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This is to certifg that the

thesis entitled

ELECTRON SPIN-LATT ICE RELAXATION

OF Cr(CN)5No3' IN KBr AND KCl

presented bg

George T. Johnston III

has been accepted towards fulfillment
of the requirements for

___Ph‘_D‘___ degree in

\Am

0 ' ‘Major professor

Physics

Date — l5" (9

0-169

 

 

 

 

ABSTRACT

ELECTRON SPIN—LATTICE RELAXATION

OF Cr(CN)5No3‘ IN KBr AND KCl
by George T. Johnston III

Electron spin—lattice relaxation times T1 of
Cr(CN)5NO3— substitutional in KBr and KCl have been
measured for l < T < 1500K. Spin echo techniques employ—
ing either picket or single-pulse saturation methods
were used. From the temperature dependence of the ob—

served T values, we conclude that at least three relaxa—

1
tion mechanisms are operative: an anisotropic direct
process at T < 50K, an isotropic Raman process at inter—
mediate temperatures, and an anisotropic local mode pro—
cess at T > 650K. The functional form of the relaxa—
tion rate is %1 = AT + BT9J8(e/T) + C exp(—eo/T), where
e is the Debye temperature and was taken as l70°K for
KBr and 230°K for KCl. Empirical values of A, B, C,

and 60 were determined by least squares computer analysis.
Published energy level assignments preclude the possi—
bility of an Orbach process. Therefore, since the para-

meter 90 is 575:300K for KBr and 610:3OOK for KCl, in

agreement with published Cr—CN stretch frequencies, we

George T. Johnston III

conclude that the exponential process occurs through the
interaction of lattice phonons with vibrational modes
localized in the Cr(CN)5NO3_. An anomaly in the T de-
pendence of T1 for KBr has been tentatively explained

as a driven-mode process, in which relaxation is effected

by low frequency (<20 cm-l) bending modes of the complex.

ELECTRON SPIN—LATTICE RELAXATION

OF Cr(CN)5No3‘ IN KBr AND KCl
By

George T. Johnston III

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Physics

1967

DEDICATION
This thesis is dedicated to my parents and especially
to my wife, whose help and understanding helped to make

this work possible.

ii

ACKNOWLEDGMENTS

Professor J. A. Cowen, by his encouragement and en-
thusiasm and patience, has helped to make my undergraduate
and graduate study at Michigan State University interesting
and exciting. Professor R. D. Spence has provided valuable
intuitive insight into many theoretical problems associated
with the preparation of this thesis. Mel Olman and many
other graduate students of the infrared spectroscopy group
have given invaluable aid with computer programming. The
good humor and fellowship of Charles Taylor have added
immeasurably to the enrichment of my graduate study. Without
the able assistance of the personnel of the machine and
electronic shops, particularly Ernie Brandt and Nick Rutter,
this work would have been impossible. Financial support
of the National Science Foundation and the Army Research

Office (Durham) was appreciated.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS
LIST OF TABLES
LIST OF FIGURES
LIST OF APPENDICES
Chapter

I. INTRODUCTION

History
Summary of Theory

II. THEORY

Theoretical Background

Direct Process

Raman Process

Orbach Process

Driven- Mode Process

Local Mode Process

Anisotropy of Relaxation Time.

III. EXPERIMENTAL APPARATUS AND TECHNIQUES

General Description

Spin Echo Techniques
Vacuum Can Cryostat

Crystal Growing

IV. SUMMARY AND IMPLICATIONS OF PREVIOUS RESULTS

ESR Spectrium
Infrared and Optical Spectra

V. RESULTS AND CONCLUSIONS

General Considerations

Relaxation Results —— KBr
Discussion —- KBr .
Results and Discussion —— .KCl
Summary

iv

Page
iii
vi
vii

Viii

f\)l—’

l6
19
2A
2A
26

32
38

38
HO

43
“5

A6
A9

A9
50
53
62
68

 

REFERENCES

APPENDICES

Page
69

72

LIST OF TABLES
Table Page
I. Effect of lattice on vibrational frequencies

of Cr(CN)5No3‘ . . . . . . . . . . . . . . . . . 48

II. Least squares fit of % = AT + BT9J8(l7O/T)
l

+ c eXp(—575/T) to relaxation of Cr(CN)5No3‘

in KBr . . . . . . . . . . . . . . . . . . . . . 52
III. Least squares fit of % = AT + BT9J8(23O/T)
l
+ C exp(-6lO/T) to relaxation of Cr(CN)5NO3_

in KCl . . . . . . . . . . . . . . . . . . . . . 65

vi

Figure

CDNONU'IJZ‘LUN

IO.

ll.

l2.

13.

1“.

LIST OF FIGURES

Octahedrally-coordinated paramagnetic ion

The direct process

Energy level diagram for a Kramers system

The Raman process

The local mode process

Cylindrical cavity

Vacuum can cryostat

Temperature dependence of T1 for g“ and gL lines
in KBr, with fits to % = AT + BT9J8(l7O/T)

+ C exp(—575/T) i
Temperature dependence of I1
with fit to % = AT + BT9J8(I70/T)

for g“ line in KBr,

for g,L line in KBr,

Temperature dependence of T1
with fit to % = AT + BT9J8(l7O/T)

Comparison oflthermal paths for the two types
of apparatus

Temperature dependence of T for g” and gL lines

1
in KCl, with fits to % = AT + BT9J8(230/T)
+ c exp(—6lO/T) I

Temperature dependence of I for gL line in KCl,

1
with fit to % = AT + BT9J8(230/T)
Temperature dEpendence of I1 for g” line in KCl,
with fit to % = AT + BT9J8(230/T)
1

vii

Page
17
18
2O
29
39
1:2
51
54

55

61

63

66

67

LIST OF APPENDICES

Appendix Page
I. Data for KBr . . . . . . . . . . . . . . . . . . 72

II. Data for KCl . . . . . . . . . . . . . . . . . . 79
III. Fortran Program TOWPLOT Used in Curve Fitting. . 88
IV. Assignment of Weights in Curve Fitting . . . . . 107

viii

I. INTRODUCTION

History

Electron spin-lattice relaxation phenomena of para-
magnetic impurities in crystalline salts has been a subject
of both experimental and theoretical interest for more than

27 in the 1930's studied relaxation

36

three decades. Gorter
phenomena using susceptibility techniques and Waller,
Van Vleck,32 and Kronigl6 established a theoretical explana-
tion of the results. With the advent of electron paramag-
netic resonance the interest in relaxation phenomena was
renewed, since resonance provided the possibility of meas-
uring relaxation times in a more direct manner. Experi—
mental resolution limited the results to A.2°K and below
until the late 1950's. Advances in experimental tech-
nology, resulting in enhanced sensitivity which allows meas-
urements at higher temperatures, have reopened interest in
Van Vleck's theory. Orbach“ has modified the theory in
order to explain a wide range of experimental results, par-
ticularly in the rare earth ions. Castle and Feldman2

have reported results up to 250°K using a saturation—
inversion technique, and the experimental technology has
reached the point at which measurements above 78°K may be

considered routine.

In order to reduce interaction between paramagnetic
centers, they are introduced into diamagnetic host crystals
in small percentages. Therefore, the paramagnetic centers
are at defect sites. Lattice vibrations, which provide the
mechanism for relaxation, are modified by a defect, so that
one might expect theories formulated for perfect crystals

3’13 was the first

need not apply to real systems. Klemens
to explicitly consider the consequences of this fact upon
the theory of electron spin—lattice relaxation. The work
reported in this thesis was undertaken in the hope of being
able to more conclusively check the validity of the Klemens
theory. We have measured the temperature dependence of I1

for the molecular complex Cr(CN)5NO3_ substitutional in

KBr and KCl, in the temperature range l°K to 1500K.

Summary of Theory

 

In order to understand relaxation time measurements,

38,10

one must understand the theory of lattice vibrations

7’26’29 The overall

and the theory of paramagnetic resonance.
aim is to calculate the characteristic time (relaxation
time) for electron spins that have absorbed energy to re-
linquish the energy to the thermal bath.

The absorption of energy by a paramagnetic electron

is governed by the resonance equation,

hv = E — E2 = gBH, (1.1)

where v is the frequency of the applied r.f. signal, h

is Planck's constant, E is the final energy of the

1

electron, E is the initial energy, g is a parameter for

2

the paramagnetic system called the spectroscopic split-

ting factor, B is the Bohr magneton, and H is the applied

magnetic field. Assume a certain group of spins with g = gl

have their resonance condition met (1.1). If the r.f.

field is turned off, then the spins will be in a non-

equilibrium state. There are several interaction paths

by which the spin system may return to thermal equili-

brium with the lattice. We shall consider two possible paths.
If the coupling of the paramagnetic centers to the lat—

tice is much stronger than the coupling among centers, then

the spin system equilibrates with the lattice in a time

characterized by I termed the spin—lattice relaxation

1’
time. A spin disposes of its excess energy (via L-S) to
the orbital magnetic moment which, in turn, couples the
energy to the lattice heat sink by means of the time varying
electric field laid down due to modulation of interionic
distances by the lattice vibrations. This is the process
which we shall study.

If there exists another set of spins with g2 # g1,
the spins with gl may undergo mutual spin flips with them in a

characteristic time denoted I called the cross relaxation

12’

time. The energy of the spin flip transitions is not pre-

cisely balanced, since gl # g2, but the dipole—dipole inter—

 

action may take up or supply the energy necessary for energy
conservation.

In the theory of spin-lattice relaxation, one considers
the effect of a strain in the immediate vicinity of a para-
magnetic ion on the spin Hamiltonian, and uses this perturba-
tion Hamiltonian to calculate the spin relaxation by the

21’2“ It is also usual

usual perturbation techniques.
to express the thermal strain as a superposition of lattice
waves in order to describe spin—phonon interaction processes.
To lowest order, the component linear in strain gives rise
to the direct, or one—phonon process, while the component
quadratic in strain gives rise to two-phonon or Raman
processes. A Raman process is also obtained by carrying
the linear strain term to second order in perturbation
theory. An excellent account of the calculations involved
is given by R. F. Vieth.33

We do not have enough information available to calcu—

late an exact expression for I in the system under study,

I
but it is still possible to extract the theoretical temper—

ature dependence of I and compare it with our measured

1
values. By expressing the lattice waves of a crystal in
simple classical form, one can obtain several quantum
mechanical operators. These operators have the effect of
creating or destroying a phonon, a process which occurs

when an electron spin absorbs energy from or relinquishes

energy to the lattice. Two systems of ions must be con-

 

sidered:

1) those which have an even number of electrons

(non-Kramers systems), and
2) those with an odd number of electrons (Kramers
systems).

Waller,36 Kronig,l6 and Van Vleck32 originally predicted
that two types of relaxation can occur for each system:
one which involves one phonon (the direct process) and one
which involves two phonons (the Raman process). The direct
process occurs at lower temperatures while the Raman process
is dominant at higher temperatures. For non—Kramers systems
we find %1 a T for the direct process and $1 a T7J6, where
Jn is a transport integral of order n, for the Raman process.
For Kramers systems the situation is a bit more complicated.
If zero order wave functions are used in the computation of
the necessary matrix elements, the direct process will
vanish and the Raman process will have a T9J8 dependence.
An applied magnetic field may, however, admix excited states
with the zero order wave functions to produce a direct pro—
cess that again is proportional to T and two Raman processes
proportional to T7J6 and T9J8. In addition, if there are
energy levels that lie near enough to the ground state,
the Raman process may go as TSJH. For either type of
system, Orbach2u has shown that if there is an accessible
electron energy level with energy A above the ground state,

and if A < E where E is the Debye energy of the lattice,

D’ D

then there may be a resonant two phonon process for which

l « exp(-A/kT).
TI
The foregoing theories were formulated for a perfect

lattice, whereas the paramagnetic ion is at a defect site.
Since the lattice vibrations are modified by the defect,
one is not justified in using the usual relation between
strain and amplitude of a lattice wave, which holds in a
perfect crystal. In view of the phenomenological nature of
the theory, one would be very hard put to isolate the error
introduced from the consequences of an incorrect choice

of the strain dependence of the spin Hamiltonian, except
when the temperature dependence of the spin—lattice relaxa—
tion is modified in some characteristic and unusual manner.

3

Klemens has shown that relaxation rates (i.e., % )
1

11

proportional to T3J and T J for non—Kramers systems,

2 10

and TSJu and T13J for Kramers systems, arise from impur-

l2
ities which have associated with them vibrational frequencies
lying within the Debye spectrum of the host lattice.

Montroll and Potts22 have pointed out that in the case
of a substitutional impurity which is lighter than the parent
atom, the character of the lattice vibrations is particu—
larly strongly modified, giving rise to a new lattice mode
localized near the impurity, and having a frequency v0
which lies above the acoustic band. Klemensl3 has predicted
%l m exp(—th/kT) for this case. Relaxation data2 on the
oxygen—divacency (Ei) center in synthetic crystalline quartz

have been interpreted in terms of this theory, but the re-

 

sults are not conclusive.

We surmised that a better test of the theory could be
realized from measurement of relaxation in a molecular com-
plex whose localized frequencies v0 were well defined and
amenable to measurement by infrared spectroscopy. An ideal
candidate seemed to be Cr(CN)5NO3- substitutional in alkali
halides. The acoustic phonon spectrum of the alkali halides
has been studied,1 so that lattice parameters involved in
the "normal" direct and Raman relaxation of paramagnetic
impurities are known. Hence we anticipated that any
anomalous results should be due to interactions localized

in the Cr(CN)5NO3_ complex.

II. THEORY

Theoretical Background

 

The phenomenological theory of electron spin—lattice
relaxation in crystalline solids has been considered by

13,21,2u

many authors. According to current hypotheses,

the spin—phonon interaction plays the central role in para—

magnetic relaxation phenomena.32’16’21

The process occurs
through modulation of the crystalline electric field by
the lattice vibrations. This time—varying electric field
interacts with the electron's orbital angular momentum,
which is coupled to the spin via L'S.

The model we shall initially use is illustrated in
Figure l, which shows a paramagnetic ion surrounded by an
octahedron of nearest neighbors that produce an electrostat—
ic field at the ion site. In the unperturbed system these
neighbors are assumed stationary. If we allow the neighbors
to vibrate (since they are part of the lattice), the crys-
talline field will be modulated, will perturb the orbital
motion of the paramagnetic electrons, and will induce spin
transitions by means of spin-orbit interaction. We shall
neglect all effects of spin—spin interaction except for
energy level broadening. That is, we assume that the spins

relax to equilibrium only by giving up energy to the lat—

tice, and not by cross relaxation effects.

,O

 

\ O
\\ .

[/‘s
' o I .
Paramagnetlc I \.,’6R
A

Ion

O— — — — .,
R
/ 2 tth nearest
./ neighbor

as
0

Figure l. Octahedrally coordinated paramagnetic
ion in the crystalline electric

field of its nearest neighbors.

10

Under this assumption we can write the total Hamil-

tonian in the form21

f? ={FL +§€O + V + 288% + ALE + BER. (2.1)

In this equation, 8 is the Bohr magneton, A is the spin—
orbit coupling parameter, S and L are the spin and orbital
angular momenta of the paramagnetic ion, H is the external
dc magnetic field, V is the energy of the ion due to the
crystalline electric field, fig is the energy of the free
ion, and‘fll represents the energy stored in the crystal
lattice due to lattice vibrations. The lattice Hamiltonian

can be writtenlu

{IL = the (a *a + A), (2.2)

where apl, ap are the phonon creation and annihilation

Operators. They have the properties that

:3
V
II

1/
(np+l)2|...np+l...>

SD)
:3
II

(np)%|...np—l...>. (2.3)

The p index represents the phonon mode and branch number,
with mode-branch frequency mp. If we designate the equili—
brium position of an atom in the lattice by I and the dis—

placement of this atom from equilibrium by urq (a = x,y, or z),

then ura can be expanded in normal lattice modes as1

1

-24'1’2 4/2 + ++
ura — [M—] g(wp) cpfl<ap+ap )cos(kor+Ap), (2.A)

11

where M is the crystal mass, ¢pd is the qth component of
the unit polarization vector for the mode-branch p, kp
is the propagation vector for mode—branch p, and Ap is
an arbitrary phase factor.

