STUDIES OF NUCLEAR MAGNETIC RELAXATION
AND MOLECULAR MOTIONS OF SOME SMALL

WLECULES IN DENSE PHASES

By

Shaw-Guang Huang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment Of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1977

Pulsed
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ABSTRACT

STUDIES OF NUCLEAR MAGNETIC RELAXATION
AND MOLECULAR MOTIONS OF SOME SMALL

‘DLECULES IN DENSE PHASES

BY

Shaw- Guang Huang

A general program has been written to perform a large variety of
pulsed NMR studies automatically. Several interfaces were constructed
to permit varying the pulsewidths, to supply a constant pulsed current

to the field-gradient coil and to alternate the radiofrequency pulses.

14 d 19

Complete N an F relaxation studies of CFSCN were accomplish-

ed and the notional anisotropy of the molecules in the liquid state
was determined. The J-diffusion model was found to be appropriate to
describe the reorientational motion of the CFSCN molecules. SeVeral

physical constants of the molecules were determined, e.g., the chemical

13

shift anisotropy and the spin-rotation constants for both the C and

19'

the P nucleus.

13C relaxation studies have been carried out for CDBr3, CFBr3 and
Cflznrz and the scalar coupling constants between 13C and 798r
determined from the dominant scalar relaxation rates in these highly
brominated compounds. In CHZBr2 the coupling ConStant d(C793r) was

fOund to be 58.7 Hz, a reasonable value compared with the reported

value 1
to be I
discuss
lined :
test 01
cussed.
Au
CF f‘r‘

“by n

4 4

degree

liquid

perform
dipolar
lotion 1
relax”:

“but fl

NMR lin
nuclei.
IOIatin
diffusi
j“? t:

orders

 

lattice

 

Shaw~Guang Huang

value for J(C35Cl). But,in both CDBr and CFBr3,J(C7gBr) was found

3
to be about twice as big as expected and this discrepancy has.been
discussed. The coupling constant J(F798r) in CFBr3 was also deter-
mined and found to be 30.5 Hz from the 19F relaxation studies. The
test of the extended diffusion model in this molecule has been dis-
cussed.

An attempt at resolving the anisotropic motions of planar

CFZCCIZ molecules in the liquid state has been made with a certain
degree of success.It was found that the reorientational motion in the
liquid and in the solid states was via different mechanisms.

The separation of various 19

F relaxation mechanisms in CF31 was
performed but difficulties were found in extracting the intramolecular
dipolar relaxation rates and therefore analysis of the reorientational
motion was fruitless. However, it was found that the spin-rotational
relaxation mechanism was via the spinning motion of the molecules

.

about the C3 axis.

A number of plastic crystals were studied by analysis of the 19?
NMR lineshapes and by measurement of the relaxation rates of the 19F
nuclei. Measurement of the 19F spin—lattice relaxatiOn rate in the
rotating frame was found to be very informative about the self-
diffusive motion in the plastic phase of these crystals. The mean
jump time between adjacent diffusion steps was determined to several
orders of magnitude. In all the plastic crystals studied the Torrey
lattice-diffusion model was found to be appropriate to describe the
self-diffusion of the molecules in the plastic solid phase. The

reorientational motion in these plastic crystals was found to be as

fast as in the liquid phases. From the 19F spin-spin relaxation rates,

the seal:

derivati‘

deteni; e

in terms

 

 

Shaw-Guang Huang

the scalar-coupled relaxation rates in someof'these fluorochlorocarbon
derivatives were extracted and the coupling constants J(F35Cl)
determined. Finally, a test of the theory of fusion was discussed

in terms of the activation parameters derived from the NMR studies.

To my wife and my parents

ii

”'1

In
encoura
He also
some of
Depart:
author.

h'ZRYTIe B:

 

 

ACKNOWLEDGEMENT

The author wisheS‘tothank Professor Max T. Rogers for his
encouragement and freedom tOlPursue his own interesting projects.
He also wishes to thank Dr. David A. Wright for initial guidance in
some of the instrumental work. Appreciation also goes to the
Department of Chemistry for offering teaching assistantships to the
author. Kindly help from Mr. Martin Rabb, Mr. Frank Bennis and Mr.

Wayne Burkhardt is also highly appreciated.

iii

TABLE OF CONTENTS

LIST OF TABLES . . . . . .

LIST OF FIGURES.

INTRODUCTION . . . . . .

THEORIES OF MOLECULAR MOTIONS IN DENSE PHASES.

I.

II.

III.

IV.

Introduction .

Correlation Functions and Transport Coefficients .

A. Velocity Auto-Correlation Function.

8. Angular Velocity Correlation Function .
C. Angular Momentum Correlation Function .
MOdels of Molecules in Dense Phases.

A. Hard Sphere MOdels. .

8. Potential Functions and the Self-Diffusion

Coefficients. . . . . . . . . . . .

Models of MOlecular Motions in Dense Phases.
A. Debye's Model of Brownian Motion.

8. Free Rotors . . . . . . . . . . ; . . . .
C. Extended Diffusion Models .

Theories of Fusion . . . .

A. Melting Transition from the Instability of the

Liquid and Solid Phases . . . .
B. Coexistence Theory.
c. Lattice Models. . . . . . . . . . . .

D. Extended Lennard-Jones-Devonshire Models.

iv

. xiii

. 10

. 13

. 13

. 16

. 17

. 18

. 22

. 23

l. Pople and Karasz's Theory of Fusion.
2. Amzel and Becka's Modification .
THEORIES OF NUCLEAR MAGNETIC RELAXATION .
I. Historical Review. .
A. The Bloch Equation and the Origin of Magnetic
Relaxation. .
B. Density Matrix Formalism for Magnetic Relaxation.
II. Mechanisms of Nuclear Magnetic Relaxation.
A. Dipolar Relaxation.
B. Quadrupolar Relaxation.
C. Scalar Relaxation .
D. Chemical Shielding Anisotropy Relaxation.
E. Spin-Rotation Relaxation. . . . . . . . . .
F. NMR Relaxation due to Anisotropic Reorientation .
III. Relaxation and Fourier Transform NMR Spectroscopy.
A. Theories in Fourier Transform NMR Spectroscopy.
B. Fourier Transform Nuclear Magnetic Relaxation

Studies . . . . . .

C. Effects of Cross-Relaxation and Cross-Correlation .

EXPERIMENTAL. . . . . . . .
I. The NMR Spectrometer . . . . . . . . . .
A. General Description of the Pulse System .
B. Multinuclear Modification .
C. The Probes. . . .
D. Computer Interfaces .
E. External Lock System. . . . . . . . . . . . .
F. Temperature Control and Measurement . . .

V

23
26
28

28

28
30
32
33
36
37
38
38
41'
43

43

45
47
51
$1
51
54
55
59
62

62

G. Heteronuclear Decoupling and the Variable Field
Lock System . . . . . . . . . . . . . . . . . . . . 63
II. Measurement of Relaxation Times. .. . . . . . . . . . . 67
A. NMRLX-- A General Program for Computer Controlled
Pulsed NMR Relaxation Studies . . . . . . . . . . . 67
B. Sample Preparation. . . . . . . . . . . . . . . . . 68
RESULTS AND DISCUSSION. . . . . . . . . . . . . . . . . . . . . . 70
I. Molecular Motion and Relaxation Studies of Some Small
Molecules in the Liquid Phase. . . . . . . . . . . . . . 70

A. Anisotropic Reorientation and 13C, 14N, 19F

Relaxation Studies in CF3CN . . . . . . . . . . . . 70
1. Introduction . . . . . . . . . . . . . . . . . 70
2. 14N Relaxation and the Tumbling Motions. . . . 71

3. 19F Relaxation and Anisotropic Reorientation.. 74
4. The Extended Diffusion Model . . . . . . . . . 90
S. 13C Relaxation and the Spin-Rotation Constant. 92
6. 13c, 19F Chemical Shifts and JCF . . . . . . . 97
B. 13C and 2D Relaxation Studies and the Motional
Anisotropy in Liquid CDBr3. . . . . . . . . . . . . 100
1., Introduction . . . . . . . . . . . . . . . . . 100
2. 2D Relaxation and the Tumbling Motion. . . . . 103
3. 13C Relaxation and Motional Anisotropy . . . . 106
c. 19F and 13c Relaxation Studies in Liquid CFBr3. . . 114
1. Introduction . . . . . . . . . . . . . . . . . 114
2. 19F Spin-Lattice Relaxation and IsotrOpic
ReOrientation. . . . . . . . . . . . . . . . . 114

3. 19F Spin-Spin Relaxation and J(F7gBr). . . . . 121

vi

II.

F.

4.
5.

13C Nuclear Magnetic Relaxation and J(C79Br) in

13

C Spin-Lattice Relaxation and J(C798r)

Extended Diffusion Models.

Liquid CH23I2 .

1.

2.

3.

Introduction .

Mechanisms 0

Di

f 13C Relaxation and J(C798r).

scussion .

Anisotropic Reorientation and 19F Relaxation

in CFZCClz.

l.

2.

3.

Introduction .

Mo

Ro

lecular Motions in the Solid Phase .

tational Motion in the Liquid Phase.

19F Relaxation Mechanisms for CF31.

NMR Studies of Molecular MOtions in Plastic

Crystalline Fluorochlorocarbons.

A.

B.

C.

Introduction.

NMR Theories Relating to Molecular Motions

in Plastic Crystals .

l.

2.

NMR Studies of Some Fluorochloroethanes CanCl6_n..

1.

Li

newidths . . . . . .

Second Moments of NMR Absorption Lines .

Spin-Lattice Relaxation Times.

Spin-Lattice Relaxation Times in the

Rotating Frame . . . . . . .

MO

a.

lecular MOtions in CFZClCC13 .
Measurement of the 19F NMR Linewidths

and Second MOments .

vii

. 124

. 125

. 131
. 131
. 133

. 136

. 138
. 138
. 139
. 145

. 147

. 152

. 152

. 154

. 154

. 156

. 158

. 160

162

. 163

. 163

b. Rotational Motion in the Plastic Phase .
c. Self-Diffusion in the Plastic Phase,
d. 19F Relaxation in the Liquid Phase
and J(F35c1)
2. Molecular Motions in CFCIZCFC12 .
a. Measurement of the 19F NMR Linewidths
and Second Moments .
b. Rotational Motion in the Plastic Phase .
c. Self-Diffusion in the Plastic Phase.
d. 19F Relaxation in the Liquid Phase
and JCF35C1)

3. Self-Diffusion Studies in CF3CC13 .

4. Spin-Lattice Relaxation Studies in CF2C1CF2C1..

D. Molecular Motions in Some Other Plastic Crystals .
l. l,l,l-Trifluoropentachloropropane (TFPCP)
2. 2,2,4,4-Tetrachlorohexafluorobutane (TCHFB)
3. l,1,2,2-Tetrachloro-3,3,4,4-tetraf1uoro-
cyclobutane (TCTFCB).
E. Discussion . . . . . . . . . . . . . .
1. Rotational Motion in Plastic Crystals .
2. Self-Diffusion in Plastic Crystals.
3. Test of the Theory of Fusion. . . . . .
SUMMARY . . . . . 1
REFERENCES.
APPENDIX. .
A. Instructions for NMRLX - A General Program for Computer

Controlled Pulsed NMR Studies.

viii

. 167

. 173

. 178

. 184

. 184

. 187

. 190

. 195

. 195

202

. 206

. 206

. 211

. 217

. 226

. 226

. 228

. 229

. 231

. 233

. 248

. 248

II.

III.

IV.

VI.

VII.

VIII.

IX.

X.

XI.

XII.

INTRODUCTION.
CONSTRUCTION OF THE PROGRAM .
LEGAL COMMANDS.
TIMING.
DISPLAY .
WEIGHTED LEAST SQUARES CURVE FITTING.
SETUPS FOR EXPERIMENTS.
(i) Pulse Adjustment.
(ii) T1 Measurement Using the Spin-Inversion-
Recovery Technique.
(iii) T1 Measurement Using the Homospoil
Technique .
(iv) Tlp Measurement .
(v) T2 Measurement.
(vi) Th Measurement.
(vii) Dsd Measurement .
USEFUL SUBROUTINES.
ERROR MESSAGES.
AUXILIARY PROGRAMS.
USEFUL PATCHES.

INTERFACING CIRCUITS. .

Instructions for Other Programs Used .

I.

II.

III.

FTNMRD.

FTNMRB.

SECMFT.

ix

248
248
249
252
254
255
257

257

259

260
261
261
262

265

266

267

268

269

271

273

273

273

274

TABLE

10

11

12

13

14

15

16

17

18

LIST OF TABLES

Some experimental measurements and the related
time correlation functions .

Pulse methods for relaxation studies .

14N spin-lattice relaxation rates in CF3CN .
19F spin-lattice relaxation rates in CF3CN .
Measurements of the density of CF3CN .
Self-diffusion coefficients of CFSCN .
Anisotropic motional parameters for CFSCN.

Comparison of some physical constants and motional

parameters of acetonitrile and the trihaloacetonitriles.

Reduced reorientational and angular momentum
correlation times for CFSCN.
Spin-rotation constants of 13C in CF3CN.

13C and 19

Comparison of F chemical shifts and JCF
values in some fluorocarbon derivatives.

Summary of the physical constants and the motional
parameters of CF3CN.

2D spin-lattice relaxation rates in CDBr3.

13C spin-lattice relaxation rates in CDBr3 .

The separation of the different contributions to the
total relaxation rates of 13C in CHBr3 .

Physical properties of CDBr3 and CDC13 .

l9F spin-lattice relaxation rates in CFBr3 .

Comparison of the theoretical and experimental values

Page

48
72
76
80
81

83

84

89

96

99

. 102

. 104

. 107

. 112

. 113

. 115

of the intermolecular dipolar relaxation rates for CFBr3.118

TABLE Page

19

19 F spin-spin relaxation rates in CFBrS . . . . . . . . . 122

20 Summary of the 13

C spin-lattice relaxation rates in

CFBr3 . . . . . . . . . . . . . . . . . . . . . . . . . . 126
21 Summary of the information used in the calculation of

T1(7gBr) and J(c793r) in CFBr3. . . . . . . . . . . . . . 128
22 Physical properties of CFBr3 used or Obtained in

this study. . . . . . . . . . . . . . . . . . . . . . . . 130-
23 Summary of the quantities employed in the calculation

of the 13C dipolar and scalar-coupled relaxation

rates in CHZBr2 . . . . . . . . . . . . . . . . . . . . . 134
24 Summary of quantities used in the calculation of

T1(F79Br) and J(c793r) for CHZBrz . . . . . . . . . . . . 137

25 19F spin-lattice relaxation rates in CF2=CC12 . . . . . . 140

26 19F spin-spin relaxation rates in CF2=CC12. . . . . . . . 141
27 19F spin-lattice relaxation rates in CF31 . . . . . . . . 148
28 Self-diffusion coefficients of CF31 . . . . . . . . . . . 149

29 Separation of the contributions to the 19

F spin-lattice

relaxation rates in CF31. . . . . . . . . . . . . . . . . 150
30 Physical constants of some fluorochlorocarbons. . . . . . 164
31 19F linewidths and second moments in CFZCICC13. . . . . . 16S
32 19F spin-lattice relaxation rates in CFZCICC13. . . . . . 168
33 19F spin-lattice relaxation rates in the rotating

frame in CFZClCCl3. . . . . . . . . . . . . . . . . . . . 174
34 19F spin-spin relaxation rates in CF2C1CC13 . . . . . . . 175

35 Summary of the calculation of the angular momentum and

the reorientational correlation times in CFZCICClS. . . . 181

xi

G.-
(I!

46

47

48

49

A1

 

 

TABLE Page
36 19F linewidths and second moments in CFClZCFC12. . . . . 18S
37 19F spin-lattice relaxation rates in CFCIZCFCIZ. . . . . 188
38 19F spin-lattice relaxation rates in the rotating

frame in CFCIZCFCIZ. . . . . . . . . . . . . . . . . g . 191
39 19F-spin-spin relaxation rates in CFCIZCFCIZ . . . . . . 192
40 19F linewidths and second moments in CFSCC13 . . . . . . 197
41 19F spin-lattice relaxation rates in the rotating

frame in CF CCl

3 3 . . . . . 198
1

42 9F spin-lattice relaxation rates in CF2C1CF2C1. . . . . 203
43 19F linewidths and second moments in CFSCIZCCCIS . . . . 207
44 19F spin-lattice relaxation rates in CF3C12CCC13 . . . . 208
45 19F spin-lattice relaxation rates, linewidths and

second moments in CF3C12CCC12CF3 . . . . . . . . . . . . 213
46 19F spin-lattice relaxation rates, linewidths and

second moments in 1,1,2,2-tetrachloro-3,3,4,4-

tetrafluorocyclobutane . . . . . . . . . . . . . . . . . 218
47 19F spin-lattice relaxation rates in the rotating frame

in l,1,2,2-tetrachloro-3,3,4,4-tetraf1uorocyclobutane... 219 .
48 19F spin-spin relaxation rates in l,1,2,2-tetrachloro-

3,3,4,4-tetrafluorocyclobutane . . . . . . . . . . . . . 220
49 Motional parameters of some plastic crystals.. . . . . . 227
50 Comparison of the parameters F and v in some

plastic crystals . . . . . . . . . . . . . . . . . . . . 230

Al Discrete times and possible errors in the NMRLX

timing routine . . . . . . . . . . . . . . . . . . . . . 253

xii

FIGURE

LIST OF FIGURES
Page
Plot of reduced reorientation correlation time versus
reduced angular momentum correlation time for various
diffusion models; solid lines A and C are predicted
by the M-diffusion model for 2:1 and 2, respectively;
solid lines B and D are predicted by the J-diffusion
model for i=1 and 2, respectively; dashed lines are
predicted by the Debye classical diffusion model
(TGTJ=I/£(£+1)kT); dotted lines are predicted by the
perturbed free-rotor model (Te=TJ/2£+1) and broken
lines are the free-rotor limit of the extended
diffusion model (T6=TJ/22). . . . . . . . . . . . . . . . . 19
Block diagram of the pulse system for 19F observation ...... 52
Block diagram of the multinuclear pulse system. . . . . . . S3
Field-gradient coils in the probe: (a) anti-
Helmholtz configuration; (b) quadrupole configuration ..... 56
The inside cross-section of the wideband probe: a,
sample tube; b, spinner; c, BNC connector for the
capacitor pad; d, H1 tuning capacitor; e, heater sensor
connector; f, N2 gas inlet; g, H1 rf cable; h, heater
sensor; i, copper-constantan thermocouple; j, decoupling
coil; k, sample coil; 1, external lock probe; m,
decoupler tuning capacitor; n, LC circuit of the probe ..... 58
Schematic diagram of the pulsed field-gradient current
supply (resistors are given in ohms, capacitors are

given in uF).. . . . . . . . . . . . . . . . . . . . . . . 60

xiii

HOPE

 

10

ll

FIGURE Page
7 Schematic diagram of the rf phase reversing circuit
(all resistors are in ohms, all capacitors are in
uF, unless denoted otherwise). . . . . . . . . . . . . . . 61
8 (a) Circuit diagram of the decoupler gating box
(all resistors are in ohms, all capacitors are
in uF).

(b) Timing diagram of the outputs . . . . . . . . . . . . 64

9 Block diagram of the heteronuclear decoupling setup. . . . 65
10 Block diagram of the variable field lock system. . . . . . 66
11 Design of the NMR sample tube . . . . . . . . . . . . . . 69

12 Plot of the 14N spin-lattice relaxation rate in CF3CN

as a function of reciprocal temperature. . . . . . . . . . 73
13 Reorientation correlation times and diffusion coeffi-

cients of CFSCN as functions of reciprocal temperature.... 75

14 Plot of the 19

F spin-lattice relaxation rate in CF3CN

as a function of reciprocal temperature. . . . . . . . . . 77
15 Plots of X'I’ xi for liquid CFSCN. For comparison, the

dashed lines are the corresponding X plots for CH3CN

and dotted lines are those for CC13CN. . . . . . . . . . . 91
16 Plot of reduced reorientational correlation time versus

reduced angular momentum correlation time for CF3CN;

solid and Open circles are results with and without

correction for Ceffe respectively. . . . . . . . . . . . . 93
17 Plot of JCF versus 5F for some fluorocarbon
derivatives. . . . . . . . . . . . . . . . . . . . . . . . 98

xiv

FIGURE

18

19

20

21

22

23

24

25

. Page
Plot of AOF versus dF for CF3X, X= CC13, F, Cl,
Br, I and CN; solid circles are in the C-F bond
coordinate system and Open circles are in the
molecular-axis coordinate system . . . . . . . . . . . . 101
Plot of the 2D spin-lattice relaxation rate in CDBr3

as a function of reciprocal temperature. . . . . . . . . 105

13

Plot of the C spin-lattice relaxation rate as a

function of reciprocal temperature. The open circles

are the Ric values calculated from the 13C data of

03

Farrar gt_al,2 ; the solid circles are the RE

values for 13C in CDBr3. . . . . . . . . . . . . . . . . 108
Plot of the 19F spin-lattice relaxation rate as a

function of reciprocal temperature; 0: Rt

1,F‘
. d . . 8r .
__. R1,F, ---. RI’F. . . . . . . . . . . . . . . . . . . 116

Plot of the reorientational correlation times as

functions of reciprocal temperature for CFBrS;

———-: Te; ---: TE . . . . . . . . . . . . . . . . . . . 120
Plot of the 19F spin-spin relaxation rate in CFBr3

. . t
as a function of reelprocal temperature; 0: R2 F ;
3

o: R56 . . . . . . . . . . . . . . . . . . . . . . . . 123

2,F°

Plot of the 13C spin-lattice relaxation rate in

CFBrS as a function of reciprocal temperature;

SC
1,C........

Plot of reduced reorientational correlation time

0: Rt -———- Rd

1,C; . 1,C; ---: R

. 127

versus reduced angular momentum correlation time

forCFBr3;O:15;0:TE.................129

XV

FIGURE

26

27

28

29

30

31

32

33

Page
Plot of the 13C spin-lattice relaxation rate in
CHZBrZ as a function of reciprocal temperature . . . . . 135
Plot of the 19F relaxation rate for CF2=CC12 as
a function of reciprocal temperature . . . . . . . . . . 142
Plot of the 19F spin-lattice relaxation rate in
CF31 as a function of reciprocal temperature . . . . . . 151
19F NMR linewidths (scale at left) and second
moments (scale at right) of CFZClCCIS. . . . . . . . . . 166
Plot of the 19F relaxation rate in CF2C1CC13 as a
function of reciprocal temperature; : R}; 0: R5;
---- aft; —: Rd; ---: Rsr; R3“. . 17o

Plot of the 19F spin-lattice relaxation rate in the

rotating frame at three different spin-locking fields

and the spin-spin relaxation rate (___J in.CF2ClCCl3.... 176
Plot of the mean jump time for self-diffusion in the

plastic phase of CFzClCCIS; o: from Rlp data; 0: from

the linewidth data using Equation (162); 0: from the
linewidth data using Equation (163); o: from the

linewidth data using Equation (164); +: from R2 data.... 177
Plot of reduced reorientational correlatiOn time

versus reduced angular momentum correlation time

for CFZCICCIS; solid lines A and C represent the

theoretical curves predicted from the M-diffusion

model for 2:1 and 2, respectively; solid lines B

and 0 represent the theoretical curves predicted

from the J-diffusion model for i=1 and 2, respectively;

FIGURE

34

35

36

37

38

39

Page
dashed lines represent the theoretical result
predicted from the Debye classical diffusion
model (TeTJ=I/£(L+1)kT); dotted lines represent
the theoretical result predicted from the per-
turbed free-rotor model (Te=TJ/2£+l) and the broken
lines represent the free-rotor limit of the
extended diffusion model (T9=TJ/22). . . . . . . . . . . 182
19F NMR linewidth (scale at left) and second
moment (scale at right) for CFClZCFClz . . . . . . . . . 186
19F spin-lattice relaxation rate (0) and the spin-spin
relaxation rate (0) in CFCIZCFClz; the stright
solid line represents the intramolecular dipolar
relaxation rate R?; -«-: the scalar spin-spin
relaxation rate Ric. . . . . . . . . . . . . . . . . . . 189
Plot of the 19F spin-lattice relaxation rate in
the rotating frame at three spin-locking fields
(10.5, 5.3 and 2.0 G) and the spin-spin relaxation
rate (0) in the plastic solid phase of CFCIZCFCIZ. . . . 193
The mean jump time derived from R10 (0), R2 (0)

and Av (A) of 19

F nucleus in the plastic solid

phase of CFCIZCFCIZ. . . . . . . . . . . . . . . . . . . 194
Plot of the 19F NMR linewidth and second moment in

CFSCCl3 versus reciprocal temperature. . . . . . . . . . 199
Plots of the 19F spin-lattice relaxation rate in

the rotating frame at three different spin-locking

fields (17.6, 8.8 and 5.7 G), and of the spin-spin

xvii

FIGURE

40

41

42

43

44

45

46

47

relaxation rate (0) versus reciprocal

temperature, for CF3CC13 .

Mean jump time for self-diffusion in the plastic
phase of CF3CC13 derived from Rlp(0), R2(+) and Av(0).
Plot of the 19F spin-lattice relaxation rate in
CFZClCFZCl versus reciprocal temperature; the solid
straight line represents the intramolecular dipolar
relaxation rate and the dotted line represents the

intermolecular dipolar relaxation rate in the solid

phase.

Plots of the 19F linewidths and the second moments
in CF3C12CCC13 versus reciprocal temperature .
Plot of the 19F spin-lattice relaxation rate in
CF3C12CCC13 versus reciprocal temperature; dashed

line represents the intramolecular dipolar relaxation

rate .

Plot of the 19F NMR linewidths and second moments

in CF3C12CCC12CF3 versus reciprocal temperature.

Plot of the

19

F spin-lattice relaxation rate in

TCHFB versus reciprocal temperature.

Plot of the 19F NMR linewidths and second moments in

C61 CF CF CCl versus
L________J

2 2 2

19F spin-lattice relaxation rates in the rotating
frame at three different spin—locking fields (12, 6.3,
and 3.2 G) and the spin-spin relaxation rate (0)

in CCI CF CF CCl
L________J

2

2

2

2

2 .

reciprocal temperature ,

xviii

Page

. 200

. 201

. 204 -

. 209

. 210

. 214

. 215

. 221

. 222

FIGURE

48

49

A1

A2

Page
19F spin-lattice relaxation rate (0) and
spin-spin relaxation rate (A) in TCTFCB; solid
circles are RZ-Rl; dashed line represents the
intramolecular dipolar relaxation rate; dotted
line represents the total spin-lattice relaxation
rate after subtracting the intermolecular
dipolar contribution . . . . . . . . . . . . . . . . . . 223
The mean jump time for self-diffusion in the
plastic phase of TCTFCB derived from R10 (0),
Ri (0) and Au (A). . . . . . . . . . . . . . . . . . . . 224
Pulse sequence used in NMRLX . . . . . . . . . . . . . . 258
Decoupler gating box circuit diagram. (Capacities

are in uF and resistances are in O). . . . . . . . . . . 272

xix

INTRODUCTION

Ever since the first nuclear magnetic resonance (NMR) experiments
were performed, the technique has been a powerful tool in the study of
the prOperties of molecules. The extreme sensitivity of the behavior
of nuclear spins to their chemical and physical environment has made
their detection not only useful for elucidating the detailed structure
of the molecules but also the interactions among the molecular
aggregates. Molecular motions, which have been studied over a long
period of time, can range from being very simple as in dilute gases to
being highly complex and intractable as in dense fluids and solids.
While the other spectrosc0pic methods can also give information about
molecular motions, NMR studies possess several advantages in deriving
motional parameters such as the anisotrOpic reorientational diffusion
constants and the angular momentum correlation times which cannot be
obtained in most other spectroscopic studies.

The success of NMR studies of molecular motion lies in the
measurement of magnetic spin relaxation times which-give an estimate
of the extent of perturbation by the neighboring spins through various
kinds of mechanisms. Very often, there are more unknowns than relation—

ships which can be obtained from the experiments. Therefore, in this

3
CF3Br, CFBr3, CDBrs and other pseudo-spherical molecules forming plastic

work only simple molecules with higher symmetry,e.g., CF CN, CF I,

CF C1 CCCIS, CF3C12CCC12CF3

2’ 3 2
and CFZCCIZCCIZCF2 were studied. The study of motions in these molecules

crystals, CFSCClS, CF ClCCl

2 3, CFCIZCFCI

was mainly to test the validity of various models such as the anisotropic

reorientational diffusion model and the extended diffusion model. In

additior

 

sphhro.
these

TE
heat ha

amount

BO

 

fictions
Theoret

s
aatEr .

ful-

('9

:
(n

of III:

2
addition, some other information like the scalar-coupling constants,
spin-rotation constants, chemical shift anisotrOpy and the relaxation
times of quadrupolar nuclei was Obtained.

The increasing application of minicomputers to control the experi-
ment has made the measurements more rapid and accurate. A considerable
amount of effort has been devoted to this as described in the Experi-
mental section of this dissertation.

Both historical and recently updated theories about molecular
motions and magnetic relaxation are reviewed and-outlined in the
Theoretical section. The working equations are then derived for use
later in the data analyses in the Results and Discussion sections.

As the plastic phase in solids has been useful as a bridge for
further understanding of the melting processes, detailed description

Of its behavior is attempted in the last part of this work.

THEORIES 0F IDLECULAR LDTIONS IN DENSE PHASES

I. Introduction

Interest in molecular motions in dense phases has been increasing
over the years. Numerous studies, both theoretical and experimental,
have been devoted to describing the microdynamic behavior of molecules
in liquids and solids. Unlike the gaseous state in which the mean free
path is large compared to the size of each molecule, the intermolecular
interaction in dense fluids where molecules are very close to each
other, is dominated by a repulsive fbrce caused by frequent collisions.
To describe the motion of molecules in such a dense phase, one has to
deal with a many-body problem which can only be attacked by using
statistical mechanical approaches with certain approximations.

At present, the theory of molecular motion in dense phases is at
a model-building level. Recently, irreversible statistical mechanics
has been developed to supply two approaches to study hydrodynamic
properties. The first one is the derivation of general master equations
from which the states of thermodynamic equilibrium, hydrodynamic
equations, transport coefficients, fluctuation and relaxation phenomena
may be deduced as appropriate solutions to the master equationslez. The
second approach is the derivation of time correlation function
expressions for transport coefficients and the relaxation parameters.
Both the master equations and time correlation fOrmulas can, in prin-
ciple, provide solutions to kinetic and transport prOperties of many-
body systems if some potential models, mathematical approximations and
other techniques, e.g., cluster diagrams, are introduced.

Since NMR and other spectroscopic experiments all study the

3

respo:

 

theory

functj

and t1

 

aaCIOS
E06161:
£Unctj

EXist:

4

response of a system coupled with an applied weak field, linear response
theory has been found to be successful in describing the experimental
results. The major conclusions of this theory can be stated as follows:
Whenever two systems are weakly coupled to each other,such as radiation
weakly coupled to matter, or molecular vibrations weakly coupled to
molecular motion, it is only necessary to know how both systems behave
in the absence of the coupling in order to describe the way in which
one system responds to the other. Furthermore, the response of one
system to the other is completely describable in terms of time correlation
functions of dynamic properties. ~

In this chapter, we first introduce the time correlation functions
and their relations to the transport coefficients which describe the
macrosc0pic behavior of molecules. Then we introduce some molecular
models, each with its characteristic shape, along with the potential
functions, and motional models usually adopted. In the last section, the

existing theories of liquid-solid transitions are described.

II. Correlation Functions and Transport Coefficients

 

Time-dependent correlation functions have been familiar for a long
time in the theory of noise and stochastic processes. In recent years,
they have become very useful in many areas of statistical physics and
'spectroscopy3’4. Correlation functions provide a concise method for
expressing the degree to which two dynamical properties are correlated
over a period of time. For two dynamical properties A and B, the

correlation function CAB(t) is defined as

CAB(t) = <A(0)'B(t)> (1)

5
where the bracket indicates an equilibrium ensemble average. When A and

B are different properties, C is called a cross-correlation function.

AB

When they are identical, CAB is called an auto-correlation function.‘
Since the response of a system to a specific weak perturbation
field is directly related to the correlation function, many experiments

have been devised to determine the specific correlation functions. For
example, IR lineshapes can be used to determine the auto-correlation
function of a unit vector along the molecular axis, neutron scattering
experiments can determine the Van Hove scattering function? which is
related to the number density auto-correlation function, and Raman
scattering can determine the auto-correlation function of the second-
order Legendre polynomial of the direction cosine of a unit vector
along the molecular axis. Table 1 shows some familiar correlation
functions and their relations to the experimental observables.

The calculation of time correlation functions might be as
difficult as the calculation of a partition function of an equilibrium
system because the complex many-body problem cannot be bypassed. While
the Monte Carlo technique6 has been applied to evaluate the multi-
dimensional integrals in the solution of equilibrium partition functions,
a similiar type of computer experiment, namely, molecular dynamics, has
been constructed by Alder and Wainwright7, who have been able to calculate
the number-density correlation function of hard-sphere and square-well

9 used a Lennard-Jones

molecules in fluids. Later, Rahman8 and Verlet
(12-6) potential function to simulate liquid argon and found it to be
very successful. More work of this type,and the general molecular

dynamics calculation of correlation functions, has been reviewed by

Berne and Harplo.

 

Tatl

correlation functions.

Table 1. Some experimental measurements and the related time

 

Experimental
observable

Time correlation
function

Dynamic parameter

 

 

Self-diffusion
coefficient

Rotational

diffusion
coefficient

IR lineshape

Raman lineshape

NMR spin-rotation
relaxation time

<3(O)-3(t)>
<Q(0)-Q(t)>
<fi(O)-fi(t)>

<P2(fi(0)'fi(t))>

<3(O)-3(t)>

+ m
0: linear velocrty
of the mass center.

0: angular velocity
about the molecular
axis.

0: unit vector on
the molecular
dipolar axis.

A

U.

3: angular momentum
about the mass
center.

\'\‘
rotatit

tiles .

VEIOCi
magnit
Scalar

then q

 

7
NMR relaxation times are related to the self-diffusion coefficients,
rotational diffhsion coefficients and angular momentum correlation
times which, in turn, are related to proper correlation functions as

discussed in the following sections.

A Velocity Auto-Correlation Function

If a molecule has a velocity 0(0) at time t=0 and a velocity u(t)
at time t, then a suitable measure of the degree of correlation of these
velocities is the scalar product u(0)~ u(t). After a sufficient time, the
magnitude and direction of u(t) will no longer be related to u(0) and the
scalar product will be zero. The velocity auto-correlation function is

then defined by
sw(t) = <u(o)ou(t)>/<u(0)2>. (2)

For a collision-free gaseous molecule, the velocity is almost constant

at all values of t, hence sw(t)=1. In a perfect Einstein solid in which

a molecule vibrates about its site at constant angular frequency m5,

sMt): cos wt. In a liquid, sw(t) has a shape of a damped oscillator.
Each correlation function has its Fourier transform or its

spectrum sMm)

SW») = [L] sectl-e‘iwtdt. (3)

2n w

 

It is easy to show that sMm) shows a peak at m=0 for ideal gases and a
peak at m=wB for ideal solids. In liquids,sw(w) is spread over a wide

frequency. We then define a diffusion coefficient Ds as the

zero-

of t?

at:

COIT

and

The .

 

8

zero-frequency intercept of 5Wm) or, in another words, the time average

of the velocity correlation function, i.e.,

D = 5M0) J Sw(t)dt. (4)

5
on

The two limits of D5 are 0 (for ideal solid) and co(for dilute gas).

B. Angular Velocity Correlation Function

 

If a molecule is rotating about its axis with angular velocity 9(0)
at time t=0 and a velocity 0(t) at time t, then the angular velocity

. . . 1‘ .
correlation function rMt) and its spectrum V(w) are defined as

 

rwtt) = <n(0)-n(t)>/<n(0)2> (5)
and
r‘Hw) = 1 fro(t)-e‘i‘”tdt. (6)
r3; ..

The rotational diffusion coefficient is then defined as'

D = 1[31(0) =f rlp(t)dt. (7)

r G
r .
Both w(t) and Dr have the same limits as for 5Mt) and 03°

C. Angular Mbmentum Correlation Function

 

If J(O) is the angular momentum of the molecule at time t=0 and it

is changed to J(t) at time t by some forces, e.g., collisions of the

a*‘

3153

mOdel

Para:

 

that
3 ft

the

 

int.

9
neighboring molecules, then the angular momentum correlation function

and its spectrum can be defined by

 

jwctl = <J(0)-J(t)>/<J(0)2> (8)
and
j‘l’(w) = 1 f j111(t)°e-imtd1:. (9)
f2? ..

In dilute gases, collisions are rare and J is kept constant for all time,
hence jw(t)=1, and jMm) peaks at m=0. In perfect solids, there is no
rotation, hence J(t)EO and jw(t)50. There is no transport coefficient
defined for this correlation function.

To further calculate these time correlation functions, a specific
model has to be chosen in order to describe the dynamic behavior of the

parameters concerned. This will be discussed in the following sections.

III. Models of Molecules in Dense Phases

 

A. Hard Sphere Models

 

The first difficulty of statistical mechanical calculations is
the unknown potential function which is so complicated in dense phases
that the exact expression is always impossible. In order to construct
a feasible potential function, people start to build models to represent
the shape and the physical properties of real molecules from which the
interaction is determined.

