOPZ A B C
No command 0 0 0 0 O
4104 l 0 l 0 0
4102 0 l 0 l 0
4l06 l l 0 0 l

 

 

 

 

The two quad NAND IC's distinguish between these states to produce
an output pulse at A, B, or C under the condition shown in the table.
The logical circuit which performs this function is shown in Figure

14.

PULSEl

 

  

 

PUlSE2

 

l=igure l4. The logical circuit for generating the third pulse.

 

 

‘0‘

W 1W.“
xii. ”059‘ 293
meat, that 0’"
a :er‘fecl'

3313* 31??

29:3qu uS‘iFS

,. “he :MH‘
\

The pro!
:arl'estf‘né‘i
m piece C
‘ris is the
:ff‘iculty
SiéttrOiTEte
'i’take tube
m in F1
:ils, mac?
m, provi
:f thermal

The co
find that 1
iii/z becau
3‘15 #1 homage
Wake twin
3136"} mar

”Witneously

90

These control pulses could equally well be generated with the
Nicolet Model 293 I/D Controller and some software modifications;
however, that option was not available at the time most of this work
was performed. More complex pulse sequences (those requiring more
than three different types of pulses) will certainly be more easily

performed using the 293 controller.

D. The Probe

The probe which was designed and built for this work was of the
conventional single-coil design (Figure l5). The coil was wound
on a piece of 8“ long, 0.250" i.d. quartz tubing (Hilmad Glass Co.).
This is the same size as the standard Varian insert so there was no
difficulty in using a standard Varian Dewar salvaged from an A56/60
spectrometer. The only modification made to the Dewar was in the gas
intake tube. The complete probe, with one side panel removed, is
shown in Figure l6. Note that the support for the field gradient
coils, machined from Delrin, also serves as a holder for the external
lock, providing the necessary electrical insulation and some amount
of thermal insulation as well.

The coil consisted of seven turns of #26 magnet wire. It was
found that if more turns were added, the coil could not be tuned to
56 MHz because the inductance was too large. In order to increase
the ”l homogeniety, and at the same time decrease the inductance
to make tuning possible, the turns were spaced three wire diameters
(0.056") apart. This was done by winding three strands of wire

simultaneously and then removing two after the coiled wire had been

 

£51 a“... a

1-1

 

 

3% 15.

91

 

‘Ll

 

 

G} + ii—
..r t E
..— .....L.

 

Figure 15. Circuit diagram of the probe.

._‘1-4 r ~-

 

.. \al.

 

WW9 l6

92

 

Figure 16. Photograph of the probe showing the placement of the
Dewar. the field gradient coils and the external lock.

 

g‘iéi 53"“
2’54! affl’
:‘75 '5’? m'
Eguau'
:5: he 5?“
32:33.": in ‘
32.312 tui
'e: m the E.‘
pi. and 0.9 '
The qual
" swag; gene
is Width
ammo
m mplifi
:‘ 3.5 psec.
For the
h We des
:‘iiec‘ by W

zzcepted 3am;

ire:

[1) A ‘
f0:
(2) Vel
ste
sut

(3) Ea:

odc

93

cemented down with Duco cement. The remaining wire was then per-
manently affixed to the quartz tube with Epoxy cement. A number of
coils were wound and inspected carefully before the Epoxy cement

was to be applied. The technique of winding three strands insured
that the spacing between turns was exact, but rather more care was
necessary in order that the pitch of the wire be constant all around
the quartz tube. The coil which was finally selected appeared per-
fect to the eye both in spacing and pitch and had dimensions of 0.250"
i.d. and 0.9 cm length.

The quality factor Q of the completed probe was measured with an
rf sweep generator and a small search coil to couple it to the probe.
The bandwidth of the probe at 56.4 MHz was measured and from this Q
was found to be 30. With a peak-to-peak voltage of 350 V from the
power amplifier (the 3E29 tubes) this resulted in a 90° pulse length
of 3.5 usec.

For the types of experiments performed in this investigation
the probe described here has several advantages over the probe sup-
plied by NMR Specialties, which was of the crossed-coil design and

accepted sample tubes up to l5 mm in diameter. These advantages

are:

(l) A large filling factor for 5 mm tubes (which were required
for reasons stated in Section V).

(2) Very short pulse lengths without needing the high power
stage of the PA (which was somewhat tempermental and
subject to expensive breakdowns).

(3) Ease of construction - no paddles are necessary and no

oddly shaped coils need be wound.

 

 

ext!

:‘ifif the 9C0
. ’ ‘ ‘

fl: WEE let
med. A

23:36 7] we

7?

:‘Tgf'fiz SE
was of 1'”
1% Dear. F
:.'se iii-3h"
H be acce

en‘s; however

2% in geni
:‘e spl'n ”5
Them” ‘
investigated
The Sim-9
:f hsadvanh
it was here
effects on t
hrs-endless a!
fe:‘.ory f
or I
is host simp'

; .
hi 1ts opt'

94

(4) Lots of empty space in the probe for such things as the

external-lock and the field-gradient coils.

This design had one disadvantage, though, that was not anticipated.
since the 90° pulse length proved to be slightly temperature dependent
and pulse lengths had to be constantly readjusted as the temperature
is varied. Adjusting the pulse length is not difficult for the two-
pulse T1 experiment but proves rather more bothersome for the Carr-

1222 sequence. This disadvantage could be removed at the

Purcel
expense of increasing the pulse length by placing the coils outside
the Dewar. For relaxation times of spin-l/2 nuclei in liquids, a
pulse length lO-ZO times the value currently attained (3.5 usec)
would be acceptable and for this case the modification should be

made; however, for solids a 100 psec pulse would not be acceptable.

Field Gradients for Diffusion Measurements - Several methods were
tried in generating field gradients for diffusion studies. The response
of a spin system under the influence of a steady221 field gradient,
and of many types of time-dependent field gradient5230'232, has been
investigated.

The simplest method, employing a steady gradient, has a number
of disadvantages which have been discussed elsewhere233; in addition
it was feared that a steady, inhomogeneous field would have adverse
effects on the stability of the external lock. This fear proved
groundless and the steady gradient method proved perfectly satis-
factory for measuring diffusion constants in liquids. The gradient
is most simply generated by changing the current in the z-shim coil

from its optimum value. This typically amounted to a change in the

 

«'9’ m ‘J‘C‘
a; :‘ffi'Si 3“

"item 3

I
' U

“I

233610 U

bi

Aim .;:'.
a‘grtatfve n
he ‘irst inv
it): me:
massed in
as shit coil
M, the c
:"Eht, res;
t. hiring:
if he gradis
this done '
lifift‘liy en.
it: this ar
3its the gr
595e, but

tat one COL

95

current to the z-shim coil of ~20 mA. A rough calculation based on

the dimensions of the coil gives a gradient G = 0.1 Gauss cm'].

The gradient across the sample volume may be considered linear in this

case due to the relative size of the z-shim coils and the sample.
Although the steady-gradient method proved satisfactory, two

alternative methods which used a pulsed field gradient were explored.

The first involved simply pulsing the z-shim coils with pulse lengths

of 1-100 msec triggered by the C output of the "pulse booSter" unit

(discussed in Section IC). Figure 17 shows the circuit for pulsing

the shim coils. Since the normal current range in these coils is

125 mA, the circuit was designed to add 25 mA to the steady state

current, resulting in a current during the pulse of between D and 50

mA. Adding a constant known current should improve the reproducibility

of the gradient, as opposed to simply turning the z-shim potentiometer

as was done in the steady gradient method, but this was not checked

carefully enough to permit drawing a conclusion. (It might be noted

that this argument is not directed toward the accuracy of the results,

since the gradient was always determined by calibration with a known

sample, but rather toward the ease of performing the experiment in

that one could dispense with these numerous calibrations.) The cur-

rent was kept extremely low to avoid damaging the shim coils. A

switch and trimpot, shown in Figure 17, were used to vary the pulse

length and a pulse of 10 msec duration produced an effective T5 of

NZ msec; the field transients produced by the pulse took ~17 msec

to die away. This performance is not adequate for measuring D,

even in liquids, although this field-gradient pulse worked very well

to destroy the magnetization in the x-y plane for the "homospoil"

96

 

.m—poo sagmn~ as» on acmggso mg» mcvcoupa
mo memos an mmpzq acmpuogmuupmwe numcwpumpampgm> a m:_uw>oca ace upzucpu us» .up «camp;

 

 

     

 

 

 

 

 

0:04 1 L630
H . .n n
>
...w ..me an 2. UN
>n+ 8+ on 21 U—
>a3>.guh
uosazs znsszm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fl

'. steriieht
1h order ‘

r: reiuce the

giant tits

. L

at and at:

“I the hii;
:‘e tisiance
:hstatce fro:
Elfin ‘cm

‘15}: gradie
t can t
it headt-

3‘9 tot] 5h

97
T] experiment234.
In order to produce a larger field gradient during the pulse
and reduce the pulse rise and fall times, a separate set of field-
gradient coils were wound, shown in Figure 16 and 18. The coils

were wound according to the calculations of Tanner230

to produce
the most linear gradient at the sample. Using the nomenclature of
his Figure l, 0.32<z/z<0.43 and D.337<r/2<0.60, where 2 is the distance
from the midplane of the magnet gap to the face of the magnet, z is
the distance from the midplane to turn i of the coil, and r is the
distance from the coil axis to turn 1. The coils were wound on a
Delrin form to reduce the amount of metal in the vicinity of the
field gradient. Each coil consisted of 15 turns of #30 wire for a
total coil resistance of 0.86 ohms. The steady-state current carry-
ing capacity of #30 wire is 10 amperes, so with wire of this gauge
the coil should handle up to 30 A at a duty cycle of 1% with no dif-
ficulty.

The power supply is a pair of 12 V lead-acid batteries and the
transistor switch to allow these batteries to discharge through the

230

coil was copied from the circuit of Tanner with transistor Q4

in his Figure 2,an MHT l808, replaced with a more common 2N4048.

Temperature Control and Measurement - Since the Dewar was of
standard Varian design, temperature regulation was readily avail-
able using the standard Varian heater-sensor and V-4343 variable
temperature controller. This system provided quite stable temperatures
as long as the indicating meter of the V-4343 was in the regulation

range. However, it was found that although the temperature was

 

 

Figure 18.

 

98

 

 

 

 

 

Details of the coils and coil form for generating a
pulsed field gradient.

I

 

5,, held t1
3'5 CT: the
‘o-ztster‘.
hi thereto
m, I". th
21' test tr

“,3. 000',
lewd». hi

1:,
hit “€331,
ed msec;
izti‘iec by
fiiih i 55
little tel
‘5 the use
{fitting pc
is chose:
3‘9 same

1?: the

$5100 t

2311155,.

' I

99

easily held to 10.50°K, the actual value of the temperature as indi-
cated on the dial could be incorrect by ¢50°K. Consequently, an
iron-constantan thermocouple was added to measure the temperature.
The thermocouple was first placed in the Plexiglas cap on top of the
probe, with the sensing end extending into the gas stream at a point
well past the receiver coil. It was found that the cap's thermal
conductivity was too great and temperatures below about l30°K were
not reproducible with the thermocouple in this position. (This,

and subsequent calibration curves discussed in this section, were
obtained by an indirect method. A second thermocouple was placed
within a sample tube and inserted into the probe and the first thermo-
couple calibrated against the second. The second was then calibrated
in the usual fashion using materials with well-known freezing or
melting points.) The second location tried for the thermocouple

was closer to the sample. The thermocouple wire was inserted through
the same hole in the Dewar as used for the leads to the rf coil so
that the sensing end projected into the gas stream about 1 cm above
the top of the coil. with the thermocouple in this position a linear
calibration curve was obtained from 100°K to 420°K.

Unfortunately, melting and critical points obtained using this
calibration did not agree exactly with the literature values, the
discrepancy becoming distressingly large at the lower temperatures.
Accordingly, the indirect calibration procedure was abandoned and a
calibration curve determined directly from experimental observations
of melting points. A number of compounds for which the true melting
point was well established were chosen; for hydrocarbons the magnetic

field was lowered to detect proton resonances and T5 was monitored

 

atmfim:
hmtnmfi
tainhm
homfimg
{Tying knot
theft

ESE'l‘ed and

Ehommt
:xefle
Them

the bi phat
has SIC)”.
Ehmab

3:5me

100

as a function of thermocouple reading. The melting point was detected
by the dramatic change in T5. The results of this calibration are
listed in Table 6 with the experimental temperatures calculated

by correcting the measured potential with mVact = mVexp -.083.and

employing known e.m.f-temperature conversion table5235.

Using the
results of this final calibration the discrepancies between the
observed and literature values of melting and critical points of
the fluorocarbons studied in this investigation were found to be
acceptable.

Thermal gradients in the probe were also measured. This was
done by placing a thermocouple in a sample tube, and the tube in the
probe, such that one Junction was inside the coil while the other
was 4 cm above the first. The gradient.given in Table 7 at different
temperatures, was not strongly dependent on the actual temperature

and was never greater than 2.20° in 4.0 cm, or 0.27° across a 0.5

cm sample. The average gradient was smaller than this, about 0.20°.

11. Measurement of Relaxation Times

A. The Computer Program

The computer-controlled measurement of relaxation times may be
divided into two sequential operations: (1) generating the correct
sequence of rf pulses and measuring the nuclear signal at the ap-
propriate times, and (2) analyzing the resultant data to determine
the relaxation time. The latter operation is of little interest

provided it is done correctly, so a full discussion of that portion

 

h

 

 

101

lable 6. A comparison of experimentally determined temperatures

with accepted literature values.

 

 

 

Compound MP cp Observeda Literature
CF3Br x -l75.5 -l74

x 70.8 67
CF3CC13 x 13.5 13.2
CF3CF3 x -98.2 -100

x 18.1 23
Neopentane x -159.6 -159.6

 

 

a = mV - 0.083.

mvact exp

'5'5 7. P5

 

102

Table 7. Measured values of temperature gradients across the sample

area of the probe for temperatures below 300° K.

k;
y

 

161m)a T(°C) TC2(mV)b Gradient (°C/cm)
-0.088 -2.3 0.035 0.175
-0.225 -5.9 0.083 0.415
-0.551 -13.2 0.078 0.390
-0.990 -26.6 0.105 0.525
4.253 -34.2 0.091 0.455
-1.772 -49.0 0.103 0.515
-2.049 -57.4 0.103 0.515
-2.527 -72.3 0.094 0.470
-2.749 -79.5 0.088 0.440
-3.096 -91.2 0.110 0.550
-3.358 -100.3 0.071 0.355
-3.600 -109.0 0.097 0.485
-3.888 -120.0 0.079 0.395

 

 

 

aC0pper-constantan, measuring the actual temperature.

bIron-constantan, leads separated by 4.00 cm, measuring the temperature
gradient within the sample area.

 

 

fit expert?
(it?) he I
(aim int

“1011 c
Lists 15 Sh:
2' estimate 1
at he mm:

9515 to S'

’93
3

then is
tWU%;t
'hh‘te 7(a),
htts of 9y
3531mm
iihthed to
3'1" pulse
'Sh' 0n the
'35 Possit
I“ the case
She-hi ”I.

'n'iSiated

103

of the experiment will be found in Appendix A. The pulse sequences
which may be used to measure various relaxation times are of some-
what more interest and will be thoroughly discussed.

A flow chart of the program for generating the time delays and
pulses is shown in Figure 19. The operator enters, via the teletype,
an estimate of the relaxation time T, the number of points desired Np,
and the number of repetitions Ns required for each point. 0n the

command to start, data are collected with N(= prNs) sequences
[6T - pulse 1 - n1 - pulse 2 - (HT) - measure signal],

where n is varied by the computer from g 0 to 4T (1 _<_ n g Np) and
T = 4T/Np; the second n may or may not be present. As seen in
Figure 7(a), if pulse 1 and pulse 2 correspond to appropriate combina-
tions of 90° and 180° rf pulses, the usual sequences for T] and T2
measurement may be obtained; when a slightly different meaning is
attached to pulse 2,the timing sequence will measure T] p. Imnediately
after pulse 8 in T1 experiments (signalled by the transition 0 V +
+5 V on the measure/wait line) the F10 is digitized at the maximum
rate possible (20 usec per point) and integrated to a preset limit.
In the case of T2 measurements, signal acquisition is delayed by the
second n1. Since the measure/wait line is normally high (+5 V)
the signal is digitized immediately after the second delay and is
integrated as in the T] experiment.

This basic timing sequence is readily modified to produce other,

sinfilar sequences. This has been done for the following cases:

 

104

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
 

CALCULATE 1']
“1
| 2580 DATA secnou 14,5
I m ‘ F—I
TIMING
L00? ”1'
use 2
AoJusr
CONTROL ”0
““an “fig.“ TIMING 51 TIMING 51
moor LOOP
res
n7 ,2
mum; ‘ T. or T. INCREMENT
‘ 4001' 7 :0
TI
MEAsuIze SIGNAL
a ADD to EXISTING DATA
res
CACUlATE
ElAPASED TIME (=
MATH F
suuoumes INTEGRATE
I SIGNAI. (=Y”)
exp.
commeo ”0

 

   

‘I’

 

DELETE POINTS DISPLAY ‘
a necucuute x; ,Iri i=1,"

 

 

  

 

 

Figure 19. Generalized flow diagram of the software for timing and
measurement control.

 

5511855
I? start as
it] rather

(2) 1h
F-Ieitand
7:513 mi

(3) 1r.
5:15'te for
“this
2.1111195”

05.

"the

'\,)

with“;
33391 in d.

The ha
Extent to p
339319159) .
hire 20h
5- This p,
matrices,
Etered to

The d1

105

(l) The 90°-t-90° sequence for measuring T] in solids may be
produced by omitting the 8T1 wait at the onset of each cycle. This
is a variant of the so-called "homospoil" T1 sequence234 in that
both start each pulse sequence with zero net magnetization (satura-
tion) rather than from equilibrium.

(2) The homospoil sequence234 is produced by omitting the
811 wait and inserting a field-gradient pulse (as described in Sec-
tion IE) prior to pulse 1.

(3) The final sequence is that used in the pulsed field-gradient
sequence for measuring the diffusion coefficient. This is simply
the two-pulse spin-echo experiment with field-gradient pulses oc-
curring symnetrically about the 180° refocusing pulse as shown in
Figure 20a. The modification necessary for this sequence, although
conceptually straightforward, is somewhat more involved and is dis-
cussed in detail in Appendix B.

The basic timing program has been modified to a somewhat greater
extent to produce multiple-pulse sequences (such as the triplet T1
sequence). The basic timing for this type of sequence is shown in
Figure 20b and a complete listing of the program is given in Appendix
B. This program should provide the basic timing for a number of
sequences, since within certain limits the subsequence may be readily
altered to include an arbitrary number of pulses and time delays.

The data which have been collected by one of the sequences dis-
cussed above are displayed on an oscilloscope at the completion of
the experiment and may be punched or typed out for analysis by hand
or by computer. However, the full benefit of an on-line computer

is only obtained when an immediate calculation for the desired parameter

 

Figure 20.

106

 

 

 

90 180
H\ t" ‘— " P’l A
‘\\\ ' ’,/r
I 1' ' fkfld gnadkun I t' I
pulses
180 90 180 90
l¢————————-t. :~||¢- I: -’| Ida-pz-—>| '

Figure 20. Basic timin for sequences involving more than two
pulses. (a the pulsed field-gradient sequence, and
(b) generalized multiple-pulse sequence.

107

(relaxation time, diffusion coefficient) is performed, in order that
the experiment may be redone if necessary. Additional subroutines
for the Nicolet 1083 have been written (Appendix A) which perform
the necessary arithmetic operations, such as baseline correction and
weighted least-squares fit to an equation of the form in M2 = At + B.
The differences between the experimental values of an M2 and those
calculated from the least-squares slope and intercept are displayed
so that spurious points may be detected and the time constant re-
calculated. The common error of a poor choice of the baseline (in-
finity value), which produces a large standard deviation (but a good
value for the relaxation time, if weights are used in the least-

squares calculation) may also be easily detected and corrected.

8. Performance of the Computer-Controlled NMR §ystem

An important consideration in interfacing a computer to an
instrument is that the computer does not restrict the capabilities
of the instrument. Computer control in this case provides two pos-
sible limitations of performance. These are the allowable intervals
between the 180° and 90° pulses, and roundoff errors in calculations.

The minimum value of T. the time between pulses, depends on
the speed of the computer and the number of commands required to
execute a timing loop. With the computer employed here I is 16
usec, and the minimum delay to the first 90° pulse is 42 usec, so
the shortest relaxation time which can be measured with reasonable
accuracy is about 100 usec, adequate for liquids. There is essen-

tially no limit on long relaxation times as the timing loop can

 

.0;
...»
rt.

’0'

,Joi
p
‘9 lilb

ab

: e55 tl
P. .
lean

;":'e as 1

'5' the 1

:31 veiu
'-‘ Me
It 15515
the were
hit. F1

3’16 em

' Adi}:
-._4‘;

The

1525an

.2"
. I—s

3131519”
The
‘1 591th

Btrhtti

33119th

QFOHeg

108

count-242usec (1200 hours). A simpler version of the program has
been written which reduces the minimun delay to 26 usec but limits T1
to less than one second.

Roundoff errors in the calculations should be completely negli-
gible as an accuracy of greater than three parts in 107 is claimed
for the Nicolet floating-point routine. Fixed-point multiplication
is double precision integral (40 bits) with, of course, no roundoff.

A more important roundoff error occurs in the time delay assigned
to the signal following the 90° pulse. This time is given an inte-
gral V6108 between zero and 4096 and in a typical experiment is r\10.3%
in error for the first point and ~0.03% in error for later points.

It would not be difficult to reduce this error by specifying the
time more exactly but this refinement was not necessary in the present
work. Field inhomogeneities and spectrometer noise still determine

the accuracy obtainable.

C. Adjustments for Various Pulse Sequences

The method for adjustment of pulse lengths and phases differs
depending on whether the spectrometer is under computer control
or NMRS Sequence Synthesizer control. The latter condition will be
considered first.

The pulse length required to give a 90° nutation is adjusted
by setting a train of n identical pulses (p-t)4m, l £.m- Under this
restriction, when p is a 90? pulse the magnetization vector ends the
sequence at equilibrium, consequently saturation effects do not

strongly affect the signal amplitude. When the pulse length is

 

 

 

 

:28?) 1
5515 l
1:15:57

wit the

re see.
:5 tevta
The
seer con
:5 usir

fleshg a

109

properly adjusted the signal appears as shown in Figure 21. A 180°
pulse is obtained in the same manner but now n is a multiple of two.
A properly adjusted pulse length results in no nuclear signal and
again the sequence ends with the magnetization vector at equilibrium.
The homogeniety of the5mn probe is not good enough that these decays
are seen when n is large, so the pulse lengths are adjusted such that
the deviation is minimized at the start of the sequence.

The phasing is most conveniently done under sequence synthe-
sizer control using a single 90° pulse to phase each channel in turn,
then using a closely spaced Carr-Purcell sequence to make the fine
phasing adjustments.

To set up the two-pulse spin-echo sequence one should first
adjust the phases under sequence synthesizer control, then switch
to computer control and adjust the pulse lengths by the slow pulse-
and-wait method. The 180°-r-90° sequence is easier, since everything
can be done under computer control. The phases are adjusted to 180°
difference by running two pulses (SE command) of 90° or less and
‘ turning the phasing knobs until the two free induction decays are
maximized and in opposite directions. Then the first pulse may be
adjusted to 180° by increasing its length until no decay is seen.
Depending on the T1 of the sample it may be necessary to wait longer
than the SE mode allows between repetitions. The second pulse need
only be set to approximately 90°, since it is easily shown that its
length is not important to the experiment as long as it is constant

throughout the measurement.

 

 

 

 

1--.- ,t .l --l

Figure

110

 

Figure 21. Correctly adjusted phasing as shown by a train of 90°
pulses.

 

F
‘-‘!K‘n( 4.21“

 

 

111
III. The External Lock

Some experiments were done without a field/frequency lock
by carefully minimizing the field drift with the V-K3506 Super
Stabilizer and using a small value for the integration limit (a
control parameter in RELAX2 described in Appendix A). However, field
drift was obviously limiting the accuracy of relaxation-time measure-
ments. Although corrections offsetting the drift, either in the time
or frequency236 domain, were considered, it seemed that a field/
frequency lock would provide the greatest benefits.

The most obvious procedure was to use the entire (unused) DP-60
console as the lock circuit. This meant that it would be necessary
to construct a probe which could operate from the V-4311 rf unit and
also be compact enough to be within, or ride piggy-back on, the pulse
probe,putting the lock sample in close proximity to the experimental
sample. Varian probes normally have three sets of coils - the trans-
mitter, receiver, and modulation coils. Winding three coils in a
small probe seemed too difficult, so an attempt was made to get by
with only two. The first configuration had separate transmitter and
receiver coils with the modulation riding on the transmitter coil
through the external modulation input on the V-43ll. The transmitter
coil was two turns of #30 wire square-wound on a 1“ section of 0.25"
o.d. thin-wall quartz tubing. The receiver coil consisted of five
turns of #36 wire wound two diameters apart on a 3 mm o.d. quartz
capillary. The smaller tube was wound with tape so that it made a

snug fit in the larger one but could still be rotated to adjust the

112

null. The capillary was filled with water doped with CuC12-2H20
to give a linewidth of N4 Hz. This configuration worked poorly
because the two coils were not sufficiently rigid with respect to
each other to minimize the leakage. Consequently the signal-to-
noise was very poor and a lock signal could not be obtained.

The second configuration that was tried was with the trans-
mitter and receiver using the same coil and a separate coil for the
modulation. A diagram of the probe circuitry is shown in Figure 22.
The transmitter-receiver coil is five turns of #36 wire wound on
the outside of the 3 mm sample tube. A small Plexiglas block was
machined (Figure 23) to support the modulation coils. These were
wound and the strands glued together while supported on a jig before
being glued to the supporting block. Winding the coils on a jig
made it a simple matter to obtain the forty-turn coil of the proper
dimensions (radius 3.5 mm, separation 5 mm), and gluing the turns
together on the jig made transferring the coils, which were rather
small, easy. The coils were deformed somewhat during the process
of gluing them to the Plexiglas support block but this made little
difference since extremely high resolution was not needed. It had
been determined that such a large number of turns was necessary in
order that the coil impedance match the output impedance of the V-3521A
modulation unit. More than 40 turns would have been desirable, as
the impedance was still too low, but with this number of turns suf-
ficient modulation power to saturate the lock signal was available.

The receiver and transmitter were coupled to the single coil
by a 1:1:1 transformer which consisted of five turns of #26 wire

wound on a ferrite toroid. The circuit for this device, shown in

 

 

113

 

 

 

 

 

 

0-1: 1 1'5_
15
(3:3 , IL__
K j I
80 ....
T "Pa-12
IL——1
:J: 1': __

Figure 22. The probe circuit for the external lock.

114

Pmcgmuxm 9: cm _..Poo Lm>mwuw&\LmHHwEmcmLu Ucm

e.u....

£35530

 

 

3.2:

mpwou copumpzuos 5o pcmsomcmeem one .mm mezmwu

 

 

 

£3
3:03.35

0.9.5»
.803 venom.

\

4

 

115

Figure 24, was copied almost exactly from the Varian hybrid box
circuit for the Varian V—4354 audio phase detector. The entire
external lock circuit is shown in Figure 25. The capacitor on the
receiver input was necessary since this line is normally at +250

VDC for the Varian preamplifier contained in the standard probe.

The preamplifier used here was a Vanguard dual-gate MOSFET (Vanguard
Electronic Labs, Hollis, New York) tuned to 60.0 MHz.

Little power was needed for such a small transmitter coil.
Generally the V-43ll rf unit would operate with all but‘h30 dB of
attenuation. The amount of leakage was critical to the SIM and should
be adjusted to less than 40 uA by varying the gain on the preamplifier.
Other than that adjustment, operating the DP-60 console as an external
lock is exactly the same as operating in HA mode.

The lock probe slides into the back of the pulse probe, as can
be seen in Figure 16. The separation between lock and sample can
be as small as 1.4 cm but the homogeneous region of the field is so
large that the S/N for the lock was adequate even when it was well
back in the field. Since the lock probe was not thermally insulated,
it would tend to some extent to follow the temperature of the ex-
perimental sample. For relaxation studies this is important only
iII that extremely high or low lock temperatures may cause the lock
sample to explode. With the lock slid to its fullest extent into the
pulse probe, it was determined that the lock sample attained a tempera-
ture of 0.1° C when the experimental sample was at a thermocouple
reading of -6.14 mV, or approximately -l70° C. In order to go to
lower temperatures while still using the lock, the lock probe must

be partially withdrawn from the pulse probe to decrease the thermal

 

 

 

116

QGNAL

I

 

 

1

R1:
POWER

 

 

IRISZK)
{N351
l/44N

Figure 24. The circuit for coupling the transmitter and receiver
in the external lock.

117

 

V-43"
IRF LNQTT

 

 
     
 

HYBRID
BCIX

 

 

 

””1 I I PROBE
MoouuTIOrj '1

Figure 25. A block diagram of the external lock circuit.

 

 

 

,.,o:p1

. au-

nnnnn

3‘ 11
I’ier 1

“'- 4.
N sh,

."IE gri

118

contact. In this manner locked operation was possible down to the
lowest attainable temperature. The stability of the field lock has
not been determined precisely but the drift is less than 12.5 Hz

in eight hours.
IV. The Raman Spectrometer

Raman spectra were obtained with a Spex model 1401 Ramalog
spectrometer equipped with a double grating monochromator, a l W
He-Ar laser operating at 5145 A, and photon counting electronics.

A diagram of the optics used in this experiment is shown in Figure
26. The third monochromator was removed for this application in
order to increase the throughput to the photomultiplier tube. The
polarization scrambler prior to the first slit is required because
the grating efficiency is polarization dependent, favoring the strong

1

IVV spectrum. Lines of‘MS cm' width were measured at a slit width

of 70 p, which corresponds to 1.5 cm'] resolution at 5145 K, while

1 resolution. The sample tem-

broader lines were recorded at 2.0 cm“
perature was not controlled, so the laser power was kept low to avoid
heating the sample by the well-known lens effect. At 350 mW, the
power setting at which all spectra were recorded, the temperature

was measured as 27° C by placing a thermocouple within the capillary
containing the sample and as close as possible to the laser beam

(«J mm). Since the Ramalog is a departmental instrument, more than
usual care was taken prior to each run to ensure that the system was

operating properly and that the throughput was maximized. Spectra
were digitized with a Varian C-1024 time-averaging computer (CAT)

 

 

119

IUU'IPAS EXCIYIWOI IIIIOI

  
  
   

SEAT" I
COllICVIOI
IIIIOI "

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

uIcIIoscon outcnvt
we" nun
1‘52 IEA. ("luflf “NClW's
nunom 3W" .1 /
I A! I M I Q .. ./
A mm o I Iamnmmmm nun IlC'Olllll watchman
...
"""um'" "° "m" mu utumt nun “m “1“,,“ 'uyo _. _/
‘- 14:) IIIIOI ASSEIIIV '
. us: cum" nun-'- III common one:
A
' ‘ Inoaouwo oImcmc moon
I, >
use: J

 

 

 

Figure 26. The optical train for the Raman scattering experiment.

120

and transferred to cards on an IBM 526 keypunch equipped with a
Varian C-1001 coupler. The basis for control of data collection with
the CAT was the output available from the Spex marker/encoder circuitry
which provided pulses every wavenumber or every 0.1 wavenumber.
Either of these pulse rates may be used for CAT address advance,
thus spectra may be digitized at a resolution of 1.0 or 0.1 wave-
numbers per point with a maximum sweep width of 1024 cm']. The
trigger for the start of a scan was obtained from the pulse per wave-
number output, with the autostop of the CAT set at l to inhibit further
trigger pulses. This experimental arrangement is shown in Figure 27.
The polarized component of a line was almost always measured in
a single scan while the depolarized component usually required averag-
ing two to nine scans for satisfactory signal-to-noise. No significant
difference was observed in the analysis of an averaged or a single-
scan spectrum.
In order to correct for the errors introduced by the Polaroid
analyzer and the polarization scrambler, the depolarized spectra
were corrected for different absolute transmissions for different
polarizations, and for leakage of the much stronger polarized com-
ponent through the analyzer. These corrections were made by com-
paring the observed depolarization ratios of the 314 cm"1 and 459

237

cm'1 lines of CCl4 with the reported values The experimentally

determined transmission correction c] and leakage factor c2

13:30] - czlexP (127)

- ex
' C11d pol

p
epol

were measured frequently and are given by 0.91:.01 and 0.0024:.0003,

 

ImUIU(1$:meZQ

‘Wln

 

121

.mmnecmmcwp cusam mcvucoume »__mpvmvu co; mzumgmaaa one 50 snowmen xuopn < .NN mesa?“

 

 

—

 

 

 

‘

 

 

 

 

 

 

 

has;

 

 

 

 

 

 

 

(O
A t
9.
s:

4 — 7.3.3 a
522.. h .
I Ilg1 wmaDm o
0&me T :6 [IQ L .

 

 

 

 

muémmhz.

‘ ‘—n- — _—-————_ _—

—.—‘

 

_h=xu¢nu
.amV=9<<<

 

Xmmm

 

 

 

u,-
.-
va-

5:
”‘93.
u.‘ o

.. I..
1'”

"NC
I! i

:1." '

’51-:

I |..

122

respectively.

A FORTRAN computer program was written to extract the reorienta-
tional correlation time from the experimental spectra and is discussed
in Appendix C. Basically, the analysis is as follows. After the
correction described above was made, the spectra were digitally
smoothed using the method of Savitzky and Golay238. The best value
for the orientational half-width was obtained by a simple iterative
procedure in which the width of the Lorentzian orientational func-
tion was varied so as to minimize the chi-squared error between the

experimental depolarized spectrum and the calculated convolved

SPeCtruIn.
Uncertainties in “or were estimated in the following manner.
The residual
" 2
R(“’or) ' gudepolh") ' Iconvolutionm” (128)

was deter‘mined as a function of mm. and was a parabola with the mini-

mum occurring at the value of (”or listed in each line in Table 24.

The uncertamty in “’or

"MC" the residual was less than twice the minimum residual. The

was taken to be the range of “’or values for

minimum residual depended on the limits of sunlnation since the dif-

ference -

(I de pol I conv) for points far in the wings was practically
1"dfiPe'ldent of the value of m. Therefore the residual was computed
out 110 four half-widths. Repeated measurement of ”or for the 221

-] .
cm “he of CDBr3 verified that the uncertainties determined by this

”3”)“ were realistic.

L‘

123
V. Sample Preparation

A . NMR Samgl es

Since measurements were to be made up to the critical point for
many samples, the ability to withstand large pressures was a primary
consideration in choosing the type of sample tube to be used. Since
smaller tubes are inherently stronger than larger ones, and vacuum
seal-offs could be made with less difficulty with smaller tubes, it
was decided to use 5 Hill sample tubes. The critical pressure in the
substituted ethanes and methanes which were to be studied is ~40
atmospheres and an unflawed standard-wall pyrex tube can withstand
that easily.

The design of the sample tubes is shown in Figure 28. The
restriction at point A serves both to reduce diffusion to the liquid-

Vapor interface, which provides an additional source of relaxation

39

through spin-rotation in the vapor phase2 , and also to restrict

the sample to the most homogeneous region of the H1 field. The seal-
Of‘f when the sample is filled on the vacuum line is done at point C,
and if the critical temperature is higher than roomtemperature an
additional seal-off is done at point B, after the sample has been
Carefully frozen in the bottom of the tube, in order that approxi-
I"ately uniform temperature be maintained at all portions of the
Sample when it is in the probe. The region of the sample which is
1"'ll'lersed in the gas flow in the probe is indicated in the figure.
This second seal-off was found to be necessary to prevent the sample,

When at higher than ambient temperature, from rapidly distilling

 

124

 

 

 

 

27%
3 C
e
l

 

 

 

 

 

U1

Figure 28. The design of 5 11m NMR sample tubes for relaxation time
measurements up to the critical temperature.

 

u'. , !
'11,.”

' v!

‘55:?!
W19. 1

125

from the bottom to the top of the tube causing bumping and large
thermal currents.

All samples were degassed with several cycles of the freeze-
pump-thaw procedure. Compounds which boiled at room temperature or
above showed little affinity for oxygen and reproducible relaxation
times could be obtained whether the degassing was done with the aid
of a diffusion pump or only a rough pump. Low-boiling compounds
showed a much greater affinity for oxygen and it was found that rough
pumping was not sufficient to remove the dissolved oxygen. All of
the compounds, however, were degassed using a diffusion pump, but
for the low-boiling compounds the pressure was monitored during the
"pump" part of the cycle and these samples were cycled until the
pressure dropped to 5 x 10'6 torr.

Since an explosion of a sample tube while it is in the probe
is to be avoided at all costs, all tubes were carefully tested after
they had been filled. Each tube was heated (inside a closed oven
or behind a protective shield) to a temperature 20° higher than it
would ever be subjected to inside the probe. If the tube survived
this telnperature for ~10 minutes it was considered acceptable. About
one tube in six exploded during testing, presumably due to flaws in

the g1ass which arose through the seal-off.

L Ramon Samples

Silees for Raman measurements were sealed into melting-point
caDillary tubes. Occasionally the laser beam caused the haloform

Samliles to decompose, as indicated by a black spot forming at that

 

 

I": 0

15::

. e.
_ he
I.

126

point on the'inner wall of the tube which was illuminated by the beam.
When this happened the sample was discarded.

C. Preparation of Materials

Deuterobromoform was prepared by agitating a mixture of CHBr3
and 020, made basic with NaCO3, at 50° C for 2 hours. This exchange
was repeated five times with fresh 020 until the proton NMR spectrum
showed deuteration to be 97% complete. The CDBr3 was distilled prior
to sealing in a capillary tube but the CDCl3 (Aldrich Chemical Com-
pany, Milwaukee, Wisconsin, 99.6+ it. purity) and the fluorocarbons
(Pierce Chemical Company, Rockford, Illinois and Columbia Organic
Chemicals Company, Inc., Columbia, South Carolina) were used with-

out further purification (except the degassing step for WW samples).

RESULTS

All relaxation times were measured at a magnetic field strength
of 14.1 kilogauss except for a few measurements of T] and T2 in CF3CC13
at 15.87 "12, which corresponds to a resonant magnetic field of 2.52
Re for fluorine nuclei. The following tables (8-21) couprise the raw data
from which the temperature dependence of the relaxation times was
determined. These tables list the thermocouple reading in milli-
volts with a reference ice-water bath, the temperature and inverse
temperature calculated as discussed in Section I.F of the Experi-
mental portion of this work, the number of points (NP) taken to define
the relaxation curve, and the weighted least-squares relaxation rate
(R) or R2) and standard deviation as calculated by the program
RELAX2 and discussed in Appendix A. For measurements of R1, the spin-
lattice relaxation rate, the decay of ‘the magnetization was monitored
for the time interval 0<t<4T1 (11 is the relaxation time) and usually
about thirty points in this time interval were collected. For measure-
ments of R2, generally more points were taken, but the time interval
was shorter. The only data points deleted from these tables were
those for which the observed temperature drift was unacceptable
(greater than two degrees). At all but the lowest temperatures, the
observed drift in temperature during the course of an experiment was

1955 than O.5° K after the system had reached thermal equilibrium.

127

 

I.

128

Measurements of Spin-Lattice Relaxation Rates

Table 8. Spin-lattice relaxatiOn rates in CF3CC13.