In what follows, it shall be assumed for simplicity
that the lattice is dispersionless and isotropic, with the
result that all phonons have the same velocity, v, and

thus we can describe the density of states by the Debye

formulall
l21TVv2 3N 1/3
v3 if vs HNV v ( )
00)) = 2.5
3N 1/3
0 if v> ENV v

Use will also be made of the fact that the average number
of phonons in mode p when the crystal is in thermal equili-
brium at temperature T is given by

'hw /kT
N = (e p —l)‘1 (2.6)

where m = 2nv .
p p
In order to show the interaction between spin and lat-

tice explicitly, we expand the crystal field potential, V,

in a power series in the normal displacements (Qf) of the

31

nearest neighbor ions:
““2le +tlfi—QQ'+... (2.7)
f BQf ff,3Q BQf, f f' ’

where the Qf's can be related to the ordinary displacements

of the neighbors (6R ) by

20

12

Qf = Eantdéth (2.8)

and the index A runs over all nearest neighbors. Now ura
is the displacement of any atom, and therefore it includes
nga as a special case. Hence 6R£a can be expanded by

means of (2.4). In doing this, we make the approximation
that the phonon wavelength is considerably greater than

the dimensions of the cluster of nearest neighbors, so that
kp°g<<l (assuming, for simplicity that the nucleus of the
paramagnetic ion is at the origin). Because the only near-
est neighbor displacements effective in modulating the

crystalline field are those relative to the spin nucleus,

equation (2.“) reduces to

1/
_ 2h 2 % I + .
éth - [M 2] 2(wp) cpg<ap+ap )Kp RQSlnAk, (2.9)
V p
where Kp is the unit vector in the kp direction, and we have

used |Ep| = wp/v. Substituting (2.8) and (2.9) into (2.7),

we find
' t
V = V0 + l VfApr + l fo Aprf qr r + .... (2.10)
fp p ff'pq p q
where
2
f av ff' 1 a v I
V=——;V =.———————;I‘=a+a,
an Zanan. p p p
;
fp 3232 2 l K A
MV2 p20. ZERO, p [Q]

 

 

13

In the foregoing we should distinguish between two diff
ferent sets of normal modes over which summations have

been made. On the one hand, in (2.7) we summed over the
relatively small number of modes (Qf) of the nearest neigh-
bor cluster, and on the other hand in (2.4) we summed over
the relatively large number (3N) of traveling wave modes

of the lattice. Utilizing (2.1), (2.2), and (2.10), we

find for the total Hamiltonian

_ A In I .
j??- Zhwp(ap ap+2) +Ԥ% + VO + 288 H

p
+ xE-s + sfcfi + zvafpr
fp p
t t
+ E vff Aprf qr r . (2.12)
ff'pq p q
We now dIVIde if) Into filattice’ fispin and Winter—

. . . I ;
action' Examining the terms, we fInd that Efihwp(ap ap+2)
has only lattice coordinates; §Q+VO+2BS'H+AL-S+BL0H
involves only paramagnetic electron coordinates;

I I
2 vapr + E , vff Aprf qr I involves mixed coord-
fp p ff pq p q
inates. The term in electron coordinates is precisely
the one that gives rise to the spin Hamiltonian which des-
cribes the energy levels that are observed in paramagnetic
resonance. In relaxation experiments, we are interested in

phonon-induced transitions between these pairs of spin

levels. Hence it seems reasonable to set

1A

_ + /
filattice — gnopmp ape.) (2.13)

—> ->
#spin ’fio + V0 + 2B§‘H + Af‘g + AE‘H (2.1”)
and to consider

_ f fp ff' fp f'q
d7. . — l V A I + 2 v A A I I (2.15)
Interaction fp p ff'pq p q

as inducing energy conserving exchanges of quanta between

and fl: Detailed calculation would involve

flspin attice'

diagonalizing {Epi to some appropriate order (the second

n
order would suffice —- this would produce the spin Hamil—
tonian), finding its energy levels En and corresponding

state vectors IWn>, and computing the appropriate matrix

elements cd‘ §fi_ between simultaneous eigenstates

nteraction

of {Epin and {Lattice' The rates for the various relaxa—
tion processes are then calculated from time—dependent
perturbation theory.

Let us rewrite (2.15) as

‘fldnteraction V1€ + V2€€ (2'16)

where 2 and E‘ are operators for the average strain due to
the lattice vibrations, and are proportional to a and 2+,
depending on whether a phonon is annihilated or created.
Our ignorance of the matrix elements of the Vf for the

system under study motivates this simplification. In spite

of this ignorance, a number of interesting conclusions about

 

15

the various relaxation processes can be drawn.

In evaluating matrix elements of the electron opera-
tors V1 and V2 of (2.16), we must consider two types of
systems:

1) those which have an even number of electrons

(non—Kramers systems), and

2). those with an odd number of electrons (Kramers

systems).

15 states that each energy level will

Kramers' theorem
always be at least twofold degenerate in the presence of
purely electric fields, provided the number of electrons

37 has shown that this degeneracy is related

is odd. Wigner
to the invariance of the system under time reversal, and
that a pair of Kramers degenerate states are time conju—
gates of one another. Kramers' theorem is important to
us because it predicts that certain matrix elements of
(2.16) vanish if no magnetic field is applied. The pre-
sence of a magnetic field will lift the Kramers degeneracy,
and give nonzero matrix elements.

The following sections of this chapter summarize the
theoretical results for various spin—lattice relaxation
mechanisms. Most of these results have been derived else—

214.33

where.

16

Direct Process

 

Figure 2 depicts a spin initially in state |b> relax-
ing to state |a> with the consequent emission of one phonon.
This method of relaxation is called the direct process.
ES is the energy difference between the spin levels and
therefore is also equal to hwq, the energy of the emitted
phonon. The direct process is usually dominant at low
temperatures. If first order time dependent perturbation
theory is applied to the linear strain term of the orbit-

1attice Hamiltonian (2.16), then for ES<<kT we have for

the direct process rate

3(ES)2kT 2
D = ;§———§—— |<a|Vl|b>| , (2.17)
npv

HIP

where p = M/V, the density of the host crystal.

If |a> and |b> are time conjugate states, then the
matrix element in (2.17) will be zero by Kramers' theorem.
Themagnetic dipole—magnetic field interactions can admix
excited state Kramers doublets into the wave functions
|a> and |b> to produce a nonzero result. To first order,
the ground state wave functions in the presence of a mag—

netic field are

 

 

.+
|b>' = |b> + 2' <nlu-HLb> |n> (2.18)
n Eb-En
+
|a>' = la) + it (nlu'fila) In) (2.19)
Ea‘En

where a is the magnetic dipole moment operator and H is the

 

1?

 

 

 

 

Figure 2. Schematic diagram for the direct process.
A single transition is made from |b>
to |a> and a phonon of energy ES is

emitted.

 

l8

 

 

 

 

 

 

 

 

 

 

 

 

|b>

|9>
If)

 

 

 

 

 

/’
,.
IttV) T\\
\\
//.
/
Iii-U) A ’\
\
\
\Q
,.r
Iii-l) A <:‘
2:, Au A,
”
l v: 442.. ""
lie 0 \\\

|b>

 

H=O Hso

Figure 3. A hypothetical energy level diagram for

1°)

a Kramers ion in a uniaxial crystal field

in the presence of spin—orbit coupling;

r, t, u, and v are odd integers.

 

l9
magnetic field. The only terms that contribute are those
which lie near to the ground state (Figure 3). If |c>
and |d> are the nearest lying time conjugate states, then
by applying time reversal arguments, as well as selection
rules on lAml, we are led to the result

2 2

 

1 12ES H kT 2
?_ = —H 5 2 |<d|p|b><a|Vl|d>| , (2.20)
D h va Acd
where
Acd = EC—Eb = Ea—Ed (2.21)

and u is the magnetic moment in the direction of H.

Raman Process

 

In the Raman process lattice phonons are scattered
inelastically from the spin system. This results in the
creation and destruction of two phonons whose difference
in energy equals the splitting of the participating spin
states. As outlined in Figure A the process we envisage
will have an initial state vector specified by a spin being
in the state |b> and the phonons (p) and (q) having occu-
pation numbers Np and Nq, respectively. The final state
of the system will find the spin in state [a>, a phonon
created of type (q), and a phonon of type (p) destroyed.
There are several mathematical approaches to such a process.

We shall consider two different methods which follow two

 

 

 

 

 

 

 

 

(a) “if.” °< |<°|V2lb> 2
I ld>
\
, \|c>
I \'
’ \
——9/<——lb> \ |b>
wp Wi— \ '1"qu

 

1a) __>‘<__Io>

(b) 42,- <>< |<o|V.|c><c|V.|b>l2

Figure A. Schematic diagrams for the Raman process.

In both cases, wq-mp = Eb-Ea.

 

21

basically dissimilar physical approaches.

Consider a process in which the quadratic strain
term of the interaction Hamiltonian (2.16) is used in first
order perturbation theory (Figure A(a)). The resulting

relaxation rate for non—Kramers systems is

 

7
l _ 9k 7 [a] 2
——- T J — |<a|V |b>| (2.22)
TR “U302Vloh7 6 T 2
where
. x zneZ
Jn(X) 3 f0 —Z-—-——2-dZ (2.23)
(e —l)

and 9 is the Debye temperature of the lattice.

Here, as in the direct process, the matrix element of
V2 will vanish if |a> and |b> are time-conjugate states,
i.e., for a Kramers system. In that case, higher lying

electron levels can be admixed by the 3-H interaction to

 

give

1 9H2k7 7 e 2

—— = T J — |<d|u|b><a|V |d>| (2.24)

T 3 2 10 7 2 6 T 2

R H p V ‘h A

cd
The Jn in (2.22) and (2.2M) are called transport integrals
23

and have been tabulated. They are essentially constant
for x>xn, where X6 = 20 and X8 = 25, and have the additional

property that

n+1 6 2

T Jn[T] + aT for T>8/2 (2.25)

This follows from the phonon population factor (2.6).

 

 

 

 

 

 

 

 

22
SinCe

e = l + x + ... if x<<l, (2.26)

we have

__L__:E_T_
ehw/kT_l hm

if kT>>hw . (2.27)

Hence at temperatures high compared to the phonon energies,
the phonon occupation numbers are proportional to T. But
the transition rates are proportional to phonon occupation
numbers. Therefore ggy two—phonon relaxation process has

the asymptotic form %— a T2 for temperatures comparable to

the maximum energy oflthe phonons involved.

Consider a relaxation process which uses linear strain
terms in second order perturbation theory. Here the elec-
tron spin in state |b> is considered to undergo virtual
(and therefore not energy—conserving) transitions to inter-
mediate states rather than making transition directly to

state |a> (Figure A(b)). For a non—Kramers system this pro-

cess is describable by

 

 

2
7 .<a|V |i><i|V '|b>
%— = 39g 10 7T7J6[%] I) * l A 1 (2.28)
R An p v n i
where A1 = Ei-Eb 2 Ei-Ea. (2.29)

For a system with an odd number of electrons, the time

conjugate nature of the Kramers doublets leads to the

31

"Van Vleck cancellation". The effect of the cancellation

is to raise the power of T in the relaxation rate:

23

l
c><C|VlIb> -2
2 a (2.30)
cd

<a|Vl|

 

r-Ill—L‘

 

9
_ 9k 9 [a]
_ T J _
:R n3p2vlo‘h7 8 T

 

A

where we have considered as intermediate states only the
Kramers doublet lying nearest to the ground doublet and have
explicitly used the time—conjugate nature of |c> and |d>.

We note that in this process it was not necessary
to invoke Zeeman admixture of excited states into the ground
state in order to achieve a nonvanishing result. The
reason is that here we have matrix elements between com—

ponents of different Kramers doublets rather than between

 

components of the same doublet.

25

Orbach and Blume have considered the Van Vleck can—
cellation term further. In the derivation of the T9J8

process it is assumed that the states |c> and |d> are split
apart from the ground state by a large energy, but for some

Kramers systems excited states lie close to the ground

doublet. This leads to

<a|V1|c><c|Vi|b> 2
. 2 (2.31)
A

cd

 

 

5
1 _ 9k 5 [a]
__._ T J _
TR n3p2vlqh7 A T

 

A rough order of magnitude criterion for this process to
dominate in the Raman region is given by Orbach and Blume

as

A[—2—] > kT, (2.32)

in addition to A<<k6, where A is the appropriate crystal

field splitting and A is the spin—orbit coupling constant.

2A

Orbach Process

 

Finn, Orbach and WolfLl first proposed a two—phonon
process which is similar to the Raman process depicted in
Figure A(b), except that the intermediate electron state
|c> lies within the lattice phonon spectrum. That is,

Ec<k8. Now the energy denominator of (2.28) displays a
resonance. In reality, the denominator does not vanish
because of lifetime broadening of the electron energy levels,
but the relaxation rate is enhanced considerably over the
normal Raman rate. When the resonance denominator is taken
into account the predicted relaxation rate for both Kramers

and non—Kramers systems contains an exponential term

' 2
<a|V |c><c|V |b>|
l A l e-A/T (2.33)

cd

.1. ..l
Orb A

 

H

where A(<k8) is the splitting of the excited state from the
ground state. This process is referred to as the Orbach

process.

Driven-Mode Process

 

In the previous discussion we have neglected the effect
of the paramagnetic impurity on the lattice vibrations of
the host crystal. That is, we have assumed that the strain
e at point I due to a lattice wave of wave vector 5 is given
by (2.“), even though that expression holds only in a

perfect crystal. In order to calculate I1 we need to know

25

the strain in the immediate vicinity of the spin site.

When the spin is associated with an impurity or defect, this
is precisely where expression (2.A) fails. We must, therefore
look more closely at the strain at a defect site.

Every defect has associated with it one or more charac-
teristic vibrational frequencies mi, which have to be com-
pared to the frequency describing the normal interatomic
bond of order wD(=k0/h), the Debye frequency. Roughly speak-
ing, we can expect two classes of behavior: (a) cases when
mi exceeds “D sufficiently, so that the vibrations at that
frequency are not propagated through the crystal, but are
localized at the defectg22’l3 (b) cases when mi falls be-
low wD so that localized modes in the usual sense are not
formed, but the defect system undergoes forced oscillations3
under the influence of lattice waves. We consider this lat-
ter case first, referring to it as the driven-mode process.

3

Klemens has shown that for an impurity-associated

frequency which is less than w the Raman relaxation rate

D,

(2.30) for a Kramers system is modified to the form

8 .
l 9 e 5 e 8' E. _g
s. °‘ T Jet-1+ CT EAR-AH} + [.1 Mil]. (2.3M

CD

I

where k0,L = hwi is the energy associated with the defect

mode. For a non—Kramers system the analogous term is

o 8 o
l 7 6 , 3 0 0 T 01
:t—DM on T J45] + c T [{J2[T-]—J2[T—l]} + [8i] J10[T—]] . (2.35)

26

LoCal Mode Process

 

We now consider in some detail the case in which fre-
quencies associated with a defect lie above the acoustic
band of the host lattice. In such a case the energy of the
modes is concentrated in the immediate vicinity of the de-
fect, so that one might eXpect it to modify very substan—
tially the spin-lattice relaxation of a paramagnetic impurity
associated with this defect site.

Since the energy of a local mode greatly exceeds the spin
energy, it is not possible to have a direct energy conserv—
ing spin-phonon interaction involving a local mode. The only
conceivable process is one in which a spin-flip is accom—
panied by the absorption and subsequent re—emission of a
localized phonon. If the local mode frequency were sharp,
this process would also violate energy conservation. The
local mode frequency is, however, broadened by anharmonic
interactions with the phonons of the lattice continuum.