Very often, molecules in liquids are considered to be a number

 

of spE

 

conseq

deDen;
A

e"

there
will e
surfa:

L -
the,‘ I;

 

 

10

of spheres interacting with each other by frequent collisions. The
consequence of a collision is the changing of the velocities which again
depends on both the inner and surface properties of contacting spheres.
For simplicity, the spheres are usually considered to be hard,i.e.,
there is no elasticity allowed. In other words, two colliding spheres
will experience an infinite potential wall upon contacting. As for the
surface properties, three different models have been popular in the
theoretical derivations:
1. Smooth Spheres: The surface of the spheres is considered to be
completely smooth, there is no friction exerted by contacting
and therefore the tangential velocity of each colliding sphere
is invariant.
2. Rough Spheres: The surface of the spheres is considered to be
completely rough, therefore two colliding spheres will experience
a reversal of their surface velocities at the point of contact.
3. Loaded Spheres: The center of mass of the sphere does not coincide
with its geometric center, therefore a collision will cause a

transfer of translational and rotational energies.

B. Potential Functions and the Self-Diffusion Coefficients

 

We now turn back to consider the appropriate potential function
which can describe the interaction among the hard sphere molecules. The
simplest one is called the impulse potential which is infinity when

spheres are in contact and is zero otherwise, or

V(r..) = , (10)
lJ 0 r..>o

11
wheretris the hard-core diameter and rij is the intersphere distance

This impulse potential has been used to calculate the self-diffusion

, 11,12
coefficient and obtain

0 = 3 (it) (11)

8nozg(o) flm

 

for equivalent spheres in a fluid. Here, n is the number density, m is
the mass and g(O) is the contact value of the pair-distribution function.
Another potential function, which is called the square-well pot-

ential, has been considered to be more realistic and still allows one

retain simplicity in calculationls. This is defined as fellows:
V(rij) = -e O <rij<o2 (12)
° rij’°2
where 01 and 02 are the repulsive and attractive diameters. Davis et al.13

have found using this potential function that four different types of

collisions have to be considered and the self-diffusion coefficients for

smooth, rough and loaded sphere models have been calculated to be14

 

 

 

3:
ns(smooth) = 3 ( kT ) ( g(ol)+R2g(oz)s)'1 (13)
8no1 nm
3 %
Ds(rough) = 2 ( kT )
8n01 um
( ff t(<ar1)+R"’g(cz)s)'l (14)
and
32
D (loaded) = 3 2 ( kT ) '
5 ZU(a)no1 nm
2 -1
(8(01)+R 3(02) 3) , (15)

 

12

where K is the reduced moment of inertia defined by 4I/moi, R is the ratio
02/01, the g's are the pair distribution functions and E, U(o) are defined

as follows:

 

 

E: exp(e/kT) - RT - “IOXZ(XZ+ RT )kexp(-x2)dx ' (16)
n n 2 2 8 .

U(a) = del d62(l+o(sin 81+sin 62))- sinelsinez, (17)
0 0

where oémaZ/ZI and a is the distance between the mass center and the
geometric center of a loaded sphere. As can be seen from Equations (13),
(14) and (15), adding the roughness will always decrease the self-
diffusion rate while increasing the loading factor will increase it. A
numerical comparison can be made by evaluating K and U(o) and it has been
shown14 that at the typical temperature of a liquid, Ds(rough) is no more
than 10% higher than Ds(smooth) and that with a physically realistic
upper limit of the loading factor a about 0.167, Ds(loaded) will only
increase over Ds(smooth) by 10.4%. Therefore, in interpreting experimental
results, the smooth hard sphere model seems to be adequate enough if a
.108 error is tolerable.

The inclusion of an attraction part to the potential has the effect
of decreasing the self-diffusion coefficient by an extent represented by
the second term in the brackets of Equations (13), (14) and (15). This
term is controlled by a parameter e/kT which in the high-temperature
limit is approximately zero. Therefore, the attraction effect can be
treated as a perturbation in the calculation of thermodynamic properties.

The use of the most popular potential function, the Lennard-Jones

6-12 potential, is more difficult than the two simple ones discussed

above

Chand

 

 

13

above. The Lennard-Jones potential function is defined as

V(r) = 4e((o/r)12-(o/r)6). (18)

Usually this function is separated into two parts, the repulsive part
and the attractive part and then the attractive part treated as a
perturbation. The use of the Lennard-Jones potential function in
equilibrium statistical thermodynamics has been reviewed by Anderson,

Chandler and Weeksls.

IV. Models of Molecular Motions in Dense Phases

 

A. Debye's Model of Brownian Motion

 

In a Brownian motion model, a molecule is described as a particle
with infinité mass m, and its motion in a fluid is described by

Langevin's equation16
m6(t) = -mcu(t) + F(t). ~ (19)

which separates the total force acting on the particle into two parts,

the first part is a frictional retardation which is proportional to the
velocity with a proportionality constant C, and the second part is a

randomly fluctuating force which arises from the impacts of surrounding
molecules. When this particle undergoes a random walk, which means the
particle does not remember its history of motion (Markoff process), the
probability of finding it at distance I and at time t, when it is at $0

at time zero, P(;,t/;b), follows the well-known Fokker-Planck equation

The .
+
’ e
er,(

Ratio

Where
COEIf;

0f th

17{hilt};

 

14

8P(r,t/r0)

 

= D -vzp(?,t/? ), (20)
s 0
at
where 05 is the diffusion coefficient defined before. Einstein had

related the diffusion coefficient D( Ds or Dr) to the friction constant

C by

D = kT/c . (21)

The solution of the Fokker-Planck equation gives the probability
P(;,t/?0) from which the time correlation functions of any dynamic
properties can be derived.

Debye16 has applied the Brownian translation model to the rotational

motion of the particles. The Fokker-Planck equation then becomes

3P(O,t/Qo) 2
. = Dr-V P(O,t/Oo), (22)

 

at

where O is the Euler angle and Dr is the rotational diffusion
coefficient. The general solution of Equation (22) can be written in terms

of the eigenfunctions of the spherical hamonics

P(O,t/Oo) = IE: czm-Y£m(O)-exp(-E2t), (23)
£,m

where E2=£(£+1)Dr and com is to be determined from the initial conditions.
The correlation function C(t) of an arbitrary dynamic parameter f(0)

which is a function of 0, can be calculated by the following equation:

C(t) = <f($'20) °f(O)>=fff(Oo)f(O) °P(QO,O)P(O,t/Oo)d00dfl, (24)

 

she

 

Eons“

time

b'he‘,

15

where P(90) is the initial distribution probability. For example, if f(Q)
is the unit vector u(t), which is related to 0(0) by 0(0) u(t)=cos 6(t),
where 6(t) is the angle between 6(0) and fi(t), the correlation function

can be written as
C(t) = <cos 6(t)> = )rcose-P(0)P(6,t)d0. (25)

Since the initial distribution probability is unity, using Equation (23)

for P(6,t), i.e., P(0,t)= Y1’0(0) e'ZDrt, Equation (25) gives
<fi(0)-fi(t)> = <cos 6(t)> = exp(-2Drt). (26)
A similiar approach can be used to obtain
<Pz(fi(0).fi(t))> = <%(3c056-1)> = expc-oort), (27)

where P2(x) is the second-order Legendre polynomial of the argument x.
In general, if the dynamic parameter can be written in terms of the
Legendre polynomial Pn(cose), the correlation function can always be
described by an exponential function or a linear combination of
exponential functions with time constants Te=(l(2+1)Dr)’1.

If one calculates the time correlation function of the angular
momentum of the molecule, one finds that the angular momentum correlation

time TJ is related to Teby

 

I
T T =

O J 6kT ' (28)

where I is the moment of inertia of the molecule.

16

Several assumptions made in this model, e.g., the infinite mass of
the melecules, stick boundary conditions or assumption that the media
are continuous, etc., have limited its applicability to certain systems.
There have been many effbrts devoted to modifying or generalizing this

17

classical treatment of the molecular rotation. Brot and Darmon have

proposed a model fer a rotor undergoing nearest neighbor jumps in an
eight-wall model of cubic symmetry. Cukier and Lakatos-Lindenberg18
have treated this diffusion model by allowing arbitrary size for the
random jumps and found the same exponential decaying shape fer the time
correlation functions. They also generalized the stochastic model by
considering the effect of finite duration of collisionslg. Berne20 has
modified Debye's relaxation time by considering the effect of long-range
interactions. An excellent review of the stochastic models of molecular

reorientation has recently be given by SteeleZI.

B. Free Rotors

The other extreme of describing molecular reorientation is the
gas-like free-rotor model. In this model, the molecules are depicted
as undergoing essentially free rOtation, governed by their moments of
inertia, but occassionally interrupted by a collision. The resulting
time correlation functions can be calculated by Equation (24) and
the relation between the reorientational correlation time and the
angular momentum correlation time can be described by

‘3
t6 . m e (29)

21,30

Steele has considered this model but used a Gaussian fUnction

17
as the short-time approximation of the correlation function to alleviate
the physically unrealistic long-time behavior of the exact free-rotor
model applied to dense phases. Atkins31 has also considered this gas-like
model using a quantum mechanical formalism and obtained a result which

has Steele's result as its classical limit.

C . Extended Diffusion Models

 

Both the classical diffusion model and the free-rotor model have
been found to suffer several defects. A more adequate theory which is
capable of representing the smooth transition from free rotation to
highly hindered rotation has been constructed first by Gordon22 and has

23-28. In this model, several

been extended by several other authors
assumptions are made. First, the molecules are assumed to undergo
a succession of impulse collisions with infinitesimal collision time.
Second, the molecules are assumed to rotate freely during the collision
,With a time much longer than the collision time. Third, during a
collis ion, molecular orientations are unchanged but the direction of the
angu1ar momentum vector and/or its magnitude is randomized after the
con-ision. It is called J-diffusion if both the direction and the magnitude
0f the angular momentum are changed by a collision. If only the direction
is randomized by the collision, it is called M-diffusion. A

The original work of Gordon only dealt with linear molecules. The
a13Plication to both spherical-top and syumetric-top molecules
has _ been done by McClungzs. The approach from the memory function
f0rmatlism has been given by Berne and Harp10 and also by Bliot and
Constant26-28 29

. McClung has given an updated review with discussions of

Mn .
“Y experimental results .

Of

Sl.‘

fro:

beCC

18

The most striking success of this model can be viewed from a plot

*
of T6, the reduced reorientational correlation time defined by
'k . . .
Te= Tel/I7kT ,where I is the moment of inertia of the molecule, versus
*

TJ

For a comparison, the plot of Tgversus T

, the reduced angular momentum correlation time defined by T3= TJ/JI/kT.

*

J

described by various models is shown in Figure l.

for a spherical-top molecule

V. Theories of Fusion

 

Although the melting point has been widely used to characterize
substances, the detailed dynamics of the melting process has never been
well understood. This is again because of the unsolvable many—body
problem encountered in describing the dynamic behavior of molecules in
dense phases. Up to now, only a limited amount of work on this subject
has been reported and the derived theories are still at a primitive
stage. Nevertheless, we shall review some of these efforts, especially
those which have been rather successfully applied to the study of

plastic crystals.

A. MeltingTransition from the Instability of the Liquid and Solid

Phases

This is a two-pronged attack on the problem of freezing. Approaching
from the solid, an old theory of Lindemann32 predicts that a solid
becomes unstable with respect to melting when the amplitude of vibration
of the molecules is about one-tenth of the lattice spacing. From the
liquid side, Alder and Wainwright7 have calculated the pair-correlation

function of a hard-sphere liquid and found that a long-range oscillatory

19

Figure 1. Plot of reduced reorientational correlation time versus
reduced angular momentum correlation time for various
diffusion models; solid lines A and C are predicted by the
M-diffusion model for 2:1 and 2, respectively; solid
lines B and D are predicted by the J-diffusion model for
i=1 and 2, respectively; dashed lines are predicted by the
Debye classical diffusion model (TBTJ=I/£(£+1)kT); dotted
lines are predicted by the perturbed free-rotor model
(Te=TJ/22+l) and broken lines are the free-rotor limit of
the extended diffusion model (Te=TJ/2£).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

21

character was evident when the volume of the liquid molecules becomes
1.65 times of that for closest packing. This has been considered as
a signal of freezing. Brot33 has also shown that including the
attractive part of the intermolecular potential will bring a drastic
amplification of the density fluctuation of a hard-sphere liquid. In

34, both molecular dynamic and

recent studies of Lennard-Jones systems
Percus-Yevick calculations show the same long-range oscillatory character
of the pair-correlation function at kT/c= 0.72 and p03: 0.85 (p is the

density and e and O are two parameters in the Lennard-Jones potential

function, see Equation (18)).

B. Coexistence Theory

 

Jancovici35 has proposed a theory in which one can use different
type of approximations for the description of liquid and solid phases
and produce separate theoretical expressions for the free energy, A,
then locate the equilibrium pressure of two states by a double tangent

method, i.e., set

_ aAsolid = _ 8Aliguid .
av T av T

where Tm is the temperture at equilibrium and V is the volume. This
theory has been adopted by Barker and Henderson who have constructed
an equation of state for the liquid and used the cell theory of solids

to produce a melting line for argon with fair success36

22

C. Lattice Models

 

There have been some simple lattice models which restrict molecules
to the sites of a two or three dimensional lattice and exhibit a freezing
transition when one of the two interwoven sublattices ofwhich the lattice
is made becomes preferentially occupied. For example, in the Lennard-
Jones and Devonshire mode137, a solid has its particles centered on the
sites of a regular 1attece(o sites) which is interlaced with another
set of sites(B sites). At zero temperature, the particles are all
centered on 0 sites representing a perfect order and a liquid is repre-
sented by a perfect disorder which has both a and 8 sites equally
occupied. All the thermodynamic functions can then be written in terms
of a parameter which is related to the energies of on and OB occupancies.
It has been found37 that the Lennard-Jones-Devonshire model agrees
amazingly well with many features of the experimental data for spherical
or near-spherical molecules.

Several variations of this simple site model have also been
. considered. Runnels38 has found that if multiple occupancy of the sites
and also simultanuous occupancy of nearest neighboring sites are dis-
allowed, the model shows a second-order transition from an ordered to a

disordered state. Bellemanssg’4

0 has also found that by adding another
assumption, i.e., the second and the third neighboring sites cannot be
simultanuously occupied, the model will show a first-order solid-liquid
phase transition.

The success of the Lennard-Jones-Devonshire model has been found
to break down when it is applied to molecular crystals which do not

follow the law of corresponding states(which is a consequence of the

Lennard-Jones 6-12 potential field). Two extensions of this model,

 

 

23
which take into account the possibility of different orientations for the
molecules, have been attempted and will be discussed in the following

section.

D. Extended Lennard-Jones-Devonshire Models

 

The failure of the Lennard-Jones-Devonshire theory when applied to
molecular crystals is because the intermolecular forces often deviate
appreciably from spherical form and the possibility of hindrance to ro-
tation in the solid and liquid states is not taken into account. A
remarkable modification of this model has been given by Pople and

Karasz41 , and later by Amzel and Becka42

, who considered order-disorder
phenomena involving both orientation and positions of molecules and
developed a theory which can successfully predict the thermodynamic
properties of melting and of solid-solid transitions. Since their

theories have been widely employed in the NMR studies of plastic

crystals, an outline of the derivation of their equations is given here.

1. Pople and Karasz's Theory of Fusion

 

Pople and Karasz41 modified the Lennard-Jones-Devonshire model by
allowing molecules with two orientations on either'an a or 8 site so that

four possibilities, al, a , 81 and 82 are to be considered. By defining

2
W as the energy difference between a and the neighboring 8 site, W' as
the energy difference between two orientations on each site, 2 as the
number of neighboring 8(or o) sites surrounding each o(or 8) site and

2' as the number of the closest equivalent sites, the partition

. 4 .
function 3 can be written as

24

N N
Z = f O = f exp(-(N W + N W' + N W')/kT), (30)

where NOB’ N and NB 8 are the number Of pairs associated with

o o
energies W and W', f is the partition function per molecule in a state
of perfect order and is a function only of molecular volume and
temperature, and the summation is over all orientations as well as all
arrangements of particles on sites.

Let Q be the fraction of molecules on a sites and S be the fraction

of molecules in orientation 1, then N , N and N can be replaced
GB alaz 8182

by Q, S, N, z and 2'. Hence

O = :2: y(Q,S) exp(-{zNWQ(l-Q)+z'WNS(l-S)(l-2Q+2Q2)}/kT), (31)
Q.3

where y(Q,S) is the number of ways to arrange particles for given Q and

 

 

 

S, i.e.,
Y(Q.S) = ( N’ )2- (NQ)’
(NQ)!(N(1-Q))! (NQS)!(NQ(1-S))!
1- u
. (N( 0)). . (32)
(N(1-Q)S)!(N(1-Q)(1-S))!
43

Using Stirling's theorem , we can maximize (1/N)ln O to obtain

Q = ( z" - z'"' 5(1-S))(2Q-1) (33)
l-Q 2kT kT

 

 

and

25
S z'W'
l-S kT

1n (1-2Q+2Q2)(25-l). (34)

By solving for Q and S, the order-disorder of molecules can be well
defined. From Equations (33) and (34), we can see that Q and S are only
dependent on two parameters, K=ZW/kT and v=z'W'/zW, as are the other
thermodynamic functions which can be calculated from the partition
function43. Several interesting results can be obtained after calculating
the pressure, reduced temperature of melting, volume change, entropy of
melting, etc:

(1) When v<0.325, two separate transitions may be distinguished.
One occurs when S=1/2, corresponding to orientational disorder, the other
when Q=l/2, corresponding to positional disorder or melting; the S=1/2
transition occurs at a lower temperature.

(2) When v30.325, only one transition, with relatively larger
entropy and volume change than case (1), is predicted.

(3) When v21, two transitions are observed again, but the S=l/2
transition occurs at a higher temperature than the Q=1/2 transition,
does.

These results successfully predict the characteristics of plastic
crystals (case (1)), brittle crystals (case(2)) and liquid crystals
(case (3)). The parameter v, which is the ratio of z'W' to zW, can
resonably be assigned a physical meaning, i.e., the ratio of the
reorientational energy barrier to the translational energy barrier of

the mOlecules in a crystal lattice.

 

26

2. Amzel and Becka's Modification

4 . . .

Amzel and Beckz 2 have conSidered an even more realistic case.
1Fhey assume the number of orientations on each site can be more than
two and can be represented by another adjustable parameter D. Using the

:5mme statistical approach, Equations (33) and (34) were modified to

 

 

 

 

 

‘loecome
Q 2 w z'W' osi
1n = ( - 31(1- ——)) (212-1) (35)
l-Q 2kT kT 20-2
23nd
Si z'W' 031 2
ln = - ln (D-l) - (l- )(l-2Q+2Q ), (36)
1-81 kT D-l

‘where Si is the fraction of molecules with orientation i.

The introduction of the parameter D leads to a few more features
for this model:

(1) When v<0.325, the orientational disorder (when S=1/D) occurs
Ioefore the positional disorder (when Q=1/2), and with a constant v, a
(larger D will shift the solid-solid transition, where the orientational
(disorder occurs, to occur at a lower temperature but will not change
‘the melting point.

(2) The threshold v value for the existence of a solid-solid
'transition increases when D increases, e.g., v=0.325 if D=2 and v=0.5

when D=60 .

(3) When D>2, both the solid-solid and the melting transitions

become first order .

(4) The entropy of a solid-solid transition increases with D and v.

27
(5) The relative volume change on melting becomes smaller than
that at a solid-solid transition when v>0.28 and D>2.
The number of orientations D for a molecule is dependent on its
symmetry. For example, D=6 for molecules with Td symmetry, D=16 for those
with C3v symmetry and D=20 for those with O3}" symmetry, etc. It is then

possible to predict the thermodynamic properties of a molecule with a

knowledge of v and its symmetry.

In NMR studies, the energy barriers for both the reorientational
and the translational motions can be determined with a reasonable
accuracy and therefore one should be able to test the validity of this

theory. This will be discussed later in our studies of plastic crystals.

THEORIES OF NUCLEAR MAGNETIC RELAXATION
I. Historical Review

A - The Bloch Equation and the origin of Magnetic Relaxation

After the first discovery of nuclear magnetic resonance by Purcell
Eiarmd his coworkers44, and independently by Bloch and his coworkers45, a
huge number of studies, both theoretical and experimental, have been
undertaken to develop the subject. In the earliest theoretical studies
0f NMR relaxation, time-dependent perturbation theory has often been
used to derive phenomenological equations of the macroscopic spin
lrlzmgnetization. Consider a system composed of N spin-8 nuclei placed in a
Si1teady magnetic field which splits the energy into two states a and B.

At thermal equilibrium, the spin populations in the on and 8 states (Na

and NB) follow Boltzman's law,

N
__a_= exp(AE/k'r) = 1+A_E.+l(AE_)2+ see... (37)

NB kT 2 kT
‘Vllere AB is the energy gap between the two levels and the second equality
is from Taylor's expansion of exp(AE/kT) . Equation (37) implies that NO
is bigger than NB at thermal equilibrium and there is a net magnetization
M along the direction of the applied field. If a random time-dependent
Perturbation field V(t) is applied to this system, a transition between
tho energy levels occurs and therefore the populations are changed. To

see the effect of V(t), time-dependent perturbation theory can be used

which gives the transition probability46 PoB as

28

29

 

POLB = 12] <eIV(O)ls><eIV(t)]e> exp(-AEt/M)dt , (38)
n

vulnere the long bar represents an emsemble average. The effect on the

sslpin populations can be expressed by

 

dNa

“—dt ‘ ”sped Napae; (39)
if we let Na: N+n and N3: N-n, then

dn _ _ "‘00

52'- -n(PaB+PBa) + N(PBa-Pa8) - — T1 , (40)

‘efliere T1: l/(PaB+P8a) and no: N(p8o'Po8)/(POB+PBO)' Equation(40) shows the
irleet populations of the spin system and therefore the net magnetization
‘\r:mries according to the first-order kinetic equation with a time constant
.1P21’ which is called the spin-lattice relaxation time.

Bloch47 has considered a classical mechanical description of the net
1:<)tal magnetization, which is represented by a vector M. In a steady
Unzignetic field R0, along the z direction, M will precess about 2 due to a

1t<>rque produced by Ho. The motion of'M can then be described by
— = (N x no). (41)

E3)? replacing n with the net magnetization Mz (Mz=yhn), Equation (40)

I><ecomes

.__=—-—————' ’ (42)

30

avIIere Mo is the net magnetization at equilibrium. Assuming that the
+

1tioansverse components of M also follow Equation (42),but with an

tacmuilibrium value of zero since the external magnetic field is along

2: axis,

___.Y_d”x: = 'JMX . (43)
dt T2

(22(3mbining Equations (41), (42) and (43), we obtain

 

+ 1 +3 k(M -M )
sz mxH)-MXMY ZO

dt 0 T2 T1

\efliich is the famous Bloch equation47

As we have seen above, the relaxation of a spin system is caused by
‘Irzandom time-dependent fields. In dense phases, these may come from
(izifferent sources which are brought in by thermal motions of the
nnxalecules with a suitable time scale, e.g., reorientation and diffusion
5L1: liquids, lattice vibrations in solids, collisions in gases and some
11<>rsional motions within molecules. These different sources of time-
<1<3pendent random fields give rise to different relaxation mechanisms
‘Vliich will be discussed later in this chapter.

13.. Density Matrix Formalism of Magnetic Relaxation

The above derivation of Bloch's equation depends on calculating
‘5}ie probability of population transitions and assumes that each spin
531:ate can be described by a population. A more general method, which is
Sidiscussed here, uses the quantum statistical density matrix formalism48-50.

ITt can be shown that the above derivation (section A) is a special case

31
avliich occurs when the density matrix is diagonal and the population of
spin states is described by the diagonal elements.
In quantum statistical language the time evolution of the statistical
ensemble can be described by a density operator p(t) satisfying the

Liouville-van Neumann equation
mm = [J-C,p(t)] = Lo(t), (45)

where the bracket stands for the commutator and JC is the total
Hamiltonian operator of the system. L is the Liouville operator. In this

formalism, any physical quantity A has its expectation value given by
<A> = Trpr(t)}. (46)

Consider a spin system embedded in a bath(lattice) formed by its
ssaarrounding molecules, the total Hamiltonian can be written as

JC = 3132+ 3c + J-CI, (47)

L

Where JCz is the Zeeman term, JCL is the lattice Hamiltonian and JCI
includes all of the interactions between spins and lattice. By realizing
that the lattice has very many degrees of freedom and its energy levels
are almost continuous so that it can be treated classically, and also that
the coupling between the spin system and the lattice can be assumed to be
Small so that the spin density operator o(t) can be obtained by tracing

0 (t) over the lattice degrees of freedom, one can obtain the equation of

ulotion of the spin density matrix element50

32

aaa.(t) = -i[Jc,,o(t)]aa. -23 (t), (48)

R 7 IO l
88! an 88 88
makiich has been used as a master equation for the derivation of NMR

‘Jreelaxation theories. Here, ROO'BB' are called the elements of the re-

laxation matrix (tetradic) and are given by

Roo'BB' = Jaee'e'(wee) * JOBa'B'(wo'B')

 

" Ga'B'g: JYBYGWYG) ' 6018; JYG'YB'(wa'Y)’ (49)
‘eflmere
_ . in. .
JGG'BB'(w) fodr e Gaa'BB'(T) (50)
Elrid
Goo'881 = (“IJCI(O)|“'><BIJC1(t)IB'>° (51)

‘VViLth a specific interaction Hamiltonian JCI, the correlation function G
<=Eln be calculated by choosing an appropriate model of molecular motion

6111d the relaxation matrix elements can be obtained.

II. Mechanisms of Nuclear Magnetic Relaxation

As we have discussed, any time-dependent fluctuating local field
Elround the spin system will contribute to the relaxation of the spins.

V'G3 outline here some of these which are important to nuclear magnetic

33

sp in relaxation .

[\H. Dipolar Relaxation

 

The Hamiltonian for dipolar interaction between two spins I1 and I2

can be written as51

2 = +

ch " Ylen T1°U°Iza (52)
inrlnere U is the dipolar coupling tensor. It is traceless and can be
(clziagonalized by a coordinate transformation. The three principal
1<2<3mponents Of the diagonalized dipolar coupling tensor are

U1 = (1-3cosze)/r3, (53)

U2 = (sinecosee'mflr3 (54)
Elrid

u3 = (sinZGe-21¢)/r3, (55)

‘Vliere r, 6 and O are the three spherical polar coordinates of the inter-
Illiclear vector in a space-fixed coordinate system. Since T1 is related
1t<3 the transition probability, which has a form similar to Equation (38),

We rewrite Equation (38) as

 

p (a) = Ll<a1A|e>|2f <oIV(0)IB><oIV(t)IB>e'iwtdt, (56)
OB n2

34
where A is the time-independent part of JCd and V is the time-dependent
part of JCd.
If Il=Iz=k, then the spin system has three energy levels for four-
1
I: _

1
2 (“182+“281

where the last two states are degenerate. It is easy to show51 that the

eigenstates Yl=a1a2, w-1=B182’ To:

spin-lattice relaxation rate R1 of this spin system is

R1 = P1,O(mo) + 2P1,-1(2“o)’ (57)

where “O is the energy gap between adjacent levels in unitsof M and P1,0,
P1”1 can be calculated by Equation (56). The integral in Equation (56)
is the Fourier transform of a correlation function of the time-dependent
part of JCd, or the power spectrum. Assuming that the molecules undergo

an isotropic Brownian motion with r, the internuclear distance, fixed,

R can be calculated to be52

 

 

 

l
3 4 2 T 4T
R1 =——*"( c + ° ). (38)
t 10 6 1+ 2 2 1+4 2 2
ro r onc more
where Tc is the correlation time of the motion. If r is also time-
dependent, it should be included in the correlation function and the
evaluation of R1 becomes more difficult. Bloembergen, Purcell and
Pound52 have considered an isotropic diffusion model and found that
R1 which deScribed the intermolecular dipolar relaxation can be
expressed by
4 2
311
R1 - IO'Y :0”, (59)
tran

 

35
where N is the number of molecules per cm3, a is the molecular radius
and D is the self-diffusion coeficient.
Torrey53 has used a random-walk model in which the molecules undergo
a random flight of distance 2 and found that
nT
R1 = 11.292y4h21(1+1) —‘—1- , (60)
tran 2
where n is the number of spins per unit volume and Td is the mean jump

time and is related to the self-diffusion coefficient by

2
_ gr > , (61)

 

where r is the jump distance.

AbragamS4 considered also an isotropic diffusion model but obtained

4 2
R1 - m . (62)

tran 503

HertzSS reconsidered Torrey's random-walk model and derived an expression

NT ‘ 2
d (1+ 2a

tran a3 SDTd

 

4
R = 3ny4M21(I+1)

1 )- ' (63)

As we can see these equations all have the same dependence on a, D and N,
but the numerical factors are all different.
In the case of two nonequivalent spins, a factor of 2/3 should be

introducedss, that is, in the extreme narrowing limit,

36

YZYZKZ
his = 4—2—— S(S+1)Tc (64)
r0t rIS
and
2
2 Y
R15 = - i 5(5+1) (RII ). (65)
3 2 1
tran YI tran

When the molecular motion is not isotropic, then Tc should be
56 . .
replaced by f(Q,D) , where Q=6,¢ are the Euler angles of the interacting
vector with respect to the molecular axis system and D is the diffusion

tensor. This will be discussed later.

B. Quadrupolar Relaxation

Nuclei with spin 8 usually have a quadrupole moment which can react

with the local electric field gradient. The interaction Hamiltonian is

JCQ = I 6(t) 1, (66)

where Q is a tensor with elements proportional to the quadrupole
2
moment Q and the electric field gradients BZV/axz, aZV/ayz, a V/axay,
54
etc. Using density matrix formalism, we can obtain , in the limit of

<<
more 1,

3 21+3 2 2 2
R = -{ -———-———9(1+n l3)(e Q/M) T . (67)
1Q 40 12(21_1) qz c

where qZ is the 2 component of the electric field gradient tensor and

37

n=(q -q )/q is the asymmetry parameter of the electric field gradient.
x y 2

C. Scalar Relaxation

 

In the study of proton relaxation times in aqueous solutions of
++ +++ . -
Mn and Gd ions, Bloembergen found that TzfiTl and is dependent on
the observing frequency. This was recognized as due to the strong scalar
coupling interaction between proton and metal ions. The Hamiltonian can

be written as
JC = I°A°S (68)

. . . 54
land it contributes to the relaxation terms

 

2 T
2A 1,S ‘
R1 = -—-3 S(S+1)———2 (69)
sc 1+A(l)IST1,S
and
2 T
A
R =_s(S+1)(T 4 1,3 ) (70)
2 3 1.8 “A 2 T2 1
5° “13 1,5
where A is the scalar coupling constant in radians/sec, T1 S is the
relaxation time of spin 8 and AmIS is the difference between the

resonance frequencies of spin I and spin 8. Spin S may be the total
spin quantum number for one or more unpaired electrons. It may also be
the spin of a fast relaxing nucleus with a large coupling constant
(e.g., Br nuclei in SnBr4S7) which will also contribute a considerable

scalar coupling term to the relaxation rate.

38

D. Chemical Shielding Anisotropy Relaxation

Chemical shielding is not necessarily isotropic but may be
represented by a tensor; the three principal components 01,02 and O3
can be different and therefore give a time-dependent modulation of the

Zeeman energy
3c = -1h(l-O)HOI + tho-81-I, (71)

where O is the average chemical shielding and :1 is the anisotropic
part of the shielding tensor. If 3' has three principal values Oi, 05

and O' the contribution to R1 will be51

3,

 

2T
R - fie +O'2+O'2) , (72)
1csa 10 1 2 l+m2T2
o c
. = . . . 5
if 0 is axially symmetric then
2 2 2 Tc
R1 = Ig-wo(Ao) 2 2 (73a)
csa 1+w T
0 C
and
' 6T
R = — w2(A0)2(8T + ——c—-). (73b)
2 9O 0 c 2
csa l+onc

E. Spin-Rotation Relaxation

 

52
In the study of proton relaxation in H2 gas, Bloemgergen et al.

found that the spin-rotation interaction makes a large contribution to

39

R1. This is given by

 

R = 2Y2(%H'ZJ(J+1) + 3H"2 J(J+1) (74)

lsr (zJ-I)(2J+3) Tc’
where H' is the field at one proton due to molecular rotation and H" is
the dipolar field due to another protons.

The first discovery of spin-rotation relaxation in liquids was by
Gutowsky and coworkers58 who found that several Freon's had shorter
relaxation times for 19F than for 1H and a different temperature
dependence. They attributed this behavior to a greater spin-rotation
interaction in the case of the fluorine nuclei and gave an expression
for Tl,sr in a linear molecule. Johnson and Waughsg’6o also estimated
that there is a significant spin-rotation relaxation contribution in
liquid methane. A few years later, Hubbard61 successfully derived a
profound theory of spin-rotation relaxation for a spherical-top
molecule undergoing an isotropic Brownian reorientational motion and a
Langevin type of motion for the angular velocity. Using standard density

matrix formalism, he obtained

ZIkT 2 T1 2 2 T12
R1 = ———§'((2CL+C”) -——7—§'+2(C“-QL)'_—7f77—- (75)
SI 9” 1+on1 1+on12

where C , C” are components of the spin-rotation tensor, T1 is the

-1 -1 -1 ‘ .
12-Tl +T2 , here T2 is the

reorientational correlation time and I is the moment of inertia. In the

angular momentum correlation time and T

limit of onl’ m T <<l and T <<T Equation (75) becomes

0 12 l 2’

IkT 2 2 ZIkT 2
R - — (2C_L+CH)T1 - 1T CeffTJ ,

(76)
1sr M2

onto

40

2 2

where we have used T for T and c: for (2C1+CII), which can also be

J 1 ff

written as

2 _ }_ 2 2 2 _ 2 2 2
c f - (3(cl|+2c )) + —(c|l-c ) _ c +§AC , (77)

cf 1_ 9 _L
where C is the average spin-rotation constant and AC is the difference
between the parallel and the perpendicular components of the spin-
rotation tensor. From Equation (77), it can be seen that Ceffzc is

always valid. In this limit, T and Te(=T2) are related by

J

T T = -- . (78)

HuntressS6 has attempted to derive an expression for the spin-
rotational relaxation rate in molecules undergoing anisotropic motion.
His derivation was later corrected and modified by Wang62 who con-
sidered both symmetric-top and asymmetric-top molecules and obtained
several equations which are summarized as follows:

(a) For a nucleus at the center of a spherical-top molecule

 

2
ZIkTC -

R1 = 2 (2D+T31) 1 2 Z—I-1(2-I-C2TJ , (79)

sr h n

where D is the rotational diffusion coefficient.
(b) For a nucleus off the center of a spherical-top molecule
R ZIkT c2 (80)
= —— T .
1sr n2 eff J

(c) For a nucleus along the symmetry axis of a symmetric-top

41

2
R kT (II llC2 [+21 CT (81)

1sr= 3M2 1.1.1.)

(d) For a nucleus off the symmetry axis of a symmetric-top molecule

2kT Tl!
R1= 372 [III(CII+ sin 226) 1+Zpifll
J. Tll
1+ZDLIL'+ 1+(DleDl|)Tll

51'
T

 

 

+I (stinzecosze)(

T
+I ((C - sin20)2+C2) 1‘ (82)

l_ l_ i_ 1+(D14D'|)qu’

where B=(C -C
lll

coordinate systems of the diffusion tensor and the spin-rotation tensor.

) and e is the angle between the z axes of the principal

(e) For a nucleus in an asymmetric-top molecule.

This is the most difficult case and only has a simpler form
under special conditions. For example, if the moment of inertia and the
rotational diffusion tensor have the same principal coordinates, if the
cross-correlation is neglected and if the motion is diffusive in every

direction, then

(83)
R=15? ::2§Zi:l icoimi

where a, i = x, y or 2.

F. NMR Relaxation due to Anisotropic Reorientation

When the molecular reorientational motion is anisotropic, the

rotational diffusion coefficient is no longer a constant, therefore

42

6
Equation (22) becomes 5

8P(O,t) =

at — (Loo-L)p(o,t). (34)‘

In a molecular coordinate system which diagonalizes O , Equation (84)

can be written as

8P(9,t) _ 2
This can be readily used to calculate the time correlation function and
therefore the correlation times. Both Woessner66 and Shimizu67 have used

9

this approach to derive the dipolar relaxation time. Huntress5 has
reviewed their calculations and applied the method to quadrupolar
relaxation. A general modification when there is motional anisotropy is to
replace Te by f(Q,D), a function of the components of the rotational
diffusion tensor and of the Euler angles of the concerned vector in a
molecular coordinate system which diagonalizes the diffusion tensor.

Some useful expressions for f(Q,D) are:

(a) For a spherical rotor
£(O,O) = 1/6D. (86)

(b) For a symmetric-top rotor

3(O -D ) 3(D -D )
f(O,D) = —l—-(1+ ———J-——LL— sin26(l+ .__J=._LL_ sin26)). (87)
601. (SleDll) 2(Dlelel)

(c) For a planar asymmetric rotor

43

2 (Ox-Dy)2 2
f(n,D) = _((Dx +D 5) cos 2O +(Dy +D s)sin O- ————————cos Osin 2O), (88)
4DR (Dz Ds)
where Ds= %-(DX+Dy +D z) and DR= 3(D xDy+DyDz+Dz Dx)°

III. Relaxation and Fourier Transform NMR Spectoscopy

 

A. Theories in Fourier Transform NMR Spectroscopy

 

‘ The frequency response function and the unit impulse response form a
Fourier transform pair. Both functions characterize the entire system
and thus should contain the same information. This idea, although well
known in information theory7o, was not applied to NMR spectroscopy until
about twelve years ago when Ernst and Anderson71 successfully demonstrated
the first Fourier transform NMR spectrometer. Several advantages have
been found for this pulse Fourier transform technique: (1) It brings a
great time saving and makes the Observation of low-sensitivity nuclei or
low-concentration spins possible. (2) More dynamical information can be
obtained since all spins are excited simultanuously. (3) Higher
resolution and more accurate frequency calibration can be easily obtained.