 

 

 

TC(mV) T(°K) lO3/T(°K) NP R1 ( seC'1 )
4.750 352.3 2.750 40 0.2105:.0025
6.780 399.5 2.503 34 0.2730:.0055
8.513 432.7 2.311 27 0.3772:.OO77
4.874. 354.4 2.744 22 0.2088e.0107
-5.558 147.4 5.784 27 0.5755e.0324
-5.559 144.7 5.911 35 0.155se.0442
-5.527 145.5 6.873 41 0.0525:.0031
-5.078 159.9 5.254 33 0.4975e.0272
-3.175 205.0 4.878 42 0.1454e.0029
-3.857 189.4 5.280 37 0.1828:.0052
-4.519 173.9 5.150 44 0.2511e.0057
-5.230 155.1 5.405 34 0.4389e.0082
-5.890 138.4 7.225 34 0.0795e.0043
-4.918 153.9 5.101 45 0.32781.0068
-1.553 237.9 4.203 21 0.1084e.0058
-2-100 228.5 4.375 32 0.12441.0029
-2.577 215.0. 4.530 31 0.1549:.0082
-3.424 199.5 5.013 20 0.1777e.0058
0.534 284.1 3.520 13 0.13721.0018
-0.150 268.7 3.722 10 0.1111e.0008
0.723 235.3 3.499 17 0.1360:.0038
0.520 231.9 3.547 14 0.13611.0026
°~°l3 271.8 3.579 14 0.113ze.0031
‘0-534 258.9 3.862 14 0.103lt.OOll
'4-‘53 248.0 4.032 13 0.1000:.0032
‘-°73 292.5 3.418 15 0.1333e.oo47
1.320 297.5 3.361 17 0.1351e.0044
2-050 312.1 3.204 15 o.1553:.0053
4.940 355.7 2.734 23 0.2201e.0115
‘ 755 305.8 3.270 15 0.1453:.0070

\—

Table 8 - Continued.

¥

¥

129

 

TC(1nV) T(°K) 103/1(°k) NP a] (sec'])
1.155 294.2 3. 399 20 0133950272
0.150 274.7 3.540 13 0123550029
4.745 352.0 2.752 14 0.2028:.0018
3.738 343.3 2.913 17 0.1782:.0029
2.995 329.4 3.035 21 0.1559:.0015
2.120 312.8 3.197 15 0.2085:.0017
1.130 294.7 3. 393 20 0.1970:.0031

-1 . 000 251.3 3.979 23 0.1005:.0005

-1.722 235.5 4.228 22 0.1081i.0046

7.208 407.3 2.455 15 0.2535220053

6.165 388.2 2.575 14 0.2475i.0048

4.532 359.0 2.785 17 0.2170:.0052

5.573 379.2 2.537 27 0.2322:.0040

1.520 301.3 3.319 24 0.1427:.0011

1.575 302.4 3.307 15 0154550010

-3.275 202.7 4.933 25 0.1353i.0086
-3.505 195.2 5.123 27 0.14031.0086
-3.970 185.9 5.350 21 0.1514:.0108
~4.272 179.8 5.552 20 0201350135
-4.51 5 174.0 5.747 24 0.2438i.0140
«I. 903 154, 3 5.085 25 0.3422220155
‘5- l 05 159.2 5.281 17 0.3374:.0154
-5.342 153.1 5.532 30 0471450303
~5. 360 152.5 5.553 22 047941.01 25
-5.548 144.9 5.901 5 0.1122:.0259
‘5- 531 145.4 5.878 25 0.0595:.0053
‘5 . 784 141.3 7.077 25 0. 04051.0082
‘5 - 500 149.1 5. 707 30 0. 07901.0086
‘5~ 237 155.9 5.414 23 049571.0315

 

:::::::::::

 

130

TabIe 9. Spin-1attice relaxation rates in CF3CC13 at 15.87 MHz.

 

T(°K) 103nm) NP R1 (sec'1 )

288.2 3.470 40 0.15661.0035
277.5 3.604 23 0.3667i.0138
280:5 3.565 22 0.4368i.0080
283.9 3.522 22 0.5135i.0097
286.7 3.488 23 0.5587i.0163
276.2 3.621 27 0.27831t.0168
272.4 3.671 22 0.22451.0088
269.2 3.715 16 0.2017i.0056
257.5 3.883 14 0.10601.0042
232.2 4.307 22 0.1021t.0051
204.0 4.902 22 0.0638i.0063
196.2 5.097 17 0.1060i.0087
260.3 3.842 24 0.1570i.0043
262.7 3.807 24 0.1629i.0027
258.5 3.868 26 0.1315i.0045
256.1 3.905 27 0.1149i.0030

 

Tab1e 10.

131

Spin-lattice relaxation rates of 2D in CDBr3

1

 

T(°K) 103/T(°K) R‘sec'
281.2 3.556 3.087i.160
283.8 3.524 3.64 1.12
289.5 3.454 2.850i.031
293.2 3.411 2.751i.047
296.5 3.373 2.636i.041
298.0 3.356 2.606:.081
300.6 3.327 2.479i.035
303.6 3.294 2.27 1.035
304.1 3.288 2.360:.032
305.0 3.279 2.317t.045
310.3 3.223 1.911i.024
333.2 3.001 1.416i.188
340.7 2.935 1.45 1.029
354.5 2.821 1.38 i.031

132

Table 11. Spin-lattice relaxation rates of 19F in CF3Br

 

 

 

room mm 103/1(°x) up 11, (sec")
-3.257 202.9 4.929 35 0.3835 1.0045
-1.437 242.4 4.125 32 0.5195 1.0085
+0258 275.7 3.514 34 0.5955 1.0125
1.205 295.2 3.388 34 0.8542 1.0117
2.430 318.7 3.138 35 1.153 1.031
2.947 328.5 3.044 22 1.315 1.012
3.075 330.8 3,023 33 1.320 1.034
3.438 337.7 2.951 23 1.575 1.033
3.500 340.5 2.937 35 1.527 1.075
3.774 344.0 2.907 31 2.022 1.071
3.985 348.0 2.874 24 3.152 1.081
2.380 317.7 3.148 35 1.123 1.018
-7.343 98.5 10.15 21 0.8713 1.0101
-7.395 95.8 10.33 27 0.5578 1.0390
-7.134 100.0 10.00 14 0.5578 1.0292
-5.945 105.3 9.407 25 0.8545 1.0389
-7.100 101.1 9.891 34 1.071 1.0241
-5.853 109.0 9.174 35 0.8688 1.0090
O~398 279.5 3.578 33 0.7441 1.0135
73-693 193.3 5.173 33 0.3225 1.0098
'5-530 119.8 8.347 15 0.4755 1.0135
73-118 205.4 4.845 35 0.3857 1.0073
‘3-061 207.7 4.815 31 0.3907 1.0082
73-975 185.7 5.355 31 0.3340 1.0082
'4-283 179.7 5.555 27 0.3386 1.0053
‘5-037 150.9 5.215 32 0.3289 1.0091
‘5-347 152.9 5.540 25 0.3498 1.0082
‘5588 145.7 5.817 31 0.3997 1.0058
‘5345 139.5 7.153 34 0.4189 1.0074
'5°275 127.3 7.855 32 0.4905 1.0085

\

Table ll - Continued.

133

 

 

 

TC(mV) T(°K) 103/T(°K) NP R] (sec-1)
-5.855 109.2 9.158 33 0.5919 1.0087
-7.000 104.4 9.579 24 0.9948 1.0151
-7.291 94.4 10.59 11 0.7451 1.0355
-7.335 92.5 10.80 21 0.5113 1.0228
-7.259 95.1 10.52 17 0.5500 1.0183
-5.590 114.8 8.711 33 051701.0082
-5.850 109.4 9.141 30 0.5551 1.0143
-5.954 105.0 9.434 20 0.8408 1.0219
-5.541 115.1 8.513 21 0.5423 1.0108
-4.780 157.4 5.974 30 0.2958 1.0035
-5.285 154.5 5.458 32 0.3151 1.0097
-5.552 147.3 5.789 33 0.3220 1.0041
-5.851 139.1 7.189 35 0.3590 1.0089
-5.040 134.1 7.457 31 0.4251 1.0118
-3.122 205.3 4.847 34 0.3852 1.0088
-2.540 219.1 4.554 34 0.4212 1.0112
-2.540 219.1 4.554 34 0.4190 1.0112
-1.855 233.4 4.284 27 0.4819 1.0130
'0-515 251.2 3.828 35 0.5048 1.0038
+0-le 275.8 3.525 34 0.5851 1.0052
0-842 288.2 3.470 33 0.7752 1.0175
1.555 302.3 3308 35 0.8955 1.0207
1-973 310.0 3.225 34 0.9995 1.0171
1-993 310.4 3.222 35 0.9959 1.0228

_‘

k

134

19

 

 

 

 

Table 12. Spin-lattice relaxation rates of F in CFZBr2

1cm) 11°10 103/1010 NP 1111115")
-5.757 142.2 7.032 35 0.6813 1.0038
-5.037 160.9 6.215 23 0.3542 1.0020
0.804 287.4 3.479 23 0.3251 1.0033
-3.893 188.8 5.297 42 0.2389 1.0031
-4.818 166.4 6.010 40 0.3532 1.0075
-5.316 153.7 6.506 41 0.6289 1.0096
-5.540 147.9 6.761 45 0.8596 1.0068
-5.597 146.4 6.831 47 1.026 1.0102
-2.588 218.0 4.587 50 0.1895 1.0020
+1.639 303.5 3.295 33 0.3603 1.0018
1.668 304.1 3.288 27 0.3717 1.0015
1.675 304.2 3.287 30 0.3754 1.0019
2.755 3.24.8 3.079 35 0.4321 1.0034
4.138 350.8 2.851 32 0.5150 1.0055
5.328 372.8 2.682 34 0.6391 1.0471
6.631 396.7 2.521 35 0.7188 1.0091
7.824 418.4 2.390 32 0.8503 1.0116
9.383 446.6 2.239 35 1.017 1.0187
7.385 410.4 2.437 35 0.8077 1.0087
4.973 366.2 2.731 32 0.5803 1.0070
-4.547 173.2 5.774 35 0.2726 1.0036
1.305 297.2 3.365 32 0.3466 1.0051
0.385 279.2 3.582 30 0.2987 1.0019
-0.356 264.4 3.782 32 0.2736 1.0033
-1.218 247.0 4.049 32 0.2393 1.0017
-2.453 220.9 4.527 31 0.1981 1.0056
_____~_‘-5.070 160.1 6.246 32 0.3500 1.0050

\

 

 

Table 13. Spin-lattice relaxation rates of 19F in CF2010013

135

 

 

 

TC(mV) T(°K) 103/1(°K) NP R](sec'])
2.304 315.2 3.153 15 0.1259 1 0119
-1.522 238.5 4.191 14 0.082951 00487
-0.552 250.5 3.839 22 0.082501 01475
0.45 280.5 3.555 14 0.088201 00087
1.722 305.2 3.277 15 0.1039 1.0019
5.413 392.7 2.545 15 0.2814 1.0394
5.155 359.8 2.704 15 0.1575 1 0072
4.105 350.2 2.855 15 0.1355 1 0110
3.104 331.4 3.018 17 0.1425 1 0013
2.225 314.8 3.177 15 0.1502 1 0037
1.808 305.9 3.259 47 0.1121 1.0045
0.857 288.5 3.455 37 0.090931 00159
0.108 278.2 3.595 47 0.083451 00035
-0.999 251.4 3.978 47 0.089591 00114

136

 

 

 

Table 14. Spin-lattice relaxation rates of 19F in CFClZCFCl2
TC(mV) T(°K) 103/T(°K) up algsec")
1.120 293.5 3.407 14 0.090271.00095
2.187 314.1 3.184 34 0.1049 1.00085
3.979 347.8 2.875 32 0.1397 1.0009
5.445 374.9 2.557 42 0.1548 1.0010
5.923 402.1 2.487 45 0.1815 1.0014
8.007 421.8 2.371 47 0.2118 1.0045
2.182 314.0 3.185 33 0.1028 1.0020
2.358 317.3 3.152 101 0.1020 1.0090
0.855 288.5 3.455 30 0.044581.0019
1.250 295.3 3.375 23 0.093121.0015
4.997 3.557 2.727 22 0.1472 1.0032
3.025 329.9 3.031 35 0.1295 1.0059
1.378 298.5 3.350 14 0.097351.0050
-0.545 258.5 3.857 12 0.050971 00054
-1.725 235.4 4.230 17 0.047271.00055
-2.975 209.7 4.773 14 0.045741.00120
-4.437 175.8 5.688 23 0 075101.00400
-5.555 144.7 5.911 10 0.5477 1.0325
-5.555 144.7 5.911 21 0.4829 1.0275

137

 

 

 

Table 15. Spin-lattice relaxation rates of 19F in CF31
TC(mV) T(°K) 103/1(°1<) NP R](sec“)
1.168 294.5 3.396 35 0.6094 1.0086
2.110 312.7 3.198 31 0.7172 1.0168
2.162 313.6 3.189 35 0.7126 1.0207
0.475 281.0 3.559 34 0.5521 1.0059
-3.623 194.7 5.136 27 0.3632 1.0050
-4.897 164.4 6.083 32 0.4534 1.0067
-5.075 160.0 6.250 32 0.4969 1.0064
-4.537 173.4 5.767 35 0.3863 1.0037
-2.726 215.0 4.651 35 0.3474 1.0056
-1.100 249.2 4.013 35 0.4358 1.0025

 

 

'11

138

 

 

 

Table 16. Spin-1attice relaxation rates.of 19F in CF3CF3
1cm) 11°10 103/1(°1<) up 111 1111:")
0.677 284.9 3.510 35 0.5132 1.0029
0.523 282.0 3.546 35 0.4840 1.0023
-4.000 186.1 5.373 15 0.2029 1.0032
-4.312 178.8 5.593 16 0.2097 1.0139
-4.312 178.8 5.593 34 0.1978 1.0061
-0.198 267.6 3.737 35 0.4022 1.0013
-1.153 248.3 4.027 35 0.3159 1.0015
-1.755 235.8 4.241 31 0.2557 1.0038
-2.406 220.0 4.505 25 0.2237 1.0035
-2.859 212.2 4.713 45 0.2080 1.0019
-3.672 193.8 5.160 43 0.1965 1.0026
-4.496 174.5 5.731 32 0.2000 1.0052
-4.610 171.6 5.828 34 0.1752 1.0018
-4.757 167.9 5.956 36 0.1541 1.0024
1.200 295.1 3.389 11 1.159 1.018
1.226 295.7 3,382 17 1.374 1.024
0.936 290.1 3.447 23 0.5859 1,0052
-4.842 165.8 6.031 34 0.1411 1.0014

 

 

._ __T1

139

II. Measurements of Spin-Spin

Relaxation Rates

 

 

 

 

Table 17. Spin-spin relaxation rates of 19F in $011'd CF3CC13
TC(mV) T(°K) 103/1(°K) NP Rz(sec")
0.768 287.6 3.489 8.31
“0.168 268.2 3.729 57.9
-1.097 249.4 4.001 450.
-1.988 230.9 4.331 2138.
-3.516 197.2 5.071 4770.
-3.089 207.0 4.831 12900.
285.2 3.506 77 9.4611 .259
280.2 3.569 77 12.53 1 .35
276.2 3.621 101 23.50 1 .55
273.3 3.659 73 59.35 1 1.89
268.2 3.729 50 70.93 1 2.05
263.2 3.799 50 109.2 1 8.2
258.2 3.873 21 346.8 1 35.2
253.2 3.949 310 433.8 1 8.5
248.2 4.029 310 679.5 1 12.3
243.2 4.112 226 1349. 1 73.
238.2 4.198 144 2163. 1106.
233.2 4.288 62 4068. 1138.
223.2 4.480 36 10740. 1843.
“—-————_____ 213.2 4.690 17 24320. 16710.
\

 

 

140

 

 

 

Tab1e 18. Spin-spin re1axation rates of 19F in liquid ($36013

TC(mV) T(°K) 103/1(°K) NP 82(sec")

0.571 284.8 3.511 35 7.875 1.241
0.750 285.5 3.490 35 5.454 1.200
0.850 288.5 3.455 55 0.18881.0052
1.133 302.5 3.305 55 0.22181.0079
1.534 303.4. 3.295 55 0.18431.0048
1.553 304.0 3.289 54 0.18731.0051
2.345 317.0 3.155 53 0.20541.0070
3.200 333.3 3.000 53 0.22391.0055
4.223 352.4 2.838 47 0.27111.o145
5.045 357.5 2.720 45 0.29051.0110
5.352 391.5 2.553 51 0.45581.0258
5.435 393.1 2.544 54 0.42501.0157
288.2 3.470 200 0.15941.0052

 

 

 

 

 

141

Table 19. Spin-spin re1axation rates of 19F in CJ-‘ZBr2

 

 

 

T(°K) 103/1(°K) Rz(sec")
301.5 3.317 0.37851.0089
325.7 3.051 0.55801.0187

{0.55571.0193
344.2 2.905

0.50401.0153
354.2 2.745 0.71381.0120
357.2 2.723 0.95681.0240

383 2.511 1.291 1.053
400 2.500 2.019 1.087

 

 

142

19

 

 

 

Table 20. Spin-spin relaxation rates of F in CF30F3

TC(n1V) 103/T(°K) R](sec'1)a R2(sec'1)

291 .7 3. 428 0.65621.0208
280. 6 3 . 564 0 . 49861 . 0263 0 . 52901 . 0090
266 . 7 3 . 750 0. 40221 . 0013 0 . 40071 . 0080
249. 9 4. 002 0. 31591 . 0015 0. 30421. 0070
227.2 4.401 0.22371.0035 0.24761.0037
214.2 4.669 0.21311.0041
195.1 5.126 0.20281.0032 0.21701.0031
191.0 5. 236 0. 20971. 0131 0.22741.0031

 

 

a
From Table 16.

 

143

Table 21. Spin-spin re1axation rates of 19F in CF2010013

¥

¥

 

 

£01111) T(°K) 103/T(°K) NP R2(sec'])
2.492 320.0 3.125 40 0.55721 .0250
2. 939 328.3 3.045 40 0.58551 .0285
3.441 337.8 2.950 40 0.51751 .0350
3.984 347.9 2.874 31 0.58591 .0475
4.535 358.1 2.793 40 0.58051 .0259
5.127 359.1 2.709 40 0.55021 .0287.
3.512 239.2 2.948 40 0.49391 .0139
4.023 348.5 2.859 40 0.48411 .0144
5.154 369.8 2 .704 40 0. 57701 .0129
6.488 394.1 2.537 34 0.54031 .0214
7.583 414.1 2.415 40 0.74341 .0317
8.547 431.5 2.317 35 0.94351 .0572
2. 467 319. 5 3.129 40 0.35351 .0129
0:313 013.20 104 5.592 1 .211
0-586 283.2 3.531 45 54.10 1 1.94
~0-015 271.2 3.587 24 139.0 1 5.4
~0-420 253.2 3.799 25 268.6 1 5.5
'0-974 251.9 3.970 34 1550.2 1 35.8
'2-623 217.3 4.502 17 5124. 1319.
+175 182.1 5.491 11 10820. 12451.
t:

 

 

 

 

magnetization

144

 

 

 

 

 

_1 1 L l
111 21) 31)
Time ( VT.)

Ffigure 29. A semi-logarithmic plot of magnetization versus time in
the 180°-r-90° T] measuring sequence, showing the typical

linearity and degree of scatter observed.

145

.pmp aucmgmwmmn

.mu—zmwg huuznxm

 

 

 

 

 

 

Nu.“ hm.~ oo.m 4,.Nfl ~m.m~ 5.0m mam.~ m_uumuu

,5.“ ~5._ mm_.o mw.m mm.m5 Fh.o_ o.1~ m_m.5 mpuumuo

51o.1~om.o mm.m «.1. 5055.1 5.5_ 151.0 mpuummu

mamgm>m omp.o

nmm.~ m~5.o ~15.o m~._1 me.m~ m.m~ 55m.5 m_ouu

55_.~ 155.o 511.o m~._4 55.m~ ..a Nem.o mPoku

51m._ mmp.o mmm.o mm. 5 mm.m_ “.mw- o¢o.5- m_umo

moFxAuwm\~euvmo AF1Eu mmzmuvw Auomvmp manopm Auovh A>EVUH uczoasou
m_oumuu

cw acovuvumoou cowmzmwwwnm—um 0;» mo acmemgamows vco acmwcmgm upmww mg» mo cowumcwsgmpmo .NN apnoh

 

81115131235351 .5

 

146

.am 55 ma 45 551.5511511 5158155505

.Ahx\_5ux ._.Nv 1x1 4.o_.m.o n ma 55158> m_u0uu

.mupzmmg kamsz

 

 

 

 

.1551_ummu

eo.~ om. He¢.¢ pnm.o mo.m omo.nn

Num.o em. H¢~.¢ epn.o NN.m wom.m1

o¢.~ o.FH m.op mo.p N¢.n mmm.ou

me.m w.pH N.mm up.~ om.m cmm.e1

m~.m m.~« m.- wo.~ mm.m nmm.m1

eo.¢ N.Nu «.5— Na.p m~.e mmp.m1

mm.u m.~« o.¢m em.p oom.¢ mem.pu

mn.w m.~w N.nm mm.p mm.~ «ma.m Nem.ou

m¢.m ~.FH m.n~ pnm.o Nwm.m mme.o

ammo.o .11 5551151:
mwpo.npmop.o umm.~ mm. Hop.m mm.o m.mm~ e~a.m omw.ou
mmoo.flommo.o omm.~ mo. Ho¢.m om.o w.onm mam.m coo.o
tomoo.«nmmo.o umc.N pm. Hom.m Pn.o emu o¢.m ¢m~.~
A_1su mmaumvu mopxaomm\~suvmo nonopm Aummv Axov Ahxgomopv A>evuh
MN» 5 h\5

 

 

11419 11 21515155155 151412515

1 mm
5. 8:9 5o “1055130505 use peopuogm upopm on» no =o_»u=_ago»un .nm uNQMH

147

.umpmpsupmu my pcowuwwmoou
cowm=5$_u mgu 505;: 5015 .ucwsrgwaxw ocum cram mmpznuozu wcu c_ warp msmgm>
cowumNPpocmms mo weapo> Poorumgoozp ecu Faucoswgoaxm we papa hmmzmx quwnzp < .om mgsmwu

 

>u 3pm.o wc bzmpmzom zo_m=us~m naumnmu
“ U u u u u u u

 

 

 

Hum umnmu.m .u m m .u. .u. n .u. n .
xux
00x
10x
. OK .
O
I
On
A. O
x
O
u x w
0
I v
0
x
A O y
x
O
. x0 .
x0
: O .
x
O
K
x0
1 xx .
O
uxx
- O K.
O
‘ H. u H W a! K N ‘0
.ou.151. 1 .1101 1. a .2101 11.0 “:2 =5 0.. 22501 102.129159 1:. 1.52m35qm515 z. 32.”: .

"1.115.. . 2181055,...1.5w“_w_mu. ”Mn” 12“: m5 . 1121. 22.11 112.1191.9 . “2.0: 9 .22_°1 1‘52u:_uua.. a. mz.ux .

1;. u .. .o- co . .

Na 2. s hzuxuuuz_ cc 0551 . »x¢_a ~120u" “m5m»u«»-wwmw@qe . huu&owxwx».»mawou<o.noummx uocm~nw.m.m.vuthwm%m
I x. U

148

Table 24. Raman linewidth measurements

 

 

 

 

 

Mblecule Freq.(cm'1) pa ”intra wora R(w0r) %Corrb
00013 650 0.017 3.3 2.87 .24 12
2256 0.088 2.8 2.70 .093 2
CDBr3 222 0.063 1.8 0.997 .023 3
521 0.011 2.0 1.29 .24 17
2250 0.13 3.3 1.07 .78 l
CF3CCT3 714 0.027 2.5 2.09 .41 7
amintr = intrinsic half-width at half-height; ”or = half-width at half-

height of the orientational component; 0 = depolarization ratio.

t’Percent intensity of the depolarized spectrum subtracted to correct

for leaked polarized light.

149

 

.MLmau 55 5=__ 5-15 NNN 51»
104 .A- 1 1V Eaguumam chc*amucmvgomg cmp~u=mso4 a new: acmcoaeou umwwsmpoa mg» mo
covpapo>coo umumpsupmu can .A. . .v Esguumam umNPgapoawv .alllv Esguumam vm~¢1opom .Fm mgamwm

m7.

 

Achcnuv .3.
m; Oh m. 0 WI Op...
— — _ — _

 

 

 

 

 

 

 

 

 

.mtmao 4o 515—
.20 pmm as» 1cm .5. 1 1v Eaguumam —m=oruapcmvgomg :uPNpcmgog a saw: ucmconsou ca~wgopoa
m a mo 533268 .3333me .55.. .A. . .v .553QO 82.2283 4.1V 5238.». 32.228 .Nm 953...
ATEUV 3
2 m o m- 2.. 2-
. -_ _ _ _ _
Ioooaoouoo’o.” \..\.a.\. .. .... ...
...u 2...
m 1,» ..\.

.1
C. ...s

4. ...s
...... .....

r ...s

4. ..s

... ....

6.... ....~~

7... ...:

I... m;

4.... ..u...

a... ....

a? ...“.s

.oooo‘
....

 

 

151

 

 

If!
.mgmao 05 51..
-su ommm mg» Lou .A- 1 1V Esguumnm pmcovumucmwgoog cownpcmgoA a gap: u:m:og£ou vm~w1mpon
wnu mo coruapo>cou cmumpaupmu ucm .A. . .v Eagpumnm vm~w5~—oamc .alluv Ezguumgm cm~wgmpom .mm mgsmwu
AVE 3 3
2 O— m o n 1 2.1
_ _ _ 5 fl
'4}. i o. o lo.
... I .l.. . 1.0.01.3... I. ...
......w I. \1 ....
1...! (lo \\ .....
...... or 3‘... ..
MI W... ..... ....
... It ...\.4. .4.
W. I ......V\ I...
..o..V4...... .~.. \
I ... .. s
I .. .. s
r m ... s
(A. ......\\
. . \
J4 .... \
lb..." .«o~ \\\
777... ... s
. .. s
/ ...
If .V..\
......7. \
.77. \

 

 

 

152

 

 

 

 

 

Flgure 34. Polarized spectrum (———J. depolarized spectrum (' ' °).
and calculated convolution of the polarized component
with a Lorentzian reorientational spectrum (' ' -).
for the 650 cm'1 line of CDC13.

 

153

 

 

 

 

J
-15 ' 0

m (cm" )

 

15

Polarized spectrum (———0, depolarized spectrum (- - -),
and calculated convolution of the polarized component
with a Lorentzian reorientational spectrum (- - -), for

the 2255 cm-1 line of 00c13.

Figure 35.

 

154

 

.mpuummo 4o 5155

 

use cps mg» 1o» .5- u 1v Esguumgm chorumpcmwgomg cmwuuzmgog m guw: acmccaeou um~wgopoa
wk» 40 copuzpo>=ou umumpzu_mu can .A. . .v Esguomam uw~_gm_oqmu .A v sagpumam umupgupom .mm «Lamp;
5.0:;nuv .3.
2 2 n o n- o... 2-

 

 

 

 

 

 

 

DISCUSSION

1. Raman and NMR Relaxation Studies of 0001, and CDBr3

A. Introduction

Nuclear quadrupole coupling constants are not directly measur-
able in the liquid phase since the rapid molecular motions broaden
the pure quadrupole resonance lines. They may be estimated from the

solid and/or gas phase values or else obtained indirectly from various

 

NMR experiments.

In the absence of strong intermolecular interactions in the
solid phase, such as hydrogen bonding or Lewis acid-base interactions,
it has been shown that quadrupole coupling constants from microwave

gas studies are about 5-10% larger than those obtained from NQR

240

spectra of solids and the liquid-phase value may then be presumed

to lie somewhere between these two. When there are important inter-

molecular interactions, as is often the case for 2D and 14N, an in-

241

vestigation has indicated that the solid-phase value should be

the best approximation unless the structure is considerably deformed

242

from that in the liquid, as in ammonia Unfortunately, deuterium

pure NQR spectra are not in general observed in solids because of the

low sensitivity at the low frequencies necessary and one must there-

fore obtain them from NMR spectra of solids243'245

be complications from molecular m0tions245.

where there may

NMR studies of solutes in liquid-crystal solvents have been
IJSed to obtain deuterium quadrupole coupling constants with un-
¢=ertainties ranging from 3-10%246'248. Alternatively. quadrupole
155

156

coupling constants may be obtained from liquid-phase measurements

of T] for the quadrupolar nucleus by use of the equation28

1L.= 1 1 3(21*3) "2 1 + DE. §E99.2 (129
T1 T10 1012(21-1)( 3“ " )Te’ )

if the angular correlation time for reorientation of the X-D bond,
16’ is known. Where the asymmetry parameter 0 is unknown it may be
taken as zero without too great error. Values of T0 have been
estimated from viscosity data using the Debye relationship 16 =

4nga3/3kT. (where 5 is the viscosity coefficient and a the molecular

)249.250

radius from viscosity data combined with structural param-

25], and, in the case of deuterium quadrupole coupling constants.

252-253

eters
from proton relaxation times of the protonated analogue For
a few molecules of special geometry it has been possible to determine

241,254

I rather accurately from NMR data

0
We assume here that the Raman correlation time from Raman line-

shape analysis can be used to obtain 19 in Equation (129) and describe

a new procedure for determining quadrupole coupling constants in

liquids which yields the coupling constants directly for molecules

with the quadrupolar nucleus on an axis of threefold or higher sym-

metry. Values of T1Q are then obtained directly from NMR data and

used to solve for e200 (Equation 129 above); 0 is identically zero

for this geometry. The analysis is less satisfactory for molecules

of lower symmetry or nuclei not on the symmetry axis, as has been

pointed out for benzene17].

As a result of the theoretical and experimental difficulties

IHentioned above, few "experimental" values of quadrupole coupling

¥

157

constants in liquids have been reported. The present method should
be useful in providing additional data. which may be used. for ex-
ample, to investigate environmental effects of deuterium quadrupole

coupling constants by comparison with recent gas phasezss'258 and

15110 phase243,244,259

values. Although the method is an indirect
one the uncertainties are reduced since we are determining the
square of the quadrupole coupling constant.

Since the quadrupolar relaxation mechanism dominates, elaborate
purification of the samples. chemically or isotopically, is not
important. Since T1Q and 16 are strictly related via Equation (129),
making both Raman and NMR measurements on the same sample will elimin-
ate the effects of impurities, other than paramagnetic ones.

In contrast to pure quadrupole resonance spectroscopy, where
quadrupole coupling constants below about 1 MHz are experimentally
inaccessible, small quadrupole coupling constants are easier to ob-
tain than large ones by the present meth0d since the NMR relaxation
time becomes longer. Of course. as the quadrupole coupling constant
approaches zero other relaxation mechanisms become important and
the interpretation becomes more complex. However, even deuterium
quadrupole coupling constants, which are among the smallest studied.
are large enough that other mechanisms do not compete as we demon-
strate here for the case of deuterobromoform. The importance of
other mechanisms may be estimated from the proton relaxation rate
(1/111 of CDBr3, which is found260’261 to be 0.0451 sec" at 25°
C. For the various other interactions which may be of importance
(intermolecular dipole—dipole, spin rotation, scalar coupling to

bromine) the Contribution to the deuterium relaxation rate will be

 

158

smaller by a factor of ~40 due to the dependence on gyromagnetic
ratio (72) or on spin-rotation interaction constant (C2). Thus, we
estimate 1/11(other) a 0.001 sec", which is certainly negligible
compared to llTlQ = 2.68 sec'].
The correlation time associated with NMR relaxation is the zeroth

moment of the rotational correlation function

00

T6 = J{<R(t)R(0)>dt . (130)
When the correlation function is exponential (i.e., the orientational
spectrum Ior(w) is Lorentzian) the correlation time is the half-
width at half-height of Ior(w)'

For symmetric-top molecules the motion can be described by two
correlation functions describing the motions parallel to and perpen-
dicular to the symmetry axis. Consequently, through Equation (130)
there are two rotational correlation times 151 and Tell. It was shown
that for symmetric molecules the Raman line of A] symmetry is affected
only by rotation perpendicular to the symmetry axis and thus gives
information only concerning TQ1,. Other lines contain information
about both parallel and perpendicular rotations, but Tell is more
difficult to extract. In the case of symmetric-top molecules with
the quadrupolar nucleus on the symmetry axis this presents no problem
as only motion perpendicular to the symmetry axis is effective in
relaxing the nuclear spins. Since both the NMR and Raman data refer
in this case to reorientation of the same vector they may be com-
pared directly, or Equation (129) may be used to examine the liquid-

phase quadrupole coupling constant.

159

8. Analysis of Raman Spectra of CDqu_and CDC13

In Figures 31 and 32 are shown the normalized polarized and
depolarized spectra, and the calculated convolution of the polarized
spectrum with the Lorentzian orientational spectrum for the 222 cm’1
and 521 cm'] lines of CDBr3. The calculated lineshape for the 222
cm'1 line matches the experimental as far into the wings as the
signal may be distinguished from the noise (m6 half-widths). The
521 cm“1 line, on the other hand, clearly does not conform to a
Lorentzian orientational spectrum. The depolarized component is too
narrow at the center and too broad in the wings. The polarized line
is slightly asymmetric on the low frequency side, while the depolarized
component shows a distinct shoulder on the high frequency side. The
presence of fine structure on both sides of the absorption makes
analysis difficult, as Bartoli and Litovitz discuss3o, but it is
not clear that the unresolved weaker components of the 521'1 cm
line are responsible for the large discrepancies between the calcu-
lated and observed line shape. The low-frequency asymmetry in the
polarized spectrum, which was presumed to be due to a "hot" band,
was successfully described by an absorption band with fractional
intensity of 0.14 and frequency offset 2.6 cm" (with reference to
the main band).

Since the Fourier inversion method of determining 19 makes
no assumptions concerning line shape, it must be used when the
convolution method fails. Figure 37 shows the correlation functions
obtained via Equation (85) for the 222 and 521 cm"1 lines plotted
on a logarithmic scale. The 521 cm'1 function is arbitrarily shifted

 

160

 

 

 

 

O 1 1 .__
‘~_-015 -—- --
«25

(C)
(b)
-11)-—- (a) -
l 1
11) 213
Time (picosec)
Figure 37.

Correlation functions calculated from CDBr3 Raman lines; (a)
uncorrected low-frequency side of the 521 cm‘ line, (b)
low-frequency side of the 521 cm'1 line corrected for the
presence of a hot band, and (c) low-frequency side of the
222 cm-1 line.

161

downward for clarity. The correlation function labeled 52la was
obtained by Fourier inversion of the low-frequency sides of both
components of the spectrum, while the one labeled 52lb was obtained
from the low-frequency sides after correcting for the presence of
the hot band. It was necessary to assume that the depolarization
ratio of the hot band is the same as that of the main band in order
to make the correction to the depolarized component. From Figure
37 it is clear that the hot-band correction brings the correlation
time for the 52l cm"1 line into closer agreement with To for the
222 cm'1 line (although this number was not calculated). However,
it is equally clear that both correlation functions derived from
the 52l cm'] line contain the same features and both deviate from
exponential (linear on the logarithmic scale) behavior at times when
the 222 cm"1 line correlation function is well described by an ex-
ponential decay.

Since all Al lines in a given molecule must show the same re-
orientational broadening in the absence of vibration-rotation coupling
(which is not possible in this case), we conclude from this discrep-
ancy between the 52l cm"1 line and the other two A1 lines in CDBr3,
that as many lines as possible should be studied for each molecule
and that conclusions (especially any conclusion that there is non-
exponential correlation behavior at long times) not be drawn on the
basis of a measurement of a single line.

The remaining Raman lines of A1 symmetry for CDBr3 and CDCl3
are all adequately fit with a Lorentzian orientational spectrum,
although smaller depolarization ratios or smaller absolute intensities

result in larger uncertainties in m These lines are listed in

O?"

 

162

Table 24with their observed intrinsic (vibrational plus slit broaden-
ing) and orientational widths, plus the percentage correction to the
depolarized line for leakage from the strong component. and the
measured depolarization ratios. Literature values of the depolariza-
tion ratios are much larger since they were measured before the advent
of the laser as a light source and consequently the scattering geom~
etry was not well defined. The 365 cm"1 line of CDCl3 is not included
in Table 24 because the chlorine isotope splitting (expected intensity
ratios 27:27:9:l) was resolved. making analysis more difficult.

The bromine isotope splitting (expected intensity ratios l:2:2:l)

was not observed for any of the CDBr3 A1 lines. Table 25 lists

the correlation times obtained from these measurements plus results
from Raman experiments in other laboratories. Table 26 lists the
observed deuterium NMR relaxation times and the Raman correlation
times, along with the quadrupole coupling constants calculated from
them by use of Equation (129). Other measurements”’236 of the 20
relaxation time in CDCl3 agree very well with our result of 1.47:b02
seconds, which was obtained at 27° C; however, the range of Raman
correlation times To for CDC13143 or CHC13141’155 is much larger

than the quoted uncé£tainties and seems to be independent of the
vibrational band or the isotopic species studied. The solid—phase
values of the coupling constants are included for comparison. It

was noted245 that the solid-state value of the quadrupole coupling
constant of CDBr3 was abnormally low, perhaps as a result of molecular

rotation.

163

Table 25. Raman correlation times reported for CDCl3 and CDBr3a

 

 

 

 

Line T6 .
Molecule (cm'l) (psec) Methodb Temp. Ref.
00013 667 1.4 2.3 c 22° c
3019 1.5 2.3 0 22° c
3019 1.971.17d FT 23° e
00013 650 l.851.3l 0 27° f
2256 1.961.14 c 27° f
2256 1.59:.03 FT RT 9
CHBr3 222 5.3 12 C 22° c
CDBr3 221 5.33:.22 0 27° f
521 4.1:].2 c 27° f
2250 5.022 c 27° f

 

 

aObtained from the relation 16 = [chmorJ'].

bC = convolution method [Equation (84)]; FT = Fourier inversion method
[Equation (85)].

cReference l4l.
dUncertainty estimated from Reference l55, Figure 2.
eReference 155.

fThis laboratory.

9Reference 143.

164

Table 26. NMR and Raman results and deuteron quadrupole coupling constants

in CDCl3 and CDBr3.

 

 

 

 

QCC(kHz)
Molecule NMR T](sec) Raman 191 (psec) Solid Liquid
00013 1.47 1.02 l.851.3l 166.9a 1582l4
1.952.14 153:7
cosr3 0.4032.005 5.33:.22 122b 177:5

 

 

aReference 244.

bReference 245.

Ii ...Iflrt [fl

 

165

C. Effect of 2D on Diffusional Motion

According to Gordon'sg3

extended diffusion model the effect of
isotopic substitution on the rotational correlation times will depend
on the ratio of moments of inertia for the axis in question. taken

to the one-half power. NMR relaxation experiments, such as T] mea-
surements of ‘4N in ammonia86 which give T1(NH3)/T](ND3) = l.42.
independent of temperature, while [I (N03)/I (NH311/2 = l.39, tend

to confirm this prediction but Raman results are less conslusive.