The anharmonic effects are expected to be strong, even at
T<<0, because most of the energy of a localized mode is
concentrated near the defect, making the amplitude of the
oscillation large. Even at low temperatures a localized
phonon can split into two lattice phonons. Klemensl2 has
shown that this process broadens the local mode frequency
by about 1%. The broadening may therefore be larger than
the spin energy (20.3 cm—1 in our experiments), so that

energy conservation is relaxed sufficiently to allow this

 

27

process to occur.

To take account quantitatively of how the anharmonic
broadening and the relaxation of the local mode permit the
spin—lattice interaction with absorption and emission of a
local phonon, we consider their combined effect to the next
order of perturbation theory. A local phonon and the spin
,combine; in the intermediate state we have the same local
phonon, but the spin is inverted, and in the final state the
spin stays inverted, and the local phonon has split by an-
harmonic interaction into two phonons of the Debye continuum.
Since there is only one intermediate state, there is a
one—to-one correspondence between the overall second order
matrix element and the matrix element for the anharmonic
interaction between a local phonon and two lattice—wave
phonons. Thus the contribution of local phonons to the
spin—lattice relaxation is related in magnitude to the
anharmonic relaxation time of the local mode.

Let us examine this process in more detail. Consider a
two stage process going from state |i> to state |f> via an
intermediate state |j>. Such a process is equivalent to
going from |i> to |f> with an effective perturbation

Hamiltonian

' _ <lelj><iiUli>
i¥eff — 2 31-3. 3 (2.36)

where U and W are the perturbation Hamiltonians linking

|i> and |j>, and |j> and |f>, respectively.

 
 

 

 

28

As initial state we take the local mode excited with one
phonon, the ion in the higher-lying spin state, and the lat-
tice phonons in equilibrium. (See Figure 5 for more explicit
description of the state vectors, including phonon occupation
numbers of the relevant modes.) As intermediate state |j>
we need to consider only one state, i.e., with one phonon
in the local mode and the spin inverted. Hence the sum
in (2.36) is reduced to a single term and its denominator is
merely the spin energy ES: gBH. The interaction represented
by <j|U|i> is a Raman process involving only localized
phonons, so that we may set <j|U|i> = <a|V2|b>eoeg where
V2 is the same as in (2.16) and 60,8; each represent matrix
elements of the strain due to one localized phonon. The
final state has the same spin configuration as the inter-
mediate state, but the local phonon is removed and two trav-
eling lattice modes are each excited by an additional phonon.
(Figure 5).

The interaction Hamiltonian W connecting states |j>
and |f> arises from the cubic anharmonicities (involving
three phonons at a time), and is the same as was used by
Klemensl2 to describe the relaxation to equilibrium of excess

energy in the local mode. We can write <f|W|j> as

1
/2
9

€[N0(N1+l)(N2+l)1% or ££(No+1>N1N21 (2.37)

depending on whether a local phonon is annihilated or cre—

ated, where N0, N1, and N2 are the occupation numbers of the

 

29

 

 

 

 

No-I do N0 N0 001—00 No
1..

Nl+l a, N| N. N.
I N

N2+| 02 N2 N2 2

"'2 "'2: 1%: s_ +-'2-

 

 

 

 

Figure 5. Schematic diagram of the local mode
process, along with the appropriate
matrix element. N0, N1 and N2 are the

number of phonons in the local mode

and traveling modes.

 

 

 

30

local mode and traveling modes. The perturbation Hamiltonian
(2.37) can be used in time dependent perturbation theory to
Show12 that the relaxation rate of the local mode is inverse—

ly proportional to Zf|<fIW|j>|2, i.e.,

wilt—J

o a 2(N1 + N2 + l), (2.38)
f

where the sum is over all final states which satisfy the
energy condition wo = ml + m2. For example, consider the
case (h/k)wo=600bK, with the Debye temperature of the host
lattice (2 200°K). The only lattice phonons which can part—
icipate in relaxing the local mode are those out in the
"tail" of the phonon distribution, i.e., those for which
(h/k)m>0D. When this restriction is taken into account,

(2.38) may be approximated by

r-tll—’

590/2T e‘90/2T + 1 . (2.39)

0:

0

+

As a further simplification, we shall neglect the exponen—
tials in (2.39) as they are small compared to unity, since in

our measurements T<l50°K. Then for the temperature range
13

of interest, Klemens obtains
2
%- = 32%— (2.u0)
° Mv

which is independent of temperature. At this point it is
clear that the Debye approximation (2.5) for the density of
phonon states is not valid for the lattice phonons involved

in the local mode process, and that for a serious quanti-

31

tative calculation we should need a more realistic model.
We can, however, draw some significant qualitative conclu-
sions using a "smeared" Debye spectrum, i.e., one for which
there are a small number of available states above th.

We View the overall spin-lattice relaxation via local-
ized modes as a two-step process in which the spin relaxes
to the local mode, and then the local mode gives up its energy
to the lattice. There is a one—to—one correspondence between
every final state in the sum (2.38) and the final state in
the overall process, since the spin energy is negligible when
compared with the phonon energies. The effective matrix
element (2.36) is the matrix element (2.38) multiplied by
[<j|U|i>/ES]. In order to calculate the rate of change of
o, the fraction of spins in state |b>, we must consider both
the process depicted in Figure 5 and its inverse. Then

do

at °‘ 0N0<N1+ l><N2 + 1) — N1N2(No + 1)(1 — 0). (2.111)

This expression vanishes at equilibrium. If 0 deviates from
equilibrium by 60, the term 60 is proportional to
N0(N1+ N2 + l) + N1N2. (2.)42)

The term in parentheses is approximately equal to unity by

the same argument as given for (2.39). Also,

N1N2 : e—hun/kT e-hwz/kT = e-‘h(w1+w1)/kT

= e—hwo/kT 2 N0 . (2.u3)

 

32
Hence (2.“2) becomes approximately 2N0. Thus the relaxation
rate for spins via the overall process involving local phonons

is of the form

 

 

 

 

 

 

 

2 ' 2
% _ 2 PM; [605%] <a|V2|b> 2 No, (2.8“)
LM Mv ES
or
% : §_§ <alV2|b> 2 e-GO/T , (2.“5)
LM ES

where the coefficient B is temperature independent, and we
have used the fact that eo/T is sufficiently great in our

measurements to permit setting No 2 e_e°/T.

As noted pre—
viously, for a Kramers system the electron matrix element
in (2.A5) vanishes for zero—order states |a>, |b>. If we
allow admixture of higher lying states by the magnetic

dipole—magnetic field interaction, the result for a Kramers

system is

 

HIP

<d|u|b><a|V2|d> 2 e'GO/T (2.46)

 

 

Anisotropy of Relaxation Time

 

We now discuss possible dependence of Il on the orienta—
tion of the applied magnetic field H, for a Kramers ion in
a crystal field of uniaxial symmetry. The relevant parts

of the theoretical relaxation rates for the direct process,

 

33

the various Raman processes, the Orbach process, the driven
mode process, and the local mode process are summarized

below:

% a |<d|fi.fi|b><a|vl|d>|2 (Acd)‘2 T <2.u7)
D
1 a +0 2 -2 7
?R |<d|u fi|b><a|v2|d>| (Acd) T J6 (2.u8)
1 ' 2 -“
?h a |<a|Vl|c><c|V1|b>| (Acd) T9J8 (2.”9)
1 ... ' 2 _u 5.
?fi |<a|Vl|c><c|Vl|b>| (Acd) T J“ (2.50)
1 _ _
% a |<a|Vl|c><c|Vl|b>|2 (Acd) A e A/T (2.51)
Orb
' _ .
% a |<a|V1|c><c|Vl|b>l2 (Acd) u {T dep } (2#52)
DM
% . |<d|fi°fi|b><a|V2|b>|2 (Acd>“2 e'9°/T (2.53)
M

The only anisotropy considered here is that due to electron
matrix elements. That is, all matrix elements of phonon
operators are assumed to be isotropic, which is equivalent
to assuming the velocity of sound is independent of direc-
tion in the lattice.

Anisotropy of T in Kramers systems may be attributed

l
to the admixture interaction fi'fi invoked in (2.18) and (2.19)
to give nonvanishing matrix elements of the orbit—lattice

Hamiltonian (2.16). To facilitate the quantitative deriva-

tion of this result we shall employ the alternative notation

 

3A

in Figure 3. That is, the states formerly referred to as
|a>, |b>, |c>, and |d> shall now be called I-%r>, |+gr>,
|—%t> and |+gt>, respectively. This notation will empha-
size the fact that states such as |ikr> are time conjugates
of one another and are composed of states of half—integral
quantum numbers. As stated earlier, this leads to
<—5r|Vl|%r> = O = <—5r|V2|%r>. The 3-H interaction is used
to admix higher-lying Kramers doublets such as |i%t> into the
ground doublet. The resulting states, expressed in the new

notation, are

 

 

|—%r>' = |—%r> + H° _l Ii%t> (2.5“)
t
+

|+%r>' = |+%r> + H <:gtiu +gr> |i%t> (2-55)
t

The magnetic dipole moment operator 3 is proportional
to an angular momentum operator, so that if <gt|fi|gr> # 0,
then <%t|fi|—%r> = O. This follows because r is an odd in—
teger and K can at most connect states differing in m value

by :1. Hence if

 

 

_; + _;
|-%r>' = |—%r> — fi~ < 22L3i2r> |—%t>, (2.56)
t
then +
1/ '/
|+%r>' = |+%r> — H- <+2ZIulflr> |+%t> . (2.57)

We take the crystallographic axis as the z—direction, and

decompose K into uz and HL' Consider the matrix element

   

 

35

A

<—%t|p|—%r> = <-gt|uZ|—%r> eZ + <—5t|uL|—%r> éL , (2.58)

where éZ and 6L are unit vectors parallel and perpendicular
to the z axis, respectively. By use of the following iden—

tities,

l%r> (2.59)

<-5t|uZI-%r> = -<%t|u

and

«mm—1m = <1/.t|u,|1/.r>, (2.60)

which follow from the properties of angular momentum opera-

tors,l8 the admixed wavefunctions (2.56) and (2.57) become

|—%r>' |-%r> + % [<%t|uZ|%r>cos¢-<%tlul|%r>Sin¢]I-%t> (2-61)
t

|+%r>' |+gr> — %£[<%t|uz %r>cos¢+<%t|uL|%r>sin¢]l%t> (2.62)
where o is the angle the external field makes with the
z axis.

From the properties of angular momentum operators,
<%t|uZ|%r> and <%t|uL|%r> cannot both be different from
zero. Consider first the case when <%t|uzl%r> # 0. Then

(2.62) and (2.63) reduce to

|—%r>' = |-gr> + §%9§9<gt|uz|gr> |—%t> (2.63)
t
|+%r>' = |+%r> — E29§9<%tluyl%r> Igt) - (2'6“)

At

36

The matrix element of the appropriate part of the linear
strain term of the orbit-lattice interaction (2.16) between

the new states equals, to first order in H,

<—%r|Vl|%r>' = —————<%t|u %r>{<—%t|V %r>—<-EPIV1|%t>}. (2.65)

1|

Since the terms in the curly brackets must each be invariant

under time reversal, we have

; y x y ;
<—2r|V1|%t> = (-l)2r+2t<%r|Vl|-%t> = (—1)2t+2r<—gt|vl|%r>. (2.66)

But for Kramers systems r is odd and kt = %r under the assump—

tion <%t|uzl%r> # 0. Hence
< 1 1 v _ 2HCOS¢ 1 1 1 1
_2rlvllér> _ ——K———<ét|uZ|6r><—firlvl|ét>. (2.67)
t

The other possibility, <gt|hL|gr> g o, by exactly the same

sort of reasoning, leads to

-2Hsin¢

At <gt|HL|gr><—%r|V ét>. (2.68)

<—%r|Vl|%r>' = l|1

We note that in practice there exist excited Kramers doub-
lets which have only matrix elements of “z with the ground
doublet |i%r>, and other excited Kramers doublets which have
only matrix elements of uL with the ground doublet. These
matrix elements may be considerably different in magnitude,
and, in addition, the splitting A of the two doublets may

be different. Therefore, from (2.47), (2.67) and (2.68),

we see that the direct process relaxation time may be strong-

ly dependent on the orientation of the external magnetic

37

field H.

Such anisotropy is not confined to the direct process,
for the angle dependence arose from the admixture of excited
Kramers doublets into the ground state by the magnetic field.
Hence this anisotropy is to be expected for any of the
processes in which the admixture has been invoked. In par—

7

ticular, we mention the T Raman process for Kramers systems,
and the local—mode process, in addition to the direct process.

A second important feature of the anisotropy discussed
above is that the form of the angle dependence is determined
by matrix elements of the admixture interaction 3-H, and is
independent of the actual type of relaxation process under
consideration. For example, if a Kramers system displays
both a direct process and a local—mode process, then the
ratio TlH/Tli is expected to be the same for both processes,
where T1“ and TlL are the relaxation times when HIIZ and
H1.Z, respectively.

The processes described by (2.“9), (2.50), (2.51),

and (2.52) contain matrix elements of the form

—A

'
|<a|Vl|c><c|Vl|b>|2(Acd) (2.69)

Such terms will be anisotropic only if the splitting A
between the ground doublet and excited doublet is itself

a function of magnetic field orientation.

III. EXPERIMENTAL APPARATUS AND TECHNIQUES

General Description

 

Most of the data to be reported herein were obtained
with the spin echo apparatus and right circular cylinder
microwave cavity configuration described by Vieth.3“

Figure 6 depicts schematically the cylindrical microwave
cavity and associated apparatus for maintaining and meas-
uring temperature. The bifilar manganin heater coil was
used to regulate temperatures between A.2°K and 80°K (with
liquid helium below the cavity), or above 78°K (with liquid
nitrogen below the cavity). The samples were mounted at
the center of the cavity in a styrofoam block. A thermo-
couple made from gold 0.02% iron and chromel passed through
the center of the bottom of the cavity and was secured with
Apiezon N grease inside a small hole drilled through the
sample. The other end of each thermocouple wire was spot—
welded to #36 copper wire, both junctions being far enough
below the cavity to be in contact with the cryogenic liquid
throughout a set of measurements. The thermocouple emf

was monitored with a Leeds—Northrop K-3 potentiometer.

5 against

The temperature was obtained from a calibration,
a platinum resistance thermometer, of sample thermocouples

from each end of the spool of gold .02% iron wire. As a

38

 

39

 

copper wires to

. F7 4’ thermocouple

wave guide

 

 

 

heater

 

A

 

 

 

t“““‘

 

 

 

 

 

 

 

 

cylindrical cavity

 

 

sampm
thermocouple
junction

 

 

 

 

 

liquid helium or
nnrogen

 

 

 

 

Figure 6. Cylindrical Cavity

 

 

H0

further check, we calibrated our particular thermocouple

in situ by replacing the sample with a Honeywell germanium
resistance thermometer. For T > 10°K the two calibrations
are in good agreement, and for T < 10°K the platinum re-
sistance thermometer has very poor temperature resolution
compared to the germanium resistor, so we used the calibra—
tion based on the latter. Some of the early measurements
above 78°K were made with a copper—constantan thermocouple,
and results obtained were in good agreement with the sub—
sequent data. Temperatures below A.2°K were obtained by
raising the liquid helium level above the cavity (for better
thermal contact between helium and sample), and pumping on
the helium with a Kinney vacuum pump at rates up to

230 cu.ft./min. Temperatures down to l.l°K (as determined

by the helium vapor pressure) were obtained in this manner.