In a Fourier transform NMR experiment, a strong rf field H1 with a

short duration is applied to the spin system; the magnetization in the

rotating frame will then precess about the effective field

Heff = — 31(6) -w) +(YH1)2);E ' (89)

where w is the rf frequency and ”i is the resonance frequency of spin i.

In order to assure that every spin is excited equally, yHl has to be

44

much greater than the entire spectral width A, i.e., yHl>>2A . If H1 is

a 900 pulse, then 7H1tp=T/2 where tp is the pulse width.‘Therefore the

pulse width has to be short enough such that

tp << 1/4A. (90)
In an actual case, a series of pulses are applied to the system and the
free induction decays (FID's) are coherently cumulated to increase the
signal-to-noise (S/N) ratio. If the pulse interval is represented by T,
then the pulse train will contain numerous sidebands of the rf carrier
frequency w, and these will be l/T Hz apart. This pulse interval T
determines the density of the sideband and therefore the resolution
and the faithfulness of the representation of the spectrum.

Since Fourier transform NMR is a computer assisted experiment, some
theories of sampling72 have to be considered. First of all, in order to
perform a Fourier transform, the transient FID has to be digitized as
discrete data points. From information theory, a sine wave must be
sampled in time at least twice during each cycle to be properly repre-
sented. Therefore , the sampling rate should be at least twice the
spectrum width to be observed. Second, since the sidebands of rf pulses
are equally distributed on both positive and negative sides of the
carrier frequency, but with opposite phases, the resulting spectrum
will have possible back-folded peaks if only one phase detector is used.
The development of the quadrature phase detecting technique74“80 and
the use of crystal filters have been able to alleviate this problem.
Third, the frequencybdependent phase error due to the time lag of

detection has to be considered. Other theories of computation in Fourier

45

transform NMR have been discussed by Cooper69.

B. Fourier Transform Nuclear Magnetic Relaxation Studies

 

The most striking advantage of the Fourier transform NMR technique
is the rapidity of data collection which has made possible the studies
of nonequilibrium states of nuclear spin systemsgl. Measurement of
relaxation times in a spin system containing more than one closely-
spaced transition was very difficult before the Fourier transform NMR
spectroscopy was developed. Vold §£_§1382 had first succeeded in
measuring the spin-lattice relaxation time of a multiline system using
a pulse Fourier transform NMR technique. It was found thereafter that
many other pulse methods can also be combined with the Fourier transform
technique to study nuclear spin relaxation. The initial work of Vold 35_
.51, using a nonselective 180o pulse to invert the system into a state
of negative spin temperature, and then observing the recovery to equili-
brium by means of Fourier transforming the FID's generated at different
times, is called the inversion-recovery technique (IRFT). The transformed
spectra are called partially relaxed spectra in which each transition

follows a recovery equation of the form

Mi = Moi(1-2exp(-t/Tli)), (91)
where M1 is the amplitude of the magnetization of spin i and T1. is its
1
characteristic spin-lattice relaxation time.
Markley EE_§1383 have devised a saturation recovery technique

(SRFT) in which the system is brought into a state of infinite spin

46

temperature by a 90° pulse, then the off-diagonal elements of the spin
density matrix are destroyed by violating the magnetic field homogeneity
and the recovery of each spin is again detected by observing the FID's

generated at .different times which follow the equation

Mi = MOi(1-exp(-t/T1i). (92)

Freeman and Hill84 have also devised a progressive saturation
technique (PSFT) to observe the dynamic equilibrium reached by means of
a repetitive pulse sequence. The cumulated equilibrium magnetizations
have amplitudes which depend on the interval between adjacent pulses in
the sequence and follow Equation (92) .

Many other new techniques have been developed to utilize the great
time saving advantage of Fourier transform NMR. Canet, Levy and Peat85
InOdified Vold's inversion-recovery technique to. develop a fast inversion-
IVectovery Fourier transform (FIRFT) method in which the long time delay
between each pulse sequence is shorter. Xaptein gtfl.86 also modified
the IRFT method by changing the observing pulse to 30° in order to do a
sing 1e-scan T1 measurement (SSFT) . Gupta87 developed a variable nutation
method in which T1 can be obtained from the dependence of signal ampli-
tude upon the nutation angle of the pulse. Other methods”.93 are all
balSed on different arrangements of the pulse sequence. Levy and Peat
have reviewed several different techniques for T1 measurement of 13C
in organic compounds.

The Fourier transform technique has also been applied to determine

the spin-spin relaxation time of each individual peak of a complex .
SPeCtrumgs. This method (called Fourier transform spin-echo spectroscopy)

120,121

miilizes a Carr-Purcell pulse sequence to generate a spin-echo

47

train and Fourier transforms the FID following the last echo maximum96'98.

However, this method cannot be used in case there is homonuclear
coupling since it tends to modulate the spin echoesgs. A number of

96-106 on the effects of this modulation of spin echoes by

studies
homonuclear coupling, and sometimes by the effects of chemical exchange,
have been reported. An excellent review has been given by Freeman and
Hill107 recently.

Several authors have constructed single-scan pulse sequences to

92,93 9

measure both T1 and T at the same time . Edzes 2 has summarized

2
these methods and given a detailed error analysis.
Table 2 gives a summary of the pulse sequences used in relaxation

studies.

C. Effects of Cross-Relaxation and Cross-Correlation

 

The existence of two or more nonequivalent nuclear spins very often

108 first observed that in

complicates the relaxation studies. Solomon
hydrofluoric acid the recovery curve of the longitudinal component of
the magnetization was nonexponential due to the cross-relaxation effect
caused by the other nonresonant spin. Noggle109 also analyzed the
cross-relaxation effect of coupled homonuclear spins in 2,3-dibromo-
thiophene. In the relaxation studies of both proton and fluorine nuclei
in fluoroform, Hubbard110 found that cross-correlation also caused a
nonexponential decaying of the magnetization. Werbelow and Marshall111
extended Hubbard's calculation and applied it to both two-spin systems
and to the more common cases which have methyl or trifluromethyl groups

attached to a large molecule. Campbell and Freeman112 have reviewed the

influence of cross-relaxation on NMR spin-lattice relaxation times and

48

Table 2. Pulse methods for relaxation studies.

 

 

Methoda Pulse3sequenceb Reference

IRFT(T1) (180x-mT-90x-FID-ST1)n 82
SRFT(T1) (QOx-mT(FGP)-90x-FID)n 83
PSFT(T1) (QOX-FID-ml’)n 84
FIRFT(T1) (180x-mr-90x-FID-T)n, T<5T1 85
SSFT(T1) 180x—(I-6-FID)m 86
Rapid PSFT(T1) (em-FID-T)n 88
cp (T2) 90x-(T-180x-T)m-FID 120
CPMGCTZ) QOX-(T-18Oy-T)m—FID 121
FFS(T1) QOX-T-90_x-T 590

Triplet
(T1,T2) 180x-(t-90x-1-180x-T-90x)m 122
180x-(t-90x-1-180_x-T-90x)m 123
lsny'(t'gox‘T'180y'T‘90—x)m 123

Quartet
(T1,T2) 180y-(t-QOX-I-180y-21-180y-I-90-x)m 92
Double Triplet 180y-(t-(90x-1-180y—t-90_x)2)m 92

 

aSee text for the meanings of the abbreviations.

bm Denotes the number of observations; n denotes the number of
scans for a signal accumulation; T denotes the variable delay;
x and y are used to specify the direction of the rf pulse applied.

49
found that in weakly-coupled spin systems there might be an error of

~10% induced in the T1 measurement and that this should be more serious

for strongly-coupled systems. Vold and Chan99 have studied transverse
cross—relaxation which sometimes causes T2 to be longer than T1. A

general study of transverse cross-relaxation in heteronuclear coupled

AXn and AXY systems has also been reported by Vold and Voldlos. Recently

a large amount of work on the theories of cross-relaxation and cross—

correlation effects in 13C NMR relaxation studies has been reported by

Grant and his coworkersllS-llg.
To see these effects, let us consider a simplest two-spin case.

Using Redfield's master equation, Equation (48), we can write

where Sii(t)=Tr{Iz_°;} and mi is the Larmor frequency of spin i. The
1

Riijj's are the elements of the relaxation matrix and are represented

by the following equations:

- l. .
R1111 ' 3J12(w1‘“2)+J12(w1)+2J12(“1+w2)+R1 (94)
R --1-J ( )+J ( )+2J ( + )+R' (95)
2222 ' 3 12 “1'“2 12 “1 12 “1 “2 2
R - R - la (w w )+2J (w +w ) (96)
1122 2211 3 12 1 2 12 1 2 ’

where Ri and R5 represent the nondipolar random field contribution to

thezrelaxation of each spin. Equation(93) can be rewritten as

50

s (t) iw +R R S (t)
[.11 ] g _ [ 1 1111 1122] [ 11 ‘] (97)
522(1) R 1w2+R 822(t) ,

2211 2222

The solution of Equation (97) can be obtained in case the initial
conditions are known. If spin 1 is observed by applying the inversion-
recovery method, then (denoting 11 for 1111, 12 for 1122 and 22 for

2222)

31.5? = -28?(b1exp(-a+t)+(l-b1)exp(-a_t)) (93)
82-53 = 28$b2(exp(-a_t)-exp(-a+t)), (99)

o . . . . .
where the 81's are the magnet1zat1ons at thermal equ111br1um and

_ 1 1 2 2 ‘1
at ' 2{R11+R22)i(Z{R11’R22) +R12) (10°)
b1 = (a+-R22)/(a+-a_) { (101)
b2 = R12/(a+-a_). (102)

It is obvious that each spin will have its magnetization decay bi-
exponentially unless the off-diagonal element of the relaxation matrix,
R1122 is zero. In two-spin systems the cross-correlation is absent, but
in three-or four-spin systems there will be spectral density functions
involving three spins included in the elements of the relaxation

matrix117

EXPERIMENTAL

I. The NMR Spectrometer

 

A. General Description of the Pulse System

 

The NMR spectrometer used in this study is composed of a Varian
DA-60 spectrometer system with 12 inch magnet, a regulated power suply
to generate a 14.1 kG field and a Varian V-K3506 Super Stablizer. To
this has been added a highly modified NMR Specialties MP-1000 pulse
system including two P-118A rf units, a P-103A power amplifier, a
P-lOOA receiver and P-llSA, P-114A power supply units, a homebuilt
wideband power amplifier, a few homebuilt tuned and wideband probes
and a Nicolet 1083 minicomputer with several homebuilt interfaces.
Observation of 19F nuclei utilizes all tuned components operating
at 56.45 MHz. The observation of other nuclei utilizes a multinuclear
system modified as described by Traficante gt_§13125 For long-time
signal averaging, a field/frequency external lock system is used. The
block diagrams of the overall system for 19F observation and multi-
nuclear observation are shown in Figures 2 and 3. The detailed des-
cription of the multinuclear system will be given in the next section.

For 19F observation, the rf source for both the transmitter and
the reference of the phase detector is a frequency synthesizer. The
rf is sent to the rf unit in which it is amplified by two buffer
amplifiers composed of two 7587 Nuvistor tetrodes in a push-pull
configuration. The outputs from the rf unit are gated by two external.

dc pulses ( 0 V to ~15 V) then sent to the P-103A power amplifier to

$1

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reach to at least 100 W. Only the low power stage which contains a
3E29 tube is used for this work and it is also gated by the same dc
pulses. The arrangement of two crossed-diode boxes and a few quarter
wavelength cables is to isolate the receiver from the transmitter so
that the preamplifier would not be saturated or damaged by the strong
rf pulses while transmitting. The signal induced from a single-coil
probe is sent to a tuned preamplifier then to the P-lOOA receiver and
then is phase detected. The detected dc signal can be further ampli-
fied or directly sent to the computer in which it is digitized and

stored.

B. Multinuclear Modification

 

The modification of the tuned pulse system for observing various
other nuclei is by means of a simple rf mixing technique. Figure 3
shows the diagram of this multinuclear system. The rf unit P-118A
is equipped with a crystal oscillator which provides the 19F Larmor
frequency 56.4 MHz. The frequency synthesizer is used to provide a
mixing frequency which is the difference between the observing fre-
quency and 56.4 MHz. The attenuator after the rf unit is used to
match its output amplitude with that of the mixing frequency. The
mixed rf is sent to two power amplifiers, an ENI-406L and a homebuilt
PA. The final output is at least 100 W which is sent to a wideband
probe with a single-coil configuration. The signal picked up is
carried by the observing frequency which is further mixed back to

56.4 MHz so that it can be detected by the tuned receiving components.

All the rf mixings are carried out by a few double balanced mixers

55

ZAD-l (from Scientific Components Corp.). Two wideband preamplifiers
are used before the receiver and an image filter is used to filter

out any higher harmonics.

C. The Probes

The advantages and disadvantages of using a crossed-coil or
single-coil-probe have been discussed by several authors73’74’126’130.
The advantages of the crossed-coil probe are the minimum rf leakage

and the optimum sensitivity attainable, but the power efficiency is less
and the mechanical complexity is greater than a single-coil probe. The
latter has, however,a serious rf leakage problem which must be overcome
by more delicate circuit designlzg. In this work, the probes used are .
all single-coil type. For 19F observation, the probe is made for 5 mm
sample tubes. It has a coil wound on a piece of 8" long, 0.25" i.d.
quartz tube with seven turns of #26 magnet wire and tuned to 56.4 MHz

131

with Q approximately equal to 30 . In order to do a few multiple—

pulse experiments which need a field-gradient pulse, for example, T1

132

measurement using the homospoil technique , modified hybrid relaxation

time measurementlss, self-diffusion coefficient measurement using the

PUISed field-gradient spin-echo techniquelM'147

, etc., two types of
field-gradient coils were constructed in the probe. The first one is

a pair of anti-Helmholtz coils with coil axes pointing against each
other along the direction of static magnetic field. Each coil consists
of 15 turns of #30 magnet wire wound on a Delrin form with the configu-
ration proposed by Tanner135 to produce the most linear gradient at the
144-146

sample (see Figure 4a). The second one is a quadrupole coil

built on a cylinder (see Figure 4b) which can slide over the sample

56

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\

.<

 

 

Figure 4. Field-gradient coils in the probe: (a) anti-Helmholtz
configuration; (b) quadrupole configuration.

S7

coil. Each quadrant consists of S strands of #30 magnet wire which can
easily be increased to provide a higher field gradient. The latter type
has several advantages over the former in that it is relatively easier
to construct, the configuration is not so critical to the linearity of
the field gradient produced, the main component of the field gradient
can be changed to a different direction simply by rotating the cylinder
and the dc resistance and the inductance of the coil are less so the
eddy current formation is minimized and a higher field gradient is
attainable for equal current supplied144’145.
The wideband probe for observing other nuclei is of the insert-
type. The coil is wound on a cylindrical quartz tube which can slide
into the probe body in which the other part of the probe circuit is
permanently built. In order to keep the impedance matched through a
wide range of the observing frequencies, 2 MHz to 25 MHz, the coil is
tapped at about one turn from the end. The exact tap point is deter-
mined by several tries until the impedance is measured to be 50 ohms.

13C, 14N and 20

The detailed circuit diagram is given in Figure 5. For
observation in this work, the insert was made by winding 24 turns of
#28 magnet wire on a 0.5" o.d. thin-wall quartz tube to give ~12 mm
coil length. The tap point was made at one turn from the top and the
impedance was measured to be from 30 to 75 ohms for frequencies from

4 to 20 MHz. Care has been exercised to obtain a uniform coil spacing
and pitch. With this insert, a capacitor pad (220 pF) is needed for14N
observation. The Q of the circuit is found to be 30 to 50 depending on

the frequency operated. With about 300 W transmitting rf the 900 pulse

width is about 8 usec for 13C, 15 usec for 2D and 30 usec forl4N.

58

 

 

? TO PREAMP

'4? :

  
    

llll lllll

  

T
f

he

Figure 5. The inside cross-section of the wideband probe: a, sample
tube; b, spinner; c, BNC connector for the capacitor pad;
d, H1 tuning capacitor; e, heater sensor connector; f, N2
gas inlet; g, H1 rf cable; h, heater sensor; i, copper-
constantan thermocouple; j, decoupling coil; k, sample-
coil; 1, external lock probe; m, decoupler tuning capaci-
tor; n, LC circuit of the probe.

59

D. Computer Interfaces

 

In order to have the computer drive the spectrometer, an interface
must be used to convert the logic levels of the triggering pulses
generated from the computer to those required by the spectrometer.
Also, since in this work the computer can only provide triggering
pulses with accurate time intervals but not the duration of the pulses
themselves, the interface should also have the function of adjusting
the pulsewidths. A well-constructed interface131 is used for these
purposes. This interface converts two I/O (input/output)pulses from
the Nicolet 1083 computer (0-3 V, 500 nsec) to two 0-15 V pulses with
pulsewidth adjustable from 1 to 100 usec. In addition, a T1/T1p switch
control can convert the second pulse to be a terminating trigger for a
long pulse activated right after the first pulse. A detailed circuit
description was given by Wright131.

In those experiments which needed field-gradient pulses, a cons-
tant current supply was used. Figure 6 shows the circuit diagram of
this unit. It contains a pulsewidth adjusting circuit based on a mono-
stable multivibrator SN 74121; the adjustable range is from 100 usec
to 10 msec. The output from pin 1 of the SN74121 is used to switch the
transistor 2N3415 off. A voltage divider is used to control the base
voltage of the transistor 2N3053, which in turn controls the current
passed through a power transistor 2N6OS7. The maximum current attain-
able is about 10 A.

In hybrid relaxation studies, the pulse sequences require the
adjacent 900 pulses to be 1800 out-of-phase with each other. This
is achieved by use of the phase shifter circuit shown in Figure 7.

The phase shifter is triggered by the same pulse, PULSEZ, as triggers

60

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62

the 900 pulse. The output from the flip-flop SN74109 is converted into
:15 V and acts as the gating pulse of a double balanced mixer which all

the rf pulses pass through before going to the power amplifiers.

E. External Lock System

 

When performing a long-time signal averaging, the magnet field has
to be very stable. This is achieved by a homebuilt external lock system
which uses the original DA-6O console as the lock circuit. The lock
probe was made of a single coil tuned at the proton frequency131. The
lock sample used is water doped with copper chloride. A more detailed

description has been given by Wright13l.

F. Temperature Control and Measurement

 

The temperature of the probe is controlled by a Varian V-4343
temperature controller and a Varian heater-sensor inserted into the
Dewar. A stream of dry N2 gas is precooled or preheated before being
sent to the probe. Temperature detection is achieved by using a
copper-constantan thermocouple with the junction lead placed about
0.5 cm below the bottom edge of the coil (see Figure 5). The actual
temperature of the sample is calibrated relative to the apparent
readout by putting another thermocouple directly into the sample tube.
The measured difference was found to be less than 0.50K. The accuracy
of the temperature measurement was within i0.5°K except at very low

temperature ( below -100°C) where it was iloK.

63

G. Heteronuclear Decoupling and the Variable Field Lock System

 

Heteronuclear decoupling in 13C NMR provides many important

148'151. In relaxation studies, the cross-relaxation effect

advantages

due to dipolar interaction between nuclei can be eliminated by the
113

decoupling technique . In addition, the nuclear Overhauser effect

(NOE) can be measured148’149

and the dipolar relaxation rate can be
estimated.

The requirements for heteronuclear decoupling are normally a
decoupling coil built over the sample coil and a decoupling rf source.
In this work, the decoupling coil is a Helmholtz coil wound on a 0.75"
o.d. quartz tube which is slid over the observing coil. In order to

do a dynamic Overhauser effect experiment152

, a decoupling box is made
which provides a band of closely-spaced pulses to trigger the de-
coupling rf. The purpose of using closely-spaced pulses rather than a
cw pulse is to minimize the duty cycle of the rf power amplifier P103A.
Figure 8a shows the schematic of this decoupling box. The dual mono-
stable multivibrator SN74123 acts as a self-triggering pulser which is
turned on and off by the output of flip-flop SN74107 which is in turn
controlled by two input pulses. The final output.pulses are all ampli-
fied by common-collector transistor amplifiers which have a voltage
gain of approximately unity but a large current gain. The timing
diagram is also shown in Figure 8b.

The modified setup for the heteronuclear decoupling experiment
is shown in Figure 9. An additional frequency synthesizer is needed
for the decoupling rf source. For high power decoupling, better rf

isolation can be achieved by the time-shared mode provided by the

decoupling box which blanks the receiver while the decoupling rf is

64

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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PULSE B A C B A

C
Pulse .
Sequenccl T k 1: . T K I

I’

01.22:: GNU H H H H ,1; II
V 08': —1' flflfl H ...................... r

 

 

 

 

 

 

 

 

 

 

 

Figure 8. (a) Circuit diagram of the decoupler gating box (all'

resistors are in ohms, all capacitors are in uF).
(b) Timing diagram of the outputs.

6S

 

PULSE A Pulse width to normal trans-
controller mission line

 

 

 

to receiver

 

 

 

 

 

 

 

 

 

 

 

 

 

PULSE B V “"c
Decoupler P Receiver
PULSE C Gating Box r
Gate
RI" I PA
Gate Cato
signal
Decoupler L R to decoupler
\ 1 ———o
RF Source DBJ P°A' coil

 

 

 

 

 

 

Figure 9. Block diagram of the heteronuclear decoupling setup.

 

V 4311

 

RF UNIT '

 

 

 

 

[PR5 Mi

 

Figure 10.

66

 

 

 

 

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TIME
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BOX

MOD

 

 

Block diagram of the variable field lock system.

67
transmitting.

It has been demonstratedlss’154

that by varying the frequency
information about the contribution from the chemical shift anisotropy
and from the scalar coupling to the total relaxation rate can be
obtained. In order to make the varied field stable enough to do long-
time signal averaging, a field/frequency lock system was also construct-
ed. It consists of a wideband probe with a tapped coil tunable from S

to 30 MHz (aided by an external capacitor pad), a lock sample contain-
ing HZO/DZO (l;4 ratio by volume) doped with nickel chloride and a

frequency mixing network similar to that used in multinuclear observa-

tion. Figure 10 shows the diagram of the variable field lock setup.

. 11. Measurement of the Relaxation Times

 

A. NMRLX-- A General Program for Computer Controlled Pulsed NMR

 

Relaxation Studies

 

All the relaxation studies in this work were performed by using a
general program NMRLX written for a Nicolet 1080 series minicomputer.
The detailed instructions for using this program are given in
Appendix A.

For linewidth and second moment measurements, program SECMFT was
used. The instructions are given in Appendix B.

Several authors have studied the systematic errors in NMR relaxa-
tion studies caused by the static magnetic field inhomogeneitylss, the

H1 inhomogeneity156-159, the rf phase error160 and the rf pulse

161

defects . Since NMRLX provides a suitable way to determine the rf

pulsewidths and the phases with well-adjusted field homogeneity, the

68
measured relaxation rates in this work are accurate to :38 for T1
values and 15% for T2 values except for some 13C Tl's which were only
measured with ~10% confidence due to the small S/N ratio (~20)

obtained. The reproducibility was found to be normally within 5%.

8. Sample Preparation

 

All the samples were commercially made except CDBrS which was
prepared previously131. The samples were all purified by one or two
vacuum distillations (or sublimations in case it was solid at room
temperature) after being carefully degassed with several freeze-
pump-thaw cycles. After purification, the sample was distilled or
sublimed into the sample tube until the proper amount was received then
the tube was flame sealed.

The sample tubes were specially designed using standard-wall
Pyrex tubing. Since most of the compounds were studied from the solid
phase to about the critical point, the pressure inside the tube varied
from 0 to ~40 bars. In order to avoid an explosion while making a
measurement, each sealed tube was tested at 10 to 20° above the highest
operating temperature in a shielded hood. Figure 11 shows the design
of sample tubes. The observing chamber was made to be spherical in
shape and the size was no bigger than the coil dimension so that all
of the sample compound was inside the coil. Above the observing
chamber, a narrow pathway was also incorporated to prevent extensive
convection and diffusion to the gas-liquid.interface. For some com-
pounds, a second seal-off was necessary to avoid sample rapidly
distilling from the bottom to the top of the tube and so causing

bumping and large thermal currents.

 

 

 

 

 

 

 

 

 

1 N2 in

41

Figure 11. Design of the NMR sample tube.

RESULTS AND DISCUSSION

1. Melecular Motion and Relaxation Studies of Some

 

Small Molecules in the Liquid Phase

 

A. Anisotropic Reorientation and 13C, 14N,19F Relaxation Studies in CF3§N_

 

1. Introduction

Due to the complexity of theories describing molecular motions in
dense liquid phases,experimental evidence has usually been obtained from
studies of simple and highly symmetric molecules which can largely
simplify the theoretical derivations and also decrease the errors due to
the approximations usually needed for larger and more unsymmetrical
molecules. Many of the NMR relaxation studies have been done on spherical-
top, linear and symmetric-top molecules for these reasons. For spherical
and linear molecules, molecular motions in the liquid phase are mostly
isotropic. However, in symmetric-top molecules the motional anisotropy
might be large for those with a strong dipole moment and highly aniso-
162,163 164

and CHSI . In these

molecules, two parameters are needed to describe the reorientational

tropic molecular shape, e.g., CHSCN

motion, therefore at least two nuclei at different locations should be
studied68. Since, for spin 8 nuclei there might be more than one contri-
bution to the total relaxation rate, the best choices will be the
quadrupolar nuclei in which there is only one dominant relaxation
mechanism to be considered. Unlike CH3CN,and its chlorine derivative
CC13CN, trifluoroacetonitrile CF3CN does not possess a second quadru-

polar nucleus. Therefore the elucidation of reorientational motions in

70.

71

CF CN depends on success in separating the different relaxation

3
mechanisms of the 19F nucleus. We report here our attempts to dothis in

Sections 2 and 3.The separated spin-rotational relaxation rate was then
used to test the validity of extended diffusion mode1522-24, which are
discussed in Section 4. A limited 13C relaxation study was carried out

and is reported in Section 5. Finally, the relation between 13C-19

F
. 13 19 . . . .
coupling constants and the C or F chemical shifts of some Similar

compounds and their derivatives is discussed in the last Section.

4
2. 1 N Relaxation and the TUmbling Motions

The measured 14N spin-lattice relaxation rates in liquid CF3CN are
listed in Table 3. These values are also plotted against 1/T in Figure
12. From the linear dependence of R1,N on 1/T the correlation times,
and therefore the rotational diffusion coefficients, of the C3 symmetry
axis can be obtained by using Equation (67). For 14N in CF3CN, this can

be simplified to

Q _ 2 .
R1,N _ 1.233(Q.C.C.) T6 , (103)

since the asymmetry of the electric field gradient around triplly-bonded
14N is zero. Q.C.C. stands for the quadrupole coupling constant equ/h.

From a microwave study of gaseous CF3CN, the Q.C.C. of 14N was found to

be -4.70 MHzl66. Several authorsl62’167'169

of 14N, 17O and 2

have argued that the Q.C.C.'s
D in liquids are usually very close to the values found
in the solid phase but are smaller than those in the gaseous phase by m6
to 14%169. Therefore, a value of -4.2 MHz (10% lower than -4.70 MHz) was

used for liquid CF3CN. It should be noted that using other values might

Table 3. 14

72

N spin-lattice relaxation rates in CFSCN.

 

 

T lOOO/T R1 T lOOO/T R1

1°C) (°x'1) (sec 1) 1°C) (°x'1 (sec-1)
12.0 75.51 222111 -33.0 4.17 404:8
-1.0 3.68 276:13 -34.0 4.18 396:13
-2.0 3.69 277:4 -44.0 4.37 452:12
-18.0 3.92 314:4 -88.0 5.40 908:34
-20.0 3.95 315:7 26.0 3.34 198:10
-32.0 4.15 350:8 -92.0 5.52 904:70
-47.0 4.42 449:8 -93.0 5.56 1021:81
-48.0 4.44 461:7 -103.0 5.88 1172:55
-61.0 4.72 543:8 -ll8.0 6.45 2089:90
-62.0 4.74 547:6 -24.0 4.02 337:10
-65.0 4.81 567:7 -3.0 3.70 267:7
-75.0 5.05 675:11 3.0 3.62 219:13
-88.0 5.40 834:17 8.0 3.56 196:6
-86.0 5.35 779:14 -5.0 3.73 225:20
-106.0 5.99 1228:34 ~123.0 6.67 1818:101
-115.0 6.33 1539:52 -107.0 6.02 1023:98
-123.0 6.67 2667:70 -93.0 5.56 746:14
-129.0 6.94 3136:330 -84.0 5.29 630:30
-124.5 6.73 2286:90 -52.0 4.52 379:37
-124.0 6.71 2159:123 -73.0 5.00 662:30
-116.0 6.37 1614:31 -61.0 4.72 538:19
-114.0 6.29 1580:61 -45.0 4.38 446:17
-100.0 5.78 1046:33 -20.0 3.95 371:13
-99.0 5.75 1080:50 -6.0 3.74 310:11
-90.0 5.46 890:19 25.0 3.36 210:10
26.0 3.34 208:12 -0.5 3.67 254:11
12.0 3.51 244:10 -78.0 5.13 682:32
-6.0 3.74 320:10 -94.0 5.59 1207:97
-20.0 3.95 355:8 -107.0 6.02 1622:144
-21.0 3.97 347:14 -58.0 4.65 533:26

 

73

 

201$

10

R1 (s e c")
§

 

 

_
h

1 I l

 

 

4 5 6 7 8
1000/T

Figure 12. Plot of the 14N spin-lattice relaxation rate in CF3CN
as a function of reciprocal temperature.

74

change the re '5 calculated, but the activation energy AEL would not be

1
affected. The calculated 1 's and D 's ( D = ———-) are lotted in
. 61- _L ‘L 6T6 P

Figure 13 and the activation energy is found to be 1.40 kcal/mole.

3. 19F Relaxation and Anisotr0pic Reorientation

l9 . . . .
The measured F spin-lattice relaxation rates are listed in Table
4 and plotted against reciprocal temperature in Figure 14. It is
. . . 1 .
eVident that several mechanisms must contribute to 9F relaxation, and

we may write

d,intra+Rsr+Rcsa+Rsc

t _ d,inter
R'R * 111’

1 1 R

(104)

where the first and the second terms are the intermolecular and intra-
molecular dipolar relaxation rates, respectively, the third term is the
spin-rotational relaxation rate, the fourth term is the relaxation rate

due to the chemical shielding anisotrOpy of 19F nuclei and the last

term is the 19F scalar relaxation rate due to» the 14N nucleus.

From Equation (69), the contribution from the scalar coupling JFN

can be estimated as

sc _ ~16 2
R1 - 4.QXl0 JFNR1,N' (105)

Very few J values have been reported but, since R is about the

FN 1,N
order of 103nsec'l, Ric is negligibly small and can be completely

ignored.

csa

R1

can be estimated from Equation (73a) which is, in this case,

7S

 

 

 

 

50-
79.1.
DI
1
‘1 o.
"'5
F5 701
A 9.
l a
V
g?
1 .—
0.5—
1 4 J L 1 1
4 5 6 7 8
1000/1'

Figure 13. Reorientational correlation times and diffusion coeffi-
cients of CF3CN as functions of reciprocal temperature.

Table 4. 19

76

 

 

F Spin-lattice relaxation rates in CFSCN.
T 1000/T Rl T 1000/T R1
1°C) (°x‘1) (sec' 1 (°C) (°K'1) (sec 1)
25.0 3.36 0.724: .010 -116.0 6.37 0.208: .002
21.0 3.40 0.709: .003 -120.0 6.54 0.225: .003
12.0 3.51 0.621: .009 ~126.0 6.80 0.260: .002
0.0 3.66 0.517: .005 -131.0 7.04 0.292: .006
-11.0 3.82 0.455: .008 -131.5 7.07 0.309: .006
-20.0 3.95 0.412: .007 -135.5 7.27 0.373: .011
-29.0 4.10 0.371: .006 -l41.0 7.58 0.452: .007
-38.0 4.26 0.322: .002 -140.5 7.55 0.438: .006
-37.5 4.25 0.323: .003 -145.5 7.84 0.572: .004
-52.0 4.52 0.271: .011 -148.0 8.00 0.698: .007
-53.0 4.54 0.268: .004 -151.0 8.20 0.896: .020
-57.0 4.63 0.258: .003 -153.0 8.33 1.027: .020
-62.0 4.74 0.247: .005 -154.5 8.44 1.172: .009
-66.0 4.83 0.241: .003 -156.0 8.55 1.301: .009
-68.5 4.89 0.237: .002 -157.0 8.62 1.452: .012
-73.5 5.01 0.230: .003 -158.0 8.70 1.550: .020
-77.0 5.10 0.213: .002 -158.5 8.73 1.630: .013
-77.5 5.12 0.205: .002 -159.0 8.77 1.630: .008
-82.0 5.24 0.199: .004
-88.0 5.40 0.192: .003
-92.0 5.52 0.185: .002
-94.0 5.59 0.190: .002
-96.0 5.65 0.189: .001
-99.0 5.75 0.189: .001
-103.0 5.88 0.190: .001
-lO7.0 6.02 0.193: .002
-1ll.0 6.17 0.200: .001

 

 

0.5 *-

 

nbemv .m m.

0.05 ~

Pi

77

 

 

0.05 -

 

 

 

l 1 l

6 7 8
4 5 1000/1

Figure 14. Plot of the 19F spin-lattice relaxation rate in CF3CN
' as a function of reciprocal temperature.

1.

 

For
cont

buti

Thel

74% 1

whet

 

EqUa

tOt

 

”he

(0.

78

4 2
RC53 = 1.68x10 A0 1 . (106)
1 ppm9

For Aoppi~200 ppm, Risa would become ~58 of the intramolecular dipolar

contribution. Since A0 is not exactly known, we will ignore this contri-
bution for the time being and discuss it later.

The intermolecular dipolar relaxation rate can be estimated from the
densities and the self-diffusion coefficients as summarized in Equations
(59), (60), (62) and (63). Gillen17o has discussed the use of different

numerical factors and proposed that the best one is

d,inter _ wv‘fizN

R
1 4Dsa

(107)
The molecular radius a can be estimated from the density by assuming
74% of space is occupied by molecules and so

a = (0.74 M) 1/3 (108)

4WNod
where d is the liquid density and No is Avogadro's number. The N in
Equation (107) is the number of spins per unit volume and is also related

to the density d by

Nodno

g ________ , 109
Mol.Wt. ( )

where no is the number of spins per molecule.
In order to take the pressure effect on the density into account
(the pressure in the sample tubes can vary from ~0 bar at the melting

point to ~40 bars at the highest temperature studied which is ~10°C

 

 

belo‘
dire:
weig!

in T:

visc<
may c
There
spin-
Equat
plot!
rates

dipol

their
deper
have
the 1
low (

P10t1

Corre

 

 

Equa-

wheT

 

79
below the critical point of CF3CN), we have measured the liquid density
directly in the sample tube by carefully calibrating its volume and the
weight of the sample sealed in it; these density measurements are listed
in Table 5.

The self-diffusion coefficient can be estimated from the shear
viscosity of the liquid sample but it has been found that Ds so obtained
may deviate from the direct experimental result by 20% or more171
Therefore, we have directly measured Ds by the pulsed field-gradient

13
spin-echo technique 8 and these values are listed in Table 6. Using

d,inter
1

plotted in Figure 14. By subtracting R

Equations (107), (108) and (109), R values were calculated and are

d,inter
1

rates, we can obtain curves representing the sums of intramolecular

from the total relaxation

dipolar and spin-rotational relaxation rates.

d,intra and Rsr

their opposite temperature dependence. Since the low temperature linear

The separation of R can be achieved by making use of

dependence of relaxation rates is not available for this compound, we

*

have assumed that the reduced reorientational correlation time 16 at

d,intra so that the

1

r
form a straight line; this best fit is

the critical temperature is ~0.7 and have adjusted R

5

low temperature values of R1

plotted in Figure 14.