For example, one would expect that deuteration of methyl iodide

would strong affect Tell but have very little effect on 161: Gold-

berg and Pershan142

examined Raman lines sensitive to to for both

of these molecules and. although they claimed reasonable agreement
for the different A1 lines. their results were somewhat inconclusive
in that 191 for CD31 and CH3I agreed within experimental error for
the v3 line but not for the v2 line. This may result from the
smaller depolarization ratio for the v2 line, evident in their Figure
l, but they do not give uncertainties on a line-by-line basis. Gillen
and Griffiths77 observed that Tel in c606 is 10% larger than 191.

in C6H6, which is approximately correct for the change in the'mnment
of inertia for that axis. In view of the utility of joint NMR-

Raman experiments involving a quadrupolar nucleus such as deuterium.
both to obtain coupling constants and to determine complete diffu-
sion tensors77’81’173, it would be useful to investigate more fully
the effect of deuteration, especially in light of the discrepancies
in the Raman Igl.0f CHCl3 and NMR 10 of CDCl3 observed by Campbell

1
and JonasIss.

 

 

(‘f

166

D. Deuterium Quadrupole CouplingConstants

Table 27 lists the reported 20 quadrupole coupling constants
for molecules of the type DCX3, plus other relevant data. To a good
approximation the contributions to the total field gradient. QFOt.
may be considered to arise wholly from the directly-bonded nucleus
(qnuc) and the electrons in the bonding orbital (qel). There are two
reasons for this. First. the contributions to the field gradient

depend on r.3

. so directly-bonded nuclei are much more important than
more distant nuclei and electrons. Second, the two contributions
qnuc and qe] for more distant atoms tend to cancel as can be seen
in theoretical calculations of the field gradient in polyatomic
moleculeszsz.

The electronic contribution is calculated from the relation
qel = qtot - qnuc where qnuc . 221/r2”. Z1 is the nuclear charge
(= +6), and rCH is the carbon-hydrogen bond length. We assume here
that rC-H E rC-D since these values are usually equal within the
experimental errors .

Salem263 has demonstrated that the relationship k'» eqtot
exists for diatomic molecules, where k is the force constant of the
bond. That relationship seems to be appropriate for polyatomic

wellzm’26$ and can be seen in Table 27 to be valid for

molecules as
this series, although the proportionality constant (in a.u.) is some-
what larger than the 0.70 predicted by the covalent model. He note
that if the C-0 vibrations result solely in motion of the deuteron.
while the rest of the nuclei and the electron cloud remain fixed.

then the theory predicts k = eq. Various values of k have been reported

167

 

.1. .»Po>.aumnmmc .mom wen new mmuemgmmmm seem
one mgmzo use mpuzu Lo» mmapo> “Aewmpv momp .oo Ammm amvmgmg .memsh .mocoa .4 .h .o new .so:w>ogw .m .m
.mc54 .< .a 552% mazu toe one» we” “Fumpv own ._N .maga ._oz .»~_=a .a sage we 3:9 toe x Co a=F~> «new

.xpmsvuumamae .ouu ecu New ._NN ”mocoememm gate a.” memxo we» m_u=w .mazu tow mace» "Ammmpv
m3. .mwz “conned .uom 33m .095 .comanzh .3 .1 EB vacm .m .0 .o 50...... my :0 Lo..— 21 L we 2:5; 2:6
.xsoz acmmusn ogu sage mp msmou

gcwlmapm> on» open: .»~o>wuumammc .cem new mmu mmuemgommm scum use mpuou new menu com mo=~u> “Aonmpv
moo. .mm .maga .sago .a ..agoaem_g .3 new .emacmzz .m .a .ameoz .u .m gone a, «rag so» caNo co «spas ween

H.AomaPV mm. .mm .m»:a .sogu
.au mvoguoe «cosmeewu $30» an uw>wgmv omega scam mmeogh wen avgow x: umuumme x mmapu> zuw>wuamococuumpun

.HAmnmpv vac qmw .uuw~..>mx .mxga .vam>,.4 .: one apex .> .mu «Eu “mno— x mum.m n o mcwm: umum_:upmuw

 

 

m.~ com.o wo.m mmoo.ummo~.o memo.w-mp.p1 n¢m¢._ No.umoo.p mm sup msmcu
o.m mou.o oo.m Nooo.upnew.o cmoo.wmmwc.—s cumm.~ eco.woop.p —. H m.om_ mpuoo
mm.m nwu.o oo.m omoo.wm~m~.o @nmo.umomo.—1 mm¢m.~ ~o.nmmo.— o.~« w.oup mmau
m~.~ mem.o mm.m mmw~.o ammo.pu mmmm.— vpmo.~ emu. Hm¢.Pm~ mzou

 

._x 2.3 mafia airs. 33,... 3.: £5. waves. 2322
.mW mopxx uacc .

 

 

.concmu
um~vupgaxg new so; mgauusgpm wen museumcou muse; .mpcuumcou mcvpaaou ewmzpma mcowumpoc use .NN mpnmh

168

for 00013 and c03r3255"268 but the ratio k(CDCl3)/k(CDBr3) a 1.02.
Bersohn269 has presented a qualitative argument to show that the
deuterium quadrupole coupling constant should decrease with an increase
in substituent electronegativity. His argument considers only the
charge distribution in the C-0 bonding orbital but it is clear from
Table 27 that a more important effect in determining the quadrupole
coupling constant is the C—D bond length. The value of rC-D for
CDBr3 is somewhat suspect because of the large uncertainty ( 0.02

270.

K) associated with the early microwave determination In the case

27] rC-H = 1.073:0.02 X was later redetermined272

of CHC13 the early value
and the more precise value rC-H = 1.10010.0004 K found. It seems
likely that the value of rC-D in CDBr3 should also be larger and we
prefer to use the same value (1.100 K) as found for CHC13, which
leads to q"“c = 1.3350 and qe‘ = -1.0735:.0122. In all of these

1

molecules the uncertainties in q8 are dominated by the uncertainties

in the C-H bond length.

E. The Rotational Diffusion Tensor in CDBr3

From the correlation function obtained from the 222 cm'] line
it can be seen that, at 27° C. rotational motion perpendicular to the
SYWMetry axis is well described by a diffusional process. Additional
infbrmation on molecular motion in CDBr3 can be obtained from studies
of the temperature dependence of NMR relaxation times. Table 10
gives relaxation data for the deuterium nucleus, and Farrar £3 31.239
have studied the temperature dependent 13C relaxation in 60% enriched

‘3CHBr3, which is given in Table 28. Ordinarily, to extract both

 

169

Table 28. The temperature dependence of 13C spin-lattice relaxation

 

 

 

in ‘3CHBr3
R1.tota1a T011012”) R1 .ddc R1 ,scd
T(°C) 103/T(°K) (sec'l) (sec) . (sec-1) (sec")
124 2.52 0.90 2.01 0.0362 0.864
107 2.63 0.86 2.28 0.0411 0.819
90 2.75 0.79 2.62 0.0472 0.743
69.3 2.92 0.70 3.20 0.0576 0.642
50.4 3.09 0.61 3.89 0.0700 0.536
30.8 3.29 0.61 4.91 0.0884 0.518
10.9 3.52 0.61 6.42 0.116 0.490

 

 

aReference 239, uncertainty 13%.
bFrom 2D re1axation.

c 10
dn

1,sc = Tl,total ' Rl,dd.

170

191 and 19" one must measure relaxation rates of two nuclei. one on
and one off the symmetry axis. However 13CHBr3 is a special case

in that the carbon relaxation rate at high temperatures is dominated
by scalar coupling to the three bromines and consequently 13C relaxa~
tion data is sensitive to the relaxation rate of the bromines. The
value of this is that one can indirectly measure the bromine relaxa-
tion time. which is inaccessible to direct measurement by virtue of

thelarge quadrupole coupling constant.

Evaluation ofgg - The perpendicular correlation time may be
obtained from 2D relaxation data as long as hydrogen bonding is not
appreciable. which would make the 2D quadrupole coupling constant
temperature dependent. Vanderl-lart236 has recently summarized the
data for the case of CDCl3 and concluded that hydrogen bonding is at
most of minor importance. The experimental data which supported this
conclusion for CDCl3 are not all known for CDBr3, but NMR proton
chemical shifts upon dilution in an inert solvent are very similar

273. If the 2D quadrupole 6°"P“"9'

for both chloroform and bromoform
constant is temperature independent then 191 may be calculated, and

the best fit to the experimental points gives

qu = 4.842 x 10"“exp(2.80 kcal mol’llRT). (131)
Separation of the 13C Relaxation Mechanisms in ‘3CHBr3 - 0f

 

the various mechanisms which can contribute to l/T], we can calculate
exactly only the intramolecular dipole-dipole contribution. we cal-

culate this from the 191 values determined from the 2D relaxation

 

171

measurements by use of the relation

, 222-5
1/T1,1ntra ” YcYHrCHtel.

= 1.801 x 1010.6]: (132)

where rCH = 1.100 x 10"8 cm. as discussed previously. The contribu-

tion from the dipole-dipole interaction with the three bromines may

be neglected since the product Ygrngr is about 0.002 that of yfirafi.

The dipolar contribution at 25° C calculated from Equation (2), 0.095

1 1

. agrees well with the value 0.10 sec‘ obtained from the nuclear

239.

sec
Overhauser enhancement In Figure 38 are plotted the relaxation
rates of 13C in 13CHBr3 and 2D in CDBr3, plus the intramolecular
dipole-dipole contribution to the total 13C relaxation rate as cal-
culated from Equations 132 and 131. and the sum of the remaining
contributions to the 13C relaxation rate, Rl,total - R1,dd' 0f the
remaining possible mechanisms for 13C relaxation. we neglect the

intermolecular D-D contribution. as is the usual procedure in 13C

224

relaxation studies He also neglect the spin-rotation mechanism,

275 and

since for chloroform it was at most 10% of the total rate
should be less for a more massive molecule. Relaxation through chemical
shift anisotropy is also negligible at the low field used in the

130 studies (14.1 k6). An anisotropy of 200 ppm results in
(1H,)csa . 2.4 x 107t61 . (133)

which is clearly negligible with respect to 1/T1,intra‘ The final

172

 

117-

020

R,(sec”)

(Ll-‘

 

 

 

 

10’/1('x)

Figure 38. Spin-lattice relaxation rates as a function of temperature
for 20 in CDBr3(o), and c in 3CHBr3 ( c1 ). Also shgwn
are the calculated dipole-dipole contribution to tbs C
relaxation rate (- - -), and [R1,tota1113c)'al,dd( C

11(0).

173

possibility is re1axation through scalar coupling to the bromines,

1 239

which Farrar et 3 concluded is the dominant relaxation mechanism

in CHBr3. The expression for this mechanism is

2
UNI)“ ' 3%" “STU-114;? (134)

where A - J/Zn is the scalar coupling constant in radians per second,

5 is the spin of the other nucleus (Dr in this case). I is the relaxa-
tion time of the 5 spin, and Am is the difference in Larmor frequencies
of the two spins. This expression is further complicated in this

case by the presence of two isotopes of bromine. each of ~50% abun-
dance. The total rate for the scalar-coupled interaction is then the
sum of the individual rates for the various combinations present

multiplied by their fractional abundance.

818181

1,sc 797979) + 3 (R798181+ R791381)’ (‘35)

+ R1,sc 1,sc

(1/T,)§°‘=1‘;(R
where the superscripts indicate the isotopic species present and
C B C
R1?“ ' (”Tl,sc’A+n/Tl.sc) + ("T1,“) ' (‘36)
From Equations (135) and (136) we obtain

unfit“ 3% [l/T1’5c(798r) + l/T"sc(mBr)].(l37)

For values of the correlation time on the order of 5 x 10'12.

2 T~2 31.

M212 << 1 for 798r; however, for 818n.m Therefore the

 

174

total scalar-coupled rate is

2 79

2A2 81Br ( 1) 111813?) (138)
+ -—1§—-1-s s+ .
1+Am?1¥(8‘8r)

The number of unknowns may be reduced by the relations 779/781 =
A(798r)/A(3‘8r) = 0.928, and (fig/0g1 . T](8]Br)/T](798r) - 1.474. but
there are still more unknowns than equations. Therefore we cannot cal-
culate (1/T1,sc)t°t or, conversely, obtain (1/T1,sc)t0t from the total
rate and then calculate T1(798r) and 16. What we can do, though. is
vary the values and activation energy of 16 in order to best fit the
experimental data. This may be readily done graphically. In Figure
39 we plot as functions of inverse temperature (l/T](798r))/§§E-S(S+l).
(l/T](8]Br))/-——gAE———-S(S+1), and (1/Tl’sc)t°t/3§— 5(s+1). (where A =

3(0 928)2
79 °
A( Br)) for the case of isotropic rotational diffusion. We see that

tot
(1/T1.5c)
log scale) with activation energy Eic - Egd. while at intermediate tem~

at very high and very low temperatures is linear (on the

peratures the curve smoothly varies from one linear region to the next
with an inflection point occurring at szT¥(8]Br) - l. which occurs at
317° K. The effect of anisotropic reorientation will be that this

inflection point will occur at higher or lower temperature and the

 

high- and low-temperature activation energies will have a value different

than Egd. He graphically vary Ezc and 16 by rotating this curve
and attempting to fit any section of it to the experimental data.
Obviously the right-most half of the curve is not suitable. no

matter what Egc is chosen, as the curvature is in the wrong sense.

175

 

Ill

lJ

.”.l

 

(10"sec), R,sc/-§—A25(S+]) (sec)

  

 

 

 

s 104 _-
' 1-
51 F
P - I / .
I .’. ‘ \
. \
i_ I I .... \ \\ —
.' \ \
I “-te \ \
’1' 1. ‘\\ ‘\
L. I’ \ ..
/ \
I \
"50L x1 4 1
2.0 3.0 4.0 5.0
10’/T('1<)

C relaxation rate in 13CHBr3 from scalar coupling

Figure 39. The temperature dependence of the contribution to the
to 79Br and 318r. assuming isotropic reorientation.

 

176

Therefore we can conclude that Ezc'3_Egd and consequently Eall'i
E813 which is what has been found for other symmetric tops. The
curve fits the experimental data reasonably well with values at one
extreme of Egc = Egd and JCBr = 91 Hz (isotropic reorientation),
and at the other extreme Ezc = 1.59 kcal/mole and JCBr = 55 H2
(from which ”ll/91.: 1.49 at 20° C). The comparison of theory

with experiment in these two cases is shown in Figure 40. ..

Discussion of the ExperimentallygDerived Diffusion Tensor - The

uncertainties associated with the activation energy for the spinning

 

motion and the ratio DlllDl are so large that a detailed discussion
would be out of place. It is, however. worthwhile to call attention
to two points. The first is the comparison of bromoform with chloro-
form, summarized in Table 29. Huntress72 found that for CDC13 at
20° 0 0l l/0l =- 1.98, while the x-test72 gave that the tumbling
motion is diffusional but the spinning motion is approaching free-
rotor behavior. In the case of bromoform all that can be said is
that Eaill‘S'Eatl: Raman lineshape studies reported previously show
clearly that the tumbling motion is diffusional, so XL.= 4.1 and x'l
lies between 4.1 and 2.1. The second point to be made is concerning
JCBr' Farrar gt 213239 have argued that JCI in CH3I must not be
greater than 60 Hz by virtue of the very minor importance of scalar
relaxation for 13C in 13C1131. Since 13C-Br coupling constants

would be expected to be smaller than 13C-I couplings an anisotropic

reorientation is favored. as it results in a smaller JCBr'

177

 

111,“ (sec")

 

 

 

 

1 1 1
2.5 3.0 3.5
10’ K/T

Figure 40. The temperature dependence of the contribution to the
13C relaxation rate in 13CHBr3 from scalar coupling to
bromine predicted by (a) isotropic reorientation with
JCBr = 91 Hz (———), and (b) anisotropic reorientation with
E = 1.59 kcal/mole, JCBr = 55 Hz and DIIIDL = 1.49

a,sc
at 20° C (- - -).

E-

 

178

Table 29. Physical properties of CDBr3 and CDCl3

 

 

 

Physical Property CDBr3 C0013
11. g-cmz 685-10'40 264°10'40a
. -a
Ilé, g-cm2_ 1202.10 4° 498-10 4°
(e qQ)D,kHz 177 167b
(e2q0)79 . MHz. 550c 79c
Br
d
rc_D. A 1.100 1.100
a
131' 4.1 2.2
* 4.1-2.1 0.84a
Te
II a
Ea, 2.80 1.6
E 2.80-1.6 1.2a
a.ll

 

 

aReference 72.
bReference 244.
cReference 240.

dReference 272.

 

179

II. NMR and Raman Studies of CF3§§13

Very little physical data have been reported for CF3CC13. the
most symmetrical of the fluorochloroethanes. But from the melting
point (13.2° C as compared to -35° C for its unsymmetrical isomer).

high vapor pressure in the solid276

and approximately spherical shape,
this substance may be expected to form a plastic phase in the solid. hfi
If this can be shown to be the case, then the usual temperature-

dependent relaxation time study may be enhanced in two ways: (1) ,.

 

If the rotational motion in the solid and liquid occurs by the same
process. that is, if there isn't a distinct rotational transition

at the melting point. then the effective range for studying rotational
motion is extended down to the phase transition, and the separation
of the total spin-lattice relaxation rate into contributions from
different mechanisms is easier, and (2) The contribution to T1

from translational diffusion in the solid may be separated. from
which the diffusion coefficient may be determined.

From Figure 41 it can be seen that the plastic crystalline phase
extends down to 157° K, at which temperature the relaxation time
exhibits a discontinuous and reversible change. This discontinuity
presumably signals a phase change in which the high temperature form
has the cubic symmetry typical of plastic solids. and permits either
rotation about a molecular axis or isotropic rotation. The low
value measured for the second moment in the high-temperature phase
is consistent only with isotropic rotation. as will be discussed.

The most striking feature of the relaxation curve is that the

melting transition causes very little change in T]. at least at 56 MHz.

180

 

._oomee ee.em eee ewaee. e? ea. ea eaeeexepee ee_eee_-eeam ..e ee=e_e

 

 

 

3.: >2
“an Age new five. new new
_ a . . q
1 l “0.0
1 l
1' IIL
e u
i. .1 en.
I.”
. ...
3.
1| ..1 in
II. l
r. .2

no

 

 

181

This behavior is common to all members of the series CZFnCIG-n
which were studied in the course of this investigation. Since this
is so, the results will be presented in terms of the translational
and rotational contributions to Ti]

studied.

over the whole temperature range

A. Determination of the Second Moment

For the case of isotropic rotation the intramolecular contribu-
tion to the second moment averages to zero. The total second moment

then arises from intermolecular contributions and can be quite simply

calculated a5277

-6
M2 = 358.1 No r

= 3581 Noflla's , (139)

where N0 is the number of spins per molecule and the lattice sum

-5 278
Nla

(fcc) cubic lattices as 23.045 a'6 and 115.631 a'6, respectively.

has been calculated for body-centered (bcc) and face-centered

Unfortunately the crystal structure of CF3CC13 is not known.
By analogy with similar compounds which form a plastic phase279.
the structure is almost certainly cubic. but it is not possible to
distinguish between face-centered or body-centered. The shape of
this molecule might be considered to be slightly distorted from that
of carbon tetrachloride, which would imply fcc symmetry. 0n the
other hand, the majority of the substituted ethanes which form a

plastic phase, and for which the crystal structure is known, have a

 

182

bcc structure.

However, this ambiguity is not serious, as Resing188 has shown
that the reduced second moment is the same within 0.5% for bcc and
fcc structures of the same density. We measure the density of
CF3CC13 at -10° C as 1.71 g/cm3. This corresponds to a lattice
constant of 9.0 K (cm4 has a = 8.34 K) or 7.14 K (MeZClC-CC13 has
a = 7.4 A) for fee and bcc lattices, respectively. The correspond- at
ing second moments are 0.234 62 and 0.235 62. Ne experimentally
obtain a value of 0.25{: :3g} 62 just prior to the rapid drop in M2 4'

 

which signals the onset of rapid translational diffusion.

The uncertainty in our experimental value is large for a number of
reasons. so it is reassuring to find agreement as good as this.

Since neither the density-derived M2 nor the directly-measured M2

are very precise we will choose a value which gives the best agreenent
for the Td values from the T1 and T2 measurements. This adjusted

value, though, must certainly be in the neighborhood of 0.24 62.

B. Translational Diffusion in the Solid

In the solid, the relaxation function is linear on the £n(T;])
vs, l/T(°K) plot only for the lowest temperatures, showing a distinct
curvature to shorter relaxation times as the melting point is ap-
proached. In order to understand the origin of this curvature more
clearly, T2 measurements at 56.4 MHz and T] measurements at 56.4
and 15.87 MHz were performed.» The results of all of these measure-
ments in the solid (T1,TZ at 56.4 MHz, T] at 15.87 MHz) are shown
in Figure 42. The frequency dependence of T1 near the melting point.
and the rapid increase in T2.are both indicative of rapid translational

183

 

 

 

    

 

 

 

1 1 1 1 1 1
10‘h— .4
103- -
102 _ -—
F; .
.— —-1
T410 r-
....
10°~ -
I 1‘,“(16 MI-iz)
-. L.
1() F- 1 -1 '-
L 1 1 1 1 1
4.0 5.0 6.0
1o’/1(°K)

Figure 42. The temperature dependence of spin-spin relaxation and the
temperature and frequency dependence of spin-lattice relaxa-
tion in solid CF3CC13.

Efl

 

184

diffusion. The temperature dependence of T2, and T1 at 15.87 MHz

near the melting point, are similar (AE are-l3.7:0.5 and 12.810.5
kcal/mol, respectively), supporting this conclusion. Further evidence
is given by linewidth measurements, both high-resolution (on a Varian
A56/60) and pulse FT, which show a rapid narrowing of the width from
~0.8 gauss at 208° K to a value not much greater than the width in

the liquid phase at 283° K, two degrees below the melting point. We
do not observe the "pre-melting" phenomena reported in other

studies‘90°]94:202.280

on plastic solids. Even though the observed
linewidths in the solid and liquid phases are very similar near the
melting point this effect, if it occurred, could easily be detected
as the solid-liquid chemical shift is 33.9 Hz and this resolution is
easily achieved during pulse experiments (although the sample is not
spun).

. So there is no doubt that translational diffusion narrows the
resonance line and shortens T1 at 15.87 MHz, but does it account for

all the curvature at 56.4 MHz? He decide this by separating the

translational contribution from the total relaxation rate at the two

frequencies:
g 56 56
Rl,other Rl,total ' Rl,trans (1403)
and
15 _ 15 15
Rl,other ' Rl,total ' Rl,trans ° (140b)

In order to solve these equations from the experimental knowledge

 

185

of R1 at 56 and 15 MHz we need two more relationships. From the

56 _ 15
discussion of rotational diffusion we have that Rl,other - Rl,other'
To get the second relationship we must choose a model for trans-

186,187

lational diffusion. Torrey considered a diffusion mechanism

in which the molecules execute a random walk from a lattice site to
a neighboring one with an average jump frequency 1/Td. In terms of
the spectral density functions 6 for which numerical results are

188

available , the relaxation rates due to diffusion are

'1" .. §gLfnzm+n FE;— y[G(k,y) + 4G(k,2y)] (141a)
(1)

i.

12 = 9511 .4121(1+1);3—;';3—— y[%c(k.o) + 360.1) + 60.29)]. (1411:)
(A)

01“

TIT = $112,. td[G(k.y) + 4611.211] (Ma)
1 _ 2 3 5
‘f‘ — y n.” Td[§G(k.0) + 5600) + Gik.2y)]. (Mb)

2

where n is the number of spins per unit volume, k and t are constants
depending on the crystal structure, y 3.2;93 y is the magnetogyric
ratio, m is the resonance frequency. and M2" is the reduced second
mament, measured between the phase transition and the onset of rapid
translational diffusion. Use of the second moment in this manner

has been justified by Resing‘gz.

In the temperature range between the phase transition and the

11 minimum (which is not observed here, as the sample melts while

 

186
T1 is decreasing) the expression for T2 reduces to

‘TL = §YZ M2r6(k, 0)rd . (143)
He adjust M2" to give agreement for Td calculated from T2 and T15
data, and adjust Rl,other such that Equations (140a) and (l40b)
are satisfied. For the initial value of M2'. we choose the average
of our experimental values, 0.24 02. The justification for this
procedure is that the temperature dependence of T15 and T2 show the
same process to be active in both; also the total adjustment results
in an optimized value of "Zr = 0.22 62. which is certainly not a
major change. Correlation times Td obtained from this procedure,
along with 7d calculated from linewidth measurements (Figure 43)

by Equation (124).are shown in Figure 44. Table 30 gives the values
of rd calculated from the T] and T2 measurements, plus the calculated
contribution of relaxation through translational diffusion to the
total relaxation rate at 56 MHz. This adjustment leaves some cur-

vature in the temperature dependence of 1;]

, shown in Figure 41 as
the dotted portion. but if all the experimental curvature as the
melting point is approached is attributed to translational diffusion
Equation (140) cannot be satisfied, Td values do not agree with the
values from linewidth measurements. and the temperature dependence
of Id is non-linear. He also note that the presence of this extra
curvature, indicative of another relaxation mechanism acting in the
solid, does not depend on the reliability of the second moment value.

Determining Td from the frequency dependence of T] gives a result

which is practically independent of the value of "Zr“

 

187

 

 

 

 

 

1: 0.5 — _.
3
O
O
.2
:15
6
.g

0.0 —— __

l l
4.0 5.0
1o’/1(°K)

Figure 43. The temperature dependence of the width of the 19F
resonance line in solid CF3CC13.

188

 

 

 

 

 

1 l
1(75r' "
1?

3
..‘D
.81 -.
u 1.’ '(16 MHz)
C) T;'
in flnmnwhdih
10"- "1
l
1 1 1
115 41) 4&5
10°/1(°x)

Figure 44. The temperature dependence of the mean jump time for
translational diffusion in solid CF3CCl3.

 

189

Table 30. The contribution of translational diffusion to spin-lattice
relaxation in solid CF3CC13 and the derived values of the
mean jump time. Td.

 

 

 

IIT R1§totala R1§totalb R1ftrans "d.108 R1ftransc
3.495 0. 5587 0.143 0.445 5. 50 0. 0391
3.53 0.5135 0.138 0.403 7.54 0.0334
3 . 57 0. 4358 0. 132 0 . 338 9 .15 0. 0309
3.505 0.3557 0.125 0.257 12.1 0.0210
3.52 0.2783 0.124 0.1741 18.2 0.0139
3.57 0.2245 0.118 0.118 27.0 0.0088
3.718 0.2017 0.114 0.098 33.1 0.0080
3.81 0.1529 0.107 0. 059 55.0 0.0058
3.84 0.1570 0.105 0.055 58.0 0.0055

 

 

aExperimental points.

bSmoothed values.

cCalculated from Td'

1“

 

190

He will demonstrate in the next section how this curvature is
consistent with the presence of spin-rotational relaxation in the
solid. He will show that the rotational relaxation rate, Rl,total -
Rl,trans’ varies smoothly across the melting transition, supporting
the assumption that rotational motion in this molecule is unaffected

by freezing.

C. Translational Diffusion in the Liquid

When the diffusion equation is solved without requiring the
molecules to be at fixed lattice points,the spectral density functions
thus obtained are appropriate for translational diffusion in a liquid,

and the contribution to T1 i528

_ nN 2 4

Rl,trans ' a ’ (144)

where the effect of the chlorine moment may be neglected since
Yleél leO. In this equation N is the number of spins per unit
volume, D is the translational diffusion coefficient, and a is the
molecular radius. The fact that the spins are not located at the
centers of the molecules will cause a small change to this equation181
which we ignore. In order to use Equation (144) we must determine
the diffusion coefficient. which can be measured directlyzz} or
estimated from the macroscopic viscosity. We consider the latter
method first.

The Stokes-Einstein relationship between the translational

diffusion coefficient and the shear viscosity

 

191

 

D , k1

is obtained from hydrodynamic theory. If we estimate the molecular
radius a by considering a liquid to consist of close-packed spheres

occupying 74% of the total volume this becomes

- -9 1 1/3
0 - 2.20 x 10 5‘11!) . (146)

Table 31 gives the reported values of 0282, "283’ N, and D calculated

from Equation (146) for a temperature of 25° C. Table 31 in addition
lists the values of 0 measured by the two-pulse spin-echo method22].
The static field gradient in the spin-echo method was produced by
changing the current to the x-shim coils and the system was cali-

brated by measuring the echo attenuation in a fluorocarbon sample

131

for which D, T], and T2 were reported accurately The gradient

was determined to be 0.136 g/cm and the diffusion constants are
believed to be accurate to 210%. The results of this calibration

are given in Table 22. Using the directly measured values of the diffu-

sion coefficient and the known values of the density we calculate values of

R1 trans listed in Tables 31 and 33. Figure 41 shows this correction

made to the total relaxation rate. For comparison the contribution

R1 trans calculated from the macroscopic viscosity is 0.00695 at

l/T = 0.00335. Clearly the agreement is very good. It is also
gratifying to observe that after making the corrections for transla-

tional diffusion in the solid and liquid by completely independent

56
1,0ther

shows no discontinuities, or changes in slope, through the melting

experimental procedures, the relaxation rate R (Equation 140a)

 

192

Table 31. The contribution of translational diffusion to spin-lattice
relaxation in liquid CF3CCl3 calculated from the self-
diffusion coefficient or the macroscopic viscosity.

 

 

 

D(cm2/sec) pb(g/cm3) N-lD22 T(°C) 103/T(°K) R1 trans‘sec-1)
0.902a.094 1.61 1.55 11.6 3.51 0.0147 1.0015
1.72 ..41 1.55 1.50 28.0 3.32 0.00747t.0018
r
2.97 3.22 1.52 1.47 50.4 3.09 0.00424:.003l
1.85a 1.50 25.0 3.35 0.00695

 

 

 

aObtained by use of Equation (146); n is from Reference 283.
bReference 282.

193

Table 32. Physical properties of CF3CC13

 

 

p25

n25

2r

655-10'409 cm2
448-10'40 9 cm2
+298 ppm

476 ppm

1.79 kHz

1.559 g cm'3
0.720 centipoise

0.22 62

 

 

 

Table 33. Smoothed re1axation rates in CF3CCl3

 

 

 

“17'!
|

 

 

 

IIT Rl,total Rl,transa Rl,ddb R1,sr 13d ‘38

2.31 0.379 ---- 0.0142 0.355 1.15 0.558
2.50 0.274 ---- 0.0158 0.257 1.31 0.409
2.75 0.213 ---- 0.0207 0.192 1.53 0.321
3.00 0.174 0.004 0.0254 0.145 1.80 0.253
3.25 0.148 0.011 0.0312 0.105 2.13 0.192
3.50 0.129 0.015 0.0387 0.075 2.54 0.141
3.75 0.112 0.010 0.0475 0.054 3.02 0.105
4.00 0.103 0.004 0.0585 0.040 3.50 0.081
4.25 0.110 ---- 0.0722 0.038 4.30 0.079
4.50 0.122 ---- 0.0889 0.0331 5.15 0.071
4.75 0.138 ---- 0.110 0.028 5.20 0.051
5.00 0.152 ---- 0.135 0.027 7.42 0.051
5.25 0.190 ---- 0.155 0.024 8.90 0.055
5.50 0.224 ---- 0.203 0.021

5.75 0.259 ---- 0.251 0.018

5.00 0.323 ---- 0.310 0.013

5.25 0.391 ---- 0.382 0.009

5.50 0.478 ---- 0.459 ----

5.75 0.585 ---- 0.583 ----

15 MHz

aFrom self-diffusion measurements in the liquid and R]
ments in the solid.
b

Extrapolated from low temperature linear region and room temperature
Raman T .
91

. R2 measure-

c - —
R1,sr ' Rl,total ' Rl,trans Rl.dd°

d , -11 1/2
T; ‘ 70519.10 Rl’dd(kT/I) o

e _ . -10 , 1/2
13 - 6.162 10 Rl,sr/T (kT/I) .

195

56
point. This behavior of Rl,other

as molecular rotation is unaffected through the melting point.

is entirely to be expected as long

D. Rotational Motion

The relaxation data shown in Figure 41 make it obvious that
completely different mechanisms dominate the relaxation rate at the
two temperature extremes. We identify these mechanisms as follows.
An R1 which increases with increasing temperature is indicative of

relaxation through the spin-rotational (R ) or scalar coupling

1,sr
(Rl,sc) interactions. From the difference RZ'R1° and the rotational
correlation time determined by Raman lineshape analysis, it will be
shown that JFCl = 1.7 Hz and that Rl,sc is completely negligible.
Therefore, the mechanism dominant at high temperatures is spin-
rotation. At the other temperature extreme R1 increases with de-
creasing temperature. At low temperatures in the solid translational
diffusion is not an important contribution, while at other tempera-
tures it is of minor importance at 56 MHz as discussed previously;
the dashed line in Figure 41 gives R1 corrected for this effect.
Other mechanisms which give the right temperature dependence at low
temperatures are the intramolecular dipole-dipole interaction (R1,dd)
and relaxation through anisotropic chemical shift coupled with

molecular rotation (R1 csa). He now proceed to calculate the inter-

action constants for these mechanisms.

Relaxation through Intramolecular Dipole-Dipole Interaction - For
isotropic rotational diffusion the intramolecular contribution to the

fluorine relaxation is given by28

 

196

_ 2 2 3 2 -6 - 2 -5
F' c

The second term is completely negligible and, using the intermolecular

distances of Hard and Uard284.

R = 1.33 x 1010 . (148)

l,dd Te
It will be shown in a later section that the reorientation is iso-
tropic within experimental error and is diffusional over all but the
highest temperatures studied. therefore Equation (148) is appropriate

to describe Rl,dd'

Relaxation through Chemical Shift Anisotr0py - Again employing

 

the assumption of isotropic rotational diffusion, the contribution

to R1 of the anisotropic shielding-molecular rotation interaction
28
is

2 2 2
R1 ,csa ‘ 15 M w Ta (”9’

for axially symmetric molecules. The chemical shift anisotropy
00 = all - oi refers to the molecular symmetry axis and has been
measured285 as 276 t 3 ppm for CF3CC13. Accordingly, the relaxation

rate is calculated as

56 _ 8
RI’Csa - 2.92 x 10 re . (150a)
15 = 7
Rl’csa 2.31 x 10 .6 . (150b)

 

197

Comparing these rates with the dipole-dipole rate it can be seen
that R1 csa is completely negligible at 15 MHz and is just on the
edge of detectability (2% of R1 dd) at 56 MHz.

Relaxation through Spin-Rotation Interaction Hubbard49 ob-
tained an expression for the contribution to R1 from the spin-rota-
tion interaction for the case of nuclei occupying identical positions
in a spherical top molecule undergoing isotr0pic rotational diffusion
(Equation (105)). Wang286 calculated the relaxation rate for the more
general case of anisotropic reorientation with a non-diagonal spin-

rotation tensor and obtained

R1,Sr = 25%{111(c1| ‘1' BS1026)2 % '1‘ 1182(511126C0526)
+1.11]. +11! 1111111
‘1' Ii[(Ci-BS11120)2 ‘1' 012.1%} , (151)

where the 1's refer to TJ and the D's to 1/616. and B = 91.- CII
with C11 and CL the principal components of the spin-rotation tensor
along and perpendicular to the bond axis.

For the case of CF3CC13 (DltlfDltll,Dllrll<<l) Equation (151)

reduces to

2
%{[III(CII+ sin 2e)2 + 11(82 sin ecosze)]1||

1,sr= 3:2

2 2

+ Ii[(Cl-Bsin28)2 + ci+ s sin 8C05201'll} . (152)

 

 

198

Since Cl and CII are not known. and calculating them from
Ad requires neglecting off-diagonal elements, Equation (105) will
be used initially and the attempt to calculate the components of the
spin-rotation tensor will be discussed later. Thus,

9

R],sr = 1.62 x 10 Ttsr , (153)

or, since this equation will actually be used to calculate Tsr,

after determining

R1,sr ‘ R1.tota1 ' R1,66 ' R1,trans ‘ R1,csa . (154)

we have

R1

——f§5. (155)

Tsr = 6.16 x 10"10
In Figure 16 are plotted the reduced correlation times “[3
and 13 (1* = T(kT/I)]/2) obtained from Equations (148) and (155)
for comparison with the predictions of Gordon's extended diffusion
model as applied to spherical top moleculesgS. It can be seen that
the data obtained through an isotropic diffusion model and an "ef-
fective“ spin-rotation interaction constant (ZCifCII) conform to the
general features of the model and lie between the M- and J-diffusion
limits. The fact that the data lie closer to the M-diffusion limit,
which is physically unrealistic for liquids in that it does not allow
the magnitude of the angular momentum vector to change with collisions,

indicates that the simplifying assumptions which went into the

 

10.

1.0

Figure

199

 

 

 

 

_ 1 1 l T l
a
F . -—I
- e _.
.. e _
e
- -1
e
- o -n
e
_. 0 __
o
\
\ \
\ \ C
" \\ \\ .—
\ \
. \ - \
1 1 \ 1 1 1 \
0.1
1;.

45. A comparison of the experimental relations between T;
and TJ with the predictions of the extended diffusion

model.

 

200

derivation of Equations (148) and (155) are not permissible for CF3CC13.
The reason for this is that in CF3CCl3, contrary to other molecules
which have been studied in terms of the extended diffusion model
(0014125, 01036130, 052‘12. and cc13r‘3‘) and have been found to

follow J-diffusion, the spin-rotation tensor is not diagonal in the
molecular symnetry axis and. consequently. off-diagonal elements must

be included in the calculation of R1 However before making this

.sr'
calculation we will show that, while in the case of CF3CC13 it may
be assumed that the reorientational motion is isotropic from the

spherical shape of the molecule, this fact can also be demonstrated

experimentally.

E. Demonstration of Isotropic Motion from Raman Data

If it is assumed that the temperature dependence of ‘0 is given
by an Arrhenius-type equation throughout the temperature range
studied, then To in the liquid may be extrapolated from the values
obtained in the solid. This correlation time is an “effective"
correlation time 15 which, for a symmetric-top molecule, depends on
T Land Tell as previously discussed. In order to separate the parallel
and perpendicular motion the reorientational broadening of a Raman
line of A] symmetry. which is sensitive only to 19L. is measured.
Although CF3CC13 has five A1 lines only one. at 714 cm", is of suf-
ficient intensity and also devoid of fine structure. The parameters
appropriate to the Raman experiment are given in Table 24, and Figure

40 shows the polarized and depolarized bandshape with the calculated

convolution of the polarized band with a Lorentzian orientational

 

201

component of half-width ”or = 2.08 cm']. Employing the relation

Tel

From Table 33 we have that 15 = 2.20 psec at 27° C., and conclude

= (2nowor)'], a value Tel = 2.54:0.81 psec is obtained at 27° C.

from this that T21 = Tell = Té, at least at 27° C. For comparison,
the relaxation time calculated using the Debye relation Tc = 4nna3/
3kT is 25.5 psec at 27° C., and the Hill modelél gives Tc = 5.10

psec.

III. Relaxation of 19F in CF3Br and crzgrz

Spin-lattice relaxation times were measured for CF3Br and CF28r2
from the melting point to the critical point (these were 99° and 340° K
for CF3Br (Figure 46) and 131° and 471° K for CFZBrZ (Figure 47).
respectively). Both of these curves show the same general features.
a minimum in the rate at intermediate temperatures, indicative of
dipole-dipole dominated relaxation at low temperatures and spin-
rotation dominated re1axation as the critical point is approached.
The most obvious difference in relaxation for these two molecules is
the much more pronounced curvature in R] of CF3Br at high temperatures.
This may be presumed to arise from the increasingly important free
rotation about the symmetry axis (spinning) which cannot occur in the
asymmetric top CFzBrz. In the case of fluorochlorocarbons additional
inf0rmation could be obtained from measurement of R2, the spin-spin
rel xation rate, due to scalar coupling with the chlorine; however,
the much larger bromine quadrupole coupling constants in CF38r and
CFZBrZ (1500 MH2240 240).

versu54570 MHz along with a reasonable

guess as to the scalar coupling constant (JFBr = 20-40 Hz_!§.