Spin Echo Techniques

 

For temperatures above 200K we used the conventional

3A

(w/2,w/2,n) pulse sequence described by Vieth for produc—

936

tion of electron spin echoes. At lower temperatures,

where r > 100 msec, we found that cross relaxation effects

1
dominated r1 effects to such an extent that this pulse
sequence was not usable. Instead, we employed a variation

of the "picket" technique23 in which the initial n/2 pulse

is replaced by a string of pulses. The basic idea of the

 

 

 

 

 

 

 

 

Al

picket technique is that the total time during which power

is applied to the system is long, so that even if some energy
from one pulse is lost by cross relaxation the resonance
transition will be pumped many times. The result is that

the cross—relaxation transitions are well saturated, so that
no more energy is lost by such mechanisms, and the system
returns to equilibrium in a time characteristic of the spin-
lattice interaction rather than cross relaxation inter—

actions.

Vacuum Can Cryostat

 

As will be discussed more thoroughly later, some of our
low temperature data seemed to imply that the thermocouple
junction at the sample in Figure 6 was being cooled by con—
duction to the helium bath, so that the temperature of the
sample was greater than that of the thermocouple junction.

A vacuum can cryostat (Figure 7) was constructed to test
these ideas. In this apparatus the temperature sensor,
a germanium resistance thermometer, was embedded into the

wall of a brass TE rectangular microwave cavity around

101
which a six turn bifilar manganin heater coil was wound.
The sample was attached to the inner wall of the cavity

with Apiezon N grease and the remainder of the cavity was

filled with styrofoam to hold the sample in place.

  

 
 

Figure 7.

H2

wave guide

exchange go 3 tube

lid of can

 

 

 

w}

Keg
[—

ihermometer well

 

 

 

 

 

 

 

.heater

Vacuum can cryostat (left), showing the heater
leads wrapped around a brass rod in thermal con-
tact with lid of can. The brass can, normally
soft soldered to the lid has been removed.

At right is a cutaway View of the rectangular
cavity, showing placement of sample with re-

spect to the germanium resistance thermometer.

“3

Operation of the apparatus is as follows:

1) Evacuate the can to a pressure of about 10‘5 mm Hg.

2) Precool to liquid nitrogen temperature.

3) Introduce helium exchange gas into the can at a
pressure of about 2.5 mm Hg.

4) Transfer liquid helium into the dewar.

5) When the germanium resistance thermometer indicates
the cavity has cooled to A.2°K, remove the ex—
change gas. We were able to pump the exchange gas
pressure down below 10p without observing any
temperature change, and therefore we conclude that
any heat leak into the cryostat is minimal.

By applying current to the heater we were able to control

the temperature very well between “.2 and l5°K.

Crystal Growing

 

Alkali halide crystals are normally grown from the
melt, but the substitutional Cr(CN)5No3‘ complex used in
our experiments was not stable at temperatures above 120°C,
so all samples were grown by evaporation of aqueous solu-
tions. Most alkali halides are difficult to grow from water
solution. We were able to grow only KBr and KCl crystals
large enough for our experiments, since we needed samples
approximately 6 mm on a side.

Most crystals used in the measurements were grown at

 

 

AA

atmospheric pressure and a temperature of 10:200. A few
were grown at 211200. Various concentrations of the para-
magnetic impurity were used, in order to obtain sufficient
signal strength at high temperatures and to reduce cross
relaxation at low temperatures. Solutions containing
0.001, 0.01, 0.1, and 1.0 mole percent of K3[Cr(CN)5NO]

in KBr and KCl were used.

Chemical analysis showed that crystals grown from the
0.001 mole percent KBr solution were about 20 times as
concentrated as the solution itself. Because the substi—
tutional complex was yellow, it was possible to get a quali-
tative idea of the concentration from the shade of yellow
of a particular crystal, and nearly every KBr crystal appeared
more concentrated than the solution from which it was grown.
The coloring of any particular KBr crystal appeared homo-
geneous, indicating there were no appreciable concentra—
tion gradients of the Cr(CN)5N03- complex in KBr.

Almost every KCl crystal, on the other hand, had a
yellowish spot at the center, and many crystals had visible
concentration gradients. We were unable to obtain a sat—
isfactory KCl crystal of intermediate concentration, so our
KCl measurements were confined to the highest (>25°K) and
lowest (<A.2°K) temperatures.

Optically, the samples were fairly transparent and
neither the KBr nor KCl showed visible signs of trapped

water.

IV. SUMMARY AND IMPLICATION

OF PREVIOUS RESULTS

ESR Spectrum

 

Single crystals of KBr and K01 with substitutional
K3[Cr(CN)5NO]were grown from aqueous solution. The com-
plex Cr(CN)5NO3— goes into the alkali halide lattices as
a unit, with the Cr+ replacing a K+ and each of the CN-
and NO+ groups replacing the appropriate halide ion, so that
the Cr+ is octahedrally coordianted (Figure 1), but resides
in a crystal field of tetragonal symmetry. That the com—
plex retains its identity in KBr and KCl has been estab—
lished by ESR and IR measurements.17

The ESR spectrum of CR(CN)5N03_ in KBr and KCl has axial
symmetry, with the g“ and gL lines split into triplets by
transferred hyperfine interaction with the NlLl of the NO+.

The symmetry axis is the ON-Cr-CN axis of the complex.

The g factor is nearly isotropic with g“ = 1.9722 and
gL = 2.00MB in KBr, and g“ = 1.9722 and gl = 2.00AA in KCl.
N1“ hyperfine splitting constants (in gauss) are An = 2.89

and Al = 7.10 in KBr, and A“ = 2.Al and AL = 7.10 in KCl.

During measurement of the ESR spectra, Kuska and Rogers
observed that more microwave power was needed to saturate
the gl.transitions than to saturate the g“ transitions,

45

 

46

suggesting that the relaxation times of the two might be

different.

Infrared and Optical Spectra

 

The Debye temperatures of KBr and KCl shall be taken

l These values indicate

as 170°K and 2300K, respectively.
that we should not expect any relaxation process which re-
quires lattice phonons having energies which correspond to
temperatures much higher than 3000K.

On the basis of optical and infrared spectra, Manoharan19

has shown that the splitting A between the ground state

t
Kramers doublet and the first excited doublet of

1. Now 1 cm-1

Cr(CN)5No3‘ is approximately 12,600 cm'
: 1.UA°K, so this corresponds to a temperature of 18,200°K.
Therefore, we discard the Orbach process (2.33) and the T5
Raman process (2.31) of Orbach and Blume, since these pro—
cesses require that an excited doublet lie close to the
ground state.

Far-infrared spectra of Cr(CN)5No3’ in KBr and KCl
yield the results summarized in Table I, where each band
is labeled according to its absorption mechanism (e.g., Cr-
CN stretch represents the vibration along the Cr—CN band).
The presence of strong Cr-CN stretch bands near A00 cm-l

makes this complex an ideal candidate for a local mode

relaxation process, for the frequency is low enough to

 

47

satisfy mo = ml + w2, and distortion of the Cr—CN bond should
be very effective in modulating the crystal field at the
Cr+ site. The IR frequencies of Cr(CN)5No3‘ in KBr and
KCl lattices are not markedly different from the corres-
ponding values measured for K3Cr(CN)5NO in a Nujol mull
(Table I). In addition to the ESR spectrum, this is further
evidence that the Cr(CN)5NO3_ complex remains intact in

the lattices.

 

 

48

Table I. Effect of lattice on IR frequencies of Cr(CN)5NO3_

 

 

 

Stretching Frequencies (in cm-l)
Host N—O C-N Cr-NO Cr-CN
Mull8 1630 vs 2120 s 616 m 428 s
2073 VW 397 s
346 s
305 w
291 W
17
KBr 1656 s 2101 s 434
1634 s 2123 w 401
350
17
KCl 1707 S 2103 S 428
1685 s 2112 m 397
351
347

 

Abbreviations: 8, strong; m, medium; w, weak; v, very.

V. RESULTS AND CONCLUSIONS

General Considerations

 

Before a discussion of individual results for the
electron spin—lattice relaxation of Cr(CN)5No3‘ in KBr
and KCl, some general features common to both cases will
be presented. From the fact that our crystals exhibited
ESR spectra similar to the published results, we conclude
that the complex retained its identity in the lattices.
All our relaxation measurements were made with the spin
echo spectrometer operating at a microwave frequency of
approximately 9.4 GHz. Analysis of the data followed the

h35 except that instead

same procedure as described by Viet
of measuring the spin echo amplitudes from photographs of
oscilloscope traces, we visually determined the amplitudes
directly from the oscilloscope by the use of a calibrated
grid over the face of the cathode ray tube.

At T = 78°K all three lines of the g” hyperfine triplet
had the same relaxation time which was twice that of the Tl

common to the three gL lines. At 4.2°K, any variation of

T among the lines of a particular triplet was attributed

l

to effects, i.e., diffusion of energy from one com-

Tl2
ponent of the triplet into the other components. The center

line of each triplet was used for determining the temperature

49

 

50

dependence of the relaxation time. Hence, for brevity,
we shall refer to "the g” line" or "the g‘L line", meaning
the central component of the appropriate triplet.

For T < 5°K and for T > 600K, T for the gll line is

1
about twice that for the gi line. At any particular inter—
mediate temperature both lines have the same relaxation

time. The estimated error in the measured values is 10%.

Relaxation Results —— KBr

 

The results of our measurements of T1 as a function
of T for the g” and gL lines of Cr(CN)5NO3_ in KBr are shown
in Figure 8, the data being tabulated in Appendix I. The

first striking feature is that 1 depends on the orienta—

1
tion of the external magnetic field. That is, the El
line relaxes faster than the g” line, except at intermediate
temperatures.

As was shown in Chapter II, a variety of functional

forms for the temperature dependence of I may be expected.

1
The curves in Figure 8 represent the best fits (in the least
squares sense) of the function

i = AT + BT9J8[l19] + Ce'575/T (5.1)
T1 T

to the data for the g” line and the EL line, where J8(x)

is as defined in (2.23). The resultant A, B, and C values

appear in Table II. These fits were obtained with the

 

 

I I l I III'

‘~

 

 

 

 

 

 

 

(seconds)

Tl

 

 

 

 

 

 

 

 

 

4;
IC):

.4
IO

-5
IO
K58 1' l ,l l llll l. 1 lil llllr

I no Temperature (°K) IOO zoo

Figure 8. Data for g" (a) and 5l (+) lines in KBr, with

fits to % = AT + BT9J8(170/T) + c exp(—575/T)
1

 

52

 

Table II. Least squares fit
+ C exp(—575/T) to

of % = AT + BT9J8(170/T)

relaxation of Cr(CN)5N03‘ in KBr

 

Line A

B C

 

g“ 0.112i0.006 (3.67:0.

3L O.l82i0.007 (4.23i0.

16) x 10‘15 (3.32:0.56) x 106

16) x 10‘15 (2.66:0.11) x 107

 

   

53

MSU Control Data 3600 computer, using the 3600 Fortran pro—

gram TOWPLOT, listed in Appendix III.

Discussion -— KBr

 

We shall now attempt to justify the particular func-
tional form (5.1) chosen for temperature dependence of T1.
As was pointed out in Chapter IV, the Orbach process (2.33)
and the Orbach and Blume process (2.31) can be discarded
on the basis of the energy level assignments of Manoharan.19

The direct process (2.20) is dominant at low tempera-
tures (Figure 8). It has been shown in Chapter II that the
current theory allows for anisotrOpy of the direct process
relaxation rate, and Figure 8 indicates that in the temp—
erature range 1 < T < 50K, the T1 values for the g” and
51 lines differ by a factor of two.

The theory predicts that the Raman relaxation rate

should have the form

2

% a T , for T > 9/2. (5.2)

But for T > 900K, T is a much more rapidly varying func—

1
tion of temperature than given by (5.2). The only allowed
process which satisfies this criterion is the local mode
process. In Figures 9 and 10 we have replotted the data

for the 51 and gH lines, respectively. In addition, each

figure has a least squares computer fit to the direct and

 

'0'? l IIIIII I 11111”
O I+‘:E\'Ll‘
'0 + \ \
fl.
I?
..l‘
IO '
-2?
IO '
’3
“O
8
o «6,
3 '0 l
t:
-4
IO
-5
IO
‘0-6 l‘ I l l H I I I LIIJ
I to Temperature (°K) IOO
Figure 9. Data for gL line in KBr, with fit to

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

T1

= AT + BT9J8(170/T)

 

 

 

 

I IIIIIII

 

 

K)

 

J

 

l0 §

DD
00

 

 

I J
I v 54:
I

 

 

(seconds)

 

 

 

 

 

 

 

 

‘65______‘__--I'..-__-L..__.[_ [__-l l l I . l l l I l l l I
l to Temperature (°K) ICC 200

 

Figure 10. Data for g" line in KBr, with fit to

i = AT + BT9J8(170/T)

T1

 

   

56

T9J8 Raman processes. That is, the fits have the same form
as those used in Figure 8 except that the exponential term
(i.e., the local mode process) is deleted. For both the

gll and 5L lines the fit is clearly better when the local
mode process is included. Anisotropy is observed in the
local mode process, as in the direct process. At the very
highest temperatures for which we have data on both lines,

the ratio of T values for the g‘I and gL lines is slightly

1
greater than the factor of two observed in the direct
process, but the qualitative features of the theory discussed
in Chapter II are borne out very well.

The empirical value of 00 in (5.1) was obtained by

fitting

10
% = dT + BT9J8[—%—]-+ ye

l

'e°/T (5.3)

for various 00 values, and observing which value gave the
minimum standard error of fit, i.e., the value of 00 for

which

.1.
T

IIMZ

 

l 2
1 1calci

i l .obs.
1

was minimum, where W1 is a weighting factor determined as
in Appendix IV, and N is the number of data points. For
the 31 line, 60 = 550i25°K gave the best fit and for the
g" line the best value of 90 was 600:500K. Neither of

these values is exactly equal to one of the reported

 

 

57

infrared frequencies (Table II), but in fact the three
IR lines overlap and form a broad band.20 Therefore, for
convenience in comparing the A, B and C values for the gll
and g‘L lines, we feel justified in using 00 = 575°K in (5.1).
Now the question of how to account for the data at
intermediate temperatures (10 < T < 500K) arises. We dis—
card the driven-mode process temporarily, since there is
no spectroscopic evidence for resonant modes of the
Cr(CN)5NO3— which lie within the Debye spectrum of KBr.
This point will be discussed more fully later. Neglecting
the driven—mode process, we are left with two possible
Raman processes, (2.24) and (2.30), which have temperature

dependences T7 9

J6 and T J8 respectively. We choose the lat-
ter for a number of reasons. In the first place, T1 is
isotropic for 10 < T < 500K, whereas the theoretical dis-
cussion shows that the T7J6 process should exhibit the same
form of anisotrOpy as the direct process. Also, if the

local mode process (2.46) is written in the form

 

 

 

 

 

 

3 _
§ 2 mo hm; (ES>‘202 e 9°/T , (5.5)
L Mv
where
<d|fi-fi|b><a|v2|d>
0 E A , (5.6)
cd
then the Raman process (2.24) becomes
1 =————9 M 7.1 9—02 (5 7)
TR 3210i?— 6T ° °
n p v

 

   

58

From the computer fit (Table II) we have for the g_L line,

i a 2.66 x 107 e‘9°/T . (5.8)
TL

Comparing (5.5) and (5.8) leads to

02 z 2.21 x 10"33 , (5.9)

where we have used v = 3.56 x 105cm/sec,10 p = 2.75gm/cm3,
M = 1/2(MBr + MK) 2 10—22gm, and mo = 400cm‘l. Substituting
(5.9) into (5.7), we obtain
1 —90 kT 7 0
¥ 2 2.50 x 10 E§_] J6[—] . (5.10).
R
Throughout the temperature range 10 < T < 500K, the relaxa—
tion rates predicted by (5.10) are 100 to 1000 times
greater than the observed values. Hence we conclude
again that the T7J6 Raman process is not operative in the
system under study, for if it were present the relaxation
rates should be much greater than they are observed to be
at intermediate temperatures.