The intramolecular dipolar relaxation rates are related to the
correlation times of the motion of 19F-lgF internuclear vectors by
Equation (58) which can be written, for this particular compound, as

d,intra 10

R1 = 1.31X10 Te , (110)

where T6 is related to the diffusion constants by

Ta

 

 

 

80

 

 

 

Table 5. Measurements of the density of CFSCN.

c d
T v, v: 93 W8 w, d,
(OC) (m1) (m1) (torr) (g) (g) (g cm'3)
18.0 0.755 0.6011 28,840 0.0913 0.6763 0.8958
-14.0 0.670 0.6861 10,965 0.0444 0.7232 1.079
-9.0 0.685 0.6711 12,882 0.0500 0.7176 1.048
-28.0 0.654 0.7021 6,607 0.0289 0.7387 1.130
-25.0 0.660 0.6961 7,413 0.0317 0.7359 1.115
—21.0 0.665 0.6911 8,511 0.0356 0.7321 1.101
-14.0 0.670 0.6861 10,965 0.0444 0.7232 1.079
-6.5 0.692 0.6641 14,454 0.0550 0.7126 1.030
3.0 0.710 0.6461 19,055 0.0682 0.6994 0.985
5.0 0.720 0.6361 19,953 0.0698 0.6978 0.969
8.0 0.727 0.6291 21,878 0.0748 0.6928 0.953
11.0 0.735 0.6211 23,988 0.0800 0.6876 0.936
14.0 0.747 0.6091 26,303 0.0851 0.6825 0.914
-60.0 0.600 0.7561 1,622 0.0088 0.7588 1.265
-53.3 0.605 0.7511 2,291 0.0119 0.7557 1.249
-48.0 0.617 0.7391 2,884 0.0144 0.7532 1.221
-46.0 0.622 0.7341 3,162 0.0156 0.7520 1.209
-43.0 0.627 0.7291 3,548 0.0172 0.7504 1.197
-41.0 0.632 0.7241 3,890 0.0185 0.7491 1.185
-75.0 0.571 0.7851 780 0.0043 0.7633 1.337
-160.0 0.490 0.8661 0.112 0.0000 0.7676 1.566
aVg = vtotal - Vi, where vtotal = 1.3561 ml (volume of the tube).

blog Pv(torr) = 7.900 - lOOO/T, from Reference 302.

cW = 95 P V 62.3T.
8 v g/

d = -

W, Wt Wg

, where Wt= 0.7676 g.(total weight of the sample).

Table

 

 

 

81

Table 6. Self-diffusion coefficients of CFSCN.

 

 

a b

T 1000/T g 05d

0 o ‘1 -5 2
( C) ( K ) (G [Cm) (10 cm /sec)
17.0 3.45 150 4.37

5.0 3.60 158 3.61
-6.0 3.74 170 2.98
-21.0 3.97 185 2.44
-37.0 4.24 208 1.59
-51.0 4.50 232 1.29
-71.0 4.95 282 0.800
-73.0 5.00 288 0.785
-87.0 5.38 337 0.519
-103.0 5.88 418 0.310
-104.0 5.92 422 0.294
-118.0 6.45 531 0.171
-131.0 7.04 684 0.0868
-136.0 7.30 760 0.0713

 

8Magnetic field gradients calibrated with CFClS.

13See Appendix A for the method of calculating Dsd’

82

= (360529-1)2 + 35in26cosze + 3sin40

4 +
2401- SleDll (201-49'!)

 

 

 

T0 (111)
. . . . . l9 . .
which is essentially the same as Equation (87). For the F nuclei in

CFSCN 0:900, therefore

1 3
T0 ‘ 2710f 4(20r40ll) ’ Tei

 

'f(r), (112)

where f(r) is a function of r, the anisotropy of the reorientational
motion defined by

r = DI I/DL = Te-L/Te . (113)

By using Di or 161 obtained from 14N relaxation studies, we obtained
D11 and T6 which are shown in Table 7 and are plotted against the
reciprocal temperature in Figure 13. The activation energy for the
spinning motion (reorientation about the C3 symmetry axis) was found

to be ~0.89 kcal/mole which is smaller than AE1f~1.40 kcal/mole. This
implies that the reorientation about the symmetry axis is easier than
that about an axis perpendicular to the symmetry axis. From Table 6 one
can also see that the anisotropy of the rotational motion is temperature
dependent, varying from ~6 at -148°C to ~l.8 at room temperature.
Similar phenomena have also been found in other symmetric-top molecules
Sl63-165,172.

which have the dipole moment along the symmetry axi

It is of interest to compare the above result for CF CN with those

3
for CHSCN and CC13CN165. Table 8 shows some pertinent physical constants
and motional parameters for these compounds. CHSCN has been widely
162,163,173 174-176

studied be NMR , Raman spectroscopy and dielectric

83

 

 

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86

relaxation measurement177. The dipole moment u=3.38 D . and the'an-
isotropic molecular shape (ll/III~10),of CHSCN make its reorientational
motion very strongly direction dependent. At room temperature, the
spinning motion is about ten times faster than the tumbling motion. For
CCISCN, the dipole moment of 1.99 D and a nearly spherical molecular
shape make the motional anisotropy much smaller ( about 2 at room tem-
perature). For CF

3

the anisotropy of the reorientational motion is about 1.78 at room

CN the dipole moment is 1.33 D and Il/Ill~2.0 so

temperature. It can be concluded that not only the molecular shape but
also the dipole moment of a molecule determine the reorientational motion.
Different activation energies for parallel and perpendicular reorienta-
tion are also evident in these compounds. While the activation energies
AEL for CHSCN and CC13CN are found to be in accord with those obtained
from Raman and dielectric relaxation studies and the predicted values
from microviscosity theory211, this comparison is not available for
CF3CN since the other studies have not yet been made. The smaller 0E1-
(1.40 kcal/mole) for CF3CN compared to that of the lighter analog CHSCN
(AE1~2 kcal/mole) is probably due to the much smaller dipole moment of
CF3CN. Since the dipolar "head-to-tail" attraction tends to align the

molecules in a unique direction and interfere with the tumbling motion

of the molecules, this effect should be much stronger in CH CN than in

3
CF3CN.

An overwhelming contribution of the spin-rotation interaction to
the relaxation rates is typical in small fluorocarbon;131’171’178'184.
Since in CFSCN the 19F nuclei are not on the symmetry axis, the princi—
pal coordinates which diagonalize the spin-rotation tensor will not

diagonalize the diffusion tensor or the moment of inertia tensor 0f the

87

molecule. Therefore, Equation (82) should be used for relating Rfr to
the motional parameters. Unfortunately, there are too many unknown
variables in Equation (82) and it becomes almost impossible to use.
Despite this, we will attempt to use a simplified equation (Equation
(76)) to derive the overall angular momentum correlation times. Equation
(76) can be written as

8n2 2 39

sr _ k _ 2
R - ( ”2 )TIavCefftJ _ 9.8x10 TIavCeffrJ , (114)

1

 

where Iav is the average moment of inertia and Ceff is the effective
spin-rotation constant in Hz. The spin-rotation constant can be obtained .
from the molecular beam experiments or can be estimated from the para-

magnetic chemical shielding. Flygare185 and Deverall186 have derived an

equation to relate the spin-rotation constant with the paramagnetic part

187

of the chemical shielding based on Ramsey's equation . A general form

of this equation is as follows:

06 = Cabs - 0d = K-loca , (115)

where cabs is the absolute chemical shielding of the observed nucleus,

°d is the diamagnetic chemical shielding and K is a nuclear constant

30 19

which is 2.09X10 for F. 0d for 19F has been calculated by the atom-

88

dipole method1 and many different values have been obtained but the

1
most usually used seems to be 471 ppm 89. Thereforel71,

29
- 471 = 6.97 10 2C I C
to... 1,... " ‘ 11* IIIII)

29 ’
W 6.97x10 3IavCo. (116)

88

19

The F chemical shift of CF CN was measured and found to be 59.7 ppm

3
relative to CFCl3 which has oab§~195.6 ppm188. Therefore, the spin-
rotation constant C, can be calculated from Equation (116) and we
obtain Co .-4,29 kHz. Using this value for Ceff’ TJ'S can be obtained
and are listed in Table 9.

Since rJ is the correlation time describing the motion of the
angular momentum, which is changed each time a collision occurs, 1/1J
can be considered as the collision frequency. On the other hand, the
reorientational correlation time recan be considered as the time needed
for a molecule to reorient one radian. It can be seen from Table 8 that
at low temperature, e.g., 1540K (~119°C), a molecule will undergo a
collision once for every 20 reorientation. Therefore, the diffusion
limit or small-diffusion-step approximation is probably suitable for
describing the molecular motion at this temperature. At higher tempera-
tures, e.g., 2980K (25°C), a molecule has rotated 600 before it under-
goes a collision. Therefore, a strong inertial effect should be consi-
dered in describing the rotational motion of this molecule at higher
temperatures.

Another way to test the inertial effect is by comparing the experi-
mentally derived reorientational correlation time with the theoretical
reorientational correlation time of free gaseous molecules rf{~%/I;7ET .
This test has been called the X test190 and has been widely used to test
the applicability of the small-step diffusion model. x1 is defined by

Te.
- —-L w (117)

X1 3
SJIi/RT

where i stands for components x,y,z,or II and l_in this case.

b _ .

89

 

 

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90

Figure 15 shows the plots of xi versus reciprocal temperature for both
spinning and tumbling motions. In order to be in the diffusion limit,

xi has to be much greater than 1. From Figure 15 we find that the re-
orientation about the symmetry axis is about completely inertial except
at very low temperature (~-l40°C); on the other hand, the tumbling
motion of the molecules changes from the diffusion limit at temperatures
below the normal boiling point (~-68°C) to the inertial limit at higher
temperatures. Therefore, it can be concluded that the classical diffusion
model can not be applied to this molecule except below the boiling point.

. . . 165
Similar conclusions have also been drawn for several other compounds .

4. The Extended Diffusion Model

For the purpose of bridging the classical diffusion limit and the
free-rotor limit, Gordon22 has developed an extended diffusion model
which has been discussed in the Theoretical part. Several tests have
been made with small molecules191 and it was found that most of the
molecules followed J-diffusion which is more reasonable in real
liquids.

In Figure 16, we plot the reduced overall reorientational correlation
times 1; versus the reduced angular momentum correlation times 1;,which
are obtained from the spin-rotational relaxation rates by using the
spherical-top approximation. The result shows a line lying between the
J-diffusion and M-diffusion curves at low temperatures and approaching
J-diffusion when the temperature gets higher. Since we have assumed Ceff
to be the same as CO, this might give a considerable error when the

anisotropy in the spin-rotation constants is high. In fact, Ceff is

always higher than Ca unless the anisotropy in C is zero. In order to

91

 

 

 

 

I l l
4 6 8
1000/T

 

Figure 15. Plots of x , xi for liquid CF3CN. For comparison, the
dashed lines are the corresponding x plots for CH3CN and
the dotted lines are those for CC13CN.

92
make this correction, we considered the original derivation of Equation
(115) by Flygarelss, who obtained the following equation for each com-

ponent of the chemical shielding and the spin-rotation tensors,

o = K-I -C

88' 8 88' ’ (118)

where g=x, y or 2. Montana 35 al,192 have studied the 19F chemical
shielding anisotropies of CF4, CF3C1, CF3Br, CF31 and CF3CC13 in smectic
liquid crystal solutions and obtained the three principal components of
the chemical shieldings in the molecular axis system. A very important
conclusion they drew was that the substitution of an X atom for an F
atom in CF4 had moved the principal axis of the chemical shielding
tensor from the C-F bond to another axis by an amount which increased in
the order Cl<Br<I. By comparing the C-X bond characters in CF3CN with
those in other CFSX compounds (see Section 6), it is seen that the
‘shielding pattern of 19F in CF3CN should be very close to that in CF4.
From the well-behaved trend which will be discussed later, we estimated
oll~215 ppm and ol~263 ppm for 19F in CFSCN. Using these values, we
found Ceff ~-5.S8 kHz and the 13 values are decreased by a factor of 1.7.
* versus 1* follows that pre-

0 J
dicted by the J-diffusion model very well (see Figure 16).

By making this correction, the plot of r

5. 13C Relaxation and Spin-Rotation Constants

We have measured the spinelattice relaxation rates of both 13C

nuclei in CFSCN at two different temperatures. For the methyl carbon,

1 1 at ~14OC; for the cyanide

1

at 28°C and 0.030 sec-
1

3120.034 sec”

carbon, it is 0.012 sec- at 28°C and' 0.0077 sec' at -14°C.

 

93

 

 

.xHo>Huoom
-mon .mmou you :ofiuoonnoo «segue: one an“: mafiamon one moHono some new pfiaom ”Zummo you oawu
newumfiounoo emueoaoa uofismem poospon msmno> oawu eofiuefiounoo Hoeowumpeowuoou pouncon mo uoHe .0" onswwm

 

 

 

_ nd .0. «.0 .6 nod 36
_ 7 _ 1 _ _

 

 

94
For the methyl carbon, only dipolar and spin-rotation interactions
are important relaxation mechanisms. The dipolar relaxation rate R: due
to three fluorine nuclei can be estimated as

d__ 222-6
R1 - SYCYFH rCFTe , (119)

where is the internuclear distance between C and F, which is

rCF
1.33 3193, and 16 is described by Equation (111) with 0 for the C-F
bond 69.3o 193; therefore, re~0.58 psec at 28°C and ~0.81 psec at -l4°C.
From Equation (119) the dipolar relaxation rate of the methyl carbon

1

was found to be 0.0101 sec.1 at 28°C and 0.0140 sec" at -14°C.

The spin-rotational relaxation rates are therefore equal to 0.024 sec"1
at 28°C and 0.016 sec‘1 at -1400.

Since both 13C nuclei are on the symmetry axis of the molecule,
Equation (81) can be used to relate Rir to the angular momentum correla-
tion times. Equation (81) can be rewritten as

Rsr - sflsz (I c2 T +21 02 r ) (120)
1 ’ "‘E?‘ J J '
an H" H H 1
where the spin-rotation constants CII and C1 are now in unit of Hz.
In order to obtain Cll and Cl, another condition is needed. This can be

obtained from the 13C chemical shielding by using Equation (115). For a

symmetric-top molecule, Equation (115) can be written as

. - .1

30 13

where K is 7.79xlO for C. We have measured the chemical shifts of

95

both 13C nuclei in CFSCN. These are 129.0 ppm and 130.9 ppm down field
from methyl iodide for the methyl and the cyanide carbons, respectively.
Using Equations (120) and (121), and also assuming that TJ can be re-
lated to 10 by Hubbard's relation (Equation (28)), we obtain for each
temperature two possible sets of values for C11 and CL of both the
methyl and the cyanide carbons. These are listed in Table 10. Since, for
cyanide carbon, the cylindrical electron cloud in the C-N triple bond
should make the chemical shielding highly anisotropic, the sets with
Cl] positive and CL negative seem to be more reasonable. The chemical
shielding anisotropies corresponding to the chosen C values are 380 ppm
and 330 ppm at -l4°C and 28°C, respectively. On the other hand, the
methyl carbon should not have too large a chemical shielding anisotropy
and therefore the sets of values with both C11 and Cl negative will be
more resonable. Making this choice the chemical shielding anisotropies
of the methyl carbon are -140 ppm and -160 ppm at -14°C and 28°C, res-
pectively.

In the calculations for the cyanide carbon, we have neglected the
contribution from the chemical shielding anisotropy to the total
relaxation rates. This can be:justified as a good assumption by using

Ao~360 ppm in Equation (73a) which gives

csa _ 2 2

-—w 1
l 15 °

(Ao)21'e < 0.00016 sec' .

R (122)
This value is about 2% of the total relaxation rate and therefore can
be neglected at the temperature studied. It should, however, be noted
that at lower temperatures the chemical shielding anisotropy might give

a considerable contribution to the cyanide carbon relaxation rates.

96

Table 10. Spin-rotation constants of 13C in CF

 

 

r (0C) C|l(kHz) cl(kHz)

13CF3 28.0 -2.575 -0.588
28.0 1.185 -1.552

.14.0 -2.470 -0.615

—14.0 1.763 -1.700

13CN 28.0 -1.721 -0.433
28.0 0.748 -l.066

-14.0 -1.593 -0.465

-14.0 1.098 -1.155

g

8There are two possible sets of values for each temperature.

(See text for explanation).

r-r‘

 

 

 

97

6. 130, 19F Chemical Shifts and JCF

Finding the trends in chemical shifts and coupling constants has been

150,195

a traditional subject for NMR studies . Muller and Carr196 have

found a linear relationship between the 19F chemical shifts 6 and the

F
13 _19

C F coupling constants JCF of several families of related compounds

including tetrahalomethanes. DeMarco et_al,197 have also tried to

rationalize the 6 6 and JC

C’ F F

concluded that the substitution of a less electronegative halide on one

values in some trifluoromethyl halides and

of the fluorines in CF4 tends to increase the 13C chemical shielding and

19

the carbon- fluorine coupling constant J but to decrease the F

CF
chemical shielding. The increase in JCF is evidently due to increasing
multiple bonding between carbon and fluorine atoms which will also
decrease the bonding polarity and therefore the chemical shielding of
19F nucleus.

In Figure 17 we plot JCF versus 6F for all the fluorohalomethanes
studied in this work (the parameters used are listed in Table 11). It
is obvious that CFSCN lies on the same line as all the trifluoromethyl
halides and is very close to CF4. This is evidently due to the smaller
mass of the CN group and its greater electronegativity which make it
behave somewhat like a single fluorine atom.

The effect of substitution of a fluorine atom by another group X

was also observed in the anisotropy of 19F chemical shielding. Montana

g§_gl,192 found that the anisotropies of 19F chemical shieldings in
CFSX molecules increased in the order CFSCI, -13.4:0.5 ppm < CF3Br,
2.7:0.3 ppm < CF31, 16.8:0.2 ppm (in the molecular axis systems). They

also found that the component of the chemical shielding along an axis

parallel to the symmetry axis of the molecule all was fairly constant

98

 

 

l l 1 1 1 l l

 

 

 

0 40 80

8F ("”0

Figure 17. Plot of JCF versus 6F for some fluorocarbon derivatives.

Table 11. Comparison of
in some fluorocarbon derivatives.

13

99

C and 19

F chemical shifts and JCF values

 

 

Compound JCF(Hz) JCCF(Hz) 6C(ppm) 6F(ppm)
CF3C1 299(a) 125.5(f) 33(a)
CF3Br 324(a) 112.7(f) 21(a)

323 20.7
CF31 344 78.2(f) 5(a)
5.29
*CF3CN 266 129.0(d) 59.7
CFECN -56 130.9(d)
CF3CC13 283(c) 120.1(f) 62(f)
Cpgcc13 -43.l(c) 91.0(f)
CF4 260(g) 119.9(f) 62(f)
257(a) 69(8)
CFC13 337 137.7(d) 0
CFBr3 371 67.1(d) -7(e)
372(b)
CF2012 325(a) 8(a)
CFzBrz 358(a) -7(a)
cpz=c012 289 -44.2 90
289.9(c) -43.7(c) 88.5(c)

 

(a) Reference 196.

(b) P. C. Lauterbur, "Determination of Organic Structure by Physical
Methods", F. C. Nachod and W. 0. Phillips, Ed., Academic Press,
New York (1962).

(c) G. V. D. Tiers, J Phys. Soc. Jap. 15, 354 (1960).

(d) Measured relative to CH3I.

(e) c. Filipovich and G. v. 0. Tiers, J. Phys. Chem. 63, 761 (1959).

(f) Reference 197.

(g) E. L. Motell and G. E. Maciel, J. Magn. Resonance, Z; 330 (1972).

f0)

18)

sh:

HE

+1.

pe:

U 1

0):

Se:

me'

1‘8

re

1}

 

 

 

100

19

for F in these molecules. In Figure 18 we plot Ao(both in the molecu-

lar axis system and in the C-F bond system) versus the absolute chemical

shielding of 19

F. In view of the well-behaved trends shown in Figure 18,
we can predict the chemical shielding anisotropy of 19F nuclei in CF3CN
by interpolation. The Ao values obtained in this way are -48 ppm and
+142 ppm for the molecular axis system and the C-F bond system, res-
pectively. From the anisotropies so obtained we found for CFSCN,
0I|=215 ppm, ols263 ppm in the molecular axis system.and oll=360 ppm,
olsl99 ppm in the C-F bond system. These results have been used in
Section 4 for calculating the effective spin-rotation constant Ceff'

Table 12 summarizes the physical constants and the motional para-
meters obtained from this study of CF3CN.

B. 13C and 20 Relaxation Studies and the Motional Anisotropy in

 

Liquid CDBr3

 

1. Introduction

As has been discussed before, for a symmetric-top molecule the
reorientational motion can be completely described by observing the
relaxation of two quadrupolar nuclei occupying two different sites in
the molecule68. It seemed that CDBr3 would be suitable for this study
but, unfortunately, the NMR relaxation rate of bromine nuclei is too
fast to be measured. Therefore, we still have to obtain a second rela-
tionship using the relaxation of the 13C nucleus, which is again com-
plicated by contributions from several different relaxation mechanisms.

Recnetly, a number of Raman spectroscopic studies of the molecular

131,174,176

motions in liquid bromoform have been reported . Bartoli and

101

 

100'-

40' (PPM)

 

 

Figure 18.

 

100

0 60
89 (PPM)

Plot of Act versus 5F for CF3X, X= CC13, F, Cl, Br, I,
i

and CN; so d circles are in the C- F bond coordinate

system and open circles are in the molecular coordinate
system.

 

Tab

 

 

102

Table 12. Summary of the physical constants and the motional
parameters of CF3CN.

 

 

Physical Constant Value Note

AEII 0.89 kcal/mole

AEL. 1.40 kcal/mole

CO(19F) -4.29 kHz

ceff(198) -S.58 kHz

c||(19p) -8.34 kHz

CL(19F) -3.46 kHz

00(19F) -48 ppm a
142 ppm b

Cll(13CF3) -0.60 kHz c

01(13CF3) -l.63 kHz c

Cll(13CN) -l.66 kHz c

Cl(13CN) 0.92 kHz c

00(13CF3) -150 ppm c

00(13CN) , -360 ppm c

 

a. In the molecular axis coordinate system.
b. In the C-F bond coordinate system.

c. Average of the values at two temperatures.

ti
to
us
:10
ca

me

hr

in

 

US'

 

C1.

 

 

103

Litovitz174 have analyzed the depolarized Raman lineshape of bromoform

and found the correlation time of the tumbling motion relat 22°C to be

5.3:2 psec. Wright131 has also analyzed the A1 lines of the Raman

76 have

spectrum and obtained a similar result. Peterson and Griffiths1
made a variable-temperature Raman spectroscopic study on this compound
and obtained an activation energy AE ~1.7 kcal/mole for the tumbling
motion. This activation energy can also be obtained by measuring the
relaxation times of 2D in deuterated bromoform. Unfortunately, the
activation energy AEl obtained from the previous NMR relaxation study131
does not agree with that obtained from the Raman work.

Farrar gt_§l,203 have measured the 13C nuclear magnetic relaxation
times of CHBr3 and found that the dominant relaxation mechanism was due

to the scalar coupling between 13

3

C and bromine nuclei. An attempt at

C and the 2D relaxation time measurements to study the
31

using the 1
motional anisotropy of CDBr3 has been made1 . We report here our
carefully measured 13C and 20 relaxation rates and the motional para-
meters derived from them.

2. 20 Relaxation and the Tumbling Motion

The measured 20 spin—lattice relaxation rates in 97% deuterated
bromoform131, which had been carefully purified and degassed, are listed
in Table 13 and are plotted versus reciprocal temperature in Figure 19.
Using the reported quadrupole coupling constant 177:5 kHzlgs, we cal-
culate the correlation time of the tumbling motion to be

1

= 3.30x10' 3exp(1.67 kcal mole‘l/RT). (123)

Tel-

104

Table 13. 2D spin-lattice relaxation rates in CDBr3.

 

 

T (0C) lOOO/T (°x‘1) R1 (sec‘l)
9.0 3.55 2.93:0.13
12.0 3.51 2.73:0.12
15.0 3.47 2.62:0.14
19.0 3.42 2.67:0.14
21.0 3.40 2.58:0.09
22.0 3.39 2.57:0.11
24.0 3.37 2.43:0.10
30.0 3.30 2.37:0.08
31.5 3.28 2.34:0.10
37.0 3.22 2.17:0.12
37.5 3.21 2.07:0.09
39.0 3.20 2.22:0.09
46.5 3.13 2.02:0.10
48.0 3.12 2.02:0.12
53.0 3.07 2.03:0.08
54.5 3.05 1.98:0.10
56.0 3.04 1.92:0.07
66.5 2.94 1.66:0.08
77.0 2.86 1.70:0.09
78.0 2.85 1.52:0.08
87.0 2.78 1.53:0.08
90.0 2.75 1.52:0.07
106.5 2.64 1.40:0.05
118.0 2.56 1.29:0.08
126.0 2.51 1.24:0.06
142.0 2.41 1.15:0.10
145.0 2.39 1.14:0.06
152.0 2.35 1.17:0.04
175.0 2.23 1.01:0.05

 

105

 

 

 

 

2 3 4
1000/T

 

Figure 19. Plot of the 2D spin-lattice relaxation rate in CDBr3 as
a function of reciprocal temperature.

thl

l4

ni

th

ti

in

1‘0

th

CE

 

106

Our measured activation energy AE =l.67 kcal/mole agrees very well with

the result from Raman work176 AElsl.7 kcal/mole.

3
3. 1 C Relaxation and Motional Anisotropy

The measured 13C spin-lattice relaxation rates are listed in Table
14 and plotted in Figure 20. Usually several different relaxation mecha-
nisms have to be considered for the 13C nucleus. In this compound only
the scalar coupled relaxation is important. The spin-rotational relaxa-
tion should be negligible since CDBr3 is much more massive than CDC13
in which only ~10% of the relaxation rate is contributed by the spin-
rotational interactionlgg. The intermolecular dipolar contribution to

13

the C relaxation is always neglected since the carbon atom lies at the

center of the molecule. The intramolecular dipolar contributions from
2D and the three bromine nuclei can be estimated by the following

equation
d 2 2 -6 2 -6 _ 8
R1 ’ ” YCcYDrCD+YBrrCBr)TB ’ 6‘10 T9 ’ (124)

where we have used rcd~1.07 X and rCB
1

is about 0.003 sec' which is clearly negligible with respect to the

£~1.93 R 200; with re~5 psec, R:

total relaxation rates. Relaxation through chemical shielding aniso-
tropy is also negligible at the low field (14.1 kG) studied; an aniso-

tropy of 200 ppm results in

C53

R1

7
= 2.4XIO Ta, (125)

which is even smaller than Rf.

107

13

Table 14. C spin-lattice relaxation rates in CDBr3.

 

 

r (°C) 1000/T (°x'1) R1 (sec-1)
167 2.27 0.72:0.07
132 2.47 0.64:0.06
101 2.67 0.55:0.05
74 2.88 0.50:0.05
49 3.10 0.45:0.05
34 3.26 0.44:0.04
22 3.39 0.44:0.05
13 3.50 0.4210.05

9 3.55 0.40:0.04

 

108

 

 

 

o
8" .0 ..
.‘o
6" “.0 _
A
'7
'8 o ..
m "
v
m
4- -
l

 

2 3 ' 4
1000/T

Figure 20. Plot of the 13C spin-lattice relaxation rate as a

function of reciprocal temperature. The open circles
are the Ric values calculated from the 13C data of
Farra§_ et a120 3; the solid circles are the R} values
for C_ in CDBr3.

COLL

M

 

r11

 

109

The existence of two bromine isotopes might complicate the scalar

13

coupled relaxation of the C nucleus. The decay of 13C magnetization

Mz due to different combinations of bromine isotopes follows

. 1 sc sc 3 sc
M = M -— - -tR ._ -
z 0{ 8 eXp( tR1,818181 1.797979)+ 3 exP( tR1,818179
SC
'tR1,817979)} ’ (126)

where the second subscripts represent the three isotopic species A, B, C

and

5c = sc sc sc
R1,ABC R1,A + R1,B + R1,C . (127)
Equation (126) implies that the 13C relaxation could be biexponential ,
but by a simple mathematical rearrangement of Equation (126) we can show
that the 13C relaxation due to scalar coupling with the three bromine
nuclei can be described approximately by a single exponential function

with a time constant

sc _ 3

1 - E(RSc + 85° ). (128)

R 1,81 1,79

Using Equation (69) and also by noticing that Aw212<<1 for 79Br, Equation

(128) can be written as

 

81
T ( Br)
Ric = lé-{A2(C798r) T1(7gBr) + 212 81 }. (129)
4 1+Aw T1( Br)

Since A(C798r)/A(C81Br) = 7(79Br)/y(818r) = 0.928 and T1(8lBr)/T1(798r)

 

wh

an

w}

of

In

 

 

110
79 81 . . . .
= Q( Br)/Q( Br) = 1.433, Equation (129) can be further Simplified to

1.71T1(798r)

}.
l+1.08XI014Ti(7gBr)

Ric = 148.0 J2(C798r){ T1(798r) +

 

(130)

79 79
where J(C Br)= A(C Br)/21ris the scalar coupling constant between 13C

and 798r in unit of Hz and T1(798r) can be expressed by

- 2 '
T11(798r) = 3.95 (Q.C.C.) T6 = 1.43Xl01819, (131)

where we have used 601.1 MHz201 for the quadrupole coupling constant

7 . . . . . . . .
of 9Br. re 15 the correlation time describing the motion of the princ1-

pal axis of the electric field gradient tensor which can be assumed to
be along the C-Br chemical bond. From Equation (111) and with 0~7l.9°200,

19 can be written as

1.57 1.84
5+r + 1+2r) - 1913(r)’

 

T6 = Te‘L(0.126 + (132)
where r is the reorientational anisotrOpy defined by Equation (113).
Since there are two unknowns in Equation (130), the exact solution is
not possible without any other conditions. However, since the relvalues
have been directly obtained from the deuterium relaxation studies, we

can assume different values for the activation energy of the spinning

79 Rsc

motion AEII and the coupling constant J(C Br) and calculate 1 values
at different temperatures from Equation(l30). We then try to fit these
values to those measured experimentally by varying AEII or, alternative-

ly, use the measured Ric values to calculate J(C798r) by assuming

different values for 08". In this way it was found that the best fit

111

of the Rsc values using Equation (130) required that AEII=O.9 kcal/mole

1
and J(c798r)=105 Hz.
203 13 . .
Farrar, et al. have measured the C relaxation rates in CHBr3

at several different temperatures. By subtracting the dipolar contribu-
tion from the proton in CHBr3 using our measured rel values,we find

that their relaxation rates Ric are a little higher than our results for
CDBr3 (see Table 15 and Figure 20). By using the same technique described
above for fitting AEII and J(C798r) using Equation (130), we find that
AE||~1.35 kcal/mole and J(C798r)~121 Hz in order to provide the best fit
to Farrar's data.

In view of the crude approximations made, and the unknown validity of
our fitting method, very precise conclusions cannot be made from this
study. However, we can compare our result with that for CDCl3 for which
the motional parameters were very well determined by measuring the 2D

172

and 35Cl relaxation times and Table 16 summarizes these results.

Since the geometrical anisotropies of these two compounds are comparable,
one would expect that the anisotropies of the molecular reorientations
should also be comparable and this is indeed true. Another point that
can be noted is the value of the scalar coupling constant J(C798r).

Shoup and Farrar199 have found J(CSSCI) in CDCl to be ~23 Hz. Since

3
J(c35c1)/J(c798r) ~ y(3501)/y(798r), this would give J(c798r) about

59 Hz which is much smaller than our eStimated values. Briguet et_al?02
have found J(C798r) in CHBr3 to be ~41 Hz, but their measured Ric at
room temperature was about 0.14 sec'1 which is much lower than our

value and the other reported valueslss’199

. Although a higher aniso-
tropy of the reorientational motion will give a lower calculated J(C798r),

we found that the lowest possible J(C798r) (about 90 Hz for AE|'=0 with

* 112

Table 15. The separation of the different contributions to the total
relaxation rates of 13C in CHBr3.

 

 

. b c d
a t d sc
1' 1000/1‘ 19L R1 R1 R1
(OK) (OK'I) (psec) (sec'l) (sec-1) (sec-1)
124 2.52 2.72 0.90 0.0653 0.835
107 2.63 2.97 0.86 0.0713 0.789
90 2.75 3.30 0.97 0.0792 0.711
69 2.92 3.82 0.70 0.0917 0.608
50 3.09 4.37 0.61 0.105 0.505
31 3.29 5.17 0.61 0.124 0.486
11 3.52 6.25 0.61 0.150 0.460

 

aCalculated from Equation (123).

bFrom reference 203.

c d- 10
R1-2.4OXI0 161:

d sc _ t d
R1 - Rl - R1.

113

Table 16. Physical properties of CDBr and CDCl .

3 3

 

 

Physical Property CDBr3 CDCl3
Ii (10'40g cmz) 685a 264b
Ill (10’40g cmz) 13383 498b

d
(Q.C.C.)D (kHz) 177c 167
(Q.C.C.)x (MHz) 601e 79f
O a g
rCD (A) 1.07 1.100
AE k 9h 7b
11 ( cal/mole) 0.. _ 0.
1.351
AEi (kcal/mole) 1.67h 1.6b
1.73
r (at 20°C)k 3.08 1.88b
1.81

 

aCalculated from the geometry in Reference 200 .

bReference 172.
cReference 198.

d

8Reference 201.

f

J. L. Ragle, G. Minott and M. Mokarram, J.

Chem. Phys. 69, 3184 (1974).

E. A. C. Lucken,"Nuclear Quadrupole Coupling Constants", Academic

Press, New York (1969).

gM. Jen and 0. R. Lide, Jr., J. Chem. Phys. 36, 2525 (1962).

hFrom this work.

iFrom the 13

jReference 176.
k

C data of Farrar g; 31.

203

r is the reorientational anisotropy (Equation (113)).

114

our 13C data and is ~98 Hz with Farrar's 130 data) was still high. This

is probably an anomalous property of highly brominated methanes. We have
not been able to discover any reasonable alteration in the data analysis
method which would reduce the calculated J(C798r) by a factor of two or

three§01.

 

l9 13
C. F and C Relaxation Studies in Liquid CFBr3

1. Introduction

A complete determination of the reorientational anisotropy is almost
impossible in CFBr3,which does not have any observable quadrupolar
nucleus. However, from the small dipole moment (0.58 D206) and the
fairly spherical shape of the molecule,the molecular reorientation can
be assumed to be isotropic. This has been found to be true in CFC13171
which has a dipole moment ~0.46 D and also has a nearly spherical
molecular shape. With this assumption, the 19F and 13C relaxation rates
in liquid CFBr3 can be analyzed to provide estimates of a few physical
constants of this molecule which cannot be obtained otherwise, e.g., the
scalar coupling constants between fluorine and bromine,and between carbon
and bromine,and the spin-lattice relaxation times of the bromine nuclei.

19

2. F Spin-Lattice Relaxation and Isotropic Reorientation

The measured 19F spin-lattice relaxation rates R; F are listed in
3
Table 17 and plotted against reciprocal temperature in Figure 21. A
dominant spin-rotational relaxation is evident from the negative slope

of the plot at high temperatures. At low temperatures the dominating

Ta

.

 

.071623345681111

‘

115

 

 

Table 17.19F spin-lattice relaxation rates in CFBrS.

T 1000/T R

(°C) (°x‘1 (sec'l)
-75.0 5.05 0.248:0.009
-67.0 4.85 0.208:0.007
-66.0 4.83 0.213:0.003
~59.0 4.67 0.177:0.004
-51.0 4.50 0.162:0.004
-50.0 4.48 0.155:0.006
-4l.5 4.32 0.148:0.003
-42.0 4.33 0.144:0.005
-3l.0 4.13 0.133:0.002
-21.0 3.97 0.121:0.002
~18.0 3.92 0.125:0.005
-17.0 3.91 0.122:0.002
-ll.0 3.82 0.120:0.002
-9.0 3.79 0.123:0.006
-1.0 3.68 0.120:0.003
0.0 3.66 0.125:0.002
7.0 3.57 0.130:0.006
18.0 3.44 0.130:0.002
6.0 3.58 0.124:0.001
27.0 3.33 0.138:0.004
36.0 3.24 0.155:0.009
37.0 3.23 0.142:0.002
46.0 3.13 0.156:0.004
57.0 3.03 0.170:0.006
69.0 2.92 0.188:0.006
85.0 2.79 0.203:0.006
104.0 2.65 0.233:0.007
122.0 2.53 0.259:0.009
146.0 2.39 0.319:0.016
170.0 2.26 0.370:0.014

 

116

 

0.5 — 4

I :1
)1.

RHSEC

O
. O
H

0.05

 

 

 

l
3

 

1
lOOO/TtoK) 4 5

Figure 21. Plot of the 19F spin-lattice relaxation rate as a
function of reciprocal temperature; 0: R1,F3 __3 R

---: RffF.

d .
1,F’

1‘8

in

DE

85

 

 

117

relaxation mechanism is probably that due to the intermolecular dipolar
interaction. The other relaxation mechanisms can be easily shown to be
negligible. The intermolecular dipolar relaxation rate in CFBr3 can be
estimated by use of Equation (133)

d "HZYgN 2 2
R = ————-'(YF + IOYBr). (133)

l,F 405a

where a, the molecular radius, and N, the number of molecules per unit V
volume, can be estimated from the density by using Equations (108) and
(109); 05’ the self-diffusion coefficient, can be estimated from the

shear viscosity n of liquid CFBr3 by the equation

 

s = 60::fr ’ (134)
where fr=0.5 is a correction factor from the microviscosity theory211
Gillen gt_al,17l have reviewed several studies and found that the
intermolecular dipolar relaxation rate calculated from Equation (133)
did not agree with the directly measured values from the isotopic

163’164; however, the activation energies obtained from

dilution method
both methods were essentially the same. Therefore, instead of directly
using the calculated values from Equation (133), we use the same
activation energy, which was found to be 2.97 kcal/mole, and vary the
pre-exponential factor so that the spin-rotational relaxation rates
Rifp ( RifF=R§’F-Rg,p) at low temperatures follow an Arrhenius relation
with the temperature. The best values of Rg,F and Rifp obtained by

this method of fitting are listed in Table 18 together with the

theoretically calculated R? F values from Equation (133).