 

R, (sec")

‘ 202

 

 

 

 

 

..ii 1 l T r 1 l 1_
' ce
l
J
5 MP. I
I; 1
I ..
1.0 - . ' -
. l '. o i ‘
C 'o‘
__ . ..-
o I q
— '00 O -i
— 00 o 0 o o _-
0!|-- ._a
+— I
1 1 1 1 1 1 1 1"
31) 51) a 71) 91)
lO/T(’K)
Figure 46. The temperature dependence of spin-lattice relaxation in

CF Br, and the calculated contribution from translational
di1fusion.

 

 

Figure 47.

203

The temperature dependence of spin-lattice and spin-soul
relaxation in CFZBrz, showing a separation into different
contributions based on the calculated contribution from
translational diffusion and equal but opposite tempera-
ture dependence of Rl,dd and Rl.sr' The dipole-dipole
relaxation rate predicted from dielectric relaxation data
is given by (In.

204

 

 

 

1

1:..d

_

_

_::~—_

_

_

:1:__—

 

 

:____#_

nu
A7003 .z

:_.:__ _

0.

 

5.0 7.0
10’/1('k)

31)

Figure 47

205

JFCl = 10-15 H2131’287). result in a scalar coupling contribution which
is very minor (Rl,sc = 0.025 sec") in the fluorobromocarbons. This
has been verified by measuring T2 in CFZBrz (Figure 47), from which

the value R = 0.028 is obtained at 286° K. The much greater cur-

1,sc
vature evident in the temperature dependence of T2 as opposed to T]
will be discussed later.

Using the same arguments as applied to CF3CC13 (but noting here
that the total rate 1S'M3 times larger), we conclude that the possibly
important relaxation mechanisms are spin-rotation and the inter-
molecular and intramolecular dipole-dipole interactions. The main
problem in these instances is separating the contributions from the
two dipole-dipole interactions.

To estimate the diffusion coefficient and the intermolecular

contribution to the total rate in these two molecules we use Equation

(144)

2 4

, nNa 1
Rl,trans a

and (146)
., -9 1 1/3
0 - 2.20 x 10 34%) .

The densities and viscosities of these two fluids have been measured
from -80° C to +40° (CF3Br)288 and -70° - +20° (CF28r2)289’290.
These experimental data, plus the results of calculations using Equa-

tions (144) and (146). are shown in Tables 34 and 35 for CF3Br and

 

206

Table 34. Calculation of intermolecular contribution to T1 in CF38r

 

 

 

103/1(°x) T°C o 9 en'3 n cp 0x10+5 '71, ,inter
a sec

518 -80 2.05 0.58 1.75 0.00306
469 -60 1.98 0.400 2.78 0.00864
429 -40 1.90 0.295 4.06 0.00568
395 -20 1.82 0 235 5.46 0.00404
366 0 1.71 0.200 6.77 0.00306
341 20 1.58 0.165 8.58
319 40 1.41 0.150* 9.71

 

 

 

h.
In“.

207

Table 35. Calculation of intermolecular contribution to T] in CFZBrZ

 

 

 

103/T(.°K) T°C gp:m'3 3: $222351 T1].in_. rd
sec

4.92 -70 2.584 1.594 0.635 0.02113
4.69 -60 2.551 1.346 0.7858 0.01679
4.48 -50 2.517 1.152 0.9570 0.01354
4.29 -40 2.484 1.003 1.143 0.01113
4.11 -30 2.450 0.08796 1.354 0.009233
3.95 -20 2.415 0.7788 1.584 0.007743
3.80 -10 2.380 0.6969 1.831 0.006567
3.66 0 2.344

3.53 10 2.307 0.5664 2.399 0.004808
3.41 20 2.268 0.5130 2.727 0.004135

 

 

 

a Reference 290.

p . 2.3440 -3.67-10-3t-5.75-10-5t2

-3.3-10'3t3

b Reference 289.

c. _ . -9T 1/3 =
0- 2.2010 3%) 1111 221.82

(IT? = 11550 - (Fa—H919) n spins/molecule

208

CFZBrz, respectively, and have been subtracted from the total re1axa-
tion rate in Figure 47.

(In the case of CFZBrz the intermolecular dipole-dipole mechanism
can be seen in Figure 47 to be a minor contribution to the total rate.
The total rate may be then separated into the spin-rotational and
intramolecular dipole-dipole contribution. with equal and opposite
temperature dependence except for a slight curvature in the spin- ..
rotational contribution at high temperatures. This separation is

also shown in Figure 46 and the separate contributions to the total

 

rate are given in Table 35. Miller and Smyth63 have measured the
dielectric relaxation times at 0° and 20° C as 2.60 and 2.28 psec,
respectively; if we assume that the reorientation is isotropic and
diffusional, and remembering that TNMR = Tdiel/3’ then Rl,dd

= .0058 at 0° C and Rl,dd = .0051 at 20°. The intramolecular dipole-
dipole contribution calculated from dielectric relaxation is shown

in Figure 46. The two points are not really sufficient to determine
the temperature dependence but the dielectric-derived Te values are
clearly too small by a factor ofins to account for the NMR relaxa-
tion. At 20° C the motion is probably not fully diffusional. so the
ratio of three between Tgl and 76,2 should be replaced by a factor
of1~2.4 which is appropriate for a spherical molecule undergoing iso-
tropic orientation of finite-step size. This is obviously not the
whole answer. In order to discuss anisotropy effects. we first
consider Figure 48 and the different vectors studied by NMR and
dielectric relaxation. If the coordinate system is chosen as in
Figure 48. the NMR experiments are insensitive to motion about the

y axis while dielectric measurements are insensitive to motion about

2091

 

 

 

Figure 48. Choice of axes for the CFzBrz molecule.

 

 

 

210

the z axis. However, Izz 2 Iyy >>Ixx and. since most discussions of
anisotropic reorientation relate 1011 to 1;}, it seems that even if
the motion in CFZBr2 is found to be highly anisotropic, the correla-
tion times for motion about the y and z axis should be very similar.
Since the CFzBr2 samples were not degassed with the care accorded
the CF3Br samples, there exists the unfortunate possibility that the
CFzBr2 samples still contained a small amount of dissolved oxygen.
In the case of CF3Br, the values of Rl,trans were calculated by
use of Equations (144) and (146) which employ the reported density

and viscosity data204.

The results do not agree with the measured
relaxation times, as can be seen from Figure 46. The calculated
values of Rl,trans become equal to the total rate at the melting
point. which is of course not possible. In fact, by analogy with
CFZBrZ, R1,dd should be an appreciable fraction of the total rate at
low temperatures. In an attempt to resolve this problem, the self-
diffusion coefficient was experimentally determined in the low-
temperature region (Table 23). The experimental results are plotted
along with the ("theoretical") viscosity-derived diffusion coef-
ficient in Figure 49. The scatter in the points is embarrassingly
large; however, the experimental data clearly do not fit the "theo-
retical" curve. Since the field gradient used to measure D was not
directly determined at the lowest temperatures, there exists the
possibility that the gradient is temperature dependent. Since the
echo attenuation depends on 062, an increase in G by a factor of three
will account for the discrepancy. Clearly, further work is needed

to understand the low-temperature re1axation behavior in CF3Br, but

it seems likely from the other fluorocarbons studied that the true

 

 

 

 

 

 

101)
)II- -II
1— 'fi
- -i
)I- .—
11)- .3
a _. I
8
1E '1
‘9’ '- -i
on 1.- q
52 e. -e
0.1 1— fl
”' l l l l 1
31) 51) a 71) 91)
10/1‘(°K)
Figure 49. The temperature dependence of the self-diffusion co-

efficient (D ) measured by the two-pulse, steady gradient,

spin-echo me§hod, compared with the temperature dependence

of DE as derived from the macroscopic viscosity and density.
F

for 3Br.

 

212

size of the intermolecular contribution to R1 lies between the two

possibilities discussed here.
IV. Other Systems

A number of other fluorocarbons were investigated with varying
degrees of thoroughness. In general, the reason for not analyzing L.
these systems to the extent which CF3CC13 has been examined is the

much lower symmetry of these other molecules, making both the Raman ..

 

and NMR analysis much more complex. Figure 50 shows T1 in the liquid
and solid and T2 in the liquid as functions of temperature for CF2C1C013;
Figure 51 shows the same parameters for CFC12CFC12, and Figure 52
shows T1 in liquid and solid CFClZCFClz. Only in the case of per-
fluoroethane were the measurements done at temperatures up to the
critical point. Figure 53 shows the temperature dependence of T1

in liquid CF31, and Figure 54 shows T2 in the solid for CFZClCCl3.

It is immediately obvious that the behavior of these systems is very
similar to that of CF3CC13 in all respects. T1 and T2 are dominated
by spin-rotation and there is little change in R1 across the melting
point, indicating that the contribution of R1,trans is minor both in
the liquid and solid phase. The curvature of T] with temperature in
the solid probably indicates a spin-rotational contribution, although
the effect of Tl,trans must be determined, as was done in CF3CC13,

to verify this. Furthermore. all three systems form plastic phases
and show a rapid drop of R2 as the melting point is approached in

the solid from translational diffusion. From the difference R2 - R1
in the liquid, information about JFCl and the rotational correlation

213

 

-q

'3
__ _. ..-c>-..._._...-cf::115/£f£}/S>//£T’ .-
l I I 1 I I_1

11)

 

I

R,, R2 (sec")

.3
“o
l

051

l 1 1 l l
.0—
.J 1 1 l l

 

 

1 1
41) 51)

101/T(°K)

 

21)

9°
c>

Figure 50. The temperature dependence of the spin-lattice relaxation
rate of 19F in liquid and solid 0F201cc13 and the spin-

spin relaxation rate in the liquid.

214

 

‘LO

111111

1

1

1

R, (sec")
1 1 1 111

1

1

11C"

11111

 

    

111111

1

l

J

 

l l I l

h_11 111

 

Figure 51.

411 61)
10"/1(°k)

The temperature dependence of the spin-lattice relaxation
rate of 19F in solid and liquid croich012.

Ind

"“fl‘_. 'I.‘ "

 

215

 

 

 

 

1.0 - -

h— _—

— —-1

- -1

y— -—-1

’3 " "‘

3:

7.? _ ._.
71:

0.1 - ._1

+- a

pn— _-

.. —1

- "1

-— .—+

1 1 1
4.0 5.0 6.0
IDs/T(°K)

F1 gure 52. The temperature dependence of the spin-lattice and spin-

spin relaxation rate of 19F in CF3CF3.

 

 

216

 

 

 

 

 

. 1 l T
1.0 *— .1
I -1
fl
— —J
p— _-
1— —1
r"
U
3
V F... _
a.
0.1 -— -
-- -1
._. ..q
— -1
l J l 1
3.0 4.0 5.0 6.0
103nm)
Figure 53. The temperature dependence of the spin-lattice re1axation
rate of 9F in liquid CF31.

 

217

 

 

R2 (sec-')

 

 

 

l l l
.41) 51) 65)

IOVTrK)

 

Figure 54. The temperature dependence of the spin-Spin relaxation rate
of 19F in solid crzc1cc13.

 

218

time should be obtained; this will be discussed later. However, note
that experimentally T1 = T2 in the liquid phase of CF3CF3, indicating
that the T2 experiments, which are much more susceptible to error,

are measuring the true T2 in these systems.
V. Spin-Spin Relaxation

A. Anomalous T, Behavior

For all but one case in which the temperature dependence of T2

 

in the liquid was measured the results were anomalous in that the
linear dependence on T6, expected from Equation (98) for R2 - R].

was not observed. The spin-spin relaxation rate invariably increased
faster than theory predicts as the critical point is approached.

This curvature can be seen in Figure 55 (which plots £n(R2 - R1)
versus Tc/T, where Tc is the critical temperature) to vary from a

very slight effect for CFZClCCl3 to a very large effect for most other
systems. Although the uncertainty for many of the values is large
due to the near equality of T1 and T2, the curvature is greater than
this uncertainty would allow. In order to learn more about possible
experimental errors in determining T2, measurements were made on CF3CF3,
for which T1 = T2 due to the absence of the scalar coupling inter-
action. Results of these measurements are shown in Figure 52, and it
can be seen that, indeed, T1 = T2 in CF3CF3. Although longer Tz's
are measured with perhaps a slight bias toward too short a value,

all Tz's obtained are essentialliy in agreement with observed TI's.

This small systematic error for long Tz's is very probably the cause

219

 

 

R, - R. (sec")

 

 

 

 

1
LG - _.
E -+
L- .2
h" A
0.1 - ._
. L.- _-
-- A
h ‘i
h- --4
— A

1 1

Lo 1.5 2.0
1.:(‘10/1

Figure 55. The temperature dependence of the difference R -R for
various fluorohalocarbons. (o) CF3CCl3, (D) EFzgrz,
and (o) CFZClCCl3. -

220

of the extra curvature in the AR versus Tc/T plot (Figure 55) for
CF3CCl3. No explanation is known for this anomalous behavior of T2,

but it is clear that AR cannot be used to calculate 19, and it is

also clear that the value of JFx derived will depend on the temperature

at which T2 is measured.

VI. 19F Spin-Rotation Tensors

In order to obtain quantitative information concerning molecular

motion from nuclear relaxation times of spin l/2 nuclei, the spin-

rotation tensor must usually be known. This is particularly important

for 19F relaxation data, since spin-rotational relaxation often
dominates the relaxation rate, as was found for the compounds studied
here.

There are three methods in use for determining spin-rotation
tensors (actually, usually only a spin-rotation constant). For a
number of simple molecules, C has been measured from molecular beam

(MB) experimentszg]

and, more recently, molecular beam results have
been reported for larger molecule5255’256’292'296. The number which
comes out of this type of experiment is an average value CMB =
-%-(2§1 + CII) although the full tensor can, in theory. be determined
by analyzing results for different rotational states. Molecular beam
measurements provide probably the most reliable technique for measur-
ing spin-rotation tensors, but are clearly not applicable to large.
complex molecules.

The second method utilizes the relationship between the spin-

rotation tensor and the paramagnetic component of the chemical

 

 

 

221

shift297’298. If we decompose the shielding tensor into diamagnetic

and paramagnetic terms

0 = o + op (157)

then, as Ramsey show5297.

3
(loge2 . Lopezgz i- C or 158
° m'?r."" ._.? 2r... mgw ’

 

where the Ijj are the principal moments of inertia and the other terms
have the usual meaning with k and t referring to nuclei and i to the
electrons. The procedure is to establish an absolute shielding scale
by comparing known spin-rotation constants with chemical shifts

(such as in HF) to obtain a value for the diamagnetic terms. A
number of workerszgg’300 have considered this problem; for 19F shifts

the sum of the first two terms in Equation (l58) is assigned the value

47l ppm and

_ -6 -6
aav - 471 x lo + 0.2091 x10 2:: cjjxjj. (159)

If we replace 113 with the average value of the inertia tensor Iav

then

6

cav= 471 x 10'5 + 0.2091 x 10' (l60)

COIBV'

and CO is the average spin-rotation constant from chemical shift

data.

222

300 have shown that, although the

Furthermore, Chan and Dubin
first two terms in Equation (l58) are not in general isotropic. to a
good approximation their sum is. thus we can write for the individual

components of o

o- = 471 x 10'6 + 0.2091 x 10'5c..1

.1 3.1 31' (m)

and, consequently, if the trace and anisotropy of a symmetric shield-
ing tensor are known then the individual components of C can be
calculated. If just the trace is known then the spin-rotation constant
calculated is of the same form as from the molecular beam experiment:
%-%ufl+cur

The final method of obtaining C is by estimating it from NMR
relaxation times. Other workers76’299’301 have employed Equation
(lOS) and estimated TJ by various means which include rotational
diffusion; however, it should also prove possible to use the value of
R1 at the critical point and make the assumption 13 - l.0. Since
the largest deficiency of the NMR method for calculating C is the
uncertainty in the correct value of TJ, this new approach should prove
at least as good as previous methods. This argument is supported by
two points: (l) The extended diffusion model. which predicts that
13 - l.0 at the critical point, has been found to give an excellent
description of the reorientational process in most cases which have
been carefully studied (see page 60) and (2) at the critical point
contributions to the rate by other mechanisms may always be neglected
for fluorocarbons from which paramagnetic impurities have been removed.

The spin-rotation constant measured by the NMR method is given

1..

 

223

2 g 2 2
by ceff 2C .1. + C 11’ which is related to CO or Cl;B by
2 _ 2 2 2

2
where Cav - CO = CMB' Accordingly, Ceff 2 Civ, and if the anisotropy

in C is large CO (or CMB) values will not be useful in interpreting
NMR relaxation data. L
Table 36 lists the majority of molecules for which the 19F

spin-rotation constant is known. It can be seen that agreement is ,

 

good between Co and CMB for those molecules for which both are known.
298

Contrary to previous discussions ,agreement of CO or CMB with ceff
is also generally good. The two exceptions to this are the molecules
OPF3 and AsF3; it may be pointed out that Ceff calculated here is
for the C tensor in the inertial coordinate system and, as discussed
previously, this includes the off-diagonal terms which are neglected
in cav'

The two molecules CH3F and CHF3 have been intensively studiedzoa’
209’213’219'225; in the case of CH3F the 19F shift anisotropy of -66

ppm220 seems208 to be the correct value, giving CII = -38.5 kHz and
C

i
and 4 kHz, respectively, than do the Cll and Cl values from the liquid
221

= +3.67 kHz, which agree better with the MB values208 of -Sl kHz

crystal determination of A0 8 -157 ppm. In the case of CHF3
the chemical shift-derived value of CO may be used to reject two of

the molecular beam measurements as inaccurate. In this molecule ceff
may be expected to be larger than Cav as the 19F spin-rotation tensor
is not diagonal in the inertial axis system. From a consideration of

these two molecules it is clear that there will be a great deal of

224

 

 

 

 

 

---- m.m~m m.m~ mo.¢ s~-amoa
mp” npm ---- m.aw~ om." om.m -.~ .ewaa
5 2m 9:3- as - «M3
aom.¢- m_o.m - mm. m~_ m.mm- m.em~ mw.ame.m m.e - .a awe
~._m o ---- NNN- em_ mm_- x a
om.m
e.m¢_ ~.m¢_ ---- omN c._~ _m.o m~.o - memo
we.c- mo.~ .
mm._ m~._- mpm mFm ---- o.mm_ m.o_ _o.¢ mo.m - emam
N.m- e.em-
a- N; a: a: ---- 8- w NN ---- e. 2- games
«A..- m¢.m -
-.m- o~.o - eem mac ---- o.mm. m~.m- ---- mm.m - mam.uu
o.o_
mm - m.»
mp.m- «Pane H omp m¢_ mop saw ---- ¢.% m.m - emazu
mac n mm mm so - . _m-u_-u m
am¢.m «a.ua- on em.m amp- was ---- o.¢u 0 Nu. + we :0
on, u m.mm ---- ---- m.m.- umaz
as __u As __H azoq noa=_omaao eeau mxu ecu a_=uo_°=
.35 Emma Lupaoopos 59$ vac—._.tmuou mu wwwwumgvmummmmmhwsmmeésmw meowuhwwohnwuspmgwum .mm 03¢...

mp

225

 

.Auempv omen «mm .maga .sagu .a .aowea .u ecu xeaca .a .xs
.mu oucmsmmoma
.omp mocmgmyomx

.xpm>wwwwnmmg .mmp mucmgmwwm vcm Ammmpv «mm qmm .mxgm .5059 .w .vsmnnaz .m .m can smewmcm: .: .w
scum one 0 ea mm:—~> on» use .uom aucocmwom see; m? u $0 :o*au=_sgoaou econ ga—auopoe one . mu coma

..m oucaguuomw

.om_ mucmgmvomg

.pmp mucmgmewmm

. .zpu>.uuoammg .~om wen mom moucogamom scum use mopnongmvca
mega—apgm on» can .xpm>ppuoammg .omu use .oom .mmm «mocagmwwa sage use aunt Emma m—zumpos on» .mmzu .8“.9F

.A~e_v copumaam seem umpupzupmu mew: sence» cowumuogucwam as“ ea mucocoaeoum

.apo>wuuoamog .Amaueguepuv
mom muzmcmwmm new Apmumxcu uvzaP—v won mucmgmwmm soc» use xaoguom—cu mcpupmvgm as» we mucmemeammms
o3» on» use .Nmm mucmgmmmm cw vocpsgmumu «so: cameo» cowuapog-:wam «nu ma mucmcoaeou mg» .mmxu some

.1. .uwgumsEXmo an on Auom mucmgmemmv vane; we: gonna» acmvpmpzm
use .Aommpv npm .mn .>m¢ .nga .xvgou .3 new couvgmgm .a sage one uz Loy mcopmsmcaa pacauuagumu

.okm_ xto> 3»: .xm..= .ca~ax=a> .a .a use emacza .z .u ..mpecgm

Pau_smgu «:2 am_ napeoaam cc cc.aupcgsou. sate «an .apacmtaecwu cmaauwucv mmmyca .mmapm> mcvupmvgmn
.AooFV =o_uuzom seem noun—aupmu mma—m> no .saa we nave: cw use mmapu> mcvupmwgm .msu m

oeuop we were: cw men erases. mo mucosos new ~zx mo wave: cw cm>wm wen mucupmcou copumaogucwgm P~<m

226

uncertainty in any discussion of spin-rotation tensors in that reliable
dataare as yet difficult to obtain. If we consider CH and Cl in
molecules for which seemingly complete and reliable information is
available (c1o3F‘30, cc13r‘3‘, and CH3F208), it is at least possible

to consider that, while small moments of inertia I1. produce corres-

pondingly large values of Ci, the product C111 is not isotropic as

has been suggested for 13C spin-rotation tensors226 F-

In Table 37 we calculate Ceff from the spin-lattice relaxation

 

rate at the critical point using Equation(105) with the assumption '
13 = 1.0. Since the relaxation rate at the critical temperature was
extrapolated from the rate at lower temperatures for some of these
compounds, the result is only approximate, however, for both these
compounds and those for which the spin-lattice relaxation rate was
measured at or very near the critical point (CF3CCT3, CF3CF3, CF3Br
and CFZBrZ), the agreement of ceff with CO is generally good. The
.molecules for which the discrepancy between ceff and CO is the largest,
CF3Br and CF31, are also the only ones listed in Table 37 for which
the effects of anisotropic motion should be important (compare Figures
46;} and 47, the temperature dependent relaxation rates in CF38r and
CFl’zBrz). Although no further calculations were performed for these
tum) compounds, due to the discrepancy in the size of Rl,trans in liquid
6738 r and to a lack of data for CF31, if the relation TJH/TJ = 1.1!!”
\S \f':a]id13], then 13 in Table 37, which was calculated using an average
mome‘t of inertia, will be larger than 1316110 smaller than 13”.
In tI-—1e case of CF3CCl3, the very good agreement of ceff with CO leads

one t 4, question the tentative conclusion reached earlier concerning

227

 

 

 

'Table 37. Effective 19F spin-rotation constants obtained from the
spin-lattice relaxation rate at the critical point with
the assumption TJ = 1.0.
R1
Molecule lav c a mom)b (sec-1) Tc(°K) 13¢ c§;$
CF38r 317 -3.83 2.195 1.63 340.2 1.28 3.39
CF31 416 -3.10 2.553 21.4 N400 1.47 2.56
CFZClZ 320 -4.00 1.994 1.52 385 0.96 3.97
CFZBrZ 480 -2.81 2.69 ~1.2 471 0.80 3.14
CFCl3 394 -3.30 2.37 1.43 454.5 0.940 3.40
C103F 156 -l7.4 2.161 10.0 368 1.06 16.9
CF3CCl3 517 -l.79 6.162 «0.7 482.2 1.0 1.8
CF3CF3 297 -3.00 3.819 0.730 292.2 1.11 2.85
CFZClCCl3 ~0.7
CFClZCFClZ mo.7
CFZClCFZCl «10.8

 

 

aSpin-rotation constants are given in un ts of kHz and moments of

inertia are given in units of 1040

2
b. -40
.4 = ——§!-——§-= 1.0207o1o -
8 kTIC

(5,111”

'11

it _ R]
I’d-“'1;
c
d1; 5.2-
esrjfi’ TJ

.l_.

1c2 '

228

the validity of Equation (105) in describing spin-rotational re1axa-
tion for this -CF3 containing molecule. Although thereare no data

to support any adjustment of 16 values, it is evident from the results
of Table 37 that the data of Figure 45 will better fit the J-diffusion
model if re values are adjusted rather than TJ values. In order to
resolve this point it would be helpful to have either temperature-

dependent Raman results or temperature-dependent chlorine NMR re1axa-

ti on times.

SUMMARY

An interface between an NMR pulse spectrometer and a Nicolet
1083 computer was constructed, and programs were written to allow
computer-controlled data collection and analysis for most simple
pulse sequences. A 5 mm, variable temperature. single-coil NMR
probe was constructed which operated at 56 MHz and contained a pair
of coils wound in an anti-Helmholtz configuration for use in the pulsed
field-gradient, spin-echo experiment. Three approaches were tried in
order to measure diffusion coefficients in liquids, the steady gradient
technique employing the magnet's shim coils, the pulsed gradient
technique employing the magnet's shim coils, and the pulsed gradient
technique employing a hand wound set of gradient coils within the probe.
The steady gradient technique proved satisfactory when sufficient
calibrations of the size of the gradient were performed.

An external field/frequency lock was constructed using the
Varian DP-60 console as the spectrometer monitoring the lock signal,
and building a new probe, containing a doped water sample, which
could be inserted in the back of the pulse probe and also easily
transferred from one pulse probe to another.

A simpler interface was constructed which allowed Raman spectra
to be digitized and punched onto IBM cards for analysis on the CDC
6500 computer. A FORTRAN computer program was written which calculated
the reorientational broadening of Raman A] lines. from which the
rotational correlation time was calculated.

Deuterium quadrupole coupling constants have been obtained for

two symmetric-top molecules in the liquid phase by combining NMR

229

230

relaxation data with Raman line-shape analysis of bands of A1 sym-
metry. The Raman lines have been corrected for vibrational and instru-
mental broadening by comparing the polarized and depolarized components
of a single line. More than one A1 line has been studied for each
molecule. The previously uncertain deuterium quadrupole coupling
constant for CDBr3 has been determined in this work to be 177:5 kHz

and deuterium coupling constants in CDX3 molecules are discussed.

The procedure described here provides a different method for obtaining
nuclear quadrupole coupling constants.in the liquid phase. The tem-

2D spin-lattice relaxation rate was used

perature dependence of the
to analyze published 13C relaxation data for CHBr3, and limits were
placed on the anisotropy of rotational motion. which at 20° C were
1'5<DIIIQL. >l.0. Motion in liquid CDBr3 was compared to motion in
the much more intensively studied liquid CDCl3 system.

Spin-lattice and spin-spin relaxation rates at 56 MHz were mea-
sured in CFBCCl3 from 141° K to 432° K. More limited measurements
were made of T] at 15.87 MHz and of the self-diffusion coefficient in
the liquid phase. A phase transition was observed in the solid at
147° K from a discontinuity in the T1 data.

By means of linewidth. T2.and variable-field T1 measurements,
spin-lattice relaxation in plastic crystalline CF3CC13 was found
to have contributions from intramolecular dipole-dipole interactions,
translational diffusion, and spin-rotation. The activation energies
for these processes were determined to be 1.8 kcal/mole, 12.9 kcal/
mole, and m—l.8 kcal/mole. respectively. At 56 MHz. spin-rotation
was found to be more important in the solid than translational

diffusion. The value of the translational diffusion coefficient

 

231

at the melting point was determined to be 2.1 x 10'8 or 1.3 x 10'8

2 1

cm sec' , depending on whether the crystal structure is fcc or

bcc.

The separation of the terms contributing to the liquid phase
19F relaxation in CF3CC13 was made on the basis of self-diffusion
measurements and a single room temperature Raman measurement coupled
with the temperature dependence of the rotational correlation time
determined from the solid phase data. The rotational motion was
discussed in terms of Gordon‘s extended diffusion model and it was
found that the Hubbard relation for isotropic reorientation predicted
angular momentum correlation times which were in approximate agree-
ment with the diffusion models, indicating that off-diagonal elements
of the spin-rotation tensor are small in CF3CC13.

Spin-lattice re1axation rates were measured from the melting
point to the critical point for CF3Br and CFZBrZ; also the self-
diffusion coefficient was measured in liquid CF3Br from room temperature
to the melting point. The separation of Rl,total into the contribu-
tions from various mechanisms was discussed but was not quantitatively
successful due to difficulties in obtaining satisfactory values for
the intermolecular dipole-dipole relaxation rate.

Spin-lattice and spin-spin relaxation rates were measured over
a limited temperature range for CFZClCCl3, CFClZCFClZ, CF3CF3, and
CF31. The use of the difference Rz-R1 to obtain the scalar coupling
constant JFX’ where X is the other halogen, was discussed, but it
was not, in general, possible to obtain reliable values. These
substituted ethanes were found to behave very similarly to CF3CC13

in that spin rotation dominated the liquid range, with translational

232

diffusion making only a minor contribution to the total spin-lattice
relaxation rate (R1) both in the liquid and solid, at 56 MHz. Also
the presence of a minimum in R] in the solid indicated the probable
presence of a spin rotational relaxation mechanism.

"Effective" spin-rotation interaction constants were calculated
for a number of compounds where experimental data were available
from the relaxation rates at the critical point, and were compared
with chemical shielding-derived values. The agreement was found
generally to be good, indicating that off-diagonal elements of the
spin-rotation tensor were small for these compounds and that the

motion was roughly isotropic.

REFERENCES

 

d
.

(a)

10.
11.

12.

13.
14.
15.

16.

17.

18.
19.
20.

REFERENCES

R. Zwanzig, Ann. Rev. Phys. Chem. 16, 67 (1965).
R. G. Gordon, Advan. Magn. Resonance 3, 1 (1969).

B. J. Berne,"Physical Chemistry, An Advanced Treatise", Vol. 8,
Academic Press, New York (1971).

8. J. Berne and G. D. Harpe, Advan. Chem. Phys. 12, 63
(1970).

H. A. Steele, "Transport Phenomena in Fluids", M. Dekker, New
York (1969).

N. Hax, "Selected Papers on Noise and Stochastic Processes“,
Dover, New York (1954).

J. L. Doob, Ann. Math. 43, 351 (1942).

B. Callen and T. A. Helton, Phys. Rev. 83, 34 (1951).

G. Gordon, J. Chem. Phys. 43, 1307 (1965).

K. Green and J. G. Powles, Proc. Phys. Soc. 85, 87 (1965).

 

UPI

M. Bloom, F. Bridges, and H. N. Hardy, Can. J. Phys. 45, 3533
1967 .

For a discussion of correlation functions for dilute gases see
J. M. Deutch and I. Oppenheim, Advan. Magn. Resonance g, 225 (1966).

P. Debye, “Polar Molecules", Dover, New York (1945).
Cr". Hu and R. Zwanzig, J. Chem. Phys. 69, 4354 (1974).

R. I.)Cukier and K. Lakatos-Lindenberg, J. Chem. Phys. 51, 3427
1972

E. N. Ivanov, Zh. Eksp. Teor. Fiz. 45, 1509 (1963)[Sov. Phys-
JETP lg, 1041 (1964)].

Egelstaff, "An Introduction to the Liquid State", Ch. 10, Academic
Press, New York (1967).

F. Perrin, J. Phys. Radium §, 497 (1934); Z, 1(1936).
L. D. Favro, Phys. Rev. 112, 53 (1960).
D. E. Hoessner, J. Chem. Phys. 31, 647 (1962).

233

21.

22.

23.
24.
25.

26.
27.

28.

29.
30.
31.
32.

33.
34.

35.
36.
37.
38.
39.
40.
41.
42.
43.

234

H. T. Huntress, Advan. Magn. Resonance 4, l (1970).

M. E. Rose "Elementary Theory of Angular Momentum“, Wiley, New
York (1957).

R. K. Hangsness and F. Bloch, Phys. Rev. 89, 728 (1953).

F. B1och, Phys. Rev. 195, 1206 (1957).

A. G. Redfield, Advan. Magn. Resonance 1, l (1965).”

R. Kubo and K. Tomita, J. Phys. Soc. Japan 9, 888 (1954).

Jighsiéfeutch and I. Oppenheim, Advan. Magn. Resonance _3, 43

A. Abra am, "The Principles of Nuclear Magnetism", Oxford UP,
London 1961).

R. A. Valiev, Opt. Spectry (USSR)(English Transl.) 13, 282 (1961).
F. J. Bartoli and T. A. Litovitz, J. Chem. Phys. 56, 404 (1972).
S. Bratos and E. Marechal, Phys. Rev. 54, 1078 (1971).

S. Sykora, U.S. Clearinghouse Fed. Sci. Tech. Inf. No. 724330
1971 .

R. Loudon, Advan. Phys. 13, 423 (1964).

N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 73,
679 (1948).

N. Bloembergen, J. Chem. Phys. 21, 572 (1957).
. Hinter, Compt. Rend. 249, 1346 (1959).
. R. Sharp, J. Chem. Phys. 51, 5321(1972).
. B1oom, Physica 23, 237 (1957).
. S. Johnson and J. S. Haugh, J. Chem. Phys. 35, 2020 (1961).

. Bloom, Can. J. Phys. 49, 289 (1962).
. S. Johnson and J. S. Haugh, J. Chem. Phys. 36, 2266 (1962).

J
R
M
C
M. Lipsicas and M. Bloom, Can. J. Phys. 39, 881 (1961).
M
C
H. P. Haas, G. Seidal, and N. J. Poulis, Physica 26, 834 (1960).

235

44. H. S. Gutowsky and D. E. Noessner, Phys. Rev. 1_O_4_, 843 (1956).

45. H. S. Gutowsky, I. J. Lawrenson, and K. Shimomura, Phys. Rev.
Lett. 3, 349 (1961).

46. R. J. C. Brown, H. S. Gutowsky, and K. Shimomura, J. Chem. Phys.
38, 76 (1963).

47. . Solomon and N. Bloembergen, J. Chem. Phys. _2_5, 261 (1956).

H

. M. McConnell and C. H. Holm, J. Chem. Phys. 33, 1289 (1956).
. S. Hubbard, Phys. Rev. 131, 1155 (1963). ..
. Blinc and G. Lahajnar, Phys. Rev. Lett. 1_9, 685 (1967).
. R. Hackleman and P. S. Hubbard, J. Chem. Phys. 3_9, 2688 (1963). E
. Rigney and J. Virlet,J. Chem. Phys. 31, 3807 (1969).

 

. M.)Pintar, A. R. Sharp, and S. Vrscaj, Phys. Lett. 373169
1968 .

. R. Sharp and M. M. Pintar, J. Chem. Phys. 33 2428 (1970).
. Boden and R. Folland, Chem. Phys. Lett. 1_0, 167 (1971).

54.
55.
. Boden and R. Folland, M01. Phys. :21, 1123 (1971).

. Ikeda and C. A. McDowell, Chem. Phys. Lett. 1_4, 389 (1972).

H
P
R
H
p
M
(
A
N
56. N
R
58. D. H. Sawyer and J. G. Powles, M01. Phys. 31,83 (1971).
C
A
N
E
R
K
(
H
H
C
D

57.
59. . Brot and 1. Damon, Mol. Phys. _2_l_, 725 (1971).
60. . Gi‘erer and K. Hirtz, Z. Naturforsch._A_8_, 532 (1953).
61.

62.

. E. Hill, Proc. Roy. Soc. (London) 331, 149 (1954).
. N. daC. Andrade, Ph‘i1. Mag. _1l, 497, 698 (1934).

63.
64.

. C. Mi11er and C. P. Smyth, J. Chem. Phys. 34, 814 (1956).

. A. Valiev and M. M. Zaripov, Zh. Eksp. Teor. Fiz. 33, 503
1962) [Sov. Phys. - JETP 13,353 (1962)].

. Shimizu, J. Chem. Phys. 31,765 (1962).

. Shimizu, J. Chem. Phys. 4_8_, 754 (1964).

. Brot, Chem. Phys. Lett. 3, 319 (1969).

. Hallach and H. T. Huntress, Jr., J. Chem. Phys. _5_(_), 1219 (1969).

69.
70.

71.
72.
73.
74.

75.
76.
77.
78.
79.

81.
82.
83.

85.
86.

87.

88.
89.

90.

91.

236

A. A11erhand, J. Chem. Phys. 33, 3596 (1970).

H. J.)Bender and M. D. Zeidler, Ber. Bunsenges. Physik. Chem. 13, 236
1971 .

T. T. BOPD, J. Chem. Phys. 32, 3621 (1967).
. T. Huntress, J. Phys. Chem. 13, 103 (1969).

K:

. T. Gi11en and J. H. Nogg1e, J. Chem. Phys. 33, 801 (1970).

U

. E. Hoessner, B. B. Snowden, Jr., and E. T. Strom, Mol. Phys.
, 265 (1968). 21

Jonas and T. M. DiGennaro, J. Chem. Phys. 39, 2392 (1969).
. Gillen, J. Chem. Phys. 33, 1573 (1972).

._0
I4.

Gi11en and J. E. Griffiths, Chem. Phys. Lett. 12, 359 (1972).
. Kintzinger and J. M. Lehn, Mol. Phys. 33, 273 (1971).
Kintzinger and J. M. Lehn, Mol. Phys. 31, 491 (1974).

~4:4 :0 'u -4 -4

Gillen, M. Schwartz and J. H. Noggle, M01. Phys. 39, 899

_l
\D
d
V

Griffiths, Chem. Phys. Lett. 31, 354 (1973).
Steele, J. Chem. Phys. 33, 2404 (1963).
. Steele, J. Chem. Phys. 33, 2411 (1963).

. B. Moniz, H. A. Steele and J. A. Dixon, J. Chem. Phys. 33,
2418 (1963).

P. H. Atkins, M01. Phys. 11, 321 (1969).

P. N. Atkins, A. Loewenstein, and Y. Margalit, Mol. Phys. 11,
329 (1969).

B. D. Nageswara Rao and P. K. Mishra, Chem. Phys. Lett. 31,
592 (1974).

G. Birnbaum and H. Ho, Chem. Phys. Lett. 3, 334 (1970).

=a=a=e~xeerz<e
>221"

P. V. Huong, M. Corizi, and M. Perrot, Chem. Phys. Lett. 1,
189 (1970).

J. P. Perchard, H. F. Murphy, and H. J. Bernstein, Chem. Phys.
Lett. 3, 559 (1971).

G. Birnbaum, Mol. Phys. 33, 241 (1973).

92.
93.
94.
95.
96.
97.
98.
99.
100.
101.

102.
103.
104.
105.
106.
107.
108.

109.
110.
111.
112.

113.

114.
115.
116.

237

. H. Glarum, Mol. Phys. 11, 1139 (1974).