As pointed out in Chapter II, the T9J8 Raman process
is not expected to exhibit any anisotropy unless the
splittings A between the ground doublet and excited doub—
lets are dependent on the orientation of H. Since the split—
tings A are thousands of cm-1 for all levels, the effect
of H upon them is expected to be negligible. This is veri—

fied by the observed isotropy of T for 10 < T < 500K.

1

 

59

In the temperature range 7 < T < 200K, the theoreti-
cal relaxation rate (5.1) does not fit the data nearly so
well as at other temperatures. When this anomaly was first
observed it was tentatively explained in terms of poor thermal
contact between the sample and thermocouple, the idea
being that the thermocouple junction (at the sample) was
cooled by conduction along the thermocouple wires running
to the helium bath (Figure 6). If this were the case,
then the sample temperature would be greater than that indi—
cated by the thermocouple. This, indeed, is consistent with
the data obtained (Figure 8).

Such an effect might also be due to thermal gradients
in the crystal. The slow relaxation rates of the
Cr(CN)5N03_ at low temperatures require crystals having
low concentrations of the paramagnetic impurity in order

to reduce T effects. Hence crystals of relatively large

12
volume are necessary in order to obtain sufficient signal
amplitude at the lower temperatures. The right circular
cylinder cavity required a crystal which was =6 mm on a
side. Such a large crystal was deemed likely to contain
thermal gradients. In particular, the surface of the crys—
tal would be hotter than the interior since heat was
applied to the cavity by the manganin heater above the
cavity (Figure 6).

The vacuum can cryostat (Figure 7), was constructed to

test these ideas. The thermal path from heater to sample

 

60

for this configuration is compared in Figure 11 with that
for the thermocouple and cylindrical cavity. The sample
is suspended at the center of the cylindrical cavity by

a block of styrofoam, whereas in the rectangular one the
sample and thermometer are heat sunk to the relatively
massive brass cavity. The most important point is that
from the geometry we can conclude that the sample temp—
erature is never likely to be greater than the ther-
mometer temperature whenever power is being applied to the
heater on the rectangular cavity. Hence the only type of
anomaly we can envision from such apparatus is a shift
opposite to that for the cylindrical cavity, i.e., the

measured temperature for a given T might appear higher

1
than the actual temperature, but is not likely to appear
lower. But the results obtained from the two different
configurations agree, within experimental error. This
leads to the conclusion that the anomalous T dependence

of T for 7 < T < 20°K is a property of the sample and not

1
merely a thermal effect in either piece of apparatus, since
the considerations outlined above indicate that the results
should be very different in the two cases if the effect
were only a thermal one.

At present we do not feel that a satisfactory explana-

tion of the temperature dependence of T in the range

1

7 < T < 20°K can be given. The only speculation we can

offer is that the Cr(CN)5NO3— complex may have some very

 

 

heater

 

 

 

61

 

 

 

 

 

  

thermocouple

bath

 

 

 

 

 

 

 

 

(a) Thermal path for cylindrical cavity

 

 

heater —’\N--§

 

 

 

bath

 

 

thermometer

\V

 

sample

 

 

 

 

 

 

 

 

(b) Thermal path for rectangular cavity

Figure 11.

Comparison of the thermal paths for the two

types of apparatus used.

The

present weak thermal contact,

inversely proportional to the

ductance.

The straight lines

of good thermal contact.

wavy lines re—
the length being

estimated con-

represent paths

 

 

62

low frequency bending modes. If so, then we might have
the driven-mode process (2.35). Explicit test of this
requires knowledge of mi, the frequency of the appropriate
mode of the complex. Addition of a driven—mode process

did not improve the computer fit unless hwi was less than

20 cm‘l. For 0r(CN)5N03’ in alkali halides, infrared meas-
urements17 have been made down to energies only as low as
250 cm-1. However, Gans8 et al report a series of IR

spectra of K3Cr(CN)5NO and [(CH3)uN3] [Cr(CN)5NO] down to

80 cm_l. The lowest energy absorption bands they observed

were 126 cm"1 in the latter. When compared with the Debye

l and 160 cm.1 for KBr and KCl, respect—

energies of 118 cm-
ively, these bending mode energies are too high for the
driven-mode process to fit the data significantly better
than does equation (5.1). Therefore, until more IR data
are available, the applicability of the driven-mode process

to the case at hand must remain an open question.

Results and Discussion -— KCl

 

The qualitative results for the relaxation of
Cr(CN)5NO3— in KCl are similar to those in KBr. Data for
the g" and gL lines are tabulated in Appendix II and are

plotted in Figure 12 along with least squares fits of

l -6lO/T

T1

_ 9 230
-AT+BT J8[—,IT-] + C8 (5.11)

 

IO

 

63

 

"IIIMI III I I I I I III

 

 

 

 

 

(seconds)

TI

 

 

 

 

 

 

 

Figure 12.

-5 - .
‘0 '-_--- -__.J____I_-I.._I

II IL I, I I LLIIII'

 

 

 

...—__._-_.,.. - . ,

IO Temperature (°K) IOO zoo

Data for g" (B) and gL (+) lines in KCl, with

fits to

1

AT + BT9J8(230/T) + c exp(-610/T)

 

64

to each set of data. The resultant A, B and C values appear
in Table III. The quantity 00 = 610°K in (5.11) is a com—

promise between 00 = 630i25°K for the gL line and

00 590i50°K for the g” line. Our chosen value of

90

610°K in (5.11) falls within the Cr—CN stretching
band (Table I).

There is considerable scatter in the low temperature
data, due to the small sizes of samples available, but
anisotropy of the direct process was observed. Our ina—
bility to grow sufficiently large low—concentration KCl
crystals prevented our obtaining data in the range
4.2 < T < 250K. This precluded looking for the anomaly
of the T dependence, observed in KBr. An isotropic Raman
process is observed in KCl, as in KBr. From Figures 13 and
14 we see that the direct and Raman processes alone do not
adequately describe the data for T > 650K, whereas Figure 12
indicates that the local mode process fits the high temper-

ature data very well.

 

 

65

 

 

 

Table III. Least squares fit of % = AT + BT9J8(230/T)
l
+ Ce-610/T to relaxation of Cr(CN)5NO3- in KCl.
Line A B C
6 + ‘17 + L; 6
g” . 35_.O3O (6.29i.2l)XlO (6.12_.3 )XlO
1.39:.07 (7.54i.28)x10"l7 (2.01:.62)x107

 

 

 

 

 

66

 

|O'T“‘”““I“l"‘Trl“Ill1*w‘“wtrt*"1*—r—r1-1ii

 

 

 

 

 

 

 

 

(seconds)

Ti

 

 

 

 

 

 

 

 

‘deflmmili: L11 1 L111. Li L 1 1 [III
to Temperature (°K) IOO zoo

Figure 13. Data for g, line in KCl, with fit to
1 AT +BT9J 8(230/T)

l

 

 

l
IO

67

'”I‘“" I"II ‘I"l I IIII'7”"WI""I"_III"’ I

 

 

F1 III

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—...___ -WHH.._-- i ...

 

 

D
13
'0
5
8 .3 - __._a _- ..-fi _ -._ —l-—
0, IO
FT'
-4.
IO
'\
"\
-5 -
IC) ‘fig
‘66 'l‘ ,LLIL,LI I IIIILIL
I I0 Temperature (°K) IOO

Figure 14.
1

T1

Data for g” line in KCl, with fit to
— = AT + BT9J8(230/T)

200

 

 

68

Summary
Electron spin—lattice relaxation times T1 of Cr(CN)5NO3—

substitutional in KBr and K01 have been measured for

1 < T < 1500K. Spin echo techniques employing either

picket or single-pulse saturation methods were used.

From the temperature dependence of the observed Tl values,

we conclude that at least three relaxation mechanisms

are operative: an anisotropic direct process at T < 50K,

an isotropic Raman process at intermediate temperatures,

and an anisotropic local mode process at T > 650K.

The functional form of the relaxation rate is

l = AT + BT9J8(e/T) + c exp(—00/T), where e is the

T
Dibye temperature and was taken as 170°K for KBr and 230°K
for KCl. Empirical values of A, B, C, and 00 were deter-
mined by least squares computer analysis. Published
energy level assignments preclude the possibility of an
Orbach process. Therefore, since the parameter 00 is
575i30°K for KBr and 610i30°K for KCl, in agreement with
published Cr-CN stretch frequencies, we conclude that

the exponential process occurs through the interaction

of lattice phonons with vibrational modes localized in

the Cr(CN)5NO3—. An anomaly in the T dependence of T1

for KBr has been tentatively explained as a driven-mode

process, in which relaxation is effected by low frequency

(<20 cmfll) bending modes of the complex.

 

10.

11.
12.

13.
14.

15.

l6.
17.

REFERENCES

M. Blackman, Handbuch der Physik 1 (part 1), 325
(1955).

J. G. Castle, Jr. and D. W. Feldman, Phys. Rev.
137. A671 (1965).

J. G. Castle, Jr., D. W. Feldman, P. G. Klemens,
and R. A. Weeks, Phys. Rev. 130, 577 (1963).

B. P. Finn, R. Orbach, and W. D. Wolf, Proc. Phys.
Soc. 11, 261 (1961).

C. L. Foiles (private communication).
E. L. Hahn, Phys. Rev. 16, 461 (1949).
D. J. E. Ingram, Spectroscopy at Radio and Micro—

wave Frequencies (Philos0phical Library, Inc.,
New York, 1956).

 

 

P. Gans, A. Sabatini, and L. Sacconi, Inorg. Chem.
5, 1877 (1966).

D. E. Kaplan, M. E. Browne, and J. A. Cowen, Rev.
Sci. Instr. 21, 1028 (1956).

C. Kittel, Introduction to Solid State Physics
(John Wiley and Sons, Inc., New York, 1956), p. 93.

 

Ibid., p. 125.

P. G. Klemens, Phys. Rev. 222, 433 (1961).

P. G. Klemens, Phys. Rev. 225, 1795 (1962).
G.

P. Klemens, Solid State Physics (Academic Press,
Inc., New York, 1958), Vol. 7, p. l.

 

H. A. Kramers, Proc. Amsterdam Acad. Sci. 33, 959
(1930).

R. de L. Kronig, Physica 6, 33 (1939).

H. A. Kuska and M. T. Rogers, J. Chem. Phys. 42,
3031I (1965).

69

 

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.
28.

29.

30.

31.

32.
33.

34.
35.

70

L. D. Landau and E. M. Lifshitz, Quantum Mechanics
(Addison-Wesley Publishing Company, Inc., Reading,
Mass.), p. 88.

 

P. T. Manoharan and H. B. Gray, Inorg. Chem. 5,
823 (1966).

P. T. Manoharan (private communication).

R. D. Mattuck and M. W. P. Strandberg, Phys. Rev.
119, 1204 (1960).

E. W. Montroll and R. B. Potts, Phys. Rev. 1 0,
525 (1955).

A. Narath, in Hyperfine Interactions, edited by
A. J. Freeman and R. B. Frankel (Academic Press,
New York, 1967), p. 287.

 

R. Orbach, Proc. Roy. Soc. A 264, 458 (1961).

R. Orbach and M. Blume, Phys. Rev. Letters 8, 478
(1962).

George E. Pake, Paramagnetlc Resonance (W. A. Benja—
min, New York, 1962), p. 84.

 

Ibid., p. 112.

W. M. Rogers and R. L. Powell, Tables of Transport
Integrals, National Bureau of Standards Circular
No. 595 (U. S. Governmental Printing Office,
Washington, D. C., 1958).

 

Charles P. Slichter, Principles of Magnetic Resonance

 

(Harper and Row, New York, 1962), p. 165.

R. G. D. Steel and J. H. Torrie, Principles and Pro—
cedures of Statistics (McGraw—Hill Book Company,
New York, 1960), p. 181.

 

 

J. H. Van Vleck, J. Chem. Phys. 1, 72 (1939).
J. H. Van Vleck, Phys. Rev. 51, 426 (1940).

R. F. Vieth, Thesis, Michigan State University
(1963). Chapter 2.

Ibid., Chapter 3.

Ibid., p. 112.

 

71

36. 1. Wallter, z. Physik 12, 370 (1932).

37. E. Wigner, Nachr. Akad. Wiss. Gottingen Math.-physik.
Kl. IIa, 546 (1932).

38. J. M. Ziman, Electrons and Phonons (University
Press, Oxford, 1960), p. 21.

 

APPENDICES

72

 
 

APPENDIX I.

DATA FOR KBr

73

 

74

GPERPENDICULAR LINE IN KBR

EXPERIMENTAL VALUES OF T1 VS Ta ALONG WITH I1 VALUtS CALCULATED FROM

LFAST SQUARES FIT 10 FQUATION (5.1).

T T1(OBS) T1(CALC) REL WT DATE
78.00 3.200-005 3.5/5-005 0.0000000014 02/08/66
86.50 2.200-005 2.056r005 0.000000000/ 02/08/66
86.50 2.000-005 2.036-005 0.0000000005 02/08/66
89.00 1.650-005 1.746-005 0.0000000004 02/08/66
89.00 1.500-005 1.746-005 0.0000000006 02/08/66
94.00 1.500-005 1.306-005 0.0000000002 02/08/66
94.00 1.300-005 1.306-005 0.0000000002 02/08/66
97.70 1.150-005 1.069.005 0.0000000002 02/08/66
97.70 1.150-005 1.069-005 0.0000000002 02/08/66
78.00 3.000-005 3.575-005 0.0000000012 12/08/65
78.00 3.100-005 3.575-005 0.0000000016 12/08/65
78.00 3.200-005 3.575-005 0.0000000014 12/08/65
78.00 3.200-005 3.5/5-005 0.0000000014 12/08/65
86.70 1.900-005 2.011-005 0.0000000005 12/08/65
94.30 1.150-005 1.285-005 0.0000000002 12/08/65
94.50 1.150-005 1.2/0-005 0.0000000002 12/08/65
99.30 9.600-006 9.832-006 0.0000000001 12/08/65

102.50 8.600-006 8.374-006 0.0000000001 12/08/65
98.50 1.060-005 1.025-005 0.0000000002 12108/65
4.20 1.500+000 1.309+000 3,032/515338 11/05/65
4.20 1,400+000 1.309+000 2.6418635841 11/05/65
3.51 1.850+000 .567+000 4.615151/511 11/05/65
3.51 1,900+000 1.567+000 4.8658810258 11/05/65
2.46 2.200+000 2.236t000 6,525/850915 11/05/65
2.13 2.250+000 2.582+000 6.8266909010 11/05/65
4.20 1,500+000 1.309+000 3.0627516638 11/05/65
6.20 6.600-001 8.852-001 0.5871406582 11/05/65
8.20 2.600v001 6.581-001 0.0911176290 11/05/65
8.40 1.900-001 6.398-001 0.0486588106 11/05/65
12.60 1.600-001 2.814.001 0.0227796622 '11/05/65
4.20 1.650+000 1.309+000 3.6696291159 11/03/65
4.20 1,570+000 1.309+000 3.322412/855 11/03/65
3.58 1.800+000 1.536+000 4.3671619205 11/03/65
3.21 2,200+000 1.713+000 6.5257850915 11/03/65
3.44 2,000+000 1.599+000 5.3915579208 11/02/65
3.44 1,550+000 1.599+000 3.2666044797 11/02/65
2.95 2,600+000 1.864+000 9.1117628960 11/02/65
2.29 2.500+000 9,402+000 7.1606656552 11/02/65
1.98 2.650+000 2.7]8+000 9.4695538898 11/02/65
45.80 5.100-004 4.121-004 0.0000006506 10/15/65
46.10 3.750-004 4.020-004 0.0000001895 10/15/65
50.40 3.100-004 2.851-004 0.0000001295 10/15/65
50.40 3.250-004 2.851-004 0.0000001424 10/15/65
53.80 1.800-004 2.193-004 0.0000000437 10/15/65
54.00 1.750-004 2.160-004 090000000415 10/15/65
59.20 1.600-004 1.452-004 0.0000000645 1U/15/65
58.80 1.350~004 1.497-004 0.0000000246 10/15/65