 

 

 

hm~0£mfi Lm~_.u.o~CELL+C.,

 

118

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eHH.o memo.o omHo.o eHo.o oee.o moo.m om.“ H.ee He.» o.o~
eeo.o NeNo.o meNo.o oNo.o emo.o meo.m oo.~ ~.oo oo.m o.o
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119

The low temperature spin-rotational relaxation rates are found
to have an activation energy about 1.42 kcal/mole and can be repre-

sented by
air = 1.343 exp(-l.42 kcal mole'l/RT). (135)
Since, at low temperatures, the molecular motion is very likely to be~

diffusive, Hubbard's relation (Equation (78)) can be used in Equation

(80) so that

2 2 2
2 4w I C
Rfr = §1_%I-IcgffTJ = 2 eff , (136)
n 3” Te

where the symbols have their usual meanings.

The 19F chemical shift in CFBrs was measured and found to be -7 ppm
relative to CFC13. From Equation (116) we calculate the spin-rotation
constant Co to be ~l.40 kHz. Using this value for Ceff in Equation (136),
we find the reorientational correlation time re to be expressed by the
equation

13exp(1.42 kcal mole'l/Rr), (137)

T9 1.62X10'
which gives 1621.75 psec at room temperature.

Miller and Smyth204 have measured the dielectric relaxation times
TB of liquid CFBrS at four temperatures. These are shown in Figure 22
together with the re values from Equation (137). The ratio TE/Te in
the temperature range where TB is available is found to be about 3.4,

which is very close to the theoretically predicted value 3.0 for

120

 

-12
N 10 SEC

 

 

 

 

1000/ T(°K)

Figure 22. Plot of the reorientational correlation times as
functions of reciprocal temperature for CFBr3;
—: Te; --‘: TE.

121

molecules undergoing diffusive reorientation.

3. 19F Spin-Spin Relaxation and J(F798r)

The measured 19F spin-spin relaxation rates R2,F are shown in Table
19 and in Figure 23. In addition to the intermolecular dipolar interac-
tion and the spin-rotational interaction, the scalar coupling between
19F and the bromine nuclei also gives a considerable contribution to the
spin-spin relaxation rates. Therefore the scalar-coupled relaxation
rates can be obtained by measuring the differences between R5 and R} as
shown in Figure 23.

From Equation (70), and since szTi>>l, the spin-spin relaxation

rates due to the scalar coupling between 19F and the three bromine

nuclei in CFBr3 can be written as

83° = 7.5 «2( J2(F798r)T1(798r) + J2(F818r)T1(818r). (138)

This equation can be further simplified by using the relations:

J(F798r)/J(F818r) = y(798r)/y(818r) = 0.927 and T1(818r)/T1(798r) =

‘(79

Q Br)/Q(8lBr) 2 1.433. Therefore,

Ric = 200.6 J2(F798r)T1(7gBr), (139)

where T1(798r) is given by

79

TI1( Br) = 3.95 (Q.C.C.)2 t9. (140)

Table 19.

122

19F spin-spin relaxation rates in CFBr3

 

 

7 (°C) 1000/T (°K'1) R2 (sec‘l)
32.0 3.28 0.195:0.008
8.0 3.56 0.179:0.017
18.0 3.44 0.183:0.016
49.0 3.10 0.224:0.012
-78.0 5.13 0.340:0.023
-68.0 4.88 0.257:o.024
-54.0 4.57 0.204:0.010
-56.0 4.60 0.204:0.009
.37.0 4.24 0.176:0.008
.24.0 4.02 0.16l:0.009
-l6.0 3.89 0.157:0.011
-3.0 3.70 o.173:0.011
-5.o 3.73 0.158:0.008
55.0 3.05 o.24o:o.005
64.0 2.97 0.278:0.012
85.5 2.79 0.288:0.015
106.5 2.64 0.40310.015
144.5 2.40 0.500:0.048

 

 

 

 

(108love? o.

123

 

0.5

 

 

 

 

 

14:13.
I
41 z
0.1 "'
G
0.1 -‘
TA 0
L)
“U
33
~ 8
o:
C
0.05- . -
O
O 'F
C
t ‘P
0.02 “ J“ "
1.
1 1
3 IOOO/T(°K) 4

Figure 23. Plot of the 19F spin-spin relaxation rate in CFBr3 as

a function of reciprocal temperature; 0 :Rt ;
. Rsc 2,F

2,F°

124

The quadrupole coupling constant for 81

found to be 518.96 Mizzo1 in the solid phase. The corresponding quadru-

Br in CFBr3 has been measured and

pole coupling constant of 798r is therefore ~621 MHz. From Equations
(139) and (140) we obtain

Ric = 1.32X10 16.179202 Br)/Te. (141)

Using re in Equation (137) and the measured Rgc, we find the coupling
constant J(F798r) to be ~30.5:l.4 Hz. This can be compared with the
J(F3 SCl) value (~11. 9+0. 4 Hz) in CFC13 obtained by Gillen et a1171.
The ratio of J(F798r)/J(F35 Cl) is found to be ~2.56 which is in
excellent agreement with the ratio of the gyromagnetic ratios of 798r
and 3501 (7(798r)/y(3SC1)=2.55).

4. 13C Spin-Lattice Relaxation and J(C798r)

13

It has been found that the scalar coupling between C and bromine

nuclei is a dominant relaxation mechanism in highly brominated methanes.
This is also true in CFBr3. By the same approach used in analyzing the

13C relaxation rates in CHBrs, we find that the only two important

mechanisms contributing to the 13C relaxation in CFBr3 are the dipolar
interaction and the scalar coupling due to the three fast-relaxing

bromine nuclei. The dipolar relaxation rate R? C can be estimated by

the following equation:

4 2

R1,C W” (Yrrcr * ISYBr rCBr)T9’ (142)

125
where we have assumed that the reorientation of the molecules is iso-
tropic so that the same correlation time T6 can be used to describe the
reorientation of both the C-F and the C-Br vectors. Table 20 summarizes
the experimental data (values of RE,C at various temperatures) along
with the calculated values of R?,C; the scalar-coupled relaxation rates
Rifc obtained from R§,C"R?,C are also given and are plotted in Figure 24.
By using Equations (129) and (140), we can then calculate the
coupling constant J(C798r). This was found to be ~124.2t7.0 Hz which is

comparable to that in CDBr3. The various quantities used in the calcu-

lation of J(C798r) and T1(798r) are summarized in Table 21.

5. Extended Diffusion Models

The spin-rotational relaxation rates can be used to obtain the

angular momentum correlation times TJ. From Figure 21 we found that

sr
Rl,F

temperatures. This is evidently due to the increasing importance of

plotted versus 1/T deviates from the Arrhenius equation at high

the inertia effect. In Figure 25, we have plotted the reduced re-
orientational correlation times 1; versus the reduced angular momentum
correlation times 13. It is not surprising that the points at low
temperatures lie so close to the curve predicted by Debye's
diffusion model since we have used this model to derive the we values.
At higher temperatures, the points are approaching the J-diffusion
curve. We conclude that this is another proof of the validity of the
extended J-diffusion model in describing the reorientations of the
molecules in the liquid state.

Table 22 summarizes the physical constants used or obtained in this

study.

126

Table 20. Summary of the 13C spin-lattice relaxation rates in CFBr

 

 

3.
T lOOO/T a at Rd b RSCC
Te 1.C 1.C 1,c
(°C) (OK-1) (psec) (sec-1) (sec-1) (sec-1)
-73 5.00 5.77 0.56:0.10 0.0249 0.535
~56 4.61 4.37 0.58:0.17 0.0189 0.561
-53 4.54 4.16 0.59:0.11 0.0180 0.572
-35 4.20 3.26 0.62:0.10 0.0141 0.606
-8 3.77 2.40 0.69:0.11 0.0104 0.680
27 3.33 1.75 0.84:0.13 0.0076 0.832
55 3.05 1.43 1.00:0.15 0.0062 0.994
94 2.72 1.13 1.45:0.15 0.0049 1.345

 

3Calculated from Equation (137).

bCalculated from Equation (142).

c sc _ t d
R1,C- R1,C'R1,Co

0.5

.0
N

0 SEC

0.05

0.02

QOI

0.005

127

 

 

 

 

“- “if-mi:

.J

.4
l I l
3 4 5

IOOO T(°K)

Figure 24. Plot of the 13C spin-lattice relaxation rate in CFBr3

: R¥,C; ---: Rifc.

as a function of reciprocal temperature; 0: Rt

1,C‘

 

128

Table 21. Summary of the information used in the calculation of

79
T1( Br) and J(C7gBr) in CFBr3.

 

 

T IOOO/T x: T1(79Br)b Rice J(C798r)c
(°C) (°x'1) (psec) ( sec) (sec-1) (Hz)
-73 5.00 5.77 0.114 0.535 136.0
-56 4.61 4.37 0.150 0.561 129.7
—53 4.54 4.16 0.158 0.572 129.2
-35 4.20 3.26 0.201 0.606 124.2
-8 3.77 2.40 0.274 0.680 118.8
27 3.33 1.75 0.375 0.832 116.4
55 3.05 1.43 0.459 0.994 116.8
94 2.72 1.13 0.581 1.345 122.3

 

Ave: 124.217.0

 

8Calculated from Equation (137).
bCalculated from Equation (140).

cCalculated from Equation (130).

129

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N . _ md .9 Nd .6. mod
— .. .../ a _ I 4 _ a —

 

 

 

 

 

130

Table 22. Physical properties of CFBr3 used or obtained in this study.

 

Physical Property Value Reference
81

(Q.C.C.)( Br) 518.96 MHz 201
-8

rCF 1.44XIO cm 205

rCBr 1 .91><10'8 cm 205

II I 1334x10'40g cmz

Ii. 785X10-4Og cm2

H 0.58 D 206

cabs 188.6 ppm

Co -1.40 kHz

79
J(F Br) 1(30.Stl.5) Hz

J(C79Br)

i(124.2i7.0) Hz

 

131

13

D. C Nuclear Magnetic Relaxation and J(C79Br) in Liquid CHQBr7

 

1. Introduction

In order to obtain further information concerning the scalar coupling
constant between carbon and bromine nuclei which cannot be directly
observed in NMR spectroscopy, we have chosen a less highly brominated
methane, methylene bromide, whose motional parameters have been obtained
recently by Sandhu207 so that the separation of different mechanisms

contributing to the 13

207

C spin-lattice relaxation rates can be achieved.
Sandhu studied the relaxation of 1H and 2D in CHZBrz and obtained
the rotational diffusion constants parallel and perpendicular to the C2
symmetry axis of the molecule. From Figure 4 of Sandhu‘s paper, and
using his best estimated deuterium quadrupole coupling constant, 181 kHz,
we obtain

1

DJ. = 2.19x10 2exp(-2.23 kcal mole‘I/RT) (143)

and

0" = 9.07x1012exp(-2.09 kcal mole'l/RT). (144)

Gordon gt_gl,208 have also studied the proton relaxation of CHZBr2 in
CS2 solution and found that the spin-rotation interaction gave a con-
siderable contribution to the proton spin-lattice relaxation rate in

dilute solutions of CHZBr2 at high temperatures. They also found the

reorientational correlation time of the H-H vector to be ~0.74 psec

at room temperature which is only about one-half the value 1.54 psec

132

found in neat CHZBrz. Also, Sandhu's relaxation studies in neat CHZBr2
showed no contribution of the spin-rotational relaxation. Evidently

the intermolecular interaction among the CHZBr2 molecules has an im-
portant effect on the reorientational motion of the molecules. Wang and

2

his co-workers 09 have also investigated the molecular motions of CHZBr2

by depolarized Rayleigh and Raman scattering experiments. Several con-
clusions were reached from their studies. First of all, they found that
the reorientational correlation time obtained from the Raman lineshape
analysis was not equal to the single-particle reorientational correlation
time, which was found to be 3.4 psec from the depolarized Rayleigh
scattering experiment at room temperature. Second, from a variable temp-
erature study of the depolarized Rayleigh scattering in neat CHZBrz,
they obtained an activation energy ~2.8 kcal/mole for the tumbling
motion of the molecules. Third, their plot of the correlation times
versus n/T showed a slope about ten times smaller than that predicted by
the Stokes-Einstein relation but was in agreement with that predicted by

210

considering Hu and Zwanzig's slip-boundary condition and modeling

CHZBr2 as an ellipsoid.

Wang's result20

9 for the room temperature reorientational corre-
lation time, 3.4 psec, was in very good agreement with Sandhu's result
(teie3.5 psec at room temperature) but the activation energy of the
tumbling motion was about 25% higher than that obtained by Sandhu (
~2.23 kcal/mole). Nevertheless, since Sandhu's studies covered a wider

13C relaxation rates will be

temperature range, the analysis of our
based on the reorientational diffusion constants expressed by Equations

(143) and (144).

133

13

2. Mechanisms of C Relaxation and J(C79Br)

Among various mechanisms of 13C spin-lattice relaxation, only the

dipolar interaction with the two protons and the scalar coupling with
the two fast-relaxing bromine nuclei should be considered. The other
mechanisms can be easily shown to be negligible.

The dipolar relaxation rate can be expressed by
R?,C = 2yéy§fi2regtfi = 5.37x1olorH, (145)
where TH is the reorientational correlation time of the C-H inter-
nuclear vector which can be estimated from the geometry of the molecule
and by using Equations (111), (143) and (144). In Table 23, we
summarize the calculations of R$,C and therefore the scalar-coupled
relaxation rates Rifc,which is the only other important relaxation
contribution and can be obtained by subtracting Rg’c from the measured
total relaxation rates R1,C' The results obtained by these calculations
are also plotted as a function of temperature in Figure 26.

The extracted scalar-coupled relaxation rates can be used to

estimate the coupling constant J(C79

Br) by the same technique used

before (see Section B). For this molecule, which contains only two

bromine nuclei, we obtain
sc

R1 0 = 10n2J2(C7gBr)T1(7gBr)(1+

1.71
1+1.02x1014Ti(7gBr)

 

) (146)
and

T1(79Br) = 3.95(Q.C.C.)21 (147)

Br’

134

Table 23. Sungry of the quantities employed in the calculation of

 

 

the C dipolar and the scalar-coupled relaxation rates in
CH Br .
2 2
t a db scc
T 1000/T R1,C TH R1,C R1,C
o - - - -
( C) (OK 1) (sec 1) (psec) (sec 1) (sec 1)
-47.5 4.42 0.33710.03S 4.93 0.265 0.072
-29.5 4.11 0.275:0.030 3.54 0.190 0.085
-5.5 3.74 0.240:0.025 2.38 0.128 0.112
28.0 3.33 0.24Si0.020 1.54 0.083 0.162
58.0 3.02 0.276t0.021 1.10 0.059 0.217
83.0 2.81 0.340i0.040 0.88 0.047 0.293

 

at" = 4.33x10‘14exp(2.13 kcal mole'I/RT).

bCalculated from Equation (14S).

sc Rt _ Rd

c 8
R1,c 1,0 1,0'

135

 

 

 

0.5 -
mp
t E
6 R1
é)
0.2 *- _
T
o sc ~‘
8 R1
cf 4‘
\ _.
001 fl —
d
R] ..
0.05— ._.
dL
I 1 1 l

 

 

 

3.0 3.5 4.0 4.5
1000/ T

Figure 26. Plot of the 13C spin-lattice relaxation rate in CHZBrZ
as a function of reciprocal temperature.

136

where TBr is the reorientational correlation time of the principal axis
of the electric field gradient around the bromine nuclei which is
assumed to be along the C—Br bond. The other symbols have their usual
meanings. Again, far can be obtained from the geometry of the molecule
and by using Equations (111), (143) and (144). From Equations (146) and
(147) and the extracted scalar coupled relaxation rates, we calculate

the coupling constant J(C79Br) to be m58.7il.4 Hz. Table 24 gives a

summary of the quantities used in this calculation.

3. Discussion

Of the three brominated methanes studied, we found that only
methylene bromide has a scalar coupling constant J(C7gBr) (58.7 Hz)
comparable to the coupling constant J(C3SC1) (23 Hz) obtained by Shoup

and Farrarl

99. The values of J(C798r) in both CDBr3 (121 Hz or 105 Hz)
and CFBr3 (124 Hz) seem to be too high if the relation JCX/JCY=Yx/YY
holds approximately. Although Freeman gt_§l?12 have estimated the

magnitude of J(CSS

C1) to be ~49 Hz from double resonance (proton) line
profile studies in connection with 35Cl wideline measurements of CHC13,
it was argued by Shoup and Farrar that Freeman's result was possibly in

error for many reasons. If Freeman's result for J(C35

Cl) was used to
estimate J(C79Br), we find the magnitude of J(C79Br) is about 125 Hz
which is very close to our results obtained for CDBr3 and CFBrz.
However, since the coupling constant J(C79Br) in CHzBr2 was obtained
by the same technique, we suspect that other factors should be taken
into account to explain this large discrepancy.

It has been well known that increasing the 0 bond electron density

between carbon and halogen atoms tends to increase the coupling

137

Table 24. Summary7 of quantities used in the calculation of T1(F7 9Br)
and J(C 7gBr) for CHzBrZ.

 

 

T 1000/T 10100 a 10100 b rgr T1(79Br) Rife J(C79Br)
(°C) (OK-1) (cmzsec'1)(cm2sec'1)(psec) (psec) (sec'l) (Hz)
-47.5 .43 8.59 .52 5.10 0.138 0 072 60.5
-29.5 .11 12.0 .17 3.62 0.194 0.085 57.5
-5.5 .74 17.7 .29 2.42 0.290 0.112 57.8

28.0 .33 27.3 .22 1.55 0.453 0.162 58.1
58.0 .02 37.8 .39 1.11 0.634 0.217 57.8
83.0 .81 47.2 .35 0.88 0.796 0.293 60.3

 

Ave: 58.711.4

 

3Calculated from Equation (144).
bCalculated from Equation (143).
chr = 4.30 exp(2.l4 kcal mole'l/RT).
dCalculated from Equation (147).

eCalculated from Equation (146).

138

150

constant JCX . Therefore, it is very likely that in tribromosubsti-

tuted methanes there is an important carbon-bromine double bond

295 has also found that the substitution of a more

character. Pauling
electronegative atom will increase the tendency to form C=X double bonds
in halomethanes. Since bromoform has one bromine atom substituting for
a less electronegative hydrogen atom of methylene bromide, a higher

9Br),

C=Br double bond character,and therefore a larger value of J(C7
should be expected. By the same reason, the substitution of a fluorine
atom for the hydrogen atom of bromoform should further increase the

magnitude of J(C79Br), which is exactly what we observed.

E. Anisotropic Reorientation and 19F Relaxation in CF7CC19

 

1. Introduction

Due to the complications of describing the molecular motions in
asymmetric-top molecules, very few such compounds have been studied.‘

Wallach and Huntress213

have attempted to resolve the anisotropic
reorientational diffusion constants in N,N-dimethylformamide by
observing the relaxation times of three quadrupolar nuclei at different
sites of the molecule. They found that rotation about an axis on the
molecular plane and perpendicular to the dipole moment vector was at
least thirty times slower than rotation about the dipole moment vector
and that rotation about the other principal axis proceeded at rates
that varied from five times (at room temperature) to twenty times (at
390°K) faster than the slowest rotation. This can be easily explained

by considering the nearly planar geometry of the molecule and its large

dipole moment (3.91 D). Rotation about the axis parallel to the dipole

139

moment does not destroy the alignment of the molecules due to the
dipolar "head-to-tail" attraction and only expels the molecules on top
and at the bottom of the molecular plane. 0n the contrary, to rotate
the molecule about an axis in the molecular plane and perpendicular to
the dipole moment not only destroys the dipolar alignment but also
needs a large free volume, therefore, it experiences the greatest
interference. The rotation about the other axis, which is perpendicular
to the molecular plane, does not need a large free volume but also
needs to counteract the dipolar interaction,and the interference should
not be so great as that experienced by the above rotation.

In CFZCCIZ, we expect that the reorientational motions should also
be very anisotropic. Unfortunately, there are not enough quadrupolar
nuclei in the molecule to give the number of conditions needed to
completely elucidate the molecular motions. Therefore, we only attempt-
ed to investigate the molecular reorientation about the F-F inter-

nuclear vector from our 19F relaxation studies.

2. Molecular Motions in the Solid Phase

The measured 19F spin-lattice and spin-spin relaxation rates are
listed in Tables 25 and 26 and are plotted versus l/T in Figure 27.
From the plots of the spin-lattice relaxation rates in Figure 27, we

found that in the solid phase the 19

F relaxation was largely by the
intramolecular dipolar interaction and the chemical shielding aniso-
trOpy mechanisms. The intramolecular dipolar relaxation rates can be

expressed by

140

 

 

Table 25. 19F spin-lattice relaxation rates in CF2=CC12.

T (°C) 1000/T (°K‘1) R1 (sec-1)
30.0 3.30 0.103:0.001
18.0 3.44 0.0932:0.0029
6.0 3.58 0.0872:0.0011

-ll.S 3.82 0.079210.0016
-l7.0 3.91 0.0764:0.0024
-25.0 4.02 0.075110.0038
—19.5 3.94 0.082910.0049
-27.S 4.07 0.073210.0024
-35.0 4.20 0.0738:0.0021
-44.0 4.37 0.071210.0024
-S4.0 4.57 0.075510.0015
-66.0 4.83 0.08410.001
-75.5 5.05 0.084:0.003
-82.5 5.26 0.0908:0.0024
-88.0 5.40 0.102:0.002
-9S.0 5.62 0.114:0.001
~100.0 5.78 0.126:0.001
-106.5 6.00 0.15310.002
-ll6.0 6.37 0.229:0.005
-119.0 6.49 0.23610.002
-123.0 6.67 0.27010.003
-124.0 6.71 0.283:0.009
-130.0 6.99 0.356:0.008
-l35.0 7.25 0.834:0.019
-l41.0 7.63 1.49610.056
~145.0 7.81 2.19:0.22
-149.0 8.06 4.57iO.63
~121.0 6.58 0.283:0.003

 

19
Table 26. F spin-spin relaxation rates in CF

141

 

 

2=Cc12.
T (°C) IOOO/T (°x'1) R2.(sec-1)
-130 6.99 1.093:0.097
-118 6.41 0.225:0.o12
-100 5.78 o.192:0.009
-95 5.62 o.192:0.010
-75 5.05 0.18910.017
-63 4.76 0.242:0.024
-42 4.33 o.312:0.031
-23 4.00 0.386:0.035
9 3.55 0.514:o.041
27 3.33 0.567:0.073

 

142

 

 

 

 

5 r—
........ .» R1
- —'- (3 R2 f
---- 0 R2- 12,
2 — .1"
.7”
0!:
1 -
of
as 9:9.
{1‘ \ rn;p. ,'
or ‘\\€'\ 0 : f.
U 'x l .'
o G’ ' f
m \\ o°\.‘ : ’
GK \\ ~51 ’W*V.‘r
Q2_ \ \ ~. .’.’.”0.
o\ O ~._.G.o . :
\ x
\\ .x"
,4.
‘ (l\ 3"
0.141., x .-
'N.. “A.\
s. .. .9 . ‘ o
”Q‘w \ ‘
\\\
\
0.05— X
0\\
\
\
l l l \ l l
4 5 6 7 8
IOOO/T

Figure 27. Plot of the 19F relaxation rate for CF2=CC12 as a function
' of reciprocal temperature.

143
d __ 3 4 2 -6
R1 "' '2- an rFF £6290): (148)

where f(n,D) has been defined in Equation (88) for a planar asymmetric-
top rotor. In this case, since the internuclear vector of two fluorine
nuclei is perpendicular to the symmetry axis of the molecule, Equation

(88) can be simplified to
f(n 0) --3—(0 +0) (149)
’ ‘40R y s’

where DR=3(DXDY+DyDz+Dsz) and D5: %{Dx+Dy+Dz)' The 2 axis is defined
to be perpendicular to the molecular plane, the x axis is defined to be
along the C2 symmetry axis and the y axis is perpendicular to the x

and z axes.

From the recently determined molecular geometry214, we obtained

rFF22.18 R and therefore

(1 D +0
R = 4.68XI09—%-—s-

(150)
1 R

The relaxation through the chemical shielding anisotropy can be
estimated by Equation (73a). For Ao=200 ppm this contribution will be
about 10% of R3. Since the actual A0 in this molecule is not known,
we will assume that this contribution is negligible. Since we only
have one equation and there are three unknowns,it is impossible to
solve the problem without any approximations. We can assume that in

the solid phase of CFZCCl , as was found213 for N,N-dimethyl-

z-Dx>>Dy’ D

formamide. Then Equation (150) can be further simplified to give

Z

144

Rd 2 4.68x109——-—l————-= 2.60X108/D , (151)
1 9(0y+02) .L

where DL is defined as the average diffusion constant for the tumbling
motion. By using Equation (151) and the measured 19F spin-lattice
relaxation rates in the solid phase at low temperatures, we can cal-

culate this average diffusion constant D1: The expression best fitting

the experimental data was found to be
0le (9.0:3.0)x1016exp(-5.22 kcal mole'l/RT), (152)

where the large uncertainty arises from having made the approximation

Dx>>D Equation (152) gives DisS.ZX109 cm2 sec‘l, or Tele32.0 psec,

y’Dz'
at a temperature just below the melting point. But when the dipolar
relaxation rate is extrapolated to room temperature Telf0.01 psec,
which is unreasonally small. This implies that the reorientational
motion in the liquid and solid phases should have different mechanisms.
The activation energy of the reorientational motion in the liquid
should be smaller than 5.22 kcal/mole since the onset of translational
motion has considerably eased reorientation of the molecules. 0n the
other hand, the frequent collisions in the liquid phase should slow
down the reorientational motion of the molecules. In the solid phase
the molecules are fixed in lattice sites and therefore reorientation

is more likely to be via random jumps among different orientations

of the molecules.

145

3. Rotational Motion in the Liquid Phase

The analysis of the 19

F relaxation in liquid CFZCCl2 is even more
difficult because several possible relaxation mechanisms are to be
considered and sufficient physical constants are not available..
However, by measuring the differences between the 19F spin-spin and
spin-lattice relaxation rates we obtained the scalar-coupled spin-spin
relaxation rates,which followed the Arrhenius equation with temperature
very well. The activation energy was found to be ~1.83 kcal/mole.

The existence of two chlorine isotopes might complete the problem.

The decay of the magnetization due to the scalar coupling relaxation

mechanism can be shown to be

I!

t = 9 sc _9_ 5° .1. 5°
gg' ig'exP('tR2,3535) * 16 exP(‘tR2,3537) * 16 exP(‘tR2.3737)’
(153)
where
Rsc - Rsc + Rsc (154)

2,AB ‘ 2,4 2,B

and each ch is given by a standard expression of the form of Equation
(70). Equation (153) implies a nonexponential decay of the transverse
magnetization. However, a simple calculation shows that this non-
exponentiality can be ignored if the relaxation rates are determined
from the initial decay of the transverse magnetization. Therefore, by

noticing that szTiS>>l, the scalar-coupled spin-spin relaxation rates

can be expressed by

146

83° = 98.7 12(935c1)T1(3501), (155)

where J2(F35C1) is the mean square value of the cis and trans coupling

constants. Tl(35Cl) can be written as
Til(35Cl) = 3.95(Q.C.C.)ZT¢, (156)

where T¢ is the correlation time describing the reorientation of the
C-Cl bond, which is assumed to be the principal axis of the electric
field gradient tensor at the chlorine nucleus. T¢ can also be written
as a function of the diffusion constants as in Equation (88).

Since the scalar coupling constant J(FSSCl) has rarely been
measured, we will try to estimate it from the magnitude of JFF in some

217 215

analogous compounds . Tiers and Kaiser216 have both measured the

coupling constants JFF in cis and trans-1,2—dif1uorodichloroethylene.
Their measured JFF(cis) and JFF(trans) values were 37.9 Hz and 130 Hz,

respectively. By noticing that the coupling constants JAx and JAY are

usually related to the gyromagnetic ratios YX’YY by JAX/JAY=YX/YY’
we can estimate J(F35C1)(cis) and J(F35Cl)(trans) to be m4 Hz and
~14 Hz, respectively.

The quadrupole coupling constant of 35C1 in CFZCCIZ is not

21
available, but since Q.C.C.(35Cl) is ~73.7 MHz in cis-CHC1=CHF 8,

~70.5 m: in cis-CHC1=CHC1219

and v»76 MHz in CF2=CFC1294, we
can estimate that it is ~75 Mix in CF2=CC12. Therefore, Equations (155)
and (156) can be combined to give

13

R = 9.4ZXIO- /T¢. (157)

147

From the measured kgc we find

1

= 7.8SXI0' 4exp(l.83 kcal mole-I/RT), (158)

‘4
which gives T¢=1.72 psec at room temperature.

F. 19F Relaxation Mechanisms for CF31

 

Trifluoromethyl iodide is a symmetric-top molecule with dipole
moment 021.0 D220 directed along the C3 symmetry axis and a fairly an-
isotropic molecular geometry (Il/I||=3.7)221. Therefore, its re-
orientational motions are expected to be very direction dependent.
Unfortunately, the relaxation time of iodine is too short to be
measured directly. Hence, information about the tumbling motion has to
be obtained by other spectroscopic techniques, e.g., a variable
temperature study of the depolarized Raman or Rayleigh scattering.

We have measured the 19

F spin-lattice relaxation rates in liquid
CF3I,which are listed in Table 27. The separation of various relaxation
mechanisms has been found to be difficult. From our measured densities
and the self-diffusion coefficients (Table 28),we estimate the inter-
molecular dipolar relaxation rates according to Equations (107), (108)
and (109), and subtract them from the total spin-lattice relaxation
rates. The results are shown in Table 29 and in Figure 28. It can be
seen from the plot in Figure 28 that spin-rotational relaxation is the
dominant relaxation mechanism over the whole liquid range. This has

made estimation of the intramolecular dipolar relaxation rates, and

therefore further separation of various contributions, impossible.

148

19

 

 

Table 27. F spin-lattice relaxation rates in CF31.
o O '1 -1

T ( C) 1000/T ( K ) R1 (sec )

3.0 3.62 0.558:0.007

4.0 3.61 0.559:0.012
-5.0 3.73 0.497:0.008
-15.0 3.88 0.48210.007
—25.0 4.03 0.435:0.007
-41.0 4.31 0.38510.004
-52.0 4.52 0.400:0.006
-63.0 4.76 0.359:0.004
-74.0 5.02 0.35410.004
-84.0 5.29 0.383:0.003
-94.0 5.59 0.37210.004
-103.0 5.88 0.37810.003
-111.0 6.17 0.420:0.006
-125.0 6.76 0.52910.005
-130.0 6.99 0.553:0.008
-137.0 7.35 0.62310.006
-135.0 7.24 0.146:0.012
-l36.0 7.29 0.134:0.012
-131.5 7.06 0.159:0.011
-127.0 6.85 0.284:0.007
-125.0 6.76 0.489:0.006
-124.0 6.71 0.45410.005
-122.0 6.62 0.459:0.003
-117.0 6.41 0.40910.004
~109.0 6.10 0.380:0.004
-102.0 5.85 0.381:0.002
-93.0 5.56 0.369:0.003
-85.0 5.32 0.355t0.003

 

149

Table 28. Self-diffusion coefficients of CF31.

 

 

T 1000/T g 10 05
(°C) (OK-1) (G cm-l) (cmzsec‘l)
3.0 3.62 161 2.02
-S.0 3.73 168 1.82
-15.0 3.88 178 1.57
-25.0 4.03 191 1.23
-41.0 4.31 215 0.97
-52.0 4.52 235 0.76
-63.0 4.76 260 0.60
-74.0 5.02 290 0.43
-84.0 5.29 325 0.332
-94.0 5.59 - 369 0.262
-103.0 5.88 419 0.223
-111.0 6.17 471 0.186
-118.0 6.45 531 0.122
~125.0 6.76 600 0.183

-130.0 6.99 670 0.070

 

. H u H n H
HoucH.um um nocuomo

.Hach soHumsum aoam wouaHsonoe
.HmoHv noHumocm aoum voumHsuHmuo
.nmoHV :oHumsdm Seam poumHsonun

.xuo: anu scum essHou «Hg» :H noaHe>m

150

 

 

 

moH.o owe.o eHm.o oH.o o¢.~ ma.~ ac.~ ma.o o.m~H-
omH.o ooe.o «Hm.o 6H.o ev.~ ma.~ mo.~ ~n.o o.mHH-
mNN.o owm.o mmH.o o~.o Nv.~ om.~ ~6.N ma.m o.moH-
mo~.o mmm.o mmmo.o Nm.o mm.~ mm.~ mm.~ cv.m o.om-
vom.o mmm.o NHmo.o om.o ~m.~ mm.~ Nm.~ Nm.v o.oe-
wvm.o omm.o mHmo.o mm.o m~.~ am.~ ve.~ we.e o.om-
«on.o uo~¢.o oc~o.o «HN.H 6H.~ om.~ amn.~ HH.¢ o.cm-
HH-uomv HH-666V HH-oomV HH-umm~aov Hn-EUV HMV Hm-ao my HH-goV Hues
H H H m
ouugpom pm cumucH.um amoH oz--oH pa 6 H\oocH a

 

.Hmmu :H moumu :oHpmmeoH ooHuumHuchm m map a» wcoHunnHHucou one we :oHuanmmom .mm oHpab

mH

151

 

 

 

#-

l 1 L

 

 

3 4 5 6 7
IOOO/T

Figure 28. Plot of the 19F spin-lattice relaxation rate in CF31
as a function of reciprocal temperature.

152

However, the small activation energy of the spin-rotational relaxation
rate (~0.6 kcal/mole) implies that this relaxation mechanism is pro-
bably due to the spinning motion of the molecules as has been fbund for

13
methyl iodide by a study of the C relaxation164.

II. NMR Studies of Molecular Motions in Plastic

 

Crystalline Fluorochlorocarbons

 

A. Introduction

 

The possibility of molecular rotation in crystal lattices seems
to have been first raised by Simon222 to account for certain solid—
solid transitions and this idea was developed by Paulingzzs. Several

years later Timmermans224

found a great number of molecules which
exhibited analogous physical properties such as being soft and waxy,
having high melting points and small liquid ranges, having high vapor
pressures and subliming easily, having low entropies of fusion and
high entropies of transition, etc. Such compounds were defined by him
as "globular molecules" and are known as "plastic crystals" nowadays.
The molecules generally have a high symmetry, or can achieve a
Spherical envelope by rotation, and are believed to be rotating in
the plastic crystal phase.

Many experimental methods have been used to investigate the
detailed structure and the dynamic properties of these molecules and
crystalszzs. Most of the earlier investigations were thermodynamic

studies in which the enthalpy or entropy change due to molecular

motions,and the temperature dependence of the heat capacity,were

153

measured226

. The transition of a rigid solid into a plastic phase can
be observed by the sharp rise in the heat capacity which occurs at the
onset of molecular rotation due to the extra amount of heat needed to
overcome the rotational energy barrier. A similar phenomenon has also
been found in studying the dielectric constants”? of these compounds.
A rapid increase of the dielectric constant signals the onset of a
solid-solid transition at which the molecules begin to reorient. In a

228, it was found that higher-

crystallographic study of plastic crystals
order reflections could hardly be seen because of their low intensity
and the fuzzy background which was caused by the randomly displaced
atoms or groups with respect to the mean geometric planes of the
crystal lattice. Many plastic crystals have been found to have a cubic
crystal structure and, in addition, the distance between molecular
centers in the plastic crystals has been found to be shorter than the
maximum diameter of the molecule. Some cooperative fluctuation in the
positiom of neighboring molecules therefore appears to occur when a
molecule undergoes reorientational motion.

The possibility of self-diffusion in plastic crystals was shown
by the high mechanical mobility229 in the high-temperature plastic
phases compared to that in the low-temperature phases of some crystals,
and later also by radiotracer technique5230.

Among all possible experimental methods, nuclear magnetic
resonance spectrosc0py was found to be the most powerful one for inves-
tigating the nature of plasticity in molecular crysta15231. Details of
the molecular motions can be studied by both NMR lineshape analysis and

232-261

by the relaxation behavior of the nuclei . We will discuss some

basic theories which are to be applied to our plastic crystal studies

154

in the following sections.

B. NMR Theories Relating to Molecular Motions in Plastic Crystals

 

1. Linewidths

Consider two spin k nuclei in a rigid lattice, the Hamiltonian

describing the dipole-dipole interaction can be written as

 

 

+
1 3,52 361-3102..)
JC = -{ - ), (159)
d 2 3 5
I‘ r

+ + . .
where "l and 02 are the magnetic dipole moments and I is the inter-
nuclear distance. From perturbation theory the eigenvalues can be found

to be (for "1:“2)

En = -2nuHo + uzr’3(3c0520-1)(2-3n2). n=0, 1. (160)

Transitions are allowed between two adjacent energy levels with energies
AB = 20{ Ho 1 g-ur'3(3cosze-1)}, . (161)

which should give two separate peaks in the spectrum. When there are n
spins all interacting with each other, these two peaks will be broadened
out to become a Gaussian envelope. As one might expect, any kind of
molecular motion in the lattice tends to modulate this dipolar inter-
action and therefore the resultant lineshape of the spectrum.
Bloembergen, Purcell and Pound52 considered the effect of stochastic

thermal motions upon the NMR absorption linewidth and obtained a simple

155

relation between the linewidth Av and the correlation time of the motion
Tc as follows:

2 . _
(Av)2 = C2 E-tan 1(2TCAV), (162)

where C is the linewidth of the rigid lattice spectrum.

Kubo and Tomita262 have introduced a quantum statistical formula-
tion into the NMR theories and have derived a similar equation for the
linewidth,

nAvr

4-ln2 2 -1 c
n C tan (4-ln2)° (163)

 

(Av)2 =

Gutowsky and Pake263 have also modified the Bloembergen-Purcell-
Pound equation and have obtained the following equation for the

correlation time TC!