G. Gordon, J. Chem. Phys. 11, 1830 (1965).

I. Cukier, J. Chem. Phys. 39, 734 (1974).

E. 0. McClung, .1. Chem. Phys. _5_1_, 3842 (1969); 24.. 3248 (1971).
E. D. McClung and H. Versmold, J. Chem. Phys. 31, 2596 (1972).
Fixman and K. Rider, J. Chem. Phys. 31,2425 (1969).

0. Mountain, J. Chem. Phys. 31, 3243 (1971).

E. D. McClung, J. Chem. Phys. 31, 5478 (1972).

Frenkel, G. H. Hegdam, and J. van der Elslen, J. Chem. Phys.
1, 2691 (1972).

.G. St. Pierre and H. A. Steele, Phys. Rev. 131, 172 (1969).
.G. St. Pierre and H. A. Steele, J. Chem. Phys. 31, 4638 (1972).
Chandler, J. Chem. Phys. 39, 3500; 3508 (1974).

. E. Larsson, J. Chem. Phys. 33, 4612 (1973).

Mori, Progru Theoret. Phys. (Kyoto) 33, 423 (1965).

. Mori, Progru Theoret. Phys. (Kyoto) 31, 399 (1965).

Kivelson, M. G. Kivelson, and I. Oppenheim, J. Chem. Phys.
2,1810 (1970).

. Keyes and D. Kivelson, J. Chem. Phys. 33, 1057 (1972).
. Kivelson and T. Keyes, J. Chem. Phys. 31, 4599 (1972).
. Kivelson, M01. Phys. 33, 321 (1974).

.H. Spiess, D. Schweitzer, U. Haberlen, and K. H. Hauser,
.Magn. Resonance 3,101 (1971).

X .G.: U U —| IU'ID I I 75 U > >

. Krynicki and J. G. Powles, Proc. Phys. Soc. (London) 86,
549 (1965).

J. G. Powles and M. Rhodes, Phys. Lett. 111, 523 (1967).
K. Krynicki and J. G. Powles, J. Magn. Resonance 3, 539 (1972).
R- 5- Morgan and J. H. Strange, Mol. Phys. _11, 397 (1969).

. E. Eagles and R. E. D. McClung, Chem. Phys. Lett. 33, 414 (1973).

 

117.

118.
119.
120.

121.
122.

123.
124.
125.

126.

127.
128.
129.

130.

131.

132.
133.

134.
135.
136.
137.
138.

238

J. H. Strange and R. E. Morgan, J. Phys. C Solid State Physics 3,
1999 (1970).

H. Boehme and M. Eisner, J. Chem. Phys. 13, 4242 (1967).
D. E. O'Reilly and E. M. Peterson, J. Chem. Phys. 13, 2872 (1968).

D. E. O'Reilly, E. M. Peterson, 0. C. Hogenbloom, and C. E.
Scheie, J. Chem. Phys. 31, 4194 (1971).

C. J. Gerritsma and N. J. Trappeniers, Physica 31, 365 (1971).

C. J. Gerritsma, P. H. Oosting, and N. J. Trappeniers, Physica
‘31, 381 (1971).

P. H. Oosting and N. J. Trappeniers, Physica 31, 395 (1971).
P. H. Oosting and N. J. Trappeniers, Physica 31, 418 (1971).

J. H. Campbell, S. J. Seymour, and J. Jonas, J. Chem. Phys. 33,
4151 (1973).

K. T. Gillen, J. H. Noggle, and T. K. Leipert, Chem. Phys. Lett.
13, 505 (1972).

R. M. Hawk and R. R. Sharp, J. Chem. Phys. 33, 1009 (1974).
R. R. Sharp, J. Chem. Phys. 33, 1149 (1974).

Discussed by T. C. Farrar, A. A. Maryott, and M. S. Malmberg,
Ann. Rev. Phys. Chem. 33, 193 (1972).

A. A. Maryott, T. C. Farrar, and M. S. Malmberg, J. Chem. Phys.
31, 64 (1971).

K. T. Gillen, D. C. Douglass, M. S. Malmberg, and A. A. Maryott,
J. Chem. Phys. 31, 5170 (1972).

R. A. Assink and J. Jonas, J. Chem. Phys. 33, 3329 (1972).

3i ngwaan, R. J. Finney, and J. Jonas, J. Chem. Phys. 33, 3223
97 .

J. E. Anderson and R. Ullman, J. Chem. Phys. 33, 4406 (1971).
J. Kushick and B. J. Berne, J. Chem. Phys. 33, 4486 (1973).
H. Shimizu, J. Chem. Phys. 13, 2453 (1965).

S. Bratos and J. Rios, Compt. Rend. 333, 90 (1969).

Y. Guissani, S. Bratos, and J. C. Leicknam, Compt. Rend. 333,
137 (1969).

239

139. S. Bratos, J. Rios, andY. Guissani, J. Chem. Phys. 33, 439

(1970) -
140. R. G. Gordon, J. Chem. Phys. 13, 3658 (1965).
141. F. J. Bartoli and T. A. Litovitz, J. Chem. Phys. E, 413 (1972).
142. H. S- Goldberg and P. S. Pershan, J. Chem. Phys. 38, 3816 (1973).
143. I. Laulichtand S. Meirman, J. Chem. Phys. _53, 2521 (1973).
144. M. Constant and R. Fanquemberque, J. Chem. Phys. 33, 4031 (1973). F
145. c. H - wang and P. A. Fleury, J. Chem. Phys. 5_3, 2243 (1970). I
146. R. B- Wright, M. Schwartz, and C. H. Wang, J. Chem. Phys. 5__8,
5125 (1973). '

 

147. Y. LeDuff, J. Chem. Phys. 5_9, 1984 (1973).
148. M. Schwartz and C. H. Wang, J. Chem. Phys. 33, 5258 (1973).

149. J P. Perchard, N. F. Murphy, and H. J. Bernstein, Mol. Phys.
23. 499 (1972).

150. J. P. Perchard, H. F. Murphy, and H. J. Bernstein, M01. Phys.
33, 51 9 (1972L

151. J. P. Perchard, H. F. Murphy, and H. J. Bernstein, M01. Phys.
_2__3, 535 (1972).

152- R- C L1v1ngston.w. G. Rothschild, andJ. J. Rush, J. Chem. Phys.
59 2493 (1973).

‘53- "- G- Rothschild, J. Chem. Phys. 32, 991 (1972).

‘54- "- Schwartz and C. H. Wang, Chem. Phys. Lett. 33. 25 (1974)-
155- J- H- Campbell and J. Jonas, Chem. Phys. Lett. 1_3, 441 (1973).
‘55- c- "- wang and R. B. Wright, J. Chem. Phys. 3_9, 1706 (1973).
157' T' T~ Hall, J. Chem. Phys. 33, 2792 (1970)-

153- “' Gs Rothschild, J. Chem. Phys. 5_5, 1402 (1971).

‘59- "" 9- Rothschild, Macromol. 1, 43 (1968).

160' (119Ele3'1’mcmlaye’ and A. Dayan, Chem. Phys. Lett. _l_2, 462

15‘ - R- E. o. McClung, J. Chem. Phys. 5_5_. 3459 (1971)-

162.
163.
164.

165.
166.
167.

168.
169.

170.

171.

172.
173.

174.

175.

176.
177.
178.
179.

180.

181.
182.
183.

240

C. Dreyfus and J. Vincent-Geisse, Chem. Phys. Lett. 31, 170 (1973).
J. Yarwood, Advan. M01. Relax. Proc. 3, 375 (1973).

J. P. Marsault, F. Marsault-Herail, and G. Lévi, M01. Phys. 33,
997 (1973).

T. E. Eagles and R. E. D. McClung, J. Chem. Phys. 33, 435 (1973).
G. Lévi and M. Chalaye, Chem. Phys. Lett. 13, 263 (1973).

?. vanKonynenburg and H. A. Steele, J. Chem. Phys. 33, 4776
1972 .

D. A. Jackson and B. Simic—Glavaski, M01. Phys. 13, 393 (1970).

H. D. Hardy, V. Volterra, and T. A. Litovitz, J. Chem. Phys.
.33, 4491 (1973).

M. J. Bird, 0. A. Jackson, and J. G. Powles, Mol. Phys. 33,
1051 (1973).

G. B. Alms, D. R. Bauer, J. I. Brauman, and R. Pecora, J. Chem.
Phys. 33, 5321 (1973).

N. G. Rothschi1d. J. Chem. Phys. 33, 3265 (1970).

J. H. Campbell, 5. J. Seymour, and J. Jonas, J. Chem. Phys. 31,
346 (1974).

G. R. Alms, D. R. Bauer, J. I. Brauman, and R. Pecora, J. Chem.
Phys. 33, 5310 (1973).

D. R. Bauer, G. R. Alms, J. I. Brauman, and R. Pecora, J. Chem.
Phys. 31, 2255 (1974).

J. P. McCullough, J. Pure and Appl. Chem. 3, 221 (1961).
L. Pauling, Phys. Rev. 33, 430 (1930).
J. Frenkel, Acta Physicochim. USSR, 3, 23 (1935).

G. B. Guthrie and J. P. McCullough, J. Phys. Chem. Solids 13,
53 (1961).

For a discussion of the history of plastic solids see J. Tinmermans,
J. Phys. Chem. Solids 13, 1 (1961).

I. Darmon and C. Brot, M01. Crys. 3, 301 (1967).
. P. Smyth, J. Phys. Chem. So1ids 13, 40 (1961).

no

. Clemett and M. Davies, Trans. Faraday Soc. 33, 1705, (1962).

 

184.
185.

186.
187.
188.
189.
190.

191.
192.
193.

194.
195.
196.
197.

198.

199.

200.
201.

202.
203.

204.
205.

241

C. Clemett and M. Davies, Trans. Faraday Soc. 33, 1718 (1962).

C. P. Slichter, "Principles of Magnetic Resonance", p. 59, Harper
and Row, New York (1963).

H. C. Torrey, Phys. Rev. 33, 962 (1953).

H. C. Torrey, Phys. Rev. 33, 690 (1954).

H. A. Resing and H. C. Torrey, Phys. Rev. 131, 1102 (1963).

N. Boden, J. Cohen, and P. P. Davis, Mol. Phys. 33, 819 (1972).

P. Bladon, N. C. Lockhart, and J. N. Sherwood, Mol. Cryst. Liquid
Cryst. 13, 315 (1973).

G. H. Smith, J. Chem. Phys. 33, 3595 (1969).
H. A. Resing, Mol. Cryst. Liquid Cryst. 3, 101 (1969).

E. 0. Stejskal, D. E. Hoessner, T. C. Farrar, and H. S. Gutowsky,
J. Chem. Phys. 31, 55 (1959).
J

. E. Anderson and H. P. Slichter, J. Chem. Phys. 11, 1922 (1964).

R. L. Jackson and J. H. Strange, Mol. Phys. 33, 313 (1971).
S. Albert and J. A. Ripmeester, J. Chem. Phys. 33, 1069 (1973).

J. M. Chezeau, J. Dufoureq, and J. H. Strange, Mol. Phys. 33,
305 (1971).

S. Albert, H. S. Gutowsky, and J. A. Ripmeester, J. Chem. Phys.
33, 1332 (1972).

R. Folland, S. M. Ross, and J. H. Strange, M01. Phys. 33, 27
(1973).

H. A. Resing, J. Chem. Phys. 33, 2575 (1962).

J. E.)Anderson and H. P. Slichter, J. Chem. Phys. 11, 3647
1966 .

G. N. Smith, J. Chem. Phys. 13, 4229 (1965).

G. A. Matzkanin, T. A. Scott, and P. J. Haigh, J. Chem. Phys.
13, 1646 (1965).

G. N. Smith, J. Chem. Phys. 13, 3483 (1966).

D. C. Look, I. J. Lowe, and J. A. Northby, J. Chem. Phys. 11,
3441 (1966).

 

206.

207.
208.
209.

210.
211.
212.
213.
214.

215.
216.
217.
218.

219.
220.
221.
222.
223.
224.
225.
226.

227.
228.
229.

242

D. J. Genin, D. E. O'Reilly, E. M. Peterson, and T. Tsang,
J. Chem. Phys. 13, 4525 (1968); 33_2786 (1969).
J

. A. Pople and F. E. Karasz, J. Phys. Chem. Solids 13, 28 (1961).
F. E. Karasz and J. A. Pople, J. Phys. Chem. Solids 33, 294 (1961).

L. M. Amzel and L. 11 Becka, J. Phys. Chem. Solids 33, 521
(1969).

R. L. Streever and H. Y. Carr, Phys. Rev. 131, 20 (1960).

A. Csaki and G. Bené, Compt. Rend. 331, 228 (1960).

Dinesh and M. 1. Rogers, Rev. Sci. Instrum. 13, 555 (1972).

D. C. Look and D. R. Locker, Rev. Sci. Instrum. 11, 250 (1970).

R. Freeman, H. D. H. Hill and R. Kaptin, J. Magn. Resonance 1,
82 (1972).

Varian Associates, Palo Alto, California.
Nicolet Instrument Corporation, Madison, Wisconsin.
C. P. Slichter and D. Ailion, Phys. Rev. 133, A1099 (1964).

C. Deverell, R. E. Morgan, and J. H. Strange, M01. Phys. 13,
553 (1970).

E. J. Hells and K. J. Abramson, J. Magn. Resonance 1, 378 (1969).
I. Solomon, Compt. Rend. 313, 1613 (1959).

H. Y. Carr and E. M. Purcell, Phys. Rev. 31, 630 (1954).

S. Meiboom and D. Gi1l, Rev. Sci. Instr. 33, 688 (1958).

A. Kumar and C. S. Johnson, Jr., J. Magn. Resonance, 1, 13 (1972).
U.Haber1en.Ph.D. Thesis, Technishe Hochschule, Stuttgart (1967).
I. J. Lowe and C. E. Tarr, J. Sci. Instr. 1, 320 (1968).

In general, any computer command which generates an output pulse
can be used; however, the Nicolet 1082 has commands and output
Jacks specifically for this purpose.

J. 5. Haugh, J. Chem. Phys. 43, 662 (1968).

D. D. Traficante, private communication.

E. 0. Becker and T. C. Farrar, "Pulse and Fourier Transform NMR",
Academic Press, New York (1971).

 

u..fl1n..l

LIWT

 

230.
231.
232.
233.

234.

235.
236.
237.

238.
239.

240.

241.
242.
243.
244.

245.

246.
247.

248. '
249.
250.
251.
252.

243

. Tanner, Rev. Sci. Instr.33, 1086 (1965).
. Tanner, J. Chem. Phys. 52, 2523 (1970L
. Tanner and E. 0. Stejskal, J. Chem. Phys. 49, 1768 (1968L

0)

J
J
J.
K
(9
G. McDonald and J. S. Leigh, Jr. , J. Mam. Resonance 3, 358

E

E

E
. 1907 Packer, C. Rees, and D. J. Tomlinson, M01. Phys. 13, 421
(1973).
Handbook of Chemistry and Physics, Chemical Rubber Company. :9

D. L. VanderHart, J. Chem. Phys. 33, 1858 (1974).

H. F. Murphy, M. V. Evans, and P. Bender, J. Chem. Phys. 47,
1836 (1967L

A. Savitzky, and M. J. E. Go1ay, Ana1. Chem. 33, 1627 (1964).

T. C. Farrar, S. J. Druck, R. R. Shoup, and E. 0. Becker, J. Am.
Chem. Soc. 31, 699 (1972).

E. A. C. Lucken,"Nuc1ear Quadrupole Coupling Constants“(Academic
Press, New York, 1969).

K. T. Gi11en and J. H. Noggle, J. Chem. Phys. 33, 4905 (1970).
J. v. Reed and P. M.Harris, J. Chem. Phys. g, 1730 (1961).
J. L. Ragle and K. L. Sherk, J. Chem. Phys. 33, 3553 (1969).

J. L. Ragle, G. Minott, and M. Mokarram, J. Chem. Phys. 60,
3184 (1974).

. Rinne and J. Depireaux,"Advances in Nuclear Quadrupo1e Resonance"
. A. S. Smith, editor, Vol. I (Heyden and Son, London, 1974).

. S. Millett and B. P. Dailey, J. Chem. Phys. 33, 3249 (1972).

M
J
F
H. J. Caspary, F. S. Millett, M. Reichbach, and B. P. Dailey,
J. Chem. Phys. _5_1_, 623 (1969).

N. Zumbulyadis and B. P. Dailey, J. Chem. Phys. 33, 4223 (1974).
Dinesh and M. T. Rogers, J. Magn. Resonance, 1, 30 (1972).

H. B. Moniz and H. S. Gutowsky, J. Chem. Phys. 33, 1155 (1963).
G. J. Jenks, J. Chem. Phys. 31, 658 (1971).

G. Bonera and A. Rigamonti. J. Chem. Phys. 13, 175 (1965).

253.

254.
255.

256.

257.
258.

259.
260.

261.

262.
263.
264.
265.

266.

267.
268.
269.
270.

271.

272.
273.
274.

244

3. G. .)Pow1es, M. Rhodes, and J. H. Strange, M01. Phys. 11, 575
1966

R. A. Assink and J. Jonas, J. Magn. Resonance 1, 347 (1971).

S. G. Kukolich, A. C. Nelson, and D. J. Ruben, J. M01. Spectry.
40, 33 (1971).

U)

.G. Kukolich, D. J. Ruben, J. H. S. Hang, and J. R. Hilliams
Chem. Phys. 58, 3155 (1973).

. H. Caselton and S. G. Kukolich, Chem. Phys. Lett. 33, 331 (1973).

J.
K
P. 8. Davies, R. M. Neumann, S. C. Holfsy, and H. Klemperer,
J. Chem. Phys. 33, 3564 (1971).

D

A.

. Schwartz and J. L. Ragle, J. Chem. Phys. 31, 429 (1974).

( M. )Pritchard and R. E. Richards, Trans. Faraday Soc. 62, 2014
1966

T. L. Pendred, A. M. Pritchard, and R. E. Richards, J. Chem. Soc.

11009 (1966).

C. H. Kern and M. Karplus, J. Chem. Phys. 13, 1062 (1965).
L. Sa1em, J. Chem. Phys 33, 1227 (1963).
J. F. Harrison, J.Chem. Phys. 13, 2379 (1968).

A. B. Anderson, N. C. Hardy, and R. G. Parr, J. Chem. Phys. 50,
3634 (1969).

J. P. Zietlow, F. F. Cleveland, and A. 8. Meister, J. Chem. Phys.
18,1076 (1950).

A. Ruoff and H. Burger, Spectrochim. Acta, 331, 989 (1970).
H. Burger and J. Cichon, Spectrochim. Acta. 313, 2191 (1971).
R. Bersohn, J. Chem. Phys. 33, 85 (1960).

Hi11iams. J. T. Cox, and H. Gordy, J. Chem. Phys. 20, 1524
1952

S. N. Ghosh, R. Trambarulo, and H. Gordy, J. Chem. Phys. 20,
605 (1952).

M. Jen and D. R. Lide, Jr., J. Chem. Phys. 33, 2525 (1962).
M. Z. Pajak, Comp. Rend. 313, 1211 (1959).
K. F. Kuhlmann, D. M. Grant, and R. K. Harris, J. Chem. Phys.

33, 3439 (1970).

275.
276.
277.
278.

279.
280.
281.
282.
283.
284.
285.

286.
287.
288.
289.
290.

291.

292.

293.
294.
295.

296.

297.

245

R. R. Shoup and T. C. Farrar, J. Magn. Resonance, Z, 48 (1972).
H. Hiraoka and J. H. Hildebrand, J. Phys. Chem. 31, 916 (1963).
G. H. Smith, J. Chem. Phys. 33, 3081 (1962).

J. E. Jones and A. E. Ingham, Proc. Roy. Soc. (London) A107,
636 (1925).

H. J. Dunning, J. Phys. Chem. Solids 13, 21 (1961).

H. Suga, M. Sugasaki, and S. Seki, M01. Cryst. 1, 377 (1966).
P. S. Hubbard, Phys. Rev. 131, 275 (1963).

H. T. Toth, Ph.D. Thesis, Michigan State University (1974).
1. E. Burke and s. 1. Chan, J. Magn. Resonance 3,. 55 (1970).
C. R. Hard and C. H. Hard, J. M01. SpecthY- 13, 289 (1964).

C. S. Yannoni, B. P. Dailey, and G. P. Ceasar, J. Chem. Phys. 31,
4020 (1971).

C. H. Hang, J. Magn. Resonance 3, 75 (1973).
R. E. J. Sears, J. Chem. Phys. 33, 983 (1972).

R. H. Gallant, Hydrocarbon Process. Petrol. Refiner. 11, 128 (1968).
R. K. Niku1'shin, Khe1odil'n Tekhn. 13, 30 (1966).

Z. I. Ge11er, R. K. Nikul' shin, and Yu. G. Zatvornitskii, Zh.

Prilk. Khim. (Leningrad) L2,1121 (1969).

C. H. Townes and A. L. Schawlow, "Microwave Spectroscopy”, McGraw-
Hill, New York (1955).

S. C. Hofsy, J. S. Muenter and H. Klemperer, J. Chem. Phys. 55,
2014 (1971).

S. G. Ku lolich, Chem. Phys. Lett. 13, 52 (1971).
S. G. Kukolich, Phys. Rev. 133, 83 (1967).

S. G. Kukolich, J. H. S. Hang, and D. J. Reuben, J. Chem. Phys.
.33, 5474 (1973).

J. M. H. Reynders, A. H. Ellenbroek and A. Dymanns, Chem. Phys.
Lett. 11, 351 (1972).

N. F. Ramsey, Phys. Rev. 13, 699 (1950).

298.
299.
300.
301.
302.
EN33.
3CflL
3CHL
306.
307 .
EH38.

murxwxm>xmoz

246

. H. Flygare, J. Chem. Phys. 41, 793 (1964).

Deverell, M01. Phys. 18, 319 (1970).

1. Chan and A. S. Dubin, J. Chem. Phys. 46, 1745 (1967).
Y. Dong and M. Bloom, Can. J. Phys. 48, 793 (1970).

8. Harris, E. Hunt, and H. Meyer, J. Chem. Phys. 42, 2851 (1965).
Hunt and H. Meyer, J. Chem. Phys. 41, 353 (1964).

A. Bernheim and T. R. Krugh, J. Am. Chem. Soc. 82, 6784 (1967).
J. Laver, Thesis, Pennsylvania State University, 1967.

L. Armstrong and J. A. Courtney, Can. J. Phys. 59, 1262 (1974).

. M. Crapo and G. H. Flynn, J. Chem. Phys. 43, 1443 (1965).

R. Lyerla, Jr., D. M. Grant, and C. H. Hang, J. Chem. Phys.
4676 (1971).

 

APPENDIX A

APPENDIX A

RELAX2 - Measurement of Relaxation Times

This program provides computer-controlled timing and data acquisi-

ti on for two-pulse experiments. In addition it provides operator

 

directed data analysis of the exponential curves whether generated by
these experiments or by externally timed multiple-pulse (e.g. spin- H
echo, triplet T1) trains.

If the data are collected automatically (two- pulse experiment) they I F

are! inmediately ready for analysis as described in Section 4. If,
however, the collection is manual (externally timed pulse trains),
then further steps must be taken before the analysis may be performed.
These steps are performed using the DISPLAY routines as described in
Section 3.

l . Comands and Constants

 

Operator control of the program is via the subroutine SERVO,
which accepts two character alphaneumeric comands and checks them
against internal lists of legal comnands. As described in the SERVO
listing, there are three types of legal conmands and consequently three
Possible responses. These are:

C. Execute the operation requested
R. Accept a real constant*

I. Accept an integer constant*

”115 ‘1 he the Floating Point Package FLIP routine.

¥

247

248

Legal conlnands are listed on page 256. Also noted is whether they

are of type C, R, or I.

2. Acqui ri nLData

a. Multiple-Pulse Experiments

In this mode of operation the computer functions merely as a
signal averager during data acquisition. The trigger is set on
external and the autostop limit is put to l. The number of sweeps
desired is set with the NS conmand. The dwell time selected on the
main frame thumbwheel switches must be entered with the dwell (DH)
conmand. After each sweep the displayed memory is baseline corrected;
if this is not desired change address 1517;2000135/JMS STATHD to
l 544/JMP 1544.

b. Two- Pulse Experiments
In this mode the program calculates the discrete data directly
from the input parameters and the signal from the spectrometer. The
main frame controls are usually set as follows:
Trigger - auto recur
Block size - lK
Autostop - 1
Dwell time - 20 psec
Delay - off
The input parameters are T1, T2 (necessary only for spin-echo experi-

ments ) a NP, NS and IL. Dataare collected with N (=NPxNS) sequences.

[8T - pulse 1 - m - pulse 2 - (n1) - measure signal]

 

249

where m is varied by the computer from :0 to 4T1 (or 2T2) and the
second n1 may or may not be present depending on whether 01 or 62

was typed. The integration limit, IL, is the number of points of the
digitized FID which are sumed to obtain yi, the intensity at time

xi . For maximum S/N it should be set roughly equal to T} the

rate of the FID+. However any value of IL less than the displayed
memory size will work. The time span over which the decay is determined
may be varied from 4T by changing the value of the multiplicative
constant CONl address 1154. The current value of 1720448 (=1 ,000,000/
1610) gives the time span of 4T. The wait between successive two-pulse

sequences may be reduced from 8T1 to 41] by changing address 1040;

5023/RASH3 to 5024/RASH4.

3. Display Routines

After an experiment has been performed and the receiver output
digitized and stored in memory it is usually necessary to separate
the useful data from the rest of the receiver output. For example,
for spin 1/2 nuclei in liquids the Carr-Purcell pulse spacing is
usually large enough that the echoes are well resolved. In this
case the amplitude of each echo is the useful data while the rest is
superfluous. The display routines allow this separation to be easily
made.

These routines select points from the displayed (05) section of

memory using one of six methods (see Data Transfer). The points must

 

f-
foetnlember that IL gives the number of points in the digitized FID,
the decay time.

250

occur at regular intervals to be transferred in one operation. The
data thus selected* may then be fitted to an exponential decay as
described in Section 4.

a. Data Selection - Choming the Right Interval

Intensified points will always appear in the displayed (05)
memory unless they have been suppressed by setting DL=0. The interval
between intensified points is IN-IN/FR, where IN and FR are integer
constants which may be given any positive value. Let the displayed

memory size be 8. There are four cases of interest:

1. .5358, All intensified points have the separation IN.

2. INgfggg, All intensified points have separation IN except
every (FR/IN)th point which has separation IN-l from the
preceding intensified point.

3. ffiglfl, All intensified points have separation IN-IN/FR.

4. ffiffllfl, Only n points are intensified, all with separation

IN.

Cases 1 and 2 are generally useful, since either the desired
points are spaced evenly with an integral spacing (case 1) or they
are spaced evenly with a non-integral spacing (case 2). Cases 2 and
3 are redundant. Case 4 is occasionally a nuisance but is useful when

Points after a certain time are not desired.

The point at which the train of intensified points begins is

deterflfined by the delay (DL) constant.

\

*
He r- eafizer referred to as the discrete data set or discrete set.

251

b. Data Transfer

Points in the displayed memory which are intensified by means
of the commands IN, FR and DL are transferred to another section of
memory (2000-3777 currently)for further processing by the Data Trans-
ffier (DT) or Add Data (AD) commands. There are three options for pre-
conditioning the points to be transferred and two options for storage

crf these points. The preconditioning options are:

l. Appending an M to the command (DTM or ADM) initiates a search
over a predetermined range (RA) and the transfer of that point
which has the maximum (algebraic) value.

2. Appending an A to the command (DTA or ADA) averages the 2n+l
points centered about the intensified point (n=RA) and transfers
this average multiplied bY'flfilu

3. Setting RA=1 and appending either A or M transfers the

intensified points exactly.
The storage options are:

l. The OT command writes the new data over the old, and only
those points which were transferred with the last command
are available for further manipulation.

2. The AD command adds data to the existing discrete set. If
there is no discrete data set as yet the OT and AD commands

are equivalent.

The M transfers are useful when the echoes are clearly defined
and s'591r1a1-to-noise is good. M transfers are also convenient when

the interval between intensified points does not exactly equal the

252

interval between echoes. An A transfer will be preferable when the
pulse repetition rate is such that clear echoes do not form, or when
the decay is noisy. AD transfers are a remnant from manual data collec-

tion for 2-pulse T1 measurements but may be of some use in combining

two data sets .

4. The Discrete Data Set

The discrete data set consists of precisely those points* which

 

are to be fitted to an exponential decay, and can be displayed on the
scope by the 02 comnand. Before performing the least-square analysis

points which are obviously in error may be removed by the Delete

Point (DP) comnand.
If the baseline is not already at zero the Baseline Correction (BC)

conmand will subtract the average value of the last BL points from

the rest of the discrete set.

a. Fitting:to an Exponential Decay

The L5 comnand will take the logarithm of each point? in the
discrete set and fit the resulting data to a straight line. A weighted
least-squares method is used with each point weighted by its magnitude
in the exponential. This is done because we really want to give each

Paint equal weight so that an uncertainty for any point contributed
6x

equally to the uncertainty in the slope. But 1nx1 - 7:1- and small values

 

 

*
Stored in memory as x], y]; x2, y2; xn, yn DTAl=x], DTAN=yn+1.

_f . .
”"7 '11 plied by a scaling factor, FACTR, equal to 1024.

 

253

of x are unduly important after the curve has been linearized. So
multiplying each point by a weight of y1 removes the erroneous im-
portance the linearization process gives to small values. The weights
are stored sequentially following yNP' After the fitting process the
scope displays the deviations from theoretical (straight) line. If
there is systematic deviation or if some points are obviously erroneous
the exponential data may be regained with the EX comnand. Then the
baseline may be adjusted (AC) or the offensive points removed (DP)
and the slope recalculated (LS). Baseline adjustment is most useful
in T1 measurements, when the points taken as baseline (BC) may not be
equilibrium values.

If the fitting procedure encounters points which are negative
whether through excessive noise or operator error, an error message
is typed but the calculation proceeds. This usually results in a
reasonably accurate slope but large standard deviation. The offending
point may be easily identified in the display. No matter what the
display looks like at this point the data can be recalled with the EX

comnand and the appropriate corrections made. The equation used to

determine the time constant is

 

 

2 2 2 2 2
T1 (szx )2 - szzwzx2
i i i i i
51
where "i = TS: S1. = signal amplitude at point i
The Standard deviation is calculated as
-l/2
(n xi 2
2(T— + b ' Y1)
_ 2 i 'l
O " T-T—x— (A2)
max min N - l J

 

 

254

 

szx 2w2x y - ngx2£wgy
- i i i i i 1 i 1 1
When b "‘ T 2 2 2 2 ° (A3)

If the data points are distributed about their true values in a
Gaussian distribution then the true decay rate will have a 95% probability

of lying in the interval +%-: 20. The teletype output will consist of

il-and a, then T] and o', where o' = T120.
1

5. Other Features

PRINT (PR) The points which are contained in the discrete data
set may be printed out in the sequence xi,y1,where x1 is the time in
seconds from the 180° pulse and yi is the magnitude of the ith point.

UP, DOHN and 01 (UP, DN,Dl) Since the memory reserved for the
discrete data is larger than the typical data set more than one set
may be stored. This is done by using the UP command. To avoid needing
a directory to these data sets it is assumed that they are all the
same size (NP points). By means of the UP and DN commands one may
shift from displaying the nth discrete set to the (n+l)th and the
(n-l)th set, respectively. The 01 command allows the first data set
to be viewed. Only the data viewed can be manipulated as described
in Section 3 but note that a least squares analysis of set n will
destroy set n+1.

PAGE 1 (Pl) For convenience in permanently storing teletype
output a division and sequential page number is typed every 11"

(1008 lines). The P1 command initializes this line count.

Multiple—pulse experiments

255

 

 

A DS/examine data section; J
select block size

 

[“2M/zero it if necessary,
i

 

Two—pulse experiments

 

 

‘ GO/measure, adding I Set parameters T1, TZI
NS, iNP _

together NS sweeps

 

 

A DT/transfer selected 5
points by DL, IN, FR

experiment
experiment

 

(Ft/2:2;
/

_ BC/adjust baseline]

1

 

 

{DP/delete offending points]

1

 

 

 

{

 

 

AC/add a constant

 

 

g

 

$[ES/perform least squares

 

 

4yes

no

 

 

 

.‘ lEX/recall data]
L .

 

t.—

Figure A1. A flow chart for the operation of RELAX2.

or]

 

#i Publish

 

 

256

Table A1. Legal Commands for RELAX2

Commands are listed grouped according to that part of the program to
which they are relevant.

Displayed Memory

ZM - zero memory (c)

05 - display data section selected by pushbuttons (C)

IN - interval between points (1)

FR - fraction of interval (I)

DL - delay to lst intensified point (I)

DT(A, M - data transfer intensified points (C)

AD(A, M - data transfer intensified points (C)

RA - range over which to search for maximum (M) or average (A)(I)

Discrete Set

Data

Data

02 - display current discrete set (C)

UP - display next higher discrete set (C)
,DN - display next lower discrete set (C)
01 - display lst discrete set (C)
Manipulation

DH - dwell time in seconds (R)

BC - subtract average of last BL points from rest of data (C)
BL - number of points to be taken as baseline (I)

LS - perform weighted least squares (C)

Ex - recall data prior to LS calculation (C)

AC - add a constant (BA) to data (C)

BA - constant to be added to data (1)

DP - delete point (I)

Acquisition

GO - jump to measure mode and take NS Autostop sweeps. (C)
Gl - perform T1 180°-t-90° experiment (C)

G2 - perform T2 90"-'r-180°-r experiment (C)

NS - number of2 sweeps to take with GO, G1 or oz (1)

NP - number of points to take with 61 and 62 (I)

ll - the estimate of T] (R

T2 - the estimate of T2
IL - integration limit2 for T](I)
SE - set up mode produces a ong series of pulses at a convenient
repetition rate
Miscellaneous

Pl - reset page counter (C)
PR - print out data in discrete set (C)
MD - call monitor (C)

257

 

 

 

 

 

 

.~x<4mm com ans ocou .N< mesmwm
alwaannmsaalmo<xu<m HzHom uz~a<ogm "a mums
comm onmm
(in was g3 23 23
fi fi mmm<=omuam<mq ”m omaa
”In "mum ow
new ou muouaaoo
owoMOun auuuooaou
.3st 5.3 meadows _ “N ones
used new” cans,sn~fl one -
c>mmm— 09m — _ Ha . ngmman d «H owwm

m: ou— nuoucwoa
owuu0uo >uuuooa0u

258

Table A2. Useful addresses

Address - - - - Contents

577 2000 ILLIMIT; absolute lower limit discrete data set
600 /DTAl; current lower limit

601 IOTAN; current upper limit

602 3776 /ULIMIT; absolute upper limit

5646 20000 /FACTR; multiplicative factor for least squares.

Table A3. Error message;

NO MORE ROOM - Discrete data memory full
RA T00 LARGE - obvious

BL T00 LARGE - obvious

A OR M MUST FOLLOH DT - obvious

SET RA - 2**N - obvious

#X's.NE.#Y's - should never occur

NEG POINT - Discrete point set should contain only positive numbers
during LS

FPP ERROR FLAG - should never occur.

 

259

#0/ DATA HANDLING AND DISPLAY ROUTINES
0 44453 RDTTY
1 2001573 STARTadfls SERVO
2 5431 REALS
3 5435 COMMANDS
4 5462 INTEGERS
5 5474 CDEST
IMATH ROUTINE ENTRIES
6 3001765 CALC1:JMS OMASTER
7 577 DTA1
10 600 DTAN
11 155 JMP DSPLY2

IDISPLAY 01
12 2000135
13 2165771
14 2110153
15 2405770
16 2110624
17 2405772
20 2110627
21 2405773
22 2110602
23 2405776
'24 2110623
25 2405775
26 170000
27 214001

30 3111776
31 105000
32 4012
33 2103771
34 41
35 2707773
36 51
37 2025771
40 50
41 2707775
42 46
43 2705772
44 2110623
45 2405775
46 2707772
47 51
50 2000060
51 4014
52 2125776
53 2707770

D1IJMS STATVD
ZERM BLOCK
"EMA STAT2
ACCM DECK
MEMA INTV
ACCM INTDEC
MEMA DELAY
ACCN DELDEC
"EMA DATA1
ACCM DATAP.
"EMA INTADJ
ACCH ADJDEC
238A

TACXD

D2aMEMA ODAIAP
VDSH

TACYD , _
MEMZ BLOCK

JMP D11

HMOMZ DELDEC
JMP D5

ONE" BLOCK

JMP D6 '
D11aMMOMZ ADJDE
JMP D4

MMOM INTDEC-

'MEMA INTADJ‘

ACCM ADJDEC
D43MM0MZ INTDEC
JMP D5

D63JMS INT!
DSAINCXD

MPOM DATAP

'Mwowz pzcn

 

 

54
55

56
57
60
61
62
63
64
65
66
67
70
71
72

30
6454

12
1

110300
2404614
3111776

105000

4012
2706614
63
2110624
2405772
1000060

IDATA TRANSFER

73
74
75
76
77
100
101
102
103
104
105
106

107
110
111
112
113
114
115
116
117
120
121
122
123
124
125

126
127
130
131

2110577.

2404600
2110050
2404612
110107
2510575
2404050
110126
2404056
2001613
2404621
12

0
2000443
2110611
2470514
3404600
2124600
2110607
3404600
2134600
2330601

5144

554
2110624
2405772
1000107

2102610
2000261
2110612
2404050

260

JMP D2
TTYRF

D73 4311’ D1
JMP.3TART

INT230
"EMA (300
ACCM TEMP1

II or 11025 021 HT 020 POINT

D33MEMA ODATAP

VDSR

TACYD

MMOMZ TEMPI'
0"? D3

MEMA INTV
ACCM INTDEC
JMP OINTZ

DT1OHEHR DTR1
ACCM DTAN
HERA D6

ACCM D6STOR
MEMA (DTRANS
A+MA CNSTI
ACCH D6
HERA (D72
ACCM D7

JHS ECRO
ACCM VORD1
JMP D1

DTRANSLO
JMS DPREP
MEMA MAXP
A-MA ONERT
ACCM ODTAN
UPON DTAN
HERA YMAX
ACCM ODTAN
NPOMA DTAN
M-AA ULIMIT
EXCT A019
JNP ERRX3
MEMA INTV
ACCM INTDEC
JMP ODTRANS

.IRESTORES INTERVAL COUNTER

IDATA TRANS FOR INTZ PTS

70213 4002235 or 30020011»:
-zpnovxnss ron 2x111vor6

zor 105120011005 AFTER
xsroas 4 on n 0000400

I Y

IPREVENTS DATA OVERFLOU

IRESTORES INTERVAL COUNTER

012.0202 ADDENT

JNS ADDATA
MEMA D6STOR
ACCH D6

Inssrones 0L0'1wsrnuc110w

132

133 2404056 ACCM D7

134

110012

155

ISTATUS UORD

135

136
137

140
141
142
143
144
145
146
147
150
151
152

153
154

IDISPLAY

155
156
157
160
161
162
163

164
165
166
167
170
171
172
173
174
175
176
177
200
201
202
203
204
205

ITYPE 3
206

5012
2510154
2404602

44034
210000
12360
162000
110400

5006
2404153
1000135

0
100000

02
2110577
2332600
5144

1
2404615
2110577
2405776

3111776
5002
214001
2125776
110300
2404614
3111776
105000
4012
2706614
172
2125776
2704615
2706615
164
6454
155

1

44034 STATUS
10017 ANDA (17

261

MEMA (D1
IDITTO
JMP DSPLYR

STATUD30

[IN STAT2
LASH 12
A+MA STATS
ACCM DATA1
STATUS

ACPA

ANDAZ (360
ZERZ

MEMA (400
LASH 6

ACCM STATS
UMP OSTATVD

STAszfl .
STAT33100000

DSPLY23MEHA DTA1
M-AZA DTAN

EXCT A019

JMP START

ACCM TEMP2

MEMA DTA1

ACCM DATA?