 

 

 

 

63.20
63.00
72.20
72.20
75.60
76.40
76.50
79.90
79.80
77.50
77.30
73.80
73.50
70.50

4.20
17.60
17.60
19.20
19.50
20.60
20.60
22.80
22.80
24.80
24.80
26.60
26.60
28.20
28.20
29.70
29.70
31.80
32.00
33.40
33.40
38.20
38.20
42.50
42.30
46.60
46.60
46.20
50.30
50.30
56.60
56.60
60.10
59.90
64.00
63.80
68.60
68.60
78.00
78.00

1.060.004
9.700-005
5.400-005
5.700-005
4.600-005
4.700-005
4.000-005
2.550~005
2.750-005
3.100-005
3.300-005
5.200-005
4.500-005
6.200-005
7.900-001
2.800-002
2.000-002
1.700-002
1.800-002
1.450-002
1.600-002
9.000-003
8.500-003
6.000-003
5.700-003
4.250-003
5.750-003
3.200-003
2.800-003
2.500-003
2.700-003
2.000-003
2.000-003
1.4502003
1.550-003
6.500-004
5.400-004
6.500-004
6.700-004
3.700-004
6.000-004
4.000-004
3.000-004
3.200-004
1.600-004
1.700-004
1.620-004
1.500-004
9.700-005
1.070-004
7.200'005
7.200-005
2.850-005
2.600-005

75

1.070-004
1.086-004
5.424-005
5.4?4-005
4.2560005
4.001-005
3.973-005
3.136.005
3.157-005
3.702-005
3.755-005
4.824-005
4.931-005
6.245-005
1.309+000
4.843-002
4.843-002
2.819-002
2.561v002
1.861-002
1.831-002
1.005-002
1.005-002
6.274~003
6.2/4-003
4.322-005
4.322-003
3.?13-003
302I3'005
2.496-003
7.496-003
1.815-003
1.764-003
1.458-003
1.458-003
5.579-004
5.579~004
3.858-004
3.858-004
3.987-004
2.8/3-004
2.873-004
1.771-004
1.7710004
1.3779004
1.006.004
1.022-004
7.094-005
7.094.005
3.575.005
3.575-005

0.0000000151
0.0000000127
0.0000000039
0.0000000044
0.0000000029
0.0000000050
0.0000000022
0.0000000009
0.0000000010
0.0000000016
0.DUUDUOUOIb
0,0000000036
0.0000000027
0.0000000092
0.8412178255
0.0010561454
0.0005591556
0.0003899401
0.000466/162
0.0002866908
0.0006450597
0.0001091790
0.0000976850
0.0000485240
0.000043/929
0.0000246463
0.0000445646
0.0000138024
0.00001096/5
0.0000084243
0.0000098261
0.0000056916
0.0000056916
0.0000028669
0.0000032683
0.0000002695
0.0000009511
0,0000005695
0.0000006051
0.0000001845
0.0000004852
0.0000002197
0.0000001216
0.0000001380
0.0000000345
0.0000000390
0.0000000334
0.0000000503
0.0000000127
0.0000000194
0.0000000070
0.00000000/0
0.0000000011
0.0000000009

10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
1U/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
1U/l3/65
1U/13/65
1U/13/65
10/13/65
10/13/65
10/15/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
10/13/65
lU/13/65
10/13/65
10/13/65
0//23/64
0//23/64

 

 

 

 

80.00
85.50
89.00
94.10
98.20
102.00
102.80
105.00
107.30
110.00
111.60
114.30
123.00
128.50
135.00
141.20
145.00
150.00
150.00
79.50
79.50
70.00
75.00
79.00
79.00
87.00
94.50
99.50
106.50
113.00
113.00
121.00
127.00
4.20
4.20
4.20
3.57
3.01
3.01
2.45
2.41
1.93
1.93
1.56
1.56
6.00
5.80
8.40
8.40
8.95
9.05
10.00
11.70
13.05

3.800-005
2.500-005
1.700-005
300-005
.080-005
.000'000
.750-006
500-006
000'006
.500-006
000-006
.400-006
.000-006
.000-006
.200-006
600-006
.900-006
700-006
.300-006
.950-005
.100-005
.000-005
.300-005
.400-005
.800-005
700-005
.100-005
.400-006
.000-006
.800v006
.200-006
.300-006
.800-006
270+000
.250+000
360+000
280+000
.760+000
950+000
.000+000
.450+000
.520+000
.350+000
.700+000
850+000
.150+000
.200+000
.400-001
.700-001
.700-001
2.500-001
2.200-001
9.700-002
8.000-002

O

QHvRJHFJDGNJACN£>0<)‘J\Jm‘OH*H

O

O

76

3.114P005
2.168-005
1.746-005
1.299-005
1.0419005
8.583-006
8.253-006
7.429-006
6.681-006
5.926-006
5.533-006
4.946-006
3.5519006
2.941-006
2.397-006
2.005-006
1.810-006
1.593-006
1.593-006
3.2239005
3.2239005
6.387e005
4.423-005
3.335-005
3.335-005
1.974.005
1.270.005
9.7319006
6.9299006
5.218.006
5.218-006
3.818-006
3.092-006
1.309+000
1.309+000
1.309+000
1.541+000
1.827+000
1.827+000
2.245+000
2,282+000
2,850+000
2.850+000
3.525+000
3.525+000
9.152-001
9.471-001
6.398-001
6.398-001
5.916-001
5.831-001
5.032-001
3.575-001
2.458-001

0.0000000019
0.0000000008
0,0000000004
0.0000000002
0.0000000002
0.0000000001
0.0000000001
090000000001
0.0000000001
0.0000000001
0.0000000000
0.0000000000
0.0000000000
0.00U0000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
0.0000000012
0.0000000013
0.0000000049
0.0000000025
0.0000000008
0.0000000011
0.0000000004
0.0000000002
0.0000000001
0.000000U000
0.0000000000
0.0000000000
0.0000000000
0.0000000000
2.1740109450
2.1060773121
2.4930563853
2.2083821268
4.1/52224586
5.125349/542
5.3915579268
8.090/066138
8.559637364/
7.4437196626
9.8261143216

10.9482323148

1.7825858396
1.94U9908537
0.0776384341
0.0389540060
0.0962611432
0.0842430926
0.0652678509
0.0126922921
0.0066264927

0//24/64
0//24/64
0//24/64
07/24/64
07/24/64
0//24/64
0//24/64
07/24/64
07/24/64
07/24/64
07/24/64
0//24/64
01/24/64
07/24/64
0//24/64
07/24/64
0//24/64
0//24/64
O/l24/64
0//27/64
0//27/64
0//30/64
0//30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/60/64
09/30/64
09/60/64
09/30/64
09/21/66
06/21/66
09/21/66
06/21/66
06/21/66
06/21/66
06/21/66
00/21/66
05/21/66
06/21/66
09/21/66
09/21/66
06/21/66
06/21/66
05/21/66
00/21/66
09/21/66
06/21/66
06/21/66
06/21/66
06/21/66

 

 

 

 

 

77

GPARALLEL LINE IN KRR

EXPERIMENTAL VALUES 0? 71 V5 T: ALONG WITH 71 VALUES CALCULATED FROM
LEAST SQUARES FIT TO EQUATION (5.1):

7 T1(OBS) 71(CALC) REL HT DATE
4.20 2.8309000 2.1246000 5.4593390386 03/21/66
4.20 2.8609000 2.1244000 5.5367755047 03/21/66
3.57 3.3009000 2.4999000 7.4232666817 03/21/66
3.57 2.8609000 9.4999000 5.5756982355 03/21/66
3.00 3.6006000 2.9749000 8.8343010822 03/21/66
1.93 4.3009000 4.6229000 12.6036755414 03/21/66
1.93 3.900.000 4.6229000 10.3680339092 03/21/66
1.56 4.5006000 6.7199000 13.8035954414 03/21/66
1.10 5.5006000 0.1109000 20.6201857827 03/21/66
6.00 2.3009000 1.4849000 3.6059762906 03/21/66
5.80 1.7809000 1.5359000 2.1597684838 03/21/66
7.35 8.2009001 1.2009000 0.4503475346 03/21/66
7.35 1.1009000 1.2009000 0.8248074313 03/21/66
8.10 9.4009001 1.0759000 0.6023139226 03/21/66
8.40 5.5006001 1.0239000 0.2062018578 03/21/66
8.40 6.5009001 1.0209000 0.2880009419 03/21/66
9.05 4.4009001 9.3059001 0.1319691890 03/21/66
9.05 5.2009001 9.3059001 0.1843206028 03/21/66

10.00 3.3009001 7.8889001 0.0742326688 03/21/66
10.00 3.9009001 7.8889001 0.1036803391 03/21/66
11.70 3.4009001 5.2829001 0.0787997843 03/21/66
13.00 2.2008001 0.4709001 0.0329922973 03/21/66
78.00 8.2008005 0.4519005 0.0000000046 02/08/66
78.00 8.0008005 0.4519005 0.0000000044 02/08/66
86.00 5.2009005 6.0219005 0.0000000018 02/08/66
09.00 4.8009005 9.3329005 0.0000000016 02/08/66
69.00 4.600-005 5.3329005 0.0000000014 02/08/66
94.50 3.7009005 4.3029005 0.0000000009 02/08/66
94.50 3.8009005 4.3029005 0.0000000010 02/08/66
78.00 1.0209004 0.4519005 0.0000000071 12/08/65
78.00 1.0609004 0.4519005 0.0000000077 12/08/65
78.00 9.8009005 0.4519005 0.0000000065 12/08/65
86.50 6.4000005 6.8999005 0.0000000028 12/08/65
86.00 6.0009005 6.021”005 0.0000000025 12/08/65
93.70 4.350.005 4.436'005 0.0000000013 12/08/65
93.50 4.7009005 4.4709005 0.0000000015 12/08/65
98.70 3.9009005 3.6789005 0.0000000010 12/08/65
4.20 2.4009000 2.124+000 3.9263560366 11/05/65
3.52 3.2009000 9.5349000 6.9801885096 11/05/65
3.51 3.2009000 9.5429000 6.9801885096 11/05/65
2.46 3.300.000 3.6264000 7.4232668817 11/05/65
7.00 7.1009001 1.2659000 0.3436243191 11/05/65
7.30 6.1009001 1.2099000 0.2536453266 11/05/65
9.70 3.6006001 0.3389001 0.0803430108 11/05/65
11.20 3.4009001 4.0479001 0.0707997843 11/05/65
4.20 2.3506000 9.1249000 3.7644620160 11/03/65
4.20 2.3506000 9.1244000 3.7644620160 11/03/65

3.57

3.20
45.10
45.00
46.10
50.10
50.20
54.70
54,10
58,40
58.50
63.50
63.30
67.00
67.00
72.30
72.30
76.00
75.80
30.20
00.20
77.30
77.30
74.00
73.90
70.00
67.30
14.00
14.00
15.90
15.90
16.50
17.80
17.80
19.80
19.80
19.80
21.20
21.20
22.70
72.70
24.60
24.60
26.20
25.80
32.70
32.40
32.40
35.50
34070
34.70
39.00
39000
39.00

2.780*000
3.000‘000
6.5009004
603803004
5.4000004
4.3009004
4.0009004
3.350I004
3.3009004
2.6009004
2.5009004
1.6509004
1.6009004
1.6202004
1.0502004
102903004
1.2505004
1.1205004
100305064
7.0009005
8.6003005
1.2509004
100309004
9.6008005
100305004
1.1403004
1.4505004
7.000.002
1.2008001
3.6009002
3.6005002
2.8005002
2.5000002
2.1009002
1.8009002
1.7009002
1.6009002
8.0008003
9.5000003
1.1408002
9.2009003
4.8009003
5.0000003
6.1009003
4.7009003
1.8009003
1.8009003
1.7509003
1.3009003
1.4009003
1.5009003
101“0“003
1.0000003
1.0000003

78

2.4999000
7.780‘000
5.1889004
5.229'004
4.803’004
3.613'004
3.589'004
2.703”004
2.802'004
7.187'004
,9175'004
106699004
1.686'004
1.403*004
1.4039004
19°92'004
1.092'004
0.2319005
9.314.005
79681'005
7.681'005
0.7159005
H.715‘005
1.010‘004
1.015'004
1.2159004
1.382'004
993849001
9.384.001
1.120'001
1.120'001
H.349”002
qo430'002
8.430900?
9.752”002
?.752’002
9.752’002
10805"002
1.805'00?
10202'00?
1.202900?
7.618"003
70618'003
$0423"003
3.883'003
1.851’003
10929'003
1.929’003
10299'003
1.430”003
1.430’003
l.893”004
8.893°004
0.8939004

5.2681336795
6.1349313073
0.0000002880
0.0000002749
0.0000001988
0.0000001260
0.0000001091
0.0000000765
0.0000000742
0.0000000461
0.0000000426
0.0000000186
0.0000000175
0.0000000179
0.0000000075
0.0000000112
0.0000000107
0.0000000086
0.0000000072
0.0000000033
0.0000000050
0.0000000107
0.0000000072
0.0000000065
0.0000000072
0.0000000089
0.0000000143
0.0033401293
0.0098158901
0.0008834301
0.0008834301
0.0005344207
0.0004260369
0.0003006116
0.0002208575
0.0001969995
0.0001745047
0.0000436262
0.0000615197
0.0000885804
0.0000576956
0.0000157054
0.0000170415
0.0000253645
0.0000150578
0.0000022086
0.0000022086
0.0000020876
0.0000011520
0.0000013361
0.0000015337
0.0000008248
0.0000006817
0.0000006817

11/03/65
11/03/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10115/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10/15/65
10115/65
10/15/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65
10/14/65

 

46.40
46.40
50.80
50080
55.50
55.50
79.00
79.00
79.00
87.00
07.00
94.50
09.50
106.50
106.50
113.00
113.00
121.00
70.00
75.00
75.00
79.50
79.50
82.00
02.00
86.50
06.50
90.40
976.00
101.00
101.00
105.20
111.00
30.00
94.00
105.00
78.00
85.80
94.60
98.00
100.00
104.70
110.20

5.3009004
5.5000004
4.2002004
4.1009004
2.5500004
2.7508004
6.1002005
5.3000005
5.0009005
3.1002005
3.5.9.005
2.800-005
2.3809005
1.8002005
2.0009005
1.4009005
1.6009005
9.6002006
1.1”09004
9.5002005
7.0009005
7.4009005
8.0002005
6.5009005
7.400I005
4.8002005
50400’005
4.2002005
303009005
3.1009005
3.2002005
2.8003005
2.5008005
7.4009005
4.2003005
4.5002005
6.1009005
7.2009005
5.0008005
4.5002005
3.300.005
3.7000005
3.550a005

79

406969004
4.696'004
3.449.004
3.4499004
9.579-004
7.579’004
8.090'005
0.090’005
0.0909005
5.780.005
4.780-005
493029005
3.572-005
?o794'005
79794.005
?0258'005
?9258'005
1.771~005
0.215.004
0.654'005
9.654’005
799177005
7.917.009
7.113’005
79113-005
50899'005
4.899-005
R9044’005
4.065'005
3.384'005
39384’005
2.921'005
9.4079005
7.748'005
4.385’005
70941”005
0.451'005
0.070"005
4.2869005
3.7740005
30503’005
2.971’005
9.471'005

0.0000001915
0.0000002062
0.0000001202
0.0000001146
0.0000000443
0.0000000516
0.0000000025
0.0000000019
0.0000000017
0.0000000007
0.0000000008
0.0000000005
0.0000000004
0.0000000002
0.0000000003
0.0000000001
0.0000000002
0.0000000001
0.0000000052
0.0000000002
0.0000000033
0.0000000037
0.0000000044
0.0000000029
0.0000000037
0.0000000016
0.0000000020
0.0000000012

0.0000000007'

0.0000000007
0.0000000007
0.0000000005
0.0000000004
0.0000000037
0.0000000012
0.0000000014
0.0000000025
0.0000000035
0.0000000017
0.0000000014
0.0000000007
0.0000000009
0.0000000009

10/14/65
t0/14/65
10/14/65
10/14/65
10/14/65
10/14/65
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
09/30/64
07/30/64
07/30/64
07/30/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/27/64
07/24/64
07/24/64
07/24/64
07/23/64
07/23/64
07/23/64
07/23/64
07/23/64
07/23/64
07/23/64

APPENDIX II.