_ 8'ln2

((4v12-821}
C - ZwAv

2(C2-B2)

 

T tan { (164)
where B is the linewidth after the motional transition.

Recently, Hendrickson and Bray264 have used a simple model of
molecular motion which is based on the partition of two states, non-
excited and excited, and have considered the linewidth as an arithmetic
average of those in these two states. They obtained

E
) = - —§-— ln( 1-- -l-). (165)

ln( —l-- 1-
Av C kT B Av

As we shall see, these equations give estimates of the correlation

156

time of the same order of magnitude, and also give the same activation

energy.

2. Second Moments of NMR Absorption Lines

Theoretical calculation of the second moments of NMR absorption
lines has been successfully done by Van Vleck26s. For a polycrystalline

specimen,
3 1 -
M2 = .5472”? ; I(I+1)E rjg
4 2 2 1 -6
* 13'an Ifufdkzrjf ' (166)

where the first term and the second term are the contributions from the
homonuclear and the heteronuclear dipolar interactions, respectively,
and n is the number of equivalent nuclei in a unit cell. Since the
indices j, k and f run over every nucleus in the lattice, it is more

convenient to write the second moment as the sum of two terms,

M = M

2 (167)

2,intra + M2,inter’
where the first term is the contribution from the nuclei within the
molecule and can be readily calculated if the molecular geometry is
known. The second term contains long-range interactions from nuclei
at remote lattice sites and is more difficult to evaluate. Andrew266
has considered the intermolecular contribution from remote nuclei with
rij>M as a continuous distribution of spins. M, the truncating radius,
depends on the lattice structure and 5 X has been proposed as a proper

157

value. With this choice, the intermolecular contribution to the second

moment of fluorine nuclei can be estimated by

-6 4nN
M . = 317.2 Z N. Z n. 2 r. + 317.2 -——, (168)
2,1nter i 1 j J k Jk 3M3V

slh‘

where n is the number of fluorine nuclei in a molecule, Ni is the
number of molecules on the i£h_nearest neighbor sites, nj is the number
of jth_fluorine nuclei in the origin molecule which are equivalent to
an ith_neighbor, rjk is the distance between fluorine j of the origin
molecule and fluorine k on the nearest neighbor, N is the number of
fluorine nuclei per unit cell and V is the volume of the unit cell in
cubic Angstrons.

When the molecules in a lattice undergo an isotrOpic reorientation-
al motion, the intramolecular part of the second moment vanishes and
the intermolecular part can be simplified to

= 317.2 n2 Niric’, (169)

1

M2,inter

where E Nir;6 has been calculated to be 29.045 a.6 and 115.631 a.6 ( a
is the lattice constant) for body-centered cubic and face-centered
cubic crystal structures, respectively. When self-diffusion occurs in
the plastic phase, this intermolecular contribution to the second
moment will be averaged out and the linewidth will approach that in
the liquid phase.

The effect of lattice vibration on the NMR second moment has

267,268

been discussed by Shmueli, Polak and Sheinblatt . From their

calculations and comparison with the experimental results for a few

158

compounds, it was found that these vibrations of the lattice tend to
decrease the intramolecular part, but increase the intermoleculat part,
of the rigid lattice second moment and the total effect does not

exceed 12%.

3. Spin-Lattice Relaxation Times

Since the linewidth is only sensitive to those molecular motions
with correlation frequencies of the order of the linewidth in the solid
phase, i.e., 103 to 105 Hz, only slow motions of the molecules can be
studied by linewidth measurement. Spin-lattice relaxation times, on the
other hand, have a much shorter time scale, namely, the reciprocal of
the Larmor frequency, and are therefore more sensitive to faster
molecular motions. Bloembergen, Purcell and Pound52 have derived a
general equation for homonuclear dipolar relaxation in terms of the

power spectra of the nuclear position functions which are random.

functions of time:
3 4 2 1
R1 = 27 h I(I+l) J1(w°)+ 552(2wo) , (170)

where the spectral .densitiesTJ1(u) are the Fourier transform of the

correlation functions

1-3cos20..(t)

 

Go(t) = < X | 3 1’ }> (171)
j rij(t)
sinze..(t)ex (21¢..(t))
Gl(t) = < Z 13 p *1. !> (172)

 

. 3
J rij(t)

and

159

sine . . (t) cose . . (t) exp(-i¢ . . (t))
62(t) a < Z | ' 13 11 13 i)
J

 

3 , (173)
rij (t)
in which r, 0, and 0 are the spherical polar coordinates of spin j
relative to spin i. If the molecules undergo a rotational Brownian
motion, the internuclear distance is fixed and the correlation functions
can be expressed by exponential functions with a time constant Tc, which

is the reorientational correlation time. Therefore Equation (170) becomes

 

T 4Tc
R1 = 1% Y4MZI(I+1)( (2: 2 + j?) i—Z Z ri?’ (174)
l+morc 1+4moTc i jfii

where indices 1 and j run over all equivalent nuclei in the lattice.
By combining this equation with the second moment expression, a more

convenient equation can be derived, that is,

 

R --3 2M ( 1° 4 41° ) (175)
l ' 3 7 2r 1+ 2 2 1+4 2 2 '
“oTc onc
where M21. is the second moment modulated by the molecular motion
considered.
When the molecules undergo a translational motion,rij will also

53’269 has considered a random-

be a random function of time. Torrey
flight model for the self-diffusion in a crystal lattice and derived

an equation for the spin-lattice relaxation rate as follows:

= 8fly4H2I(I+1)nc

Skslsw

 

R 9(k.y). (176)

1

where n is the number of molecules per unit volume, c the ratio of the

number of nuclei to the number of lattice sites, k the ratio of the

160

closest possible distance a of two approaching nuclei to the length of
each diffusion jump t; y is equal to %wtd and rd, the mean time between
adjacent diffusion jumps, is related to the self-diffusion coefficient
Ds by Equation (61). w(k,y) has been calculated for various values of y
and for both bcc and fcc lattices.

Equation (176) can also be written in terms of the second moment

as follows:
2
R, = v 112de {C(k.y)+4cck.2y)}, (177)

where M2d is the second moment modulated by the self-diffusion,and the
6 functions are calculated and tabulated by Torrey53’269. It should be
noted that by writing the spin-lattice relaxation rate in the form of

Equation (176) or (177), the contributions from both the homonuclear

and the heteronuclear interactions have been considered.

4. Spin-Lattice Relaxation Times in the Rotating Frame

_ In deriving the Bloembergen-Purcell-Pound equations, perturbation
theory was used since the fluctuating dipolar field is much weaker
than the static field (~10 kG) applied. When the static field is low
enough that the dipolar interaction becomes comparable to it, the
perturbation treatment of Bloembergen-Purcell-Pound is no longer
valid. Slichter and Ailion27o'273 first considered this strong-collision
case and devised an experimental technique, namely, adiabatic de-

magnetization in the rotating frame271

, to measure the spin-lattice
relaxation time in the presence of a moderately strong field H1 after

the static field Ho has been pulsed off resonance. The measured

161

relaxation rate Rlp was found to be

Hi+aH§ H;
R = R'(—-—) + R (——). (173)
10 1 2 2 c 2 2
H1+HD H1+HD

where HD is the local dipolar field strength, Ri is the spin-lattice
relaxation rate due to all mechanisms other than molecular jumping in
the lattice, a is a number which determines the effect of the spin-
lattice relaxation on the dipolar relaxation and Rc is the dipolar
relaxation rate due to the molecular jumping.

The technique of adiabatic demagnetization in the rotating frame
was found to be inadequate for measuring short T1p values. An alterna-
tive method has been developed by Hartmann and Hahn274. In this method,
the alignment of the magnetization along H1 is obtained by a 900 pulse
and followed immediately by a 900 phase-shifted H1 pulse with variable
duration in order to measure the decaying magnetization at different
times. Several other methods have also been developed, each with its
own advantages and disadvantage5275.

When the applied spin-locking pulse field H1 is much smaller than
the local dipolar field HD, perturbation theory can still be used.

276 277

Look and Lowe , and also Jones , have derived an equation for R1

in this weak-collision case. For molecules undergoing an isotropic

random motion with a correlation time T Rlp can be written as

c!

5 T T 3 1’
_. 22.—:2... c ), (179)
2 l+motc 1+4wotc 2

O

 

 

R19 3 K(
where K is a nuclear constant and w1=yH1. In the case of Tcwo>>1,

Equation (179) becomes

162

3 Tc
R19 3K :77 . (180)
m17c

From the above equation, it can be seen that R1p is sensitive to
molecular motions with a time scale of the order of 1/m1. By adjusting
the spin-locking field strength H1, a wide range of Tc values can be
determined. Accordingly, this should be very useful in studying the
self-diffusion of the molecules in plastic crystals. Using Torrey's

lattice-diffusion m0d8153’269,

245

R10 of a nucleus in a plastic solid

phase was found to be

- §_ Td 5 rd
R10 " 7.2"2de1 2 50921013143 60,603)

+G(k,zuo%9)}, (181)

where the symbols are defined as before. When the diffusion is slow,

such that mon>>1, Equation (181) becomes
R1p . g-szzdrd 6(k,261;9) (182)
and R1p has a maximum value when wltd=0.86.
C. NMR Studies of Some Fluorochloroethanes ngnglé-n
Due to their inert chemical properties, some of the fluorochloro-
ethanes have been widely used as refrigerants and heat transfer fluids

as well as for other industrial purposes. For many years, NMR workers

have been interested in studies of the internal rotation of the molecules

163

279’291. However, very few people have tried to investi-

in these Freons
gate the overall molecular motions in both solid and liquid phases of
these compounds, many of which form an interesting plastic phase in the
solid state.

In Table 30, we list some physical properties of these compounds.
It can be seen that C2F6, CF3CC13, CFZClCCls, CFClZCFClz and CFClZCCls
all have a very small liquid range (less than 55°C) which is one of
the characteristic prOperties of plastic crystalline substances. In the
following sections, we report our studies of molecular motions in some

of these fluorochloroethanes by means of 19F NMR lineshape analysis

and relaxation rate measurements.

1. Molecular Motions in CFZCICCl3

 

19

a. Measurements of the P NMR Linewidths and Second Moments

 

The linewidths and second moments of the 19

F absorption in
CFZClCCl3 at different temperatures are listed in Table 31 and plotted
in Figure 29. A solid-solid phase transition at ~-115°C is very

obvious since both the linewidth and the second moment drop rapidly at
that temperature. After this transition point, the second moment
reaches a plateau with a value of ~0.15 62. At temperatures higher than
-60°C, both the linewidth and the second moment decrease slowly to zero.
This is evidently due to the increasing self-diffusive motion which
further averages out the intermolecular dipolar interaction. Experi-
mentally, we found that the lineshape was changing from a Gaussian

to a Lorentzian shape as the temperature increased in this region and

the second moment became zero far below the melting point.

164

a
Table 30. Physical constants of some fluorochlorocarbons.

 

 

Compound Transition Meltigg Boiligg Critical

point point point point
*cpscp3 -101 -78 33.3
CF3CF2C1 -192.8 -106 -38 80.1
CF3CFC12 -56.6 3.6
CFZCICFZCI -94 3.6 145.7
*CF3c013 -118° 14.9 45.9 210c
CFZCICFCIZ -35 47.6 214
*CFClZCFClz 26.5 91 278
*cpzmca3 -115d 40.6 91
*CFCIZCC13 100 136.8
*
CF3C12CCC13 109.1 153.1
*CF3C12CCC12CF3 83 131

d

* -
CFZCCIZCClZCFZ 70 84.8 131.6

 

 

aAll temperatures are in °C.
bReference 278.

cReference 131.

dThis work.

*Plastic crystal.

165

 

 

Table 31. 19F linewidths and second moments in CFZClCClS.
T 1000/T Ava M2
(°C) (°x 1 (Hz) (62)
-43.5 4.36 1465 0.040
-51.0 4.50 3051 0.141
-56.0 4.61 4040 0.141
-62.0 4.74 4330 0.164
-68.0 4.88 4444 0.153
-73.0 5.00 4786 0.142
-78.0 5.13 4712 0.148
-86.0 5.35 4932 0.158
-94.0 5.59 5249 0.150
-100.0 5.78 5128 0.147
-108.0 6.06 5200 0.184
-llS.0 6.33 5670 0.207
-121.0 6.58 7790 0.592
-127.0 6.85 7373 .0.960
-116.0 6.37 8300 0.875
-llS.0 6.33 5740 0.785
-ll3.0 6.25 5029 0.157
-34.0 4.18 635 0.027
-33.0 4.17 635 0.013
-18.0 3.92 125 0.001

 

aAv(Hz) = 4007.4 X Av(G).

166

 

 

 

 

2
O
o O A V
—0.9
0 M2
10.8
1.5—
-0.7
O

O -0.6

1:? .2."

U) U)

a 8
3| — ~41-0.5

3 N

<1 2
0.4

D
0.3
0.51—

—0.2
‘001

l I l

 

 

-120 -100 -80 -60
t.°c
19

-4O

 

Figure 29. F NMR linewidths (scale at left) and second moments

(scale at right) of CFZClCClS.

 

167

From the linewidth narrowing in the temperature region from -70°C
to ~0°C, the mean jump time Id can be obtained by using Equation (162)
or (163) or (164). These results are plotted in Figure 32 and will be
discussed later.

The second moment observed in the plateau region (see Figure 29)
is the contribution from the intermolecular dipolar interaction,which
can also be calculated theoretically by using Equation (169).
Unfortunately, no crystal structure has been reported for this
compound, therefore, we can only estimate the lattice constant from
the density by assuming either the body-centered or face-centered
cubic structure for the crystal. We have measured the density d of
this compound at room temperature and found it to be ~1.719 g cm's,
hence, the lattice constant a will be 37.33 X if the crystal structure
is body-centered cubic and ~9.24 X if it is face-centered cubic.
Therefore, the second moment M2d due to the intermolecular dipolar
interaction of 19F nuclei can be calculated to be @0.119 G2 (bcc)
or ~0.118 G2 (fcc). The contribution from chlorine nuclei can be
estimated to be about 4.5% of the above values, hence the total
intermolecular second moment is ~0.124 62 which is in good agreement

with the experimental result.

b. Rotational Motion in the Plastic Phase

 

The spin-lattice relaxation rates have been measured over a
wide range of temperatures. These are listed in Table 32 and plotted
in Figure 30. A discontinuity in the measured spin-lattice relaxation
rates signals a solid-solid phase transition which occurs at N-IISOC;

this agrees with similar discontinuities observed in the linewidth

Table 32.

19

168

F spin-lattice relaxation rates in CF2C1CC13.

 

 

T 1000/T R1
(°C) (°x'1) (sec-1)
-60.0 4.69 0.1087:0.0023
-65.0 4.81 0.134410.0034
-79.0 5.15 0.1518:0.0014
-73.5 5.01 0.125810.0048
-73.0 5.00 0.125710.0036
-69.0 4.90 0.125410.0044
-67.0 4.85 0.1320:0.0040
-62.0 4.74 0.106010.0030
-77.0 5.10 0.1560i0.0050
-86.0 5.35 0.1850:0.0030
-58.0 4.65 0.102010.0020
~50.0 4.48 0.0938:0.0022
-42.0 4.33 0.0933:0.0030
-33.0 4.17 0.0752:0.0018
-22.0 3.98 0.0778:0.0011
-43.0 4.35 0.0913:0.001S
-l3.0 3.85 0.0861:0.0010
-2.0 3.69 0.095510.0015
6.0 3.58 0.0922:0.0018
-2.5 3.70 0.0861:0.0014
16.0 3.46 0.094410.0036
35.0 3.25 0.1300:0.0060
41.0 3.18 0.1340:0.0070
-6.0 3.74 0.078110.0016
-11.5 3.82 0.0805:0.0024
-18.S 3.93 0.0860:0.0021
-25.0 4.03 0.0900:0.0028
-31.0 4.13 0.0935:0.0021
-32.0 4.15 0.0863:0.0012
-38.0 4.26 0.088510.0020
-43.0 4.35 0.0945:0.0027
-43.5 4.36 0.0917:0.0016
-51.0 4.50 0.103210.0048
-56.0 4.61 0.108110.0112
-62.0 4.74 0.1300:0.0046
~68.0 4.88 0.145010.0070
-73.0 5.00 0.167010.0063
-78.0 5.13 0.1930:0.0090
-86.0 5.35 0.2610:0.0090
-94.0 5.59 0.5720:0.0270

 

Table 32. (cont'd)

169

 

 

T IOOO/T R1
(°C) (°x'1) (sec‘l)
-100.0 5.78 0.4150i0.0110
-108.0 6.06 0.5720:0.0270
-115.0 6.33 0.7700:0.0570
-121.0 6.58 0.1680i0.0170
-127.0 6.85 0.129010.0040
-136.0 7.30 0.0518:0.0040
-144.0 7.75 0.0203:0.0037
-116.0 6.37 0.2770i0.0090
-120.0 6.54 0.1980i0.0120
-117.0 6.41 0.1560:0.0110
-117.5 6.43 0.1670i0.0140
-115.0 6.33 0.1530:0.0110
-113.0 6.25 0.6400:0.0120
-12.0 3.83 0.0868i0.0021
23.0 3.38 0.094910.003S
29.0 3.31 0.1153i0.0031
17.0 3.45 0.0978i0.0022
11.0 3.52 0.0986:0.001S
4.0 3.61 0.0894i0.0012
-3.0 3.70 0.0882:0.0016
-21.0 3.97 0.0894:0.0030
-22.0 3.98 0.0849i0.0012
63.0 2.98 0.1480:0.0083
80.0 2.83 0.1560i0.0090
103.0 2.66 0.1820:0.0130
132.0 2.47 0.224010.0220
170.0 2.26 0.2850:0.0170
207.0 2.08 0.380010.0290

 

170

 

 

 

\ t
. 0
1—‘., ,.
\
9.
.. -\ . :
2, i
05- 3 ‘\ r
v—A 1.. q? n?‘p E
v 9. . ..
| I
GE I 1
0-2C . ' - 1 '\
\. '. I . 1 0‘
\. . ' . e \
\ . ' \
\. ',' 0 ~ *
\I . \
0.1 P '\. e . . . . . \\
\ I.
\. 0. e .
\-
\
.\
005- x
\e
.\O
\0
\e
\e
\e
002% \
\
l l J L l
2 3 4 5 6 7
IOOO/T
Figure 30. Plot of the 19F relaxationrates in CFZClCClg as a

functirim of reciprogal temperature. 0 : R}; 0: R5;

---. are. _ e -0- 51'- one. SC
. R1 , . R1, . R1 , . 32

 

171

and the second moment measurements.

A few R1 values obtained at temperatures below the transition
point imply the existence of a molecular motion with an activation
energy about 4.1:0.2 kcal/mole. This is probably the hindered internal
rotation of the CFZCl group of the molecule with respect to the pseudo
C3 symmetry axis along the carbon-carbon chemical bond. To describe
this motion, we consider only the intramolecular dipolar interaction
between two fluorine nuclei separated by a distance r¢2.l6 2 (assuming
rCF-z 1.33 X and _/_ FCF=108° as in szcmpzazgz). The spin-lattice

relaxation rate due to this slow motion, which has morc>>1, can be

derived from Equation (174) and is found to be

1.r. _ 3 4 2 -6 1 ~ -8
R1 - 'S—Y n 1' T - 2.09x10 /Tc. (183)
“01c

Therefore, the correlation time T of this internal rotation is

C

calculated to be

Tc 2 1.16XI0-13exp(4.1 kcal mole'I/RT). (184)
In the temperature range between -115°C and -63°C, the spin-lattice
relaxation rates show a fairly linear dependence on l/T and have an
activation energy ~2.S6:0.10 kcal/mole. The motion which produces the
largest contribution to the relaxation is the overall tumbling motion
of the molecules with a correlation time Tr short snough so that
'onr<<1. From Equation (175), the spin-lattice relaxation rate in this

temperature region can be written as

172

(185)

where M21. is the dipolar second moment averaged out by this overall
tumbling motion and can be estimated from Equation (166) by considering

only the intramolecular contribution. We obtained M2r=3.1 G2 and
Rr ~ 6 55 109 186
1-... .r. (1

Therefore, the correlation time T1. is calculated to be

1

T 2 3.6QXIO- 4exp(2.56 kcal mole-I/RT), (187)

r
which gives rr=2.25 psec at the melting point.

As the temperature increases to higher than -63°C, the relaxation
rates start to deviate from the linear plot of the intramolecular
relaxation rates versus reciprocal temperature. This is a result of
contributions from additional relaxation mechanisms such as the self-
diffusion of the molecules and the spin-rotational interaction. In
order to separate these two contributions, we first try to estimate the
effect of self-diffusion. Assuming that the lattice-diffusion model is
valid and that wOTd>>1, so that G(k,y)+4G(k,2y)=0.5618/y269, the spin-
lattice relaxation rate due to the self-diffusion can be calculated
from Equation (177) to be

8 M26
‘td'

R: = 1.12 72M -—————-= 1.13x10'

word/2

(188)

Using the measured M2d=0.15 62, Equation (188) becomes

173

Rf = 1.69><10'g l—. (189)
T6

We will discuss this later after the values of Td are obtained.

c. Self-Diffusion in the Plastic Phase

Since the spin-lattice relaxation in the rotating frame has the
right time scale to describe the self-diffusive motion in plastic
crystals, we measured Rlp values at three different spin-locking fields
(22.5 G, 5.7 G and 4.5 G). These are shown in Table 33 and are plotted
in Figure 31. The maxima of the curves in Figure 31 occur when wle
20.816. Using Equation (182), the second moment modulated by the
translational motion can be obtained from these maxima. This is found to

2 2 at the three different

have the values 0.154 62, 0.151 G and 0.156 G
spin-locking fields used. The agreement of these values with the
measured one ( 0.15 62) is very good.

Assuming that the molecules undergo a self-diffusion via vacancy
migration, the mean jump time Td can be derived from the measured R1
values using Equation (182). These are plotted versus reciprocal
temperature in Figure 32 together with those derived from the linewidths,
and from the spin-spin relaxation rates,listed in Table 34. As can be
seen from this plot, the mean jump times derived from the linewidths
depend on whether Equation (162), Equation (163) or Equation (164) is
used and can differ by a factor of two to ten. Also, the low temperature
values deviate from linearity. Therefore,reliable information about

the self-diffusive motion cannot be obtained solely from linewidth

measurements. The best least-squares fit to these Td values can be

174

Table 33. 19F spin-lattice relaxation rates in the rotating frame

in CFZClCClS.

 

 

T 1000/T R1p ug
(°C) (°x'1) (sec‘l) (kHz)
29.0 3.31 5.5410.11 90.0
15.0 3,47 12.98i0.15 90.0
6.0 3.58 22.37i0.05 90.0
-2.0 3.69 32.8910.49 90.0
-10.0 3.80 36.77iO.30 90.0
-17.0 3.91 28.5010.41 90.0
-23.0 4.00 20.54i0.39 90.0
-30.0 4.12 13.55i0.34 90.0
-36.0 4.22 9.820i0.23 90.0
-42.0 4.33 6.160i0.24 90.0
28.0 3.32 6.820t0.20 27.0
17.0 3.45 11.36i0.33 27.0
8.0 3.56 24.92i0.49 27.0
-1.0 3.68 52.77to.51 27.0
-9.0 3.79 86.4811.35 27.0
-16.0 3.89 114.9:2.5 27.0
-23.5 4.01 114.1i1.8 27.0
-30.0 4.12 92.10i1.13 27.0
-36.0 4.22 61.00:].03 27.0
-43.0 4.35 40.51i1.27 27.0
-53.0 4.54 . 21.87i0.77 27.0
20.0 3.41 11.05:0.58 18.0
11.0 3.52 22.16:0.28 18.0
1.0 3.65 47.0810.89 18.0
-7.0 3.76 91.96il.29 18.0
-15.0 3.88 154.2:2.6 18.0
-22.0 3.98 188.1:3.5 18.0
-29.0 4.10 180.314.1 18.0
-37.0 4.24 125.212.2 18.0
-43.0 4.35 74.51il.49 18.0
-S0.0 4.48 40.41:0.93 18.0
-55.5 4.60 33.0211.57 18.0
-60.0 4.69 20.6811.26 18.0

 

8H2(kHz) = 4007.4XH2(G).

175

Table 34. 19F spin-spin relaxation rates in CF2C1CC13.

 

 

T IOOO/T R2
(°C) (°K“) (sec'l)
207.0 2.08 2.89:0.23
170.0 2.26 0.788:0.046
132.0 2.47 0.526t0.041
103.0 2.66 0.410i0.026
80.0 2.83 0.37810.023
63.0 2.98 0.301i0.022
49.5 3.10 0.270i0.006
34.0 3.26 4.18:0.26
26.0 3.34 10.36i0.98
12.0 3.51 45.5:2.8
1.0 3.65 133:5
-6.0 3.74 372:29
-18.0 3.92 393130
-33.0 4.17 19951210

-43.5 4.36 46021418

 

 

176

 

 

 

 

 

I‘D‘N3i
\
\
o’ '6. 3
Im '" .30 \\
o, \x
. \
\
\

A o G \
To . \

a: ' Q

a, g \ 4.5 G.
‘1; \. '. \
(f b, 9 b

.\ .2 \\
\ '. \
9,. 6.76
I o —- ‘y
K.
‘. 225 G.
'\
' L 1
3 4 5
IOOO/ T

Figure 31. Plot of the 19F spin-lattice relaxation rate in the
rotating frame at three different spin-locking fields,and
the spin-spin relaxation rate (———),in CF2C1CC13.

177

 

 

 

 

 

 

o
o
o
16 - j
o
16‘— 4
53
.3
.5 .
IO t- .1
i
10‘- 4
-7
'0 H -<
1
3

 

IOOO/T

Figure 32. Plot of the mean jump time for self-diffusion in the
plastic phase of CFZClCCl3; e : from R19 data; 0: from
the linewidth data using Equation (162); o: from the
linewidth data using Equation (163); 0: from the line-
width data using Equation (164); +: from R2 data.

178

written as
-16 -1
Td = 5.8XI0 exP(11.6 kcal mole /RT), (190)
which gives -rd=*6.7><10'8 sec at the melting point. For a monovacancy

diffusion mechanism, the mean jump distance <r> is the distance to the

. . . /' .
closest lattice Sites and 15 equal to —%-a for a body-centered cubic

structure and 2%; for a face-centered cubic structure. Using Equation
2
(61), the self-diffusion coefficient at the melting point is

~1.OXI0-8 cm2 sec'1 or '\al.06><10-8 cmzsec'l, depending on whether the

crystal structure is body-centered cubic or face-centered cubic.
With the Td values expressed by Equation (190), we can estimate

the contribution to the spin-lattice relaxation rate from the inter-

molecular dipolar interaction. This can be readily done by combining

Equation (189) and Equation (190) to give
5 6 1 ‘1
R1 = 2.QZXIO exp(-11.6 kcal mole /RT). (191)

The calculated values of R: are about 17% of the total spin-lattice
relaxation rate near the melting point and become negligible at
temperatures below 0°C.

19 3
d. F Relaxation in the Liquid Phase and J(F sC1)

In the liquid phase of CF2C1CC13 the molecular motions are
expected to be very isotropic in view of the near-spherical molecular

shape (the three principal components of the moment of inertia are

179

"4°g cm2). From Figure 30, it

74SXI0'40g cmz, 851x10'4°g cm2 and 957X10
can be seen that the spin-rotational interaction dominates both the
spin-lattice and the spin-spin relaxation. The difference between the
spin-spin and the spin-lattice relaxation rates is due to different
contributions from the scalar-coupling interaction, which has a
negligible effect in determining the spin-lattice relaxation rate but
is important for the spin-spin relaxation rate. Therefore, we can
write

3 2 3S

2 37 37 35
Rz-Rl = age = 10n2( %-J (F Cl)T1( Cl)+z-J (F Cl)T1( 01))

= 95.7 J2(F35C1)T1(°5Cl), (192)

where we have used Equation (70) for each isotope and relate the
coupling constants and the spin-lattice relaxation times of two
chlorine isotopes by their gyromagnetic ratios and their nuclear
quadrupole moments, respectively. The chlorine spin-lattice relaxa-
tion time can be calculated from the quadrupole coupling constant

and the reorientational correlation time of the molecule as

described by Equation (67). Although the quadrupole coupling constant
of 35C1 in this compound has not been reported, a value of 78 MHz,
which lies between those found in CFZCl2 and CF3C1, should be an
adequate estimate since the chlorine quadrupole coupling constants

in other chloromethanes and chloroethanes do not vary appreciably219.
We assume that the reorientational correlation time can be estimated

from Equation (187), since for plastic crystals it is very likely

that the reorientational motion of the molecules in both liquid and

180

plastic solid phases follows the same mechanism. Therefore, we obtain
-1 35 -1
T1 ( C1) = 780 exp(2.56 kcal mole /RT) (193)

and the coupling constant J(F35C1) can be calculated from Equation (192)
and Equation (193) giving J(F35C1)28.1 Hz.

Since the necessary physical constants have not been reported for
this compound, the contribution of the intermolecular dipolar inter-

19

action to the F relaxation in the liquid phase was not evaluated.

However, as in CFSCCIS, which has been previously studied131, this
contribution should be minor but just enough to smooth out the
relaxation rate plot shown in Figure 30. The significance behind this
is that the reorientational motions in both liquid and plastic solid
phases are via the same mechanism. The resultant relaxation rates,
after subtracting the intermolecular contributions show two com-
peting relaxation contributions with opposite temperature dependence.
These are obviously those from the spin-rotational and the intra-
molecular dipolar interactions. The spin-rotational relaxation rates
can be readily separated since the intramolecular dipolar relaxation
rates are available from Equations(186) and (187). Assuming that the
molecules are spherical-top rotors, the spin-rotational relaxation
rates can be expressed by Equation (80). We have calculated the
average moment of inertia to be 8SOX10‘4°g cm2 and the effective

spin-rotation constant Ceff to be ~1.17 kHz from the 19F chemical

shielding (66.8 ppm downfield from CFC13). Therefore,

sr
R1

R

1.14X109TTJ (194)

181

HewHV :oHuascm eH Hemoe

a
.HemHU :eHuesem segue

 

 

MN0.0 0.vN vw.H H.vv mv0.0 000.0 m.m NmH
wmo.0 N.MH mh.H m.MN 000.0 mH0.0 0.m 00m
0m0.0 mn.h 00.H N.NH 000.0 mN0.0 m.v NNN
000.0 00.v hm.H mm.0 HmH.0 mv0.0 0.¢ 0mm
mH.0 0N.N hv.n mm.n vNN.0 n50.0 m.n 0mm
vm.0 0N.H 0m.H 0b.H mm.0 mNH.0 0.n nnm
Hm.0 mm.0 mm.~ 0N.H 0v.0 BH.0 h.N 05m
wm.0 «5.0 QN.~ Nm.0 he.0 mHN.0 m.N 00v
«v.0 50.0 NN.H Hw.0 0m.0 v~.0 v.N hfiv
00.0 00.0 0H.H Hh.0 nm.0 m0N.0 n.N mmv
Nm.0 vm.0 0H.“ m0.0 00.0 Hm.0 N.N vmv
00.0 wv.0 QH.H mm.0 00.0 hn.0 H.N 059
05.0 vv.0 HH.H 0v.0 vw.0 00.0 0.N 00m
noommv noommv noommv nHtoomv AHuxoV axov

me me xHex\HV Me. eee ewe e\oooH a

 

.nHuu Hu~au eH meaHe
eoHumHouuou HecoHuapcoHnoou on» use emueoeoa ueHmmee 0gp mo :oHumHmoHeu on» we Aheaamm .mm oHnmh

182

Figure 33. Plot of reduced reorientational correlation time versus
reduced angular momentum correlation time for CFZClCCl3;
solid lines A and C represent the theoretical curves
predicted from the M-diffusion model for i=1 and 2,res-
pectively; solid lines B and D represent the theoretical
curves predicted from the J—diffusion model for i=1 and 2,
respectively; dashed lines represent the theoretical
result predicted from the Debye classical diffusion model
(TeTJ=I/2(2+l)kT); dotted lines represent the theoretical
result predicted from the perturbed free-rotor model
(T9=TJ/22+l) and the broken lines represent the free-rotor
limit of the extended diffusion model (T9=TJ/2£).

183

0.0 t «.0 H0 00.0

 

 

 

 

 

 

 

 

184

and the effective angular momentum correlation time TJ can be
estimated and is shown in Table 35. In Figure 33 we plot the reduced
reorientational correlation times T5 versus the reduced angular
momentum correlation times 13. It is obvious that the experimental

data points do not follow the J-diffusion or the classical diffusion

model.

2. Molecular Motions in CFCIZCFCIZ

 

a. Measurement of the 19F NMR Linewidths and Second Moments

 

CFClzCFClz is an isomer of CFZCICClS. Both compounds have the
same boiling point but the melting point of CFC12CFC12 is lower, which
gives a wider liquid range and therefore a smaller plasticity is

9F NMR linewidths and the second moments are

expected. The measured 1
listed in Table 36 and are both plotted versus reciprocal temperature
in Figure 34. No obvious phase transition was observed above -130°C.
The intermolecular dipolar second moment was found to be ~0.18 G2
which decreased to zero gradually at temperatures above -90°C. The
theoretical estimate of this intermolecular dipolar second moment was
~0.lZ7 Gz( if the crystal has a body-centered cubic structure) or
~0.125 G2( if the crystal has a face-centered cubic structure). These
values are obtained using the measured solid densityof 1.735 g cm"3 at
5°C, which givesa lattice constant of 7.31 X or 9.21 X,depending on

whether the crystal structure is body-centered cubic or face-centered

cubic.

185

 

 

Table 36. 19F linewidths and second moments in CFClZCFClz.

T 1000/T 00' M2
(°C) (°x'1 (Hz) (62)
23.0 3.38 14.6 0

7.0 3.57 26.0 0

0.0 3.66 38.0 0

10.0 3.53 24.0 0
-2.0 3.69 41.0 0
-10.0 3.80 60.0 0
-ll.0 3.82 63.0 0.0003
-22.0 3.98 107.0 0.0008
-23.0 4.00 102.0 0.0009
-32.0 4.15 224.0 0.0027
-33.0 4.17 229.0 0.0026
-39.0 4.27 395.0 0.0048
-40.0 4.29 419.0 0.0062
-43.0 4.35 585.0 0.013
-46.0 4.40 779.0 0.015
-53.0 4.54 1374.0 0.053
-S8.0 4.65 2515.0 0.100
-64.0 4.78 3370 0.126
-72.0 4.98 4035 0.155
-77.0 5.10 4300 0.165
-83.0 5.26 4540 0.180
-90.0 5.46 4675 0.180
-96.0 5.65 4690 0.180
-106.0 5.99 5020 0.200
-107.0 6.02 4950 0.190
-llS.0 6.33 4830 0.180
-122.0 6.62 4950 0.190

 

186

 

0 AI'
(9 0 M2
I .- -—0.2
3? .6"
2
£5 <
a 43
Q N
:5
0.5" “'10.!

 

 

 

-100 -50 o
TEMP.(°C)

Figure 34. 19F NMR linewidth (scale at left) and second moment
(scale at right) for CFClZCFClz.

187

b. Rotational Motion in the Plastic Solid Phase

 

The measured 19F spin-lattice relaxation rates are listed
in Table 37 and plotted versus reciprocal temperature in Figure 35.
At low temperatures the plot shows fairly good linearity with a slope
~3.l6 kcal/mole. The most probable interactions contributing to
relaxation in this temperature range are the intramolecular dipolar
interaction and the interaction through the chemical shielding aniso-
tropy. The intramolecular dipolar relaxation rate can be expressed by
Equation (175). The intramolecular dipolar second moment M2r can be
claculated from the geometry of the molecule by assuming a transfform,
which was found to be ~16515 cal/mole more stable than the gauche-form
in CFC13 solution287. We calculate MZr to be ~0.23 G2 which gives

d
R

8
l = 4.SX10 Tr. (195)

By assuming that the overall tumbling motion of the molecules is
isotropic, we can use the same Tr to evaluate the relaxation rate due
to the chemical shielding anisotropy A0; the latter has been measured
in polycrystalline CFClZCFCl2 and found to be ~240 ppm293 (in the

C-F bond axis system). This gives the relationship

C58

8
1 = 9.6x10 Tr. (196)

R
Therefore, the low temperature linear part of the spin-lattice
relaxation rate plot can be expressed by the sum of Equations (195)
and (196), and the reorientational correlation time Tr can be

determined to be

188

 

 

Table 37. 19F spin-lattice relaxation rates in CFClZCFClz.
T 1000/T R1

(°C) (°K'1) (sec‘l)
-132.0 7.09 0.700:0.123
-13l.0 7.04 0.51310.089
-128.0 6.90 0.566:0.082
-122.0 6.62 0.412:0.042
-121.0 6.58 0.30210.024
-114.0 6.29 0.276:0.015
-113.0 6.25 0.180:0.018
-104.0 5.92 0.122:0.010
-103.0 5.88 0.100:0.017
-97.0 5.68 0.075:0.004
-88.5 5.42 (0.07910.006
-80.5 5.19 0.05410.006
-73.0 5.00 0.061910.0033
-63.0 4.76 0.0457:0.0012
-S7.0 4.63 0.0503:0.0035
-37.0 4.24 0.0473:0.0010
-14.0 3.86 0.0510:0.0010
2.0 3.64 0.0622:0.0025
23.0 3.38 0.0931:0.0015
25.5 3.35 0.097410.0060
41.0 3.18 0.1028:0.0020
44.0 3.15 0.1020:0.0090
57.0 3.03 0.1295:0.0059
75.0 2.87 0.1397:0.0010
102.0 2.67 0.155:0.001
129.0 2.49 0.18210.002
149.0 2.37 0.21210.005

 

189

 

 

 

\Q
.90 ME
I" 1&3) I
.\c;
'3.
I ‘-
'\
| °\.
45" : ‘-\‘ I.
I ‘4‘
L) I .
m .
U° I
\_z '
(I , /
'/
OJ " 0
I
I II
I
I
I I I I l
2 3 4 . 5 6 7
IOOO/T
Figure 35. 19F spin-lattice relaxation rate (0) and the spin-spin

relaxation rate (0) in CFCIZCFCIZ; the straight solid
line represents the intramolecular dipolar relaxation
rate R9; --- : the scalar spin-spin relaxation rate Rgc.