023.0204 004142.
LASR 2 IX AXIS SCALE
140x0

0200 0414?

0204 (300

4cc0 12091
024.0204 004142
v050

14010 .
00o0z 1:021

002 024

0200 04142

0000 10022
00007. 12022

302 023

11125

JMP DSPLYE

J“? 51421

TYPE330

[GETS STATUS UORD 2 STORED
[ADDRESS IN STATI AND READ“

 

207
210
211
212
213
214
215
216
217
220
221
222
223
224
225

226

227
230
231
232
233
234
235
236
237
240
241
242
243
244
245
246
247
250
251
252

3110206
2404266
2124206
3112266
162000
252
2024605
110003
2404604
3110266
5104
2164605
10077
470040
5144

510240
462312
162000
235
462315
237
2102605
470100
2001624
3110266
5066
3404266
2706604
223
3110266
5062
3404266
2124266
212
1000206

262

MENA OTYPES
ACCM TSPTR
"POM TYPES
T310MEMAZ .TSPTR
ZERZ

J"? 733

ONE" TO

MINA (3 I 3 CHARS PER “OR
ACCM FROM .
"EMA OTSPTR
SKIP A019
ZERH T0
T320ANDA (77
A-MA (40
EXCT AC19

[GETS ADDRESS OF WORDS

/ NO. LESS THAN 40

510100 A+MA (100

4+04 1240 x20

4-02 (312 I 0210 7
zzaz

002 134

4-0z c315 / 0210 7

002 135 [NOT 02 02 L2
134.0200 10
4-04 (100-
135.005 112:
0204 013212
2L50 6

4000 013212
00000 2200
002 132

0204 013212
2L50 2 - 120510225 9020 10 0210104L
4000 013212

0200 13212

002 131 . .

133.002 011223

[ITS A OR OR LE

x400 0414 2001122

253
254
255
256
257
260

261
262
263
264
265

2110577
2404613
2110600
2404577
2024610

73

0
2110613
2404577
2164610
1000261

ADD10MENA DTA1
ACCM OLDD1
1401A DTAN

ACCM DTA1

ONEM ADDENT
JMP DT1

ADDATA00
"EMA OLDDI
ACCM DTA1
ZERM ADDENT
JMP OADDATA

 

263

x24005 20205 202 11223.

266
267
270
271
272
273
274
275
276
277
300
301
302
303
304
305
306
307
310
311
312

313
314
315
316
317
320
321
322
323

324
325
326
327
330
331
332
333
334
335
336
337
340
341
342
343
344
345

346

0

0
3110267
2405770
3111770
2405770

2124267'

3110267
2405771
2124267

110007
2404614
2111770

5063 .

5043
2404621
2704614
2012573

313
2110621
304

2714614
2404615
2110346
2102615
162000
324
5066
2704615
316

2405772
2111770
2404621
10007
510020
2515772
2102614
162000
343
5046
2405772
2110621
5063
2704614
326
2111772
3405771
1000267

2404040

'4000

TSPTRJU
PACK00

"EMA OPACK
ACCM NUMBER
MEHA ONUMBER
NUMBER
PACK
OPACK
ADDRESS
PACK.
(7

[ADDRESS

[VALUE
MPO"
"EMA
ACCM
MPOM
MEMA
ACCM TEHPI
MEMA NUMBER
RLSH 3
930LL3H 3
ACCM UORDl
”MOM TEMP1
ANDAZ MASKS
UMP “A
MINA UORD1
J"? 93

[ADDRESS

940HHOHA TINP1
ACCN TEMPE
"EMA BLNKUD
UQJMENZ TEMPS

"ZERZ

JMP US
RLSR 6
MMOM TEMPB
JMP U2

HSaAcCM PAKUUD
MEMA NUMBER.
HloACCM WORD!-
ANDA (1

A+MA (20

A+MMA PAKDUD
01:02 TEMP!
:83:

JMP v6

LLSH 6

ACCM PAKDVD
MEMA WORD!
RLSH 3

014011 12021

JMP 91

U6:MEMA PAKDVD
ACCH OADDRESS
JMP OPACK

2LNKUD02404040

 

264

.svsaLxus 0022201100

347
350
351
352
353
354
355
356

357

360
361
362
363
364
365
366
367
370
371
372
373
374
375
376
377
400

401
402
403
404
405
406
407
410
411
412

XSHIFTS
413
414
015
416
417
420

421
091".
423
424
425
7&31113‘1'5

2110622

5144
2234622
2404614
2110600
2470577

5021
2472622

562
170000
3001766
2704600
3110600
3001767
3001750
2704600
2704600
2706614
364
2124600
2110622
3001767
3001752
3001761
401

2130577
2405773
3111763
3325773
2135773
2322600
162000
155
2125773
403

DISPL 2
2110600
2404577
2110630

5001
2510600
2330601

5144

426
2330601
2404600

155

DOWN ONE

.ZERA

BSLN0MEMA BASEL
EXCT A019
ANGMA BASEL
A00" TE"P1

"EMA DTAN

A-"A DTA1

RASH 1

A-"AZ BASEL

[BASEL MUST BE POSITIVE

[BASEL TOO LARGE?

5144‘ EXCT A019

JMP ERRXS [YES
[ZEROES PAC
JMS OIFLOC

""O" DTAN

BS1:"E"A ODTAN

JMS DIPLOR

JMS OFADD

""0" DTAN

"MO" DTAN

"MOMZ TEMP!

JMP BS1

"PO" DTAN

"EMA BASEL

JMS OIFLOR

JMS OFDIV

JMS OFIX

JMP BSLN2

BSLN20MPOA DTA1
A00" PNTR2
BS2:"E"A 0FAO"-
M-A" 0PNTR2
"POMA PNTR2
M-AZ DTAN

ZERZ

JMP DSPLYS
"PO" PNTR2

J"? 282

UP ONE BLOCK OF DATA

.UPJMEMA DTAN
A00" DTA1
UP10MEMA NPOINTS
LASH 1
A+MA DTAN
M-AA ULIMIT
EXCT A019
JMP DOWN
M-AA ULIMIT
A00" DTAN
JMP DSPLY2

/ou15102 ALLOWED 2201007

[YES

 

 

EL_

 

426
427
430
431
~432
433

434

435
436
437
440

2110577
2404600
2110630

5001
2330577
2470576

413
2510576
2404577

155

5144'

265

DOVNIMEMA DTA1
ACCM DTAN
"EMA NPOINTS
LASR 1
M-AA DTA1
A-MA LLIMIT
EXCT A019
UMP UP
A+MA LLIMIT
ACCM DTA1
JMP DSPLY3_

[OUTSIDE ALLOWED REGION?

[YES

[GOES BACK TO TRE FIRST DISPLAY
441 2110576 FIRST0"E"A LLIMIT
442 2404577 A00" DTA1
443 2404600 A00" DTAN

415 J"? UP1

444

445

446 2110626 "EMA RANGE
447 405021

450
451
452
453
454
455

456

457

460
461
462
463
464
465

‘466
467
470
471
472
473
474
475

476
477
500
501
502
503
504

0 DPREP00

2404616
2511776
2404605
2110616
2331776
2404604

2110621

162000
466
462315
565
2000530
1000445

2164607
5210
3110604
2504607
2134604
2330605
422000
470

5141
557
110010
2404620
2164617
2110626
2002527

462301

[RANGE IS T I 0? POINTS
R15" 1 [OR SEARCHED
A00" RANGE1
A+"A DATAP
A00" TO

"EMA RANGE1
M-AA DATAP

4000 2200

"EMA UORDI
A-"Z (301
ZERZ ‘
JMP SIGAVE
A-"Z (315
JMP ERRX6
JMS "AX
JMP CDPREP

[STORAGE FOR A OR M CO""AND
[ A7

[ "7
.[NEITHER

SIGAVE02ERM'YMAX
0LL

SIG10MEMA OERO"
A+"" YMAX
"POMA FRO"
M-AA TO
APOZ

JMP SIG1

[SUMS N POINTS

[DONE7

EXCT L

JMP ERRX4
"EMA (10
A00" OOPS
ZER" COUNT
"EMA RANGE
SIG20ANDZ "ASKS

[TESTS FOR OVERFLOV
[PREVENTS ENDLESS LOOPING

'[3777776

 

 

505
506
507
510
511
512
513
514
515
516
517
520
521
522
523
524
525

526
527
530
531
532
533
534
535
'536
537
540
541
542
543
544
545
546
547
550
551
552
553

554
555
556
557
560
561
562
563
564
565
566
567

513
2110526
2510617
2404521
2110607

521
2124617
2706620

162000

570

5061
504
5021
2404607
2111776
2404611
1000445

5020
3777776
0
2110604
2404606
2404611
3110606
2404607
2134606
2330605
422000
162000
1000530
3110606
2330607

5104

536
2110606
2404611
3110611
2404607

536

5366
126
2000206
5375
126
2000206
5403

1
2000206
5411
126

266

ON? 5184
MEMA SHIFT
A+MA COUNT
ACCM 5108
"E"A YHAX
J"? 5163
81641090" COUNT
MMOMZ OOPS
238:

J"? ERRXT
RLSH 1

J0? 5162
5163:2ASH 1
ACCM YMAX
"EMA DATAP
ACCM MAX?
J"? ODPREP

SHIFT05020
"ASK503777776
MAX00
"E"A FRO"
A00" XMAX
A00" "AXP
"E"A OXMAX
A00" YMAX
"10MPOMA XMAX
M-AA TO
APOZ
ZERZ
JMP IMAX
"E"A OXMAX
M-AA YMAX
SKIP A019
JMP "1
"E"A )("AX
A00" "AXP
"E"A OMAXP
A00" YMAX
JMP "1

22223.005 11203
222023

002 012
22224.005 11223
222024

002 012 .
222xs1005 11223
222025

002 51421 .
ERRX60JMS 11223
222026

002 012

[STORES CODE FOR "RASH"

[NO

x020 1 LARGER 1240 0007

[NO

[YES

 

267

 

570 2000206 ERRX70 JMS TYPES

571 5422 ERROR7

572 126 JMP DT2

573 700000 MASK30700000

574 100000 ONEHT0100000

575 2000000 CNST102000000

576 2000 LLIMIT02000 IABSOLUTE LOWER LIMIT
577 2000 DTA102000

600 0 DTAN00 .

601 3776 ULIMIT03776 [ABSOLUTE UPPER LIMIT
602 100000 DATA10100000

603 107777 DATAN0107777

604 0 FROM00

605 0 T000

606 0 XMAX00

607 0 YMAX00

610 0 ADDENT00 [ENTRY MARKER MUST BE SAVED
611 0 MAXP00

612 0 D6STOR00

613 0 OLDD100

614 0 TEMP100

615 0 TEMP200

616 0 RANGE100

617 0 COUNT00

620 0 OOP500

621 0 WORD100

[THE FOLLOWING LOCATIONS MUST BE SAVED
[AS TREY ARE THE CONSTANTS SET VIA SERVO

622 1 24520.1 120-0 02 215 14020 45 045
623 5000 101400.5000 [FR-FRACTIONAL 2421 or
624 500 101v.500 [IN-INTERVAL 221022020
625 400 101011.400 /10-10120241100 01011
626 0 24002.0 [RA-SEARCHING 1012v0 202
627 02041.0 [BL-DELAY 10 151 1012051
630 0 0201015.0 /02
631 0 0502225.0 105
632 0 2050121.0 [BA-AMT 24520102 15 501r120
633 0 11251.0
634 0 0
635 0 12251.0
636 0 0
01770 - -
[THE 200000102 000411005 422 0520 0001 BY 2400
1770 0 000022.0
1771 0 4002255.0
1772 0 240000.0
/400 10 s 15 0520 21 2500
1773 0 20122.0
41770
I102 200 0100 422 0520 BY 0152041 01
' 1770 0 0202.0

268

1771 0 BLOCK00
1772 0 INTDEOo0
1773 0 DELDECJO
1774 0 DLPDE020
1775 0 ADJDECJO
1776 0 DATAP00

*7600

DEMON:

[THE FOLLOWING ARE ADDRESSES IN T1 PROGRAM
#1230
GO“T1:
01236
GO'TQJ
#1075
ZRMEM:
#1162
EXPNT:
#1203
SETUP:
#1212
ADCNST:
[THE FOLLOWINO.ARE ADDRESSES IN SERVO
21573 '
1573 0 SERVOJ0
#1613
1613 0 ECHOOO
41624
1624 '0 TTPE00
[THE FOLLOWING ARE ADDRESSES IN ETO
#1310
PRINT:
. #1447
PAGEI:
#1452
DLPNT:
41515
MEASURE:
[THIS IS ALL FROM LEAST'SOUARES
#4017
4017 2000 DUELL22000
4020 1000000 1000000

41750 _

1750 7245 2400.7245
1751 7416 FMULT.?416
1752 7461 2010.7461
1753 6736 FLIP.6736
1754 7062 02140.7062
1755 7050 02142.7050
1756 7026 240242.7026
1757 7074 20140.7074
1760 7534 20041.7534
1761 7541 112.7541
1762 7572 2402.7572

F“

 

 

1763 ' 7573

1764 7574

1765 4000

1766 ' 4021

1767 4086
45366

269

FACM27573.
FACML27574
MASTER24000
IELOC:4021
IFLOR04026

5366 3171640 ERRORS:3171640

5367

5370 2400522 2400522

5371 3171722 3171722

5372 404015 0404015

5373 2121215 2121215

5374 0 0

5375 3012240 22202493012240
5376 3172440 3172440

5377 3144017 3144017

5400 3072201 3072201

5401 2121505 2121505

5402 0 0

5403 3140240 ERROR503140240
5404 3172440 3172440

5405 3144017 3144017

5406 3072201 3072201

5407 2121505 2121505

5410 0 0

5411 2400140 ERROR602400140
5412 2402217 2402217

5413 154015 0154015

5414 3242325 3242325

5415 3170640 3170640

5416 3171414 3171414

5417 3044027 3044027

5420 2121524 2121524

5421 0 0

5422 3052340 22202713052340
5423 3224024 3224024

5424 2754001 2754001

5425 2526240 2526240

5426 2401652 2401652

[TABLE OF LEGAL INSTRUCTIONS

5427 2121540 2121540

5430 0 0

5431 304327 REALSJ304327

5432 324261 324261

5433 324262 324262

5434 0 0

5435 304323 COMMANDSJ304323

5436 315317 315317

5437 301304 301304

5440 304320 304320

5441 314323 314323

5442 302303 302303

171540 0171540

)00 0022 2000

[RA TOO LARGE

/ BL TOO LARGE

[A 02 M MUST FOLLOW DT

[SET 24 - 2440

[D0
[T1
[T2

[05
[MO
[AD
[DP
[L5
[BC

270

5443 304324 304324
5444 332315 332315
5445 325320 325320
5446 304316 304316
5447 323305 323305
5450 304261 304261
5451 304262 304262
5452 320261 320261
5453 307317 307317
5454 307261 ,307261
5455 307262 307262
5456 305330 305330
5457 301303 301303
5460 320322 320322
5461 0 0
5462 322301 10120225.322301
5463 304314 304314
5464 311316 311316
5465 306322 306322
5466 302314 302314
5467 316323 316323
5470 316320 316320
5471 311314 311314
5472 302301 302301
5473 0 0
x1420: or 025110411005
5474 4017 00251.00200
5475 633 11251
5476 635 12251
5477 12 01
5500 7600 02000
5501 253 4001
5502 1452 00201
5503 6 04001
5504 347 2500
5505 73 011
5506 1075 22020
5507 413 02
5510 426 0000
5511 1203 52102
5512 441 11251
5513 155 052072
5514 1447 24021
5515 1515 0245022
5516 1230 00-11
5517 1236 00-12
5520 1162 20201
5521 1212 400051
5522 1310 22101
5523 626 24002
5524 627 02041
5525 624 101v
5526 623 101403

[DT
[ZM
[UP
[ON
[SE
[D1
[D2
[P1
[00
[G1
[G2
[EX
[AC
[PR

[RA
[DL
[1N
[FR
[BL
[NS
[NP
[IL
[BA

 

5527
5530
5531
5532
5533

622

631'

630
625
632

271

BASEL
NSVEEPS
NPOINTS
ILIMIT1
BLSRIFT

 

4640/

272

00020122 0001200 202 100 20052 22222102015

SETM-4306

PULSE1-4102
PULSE2=4104
SENSE136112

640
641
642
643
644
645
646

647
650
651
652
653
654
655
656
657
660
661
662

663
664
665
666
667
670
671
672
673
674
675
676
677
700
701
702
703

704
705
706
707
710
711

2110577
2404600
2110630
2405770
2110631
2404615
2164616

3001754
633
2111154
3001767
3001751
110100
3001767
3001751
2000135
2110153
3001767
3001752

3001253
3001761
3111763
2405775
3001254
2103250
2001215
3001757
1251
2111154
3001767
3001752
110020
3001767
3001752
3001757
4017

2110630
3001767
3001754

633
3001752

2103250

BEGINpMEMA LLIMIT [LOVER LIMIT DATA STORE

ACCM PNTR2

.0204 0201015

ACCM‘NMPTS1
MEMA NSUEEPS
ACCM SHDEC
ZERM TIME

MS QBETAC
T1EST

MEMA CONI
JMS DIPLOR
JMS OFMULT
MEMA (100
JMS OIFLOR'
MS OFMULT
JMS STATVDN
MEMA STAT2
JMS OIFLOR
JMS OFDIV

JMS OPACTEM
JMS OFIX
MEMA .FACM
ACCM IDVELL
JMS OTEMFAC
MEMZ TIFLAG
JMS.SCALE
JMS OPUTAC
DVELL'

MEMA CONI’
JMS OIFLOR
JMS OPDIV
MEMA (20
JMS OIFLOR .
JMS OFDIV
JMS OPUTAC
DUELL2

MEMA NPOINTS
JMS OIFLOR
JMS IGETAC
TIEST

JMS OFDIV
MEMZ T1FLAO

[NUMBER OF POINTS DESIRE

[NUMBER OF SWEEPS PER PO.

[DELAY TIME BETWEEN

[THE ESTIMATE OF T1
[10.46/16

[16 TIMES 4 (OCTAL)

/2240001 200025122

'[SOLEY FOR 8T1 VAIT
[REPLACE T1.WITH T2

x000 010102 21 10:46

x00 200 10 5200005

'1‘

 

712
713
714
715
716
717
720
721
722
723
724
725
726
727
730

731
732
733
734
735
736
737
740
741
742
743
744
745
746
747
750
751
752
753
754
755
756
757
760
761
762
763
764
765
766
767
770
771
772
773
774
775

2001215
2111154
3001767
3001751
3001761
3111763
2405116
110004
2103250
5021
2001113
2404617
3111764
2011156
5042

2514617
2404616
2001100
2110620
2405774
1035
2111774
2404612
2164614
2110616
4102
2001051
4104
2103250
1225
410000

410000

4306
5220
6454
162000
1
2706615
1035
2110612
3001766
2110621
3001767

3001751-

3001755

1152
3001750
3001755

1251
3001752
2111155
3505762

273

JMS SCALE
MEMA CON!
JMS OIFLOR
JMS OFMULT
JMS OFIX
MEMA OFACM
ACCM MULTI
MEMA (4
MEMZ TIFLAG
RASH 1

JMS MULTIPLY

ACCM INTRVL
MEMA OFACML
ANDA MASKI
LLSH 2

A+MMA INTRVL

ACCM TIME
JMS ZERMEM
MEMA PIHI
ACCM HITIME
JMP AGAIN

[SINCE T2 GOES THRU TIMER T1
[ T1.(104*6)[4N 'CYCLES

-/3000000

 

. I11110¢¢61100 L"
[IN 00113 02 16 Mtcaosxcs

 

PULSE: MEMA HI TIME

ACCM UAITB
ZERM COUNT2
MEMA TIME
PULSEI

JMS TIMER
PULSE2
MEMZ TIFLAG
JMP ECHO
ACCA

ACCA
P2:SETM'
STOP

TTYRE

ZERZ

JMP START
MMOMZ SVDEC
JMP AGAIN
MEMA UAITB

005 01rLocw'

MEMA COUNT
JMS OIFLOR
JMS "‘11th
JMS OGETAR
CORRECTION
JMS OFADD

JMS OGETAR
DUELL

JMS OFDIV

MEMA 0002'
A+MM OFACE

/12-0211 202 102.2000 .

[THE 52052 2002 0025 0222

.A/2001222 50222 0110 101
/00

[MINIMUM PULSE SEP POSSIBLE

110000
I 0021122? 01 10

776

777
1000
1001
1002
1003
1004
1005
1006
1007
1010
1011
1012
1013
1014
1015
1016
1017
1020
1021
1022
1023
1024
1025
1026
1027
1030
1031
1032
1033

1034

1035
1036
1037
1040
1041
1042
1043
1044
1045
1046
1047
1050

1051
1052
1053
1054
1055
1056

3001761
3111763
2001066
2000135
2110602
3405711
2510625
2405161
3111771
2504613
2135771
2323161
' 1006
2110613
2001066
2164613
2110617
2514616
5104
1027
405021
2404616
2111774
5001
2405774
2707770
162000
155
2110631
2404615

2001100

2131775
2405116
2110153
5023
2001113
2404611
2110620
2404612
2110611
2164614
2001051
1737

~ -0
2404011
2104021
2134021
2332011

5144

274

JMS OFIX

MEMA IFACM
JMS DTRANS
JMS STATUD
MEMA DATAI
ACCM PNTR
A+HA ILIMIT
ACCM LAST
811MEMA OPNTR
A+MM SUM .
MPOMA PNTR
M-AZ LAST

JMP S1

MEMA SIM

JMS DTRANS I Y
ZERM SUM
MEMA.INTRVL
A+MMA TIME
SKIP ACI9
JMP T11
RISR 1

ACCM TIME
MEMA RITIME
LASH 1

ACCM HITIME [DOUBLES HIGH WORD
111.0000: NMPTSI [DONET

ZERZ

JMP DSPLYB
MEMA NSVEEPS
ACCM SVDEC

[100033

[HALVES LOU VORD

[YES

JMS ZERMEM
AGA IN: MPOA I DUELL
ACCM MULTI
MEMA STAT2‘
RASH 3

JMS MULTIPLY
ACCM VAIT
MEMA PIHI
ACCM UAITE-
MEMA VAIT'
ZERM COUNT2
JMS TIMER
JMP PULSE

TIMERpO

ACCM WAIT
TIB:ZERM COUNT
T110MPOMA COUNT
M'AZA WAIT

EXCT AC19

[ 2 DVELL<STAT2)[16'

( 104112).<00001).10/0021.L-x

x1002LL 040 02 .21. 1

 

1057
1060

1061
1062
1063
1064
1065

1066
1067
1070
1071
1072
1073
1074

1075
1076

1077
1100

1101
1102
1103
1104
1105
1106
1107
1110
1111
1112

1113
1114
1115
1116
1117
1120
1121
1122
1123
1124

1061
1054

2134614
2332612
5144
1001051
1053

'0
3404600
2134600
2332601

5144
554
1001066

110012
2405100

162000

2000135
2110602
2405776
2510153
2404603
3165776
2135776
2322603

1106

,1001100

0
4354
505320
0
2404620
4343
2404011
2025110
2102020
1133

1125 5144.'EXCT AC19

1126 1133
1127 2111116
1130 2404620
1131 2110611
1132 1001113

[LONGTIME HALVES

1133 405021
1134 2404611
1135 2110620

275.

JMP TIMER2
JMP TI1

TIMER20MPOMA COUNT2
M-AZA UAIT2

EXCT AC19

JMP OTIMER

JMP T12

DTRAN500
ACCM OPNTR2
MPOMA'PNTRE
M-AZA ULIMIT
EXCT A019
JMP-ERRX3
JMP ODTRANS

22020.022141052211 124025 005 30 111111
2000 222020 1522020.0A0 22 2012220
2222 ~ 10110 4.002 0000400
22202010 . -

005 512100

‘MEMA DATAI

ACCM DATAP
A+MA STAT2

ACCM DATAN
ZMI:ZERM ODATAP
MPOMA.DATAP ’
M-AZ DATAN

JMP 2M1

JMP OZERMEM

0021122110
14000
0021

,MULT100

ACCM P1HI
THQAC

ACCM WAIT
ONEM MULTI
M12MEMZ PIHI _ .
JMP LONGTIME [PRODUCT .GT020420

JMP 20001102 10011 15 100 24202

MEMA MULT1 [LONGTIME MAKES THIS LARG.
ACCM PIHI. [NOV 1 IF ORIGIONALLY 0
MEMA WAIT

JMP OMULTIPLY

LOU WORD: DOUBLES HIGH VORD
LONGTIMEoRISH 1
ACCM WAIT
MEMA PIHI

[WAIT/2

‘
In-

 

 

1136 5110
1137 1142
1140 2111157
1141 2504611
1142 2110620
1143 405021
1144 2404620
1145 2111116
1146 5001
1147 2405116
1150 2110611
1151 1123

1152 4000
1153 1100000
1154 172044
1155 10000
1156 3000000
1157 2000000

276

SKIP ACO [BIT 0 ' 17

JMP L1

MEMA CON3

A+MM WAIT IVES: ADD IT TO UAIT
LlaMEMA P1HI

RISH 1

ACCM PIHI

'MEMA MULT1

LASH 1
ACCM MULTI
MEMA WAIT
UMP M1

[MULTI * 2

CORRECTION04000
CORRM01100000
CON11172044
CON2010000
MASK113000000

CON302000000 [TRANSFER BIT FOR LONGT

1100 '7040 0004.1040

1161 100033

1162 2110577
1163 2332600
1164 5144
1165 1
1166 2130577
1167 2405772
1170 2110600
1171 2405773
1172 3111773
1173 3405772
1174 2125772
1175 2125773
1176 2135772
1177 2332600
1200 5144
1201 155
1202 1172

1203 2165774
1204 2111160
1205 2404616
1206 2224615

LAST0100033

EXPNTaMEMA LLIMIT
M-AZA PNTR2

EXCT AC19

JMP START

MPOA LLIMIT [ Y1
ACCM PNTRI

MEMA PNTR2 [ U1
ACCM PNTR3

EX12MEMA OPNTR3

ACCM OPNTRI

MPOM PNTRI

MPOM PNTR3

MPOMA PNTR1

M-AZA PNTR2

EXCT AC19

JMP DSPLY2

JMP EX1

[DATA STOP

SETUP02ERM HITIME
MEMA CON4
ACCM TIME

ANGM SVDEC [JUST SOME LARGE NUMBER

1207 110764 MEMA (764

1210 2405775
1211 737

1212 2110632
1213 3225763
1214 401

ACCM IDVELL
JMP PULSE

ADCNSTaMEMA LVLADJ
ANGM OFACM
JMP BSLN2

r“

 

 

277

SCALEtfl
JMS OGETAR
TZEST

JNS OFMULT
JMS OGETAR
TIEST

JMS OFDIV
JMP OSCALE

ECH02MEMA TIME
JMS TIMER
JMP P2

[DETERMINES UHETIER OR NOT TO UAIT FOR THE ECHO

1215 0
1216 3001755
1217 635
1220 3001751
1221 3001755
1222 633
1223 3001752
1224 1001215
1225 2110616
1226 2001051
1227 752
1230 2165250
1231 2111244
1232 2405155
1233 2111246
1234 2405153
1235 640
1236 2025250
1237 2111245
1240 2405155
1241 2111247
1242 2405153
1243 640
1244 10000
1245 12000
1246 1120000
1247 1100000
1250 0
1251 0
1252 0
1253 7034
1254 7042
#611
611 0
612 0
613 0
614 0
615 0
6d6 0
617 0
620 ‘ 0
621 0
01770
1770 0
1771 0
1772 0
1773 0
1774 0
1775 0
1776 0

T1:ZERM TIFLAG
MEMA T1CONST
ACCM CONQ
MEMA TICORR
ACCM CORRM

JMP BEGIN
T20ONEM TIFLAG
MEMA TQCONST
ACCM CONE
MEMA TQCORH
ACCM CORRM

JMP BEGIN
TICONSTJIOOOG
T2CONST012000
T1C0RR21120000
T2C0R301100000
T1FLA620
DWELL20

0

FACTEM27034
TEMFA007042

UAITJO
UAIT220
5UM00
COUNTEOU
SVDECOG
TIMEoG
INTHVLoO
P1H120
COUNT20

NMPTS100
PNTR00
PNTR100
PNT3300
HITIM820
10082210
DATAPDQ

0401
BSLN2:
0135
135
#153

153 '

01
START:
#155
DSPLYz:
#12
DSPLYI:
#625
625
#630
630
631
632
633
634
635
636
#577
577
600
601
602
603
04017
4017
4020
#554
ERRXS:
01750
1750
1751
1752
1753
1754
1755
1756
1757
1760
1761
1762
1763
1764
1765
1766
1767

400

8858388

2000

0

3774
100000
107777

2000
1000000

7245
7416
7461
6736
7062
7050
7026
7074
7534
7541
7572
7573
7574
4000
4021
4026

278

STATUD20

STAT2:0

ILIMIT0400

NPOINT550
NSVEEPSaO
LVLADJ00
TIEST00

0

TEESTJG

0

LLIMITJQUCU
PNTRZ20
ULIMIT23774
DATAIJIUUUOO
DATAN0107777

DUELL222000
1000000

FADD:7245
FMULT07416
FDIV07461
FLIP26736
GETAC07062
GETA807050
FACFA807026
PUTAC07074
FLOAT27534
FIX27541
FAC507572
FACH07573
FACML27574
MASTER:4000
IFL0004021
IFLORa4026

 

 

*1350/

279

ETC (ADDITIONS TO DISPLAY ROUTINES)

SETM=4306

1350

1351

1352
1353
1354
1355
1356
1357
1360
1361
1362
1363
1364
1365

1366'

1367
1370
1371
1372
1373
1374
1375
1376
1377
1400
1401
1402
1403

2165350
2001404
2001413
2125560
2000267

1560

1556
2110206
2405770
2110266
2405771
2110604
2405772
2110605
2405773
2001422
2111770
2404206
2111771
2404266
2111772
2404604
2111773
2404605
2001404

1001351'

LCOUNT00 [ I OF LINE FEEDS THIS PAGE

0 NEWPGJC

NPIJZERM LCOUNT
JMS LINE?

JMS DASH

MPOM PAGENUM
JMS PACH
PAGENUM

PAGE
MEMA
ACCM
MEMA
ACCM
MEMA
ACCM

11223 154025 11223 000514015
02101
132012
02102
110121
02103
0204 12022
4000 02104
305 0240100
0204 02101
4000 11223
0204 02102
4000 132012
0204 02103
4000 12021
0204 02104
4000 12022
JMS 21022
302,002020

12204225 11223 000514015

[GENERATES 2 LINE FEEDS

1404
1405

0
110212

LINEF00
MEMA (212

 

 

ACCM CHAR
MEMA (2
ACCM REP
JMS TYPE'M
JMP OLINEF

1406 2405774
1407 110002
1410 2405775
1411 2001437
1412 1001404

[GENERATES THE DASHES BETWEEN PAGES

1413 0 DASH00
1414 110255 MEMA (255
1415 2405774 ACCM CHAR
1416 110010 MEMA (10
1417 2405775 ACCM REP
1420 2001437 JMS TYPE'M
1421 1001413 JMP ODASH
'1422 0 HEADING20
1423 110240 MEMA (240

1424
1425
1426
1427
1430
1431
1432
1433
1434
1435
1436

2405774
110022
2405775
2001437
110212
2001624
2000206
1554
110215
2001624
1001422

ACCM
MEMA
ACCM

280

CHAR
(22
REP

JMS TYPE'M
MEMA (212
JMS TYPE
JMS TYPES
PAGEHEAD
MEMA (215
JMS TYPE

JMP

OHEADING

[TYPES "CHAR" REP NUMBER OF TIMES
TYPE'M20
MEMA REP

1437
1440
1441
1442
1443
1444
1445
1446

1447
1450
1451

0
2111775
2405776
2111774
2001624
2707776

1442
1001437

2165560
2025351
1352

[POINT DELETE

1452
1453
1454
1455
1456
1457
1460
1461
1462
1463
1464
1465
1466
1467
1470
1471
1472
1473
1474
1475
1476
1477

1500
1501
1502

2111502
2001504
3001153
3001101
2110000
2410511
5021
3413103
102000
1512
5104
1411

110277

2001624
1
3111763
5001
2510577
2405771
750000
2511771
2405770

3111771
3405770
2125771

ACCM

REPDEC

TP13MEMA CHAR
JMS TYPE
MMOMZ REPDEC
JMP TP1

JMP OTYPE'M

PAGE1oZERM PAGENUM

ONEM

02020

JMP NPI

DLPNTaMEMA EOUALS
JMS TYPE2

JMS

OFLIP

JMS OFIX

MEMA
A-MA
RASH

A-MAZ
.ZERZ

DTAN

DTA1.

1
OFACM

JMP DP3

SHIP

AC19 [POINT f MUST BE POS

JMP DP!

MEMA

(277

JMS TYPE
JMP START
DP13MEMA OFACM

LASH
A+MA
ACCM
HTOA
A+MA
ACCM

1
DTA1
OLDP

OLDP
NEWP

022.0204 00202

ACCM
MPOM

ONEWP
OLDP

 

 

1503
1504
1505
1506
1507
1510
1511

1512
1513
1514

2125770
3111771
3405770
2125770
2135771
2322600

1500

2704600
2714600
155

[MEASURE

1515
1516
1517
1520
1521
1522
1523
1524
1525
1526
1527
1530
1531
1532
1533
1534
1535
1536
1537
1540
1541
1542
1543
1544
1545
1546
1547
1550
1551
1552
1553

1554

1555 2400507

1556
1557
1560
1561
1562
1563
01310

2130631
2405770
2000135
2111561
2405771
2165772
2110602
2510153
2405773
2705773
3111773
2515772
2707771

1526

5023
2405772
2110602
2405773
2111772
3325773
2135773

2100153,

1537
2707770
162000
12

4306
5220
6454
1517

1

3012040

0

0

1

10
275240
5535

281

MPOM NEWP

MEMA OOLDP
ACCM ONEWP
MPOM NEWP

MPOMA OLDP
M-AZ DTAN

JMP DP2

1313321111014 DTAN
MMOMA DTAN
JMP DISPE

MPOA NSWEEPS
ACCM SWPDEC
ME13dMS STATWD
MEMA AVENUM
ACCM AVEDEC

'ZERM LVLADJ

MEMA DATAI
A+MA STAT2
ACCM DATAP
08210000 DATAP
MEMA ODATAP
A+MMA LVLADJ
MMOMZ AVEDEC
JR? 022

RASH 3

ACCM LVLADJ
MEMA DATAI
ACCM DATAP
023.0204 LVLADJ
M-AM IDATAP
MPOMA DATAP
00002 STAT2
JMP 0E3
MMOMZ SVPDEC
ZERZ

JMP DSPLI
SETH

STOP

TTYRF

JMP MEI

30? START

PAGEHEAD33012040

2400507
PAGE:0
0
PAGENUM21
AVENUM010
ECUAL50275240
PFLOP25535

[SHIFTS Y DOWN TOO

[I 0F POINTS TO BE AVERAG

[CORRECTION TO BE ADDED

[DIVIDE BY 10
[THIS IS THE CORRECTION

[START CORRECTING EACH PT

[PAGE N

[FOR PRINT

 

1310
1311
1312
1313
1314

1315

1316
1317
1320
1321
1322
1323
1324
1325
1326
1327
1330
1331
1332
1333
1334
1335
1336
1337
1340
1341
1342
1343
1344
1345
1346
1347
01564
1564
01753
1753
#1761
1761
01763
1763
012
DSPLI:
#153
153
#135
135
#1
START:
*155
DISP21
#266
266
0604

2001607
2001607
2110577
2404606
2332600

1
3110606
3001766
3001755

4017
3001751
2000206

1346
3001563
2124606
3110606
3001766
3001755

14017
3001751
2000206

1346
3001563
2001607
2134606

6454

162000
155

1314
2404040

0

0
6736
7541

7573

282

PRINTaJMS CRLF
JMS CRLF

MEMA DTA1

ACCM TEMP3

PRIoM-AZA DTAN [ANY POINTS?

5144 EXCT AC19

002 51421 100
0204 012023
005 012200
005 002142
002222

005 020021
005 11223
220411

.245 022202
0200 12023
0204 012023
005 012200
305 002142
002222

.105 0201121
005 11223
220411

005 022202
.105 0222
02004 12023
11122

2222

002 01522
002 221
22041112404040
0

[Y

TYPE2: 0
FLIP06736
FIX07541

FACM07573

STAT200

STATWD00

T3PNTRa0

 

283
604 0 TEMPla0
605 0 TEMP2:0
606 0 TEMP3:0 IFOR PRINT
0631
631 1 NSVEEPS:1
0577
S77 2000 DTAI:2000
600 0 DTAN:0
601 3776 ULIMIT:3776
602 100000 DATAlal00000
603 107777 DATAN:107777
01751
1751 7416 FMULTa7416
01755
1755 7050 6214357050
IFROM LEAST-SQUARES
01766
1766 4021 1FLOC:4021
04017
4017 2000 DUE22250002000
IFROM SERVO
01607
1607 0 CRLFoO
IUSED ONLY AROUND READING
01770
1770 0 NP101:0
1771 0 NP102:0
1772 0 NP103:0
1773 0 NP10400
01770.
1770 0 0202.0
1771 0 OLDP:0
1772 0 NPTSo0
01624
1624 0 TYPE:0
01770 -
1770 0 SUPDEC:0
1771 0 AVEDEC:0
1772 0 LVLADJ:0
1773 0 DATAP:0
1774 0 CHAR:0
1775 0 REPa0
1776 0 REPDEC-0
0206
206 0 TYPE3.0
0267
267 0 PACK;0

 

 

284

*1573/ A GENERAL SUBROUTINE FOR REAL-TIME PROGRAM CON

ITHIS SURROUTINE PERFORMS THREE FUNCTIONS!
I (1) IT WILL ACCEPT ANY NUMBER OF FLOATING POINT DECF

I CONSTANTS AND PLACE THEM IN USER SELECTED ADDR
I AND
I (2) IT WILL ACCEPT TYPED COMMANDS TO JUMP TO USER
I SELCETED ADDRESSES IN THE MAIN PROGRAM
IAND

I (3) IT WILL ALSO ACCEPT AND TRANSFER DECIMAL INTEGEP

ITWO CHARACTER ALPHANUMERIC ABBREVIATIONS ARE USED FOR

I VARIOUS CONSTANTS AND COMMANDS. THESE MUST BE LISTED
ITHEIR ASCII CODES STARTING AT LOCATION REALS. THE TWO
I ACTER CODES TO BE IDENTIFIED WITH CONSTANTS MUST BE I
I A ZERO (0) SEPARATES THEM FROM THE ABBREVIATIONS FOR
I (DESIGNATED COMMANDS). SEPARATED BY ANOTHER ZREOARE
ITHE ABBREVIATIS FOR INTEGERS (DESIGNATED INTEGERS).