DATA FOR KCl

80

 

81

GPERPENDICULAR LINE IN KCL

EXPERIMENTAL VALUES 0F T1 VS To
LEAST SQUARES FIT T0 EQUATION (5.9).

T
73.90
73.90
77.20
68.90
68.70

1.77
3.57
3.38
2.83
2.38
2.04
1.67
1.30
1.10
3.83
3.53
2.98
2.49
2.04
1.64
25.20
30.00
30.00
35.00
35.00
40.30
40.30
45.00
45.00
47.70
47.60
27.30
37.00
37.20
45.10
45.10
50.30
50.30
54.70
54.70
59.90
59.80
64.60
64.60
69.10
69.00

71(009)
7.0005005
7.5009005
5.5009005
9.5009005
9.2001005
5.3005001
1.0009001
1.0009001
1.5009001
206603001
300008001
6.5009001
8.2009001
7.8009001
1.5003001
1.9009001
1.3009001
1.3505001
3.6009001
3.6009001
4.9009003
1.9009003
293003003
1.2509003
1.420'003
6.6009004
6.5009004
4.6509004
400009004
3.8009004
4.0509004
2.3309003
2.730'003
1.0109003
1.0109003
4.2009004
4.3409004
3.4003004
3.6009004
3.0009004
3.350'004
2.0509004
1.9409004
1.3509004
1.430'004
8.1009005
8.6005005

T1(CALC)
6.035’005
6.035'005
4.899-005
8.364*005
8.4769005
4.051’001
2.009'001
7.122-001
795347001
3.013'001
3.515’001
4.294-001
50516'001
60519’001
10572-001
2.031'001
2.406'001
2.880’001
3.515'001
4.373*001
9.500’003
3.855’003
3.8559003
19736'003
1.736v003
8.750’004
0.750'004
8.299-004
5.299’004
40106'004
4.145’004
6.447'003
6.325'003
1.317'003
1.283'003
59248’004
8.248.004
3.272-004
34272'004
2.293'004
2.293‘004
095567004
1.568'004
1.119’004
10119-00‘
3.253’006
0.308'005

REL NT
0.0000003104
0.0000003563
0.0000001916
0.0000005717
0.0000005362

17.7940941970
0.6334672196
0.7388761649
1.4253012440
4.4405235492
5.7012049760

26.7639900255

42.5943358438

38.5401456375
1.4253012440
2.2868166625
1.0705596011
1.1544940076
8.2097351654
8.2097351654
0.0015209548
0.0002286817
0.0003351042
0.0000989793
0.0001277323
0.0000275938
0.0000267640
0.0000136971
0.0000101355
0.0000091473
0.0000103904
0.0003439030
0.0004721168
0.0000646200
0.0000646200
0.0000111744
0.0000119317
0.0000073229
0.0000082097
0.0000057012
0.0000071091
0.0000026621
0.0000023841
0.0000011545
0.0000012954
0.0000004156
0.0000004685

ALONG WITH 71 VALUES CALCULATED FROM

DATE
04/12/67
04/12/67
04/12/67
04712/67
04/12/67
04/20/66
04/20/66
04/20/66
04/20/66
04/20/66
04/20/66
04/20/66
04/20/66
04/20/66
04/06/67
04/06/67
04/06/67
04/06/67
04/06/67
04/06/67
04/07/67
04/07/67
04/07/67
04/07/67
04/07/67
04/07/67
04/07/67
04/07/67
04/07/67
04/07/67
04/07/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67
04/10/67

l
.-
’ .
\
a
‘ no
u
‘ .
s
n
. n
u

 

72.50
72.40
79.70
78.80
78.30
78.00
77.50
79.80
82.30
88.50
95.20
98.80
77.20
77.20
87.50
87.30
87.40
86.00
85.90
84.10
84.00
91.00
91.00
92.50
92.50
94.00
94.10
94.50
94.50
96.50
96.20
99.90
102.70
103.00
103.80
104.90
94.50
97.90
98.00
101.20
101.20
105.80
105.80
107.50
107.50
109.10
110.00
110.10
112.40
112.20
112940
112.40
114.20
114.30

7.5009005
8.0009005
4.4009005
4.7009005
4.9009005
4.6009005
5.3009005
4.0009005
3.6009005
2.3002005
1.5009005
1.7009005
4.3003065
4.8009005
2.5009005
20350’005
2.4009005
2.6009005
2.8009005
2.8509005
2.7002005
2.000'005
1.9002005
1.9009005
1.8502005
1.7709005
1.7009005
1.7509005
1.9009005
1.5509005
1.5503005
1.6009005
1.3509005
102009005
1.2?09005
1.1609005
1.6009005
1.5009005
1.500P005
1.2805005
1.2308005
101‘09005
1.1408005
9.2009006
9.9009006
9.3009006
8.0009006
7.7000006
695009006
6.8009006
7.7009006
8.5009006
7.7009006
7.1008006

82

6.6049005
6.6479005
4.1999005
4.4379005
4.576'005
4.6629005
49809'005
4.174'005
3.591'005
29513’005
107567005
1.466'005
4.899'005
4.899-005
206589005
296889005
2.673'005
2.8949005
209107005
3.230’005
3.249'005
2.191'005
2.191'005
9.022'005
2.022’005
1.868’005
1.859'005
1.820'005
{.820’005
1.6439005
4.668’005
1.389’005
1.216-005
10199”005
1.156’005
1.099'005
1.820'005
1.532’005
19525'005
1.305’005
1.305'005
19°56'005
1.056”005
99793'006
9.793'006
9.139'006
0.796’006
8.759’006
7.961.006
8.026'006
7.961'006
70961’006
70402-006
7.373.006

0.0000003563
0.0000004054
0.0000001226
0.0000001399
0.0000001521
0.0000001340
0.0000001779
0.0000001014
0.0000000821
0.0000000335
0.0000000143
0.0000000183
0.0000001171
0.0000001460
0.0000000396
0.0000000350
0.0000000365
0.0000000428
0.0000000497
0.0000000515
0.0000000462
0.0000000253
0.0000000229
0.0000000229
0.0000000217
0.0000000198
0.0000000183
0.0000000194
0.0000000229
0.0000000152
0.0000000152
0.0000000162
0.0000000115
0.0000000091
0.0000000094
0.0000000085
0.0000000162
0.0000000143
0.0000000143
0.0000000104
0.0000000096
0.0000000082
0.0000000082
0.0000000054
0.0000000062
0.0000000055
0.0000000041
0.0000000038
0.0000000027
0.0000000029
0.0000000038
0.0000000046
0.0000000035
0.0000000032

04/10/67
04/10/67
03/25/67
03/25/67
03/25/67
03/25/67
03/25/67
03/25/67
03/25/67
03/25/67
03/25/67
03/25/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/27/67

 

 

 

114.30
116.10
116.10
117.80
117.80
114.50
114050
118.80
118.70
120.40
120.40
122.90
123.00
124.10
124910
125.30
125.30
125.30
127.50
127.50
128.30
128.30
129.70
129.70
131.50
131.40
131.30
131.30
127.90
127.90
131.50
131.40
133.00
133.00
134.00
154.00
135.90
135.90
136.80
137.60
187.40
140.20
140.00
142.80
142.60
144.50
144.60
147.60
147.60
149.60
152.80
152.90
154.10
154.20

7.6005006
7.2009006
7.2009006
6.0005006
6.6009006
794009006

7.5009006-

5.7009006
5.2009006
5.8009006
5.7009006
5.2002006
4.9009006
4.9009006
4.830'006
4.SOOEO06
4.3008006
4.2603006
4.5003006
4.3509006
4.6009006
4.5009006
4.0002006
3.8000006
3.7009006
3.6009006
4.0008006
3.8005006
4.3409006
4.2000006
3.7009006
4.0005006
4.0009006
3.6002006
3.9000006
4.0003006
3.8009006
3.4008006
3.9909006
3.6009006
3.7009006
3.3502006
3.3509006
2.9508006
2.7208006
2.8609006
207‘09006
2.5203006
2.4802006
2.4009006
2.2409006
2.2002006
2.2408006
2.2409006

83

7.373’006
6.8689006
6.868’006
69434’006
6.434'006
7.314’006
79314’006
6.1969006
6.219.006
5.838'006
5.8389006
5.335.006
5.316'006
5.114’006
5.114‘006
4.906’006
4.906'006
49906'006
40554'006
4.554'006
4.434’006
404349006
4.236’006
4.236’006
3-998’006
4.011’006
4.024.006
4.024'006
4.493'006
4.493-006
3.998’006
4.0119006
3.814-006
1.814’006
3.698'006
1.6989006
1.492’006
3.492‘006
3.399'006
3.320’006
3.3409006
3.031'006
3009a'006
2.8659006
2.831'006
9.736'006
997299006
7.521'006
9.3969006
2.213'006
2.208'006
2.145*006
2.140’006

0.0000000037
0.0000000033
0.0000000033
0.0000000023
0.0000000028
0.0000000035
0.0000000036
0.0000000021
0.0000000017
0.0000000021
0.0000000021
0.0000000017
0.0000000015
0.0000000015
0.0000000015
0.0000000013
0.0000000012
0.0000000011
0.0000000013
0.0000000012
0.0000000013
0.0000000013
0.0000000010
0.0000000009
0.0000000009
0.0000000008
0.0000000010
0.0000000009
0.0000000012
0.0000000011
0.0000000009
0.0000000010
n.oonooono1o
n.oooonooooa
0.0000000010
o.aooooooo1o
0.0000000009
0.0000000007
0.0000000010
0.0000000008
0.0000000009
0.0000000007
0.0000000007
0.0000000006
0.0000000005
n.0000000005
0.0000000005
0.0000000004
n.oonoooooo4
0.0000000004
0.0000000003
n.0000000005
0.0000000003
0.0000000003

03/27/67
03/27/67
03/27/67
03/27/67
03/27/67
03/29/67
03/29/67
83/29/67
03/29/67
03/29/67
03729/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/29/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03130/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/57
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03/30/67
03730/67
03130/67
03/30/67

 
 
 
 
 

 

80

GPARALLEL LINE IN KCL

EXPERIMENTAL VALUES 0: T1 vs T. ALONG WITH f1 VALUtS CALCULATED FROM
LEAST SQUARES FIT T0 EQUATION (5.9).

T T1(OBS) T1(CALC) REL wT DATE
73.90 1.090-004 9.114-005 0.0000001964 04/12767
77.20 8.100-005 7.707-005 0.0000001068 04/12/67
73.90 9.100-005 9.114-005 0.0000001648 04/12/67
69.20 1.180-004 1.171-004 0,0000002267 04/12/67
69.00 1.200-004 1.184—004 0.0000002344 04/12/67
79.40 7.700—005 6.914-005 0.0000000965 06/25/67
78.90 6.700-005 .7.085-005 0.0000000761 06/25/67
78.60 7.600-005 7.190-005 0.0000000940 06/25/67
78.50 8.000-005 7.226-005 0.0000001042 06/25/67
77.90 6.700-005 7.443-005 0.0000000761 06/25/67
77.80 7.400-005 7.480-005 0.000000U891 06/25/67
77.50 7.400-005 7.592—005 0.0000000891 06/25/67
77.50 7.200-005 7.592-005 0.0000000844 06/25/67.
79.50 7.000-005 6.881-005 0.0000000798 06/25/67
79.90 6.800-005 6.748-005 0.0000000793 06/25/67
79.70 7.000-005 6.814-005 0.0000000790 06/25/67
82.20 6.200-005 6.044-005 0.0000000626 06/25/67
81.80 6.000-005 6.160-005 0.0000000586 06/25/67
82.30 5.900-005 6.015-005 0.0000000567 06/25/67
82.60 6.600-005 6.015-005 0.0000000646 06/25/67
88.50 4.600-005 4.595-005 0.0000000344 06/25/67
88.00 4.400-005 4.628-005 0.0000000615 06/25/67
97.10 3.000-005 3.167-005 0.0000000147 06/25/67
97.00 3.220-005 3.149-005 0.0000000169 06/25/67

101.70 2.150-005 2.612-005 0.00000000/5 06/25/67
100.90 2.700-005 2.694-005 0.0000000119 06/25/67
100.20 2.900-005 7.770-005 0.000000016/ 06/25/67
106.80 2.000-005 2.409-005 0.0000000065 06/25/67
103.50 2.500-005 2.409.005 0.0000000102 06/25/67
119.90 1.650-005 1.3/4-005 0.0000000060 06/25/67
118.20 1.500-005 1.451-005 0,0000000067 06/27/67
77.20 7.500-005 7.707-005 0.0000000916 06/27/67
77.20 7.400-005 7.707-005 0.0000000891 06/27/67
87.40 4.600-005 4.754-005 0.0000000601 06/27/67
87.50 4.250-005 4.762-005 0.0000000294 06/27/67
85.80 5.000-005 5.111-005 0.0000000407 06/27/67
85.90 4.600-005 5.088-005 0.0000000644 06/27/67
84.60 5.600-005 5.401-005 0.0000000457 06/27/67
84.50 5.600-005 5.4?6-005 0.000000045/ 06/27/67
90.90 3.450-005 4.072.005 0.0000000194 06/27/67
90.80 3.800-005 4.090-005 0.0000000265 06/27/67
92.50 3.100-005 3.801-005 0.0000000196 06/27/67
92.50 3.400-005 3.801-005 0.0000000188 06/27/67
94.20 2.700-005 3.567-005 0.0000000119 06/27/67
94.30 3.400-005 3.522-005 0.0000000158 06/27/67
94.60 6.500-005 3.522-005 0.0000000199 06/27/67
96.50 2.700-005 3.215-005 0.0000000119 06/27/67

 

96.50

99.90

99.90
104.90

94.50

94.50

97.80

97.80
101.20
101.20
105.80
105.80
105.80
107.60
107.60
109.00
109.00
110.00
110.00
112.60
112.40
112.40
114.10
114.20
116.00
116.10
117.90
117.90
114.40
114.50
118.60
118.60
120.50
120.60
120.50
120.40
126.00
126.00
124.00
124.00
125.60
127.50
127.50
128.60
128.80
129.70
129.70
161.20
161.10
128.00
128.00
161.60
161.60
133.30

2.850-005
2.350-005
2.400-005
2.000-005
3.300-005
3.500-005
2.720-005
2.770-005
2.420-005
2.510-005
2.000-005
2.070-005
2.150-005
2.140-005
2.150-005
2.100-005
1.990-005
1.700-005
1.830-005
1.750-005
1.760-005
1.630-005
1.85Dv005
1.760-005
1.700-005
1.740-005
1.600-005
1.570-005
1.500-005
1.480-005
1.350-005
1.300-005
1.500-005
1.380-005
1.400-005
1.260-005
1.210-005
1.220-005
1.160-005
1.230-005
1.130-005
9.800-006
1.100-005
1.080-005
1.030-005
9.400-006
9.500-006
1.000-005
8.700-006
1.060-005
1.000-005
9.900-006
1.030-005
9.700-006

85

3.?15-005
2.803-005
?.803-005
9.311-005
3.492-005
3.492-005
3.049-005
3.049-005
2.663-005
2.663-005
2.264-005
2.264-005
2.091-005
?.091-005
1.988-005
1.9889005
1.916-005
1.915-005
1.769-005
1.763-005
1.763-005
1.663-005
1.658.005
1.555-005
1.466-005
1.646-005
1.641.005
1.433-005
1.453-005
1.348-005
1.357-005
1.348-005
1.393-005
1.247-005
1.210-005
1.?10-005
1.163.005
1.090.005
1.055-005
1.049-005
1.023-005
1.023-005
9.891-006
9.829-006
1.0749005
1.074-005
9.774-006
9.7/4-006
9.247-006