 

190

T = 5.8x10'l

r Sexp(3.l6 kcal mole'I/RT). (197)

This equation gives Tr=1.21 psec at a temperature just below the

melting point.

c. Self-Diffusion in the Plastic Phase

 

From the gradual decrease of the linewidth, we expect that
the self-diffusive motion of the molecule has reached a rate of the
order of the linewidth,~5000 Hz at temperatures higher than -100°C.
We have measured the spin-lattice relaxation rates in the rotating
frame at three different spin-locking fields, and also the spin-spin
relaxation rates. These are listed in Tables 38 and 39 and are plotted
versus reciprocal temperature in Figure 36. The mean jump times for the
self-diffusive motion can be obtained from R19, R2 and Av as before.
These are plotted in Figure 37 and show an activation energy ~11.4
kcal/mole. The best least-squares fit of these experimental data points

can be expressed by

Td = 1.9XI0-16exp(11.4 kcal mole-I/RT), (198)

8

which gives Td=4.SXl0' sec at the melting temperature. Again, by

assuming a monovacancy diffusion mechanism, the self-diffusion

8 1

coefficient at the melting temperature is ~1.48X10' cmzsec' or

1.57x10'8 cmzsec'1 depending on whether the crystal structure is
body-centered cubic or face-centered cubic.

The maximum values of R10 give estimated intermolecular dipolar

191

19 , ,
Table 38. F spin-lattice relaxation rates in the rotating frame
in CFCIZCFC12.

 

 

T 1000/T R1 H2
(°C) (°x‘1) (sec'l) (kHz)
13.0 3.50 ll.55:2.28 41.5
-1.0 3.68 26.07:0.37 41.5
-10.0 3.80 55.1510.60 41.5
-16.0 3.89 68.09:1.07 41.5
-22.0 3.98 93.14:0.78 41.5
-28.0 4.08 93.77:1.04 41.5
-34.0 4.18 77.52:0.78 41.5
-40.0 4.29 52.60:1.02 41.5
-47.0 4.42 35.65:0.88 41.5
-54.0 4.57 22.48iO.69 41.5
-S4.5 4.58 23.65i1.19 41.5
-58.0 4.65 14.87i0.84 41.5
-59.0 4.67 16.69i0.55 41.5
-62.0 4.74 16.13:].07 41.5
-6S.0 4.81 11.81i0.58 41.5
-65.0 4.81 30.63il.O8 20.8
-60.0 4.69 46.74i0.24 20.8
-54.0 4.57 62.03iZ.38 20.8
-45.0 4.38 130.2:2.4 20.8
-39.0 4.27 190.1:5.8 20.8
-34.0 4.18 202.6:8.3 20.8
-27.0 4.06 159.3:2.4 20.8
-21.0 3.97 119.9i1.3 20.8
-13.0 3.85 71.7310.69 20.8
-1.5 3.68 31.05t0.48 20.8
23.0 3.38 4.2610.19 20.8
-10.0 3.80 54.88i0.86 8.2
-18.0 3.92 113.012.2 8.2
-26.0 4.05 236.6i5.0 8.2
-32.0 4.15 371.1:13.2 8.2
-39.0 4.27 _ 521.4i20.0 8.2
-4S.0 4.38 643.2:27.2 8.2
-51.0 4.50 437.0t21.6 8.2
-56.0 4.61 217.017.9 8.2
-60.0 4.69 157.4:10.8 8.2
-72.0 4.98 47.16i1.19 8.2

 

Table 39.

19

192

F spin-spin relaxation rates in CFCl

2

CFCIZ.

 

 

T lOOO/T R2
(°C) (°K l1 (sec-1)
-35.0 4.20 556.5i50.0
-23.0 4.00 208.1:20.8
-14.0 3.86 48.2i4.9
-5.0 3.73 34.7i1.9

1.0 3.65 17.2i0.8
7.0 3.57 13.1i0.6
11.0 3.52 8.810.3
17.0 3.45 6.1110.18
20.0 3.41 4.69i0.14
21.0 3.40 4.39:0.18
23.0 3.38 0.88:0.04
24.0 3.37 0.88:0.04
22.0 3.39 1.55i0.14
21.5 3.40 3.97:0.36
29.6 3.30 0.89610.053
49.2 3.10 1.00010.053
57.5 3.02 1.053i0.092
76.0 2.86 1.145i0.127
99.4 2.68 1.33910.101
122.0 2.53 1.542i0.122

 

193

 

 

 

 

 

Ioooh _
Ioo- ~
.2)
DJ
3
0:
IO — ~
I .. _
L 1 1 I
3.5 4 4.5 5
IOOO/T

Figure 36. Plot of the 19F spin-lattice relaxation rate in the rotating
frame at three spin-locking fields (10.5, 5.3 and 2.0 G)

and the spin-spin relaxation rate (0),in the plastic solid
phase of CFCIZCFClz.

194

 

 

 

 

 

A
A
A
IC§4P A. . ‘
.
.
1
.J
g 1 1
6
IOOO/T

Figure 37. The mean jump time derived from R10 (0), R2 (0) and
Av (A) of 19F nuclei in the plastic solid phase of
CFClZCFClz.

195

second moment of 0.18 62

, 0.19 G2 and 0.23 G2 which are in good to
fair agreement with the directly measured value.

The contribution to the spin-lattice relaxation rate from the
translational motion in the plastic solid phase can also be evaluated
from the second moment M2d and the mean jump time rd. The result is
shown in Figure 35.

19

d. F Relaxation in the Liquid Phase and J(F35Cl)

 

Accurate extraction of the spin-rotational relaxation rates
was not attempted for this compound because the measurements were not
carried to high enough temperatures. Nevertheless, several measurements
of the spin-spin relaxation rate show that there is a large scalar
relaxation contribution due to the fast-relaxing chlorine nuclei. This
contribution should be about twice as big as that found in CFzClCC13
since there are two geminal chlorine nuclei coupled with each fluorine
nucleus. By using the same approach as before, we obtain J(FSSCl)

=11.7tl.7 Hz.
3. Self-Diffusion Studies in CFSCCIS

CFSCC13 is the most symmetric compound in the fluorochloroethane
series. The small liquid range (~32°C) and the relatively high melting
point (14°C compared to -36°C for its asymmetric isomer CFZClCFClz)
imply a great plasticity in its solid phase. In previous studies
advantage has been taken of these properties to separate the possible
contributions of different mechanisms to the 19F spin-lattice

31

relaxation in this compound1 . Also Gordon's extended J-diffusion

196

model was found to be appropriate to describe the reorientational
motion of the molecules in both the liquid and the plastic solid
phases. Furthermore, from Raman lineshape analysis it was found that
the reorientational motion is nearly isotropiclsl.

In order to investigate in more detail the self-diffusive motion
in the plastic phase of this compound, we have carefully measured the
linewidths and the second moments of the 19Fresonance lines and also
the spin-lattice relaxation rates in the rotating frame. These are
listed in Tables 40 and 41. In Figure 38 the linewidth and the second
moment are plotted versus reciprocal temperature. We did not reach to

131

the transition point reported in the previous study because our

digitizer (20 psec minimum dwell time) was not fast enough to follow
the decay of the magnetization at low temperatures and to obtain
enough data points for Fourier transformation. Nevertheless, a plateau
was found in both the linewidth and the second moment plots at tempera-
tures below 200°K. The second moment in this region was found to be

19

~0.34 G2. Figure 39 shows the plots of the F spin-lattice relaxation

rates in the rotating frame R at three different spin-locking fields

10
(17.6 G, 8.8 G and 5.7 G),and the 19F spin-spin relaxation rates. From
the maxima of the R1 plots, the second moments due to the intermole-

cular dipolar interaction were found to be 0.28 62, 0.29 G2 and 0.35 G2
at the three different spin-locking fields. The calculated value from

the lattice constant of the crystal derived from the density (1.71

131 2 or 0.212 G€,depending on whether the crystal

g cm'3) is ~0.213 G
structure is body-centered cubic or face-centered cubic. The agreement
with the experimental results is only fair.

Using the same method as in previous sections, the mean jump time

197

 

 

Table 40. 19F linewidths and second moments in CF3CC13.
T 1000/T Av M2

(°C) (°x‘1) (Hz) (02)
-23.5 4.01 116 0.0033
-31.0 4.13 233 0.0058
-41.5 4.32 700 0.0185
-42.0 4.33 696 0.0232
-47.0 4.42 1155 0.0833
-50.0 4.48 1648 0.110
-55.0 4.59 2516 0.145
-59.0 4.67 3521 0.195
-64.0 4.78 4857 0.253
-68.0 4.88 5633 0.322
-74.0 5.02 6054 0.354
-78.0 5.13 6286 0.332
-84.0 5.29 6567 0.340
-88.0 5.40 6604 0.341
-94.0 5.59 6750 0.336
-101.0 5.81 6720 0.341
-111.0 6.17 6726 0.345
-120.0 6.53 6770 0.340

 

198

 

 

Table 41. 19F spin-lattice relaxation rates in the rotating frame
in CFSCCls.

T 1000/T Rlp H2
(°C) (°K'1) (seC’l) (kHZ)
4.0 3.61 10.70i0.20 35.5
-7.0 3.76 32.91i0.28 35.5
-17.0 3.91 79.2311.53 35.5
-28.0 4.08 163.Zil.7 35.5
-39.0 4.27 167.112.3 35.5
-48.0 4.44 75.44tl.44 35.5
-57.0 4.63 37.54:0.67 35.5
-64.0 4.78 21.36i0.37 35.5
6.0 3.58 10.3310.18 71.0
0.0 3.66 15.5410.17 71.0
-8.0 3.77 33.47:0.48 71.0
-16.0 3.68 55.1310.99 71.0
-23.0 4.00 86.28io.82 71.0
-3l.0 4.13 79.72:0.l7 71.0
-38.0 4.26 48.33tl.45 71.0
-44.0 4.37 31.6410.62 71.0
-50.0 4.48 18.96i0.52 71.0
-S6.0 4.61 10.0310.43 71.0
5.0 3.60 11.1910.15 22.7
-2.0 3.69 22.4110.14 22.7
-12.0 3.83 55.3810.94 22.7
-22.0 3.98 138.7il.3 22.7
-31.0 4.13 248.915.0 22.7
-39.0 4.27 335.3i2.1 22.7
-46.0 4.40 225.4i4.8 22.7
-53.0 4.54 129.7i3.0 22.7
-58.0 4.65 71.39i2.55 22.7
-62.0 4.74 53.54i2.34 22.7
-67.0 4.85 30.9310.65 22.7
-72.0 4.98 24.14i0.76 22.7
-76.0 5.08 14.71t0.l4 22.7
-81.0 5.21 12.26t0.4l 22.7

 

199

 

 

 

 

O
2 - O
- 4.3
e) .g ‘9
. {I
0
LS-
O
A
8 .
2 . “-0.2
(D r-\
\_I (H
e H
4 | +- 2
3
’ N
2
I
e -0.1
0.5 r-
' o A
o 0 M2
0
/
v, 7 I 1
4 5 6
IOOO/T

Figure 38. Plot of the 19F NMR linewidth and second moment in
CF3CC13 versus reciprocal temperature.

200

 

 

 

 

 

I"o“\
:0 3“
,' a ""9. \
.. Q
IOO- _, \
Ie»‘. ‘.K d} 25‘
'\ g
f‘\ .1 .3 2°\
- \ G) \
ti 3. '3‘ 576.
m \. .. \ .
‘\_l 3" ‘i? \
C]: \\ .'ELEI(i
I o - ‘\
I7.6 G.
i L, I - I
35 4.0 4.5 5.0
IOOO/ T

Figure 39. Plots of the 19F spin-lattice relaxation rate in the
rotating frame at three different spin-locking fields
(17.6, 8.8 and 5.7 G), and of the spin-spin relaxation
rate (0) versus reciprocal temperature, for CF3CC13.

201

 

'4
IO)-

 

 

 

I

 

U]—

IOOO/T

Figure 40. Mean jump time for self-diffusion in the plastic phase of
CF3CC13 derived from R19 (0), R2(+) and Av (0) .

202

of the self-diffusive motion of the molecules in the plastic solid
phase can be estimated from the spin-lattice relaxation rates in the
rotating frame, the spin-spin relaxation rates and the linewidth
narrowing. These are plotted in Figure 40. The best least squares fit
to these experimental data points can be expressed by the equation

-17

rd = 4.7x10 exp(11.7 kcal mole'l/RT), (199)

which gives Td=4.0X10-8 sec at the melting point. The self-diffusion
coefficient Ds at the melting point was found to be 1.61><10'8 cmzsec'1
or 1.70x10'8 cmzsec'i depending on whether the crystal has a body-

centered cubic or a face-centered cubic structure.
4. Spin-Lattice Relaxation Studies in CFZClCF2§l_

The other two fluorochloroethanes, CF3CF2C1 and CFCIZCC13, also
may form a plastic phase but data were not obtained in this work since
the 19F relaxation in the former compound is complicated by the homo-
nuclear coupling and the latter compound was not available for this
work.

We have also measured the spin-lattice relaxation rate of 19F in
CFZCICFZCI, which is not expected to form a plastic solid phase. The
results are listed in Table 42 and plotted versus reciprocal temperature
in Figure 41. In the solid state, the low temperature R1 values are
linear with reciprocal temperature and the activation energy is ~1.17

kcal/mole. Translational motion begins to contribute to the spin-

lattice relaxation at a temperature about 18° below the melting point.

203

 

 

Table 42. 19F spin-lattice relaxation rates in CFZClCFZCl.
T IOOO/T R1
(°C) (°x‘1) (sec‘l)
-12.0 3.83 0.1803:0.003S
-105.0 5.95 0.1643:0.0029
-111.0 6.71 0.1506:0.0027
-121.0 6.58 0.2013:0.0036
-128.0 6.90 0.230410.0050
-134.0 7.19 0.262810.0062
-135.0 7.25 0.256010.0115
-l41.0 7.58 0.3858:0.0217
-ll7.0 6.41 0.1797:0.0024
-103.0 5.88 0.1704:0.0050
-98.0 5.71 0.1881:0.0163
-9S.0 5.62 0.2175:0.0071
-91.0 5.49 0.234810.0088
-85.0 5.32 0.250410.0072
-78.0 5.13 0.185110.0024
-68.0 4.88 0.1694:0.0017
-60.0 4.69 0.192010.0088
-57.0 4.63 0.1603:0.0025
-40.0 4.29 0.1491:0.0042
-23.0 4.00 0.175610.0036
10.0 3.53 0.221210.0073
22.0 3.39 0.2969:0.0117

 

204

 

|_ .4
H
C ..
F‘ ..
r d
1 M.P. . -.
9 : ..
'0 - ' .I ?.
LIJ 0. 0
U5 . ’ . _
V — O . i . 0
c1: ’ . ' '° .
o
O.| :- ' ‘7.
1' Z
_. 4
_- "1
l l l l

 

 

 

4 5 6 7
(000/1‘

Figure 41. Plot of the 19F spin-lattice relaxation rate in CF2C1CF2C1
versus reciprocal temperature; the solid straight line
represents the intramolecular dipolar relaxation rate and
the dotted line represents the intermolecular dipolar
relaxation rate in the solid phase.

205
The low temperature relaxation is evidently due to the intramolecular
dipolar interaction modulated by the overall tumbling motion of the

286

molecules. From the geometry of the molecule, and assuming that

the molecules are in the trans-form (which has been found to be more

292

stable than the gauche-form) we calculate the intramolecular

dipolar relaxation rate according to Equation (17S) and obtain
8‘11 = 8.8X1091'r , (200)

from which the reorientational correlation time Tr can be obtained;

it is found to obey the equation

13exp(1.17 kcal mole'l/RT), (201)

Tr = 4.6X10-
which gives Tr=12 psec at the melting point.

Due to the relatively slow reorientational motion, the spin-
rotational relaxation is not important in the solid phase. Therefore,
the curvature of the spin-lattice relaxation rates below the melting
'point must be due to the translational motion which shows an
activation energy of ~9.5 kcal/mole.

In the liquid state the molecular motion is more complicated
due to the asymmetry of the molecules and the two possible conformers

(trans and gauche) so the analysis was not attemped.

206

D. Molecular Motions in Some Other Plastic Crystals ~

 

In view of the similar plasticity found in CF3CC13 and in CC14,
it is possible that the substitution of a trifluoromethyl group for a
chlorine atom will not vary the physical characteristics of the
perchlorocarbons very much. From Table 30 we noticed that the
compounds obtained on substitution of the chlorines of C2C16 by CF3
groups (the last three compounds) still retain the high melting point
and the small liquid range of C2C16. We also noticed that the physical
appearance of these three compounds is very similar. They are all waxy,
easy to sublime and having a camphor-like odor. We have studied the
19F NMR lineshapes and the relaxation behavior of these three compounds.
Due to the more complex molecular structure, success in elucidating

details of the molecular motions varies from compound to compound.

1. l,l,1-Trifluoropentachloropropane (TFPCP)

 

This compound is formed by substituting one chlorine atom of C2Cl6
with a CF3 group. The measured 19F linewidths, the second moments and
the spin-lattice relaxation rates are listed in Tables 43 and 44..
Figure 42 shows the plots of the 19F linewidths and the second moments
versus reciprocal temperature. An obvious self-diffusion with a rate
of the order of the linewidth occurs at ~-40°C, which is about 1400
below the melting point. M2d was found to be ~0.l45 G2 which can be
'compared with the theoretical value 0.105 G2 obtained from our
measured density (1.712 g cm'3 at 25°C).

From Figure 43,the spin-lattice relaxation rate in the solid

phase shows a curvature, becoming a fairly straight line at low

207

9
F linewidths and second moments in CF3C12CCC13

 

 

Table 43.
T 1000/T Av M2
(°C) (°x'1) (Hz) (62)
0.0 3.66 494 0.0068
~8.0 3.77 693 0.0241
-16.0 3.89 1200 0.0408
-22.0 3.98 -- 0.0694
-27.0 4.06 1572 0.1150
-34.0 4.18 1954 0.1390
-37.0 4.24 3035 0.1430
-44.0 4.37 3056 0.148
29.0 3.31 132 0.0009
20.0 3.41 161 0.0020
13.0 3.50 210 0.0022
6.0 3.58 283 0.0130
0.0 3.66 459 -
—7.0 3.76 770 -
-15.0 3.88 1312 -
-17.0 3.91 1655 -
-23.0 4.00 2190 -
-20.0 3.95 1950 -
-32.0 4.15 2978 -
-26.0 4.05 2820 -
~20.0 3.95 2160 -
-15.0 3.88 1600 -
-9.0 3.79 854 -

 

Table 44

208

19F spin-lattice relaxation rates in CF3C12CCC13°

 

 

T 1000/T R1
(°C) (°x'1) (sec'l)
23.0 3.38 0.1144i0.0037
16.0 3.46 0.122010.0022
8.0 3.56 0.1229i0.0042
0.0 3.66 0.1330t0.0034
-26.0 4.05 0.1556:0.0044
-20.0 3.95 O.1647i0.0051
-9.0 3.79 0.167010.0130
-15.0 3.88 0.1710:0.0070
32.0 3.28 0.1133i0.0007
39.0 3.20 0.1116i0.0030
40.0 3.19 0.1103:0.0008
49.0 3.10 0.1078i0.0018
56.0 3.04 0.110210.0014
64.0 2.97 0.1160:0.0023
65.0 2.96 0.1164i0.0020
86.5 2.78 0.142:0.002
95.0 2.72 0.162i0.003
102.0 2.67 0.161:0.004
97.0 2.70 0.17110.006
111.5 2.60 0.139i0.003
121.0 2.54 0.14010.004
146.0 2.39 0.151i0.007
173.0 2.24 0.17010.014
159.0 2.31 0.148:0.010
133.0 2.46 0.142:0.002
-42.0 4.33 0.28510.005
-43.0 4.35 0.310i0.009
-47.0 4.42 0.319:0.012
-37.0 4.24 0.277i0.027
-35.0 4.20 0.234i0.013
-30.0 4.12 0.197i0.007
-23.0 4.00 0.175:0.008
-18.0 3.92 0.159:0.006
-9.0 3.79 0.164t0.006
-l7.0 3.91 0.18110.006
-30.0 4.12 0.176:0.006
-24.0 4.02 0.226:0.007

 

209

 

0 Av .
1.5" 0 M2 " 0.3

A I L. 3
- 0.2
H (”A
5 2
3 0
x.) V
i N
<' 2
0.5- - 0.1

 

 

50

 

Figure 42. Plots of the 19F linewidths and the second moments in
CF3C12CCC13 versus reciprocal temperature.

210

 

 

 

 

 

O. 5 '- "
1
005 .- l . ”I, . .1
L J l l I
2 3 4

IOOO/T

Figure 43. Plot of the 19F_spin-lattice relaxation rate in CF3C12CCC13
versus reciprocal temperature; dashed line represents
the intramolecular dipolar relaxation rate.

211
temperatures. The slope of this linear portion is ~2.74 kcal/mole,
which can be reasonably assigned as the activation energy of the
overall tumbling motion of the molecules. Assuming that the F-F
internuclear distance in CFSCIZCCl3 is the same as that in CF3CC13,
the intramolecular dipolar relaxation rate can be expressed by

a? z 1.33X10101r (202)

and the reorientational correlation time can be obtained as

1

T = S.SXl0- 4exp(2.74 kcal mole'l/RT), (203)

1‘

which gives Tr=2.2 psec at the melting point.
The mean jump times estimated from the linewidth narrowing can

be expressed by

13

z 4.8x10' exp(9.0 kcal mole-I/RT), (204)

T6

which gives Td=8.le0-85ec at the melting point. Using the lattice

diffusion model, this gives the self-diffusion coefficient to be

-8 2 1 2

l.OXl0 cm sec' or l.lX10-8cm sec-E depending on whether the crystal

structure is body—centered cubic or face-centered cubic.
2. 2,2, 3,3-Tfetrachlorohexafluorobutane (TCHFB)

This compound is formed by substituting two chlorine atoms of

sz6 by two CF3 groups. Weight and Roberts296 have studied the 19F

212

NMR spectra of this compound in CF2C12 solution. They found that at
-112°C the equilibrium mixture contains 8% of the trans_and 92% of the
gauche conformer. The activation energy for CF3 internal rotation was
found to be 8 kcal/mole for the gaughg_isomer and 6 kcal/mole for the
trans_isomer. They also concluded that the energy barrier restricting
the rotation about the central carbon-carbon bond should be greater
than 8 kcal/mole.

We have measured the linewidths and the second moments of the 19F
resonance lines in this compound, and also the spin-lattice relaxation
rates. These are listed in Table 45. Plots of the linewidths and the
second moments versus reciprocal temperature are shown in Figure 44.
We did not observe a solid-solid transition but it may occur at a
temperature lower than we attained (N-158°C).

Both the linewidth and the second moment begin to decrease
rapidly above -110°C,at which the self-diffusion reaches a speed of
the order of the linewidth, ~5500 Hz. The second moment just before

this decrease is ~0.25 6%,comparable to the theoretical value 0.19 G2

3 at 25°C).

estimated from the density (~1.725 g cm'
The spin-lattice relaxation rate is plotted in Figure 45. A
maximum in the low temperature solid phase is unique. Since no
transition point is detected, the low temperature relaxation is
probably due to only one type of motion, namely, the overall molecular
tumbling motion. To describe this motion, we first neglect the long

range dipolar interaction of two CF groups since they are separated

3
by five bonds. (This assumption is valid for the trans isomer but
might cause ~20% error for the gauche isomer.) Without using the fast-

motion approximation, the intramolecular dipolar relaxation rates can

213

 

 

 

Table 45. 19F spin-lattice relaxation rates, linewidths and second
moments in CF3CIZCCCIZCF3.

T 1000/T R1 Av M2
(°C) (°x‘1) (sec‘l) (Hz) (a?)
29.0 3.31 0.11310.005 239 0.00085
9.0 3.55 0.834:0.004 581 0.0054
-59.0 4.67 0.47510.017 -- --
-74.0 5.02 0.75810.052 3015 0.115
-78.5 5.14 0.95410.088 3918 0.148
-91.S 5.49 1.62:0.11 4736 0.199
-100.0 5.78 3.32:0.10 5200 0.231
-107.5 6.04 4.80:0.28 5249 0.219
-115.0 6.33 7.26:0.39 5690 0.259
-122.0 6.62 10.820.6 5505 0.252
-128.0 6.90 12.010.8 5396 0.227
-l34.0 7.19 ll.ltO.S 5676 0.259
-141.0 7.58 8.4:0.2 -- --
-150.5 8.16 3.02:0.12 -- --
-158.0 8.70 0.65:0.10 -- --
-50.0 4.48 0.323:0.022 -- --
-32.5 4.16 0.190:0.022 -- --
-13.5 3.85 0.141:0.024 -- --
-4.0 3.58 0.104:0.001 -- --
42.6 3.17 0.0964:0.0046

 

214

 

 

 

 

 

 

O : AV J
LS)" O=-M2 0.3
LO"
5?
m
3
<1
8
a
<
0.5-
l I I I
3 4 5 6 7

I000! T

Figure 44. Plot of the 19F NMR linewidths and second moments in
CF3C12CCC12CF3 versus reciprocal temperature.

215

 

l0—

R|(5£c") -

 

 

 

 

IOOO/T

Figure 45. Plot of the 19F spin-lattice relaxation rate
versus reciprocal temperature.

1n CF 3(212(3CC12CF3

216

be expressed by

 

4 _ T T
8’ =51 nzr 6( r + —-‘l—). (205)
1 5 1+m2T2 1+4m2T2

or 01'

R: reaches a maximum when worr=0.6l6 and, from Figure 45, this occurs

0
at -130°c. With r=2.l6 A, Equation (205) gives (R{)max=10.6 sec'l,
which is in very good agreement with the observed value. The difference

~1.6 see“1

is probably due to the additional contribution from the
interaction between the two CF3 groups in the same molecule. At
temperatures higher than -130°C, onr<<1 and Equation (205) can be
simplified to

6

r 4 2 -
R1 = 3 y M r Tr (206)

and the reorientational correlation time T is found to be

1‘

15exp(3.49 kcal mole-I/RT). (207)

Tr = 7.9X10-
The mean jump time can be estimated from the linewidth narrowing and

we obtain

-1 -
Td = 1.96Xl0 2exp(8.3 kcal mole 1/RT), (208)

-9 2

which gives Td=2.4SXI0-7sec,and DS=S.6XI0 cm sec"1

or 3.8x10'9cm2sec'l
depending on whether the crystal has a body-centered or face-centered

cubic structure.

217

3. 1,1,2,2-Tetrachloro-3,3,4,4-tetrafluorocyclobutane (TCTFCB)

 

This compound can also be considered as a derivative of C2C16.
But, instead of substituting chlorine atoms by CF3 groups, two chlorine
atoms are substituted by a CFZ-CFZ group to form a cyclic compound.
From the IR spectra of the vapor and of the crystalline phases, Harris

297

and Yang found that this compound has C2v symmetry, which means

that the ring is planar. No crystal structure of this compound has been

29° found that an analogous compound,

reported but Schapiro and Hoard
l,2,3,4-tetrachlorotetrafluorocyclobutane, had a body-centered cubic
crystal structure with lattice constant ~7.9510.1 X. We have measured
the density of TCTFCB and find it is ~1.715 g cm‘3 at 25°C. If TCTFCB
also has a body-centered cubic crystal structure, then the lattice
constant will be ~8.02 R.

The measured 19F linewidths, second moments, spin-lattice
relaxation rates in the laboratory and in the rotating frames,and the
spin-spin relaxation rates,are listed in Tables 46 to 48 and are
plotted in Figures 46 to 48.

The observed second moment, before its rapid decrease at tempera-
tures above -25°C, is about 0.145 G2 which is in excellent agreement
with the calculated value (0.142 62) from the lattice constant derived
from the measured density. A possible phase transition at -67°C is
observed based on the rapid increase of the second moment. The second
moments obtained from the maxima of R1p are 0.14 G2, 0.123 G2 and
0.15 Gz,which agree very well with the above values.

Figure 49 shows the claculated mean jump times for self-diffusion

in the plastic phase of this compound from R10, R2 and the linewidth

218

 

 

Table 46. 19F spin~lattice relaxation rates, linewidths and second
moments in 1,1,2,2-tetrachloro-3,3,4,4-tetraf1uoro-
cyclobutane.

T 1000/T R1 Av M2
(°C) (°K'1) (sec'l) (Hz) (62)
-12.0 3.83 0.1098:0.0029 --- --
-18.0 3.92 0.1226:0.0030 928 0.0351
-24.0 4.02 0.1347:0.0039 1490 0.0595
-32.0 4.15 0.1640:0.0020 2396 0.0875
-39.0 4.27 0.180710.0060 3711 0.1325
-46.0 4.40 0.208010.0100 4273 0.1420
-53.0 4.54 0.2800:0.0180 4861 0.1530
-60.0 4.69 0.4060:0.0080 4882 0.1630
-ll.0 3.82 0.1140:0.0040 610 0.0222
-3.0 3.70 0.0975:0.0040 259 0.0032
7.0 3.57 -- 96 0.0005
6.0 3.58 0.0858:0.0025 -- --
4.0 3.61 0.0953:0.0020 -- --
13.0 3.50 0.0863:0.0012 74 0.0007
22.0 3.39 0.0837:0.0010 34 0.0001
31.0 3.29 0.080210.0015 20 0.0000
39.0 3.20 0.0846:0.0011 14 0.0000
55.0 3.05 0.085010.0020 12 0.0000
64.0 2.97 0.0835:0.0035 -- --
71.0 2.91 0.096410.0019 -- --
74.5 2.88 0.0887:0.0010 -- --
79.0 2.84 0.0859:0.0016 -- --
87.0 2.78 0.0878:0.0013 -- --
98.0 2.70 0.0866:0.0021 -- --
110.0 2.60 0.0888:0.0020 -- --
87.0 2.78 0.0881:0.0009 -- --
115.0 2.58 0.0908:0.0011 -- --
142.0 2.41 0.1014:0.0024 -- --
181.0 2.20 0.1220:0.0050 -- --
218.0 2.04 0.1610:0.0060 -- --

 

219

 

 

Table 47. 19F spin-lattice relaxation rates in the rotating frame in

1,1,2,2-tetrachloro-3,3,4,4-tetraf1uorocyclobutane.

T (°C) 1000/T (°x'1) R1 (sec'l) H2 (kHz)
-11.0 3.82 18.810.5 47.0
-21.5 3.98 9.0010.29 47.0
-27.0 4.06 8.86:0.25 47.0
-18.0 3.92 12.010.44 47.0
-14.0 3.86 16.710.30 47.0
-6.0 3.74 28.7510.99 47.0
-1.0 3.68 41.0:2.2 47.0
6.0 3.58 60.912.2 47.0
10.0 3.53 66.8:l.7 47.0
20.0 3.41 50.811.6 47.0
28.5 3.32 26.410.9 47.0
15.0 3.47 62.612.8 47.0
18.0 3.44 64.lil.3 47.0
24.5 3.36 31.9il.0 47.0
31.0 3.29 21.610.9 47.0
35.0 3.25 17.810.6 47.0
45.0 3.14 ll.3:0.3 47.0
55.0 3.05 6.6:0.1 47.0
22.0 3.39 37.7:0.7 47.0
20.0 3.41 54.810.5 25.0
2.0 3.64 11213 25.0
7.0 3.57 10412 25.0
13.0 3.50 8311 25.0
-4.0 3.72 10112 25.0
-l4.0 3.86 63.711.5 25.0
—21.0 3.97 36.810.9 25.0
-9.0 3.79 81.111.1 25.0
-29.0 4.10 21.3:0.8 25.0
-34.0 4.18 12.610.4 25.0
28.0 3.32 32.7:0.5 25.0
-34.0 4.18 40.6:6.2 12.5
-30.0 4.12 62.114.8 12.5
-23.0 4.00 10515 12.5
-17.0 3.91 15316 12.5
-12.0 3.83 19214 12.5
-8.5 3.78 25018 12.5
-4.0 3.72 26916 12.5
-1.0 3.68 260111 12.5
4.0 3.61 231110 12.5
9.0 3.55 170:4 12.5
12.0 3.51 14317 12.5
20.0 3.41 7S.9:1.l 12.5
33.0 3.27 31.510.7 12.5
26.0 3.34 53.210.9 12.5

 

220

Table 48. 19F spin-spin relaxation rates in l,l,2,2-tetrachloro-
3,3,4,4-tetrafluorocyclobutane.

 

 

T 1000/T R2
(°C) (°x‘1) (sec‘l)
-18.0 3.92 29141310
-11.0 3.82 18411150
-3.0 3.70 814140

0.0 3.66 410127

6.0 3.58 312122

8.0 3.56 269111

10.0 3.53 228111

15.0 3.47 136110

16.0 3.44 105111

22.0 3.39 58.814.0
31.0 3.29 49.312.7
32.0 3.28 40.712.7
41.0 3.18 20.610.9
50.0 3.10 12.410.4
58.0 3.02 7.4210.30
63.0 2.98 4.6610.20
64.0 2.97 4.5910.21
71.0 2.91 2.0310.08
87.0 2.78 0.086210.0029
115.0 2.58 0.10510.005
142.0 2.41 0.11310.004
181.0 2.20 0.13910.006
218.0 2.04 0.18510.012

 

221

 

oAV
0M2

0.15
O
o ‘ ’
A 0
H | ~— “1
2 N83 0.1
0
L9 3
v g
§ v
4 N
2
.. e
0.5"“ 40.05

 

0
I.
'D
I
3
a
P

 

Figure 46. Plot of the 19F NMR linewidths and second moments in
O‘CIZCI’Z'CFZCCl2 versus reciprocal temperature.

222

 

 

 

O
' Iooot-
C)
m 1‘0
o 9 °\
I a
- \
CI
0 \
o- ‘
loo— ,0 0. °\
. a ‘. \\
4" ° °\
0 .I \. \
_C‘ i .\ o G\
. -. \
1‘11 " °\. Q
U) ‘, \ E. .
V ' I 0) °. ‘ 3.2 G
CI P \.\
.I . a
| 0_ ' 3. -6.3 6.
I, h 9
e \
I2 G.
0_I I I I
3 4

Figure 47. 19F spin-lattice relaxation rates in the rotating frame at
three different spin-locking fields (12, 6.3 and 3.2 G) and
the spin-spin relaxation rate (0), in ({ClZCFZCFLC‘Cl2 .

 

223

 

 

 

0.02 r— ‘i ll '—

 

l I ‘ I I

h
I—

 

Figure 48.

IOOO/T

19F spin-lattice relaxation rate (0) and spin-spin
relaxation rate (A) in TCTFCB; solid circles are Rz-Rl;
dashed line represents the intramolecular dipolar
relaxation rate; dotted line represents the total
spin-lattice relaxation rate after subtracting the
intermolecular dipolar contribution.

224

 

I0 " .A

i67~

 

 

 

 

IOOO/T

Figure 49. The mean jump time for self-diffusion in the plastic

phase of TCTFCB derived from R10 (0), R2 (0) and
Av (A).

225

data. The best fit of these results can be expressed by
-17 -1
Td = 4.BSXI0 exp(l3.9 kcal mole /RT), (209)

which gives rd21.67x10’°sec,and therefore D524.SXI0-°cm25ec'l,at the
melting point.

To analyze the spin-lattice relaxation data, we assume that the
molecules are planar. Using the standard bondlengths and bond angleszg?
the intramolecular dipolar relaxation rate can be expressed by

d 9 '
R1 = 8.74X10 Tr. (210)

The reorientational correlation time Tr can then be obtained from the

low temperature R1 values. We obtain

- -1
Tr = 2.18X10 14eXp(3.18 kcal mole /RT), (211)
which gives Tr=2.05 psec at the melting point. The intermolecular
dipolar relaxation rate in the plastic solid phase can be estimated
from the mean jump time for self-diffusion. This can be expressed by
the relation

5 -9

R1 - 1.64x10 /Td. (212)
In Figure 48, we subtract this contribution from the total spin-
lattice relaxation rate. The remaining relaxation rate is represented I

by the dotted line. An important contribution from the spin-rotational

226
interaction is very obvious even below the melting point.
We have observed that in the liquid phase of TCTFCB the spin-spin
relaxation rate also has an additional contribution from the scalar
coupling due to the fast-relaxing chlorine nuclei. Although this con—

SC1)

tribution is small, we can still estimate the coupling constant J(F3
from it. Using the same approach used in the previous sections, we

obtain J(FSSCI) to be ~1.710.2 Hz.