I TABLE MUST END WITH A ZERO

ITHE TABLE OF DESTINATIONS MAY: LIKE THE TABLE OF LEGAL
I ACTERS: BE LOCATED AT ANY POINT IN THE PROGRAM0 THIS
I IS MERELY A LISTING OF THE LABELS: IN THE PROPER ORDE
I WHICH THE ALPHANUMERIC COMMANDS REFER.

ITHE FOLLOWING SEQUENCE IS USED TO CALL SERVO

I ADDRESS--------CONTENTS

I 4 3000047 JMS O SERVO

I 5 50 REALS .

I 6 53 COMMANDS

I 7 55 INTEGERS

I 10 56 CDEST

/ 0

I o

/ o

ITABLE OF LEGAL CHARACTERS (ASCII CODES)

I 50 301302 REALS:30130((0IGG

I 51 302301 302301/BA
I 52 0 0

I 53 307317 COMMAND50307317 I30
I 54 0

0
I 0 INTEGERSo0
ITABLE OF DESTINATIONS
I 56 406 CDESToCONSTI IAB
I 60 127 CONST2 IRA
I 60 61 START IGO'

 

285

ITHE ACTUAL PROGRAM STARTS HERE

\\\\\\\\\\\

IIN THIS EXAMPLE: TWO CONSTANTS ARE INPUTTED BY TYPING
I'"BA" ON THEPTELETYPE:
ICONSTANTo'
I RESP.

1573
1574
1575
1576
1577
1600
1601
1602
1603
1604
1605
1606

1607
1610
1611
1612

1613
1614
1615
1616

1617
1620
1621
1622
1623

61

127

406

0
2001607
2001613

5011
2405770
2001613
2001651

110277
2001624
110003
2325573
1574

0
2111746
2001564
1001607

0
2001617
2001624
1001613

0

6454
1620
44453
1001617

THESE CONSTANTS ARE PLACED IN LOCATIONS

ASSUMING THAT ADDERSS 61 IS THE START OF THE
I CONTROL CAN BE TRANSFERED TO THIS LOCATION AT ANY T
I THERE ARE NO

44453 STARTaRDTTY

0 CONST200

0 CONST100

12.1:-

FOLLOWED BYTYPING THE DECIMAL

 

INTEGERS TO BE INPUTTED.

SERVOO0

. S I 0 MS CHI-F
JMS ECHO
LASH 11
ACCM CHAR
JMS ECHO
.JMS TBSRCH
MEMA (277
JNS TYPE ITYPES T

S20MEMA (3 IRESETS POINTER TO TABLES
M-AM SERVO ICOMMAND HAS BEEN TYPED
JMP $1

ISTORES TYPED CHAR

CRLF: 0

MEMA CONST
JMS PRINT
JMP OCRLF

ECH000

JMS READ
JMS TYPE
JMP OECHO

READ:0
R10TTYRF
JMP RI
RDTTY

JMP IREAD

1624
1625
1626
1627
1630
1631
1632
1633
1634
1635

1636

1637
1640
1641

1642
1643
1644
1645
1646
1647
1650

1651
1652
1653
1654
1655
1656
1657
1660
1661
1662
1663
1664
1665

1666

1667

1670

1671 2323770 M-AZ CHAR

0
2405571
462212
162000
2125350
6444
1631
4443
2111350
332100

5144

2001642
2111571
1001624

0
2111624
2405572
2001351
2111572
2405624
1001642

0
2505770
3111573
2405771
2125573
3111573
2405772
2125573
3111573
2405773
2125573
3111573
2405733

3113771
1677

286

"P E: 0

ACCM TEMPI

A-MZ (212

ZERZ

MPOM LCOUNT

T10TTYPF

JMP TI

PRTTY

MEMA LCOUNT

M-AZA (100 I 100 LINES PER PAGE.
EXGT A019

JMS TYB

MEMA TEMPI

JMP OTYPE

ISAVE ACC
IA LINE FEED?

TYQJE
MEMA TYPE
ACCM TH‘IPE
JMS NEWPG
MEMA TEMPE
ACCM TYPE
JMP OTYE

ISAVES RETURN ADDRESS

TBSRCH00 ITABLE SEARCHING ROUTINE
AOMM CHAR
MEMA OSERVO
ACCM TABL1
MPOM SERVO
MEMA OSERVO
ACCM TABL2
MPOM SERVO
MEMA OSERVO
ACCM TABL3
MPOM SERVO
MEMA OSERVO
ACCM DESTN'

IGETS ADDESS OF CHAR TABL

IGETS ADDRESS OF INTEGER

IGETS ADDRESS OF DESTINA

121.0204: 014221 15242va 14022 02 2242

162000 ZERZ

JMP TB2
ICOMPERE WITH CHAR JUST TYP

1672 162000 ZERZ
1673 1724 JMP DSET
1674 2125771 MPOM TABLI

1675 2125733 MPOM DESTN

1676

1677 3113772 TB20MEMAZ OTABL2

1666

JMP TB1

I SEARCH TABLE OF COM

wr~r F

 

IGETS ADDRESS ON COM TAB.

1700
1701
1702
1703
1704
1705
1706
1707

1710 3113773 TB30MEMAZ

1711 162000 ZERZ

1712
1713

1714.

1715
1716
1717
1720

1721
1722
1723

1724
1725
1726
1727
1730
1731
1732
1733
1734

1735
1736
1737
1740
1741
1742
1743
1744
1745

1746
1747
01564
1564
1565
1566

162000
‘1710
2323770
162000
1721
2125772
2125733
1677

1001651
2323770
162000
1735
2125773
2125733
1710

3111733
2405733
1001733

2111747
2001564
3111733
2405733
3001753

3111733,

3001757
0
1604

2111747
2001564
3111733
2405733
3001753
3001761
3111763
3405733

1604

212215
275240

0
2001624
5031

287

ZERZ

JMP T23
M-AZ CHAR
ZERZ

JMP DGO
MPOM TABL2
MPOM DESTN
JMP T82

OTABL3 ISEARCH TALBE OF INTI
JMP OTBSRCH
M-AZ CHAR

ZERZ

JMP DINT

MPOM TABL3

MPOM DESTN

JMP TB3

IERROR EXIT

DG00MEMA CDESTN
ACCM DESTN
JMP CDESTN

DSEToMEMA CONST2

JMS PRINT IPRINTS '

MEMA CDESTN ICONVERTS POINTER TO BEST
ACCM DESTN ITO ACTUAL DESTINATION
JMS OFLIP

MEMA ODESTN

JMS OPUTAC

DESTN00

JMP S2

DINTpMEMA CONST2

JMS PRINT I PRINTS -

MEMA IDESTN I CONVERTS POINTER TO DEST
ACCM DESTN ITO ACTUAL DESTINATION
MS OFLIP

JMS OFIX

MEMA OFACM ILOCATION OF FIXED l
ACCM ODESTN

JMP S2

CONST0212215

CONST20275240

PRINT00 IPRINTS IG-BIT WORD IN A00 A5
JMS TYPE ITWO ASCII CHARACTERS

RASH 11

 

 

1567 2001624
1570 1001564

1571
1572

01770
1770
1771
1772
1773

01350
1350
1351

01757
1757

01763
1763

01761
1761

01753
1753

7074
7573
7541

6736

288

JMS TYPE

JMP OPRINT

TEMP100 IACC DURING TYPE

TEMP200 IRETURN ADDRESS DURING TYPE

CHAR00

TABLIIU
TABL200
TABL300

LCOUNT00
NEWPGo0

PUTA007074
FACM07573
FIX07541

FLIP06736

 

#4000I

4000
4001
4002
4003
4004
4005
4006
4007
4010
4011
4012
4013
4014
4015
4016

4017
4020

4021
4022
4023
4024
4025

4026

4021,

4030
4031
4032
4033
4034
4035

289

LEAST-SQUARES FOR EXPONENTIAL DECAY

0
3110000
2405770
3111770
2405770
2124000
3110000
2405771
3111771
2405771
2000204
2000036
2000514
2124000
1000000

2000
500

0
3405766
3165744
3001755
1000021

'0
2405776
3001742
2111776
2000021
3001753
3001743
1000026

ILEAST SQUARES

4036
4037
4040
4041
4040
4043
4044

~4045
4046
4047
4050
4051
4052
4053
4054

0
2111771
2405650
2111770
2405772
2323771

162000
1000036
3165764
3111772
2000021
2103547

100
3001742
2000460

MASTER00

MEMA
ACCM
MEMA
ACCM
MPOM
MEMA
ACCM
MEMA
ACCM

OMASTER
DATA1
ODATAI
DATA1
MASTER
OMASTER
DATAN
ODATAN
DATAN

005 200

005 25042
005 001201
0200 045122
002 0045122

DWELL00002000

0000500

IFLOC00

ACCM
ZERM
JMS
JMP

IDOES ALL CALCS: PRINTS OUTPUT

OFACM I AC TO FAC

OFACML

OFLOAT
OIFLOC

IFLOR00

ACCM

DATAP

JMS OFACTEM

MEMA

DATAP

JMS IFLOC

JMS

OFACFAR

JMS ITEMFAC
JMP OIFLOR

2504200

MEMA
ACCM
MEMA
ACCM
M-AZ
ZERZ
JMP

ZERM

DATAN
CNTR3
DATA1
CNTR2
DATAN

OLSOAR

OERRF

IPRESERVES FAC

ICALCULATES A LEAST SQUARES

IOCCUPYING DATA1 TO
IX20Y21000 XNaYN

L10MEMA OCNTR2
JMS IFLOC
MEMZ ALT
JMP BRI

JMS

0240120 '

JMS WCALC

IBRANCH FOR X OR Y

1224002 202 x
x2005 0112 120240

/4 2001102 10 22041 40 101202'

IFLOATS AN INTEGER FROM AC TO
ISTORES NUMBER TOBE FLOATED

DATAN:

 

4055
4056
4057
4060
4061
4062
4063
4064
4065
4066
4067
4070
4071
4072
4073
4074
4075
4076
4077
4100
4101
4102
4103
4104
4105
4106
4107
4110
4111

4112
4113
4114
4115
4116
4117
4120
4121

4122
4123
4124
4125
4126
4127
4130
4131

4132
4133
4134
4133
’41.?!)
11137
4140
11141

2000453
3001752
4355
3001740
3001754
4355
3001743
3001761
2000453
3001752
4363
3001740
3001754
4363
2125547
2135772
2323771
47

125
3001754
1772
2125777
2000453
3001752
4357
3001740
3001754
4357
3001752
1772
3001743
3001745
2000453
3001752
4361
3001740
3001754
4361
2705547
74
2103547
1557
3001051
4367
3001752
4361
3001745
3001754
1772
3001752
4363
3001751
4367

290

005 021021
005 002142
5002
005
00s
5002
005 0120240

005 025042

005 021021

005 002142

500x50

005 02400

005 020140

500250

0200 421 /22125 22122202
22.02004 00122
0-42 04140 ‘
002 21

002 0022
2211005 020140
22001 .

0200 000

005 021021

005 002142
5001
005
00s
5001
005 002142

22001

005 0120240 / x
005 020021
005 021021
005 002142
500x1

005 02400
005 020140
500x1

0000 421
002 22
002210202 421
002 22222

005 002140
500050

005 002142
su0x1

005 020021
005 020140
22001

005 002142
500250

005 002140
500050

OFADD
OPUTAC

IDONE 7

IBRANCH FOR Y

I SUMY

IFADD
IPUTAC

IFLOPS FLIPFLOP

IERROR IF OX'S 0NE0 OY'S

4142
4143
4144
4145
4146
4147
4150
4151
4152
4153
4154
4155
4156
4157
4160
4161
4162
4163
4164
4165
4166
4167
4170
4171
4172
4173
4174
4175
4176
4177
4200
4201
4202
4203

3001745
3001754
1774
3001751
4355
3001742
3001752
4357
3001745
3001752
.1772
3001741
3001754
1772
3001743
3001761
3001752
1774
3001741
3001754
1774
3001753
3001751
1772
3001746
3001752
4017
3001746
3001754
5550
2000241
3113764
1573
1000036

ILOG ROUTINE

4204
4205
4206
4207
4210
4211
4212
4213
4214
4215
4216
4217
4220
4221
4222
4223
4224

0
2111770
2425772
2111771
2405650

353
3111772
2000502
3111772
2000021
3111766

5144
1562
3001757
2111652
3505765
3001756

291

JMS OFMULT

JMS OPUTAC

P8002

JMS 008740

SUMX

JMS OFAOTEM

JMS 008748

SUM?

JMS OFMULT

JMS 008748

P8001 I N(SUMXY)

JMS OFSUB

JMS OPUTAC

P8001

JMS OTEMFAC

JMS OFSQAR

JMS OGETAR

P8002 / NtSUMXSO)
JMS IFSUB

JMS OPUTAC ‘
P8002 IDENOMINATOR
JMS OFACFAR

JMS OGETAC

P8001 INUMERATOR

JMS OFDIV
JMS OGETAR
DUELL

JMS OPDIV
JMS OPUTAC
SLOPE

JMS DEV
MEMAZ OERRF
JMP £88X4
JMP OLSOAB

/CONVERTS SLOPE TO DECONDS

L0600

MEMA DATA1
APOM CNTR2
MEMA DATAN
ACCM CNTRG
JMP ZERO
LN1oMEMA OCNTR2
JMS NORM
MEMA OCNTRB
JMS IFLOO
MEMA OFACM
EXCT 4019
JMP £88X3
1.012: 1.015 0220
MEMA P4078
A+MM OFACE
JMS OFIX

. ._.1'1

4225
4226
4227
4230
4231
4232
4233
4234
4235
4236
4237
4240

ISTANDA80 DEVIATION

4241
4242
4243
4244
4245
4246
4247
4250
4251
4252
4253
4254
4255
4256
4257
4260
4261
4262
4263
4264
4265
4266
4267
4270
4271
4272
4273
4274
4275
4276
4277
4300
4301
4302
4303
4304
4305
4306

3111766
3405772
2135772
2323771
162000
1000204
2125772
2135650
3331734
5144
1001735
212

0
3001751
4355
3001752
4361
3001745
3001742
3001751
4363
3001752
4357
3001745
3001753
3001743
3001741
3001752
1774
3001746
3001754
5552
2111770
2405651
2111771
2405650
2000460
3001752
5550
3001751

4017.

3001745
3001753
3111651
2000021
3001745
3001752

5552
3001740
2125651

‘JMS

292

MEMA OFAOM
ACCM CONTR2
MPOMA CNTRB
M-AZ DATAN
ZERZ

JMP OLOG
MPOM CNTR2
MPOMA CNTR3
M-AA CULIMIT
EXCT AC19
JMP OERRXS
JMP LN1

IPREVENTS PROGRAM OVERVR

r1-

0EV20

JMS OGETAC
SUMX
JMS-IGETAR
SUMXY

JMS OFMULT
JMS OFACTEM
JMS OGETAC
SUMXSO

JMS OGETAR
SUM?
JMS
JMS
JMS

 

ISUMX 0 SUN Y

020021
0240242
0120240
02502

005 002142
22002

005 02010

005 020140

2

0204 04141
4000 20122
0204 04140
4000 00123
0201.005 00420
JMS 002142
52022

005 002140
00222

005 020021
.105 0240242
0204 020122
.ms1nmc
005 020021
005 002142
2

005 02400
0200 20122'

ISUNXSUNXY - SUMXAO SUMY

/00222015 202 00222

IMX

IMX + B

4307
4310
4311
4312
4313
4314
4315
4316
4317
4320
4321
4322
4323
4324
4325
4326
4327
4330
4331
4332
4333

' 4334

4335
4336
4337
4340
4341
4342
4343
4344
4345
4346
4347
4350
4351
4352

4353
4354
4355
4356
4357
4360
4361
4362
4363
4364
4365
4366
4367
4370
4371
4372

3111651
2000026
3001741
3001742
3001756
3111766
3405651
3001743
3001761
3001752
4371
3001740
3001754
4371
2135651
2323771
271
3001751

4371,

2711777
2000026
3001746
3001760
2000406
2111774
2331775
2000026
3001746
3001752
4017
3001746
131777
3505765
3001754
5554

1000241

2165777
2000373

293

0204 020122
005 12202
005 02500
005 0240120
005 0212- .
0204 02400
4000 020122
.105 0120240
005 025042
005 002142
22003

005 02400
005 020140

P8003 ISUM(Y -

MPOMA PNTRQ
0-42 DATAN
JMP DEV1
JMS OGETAC
PRODS

MMOA NON
0MSIflfiR
JMS OFDIV
JMS OFSORT
JMS M1NMAX
MEMA MIN
‘0-44 042
JMS IFLOR
JMS OFDIV
JMS OGETAR
DVELL

JMS OFDIV
MPOA (1777
A+MM OFACE
JMS OPUTAC
SIGMA

JMP 008V

ZERO0ZERM NU"
JNS Z1
SUMX00

0

SUMY00

0
SUMXY00
0
SUMX5000
0
5UM55°00
0
SUM05000
0
P800300
0

IN-l

(MX+B)) SQUARED

IX SUB N - X SUB 11'

ICONVERTS TO SECONDS
IMULTIPLY BY 2

0.2.2

 

294

4373 0 z1.0
4374 2110373 MEMA Z1
4375 2405651 ACCM PNTRB
4376 110373 MEMA (Z1
4377 2404373 ACCM Z1
4400 3165651 22.2220 020122
4401 2135651 MPOMA PNTRQ
4402 2322405 0-4z z3 IZERO 02 10 102 4002255 02 21
4403 400 002 22
4404 212 002 201
4405 4373 23.z1

/21005 0101000 400 0421000
4406 0 010042.0
4407 2111770 0204 04141
4410 2405776 4000 04142
4411 3111776 MEMA ODATAP
4412 2405773 4000 1251
4413 2405775 4000 042
4414 2125776 001.0200 04142
4415 2135776 MPOMA DATAP
4416 2323771 0-42 04140
4417 162000 ZERZ -
4420 431 002 002 ILOOK 202 010
4421 3111776 0204 004142
4422 2331773 M-AA TEST
4423 5104 SKIP 4019
4424 414 002 001
4425 3111776 MEMA ODATAP
4426 2405773 ACCM TEST
4427 2405775 ACCM MAX
4430 414 002 001
4431 2111770 MM2.MEMA DATA1
4432 2405776 ACCM 04142
4433 3111776 MEMA ODATAP
4434 2405773 ACCM TEST
4435 2405774 4000 010
4436 2125776 MM3.MPOM DATAP
4437 2135776 MPOMA DATAP
4440 2323771 M-AZ DATAN
4441 162000 ZERZ .
4442 1000406 JMP OMINMAX
4443 3111776 MEMA ODATAP
4444 2331773 M-AA TEST
4445 ' 5144 2201 4019
4446 436 002 003
4447 3111776 MEMA ODATAP
4450 2405774 ACCM M1N
4451 2405773 4000 1251.
4450 436 002 003

295

IWEIGHT MULTIPLYS EACH TERM BY W SQUARED

4453 0 021021.0
4454 3001752 005 002142
4455 1770 050
4456 3001745 .105 02001.1
4457 1000453 002 0021021
x0420024125 0020. 021021 400 500 021021 5004220
4460 0 00420.0 .
4461 3111650 0204 000123
4462 2000021 005 12200
4463 410000 4004 /00020 201 0222 005 025042
4464 3001752 005 002142
4465 4365 500550
4466 3001746 005 02010
4467 3001761 005 025042
4470 3001754 005 020140
4471 1770 050.
4472 3001752 005 002142
4473 4367 500050
4474 3001740 005 02400
4475 3001754 005 020140
4476 4367 500050 .
4477 2125650 0200 00123
4500 3001743 005 0120240
4501 1000460 002 000420
10420024125 00204212100 240102
4502 0 0020.0
4503 3405650 4000 000123
4504 2000021 005 12200
4505 410000 4004 100020 201 2222 005 025042
4506 3001752 005 002142
4507 4365 500550
4510 3001740 005 02400
4511 3001754 005 020140
4512 4365 500550
4513 1000502 002 00020
[OUTPUT
4514 0 001201.0
4515 3001736 005 011223
4516 5577 220411
4517 3001751 005 002140
4520 5550 52022
4521 2111652 0204 24012
4522 3325765 0-40 02402
4523 3001754 005 020140
4524 5550 52022
4525 2001535 005 22202
4526 3001736 JMS 011223
4527 5605 220412
4530 3001751 005 002140
4531 51004

5554

 

4532
4533
4534
4535
4536
4537
4540
4541
4542
4543
4544
4545
4546
4547
4550
4551
4552
4553
4554
4555

4556
4557
4560
4561
4562
4563
4564
4565
4566
4567
4570
4571
4572
4573
4574
4575

#5535
5535
5536
5537
5540
5541
5542
5543
5544
5545
5546

5547

5550
5551

2111652
3325765
3001754
5554
2001535
3001736
5611
2000556
2001535
3001736
5605
3001751
5554
2001535
3001737
5777
5622
3001736
5617
1000514

0
3001751
5550
3001761
3001753
3001751
5554
3001746
3001754

5554'
13001752

5550
30000
2000021
3001746
1000556

0
110015
3225763
3001747
3103763
162000
1001535
110240
3001762
1541

296

MEMA FACTR
M-AM IFACE
JMS .PUTAC
SIGMA

JMS PFLOP
JMS ITYPE3
FRMAT3

JMS INVT
JMS PFLOP
JMS OTYPE3
FRMAT2

JMS OGETAC
SIGMA

JMS PFLOP
JMS OPACK
NUM

NPTWD

JMS OTYPE3
FRMAT4

JMP OOUTPUT

INVT00

JMS .GETAC
SLOPE

JMS .FSQAR
JMS .FACFAR
JMS .GETAC
SIGMA

JMS .FDIV
JMS .PUTAC
SIGMA

JMS .GETAR
SLOPE

ONEA

JMS IFLOC
JMS .PDIV
JMP CINVT

PFLOP00
MEMA (15
ANGM
JMS .FLOP
PF10MEMZ
ZERZ
JMP .PFLOP
MEMA (240
JMS .PCHAR
JMP PFI
421.0
52022.0
0

ICARCNT

INOW SIGMA/SLOPE SOUARED

ICONVERTS SLOPE TO 1/SL
ITO SIGMA/SLOPE SOUARED

OCARCNT

ILOOP TIL 14 CHARS TYPED

 

5558
5553
5554
5555
5556

IERROR EXITS

5557
5560
5561
5568
5563

5564

5565
5566
5567
5570
5571
5578
5573
5574
5575
5576

5577
5600
5601
5608
5603
5604
5605
5606
5607
5610
5611
5618
5613
5614
5615
5616
5617
5680
5681
5688
5683
5684
5685
S686
5687

3001736

5630
1000036
3111736
8463576

1570
3001736
5636
170000
8000081
881
3001736
5648

1000036'

5570

8181815
8404040
3840188
3404005
8404075

0
3848340
3044004
3408605

0
8181815
8576140
3840188
3404005
8404075

0
8404040
3030583
8181815
8404040
3178040
3841611
3404083
8181815

0

IERROR MESSAGES

5630
5631

473043
564083

297

300

a .
SIGMA00
0

0

ERRX80JMS .TYPE3
ERROR8

JMP OLSQAR
ERRX30HEMA OTYPE3
A-MZ EX31AD

168000 ZERZ

JMP ERRX31

JMS OTYPE3
E38033
ERRX3102ERA

JMS IFLOC

JMP LN8
ERRX40JMS OTYPE3
E38084

JMP OLSQAR
EX31AD:ERRX31

FRMAT108181815
2404040
3840188
3404005
8404075

0
ERMAT803848340
3044004
3408605

0
FRMAT308181815
8576140
3840188
3404005
8404075

0
ERMAT408404040
3030583 . I
8181815 ,
NPTUD08404040
3178040
3841611
3404083
8181815

0

ERROR80473043
564083

[WHAT WAS JUST TYPED
[WAS IT ERROR37
INO: GO AHEAD

/4 02041102 400 0122
/0102 40 22202 10 200

/
SEC

ICRLFLF

I

I PO
IINT

IS
ICRLFLF

IIX'S oNEo 'Y'S

F

 

 

 

5638
5633
5634
5635
5636
5637
5640
5641
5642
5643
5644
5645
5646
5647
5650
5651
5658

#1770

1770
1771
1778
1773
1774
1775

#5770

5770
5771
5772
5773
5774
5775
5776
5777

#5734

5734
5735
5736
5737
5740
5741
5742
5743
5744
5745
5746
5747
5750
5751
S752
5753
5754
5755
5756

560516
314340
402347
0
70516
178040
841611
0
202006
220540
821782
140640
400701
0

0

0
20000

601

550

206

867
7245
7314
7034
7048
7574
7416
7461
6510
6736
7068
7050
7026
7074
7534
7541

QGGGSGQG 883898

298

560516

314340

402347

0
ERROR32070516.
172040

241611

0
ERROR42202006
220540

821722

140640

400701

0

CNTR320
PNTR250
FACTRp20000

INEG POINT

IFPP ERROR FLAG

V5900

0
PROD1J0
0
PROD230
0

DATA1J0
DATAN30
CNTRQ:0
TESTOG
MINJO
MAX10
DATAP10
NUM20

ULIMITo601
ERRX51550
TYPE33206
PACK1267
PADDo7243
FSUBa7314 .
FACTEM07034
TEMFACJ7042
FACML17574
FMULTo7416
FDIV97461
FLOPa6510
FLIP56736
GETACD7062
GETAR:7050
FACFARo7026
PUTACI7074
FLOATo7534
FIX27541

 

 

5757
5760
5761
5762
5763
5764
5765
5766

6330
6176
6352
7013
7021
7555
7578
7573

2%!

FLNJ6330
FSQRTo6176
FSQA896352
PCHARa7013
CARCNTQ7021
ERRF97555
FACEJ7S78
FACM07573

F‘-

 

 

APPENDIX B

 

APPENDIX B

Other Nicolet Programs

1. Pulsed Field Gradient Timing and Data Collection

The basic structure of the program for computer control of the
pulsed field gradient experiment, PFGRAD, is essentially identical to
RELAX2. The differences arise, naturally, in the timing section, which

is as follows:

 

PULSE.MEMA HITIME
ACCM NAITZ
ZERM COUNT2
MEMA TIME
PULSE]

JMS TIMER
FGPULSE

JMS TAUZ
PULSEZ

MEMZ TlFLAG
JMS TAUZ
FGPULSE

JMS ECHO

TAU2,0
MEMA INTRVL

JMS TIMER
JMP@TAU2

This section of code replaces locations 737-747 in RELAX2; since the
length is different the program must be reassembled. The subroutine
'TAUZ, of course, is placed at some other point in the program. All
labels have the same meaning as in RELAX2; FGPULSE generates the C
output pulse as shown in Table 5 . The timing produced by PFGRAD
was shown in Figure 20, with reference to that Figured"2 is fixed

2T
as W3?- while 1’] varies in the normal fashion from O to 2T2. This

123

sequence produces the usual e' time dependence of the echo train.

300

301

*640/ COMPUTER CONTROL FOR TRIPLET T1 SEQUENCE
RDGRED=44375
SETM=4306
RILSE1=4102
PU LSE7=4104
LOR=4341
640 2110765 BECIM1HEMA ADC2 /CODE FOR PULSEI
641 2404740 ACCM NUTATE ISETUP CHANGES THIS L0
642 2110631 VEMA NSKEEPS /NUHBER OF SUEEPS PER
643 2405216 ACCM SVDEC
644 2165217 UEGI.ZERM TIME /SUB SEQUENCE SEP
645 3001754 JMS @GETAC
646 1237 GUESS ITHE ESTIMATE OF T1 OR T2
647 2111136 MEMA CONl /10**6/16
650 3001767 JMS @IFLOR
651 3001751 JNS GFMULT
652 110100 MEMA (100 /16 TIFES 4 (OCTAL)
653 3001767 JHS @IFLOR
654 3001751 JMS @FMULT
655 2000135 JMS STATWD /
656 2110153 VEMA STAT2 /READ OUT BLOCK SIZE
657 3001767 JMS @IFLOR
660 3001752 JMS @FDIV
661 3001757 JMS @PUTAC
662 1241 DWELL
663 3001761 JMS @FIX
664 3111763 MEMA @FACM
665 2405243 ACCM IDwELL
666 2111136 MEMA CONI
667 3001767 JMS @IFLOR
670 110020 MEMA (20
671 3001766 JMS @IFLOC
672 3001751 JMS @FMULT /ONE MILLION DECIMAL
673 3001756 JMS @FACFAR
674 3001754 JMS eGETAC
675 1241 DUELL
676 3001752 JMS OFDIV
677 3001757 JMS OPUTAC
700 4017 DWELL2 IDWELL IN SECONDS
701 3001754 JHS OGETAC
702 1235 PULSEP /PULSE SEPARATION
703 110024 MEMA (24
704 3001767 JNS GIFLOR
705 3001752 JMS QFDIV IPULSEP IN UNITS OF 20 MS
706 3001761. JMS @FIX
707 3113763 MEMAZ @FACM
710 550000 AMOA /ADDS A ONE IF

711
712
713
714
715
716
717
720
721
722
723
724
725
726
727
730
731
732

733
734
735
736
737
740
741
742
743
744
745
746
747
750
751
752
753
754
755
756

757
760
761
762
763
764

765
766
767
770
771

2425225
2110630
3001767
3001754
1237
3001752
2111136
3001767
3001751
3001761
3111763
2405117
110004
2001114
2405220
3111764
2011140
5042

2515220
2405217
2111822
2405234
2001100

4102
2110577
2404600
2110630
2405231
2111234
2405214
2111225
2405226
2405227
2165215
2165230
2111217
2001010

4104

4372
2001110
44375
2325230
2707226
757

4102
4372
2001110
44375
2515230

302

APOM TAU2
MEMA NPOINTS
JMS OIFLOR
JNS @GETAC
GUESS

JMS eFoIv
MEMA CONI
JMS GIFLOR
JMS @FMULT
JMS ele
MEMA @FACM
ACCM MULTI
MEMA <4

JMS MULTIPLY
ACCM IMTRVL
MEMA eFACML
ANDA MASKI
LLSH 2

IFACM IS ZERO

I T1.(10**6)I4V =CYCL?

/3000000

 

A+1’.MA INTRVL
ACCM TIME
MEMA PIHI
ACCM HITIME
JMS ZERMEM

IT1<10**6)I4U
IIN UNITS OF 16 MICROSEC'

NUTATE1PULSE1 INUTATION PULSE
MEMA LLIMIT
ACCM PNTR2 IRESET POIVTER TO X1

MEMA NPOINTS INEW SWEEP REQUIRES
ACCM NMPTSI IRESETTING POINT COUNTER
TRIPLETaMEMA RITIME

ACCM UAIT2

MEMA TAU2

ACCM WTAU21

ACCM WTAU22

ZERM COUNT2

ZERM YSUM

MEMA TIME

JMS TIMER

PULSE2 I 90 DEGREE PULSE

ADCl:STDG
JMS BLOB
RDGRED

M-AM YSUM
MMOMZ UTAU21
JMP ADCl

IRDG REDS

ISIGHALS ARE OF OPP. PHASE

ADC21PULSEI /
ADC7;STDG
JMS BLOB
RDGRED
A+MNA YSUM

180 DEGREE PULSE

IRDG REDS

772
773

774

775

776

777
1000
1001
1002
1003
1004
1005
1006

1007
1010
1011
1012
1013
1014
1015
1016
1017

1020
1021
1022
1023
1024

1025
1026
1027
1030
1031
1032
1033

1034
1035
1036
1037
1040
1041
1042
1043
1044
1045
1046
1047

2707227
766

4104

6454
162000
1
2124600
2001025
2707231
745
2707216
740
2001034

155

0
2405213
2165224
2135224
2333213
5144
1020
1013

2135215
2333214
5144
1001010
1012

0
3504600
2134600
2332601

5144
1000544
1001025

0
2025221
2110577
2404600
2111214
3001766
2111224
3001767
3001751
2111221
3001767
3001751

303

MMOMZ UTAU22
JMP ADC7
PULSE2 I 90 DEGREE PULSE
TTYRF

ZERZ

JMP START
MPOM PNTR2
JMS DTRANS
MMOMZ NMPTSl
JMP TRIPLET
MMOMZ SUDEC
JMP NUTATE
JMS XCALC

ISKIPPING XCK)

IMORE POINTS

I 150 RE SWEEPS

JMP DSPLY2
TIMER10

ACCM WAIT
T2:ZERM COUNT
T11MPOMA COUNT
M-AZA WAIT
EXCT A019

JMP TIMER2

JMP T1

TIMER21MPOMA COUNT2
M-AZA WAIT2

EXCT AC19

JMP OTIMER

JMP T2

DTRANSaE
A+MM @PNTRZ
MPOMA PNTR2
M-AZA ULIMIT
EXCT A019
JMP @ERRX3
JMP GDTRANS

INOTE THAT PNTR2 MUST BE

XCALCa0
ONEM PTNUM
MEMA LLIMIT
ACCM PNTR2
XCIaMEMA UAIT2
JMS GIFLOC
MEMA COUNT
JMS GIFLOR
JMS @FMULT
MEMA PTNUM
JMS OIFLOR
JMS @FMULT

ICALCULATES VALUES OF X

IRESETS POINTER FOR SECO
IPASS THROUGH DISPLAY SE

I TAU1

I TAU! . N

 

1050
1051
1052
1053
1054
1055
1056
1057
1060
1061
1062
1063
1064
1065
1066
1067
1070
1071
1072
1073
1074

1075
1076
1077
1100
1101
1102
1103
1104
1105
1106
1107

1110
1111
1112
1113

114
115
116
117
120
121
122
123
1124
1125
1126
1127
1130
1131

Dung-n—u—QQ-IQ-nhg—

3001755

1134
3001750
2111225

5003
3001767
3001750
3001755

1241
3001752
2111137
3505762
3001761
3111763
2001025
2134600
2135221
2330630

5104

1040
1001034

110012
2405100
162000
0
2110577
2405244
3165244
2135244
2322601
1103
1001100

0

1112
1113
1001110

0

4354
505320
0
2405222
4343
2405213
2025117
2103222
1143
5144
1143
2111117

2405222

304

JMS @GETAR
CORRECTION
JMS @FADD I
MEMA TAU2

LASH 3

JMS OIFLOR

JMS @FADD I
JMS @GETAR
DWELL

JMS GFDIV
MEMA CON2
A+MM @FACE
JMS @FIX
MEMA @FACM
JMS DTRANS
MPOMA PNTR2

TAU1 .N + B

TAU1 o

N + 13 + 8TAU2

10000
MULT BY 16

\\

ISKIPS Y(I)

MPOMA PTNUM IFOR X(I+1)
M-AA NPOINTS IDONE?
SKIP AC19

JMP XCI INO

JMP GXCALC

ZRMEM;MEMA (DSPLYl IFAKES JMS SO THAT
ACCM ZERMEM IZERMEM CAN BE ENTER
ZERZ INITH A JMP COMMAND
ZERMEM10 IZEROES DISPLY2 SECTION
MEMA LLIMIT

ACCM DATAP

ZMI:ZERM GDATAP

MPOMA DATAP

M-AZ ULIMIT

JMP 2M1

JMP @ZERMEM

BLOB:0
JMP BBI
BBIJJMP 882
BBZ:JMP QBLOB

IEIDDLES AROUND FOR 20 MS

MUCTIPLYag
TACNQ

MULT

MULT1:0
ACCM PIHI
TMQAC

ACCM WAIT
OMEN MULTI
1'11 :MEMZ P1111
JMP LONGTIME
EXCT AC19
JMP LONGTIME
MEMA IIULTI
ACCM PIHI

IPRODUCT .GT. 2**20
INAIT IS TOO LARGE
ILONGTIME MAKES THIS LA

INOU 1 IF ORIGIONALLY 0

 

 

 

1132
1133

1134
1135
1136
1137
1140
1141
1142

ILONGTIME HALVES

1143
1144
1145
1146
1147
1150
1151
1152
1153
1154
1155
1156
1157
1160
1161

1162
1163
1164
1165
1166
1167
1170
1171
1172
1173
1174
1175
1176
1177
1200
1201
1202

1203
1204
1205
1206
1207

2111213
1001114

2000
1740000
172044
10000
3000000
2000000
100033

405021
2405213
2111222

5110
1152
2111141
2505213
2111222

405021
2405222
2111117

5001
2405117
2111213

1124

2110577
2332600
5144

' 1
2130577
2405232
2110600
8405233
3111233
3405232
2125238
2185233
2135232
2332600
5144
155
1172

110740
11777
2424740
644
2110632

305

MEMA WAIT
JMP @MULTIPLY

CORRECTIONa2000
1740000
CON13172044
CON2110000
HASK1a3000000
CON3a2000000
LASTa100033

LOW WORD: DOUBLES HIGH WORD
LONGTIMEaRISH 1
ACCM WAIT

MEMA PIHI

SKIP ACO

JMP L1

MEMA CON3

A+MM WAIT
LIJMEMA PIHI
RISH 1

ACCM PIHI

MEMA MULTI

LASH 1

ACCM MULTl

MEMA WAIT

JMP M1

IUAITI2

IBIT 0 =1?
I NO

IYES: ADD IT TO WAIT

IP1HI/2

IMULTl o 2

EXPNTaMEMA LLIMIT
M-AZA PNTR2

XCT AC19

JMP START

MPOA LLIMIT
ACCM PNTRl

MEMA PNTR2
ACCM PNTR3
EX1;MEMA OPNTR3
ACCM GPNTRl
MPOM PNTRI

MPOM PNTR3
MPOMA PNTRl
M-AZA PNTR2
EXCT AC19

JMP DSPLY2

JMP EXI

I Y1

I U1

SETUP:MEMA (NUTATE ITHE ADDRESS

ANDA (1777 IPAGE RELATIV3
APOM NUTATE IJMP TO NEXT LOC
JMP BEGI

ADCNSTaMEMA LVLADJ

ITRANSFER BIT FOR LONGT

 

 

1210 3225763

1211
1212

1213
1214
1215
1216
1217
1220
1221
1222
1823
1224

1225
1226
1227
1230
1231
1232
1233
1234
1235
1236
1237
1240
1241
1242
1243
1244
#401
ESLN:
*135
135
*153
153
*1
START:
*155
USPLYZ:
*12
IEPLYIa
#625
625
#630
630
631
632
633
634
635
636

1001212
443

8888888888

8888888888888888

400

8888888

306

ANGM @FACM
JMP QBSLNQ
BSLN21443

WAITSO
WAIT210
COUNT2a0
SWDEC:0
TIHEJQ
INTRVLJ0
PTNUMJG
PIHIaG
PNTR10
COUNTJ0

TAU2a0
WTAU2110
WTAU2230
YSUMJQ
NMPT5130
PNTRIJG
PNTRaaG
HITIM530
PULSEPI0
0
6035510
0
DWELLJZ
0
IDWELLJO
DATApafl

STATUDa0

STAT2a0

ILIMITJ400

NPOINTSa0
NSNEEPSa0
LVLADJa0
T1ESTa0

0

T2E5Ta0

0

*577
577
600
601
602
603

*4017

4017
4020

*544

ERRX3:

*1750

1750
1751
1752
1753
1754
1755
1756
1757
1760
1761
1762
1763
1764
1765
1766
1767

2000
0
3774

100000
107777

2000

1000000

7245
7416
7461
6736
7062
7050
7026
7074
7534
7541
7572
7573
7574
4000
4021
4026

307

LLIMITa2000
PNTR2a0
ULIMITa3774
DATA1:100000
DATAN:107777

DWELL232000
1000000

FADDJ7245
FMULTJ7416
FDIVa7461
FLIP16736
GETACJ7062
GETARI7056
FACFAR37026
PUTACa7074
FLOATJ7534
FIXJ7541
FACE37572
FACMa7573
FACMLa7574
MASTER)4000
IFL063402I
1FLOR34026

 

308

*4036/ SECOND MOMENT CALCULATION FROM CW LINESHAPE

ITHIS PROGRAM IS TO BE READ OVER RELAX2. IT MODIFIES NI
IAND DESTROYS THE LEAST-SAUARES SUBROUTINE. THE L5 COMM
IIS REPLACED WITH AN M2 COMMAND.