E. Discussion

 

1. Rotational Motion in Plastic Crystals

 

One of the characteristic properties of plastic crystals is the
rotational freedom of molecules in the plastic solid phase. In Table
49, we summarize the motional parameters of the plastic crystals we
studied in this work and some other compounds studied elsewhere243’246.
From the last column of this table, it can be easily seen that the
reorientational correlation times of molecules in the plastic crystals
are all of the order of picoseconds,as normally found in the liquid
phase. This implies that the molecules in the plastic phase have
already obtained complete freedom of reorientation. In non-plastic
crystals, e.g., hexamethyltetramine (HMT), Tr just below the melting
point is much longer. In CFZCICFZCl, which we expected to have little

plasticity, It just below its melting point is also longer than the

order of picoseconds.

227

Table 49. Motional parameters of some plastic crystals.

 

 

 

 

Compound AEr AEt AEi.r. 108Ds Tr
(kcal mole‘l) (cmzsec'l) (psec)
053cm3 1.8c 11.7 1.26b 1.61 2.53c
12.8C 1.5c
cpzc1c013 2.56 11.6 4.1 1.0 2.25
12. 8g

CFCIZCFCIZ 3.16 11.4 9.0f 1.5 1.21
CF c01 cc1 2.74 39.0 5.7d 0.16 2.2

3 2 3

e

CF3C12CCC12CF3 3.49 28.3 8 0.36 1.1

- 5 -- j _- j --
c C6F12 14.1 0.2
CwZCCIZCCIZpFZ 3.18 13.9 -- 4.8 2.05

k k k k
Adamantanek 3.08 36. .72 -- 0.1 1.65
33.0
CFZClCFZCIi 1.17 9.5 —- -- 12.0
k

Hexamethyl- 19.3k -- -- 10"s 7641k
tetraminei

 

aInternal rotation of CFn group in the molecule.
bValues at the melting point.

cReference 131.

dIn CF C12 solution, F. J. Weigert and W. Mahler, J. Am. Chem. Soc.
94, 5314 (1972).

eFor the gauche isomer, F. J. Weigert and J. D. Roberts, J. Am.
Chem. Soc. 20, 3577 (1968).

fIn CFC13 solution, Reference 287.
8N. w. Lufts, J. Phys. Chem. 52, 92 (1955).
hReference 289. iNon-plastic crystal. jReference 243.

1‘Reference 246. 2E. Hampton, N. C. Lockhart and J. N. Sherwood,
Chem. Phys. Lett. 21, 191 (1973).

228

2. Self-Diffusion in Plastic Crystals

 

Comparison of self-diffusion studies from NMR spectroscopy, and
from other methods such as radiotracer and plastic flow, has been the
aim of a series of papers published by Sherwood and his coworker5238-242
and by Strange and his coworker5233’237. They found that the activation
energy of the self-diffusion obtained from NMR relaxation studies
(Afigmr) is not always equal to that from radiotracer or plastic de-
formation work (A82). Since both the radiotracer and plastic deformation
techniques have a longer time scale, the measured diffusion constant
is that for the true macroscopic self-diffusion. 0n the contrary, the
NMR eXperiment has a much smaller time scale and therefore should
reflect both the microscopic and the macroscopic diffusive motions. In
those solids with high entrapy of fusion (about 2R), e.g., hexamethyl-
ethane or perfluorocyclohexane, Afigmr=AEg=2Ls, where L5 is the latent
heat of sublimation. This result is consistant with the vacancy diffu-
sion mechanism in which both the formation and the migration of a
vacancy involve an entropy change of magnitude Ls. 0n the other hand”
Afigmr was found to besmaller than A52 for those plastic crystals with a
low entropy of fusion (about R). In these crystals, the defect which
caused diffusion might involve many molecules (12 to 20) and form a
small disordered region in the lattice or a vacancy into which the
surrounding molecules have relaxed.

Since insufficient thermodynamic data are available for the
compounds studied here, we are not able to make the same comparison.

However, we find that the rate of the self-diffusive motion in the

plastic solid phase near the melting point is about the same order of

229

magnitude for all the plastic crystals we studied and also for some

300
243’246. Recently, Boden gt_gl: have discussed the

studied by others
failure of the Torrey model for describing self-diffusion in some
plastic crystals with a low entropy of fusion. They found that for
these crystals the isotropic diffusion model, which does not restrict
the mean jump distance to be equal to the lattice constant, gives a
better result. The main difference in using the isotropic diffusion
model is the greater Td (by about a factor of two) at temperatures near
the melting point than that predicted from Torrey's lattice-diffusion

model. For the compounds we studied, the Torrey model seems to be

appropriate and so is used.

3. Test of the Theory of Fusion

 

Smith232

has tried to relate the parameter T, defined as the ratio
of the rotational to the translational activation energies, to the
parameter v in the Peple-Karasz theory of fusion41 as modified by

Amzel and Becka42

. In Table 50, we summarize the comparison of the
parameters r and v and the physical constants of the plastic crystals
we studied. From this table, it can be easily seen that P and v are

very well correlated.

Table 50. Comparison of the parameters F and

230

v in some plastic

 

 

crystals.

Compound T$a TEa’d Pb c

CFSCCl3 0.65 0.33 0.14 .13
CF2C1CC13 0.62 0.37 0.21 .20
CFC12CFC12 0.59 (0.41) 0.28 .30
CFSClZCCC13 0.64 (0.34) $0.30 .16
CFSCIZCCCIZCFS 0.64 (0.34) $0.42 .16
CFZCC12CCIZEF2 0.64 0.34 0.18 .16

 

 

aT* = 0.72 T/Tb. m: melting; t:

b

r = AEr/AES.

transition; b: boiling.

CPredicted from T5 value according to the Pople-Karasz theory.

dValues in parenthesis are predicted from Reference 232.

SUMMARY

A general program has been written to perform a large variety of
pulsed NMR studies automatically. Several interfaces were constructed
to permit varying the pulsewidths, to supply a constant pulsed current
to the field-gradient coil and to alternate the radiofrequency pulses.

Complete 14N and 19

F relaxation studies of CF3CN were accomplish-
ed and the motional anisotropy of the molecules in the liquid state
was determined. The J-diffusion model was found to be appropriate to
describe the reorientational motion of the CF3CN molecules. Several
physical constants of the molecules were determined, e.g., the chemical

13

shift anisotrOpy and the spin-rotation constants for both the C and

the 19F nucleus.

13C relaxation studies have been carried out for CDBr3, CFBr3 and
CHZBrZ and the scalar coupling constants between 13C and 798r
determined from the dominant scalar relaxation rates in these highly
brominated compounds. In CHZBr2 the coupling constant J(C7gBr) was
found to be 58.7 Hz, a reasonable value compared with the reported

value for J(CSSCI). But,in both CDBr and CFBr3,J(C7gBr) was found

3
to be about twice as big as expected and this discrepancy has been
discussed. The coupling constant J(F798r) in CFBr3 was also deter-
mined and found to be 30.5 Hz from the 19F relaxation studies. The
test of the extended diffusion model in this molecule has been dis-
cussed.

An attempt at resolving the anisotropic motions of planar

CFZCCl2 molecules in the liquid state has been made with a certain

231

232

degree of success.It was found that the reorientational motion in the
liquid and in the solid states was via different mechanisms.

The separation of various 19F relaxation mechanisms in CF3I was
performed but difficulties were found in extracting the intramolecular
dipolar relaxation rates and therefore analysis of the reorientational
motion was fruitless. However, it was found that the spin~rotationa1
relaxation mechanism was via the spinning motion of the molecules
about the C3 axis.

A number of plastic crystals were studied by analysis of the 19F

NMR lineshapes and by measurement of the relaxation rates of the 19F
nuclei. Measurement of the 19F spin-lattice relaxation rate in the
rotating frame was found to be very informative about the self-
diffusive motion in the plastic phase of these crystals. The mean

jump time between adjacent diffusion steps was determined to several
orders of magnitude. In all the plastic crystals studied the Torrey
lattice-diffusion model was found to be appropriate to describe the
self-diffusion of the molecules in the plastic solid phase. The
reorientational motion in these plastic crystals was found to be as
fast as in the liquid phases. From the 19F spin-spin relaxation rates,
the scalar-coupled relaxation rates in someof'these fluorochlorocarbon
derivatives were extracted and the coupling constants J (F35Cl)

determined. Finally, a test of the theory of fusion was discussed

in terms of the activation parameters derived from the NMR studies.

’

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1273 (1973).

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39, 1332 (1972).

 

252.
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254.

255.
256.
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246

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295.

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298.

299 O

300.

301.

302.

247

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13 79

The value J ( C Br)- 30 Hz for CH Br which they report is

3
actually lower than predicted from J (ISCSSCI) and the ratio of

gyromagnetic ratios .

G. J. Jan: and S. C. Wait, Jr., J. Chem. Phys. _2_6_, 1766 (1957).

 

APPENDIX

APPENDIX

A. Instructions for NMRLX - A General Program

for Compgter Controlled Pulsed NMR Studies

I. INTRODUCTION

NMRLX is a powerful program used with a Nicolet 1080 series com-
puter and a Diablo disk system to do various on-line measurements of
nuclear relaxation times and of the self-diffusion constants of molec-.
ular motion. In these experiments, the perturbation pulses are trig-
gered by software instructions with controllable pulse intervals.’ For
single-line spin systems, transient signals are digitized, integrated
and analyzed using weighted least-squares fitting. For multiline spin
systems the free induction decays are collected and stored in the disk
for further data manipulations.

This manual describes the structure of this program and the

detailed procedures fer each type of experiment.

11. CONSTRUCTION OF THE PROGRAM

The program NMRLX is written in assembly language. It occupies
totally 4 K memory words each with 20 bits. These 4 K words are.
divided into four 1 K-word pages for ease of programming. The whole
program can be separated into three main routines: display, pulsing
and least squares, plus two embedded auxiliary standard Nicolet pro-
grams: Nicolet Debugging Program (N06-30612) and Floating Point

Package, 1972 (Nll—20823). The address assignment is shown below:

248

 

249

 

 

 

0 1112 1663 1777
1 Display 1 Dsd P01nters [
Page 1
2000 2777 3777
1 Discrete Data L Pulsing (T1,T2.T10.Th) l
Page 2
4000 4602 S446 S777
Least Squares NICOBUG Subroutines
Page 3 l I
6000 7600 7777
FPP-72 Monitor
Page 4 I l [

 

The source tapes are made in pages; each page can be assembled
independently. The binary tape of the entire program was also made.

It can be read into core by Demon II in the usual way.

III. LEGAL COMMANDS

The legal commands in NMRLX are mostly composed of two alphanu-
meric characters, only a few need one more character as an extension.
These commands are divided into three groups: the first group of com-
mands accepts parameters which are fixed to integers before being stored;
the second group of commands accepts real number parameters which are
stored in floating point format; and the third group of commands are

executing commands. All the legal commands are listed below:

Group 1:

NS -

N1 -

IL -
BL -
BA -

IN -

DL —

Group II:

D" -

T3 -

250

Number of scans for each experiment
Number of points for each experiment

Number of preceding u pulses (used in D5 measurement)

d
Number of intermediate 1 pulses (used in Dsd measurement)
Number of'points used for averaging or maximum finding in
data transfer

Number of points for integration

Number of points fer baseline correction

The number to be added to each point

Number of points as the interval between the adjacent inten-
sified points

Number of points delayed before the first intensified points

Dwell time
Estimated Spin-lattice relaxation time (used in T1 measure-
ment) or tau delay in spin-echo pulse train (used in Dsd
measurement)

Tau delay in Spin-echo pulse train (used in T2 measurement)
or time interval between the first field-gradient pulse and
the a pulse thereafter (used in D5

d
The width of field-gradient pulse (used in Dsd measurement)

measurement)

Time delay between adjacent pulse sequences

251

Group 111:

DS - Display all the data points on the sc0pe

DTA - Average the RA points around the intensified points and
transfer the average to the discrete data points storage
area

DTS - Direct transfer of the intensified points to the discrete
data points storage area without averaging or searching for
maxima

ZE - Zero the displayed memory

DTM - Search the maximum point within the RA point range around
each intensified point and transfer it to the discrete data
points storage area

DZ - Di5play the current discrete data points on the scope

D1 - Display the first discrete data points on the scope'

UP - Shift up one block of data points A

DN - Shift down one block of the discrete data points

DV - Divide the current discrete data points by the previous
discrete data points and display the result

BC - Use BL points for baseline correction

LS - Fit the data points to a straight line using weighted least-
squares method

EX - Display the pre-analyzed data points

AC - Add constant BA to each data point

DP - Delete the selected point

GO - Start Ds measurement

d

61 - Start Tl measurement

GZ - Start T2 measurement

 

252

P1 - Reset page number
PRA - Print out the data points in decimal form
PRO - Print out the data points in octal form

MO — Exit to the monitor

NG Go to NICOBUGII

£3

Start the triplet or quartet pulse experiment
HS '- Set up Homospoil Tl experiment

SE

Set up mode for pulse adjustment

IV. TIMING

The basis of timing in this program is the definite executing time
for software instructions. (Nicolet 1080 computers have two accurate
master crystal oscillators to control the timing.) Since the execu-
tion time of each direct instruction is 4 us and is 6'us for indirect
instructions (newer models of 1080 computers have 3.3 us and 4.9 us fer
direct and indirect instruction, respectively), this timing basis is
not for pulse-width-control which is usually only a few us to less
than 100 us, except for the spin-locking pulse for T1p measurement.

It is only used to control the pulse intervals which are usually longer
than a few hundred us. Due to the definite execution time and the
limited minimum number of instructions for complex programming, the
software-controlled timing has limited accuracy. Theoretically, there
is no limit of accuracy for long time delays, but for time delays
shorter than 1 ms the accuracy is limited to 1% or less and becomes
even greater for a shorter time unless a discrete value is selected.
Table I shows these discrete values and the possible errors for arbi-

trarily entered values.

Tab 1e A1 .

253

Discrete Times and Possible Errors °

the NMRLX Timing Routine

 

 

Time Entered (us) Actual Time (us) 3 Error
0 N 15 38 153 N m

16 N 31 38 22 N 138
32 N 47 38 0 N 23.4
48 N 63 54 0 N 14.3
64 N 79 70 0 N 11.4
80 N 95 86 0 N 9.5
96 N 111 102 0 N 8.1
112 N 127 118 0 N 7.1
128 N 143 134 0 N 6.3
144 N 159 150 0 N 5.7
160 N 175 166 O N 5.1
176 N 191 182 0 N 4.7_
192 N 207 198 0 N 4.3
208 N 223 214 0 N 4.0
224 N 239 230 0 N 3.9
240 N 255 246 0 N 3.5
256 N 271 262 0 N 3.3
272.N 287 278 0 N 3.1
288 N 303 294 0 N 3.0
304 N 319 310 0 N 2.8
320 N 335 326 0 N 2.7
336 N 351 342 0 N 2.6
415 N 400 406 0 N 2.2
480 N 495 486 0 N 1.8

 

254

Generally, when time entered is 16n us to (l6n+15) us, where n is
an integer not less than 3, the actual time will be (l6n+6) us and the
possible timing error will be 0 to 900/16n+15. Therefore, to avoid
serious timing errors one should always choose a value (l6n+6) us
unless the time is longer than a few milliseconds. The minimum time

which can be provided is 38 us.

V. DISPLAY

When first started, or after the command has been executed, the

program is always in display routine waiting for another command.

There are two different types of display; one is for displaying data

in memory addresses selected by push buttons on the 1080 mainframe, the
other is for displaying discrete data points stored in memory addresses
2000 to 2777.

In the first display routine, one can select equally Spaced points
to be intensified for transferring to the discrete data storage area..
The intensified points can be selected by three commands DL, NP, and
IN. The first intensified point is the (DL+l)th point, the following
points to be intensified are separated by IN points and NP is the total
number of intensified points. The intensified points can be trans-
ferred to the discrete data point storage area by command DT(A,M or S).
DTA will average 1/2RA points to the left and the right of each inten-
sified point then transfer it to the discrete data point area. DTM
will transfer the maximum point in the l/2RA point range to the left
and the right of each intensified point. DTS simply transfers the

intensified points. This is very useful for picking up spin-echo

255

maxima during the T2 experiment. Setting either DL=0 or NP=0 will
abort this function and return to normal continuous display.

The second di5play routine displays the data points in storage
area 2000-2777 which are stored in the fOrm x1, y1, x2, y2, ... xn, yn.
This routine can be terminated by hitting any character on the tele-
type and the program will go back to the beginning waiting for a new
command. Command D1 will display the first NP point block of the
storage area, DZ will display the current NP point block. UP command
'4111 shift up one block, DN will shift down one block. DV command is
used to divide each point by the corresponding point in the previous

block and replace the current block by the result.

VI. WEIGHTED LEAST-SQUARES CURVE FITTING

A general expression for the decay of magnetization is

M(t) = M e‘tfr ;
0

when t gets long M(t) approaches the equilibrium value ”0' On a
logarithmic scale small numbers are more seriously scattered and are
less important to the fitting. In order to suppress the errors induced
by this scattering of small numbers, every data point was weighted by
its own amplitude. The mathematical equations for this weighted least-
squares curve fitting are shown below:

For y1 = -R1Xi + c, where y1 = ln M(tk), Xi = ti, R1 = relaxation

rate = 1/T1

"’256

I . 2
Z"i inyiwi ' inwf Z’iwi
l/Tl 3 R1 = 2 2 2 2_ s (1)
(inwi) — {wi {xiwi

 

M(t.)

1 C
where Wi - W , 8.150,
. 1
1

2 2 2 22
. 2x1": inyiwi ’ Zyiwi inwi

 

 

 

c 2 2 2 2 2 (2)
(zxiwi) ' 2"i {xiwi
and the standard deviation,
P x. '-
chl-+ c - v.12 1’2
2 1 1 (3)
o = ,
Xnax-den L N - l
d

 

 

where N is the number of data points used for curve fitting.

When discrete data points have been collected and displayed on the
scope, they have to be baseline corrected in order to make every point
positive in magnitude. Baseline correction is via command BC which will
use the last BL points as the baseline. Command LS starts the least-
squares calculation and then prints out the slope and l/slope of the
theoretical line and the corresponding standard deviations. The scope
will show the deviation of each point from the theoretical line. Data
can be recalled by command EX and can have a constant BA added by com-
mand AC to shift the baseline up or down. The best fit can be seen
from the minimum standard deviation printed out and the uniform dis-

tribution of the deviations from the line as seen on the scOpe.

 

257

VII. SETUPS FOR EXPERIMENTS

(1) Pulse Adjustment

 

Before starting data collection, the pulse width has to be
adjusted to the desired width. This can be done by using setup mode
(SE) which can trigger an even number (determined by RA) of equivalent
pulses with interval controlled by N1.

Normally when adjusting a n/Z or n pulse, RA is set to be 4n (n is
any integer) and N1 is set to be (pulse interval/l6). The pulse inter-
val has to be long enough to see the signal and short enough to avoid
saturation of the signal after a pulse sequence. TL (delay between
each pulse sequence) has to be set long enough to avoid an excessively
heavy duty cycle for the power amplifier.

When the pulse width is adjusted to be exactly 90°, the signal on
the scape should be up-null-down-null-etc. The signal will be nulled
when the pulse width is exactly 180° and also when the RF phase is
adjusted to be 90° (along the Y axis). If the RF pulses are not
homogeneous, there will be some deviation from the uniform up-null-
down-null pattern. Therefore, this procedure also offers.a method for
testing RF pulse homogeneity.

This setup mode will always trigger pulse 1 of the computer. If
one needs to adjust the second or the third pulse, one can either con-
nect the pulse 1 cable to the second or third pulse input or use
NICOBUGII (called by command NC) to change address 3745/4102 to 4104

or 4106.

258

INA") an)

'H\§r n *

00,021) vo.(P1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b I 'k T L =
FG P
K,(P1) P2
C T L 4
S L P A
we. . 00y . no, my
6 l /f\\; W/:\\ I L] . TL
‘ ' : I 5 //\\
. 2 ,1, I
noy soIt no, 00..
e p '1 I; I;
FGP [PCT] F— T L ——_"
no, 90,I looy noy 90., so, . no, 00,,
1 ‘ h 3) 2‘) 33 «— TL —.
FGP FGP
nay ”a '”y Qxfi'w ”.100. no, 00_ ”'0 no eq-
9 I: I; I: I: I; *— TL ——9
FG P FGP
.0. I'D, 100,342.— "1001”,”, 'I0, IIOy
h .- I)‘ b '1'. I. <— T L _.
IFGPL; . [FGP j\
877 .44 * A

Figure 1. Pulse sequences used in NMRLX.

259

After the pulses have been well adjusted, one can start to set up
parameters for each type of experiment. Pulse sequences for these
experiments are shown in Fig. 1. Typical procedures are given in the

following sections.

(ii) T1 Measurement Using the Spin-Inversion-Recovery Techniqge

 

a. Adjust PULSE A to be (n,x) and PULSE B to be (n/2,-x). (x,y,z
are the laboratory coordinates, z is in the direction of HO, the static
magnetic field,and y is the direction of the coil axis.)

b. For a single-line spin system set on resonance, set:

T1 = Estimated spin-lattice relaxation time of the spin to be

measured

NP Number of points exploring the exponential decay of
magnetization, therefore number of r delays
NS = Number of scans to be averaged for each point

TL

At least 5 times the estimated T1

IL Number of points to be integrated as the amplitude of

magnetization for each I delay. It can be from 1 to the
size of the di5played memory selected by 1080 mainframe
buttons.

Then type 61 and answer "N" for the question FOR MULTILINE? immediately

after 61. The computer will use the T1 value entered to calculate r
4T
1

delays Ti = ljfiy-, i = l to NP, then trigger the pulses 180—ri-90 and
collect the transient signal (usually 20 us dwell time and 1 K memory
size are used), integrate it and store the amplitude in storage area

2000-2777. Instead of collecting NS scans with each Ti then inte-

grating the averaged signal, the program will increment r after each

 

260

scan and collect NP points first then do another scan and add the
result to the previous cumulated amplitudes stored. The advantage in
this procedure is that it minimizes the error induced by field drift
during long time averaging. Of course, this can also be achieved by
using a smaller IL. After the experiment is done the scope will dis-
play NP points with the x axis representing 1 delays and the y axis
representing amplitudes. The least-squares routine can then be used to
fit the exponential decay to a straight line and print out slope (R1),
l/slope (T1) and the standard deviations.

c. For multiline spin systems only T1, NP, NS, and TL are needed.
Answer "Y" for the question FOR MULTILINE? after which Gl will begin
data collection. The computer will first check if tracks 400-625 have
any program or data stored. If that is true, an error message NO MORE
ROOM will be printed out and stop any action thereafter. If not, the
computer will zero those tracks then start collecting FID's corre-
sponding to each 1 delay and stored them adjacently on the disk
starting at.xrack 400. The memory size and dwell time should be set on
the 1080 mainframe before data collection. After the experiment is
done the scope will show the cumulated FID of the last r delay for fur-
ther Fourier transform using the auxiliary program FTSWAP (see

description in Section X).

(iii) T1 Measurements Usigg the Homospoil Techniqge

 

a. Adjust PULSE A to be (n/2,-x) and connect PULSE C to field-
gradient pulse input.

b. Enter parameters as in (i).

261

c. Type command HS then start data collection by 61.

d. The analysis is the same as in (i).

(iv) T10 Measurement

 

a. Adjust PULSE A to be (n/2,x) and PULSE B to be (n/2,y).

b. Push in buttons marked Tlp so that PULSE 2 becomes the ter-
minating trigger of the spin-locking pulse right after PULSE 1.
Note: An occasional circuit hangup might turn the -15 v pulse to the
power amplifier always on which can be noticed on the current meter of
the 500 V plate power supply. If this occurs release the T19 button
then push it in again.

c. Enter parameters Tl, NP, NS, IL (for single-line system) and
TL as before and use the command 01 to start data collection.

d. Data analysis is the same as in (i).

(v) T2 Measurement

 

a. Adjust PULSE A to be (n/2,x) and PULSE B to be (n,y).

(Steps b-e are for a single-line system, f-h are for a multiline sys-
tem.)

b. For a single-line system set on resonance, connect cable T2 to
SWEEP CONTROL CONNECTOR so that pins X, H, F are plugged in. The other
end of the cable should be connected to a pulse inverter then to the
pulse B input of the pulse-width controller.

c. Enter T2 equal to the r delay in the pulse sequence, DW equal

to the number selected on the 1080 mainframe and set ADDRESS SELECTOR

262

to a number n which fulfills the relation T2 = DW-n and DELAY MODE to
OFF. In this case, when the MEASURE mode is in process there will be a
(n,y) pulse triggered for every n addresses advanced.

d. Enter NS to be the number of scans and TL to be at least 5T1,
then use command 62 to start data collection after answering "N" for
the question FOR MULTILINE?

e. When the experiment is done the scope will show decaying spin
echoes; the display routine can then be called to pick up the echo
maxima and transfer them to the discrete data point storage area for
least-squares analysis.

f. For a multiline system disconnect cable T2 and set the memory
size and dwell time on the 1080 mainframe as in the FTNMR experiment.

g. Enter T2 (1 delay), NP, NS, and TL as usual and answer "Y" for
the question FOR MULTILINE? after the GI command.

h. The analysis is the same as for multiline T1 measurements.
Note: In order to decrease errors from pulse misadjustment, only odd

echo FID's are collected.

(vi) Th Measurement

Th

bination of T1 and T2. For pulse sequences e-q in Fig. 1 the apparent

is defined as the hybrid relaxation time and is a linear com-
relaxation time can be expressed by

Th'l = (I-p) Tl-l + pTz-1 . (1)

 

263

where p = th/t1+2t2 (or p = 4t2/t1+4t2 for pulse sequences f or g)

and the decay of the signals follows the expressions

Mt - Moo = (MO-Moo)exp(-t/Th) ,’ (2)

where Mo is the magnetization at t = 0 and Mm is related to M6 by

1-8XPI-t1/71]
M = M

m o l-eXPl-(t1+2t2)/Th] ° ‘33

 

If pulse sequences f or g of Fig. l are used, 4t2 replaces th in
Equation (3).

The principal advantage of this technique is a great saving in
time and it should therefore be applicable to the measurement of T1_and
T2 in natural abundance 13C or 15N.

The following procedure is used to set up the Th experiment:

3. Adjust PULSE A to be (n,y) and PULSE B to be (n/2,x).

“ 264

b. Connect the phase inverter as follows:

J

rModified Part .
I ,

Phase
Invertor

 

   
   
 
 

 

 

 

 

 

 

 

 

r ---------------------- ‘n
PULSE
Controller
: To PC

PULSE C Current V Mixin to

Pulser RF Gaté—4pto Mixer Mixer
- Network
5 PA Gate .4pto PA

 

 

c. Enter the following parameters:

T1 = Estimated Th

DW = Hardware setting oanWELL TIME on the 1080 front panel
N1 = Hardware setting of DELAY TIME on the 1080 front panel
NP = Number of FID's to be collected

NS = Number of scans

d. Type the TR command to start data collection.
Note 1: Three different pulse sequences may be chosen. These are
determined by address 3601. When it contains 4104, e sequence will be
chosen; when it contains 1676, f sequence will be chosen; and when it
contains 1673, g sequence will be chosen. They can be selected by

using NICOBUGII.

 

26S

4T1
NP
routine ZRMEM and ADD. Therefore t1 = {

Note 2: t1 in Eq. (1) is plus the computer time for executing sub-

4T1
NP

34(SIZE/1024 - 1)} , where SIZE is the memory size used for collecting

+ 240 us + 96 x SIZE +

 

each FID; t2 in Eq. (I) is equal to [N1 + DW x SIZE].

e. Data analysis is the same as in (i) for multiline system.

(vii) DSd Measurement

 

The pulsed field-gradient spin-echo technique (pulse sequence h in
Fig. l) is used in this program to measure Dsd' The echo amplitudes
with the field-gradient pulse (FGP) on and off follow the relation

1n §é(%-= -v2g26205d(A-6/3) . (4)

where M(t) is the echo amplitude with PCP on, M°(t) is the echo ampli-
tude with PCP off, g is the magnitude of the field gradient, 6 is the
duration of PCP and A is the diffusion time.

To set up this experiment follow the procedures below:

a. Connect PULSE C to PG CURRENT PULSER trigger IN. Turn the
power on.

b. Enter parameters T1, T2, T3 (see Fig. l.h) N1, N2, NP, N, IL,

and TL.

Note: T1>T2>T3 and 2NP§(N2+1) have to be fulfilled.
c. Adjust FGP duration and FG current by doing a dummy run (GD
command), monitoring the PCP duration from the BNC connector and the

current from the pin test-point on the front panel of PG CURRENT PULSER

(+3 V corresponding to ~10 amps current).

 

 

266~

d. Turn the PCP trigger control switch off and run once (without
FGP) by typing GO command.
Note: A message "Tl,T2&T3 MUST BE <8 SEC" will be printed out
immediately after GO command. This was done to save memory in pro-
gramming. Actually no such experiment needs T1 longer than 8 seconds.

e. Turn the PCP trigger control switch on and run once again.
(After each run the scope should display data points collected.)

-f. Type DV command to display the result of M(t)/M°(t).

g. Call the LS routine to analyze the data. The slope will be
equal to yzgzézDsd. Therefore, Dsd = slope/yzgzoz.
Note: Due to insufficient heat insulation of the field-gradient coil,
the actual field gradient is temperature dependent. A calibration has

been done using CFC13 for which accurate Dsd values have been reported.

VIII. USEFUL SUBROUTINES

In NMRLX, several subroutines are used repeatedly. They are con-
structed rather independently and can be called for use at any time.

1. TIMER: This subroutine is actually a counting clock. The
basic logic of its construction is to count down a number stored pre-
viously in address WAIT. One count loop takes 4 instructions and there-
fore 16 us (or 13.2 us for newer models of 1080, see Nicolet 1080 manual
for instruction execution time). Since NIC 1080 has 20 bits in one word
the maximum counting time for each word is about 8 seconds. Therefore,
another address HIWAIT is used to store the number of times of counting

down the number stored in WAIT.

267

2. TMS: This subroutine is used to set up WAIT and HIWAIT. It
must be called before using the TIMER subroutine. The time entered
in seconds should be put in FAC of the Nicolet FPP-72 package before
calling TMS. A typical set up for a timer is:
JMS @GETAC
TI IVE
JMS TMS
JMS TIMER
Note: A floating-point constant CONFPI in addresses 1105 and 1106 is
used in subroutine TMS to calculate WAIT. For models with 4 us exe-
cution time it is 105/l6 (1105/4o,ooo; 1106/1,720,440). For those
with 3.3 us execution time it should be changed to 106/13.2 (1105/
42,466; 1106/1,117,676).
3. ZRMEM: A.subroutine to zero the displaying memory addresses.
4. ZRTK: A subroutine to zero the content on chosen tracks in
the disk. The first and the last track numbers to be erased should be
put into addresses TRACKI and TKEND.

5. ADD: A subroutine to add displaying memory content to the

content stored in selected tracks.

Ix. ERROR MESSAGES

NO MORE ROOM: Discrete data storage area is full or tracks on the
disk have been used.

RA T00 BIG: Assigned number of points in RA exceeds the dis-
playing memory size.

BL TOO BIG: Number of points assigned for baseline correction

exceeds total discrete data points displayed.

268

A,M OR 8 MUST FOLLOW: Other extention is used after DT.

UNEQUAL POINTS: Number of x's is not equal to the number of y's.

?? SET N2 = ODD, RA = EVEN: Even number (or odd number) has been
selected for N2 (or RA). .

0 OR A MUST FOLLOW: Other extention is used after PR.

T1<=T2 MUST FOLLOW: Obvious.

T2<=T3 MUST FOLLOW: Obvious.

2NP>N2+1 MUST FOLLOW: Obvious.

X. AUXILIARY PROGRAMS

l. FPP72: Standard Nicolet Floating-Point Package, 1972 version
(see Nicolet manual).

2. NICOBUG: Standard Nicolet Debugging Program (see Nicolet
manual).

3. FTSWAP: A modified Nicolet FTNMRII for fetching FID's stored

on the disk, doing the Fourier transforms and storing spectra back on

the disk.
Command to be used:

NT: Number of tracks for each FID (l for l K words, 2 for 2 K,
3 for 4 K and 6 for 8 K).

ST: Starting track nunber wherethe FID's were previously
stored. (It is preset to 100,400 (in octal) track number
400.)

NS: Number of FID's stored and to be transformed.

NM: Select a three-character file name for the spectra to be
stored. Series numbers from 001 to nnn will be added

automatically when spectra are stored.

 

'269

L1: Link a number of normal FTNMRII commands needed for doing
data processing, e.g., TR, EM, FT, PC, TR, PP, etc.

80: Start execution of fetching the FID's, processing of data
according to commands linked by LI, then storing spectra

onto the disk with file name selected by NM.

XI. USEFUL PATCHES

Since a large variety of pulse NMR techniques have been developed.,
it is impossible to include everything in a single program. Therefbre,
a few small patches have been written in modular form which can be read
into NMRLX to do other experiments.

1. SSFT: A.patch to do single-scan FT multiline T1 measurement
with pulse sequence 180-1-30°-FID-r-3O°-FID-r...1 The pulse sequence

and the decay of magnetization are illustrated below.

180 30 so 30 30 30
L7 igEaLT 7&1 7 T
17:61 . [a [E].
F r f

 

2. STR: A patch to do a single-line Th experiment. For single-
line spin systems there is no need to use FT. Therefore, t2 in the
pulse sequence Fig. le-g can be set short enough to ignore spin-
spin relaxation effects and the transient signal can be integrated as
in the single-line T1 experiment. The resulting decay constant will no

longer be Th’ it will simply be T1. This patch occupies 3034 to 3131.

270

3. DYNOE: A patch to measure the dynamic Overhauser effect. It
utilizes the homospoiling T1 measurement module of NMRLX and has only
two addresses modified:

3773/111760 changed to 111762
3762/4106 changed to 4104.

A decoupler gating box should be connected as follows:

PULSE A Pulse width to normal trans-
controller . mission line

to receiver

 

 

 

 

 

   

 

 

PULSE B 1V - line
Decoupler [P jFlReceiver
PULSE C Gatin Box . 7 ,
:l g . m
Gate ,

_ Signal
Decoupler to Decoupler
RF Source Coil

 

 

PULSE B is then used to trigger the decoupler and PULSE C to turn off

the decoupler. The pulse sequence is:

PULSE B A C

B A C
Pulse .
Sequence 7 TL 7'

~_—

or g 2:: ON[] Jill] :[I {1 1L*- [n.n.uuu..n..n.]

I,

Decoupling
RF

"'°3:_flfl_ HI ,, I ...................... [—

 

 

 

 

 

 

 

271

XII. INTERFACING CIRCUITS

1. Pulse booster: Convert PULSE 1 and PULSE 2 from NIC 1080

(500 nsec) to PULSE A, PULSE B and PULSE C (O to S V, 2‘Usec, SmA)
(see Reference 131 for description).

2. Pulse width controller: Convert two 0 to 5 V pulses to two 0 to
~15 V pulses with pulse width adjustable from 1 to 100 psec for
driving the NMRS MP-lOOO spectrometer (see Reference 131 for
description).

3. FG current pulser 1: Supply an externally triggered field-
gradient pulse with lOlisec to 10 msec pulse width and 0 to 10 A
current to a field-gradient coil built in the NMR probe (see
Reference 133 for description).

4. FG current pulser 2: Supply an externally tiggered field-
gradient pulse with l to 100 msec long and 25 mA current to the

z shim coils of the magnet (see Reference 131 for description).

5. rf phase inverter: Reverse the output rf phase everytime

it is triggered (see Reference 133 for description).

6. Decoupler gating box: Provide gating pulses for decoupling

rf to the power amplifier and for the receiver. (see Figure A2).

272

 

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E : 3

009 xu

 

 

 

 

 

 

0+

273

INSTRUCTIONS FOR OTHER PROGRAMS USED

1. FTNMRD - A modified Nicolet FTNMRII for triggering a pulse to do
FTNMR.

Additional commands:

NS - Number of sweeps

DL - Delay in seconds after each pulse and data collection

GO - Starts triggering perturbation pulse and collects FID.
TOTAL EXPERIMENT TIME will be printed out.

CO - Resume data collection after interruption

II. FTNMRB - A modified Nicolet FTNMRII for collecting data after a
pulse triggered by NIC-293 general pulser.

Additional comands:

P2 - Enter pulse width in “sec

DZ - Enter delay time in seconds after each pulse and data col-
lection

NC - Number of scans

GO - Start data collection after total experiment time is
printed out

CO - Resume data collection after it has been interrupted

 

 

274

III.SECMFT - This is a modified FTNMRD or FTNMRB for calculating the
second moment of the selected peak.

Additional legal commands:

GM - Gyromagnetic ratio divided by 2n
M2 - Enter second moment calculation mode and intensify the peak
selected
Subcommands of M2 mode:
N - Calculate and print out unnormalized second moment
C - Calculate and print out normalization factor
G - Calculate and print out normalized second moment

Procedure for the experiment:

1. Use normal FTNMRII functions to collect data and transform the
FID to a spectrum.

2. Select F1 and F2 to confine the peak whose second moment is
to be calculated.

3. Set cursor to the center of the peak.

4. Enter correct GM for nucleus (4259.1 for 1H and 4007.4 1F).

5. Type M2 to intensify the peak area and then type N, D and
finally G. The second moment of the right and left sides of the peak
will be printed out. (This is designed for separating the contribu-
tions from an embedded peak which causes asymmetry of the overall peak.)
Note: In order to get a correct second moment the baseline of the
selected peak has to be correctly set. Command AC of FTNMRII is very

useful for doing this.