/COMMANDS:
G - STARTS CALCULATION OF UNNORMALIZED SECOND MOMENT

I - STARTS CALCULATION OF NORMALIZATION CONSTANT
(WHICH IS THE AREA UNDER THE LINE)

0 - CONTINUES EITHER THE SECOND MOMENT OR NORMALIZ
CALCULATION.

0 - STOPS CALCULATION AND JUMPS TO DISPLAY MODE:
BRIGHTENED POINT WILL SHOW THE INTEGRATION LIM
G WAS THE PRIOR COMMAND

\‘\\~\~\\-\~\\.\‘\\.\

IALL COMMANDS MUST BE PRECEDED BY AN OCTAL NUMBER WHICH
ITHE NUMBER OF POINTS OVER WHICH THE CALCULATION IS TO B
IPERFORMED. FOR EXAMPLE: 10G MEANS CALCULATE THE SECOND
I FOR THE FIRST 10 POINTS: AND 1000 MEANS CONTINUE THIS
ICALCULATION FOR 100 MORE POINTS.

4036 44453 RDTTY

4037 2001043 STARTaJMS CRLF

4040 2000770 JMS OCTIN

4041 462303 LISTaA-MZ (303 I0

4042 162000 ZERZ

4043 123 JMP CONTINUE

4044 2405773 ACCM COMWD ISTORE END CHAR
4045 462307 A-MZ (307 I0

4046 162000 ZERZ

4047 66 JMP GO

4050 462311 A-MZ (311 II

4051 162000 ZERZ

4052 66 JMP GO IBUT JUST INTEGRATE
4053 462321 A-MZ (321 I0

4054 162000 ZERZ

4055 61 JMP DISPLY

4056 110277 ERRORaMEMA (277

4057 2001035 JMS TYPE I?

4060 37 JMP START

4061 2111774 DISPLYaMEMA COUNT

4062 3404125 ACCM EINTADJ ISETUP FOR DISPLAY WITH
4063 3404126 ACCM OINTV ICURSOR SHOWING INTEGRAT
4064 3404127 ACCM GDELAY ILIMIT

4065 1000133 JMP @DISP

 

4066
4067
4070
4071
4072
4073
4074
4075
4076
4077
4100
4101
4102
4103
4104
4105
4106
4107
4110
4111
4112
4113
4114
4115
4116
4117
4120
4121
4122

4123
4124

4125
4126
4127
4130
4131
4132
4133
*5770
5770
5771
5772
5773
S774

#6043
5043

*4770
4770

*5035
5035

2165771
2165772
3000130
3110131
2405770
2035774
2000021
3001761
3111770
2000026
3001745
3001752
5771
3001740
3001754
5771
2135770
2328132
162000
1000133
2111773
462307
2165774
2135774
2707027
74
2001043
3001747
37

2111774
74

623
624
627
135
154
153

12

88888

309

GOIZERM M2SUM
ZERM M2SUMA
JMS ESTATWD
@STAT2

MEMA
ACCM Y

ONEMA COUNT
INladMS IFLOC

JMS @FSOAR
MEMA GY
JMS IFLOR
JMS @FMULT
JMS @GETAR
M2SUM

JMS @FADD
JMS @PUTAC
M2SUM
MPOMA Y
M-AZ BLOCK
ZERZ

JMP @DISP
MEMA COMWD
A-MZ (307
ZERM COUNT

MPOMA COUNT

MMOMZ WORD
JMP 1N1
JMS CRLF
JMS OFLOP
JMP START

CONTINUEaMEMA

JMP INl

INTADJ3623
INTV3624
DELAYa627

STATwD.135

STAT23154
BLOCK1153
DISPJIQ

Y10
MQSUMIG
MQSUMAJB
COMWDJZ
COUNT:0
CRLFa0
OCTINag

TYPEa0

ISTARTING ADDRESS
IFREO FROM CENTER

 

I.GT. DISPLAYED SIZE?
IYES

IG?
INO:MUST BE I

IANY MORE THIS INCREMENT?
IYES

IM2SUM STILL IN FAC
IBACK FOR MORE

COUNT

ITHE COMMAND CHAR(BUT NOT

310

#4021

4021 0 IFLOC:0
#5021

5021 41 JMP LIST
#5761

5761 6352 FSOAR.6352
#5745

5745 7416 FMULT17416
#5751

5751 7062 GETAC.7062

5752 7050 GETAR170S0

5753 7026 FACFAR.7026

5754 7074 PUTAC:7074
#5740

5740 7245 FADD.7245
#5747

5747 6510 FLOP16510
#5503

5503 4036 4036 IDESTINATION FOR M2 COM
#5441

5441 315262 ’315262 IM2 COMMAND FOR SERVO
#4026

4026 0 IFLOR.0
#5027

5027 0 WORDa0

 

311

*5653/ RAMAN CORRELATION FUNCTION DIVISION

ITHIS PROGRAM DIVIDES THE DEPOLARIZED FOURIER INVERTED S.
IWITH THE POLARIZED: POINT BY POINT. IT IS TO BE READ IN
IRELAX2; WHICH IS NOT MODIFIED BY THIS PATCH. ISO IN THE

ISTARTING ADDRESS OF THE POL CORRELATION FUNCTION:

ANIS

ISTARTING POINT OF THE DEPOLARIZED CORRELATION FUNCTION:
IAND THE RESULT STARTS AT RESULT.

ITHE PROGRAM IS STARTED THROUTH NICOBUG (S.A.
IAFTER EXECUTION IT RETURNS TO NICOBUG.

4700) AS:
RELAX2 MAY THEN

ISTARTED (G) TO CALCULATE THE LOGARITHM OF THE RESULTING
IROTATIONAL CORRELATION FUNCTION.
5653 2111704 MEMA ISO
5654 2405702 ACCM P2
5655 2111705 MEMA ANIS
5656 2405701 ACCM P1
5657 2111706 MEMA RESULT
5660 2405703 ACCM P3
5661 3111701 DlaMEMA 0P1 IC(ANIS)
5662 2000021 JMS IFLOC
5663 3111702 MEMA 6P2 IC(ISO)
5664 2000026 JMS IFLOR
5665 3001711 JMS OFDIV
5666 2111710 MEMA CONI ISCALING FACTOR
5667 3505712 A+MM GFACE
5670 3001714 JMS @FIX
5671 3111713 MEMA OFACM
5678 3405703 ACCM 0P3 IC(ROT)
5673 2125701 MPOM P1
5674 2125703 MPOM P3
5675 2135702 MPOMA P2
5676 2323707 M-AZ END IDONE?
5677 1661 JMP D1
5700 700 JMP NBUG IYES
5701 0 P1:0
5702 0 P230
5703 0 P330
5704 100000 150:100000
5705 110000 ANIS:110000
5706 112000 RESULTa112000
5707 111777 END:111777
5710 30000 CON1a30000
5711 7461 FDIVo7461
5712 7572 FACE17572
5713 7573 FACMa7573
5714 7541 FIX.7541
#4021
4081 0 IFLOC:0

#4026

 

 

 

312

4026 0 IFLOR.0
“1700
NBUG:

 

 

APPENDIX C

 

APPENDIX C

RMANFIT - Raman Lineshape Fitting Program

RMANFIT is a program for determining the orientational broadening
of Raman vibrational bands which uses the procedure suggested by
Bartoli and Litowitz3O (their method A) of convolving the polarized
component of the line with a Lorentzian and fitting the result to
the depolarized component. The program as written converges strongly
to a solution even though the minimization routine is very primitive.
A number of improvements could be made especially for the purpose of
reducing the program size (125K) and running time OMS N-H of CM time
per spectrum). Some features of the program operation will be dis-

cussed and then possible improvements will be suggested.

1. Operation

 

All of the adjustments which one might wish to make to the data

are done through subroutines SELECT and FUDGE. Therefore one may avoid

reading in a long deck by placing the main program on a permanent
file. The procedure for running the program is then to read in SELECT
and FUDGE (and the one data card, to be discussed) with the following
control cards:

FTN(B=X)

ATTACH(TAPEZ,“the raman data PF name")

ATTACH(RMANFIT, FITPARA,PLOTll,CALNDER,INVRl)

LOAD(RMANFIT,X,FITPARA,PLOTll,CALNDER.INVRl)

EXECUTE(RMANFIT)

This presumes that the data have previously been placed on permanent

313

 

V .2...“

 

314

file. The only data whichare read in with SELECT and FUDGE are on a single
card giving the experimental polarization leakage (RHO) and the relative
efficiency of the detector for different polarizations (POLEFF).

These parameters should be determined from the relative intensities

of the various lines of CCl4 as discussed in the Experimental section.

1 then the

1

If the linewidths are greatly different from m5 cm-

initial guess of the orientational width (XHDWTH) of 2.0 cm' , and

the step size for the minimization routine (XMSTEP) of 0.5 cm'] may
be changed in SELECT by inserting a COMMON/SEARCH/XHNDTH,XMSTEP card
and then redefining these parameters in the subroutine. Also, cur-
rently all l023 point spectra are reduced to Sll points by subroutine
TRIM, with the parameter ICHOP entering as

I = Io]d(w + ICHOP).

new(w)
Consequently ICHOP determines which Sll points of the old spectrum
are saved. If a full 1023 point spectrum is to be retained, the call
card for TRIM must be deleted.

Subroutine FITPARA is used twice; both to fit the baseline and
to fit the center of the data to find the center frequency. The co-
efficients for both these least-squares fits are printed (the equation
used is F(X) = C(l)*x + C(Z) + C(3)*X*X) but the plot is suppressed
by setting IPLOT = 0 before calling FITPARA). Little is gained by
viewing these plots, and as long as both remain suppressed the sub-
routine PLOT13 need not be attached. The baseline is fit to a parabola
rather than a straight line in order to handle the case where the line

of interest is on the shoulder of another peak. The "goodness of fit"

 

 

 

315

for lines which do not require this feature is, however, slightly
worse.

The messages "UNSAT EXTERNAL PLOTl3" and "UNSAT EXTERNAL HEADING"
normally appear in the dayfile.

The correction Ipol(w) = Ipol(w) - 4Idepol(w)/3 is only made
when the depolarization ratio p < O.l. It may be desired to make this
correction regardless of the size of p, in which case the appropriate

statement must be removed from subroutine INTNCOR.

2. Suggested Improvements

The greatest problem with this program is the cost per spectrum.
The only lengthy calculation is the convolution, which is recalculated
for each iterative step and requires (6n)2 operations for a spectrum
of n points. Currently every fifth point in the spectrum is calculated
(ISKIP = 5 in MINN - but note that PLOTll also has an ISKIP). Clearly
a larger value of ISKIP would greatly decrease the running time, how-
ever, no other values were tried (except 1). Note that the procedure
for making the baseline correction in CONVOLV will only work for
values of ISKIP which are integer divisors of SlO.

Three other improvements were considered but not deemed necessary
for the present. The first was improving (or replacing) the minimiza-
tion routine since the present version will converge very slowly if
the initial guess is off by more than a few wavenumbers. The second
improvement considered was to weight the points in determining the
residual in order that the points on the baseline not be so important.

Finally, depolarization ratios are presently calculated from band

 

316

center intensities, which is not strictly correct. They are properly
calculated from areas, and when the widths are greatly different the
two methods of calculation will clearly disagree. The correct calcula-

tion could quite readily be added to subroutine INTNCOR.

 

 

J7

)0

PS

30

’40

as

50

80

A9

70

.‘3‘55‘1503‘5333‘3'33‘1‘5‘3 Dflfifixfififififinfifi‘fifififfi

3l7

PunnnAu RMANFIT1INPUT.TAPE2.DWT°UT.TADE11
Ranlqu sURRDHTINEs vanl. PLDTII. AND FTTCRV? wTTH THF FTTTING
FDnATInq HRITTFN As F(X) = C11) . (121-x 9 C(31.!DX(HPTTTFN av v.6.
WRIIF“r CHFM DFPI..M<D). THF LDCAI FITE NAMES SHOUTD RF INVRloPLOTIIo
AMW FITPARA. THIs DDDGQAM DETFDMTNFs THE DHIENTATIDMAI cDanNENT or
RAMAN llMFs USING THE CDMVDLUTIDN TECHHIDHE SUGDFSTFD RY RADTOLT AND
LlTnVlT71J.CHFM. PHY<.56.404(1°721). IT ASSUMES THAT THF'soECTpA HAVE
HEVN Pnururn TM VARTAN CAT FORMATISIX DCTAL It's/400D. TUELvE woqu/
CADD. STARTING CnLUMN 0). AND CAN RF ATTACHED Peon DFRMANFNT FILF uTTH
THr LOCAL FILF NAME T4962. THE INPUT DATA SHDuLn CONSIST OF THE
PDLAPI7FD AND THEN THF DEDDLAPTZED CDMODNEMTS or A SINGLE LINE. thH
THF CENTER FRFDUENCV OF THE LINE ROUGHLY CENTERED TM EACH SDECTDHM.
THF srnanuPF nF THF DATA nFCK sHouLD DE As FDLLDds

NHMDEQ DF DRDHDQ TD RE ANALY7ED

NJMDFP OF QPFCTQA/GODUP. NUMBER OF DOINTQISPFCTDHM

TITLE CARD Tn SPECTRUM 1

TITIF CARD Tn SPFCTQUM 2

PMT CDDNTs/sfic Foo soECTnA 1 AND 7

Nnuncp OF st692 Pop SPFCTPA 1 AND 2

A HF'K 0F Hh cAons or DATA

A SECOND HFCK 0F Rh PADDQ 0F DATAIONLY TF 1023 POINTS/SDFCTPHHT

wrorAT THIS SFDHFNCE sraoTlNfi HTTH CARD 2 As MANY TTMEs As MECEsson.
THF PHDGDAM DFTFDMINFS THE CENTFD FREQUENCY AND TdFM vAoTEs THF HIDTH
0F THF IDQFNT7TAM ORIENTATIONAL 5°ECTDUW TO OBTAIN THF REST FIT
HFTUTFN THF PDIA017FD AND DEPOLA°I7ED sDECTDA. 746 POINTS ADE MDT
UFTGHTED. THF PFSHLT(MTNIMUH NAIF-WIDTH) IS IN 74F RAMP UNIT5 45 THF
09,612“. qpccronu. Most 20195 qptxss HILL er ADTDMATTCALLV DELETED.
A 17-nnxnr sMooTH Is DDME Tn THE DEPOLA°I7ED SDECTQDM wITH THF MFTHOD
nr qAVtt/xv AND GOLAY1ANAL. CHEM. 16.16271196411. AN DUTPHT FILF
cnuqlsilun 0F THF 001.. DEPDL.. AND CDNVDLUTION Ts HDITTFN DN TA'F1
FOO ”lnTTING wlTH THF TFKTDONIX GRAPHICS TERMINAL. THF CURRENT anuhT
Is COMPATIRLF wITH THE sHHRDUTIME LDLOTIT(L.A.DAC4LA.CHEM.DEPT..WSH).
DIMFMQTON x1(1073).Y110231.C(31.UT110231.IHEA)?181.0131.CF1101
COMMON/COHV/016.10211oERPOPTAT.SCALC(81
CHMHONISDFCTI516910231
CJMMDN/VIV(J.IO?41
CJMunN/SCDAD/TGDNETTDZTT
CO””OMIA“PISCALF(R1.ICF181oQHOoPOLFFFoGAIN151
CUMMnN/OATAITDATA(IO?31
CunnoN/SFADcH/XHHDTH.XMSTEP
COMMON/TIT:E/ITTTLF(RI.MDATF(TT
Cnuunn/PLnT/LFNGTH.18090699104660ISKIP.IOVEQ.THFADanPMOFN””oA”G"
CUMMDN/rIT/MTHP.XRFG.XENO
Cowman/SYMRDL/IQYMDDL(3)
DATA 151M401I120.19!.IRCIolFNGTHISIoIBOPOE912/oTDAGF/1/.TSKT'IO/ol
OOVFR/O/oIHFAn/OIoFQPMI5.1I
DATA dT/102301.0/
TOD FDDMAT (RX.1?061
101 FORMAT 1/110X914106.?Xl)1
In? FopuAT¢//.a AFTER NDDMALTZATTON AT0.144..CENTER FQFDHENCY= 9.14.2!
Io‘THF SCALING FACTflQ= 29611.71
191 FHDVGT (64101
104 FDDwAT(13)
105 FDPM4T113.6X.I41
10A FORMAT1F6.0.1K.F6.DT
[04 rqqu1(/.1qx,oonnaoau RMANFIT READING 2012.. DFPKS WITH “0140” 90‘
1NTs DFQ SOFCTDUM91
Inn FDQHATT/oPX.°THF TITLE CARD FOR RDFCTQUM NHunEn 0.12.. DFAnc : 9.4
14101
110 FORMAT113.6X.131
1|] FDQqAr(//.o THF FOLIDHIMG DDTMTS UFDE DELETED As NDTGF SPIKE<°./o(
1105.1011K011111
I]? FORMAT(///.3x,otnour PARAMETFQS FOP THE NEXT ”ATP 0F SPFCTRA ARE“.
l/lox.oquCTRnM I°.<x.611.5.o DHT COUNTS/$609.11.! sCAMS'-/.IDX.*
7QDFCTDHM ?§.§Xgfilloqo° pMT CQHNTS/REC.OI3O. SCANRHI
CALL CALNOEPTNOATFT
xuwuiuzzo, C XMQTFD=5.
Winn12.1041 NGWDS
an [HHHHGH THIQ TODD UNTIL AlL DATA RFAD
Du ?D KGN°=1.NGD°§
RIAH (701051 NSPfinpcflpnlflTQ
NPFCK§=NGDGRD
TF1unD1NTq.En.§|11 MDFCK<=(u<°G°°oli/?
pulmt 104. NDECK<.honINT<

 

N0

Jfl

‘0

T9

318

c fin THnnnnH Tqu LnnbnNrr FOR EACH DECK or AA CADDS. EACH DFCK HILL
C CONTAIN EITHFD 1 DR 9 QDFCTDA DFDFNDTNG DN hHEVHEQ THE QPFCTRA

c CONTAIN at: no no); POINTS.

la

"n )0 KK:|oNnFCK‘
[f(VKvo') '10l‘
I‘(UPHINT<.F0.IOP3I GD TD 1%

1 HEAD(?.ID1)ITITLF

”FAG 1).!01) lHFAD?

PEAD(?oIOAT GAINIoGAlN?

PFADtfiolID) NGHFFDI.NEMFFDZ

PRINT lleo MAIN!oNQwEFDIoGAIN2oN<dFF°?
GAINu=nAIN1/<DRT¢FIDAT(N<HEFDITI‘GAIN?=GAIN?/§DDT(FTDATTNGNEFP?)T

IH PFAH¢2~100) ([DATA(K)9K=IOIO?3)
C THIS lnn° PEADD <1! ”D'MTS AT A TIME. EACH SPECTQHH IN A GDOHD HA: A
C UNIOHK VALUE or J.

00 ll JJ=|o2

J=Jjo79xx-2

NPD; =MPDIMTQ

IF¢N°P§.EO.1021) J=KK

IF(HDD§.ED.IDPW.A.JJ.ED.2) an TD 1|

IGVIO=D ‘ ICHDD=?§A % CALL QFLFCT(JonfipoolfiklP-ICHDDT
IF(I<K196EO.I) 60 TO 8

GAIM(JT=GAINI

PPINT IOQ. JoITITL‘

c 99999999999» pfiuovfi NOIQE pFAKq 99999999999999999

I

2

DLDDNTsIDATAtl) s nLDQLP=.nS s IDELFTE=0

no ? K=loHDp§

KNEHDNT=FLDAT(IDATA(K)) T XNEH<L9=AD§(ANEVDNT-DLDONT)
lthNFHSlD.ED.D) XNENSLD=DLD§LP

FUQEOLQSL°“GOOQ

IFTXNFHSL°.DT.FD<) GD TD 1

GLDQLD=XHFU§L° $ OLDPNT=XNFHDNT % 60 In 2
‘NFH<I°=DID9LP T XNEHPNT=DLD°NT * [DATA(K)=IFIX(OLODNT)
TUELETE=IDELFTEoT % [GONECIDELFTE)=K

CHNTINHE

IFTTDELFTF.HF.0) DDTNT Ill. (IGONETK).K=loIDFLETET
[F(TDELETF.DT.09) an To ll

lF(J.NE-l) CALL GHDDTHCNDPST

TFTNDDS.ED.1023) CALL TDTM(J.NDD§.ICHDPT

C ooqnoonu 'IAKE “AX VALHF Aq INITIAL GUESS you unauy 999999999999

I?

MAx=|DATA(1) A unPNT=l
DH 1) K=?oHPD§
IF(InATA(K).DE.MAX) an TD 1?
HA!=|DATA(K) 5 MDDMT=K
CONTINUE

C 9999(«9999999999rALCuLATfi HA§EL[M5999999999999

1

h

L=0 s THP=Npoq-qo

DD 1 Kzloqn

L=l°l

xl(L)=K-HDDNT

Y(L)=FLOATCIDATA(K))

00 h K:1uu.~ppq

L=LOI

Xl([)=K-HHDNY

Y‘L)=FL0AT(IDATA(KT)

NTHuzloo x XHEG=-2€.H s KEND=?R.6 x Npalon
HF=T T IoLnTso « IPDINT=0

CAI! FlTPADAtIDIDToNE-NDoXI.HT.V.C.IDDTNT)

c 9poq9¢9¢aooooaaunETFDMTMF CENTFD FREDHFNCYQ9999999999999

I;

IdHTH=70

M=0 ‘ M|D=MDDnTolunTH$ MHI=MDDNT0IHDTH
n“ H K=MLnoMHI

M=Mo|

YW) =W.0ATHHATA¢KH

XI(H)=K

TPLDT=I s NF=3 : NP=P0IHDTH01

NTHp=P¢NP $ XHFG=F|DAT(HLD) s XEND=FL0ATTHHTT
IDLDT=D

CAIL FITPADATIPLDT.NF.ND.XI.HT.V.D.IDDTNT)

r 990999995999999HAK¢ RAQFLjnr cnppEchnM9999999999999

A

CF(I)=-D(1)/(?¢n(1\) s lcchT=TFTX(cr(J))
FNHH=D(1)/(RD.'D(3)) A AHGM=.I

CALL FHDGFTFoJoKGDD)

DO 5 K:|.Nppq

K2=K-TCF(JT-] sxxsrlnATcxz)
§(JoK)=FLOAT(IDATA(K)D-C(IIOIK-Ct?l-CT1)OXK'X<

 

15"

Is;

17A

175

CUGDDUTINE

315)

c 9A9999099o9999999noougt[7e99999999999A99999999

CALL NORMIQoJoNDDSoIPf(UIASCALEtJII
PRINT l02o [CFIJIoQCALFIJI
ISCALF(J)§§CAL§(JI'GAIN(J) a-
C GET 2 <CECTDA aErnDE EMTréING IM'VCWD Ava uIHA.
IEIJ.ED.II GD TD A
CALL INTNCDDIJ.HDD<)
L=l S M:9
C Loan ?/1 DE v MATQIX rno DIDTIA.
DO 7 K=IOVDP<
V‘Lo'(’=S(LQK,
7 Vl“.<)=S(J.K)
CALL MIHN(JAHP°<I
9 [F(MPOINT§.FD.ID?1I G“ T“ 1*
C HEAn «Fann SppCTnuH Ian FIDST GII Dntvrs DF DATA.
DO 9 K=leIl
9 IDATAIKI=IDATAIxoSIz)
IA 00 ID K310“
In ITITLFIKI=IHFADPI<I
GAIVISGAIN?
II CONTINUE
20 CONTINUE
END

TD!”

QHQPDHTINF T°I“"‘”°°5'IPHOD)

C CP‘IFDTQ A 10?? 901”? QDECTDvH Tn A <1] Dn'ur qprcrcuu. THI§ [C an”?!
C NF HVLECE ICHDD IQ EFT TD D IN CMQDDHTINF CrIECT,

10

:JQonlJTl'IE

’0

P5

C)uunn/AHo/granqa,,TCFIAI
CDMMDH/OATA/IDATAIIA71)
DJ 1 K=lvqll

I IJATAIKI=IDATAIK0ICHDDI
N°°§=fill
RETURN
END

SAVEDLT

SUPDDUTINE EAVFDITIJIcuopfiouI

r sAu§§ THF DATA rnn THE TFKTDJNIY DLDTTED

“7“‘NSIFHJ| AHELI?)
HIMEAEIOM X(IO?AI
C0”JnN/V/Vl‘019?4)
COMMON/TITLEIITITLEIAI.NnATrta)
CnanN/AHo/CCALFIHI.ICFIDI.OHDoDDerE.nAINIHI
DATA LADEL/«HEDFD.6HIMTFNS/
DATA VAR/1HOHD/

loo FDHNAT (IG.(FII.«II

101 FDDHAT (33I0.I§I

10) FORMAT ((A4)!

10? FDHMAT (IEII.4))

10a FUDMAT (ZVQAIEII.6I)
TITLE=TTITLEIII $ K=l
DO 23 KK‘IQNDD?

PD XIKKI=ELDATIKKI
00 In KK=19JL
NQITE(?oIOO) NPDQoIKIJI-J31oMDDQI.(V(KK.J).J=loNDp<I
WVITE(1vl0l) LARELIIIoLAREI(2)-TITLFoK
QEALDPQ=SCALFIHa/9CALEIII
UPITF(3A10?I V49

10 WRITEI1AIO3) °EALDDP
RETURN
END

 

.320

¢U3°OUTINE N00“

SUQPfiUTINF NODV'FoNoHpPQQIC‘oQCALFI

)0

20

7g

10

15

an

49

90

<UQQOUTINE

000005130555

DIMEASIDN FI6.IO?1)
Mln=Icr-P A wnI=Icro2

SCALE=0.
DO I <=HLq.NH1

I SCAIE=<CALr.r(w.
SCALE =SC6LE/S.
“U P K=I.NDDQ

V)

1 FIM.K)=F(“AK)/SCALF

DETHDN
END

“ l 'a’l

QWRDDHTIHE'UYVNI"AHD°S)

THIG gnqpnuTTH: Tar:

Q A QTED IN HDTH DTOECTIOM< aannur

vagufi nr rowTuT Ann PALPHLATF§ THE N?» CDOTNTA HHICV 1‘
.HTCH “IVFfis THF LEAQT ropno flFYHFFH THE cnmuu'uTInu AID Tut DEDAIACI
VAIHF 1g oryroufivfin Fpnv THE DADA-«IA DITATPV“

ZED EDECTDH“. THIQ
EDD“ SHRJnnTIHF DADA

90L.

Tu: PI’QOFAIY
THAT VAIIF

THE <TE° S175 IQ DEDHCED RY DNE-HALE FOO EACH SUQCFOHCNT CYCLE AND

THE ITEQATIOH STnog

wHEN THE 599“? CHANGE§ qY LE§S THAN

.01.

THE INITIAL FSTI"FTF 0F THE HALr-NIDTH Ann THF QTFD :T7r no; 1» THF

LAQELLFD Cnuunu QEAQ
AND XNQTFD. R‘EPFCTI
COVHDN/V/VIBoIO?
COMMON/CONVIC(6.
("MHDN/AM3/SCALF

CH. ACCF<IPLE IN CHORDHTINE <ELECT.
VEIV. DEFAULT VALUES ADF 7n (7,n c
a)

IDP1IAEPHODIADoSCALClRI
IDI.TCEIH)oOHD.°DLErF

COVW‘N/SEADCH/XHHDTH.XMQTED

DIMENEIDN F(I).x

CPDTHT=XHNDTH s
Q 00 I K=191

XSTED=(K-2)¢<TED

1(QDQUTC1)
qTFP=xMSTEa % HTIMEq=l A I<KI

CALL CONVOLV(KANDOQACPOTNTATQTEOOICF(N)AMATQKIDI

dT(KI=I.

| X1(K)=CPOINTOXST
CALL PARAHDLIXI.
XIIAI8-92/(2091I
IFDHO=A s xDIr

FD
FQDODOPIAPZ.D3)

F=xlI¢I-CPDINT

AQ IHNHTH

I:5 £ VOQC:O

CALI CDNVDLVIIEDH9.NoPs.CPDINT.XDIFF.ICFIHI.H.ISKIDI
KFLAG=0. s EQRHIN=EDPDPIAI

00 P K8193

IF(EPPOR(K)-ERRHIN.LFAOI KFLAG=K
[F(EPPOR(KI-FRQMIN.LFo0) EPPMIN=FRDOQ(K)
STEDDIF8A85(XDIFF)-.Ol

IF (QTEPDIF.LT.0) KFLAG:4

7 CONTINUE

PRINT 1009 NTlflfifioXIléfioERRnpih)
IF(KFLAG.NF.OI Gfl TO 3

A CODINT=XIIAI s
1 IFIHDRE.ED.I.D.K

sTro=STEp/2. s NTINE§=NTINFSOI $
FLAD.E0.4I GD To 7

GO TO 9

M0963! S CPOINT=XIIKFLAGT S QTEP=§TE°/? $ NTINE<=NTTMF§OI $60 T“ q
7 KALL=1 S XDIFF=X1(KFLAG)-C001NT $ JL83

C‘LL CDNVOLV(KFl
DO 6 KSIONPPQ
5 V(JL9K)3C(KFLAGA

AGgMOPsofipnTNToXOTFFQICF(N)AVQKALL)

K)

PRINT 1010 NTIME§9Xl(KFLAG)0FQP0°(KFLAG’ .
103 FORMAT(/(ADXA'CYCLEOoI3Q' 0F SURPOHTINF MINN'o/IASOXQ'THE “TNIHHN

‘HALF-HIDTHI .061

1.79/060X997HE CHI-QQUADFD FD?OO:

.061197),

IO] FORMATIIICAOKA'AFTEP “0130. CYCLEQ THE VALHEQ ADF.o//AHOXAGRF§T HA
[LP-HIDTHS‘oGI3.79/0u0Xv.TH€ CHI SOHARED E99098 '06!1.7)I
CALL pLOTll¢JLoN99§I

CALL §AVEDLT(J|9
PETHQN
Fun

NDPfiofl)

cunpnutlmfi

IO

IS

?0

29

10

39

60

AS

go

«as

§URQOUTIN€

13

1Q

'Wfinf55M1C30

321

IKTNCOQ

SHJUDUTINE INTNCDDCH.NPDSI

CDQDECTC SDECTOA ACCTADIHG TD FDDHHLA

I(DEDDL.ACTDAII=I(DEDDL.EAPIIODLEFF-1(9DL.EXDTaoun

AL<n CDQDECTS THE PDLA°I7ED cornvanw Ir THF DEODLA°I7ATIDN DAT!“ 1‘
GQFATEQ THAN .1 ACCDUDING *0

a.nc~

10

1001

II
1001

IOO?

[(90L) 3 IfPOLD-h/1.T(DFDOLI

COMufiH/SC9A9/°E°(10?1)

COMHflH/SPECT/9(Aoln?1)
CONHDH/AMD/SCALEIRI.ICEIAI.OHD.DDLEEE.DAINIBI
RAUDDD:SCALEIH)/<CALE(II

SCALEIMI=<CALE(MIODDLEEF

IDFFSFT=ICF(I)-ICE(HI

IQHTETzNDoq-TARCIIDEESET)

IF(IDFESET.ED.DI GD To A

IFIIOFFSET.LT.0)GO TD ?

00 I KaloIEHIFT
PF°IK0IOFE<ETI=<IM.KI-DHD9§(IoKOIDEES‘TIO(§CA_E(II/QCALFINII
DD 7 KaloIDEFSET

SIM.KI=RED(KoIDEE§FTI

00 R KsqugHIFT

SIHoKOIOFFEET)=RE°IKOIOFESETI

60 TD 6

00 3 KSIOIQHIFT
S'“vKI=SIWoK-IOFFSETI-DHOEG(IcKI9IRCALFIIIIQCALVIHII
IUD=ISHIEToIDEE§ETol s on 0 K=IHD.I<HIET
SIN.K-IOFE§ET)=§INoKI

60 TD 6

DO 9 KilvNPPS
SINqKISSIHoKI-DHD'SIchI'ISCALEIII/SCALEINII
ICEIHI=ICEIII

CALL NDRMIqoflquoficlfiFIMIoSCDQ)
SCALEINI=QCAIEINIO<CDR

REALOPPBSCALFIMIISCALEIII

PRINT I00]. QHnopoLEEFoGAININ)oIOEESET.DApro.<CALEIN)oDEALDDP
SLD§5=(I.-§CDRIDIOO. S LO<§=IEIXI§L0§§I
IFIPEALDPP.LT..II GD T0 11

00 IO K=loNpD§

SIloKI=S(l.KI-QIMoKI91.31330DEALDDP

CALL NOPM(<QIONDP§DICF(I)DAAI

PRINT 1001

FOPNATI/lloG THF DnlAPI’ED LINE HAS CDDRECT‘D RY IIDOLI=IIDOLI-4/

I?I(DFPOLI*I

pQINT 100?. l0§§
FOQNATI/v“ AFTER IMTFNSITY COQPECTION HITH‘oTAOAOPOLAIT7ATIOM LEAK

IAGE: 'oE6.A./.TAI.¢DETECTOR EFFICIENCY: '9F§.3o/oTb2.'PU TURF cDHN
?T$/<EC= *.DII.9o/oTAA.O CENTED DEECET= O.I1./.T1A.-cnunr DFDDLADI7
3ATIDN QATID= *~E7.R.IOX.GTHE SCALING EACTDD= I.C13.7./.TTA.ATHr nr
APDLADI7ATIDN RATIO: 0.E7.S)

EOPMAT(///.¢ THIQ CDDHECTIDM ACCDHNTED EDHO.I1.¢ PFD CENT DE THE I

IDTAL LINE INTFNQITY“)

PETHDN
END

CONVOLV

In

SURDBUTXNE cnuvntVIH.MDD<.XHHUTH.IC4HGFoICFoL-ICEIOI
Dlusxstfin MDOIPAQOI

CJMHDN/EPFCT/<(fiolfi?1)
Couuqm/Cnuv/r(5.]0?1)ofiqnfiw(h)~§C8|Cl“)
HDszlCFQQDIQQ

JJ=I

DD 9 szcprY

J=KOICF§2‘ ‘J=FLOAT‘J‘-c‘do
wnocx)=1./(1,o(rI0!jI/(YHHDTHAXCHNGE)O°?I

DC In K:1¢HDPS

C(MOK)=0.

D0 1 K=loNPP§oIQVI°

DD 1 N=IoNDP<.I<KI°

KVK:wo7oICE-NoSD

C(M.I)zC‘M.K)OwnD(KVK,°§IJJ9H’

NucLTNF:5D[§y[p c Tup=NDDC-ADI<KID A HLCD3D:D.

PD

7%

30

3S

SUQRDUTIHE

)9

?D

CUPDDJTIWE

‘9

PD

‘\

322

00 3 KSIoNRSLINEoIfixTD
DLCDDDaHLchpoCIN.K)

Do A KSIHOoNPPSoISKTO
RLCDDR=8LCDPROCIMQKI
QLCDnnzHLCDRo/lo,

no a K810NDP§¢I§KID
CIN.K)=CIH.KI-DLCDDD

DO 7 K=lolSKIV

ICENT=ICF¢K~I T J=TCENT~IICENTII§KI°I'ISKIp
IF(J.EQ.1I IDCENT=ICENT

CONTINUE

IF(I<KIP.F0.1I GD Tn 9
ICENTL=IRCFNT-? S ICENTH=I9CENT97
00 H K=ICFNTLoICENTH
C(“DK,=C(HDIRCENT)

GO TD 13

IPCENT=ICF A IEDunza $ JKL=1
CALI NORMICoMoNDPSAIQCENTQSCALC‘M)I
ERQDPIHI=0.

DO 6 K=I.Nop<.I§KIP
EHQDDIHI=EQDDRIHIotscL.KI-CTH.KTIAISIL.KI-C¢H.Kiv
RETHFN

END

SNDDTH

p

SURDDHTINE QuonTHINoos)

CDVHDN/UATA/IDATA(ID73I

DI“EN§IDN NPIITI

uopggnpps-Ih

D0 1 «=1.15

NDIKo1I=IDATAIKI

90 P J=1,\‘ppq T. nn 3 K=1016

NDIKI=HPIK01)

ND(\7)=IDATA(JOIDI
NSHM:AI*H9(QIo4?°(N0(8I+NDIIDI)o1QAINpITIoMDIII)1.159(M3I5IoMDII2I

1,.279‘Np(g).qp(11\‘.139(up(a)9MD(|a))979(MDIWI0NDIIQII-6¢(N°l2I’”°
7(16)I-?l'(HDll\0N°(I7)I

IDATAIJI=NSUM/323
CONTINUE

L:Noo§ $ DD 4 K=1.uop<
IDATAILoHI=IDATAILI
L=L-1

00 9 K319“

J=N°°SOBOK-l 5 L=NDP§¢K-I
[DATAIJI=IDATA(LI
NPP§2N9P5016

DFTHQN

END

PADABOL

SAIHD’IUTINE DADQROLIXDVDADBDCI
“IVFWSIDN XII).V(1I.VIII.EDDD9I1I.E3:<Dt3)
DPIHT IOU

Kl=rIlI S ¥2=xI?\ b X3=XI3I
V1=VI1I ¢ v7=VI2I A v3=v¢1)
IMI 2 (‘l-K?I°(V?-V1I D TH? = 'X?-IAI¢IvI-V?I
ALOHA : -TH1.Tu7

DETA = TMI“(XIOA)I-TM2°(X2*X1I

D 2 -le-X?I°(!2-x1)°(x3-X1I
IEIALDHA.ED.D.DI ALOHA = I.DE-?0
[FID.EO.0.'H I) '3 l.D‘T-?0

A = ALOHA/D

R “ETA/D

C Vl-A0‘14XIvQ°XI

"QINT ‘0'. Aoqor

CHI<C=00

pPINT I02

00 l K=lo3
YIKIsAOXIKIGIIKIARDXIK)oc
EDQDHIKIszKI-VIKI
EQD§DIKI=FQDDD(KIOEDQOPIKI

I PRINT 1039 ‘(KIAVIVIAYIKIAFDDOQIKI.EDDQDIII

 

?:

10

323

Ion FDDHAT(IHocT?AoOD°TIHIZING THE DQIENTATIDNAL 4ALF-UIDTH HY THE DAD
I‘ROLIC METHODOI

lOI FORMATI///.20x.9CDFEFIcIENT9¢.//?ox.0A s 9.6I3.7./.znx.-R . .0611.
17./.20x.-c a 9.611.?)

10? EOPHAT(///.10x.°9E§HLTso./IoISXo-VARIADLEo.Taco-EXDEPIH. VALHEGoTa
150'THEDP. VALUE-cT60.0DIEEE9ENCE-.TTS.ODIEF. SOHARFn9)

IOI EDRHATIlax.GI3.7.T32.GIJ.7oTaA.DI3.7oT50o611.7.T75.GI1.7)
PETUPN
END