0.0000000162
0.0000000090
0.0000000094
0.0000000065
0.0000000177
0.00000001/7
0.0000000120
0.0000000125
0.0000000095
0.0000000106
0.0000000065
0.0000000070
0.00U0000075
0000UUUUUU/5
0.0000000075
0.00000000/2
090000000064
090000000041
0.0000000025
0.0000000030
0.0000000090
0.0000U00046
0,000000U056
0.0000000049
O.UUUUOUUO4]
0.0000000049
0.0000000042
0.0000000040
0.000000006/
0.0000000066
0.000000U060
0.0000000028
0.0000000028
0.0000000061
0.0000U00062
0.0000000026
0.0000000024
0.000000U024
0.0000000022
0.0000000025
0.0000000021
0.0000000016
0.0000000020
0.0000000019
0.000000001/
0.0000000014
0.0000000015
0.0000000016
0.0000000012
0.0000000018
0.0000000016
0.0000000016
0.000UDOUOJ7
0.0000000015

06/27/67
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06/28/67
06/28/67
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06/28/67
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06/28/67
06/28/67
09/28/67
06/28/67
06/28/67
06/28/67
06128167
06/29/67
06/29/67
06/29/67
06/29/67
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06/29/67
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06/29/67
06/29/67
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06/29/67
06/29/67
06/60/67
06/60/67
06/60/67
06/60/67
06/60/67

 

166.10
164.00
164.00
166.00
166.00
166.70
166.60
166.60
168.00
167.80
169.90
169.90
146.20
146.10
144.80
147.60
147.60
150.00
150.20
152.70
152.80
24.40
24.60
60.20
60.20
65.00
65.00
40.60
40.60
45.00
45.00
47.50
47.50
28.75
29.10
66.20
66.50
45.10
45.10
50.20
54.80
54.90
60.00
60.00
64.70
64.80
69.20
72.60
72.20
4.18
4.18
6.88
6.46
6.04

8.700-006
9.500-006
8.800-006
8.500-006
9.400-006
8.550-006
8.800-006
8.900-006
7.900-006
7.700-006
8.400-006
8.250-006
7.400-006
7.800-006
7.400-006
7.000-006
7.200-006
6.600-006
7.000-006
6.200-006
6.500-006
5.700-005
4.000~003
2.650-005
2.?00-003
1.500-006
1.850-003
1.100-003
1.020-003
6.700-004
7.700-004
2.950-004
3.950-004
2.580-003
2.650-003
1.290-003
1.2?0-003
4.840-004
4.560-004
5.100-004
3.350-004
3.500-004
2.700-004
2.700-004
2.960-004
1.500-004
1.020-004
1.070-004
1.070-004
2.950-001
2.870-001
3.400~001
2.650-001
4.000-001

86

9.298-006
9.0/2-006
9.07?-006
8.597-006
8.597-006
8.4599006
8.461-006
8.461-006
8.156-006
8.199-006
7.767-006
7.767-006
7.151-006
7.169-006
6.8789006
6.435-006
6.465.006
6.087-006
6.059-006
5.725-006
5.716-006
1.603-002
1.552~00?
4.787-003
4.787-005
?.157-003
1.07?2003
1.0/2'003
6.485-004
6.485-004
5.155-004
5.165.004
6.317-003
5.899-003
1.8]5-003
1074(1’005
6.429-004
6.422-004
4.076.004
?.865-004
2.844-004
7.014-004
2.014-004
1.511-004
1.50?-004
1.171~UD4
9.908-005
9.961-005
3.7656001
3.7659001
4.056~001
4.549-001
5.177-001

0.0000000012
0.0000000015
0.0000000016
0.00UOU00012
0.0000000014
0.0000000012
0.000000U016
0.0000000016
0.0000000010
0.0000000010
0.0000000011
0.0000000011
0.0000000009
0.000000U010
0.0000000009
0.0000000008
0.000000U008
0.0000000007
0.0000000006
0.0000000006
0.0000000006
0.0005289146
0.0002604669
0.0001146214
0.0000/8/918
0.0000669254
0.0000557159
0.0000196980
0.0000169670
0.0000076076
0.0000096520
0.000001416/
0.0000025400
0.0001086616
0.0001146214
0.0000270904
0.0000242601
0.0000036165
0.0000066851
0.0000042642
0.0000018269
0.00U0019942
0.0000011868
0.0000011868
0.0000014266
0.0000006666
0.0000001694
0.0000001864
0.0000001864
1.4167064415
1.6409100015
1.8818875568
1.1462141622
2.6046886601

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04/06/67

 

 

2.41
1.79
1.80
6.57
6.68
2.86
2.68
2.68
2.68
2.68
2.04
2.04
1.66
1.60
1.10

4.400-001
.250+000
.000-001
.000-001
.000-001
.800-001
.500-001
.400-001
.900-001
.400-001
.700-001
.500-001
..090+000
9.600-001
1,100+000

HOVAAO‘VOCACA‘OH

87

6.550-001
8.7929001
8.743-001
4.408-001
4.656-001
5.561.001
6.613.001
6.613.001
6.613.001
6.615'001
7.715.001
7.715.001
9.481°001
1.211+000

3.1516762767
25.4664126956
15.1862666416

1.4651676715

1.4651676716
15.6346436816

9.15/1085704

6.665002969/

3.9066609205

3,1516/64157

9.6519994150
14.6960/19729
19.3414412608
15.0060066820
19.69/9579921

04/06/67
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APPENDIX III.
FORTRAN PROGRAM TOWPLOT

USED IN CURVE FITTING

88

 

 

(100000000GOOOOOOOOOOOOOOOOOGOOOCOQ

89

THE FOLLOWING LISTING OF PROGRAM TONPLOT CONTAINS THE THEORETICAL
RELAXATION RATES USED TO OBTAIN FIGURE 8 AND FIGURES 9 AND 10.

PROGRAM anpLoT

THIS PROGRAM IS FOR LEAST SQUARES ANALYSIS OF SPIN5LATTICE
RELAXATION DATA ANN INCLUDES FUNCTIONS FOR EVALUATING TRANSPORT
INTEGRALS. ALSO INCLUDED ARE OPTIONS FOR PLOTTING THE OBSERVED
AND CALCULATED POINTS: AND FOR PRINTING THE RESULTS.

THF FUNCTIONAL FORM OF EACH FIT IS INSERTED INTO SUBROUTINE DEFN.
EACH FIT IS MADE Tn ALL SETS OF DATA READ IN.

THE ORDER OF DATA TO BE READ IN IS AS FOLLOWS

(1)
(2)
(5)

NUNBER OF SETS OF DATA TO BE FITTED (I2).
NUMBER OF FITS PER DATA SET (I2).
THEN FOR EACH SET OF EXPERIMENTAL POINTS THERE SHOULD BE...
(A) HEADER CARD IDENTIFYING THE EXPERIMENT (1OAB).
(B) DECK OF EXPERIMENTAL DATA: EACH CARD HAVING
TEMPERATURE, T1. AND DATE (71001: E15.1o 15X: A8)-

ALL POINTS FROM THF SAME DATE SHOULD BE‘PLACED TOGETHER IN THE

DECK:

SINCE ALL POINTS FROM A GIVEN DATE HILL BE PLOTTED

WITH A UAIOUE SYMBOL.

IF NOTPRINT = 1: EXPERIMENTAL AND CALCULATED T1 ARE NOT PRINTED.
IF NOTRLGT = 1. N0 PLOT WILL BE MADE,
NOTPRINT AND NfiTPLOT ARE DEFINED IN SURROUTINE DEFINE.

THE SCALE FACTORS FOR PLOTTING ARE SY. SX; SYY. SXX: AND ARE
DEFINED IN SUBROUTINE DEFINE.

SY AND SX ARE SCALE FACTORS FOR PLOTS OF LOG(T) AND LOG(T1).

THEY ARE 100 TIMES THE NUMBER OF INCHES PER CYCLE ON THE LL PLOT.
SYY AND SXX ARE SCALE FACTORS FOR THE PLOTTING OF SYMBOLS

DENOTI
VALUE.

NG THE EXPERIMENTAL POINTS: AND SHOULD BOTH HAVE THE SAME
THEIR VALUE IS 1000 TIMES THE DESIRED HEIGHT (IN INCHES)

OF THE SYMBOLS.

COMMON
COMMON
COMMON
COMMON
COMMON

T(500).TOH(500)6TOHIN(500)ouAT(500).wAITGSOO).NLO.NHI.NEX
L1N(500). NDATEISOO). NDP. HIPLUS. LTH. NOVAR. NVLESS
DATAISI. VECTOR(6:6). AVE(5), COENtS). SIGMCO(5).SIGMA(5)
NOEATE: KDATFISDOIaSIGY: NOIN: NNNT: INDEXI5)

SY. 9X. SYY. SXX, NOTPRINT, NOTPLOT

DIMENSION LHEADIIOI
TYPE DOUBLE COEN'SIGMA!SIGYIVECTORIDATAIwAIT
LTH = U
READ IN NUMBER OF SETS OF DATA
READ 1. NLINS ’
READ IN NUMBER OF FITS PER DATA SET

READ 1, NFITS
DO 500 K3 10 NLIMS

READ I

N READER CARD IDENTIFYING SET OF DATA

 

1RD

105
109
110
115

120

125

400
450
460
500

90

READ 10. (LHEADIL). L31.10)
READ IN NEASURED T AND T1
KDATEIi) a 1
READ 5. 7(1),an(1).VDATE(1>
DO 110 I=2.500
READ 5. T(T). TONII). NDATF(I)
IF (NDATF(I) * NnATE(I-1)> 100. 105. 100
KDAYEII) = KDATF(I-1) * 1
Go TO 1n9
KDATEII) = KDATEIIv1)
IFIYII)) 915. 115. 110
NDP = I
CONTINUE
CALCULATE HEIGHTS FOR LEAST SQUARES FIT
SUMNAT = 0.0
DC 120 I: iaNDP
TONIN(1) a 1.0 / TOW(I)
wAT(I) = Towc1> e T0u(1)
SUMNAT = SUMHAT a JATII)
AVGNT = SLMWAT I NDP
DO 125 I = 1.NnP
HAITI!) = HAT(1) / AVGNT
DU SUD NEN : 1: NFITS
PRINT 1n. (LHEAn(L). L=1:10)
CALL STEPREG
1F(NOTPRIAT9 400a 4n0; 450
CALL VARFIT
1F<NOTPLOT) 460. 46m. 500
CALL GRFTOH
CONTINUE
STOP PLOTTER
CALL pLUTIHIPLUS: 2.50 '1: ST: SX)
FORMAT(!2)
FORMAT(F1n.1. 515.1. 15x. A8)
FORMAT (1nA8)
END

 

0000000

00

0

0

00000000

10

P0

30

4O

50

99

91

SUBROUTINF DEFINE (X: Y)
COMMON TISOO).ant5n0)oTONIN(500).NATI500).UAIT(500).NLO.NHI.NEX
COMMON LIAIBOB). NDATEISOO). NDP. HIPLUS. LTH. NOVAR. NVLESS
COMMON DATAISIa VECTORI6ab). AVEIS). COENIEI. SIGMCDI5).SIGMA(5)
COMMON NOBATE. KDATFISOOIOSIGY. NOIN. NNNT. INDEXISI '
COMMON sv. sx. xvv. sxx. NOTPRINT. NOIPLo?
TYPE DOUBLE CDEN.SIGMA.SIGY.VECTORODATA.HAIT
TYPE DOUBLE u. v. w. z
sv=115. s sxszzs.

sxx a 56. s svv = 66.

NOTPRINT a n s NOTPLOT = o

AFTER STATEMENTS 1n YHRU SO THE NUMBER OF VARIABLES TO BE USED IN
IN EACH :1? IS DEFINED av A STATEMENT OF THE FORM

AOVAR a N
NHFRE N IS AN INTEGER CONSTANT EQUAL T0 (1 * THE NUMBER OF
COFFFIDIFNTS Tn 8E DETERMINED IN THE LEAST SQUARES FIT).

GD T0 <10. 20. x0. 40.50). NEX
CONTINUE

NOVAR = 4

GO TO 99
CONTINUE

NOVAR = 3

GO TO 9°
CONTINUE

GO TO 99
CONTINUE

GO To 99
CONTINUE

NVLESS = NOVAR - 1
RETURN

iéttO Obit. tottt «tit. ttttt tittw ttttt titfli tittt

ENTRY DEFN

THE PARAMETERS INPUT TO THIS SURROUTINE FROM STEPREG ARE x. THE
MEASURED TEMPERATURE AND Y. THE INVERSE OF THE MEASURED T1.

AFTER STATEMENTS 100 THRU 500 THE FUNCTIONS TO BE USED IN THE FIT
ARE DEFINED RY STATEMENTS OF THE FORM

DATAII) = BLAH BLAH

DATAIZI : RLAH

OATAISI a ETC
THF DEPENDENT VARIABLE IN THE LEAST SQUARES FIT MUST BE

00000

1ND

2ND

3ND

4ND

500

999

92

EQUATED T0 DATA(M), NHERE M = NOVAR.

NOTE THAT FOUR DUMMY DOUBLE PRECISION VARIABLES HAVE BEEN

PROVIDED FOR CONVENIENCE IN CONSTRUCTING FUNCTIONS.

THESE DUMMY VARIABLES ARE U: V: “D AND 29

GO TO (100. 200. 300, 400. 500), NEX
CONTINUE

V B TRASPRTIRA 17090 X)
U 3 Y $ 2 8 X $ N 3 Z**9 $ DATAI1)
EATAI49 = U 3 HATA(3) = DEYP(fi575./ Z)

60 TO 999
CONTINUF

V = TRNSPRT(RO 170.: X)
U 8 Y S Z = Y $ N = Z**9 $ DATAI1)
UATAI3) = U

GO TO 999
CONTINUE

GO TO 999
CONTINUE

GO TO 999
CONTINUE

RETURN
END

2

2

$

$

DATAIZ)

DATAIZ)

*N

*N

 

000000

170

125

35

40

93

SUBROUTINF VARFIT

THIS SUBROUTINE CALCULATES THE STANDARD DEVIATION. CHECKS To
SEE THAT THE SUM OR THE HEIGHTS Is EOUAL TO THE NUMBER OF DATA
POINTS. ANn CALCULATES (FROM THE FIT) A T1 VALUE FOR EACH
EXPERIMENTAL TEMPERATURE.

COMMON
COMMON
COMMON
COMMON
COMMON

TI5OO).T0NI560)ATONINISOO):NATI500)ONAITIBOO):NLOoNHI.NEX
L1N(500). NDATEISUO): NDP. HIPLUSo LTH: NOVAR. NVLESS
DATAIS): VFCTOR(6O6)Q AVE(5)n COENI5): SIGMC0(5)’S!GMA(S)
NODATEA KDATE(500’ISIGYO NOINA NNNT. INDEXIS)

SY. 5X. SYY. SXX. NOTPRINT: NOTPLOT

TYPE DOUBLE cOEN.BIOMA.SIGY.VECTOR.DATA.NAIT

TYPF DOUBLE CALC

PRINT 2%

25 FORMATciHO.AX1HT.OX7HT1(OBSI.BXBHT1ICALCI.4x12M1.0/T1 (088).
1 4X13H1.n/T1 (CALC,16XSHDEV09X6HREL NTA12X4HDATE)

SUMSQ :
SUMNAIT

DO 125

0.0
= 090
N = lgNDP

SUNHAIT = SUMHAIT + NAITIN)
CALL DEFNITIN): 0.0} S CALC = 0.0

DO 120
CALC :
DEV =
TIHCAL

K 5 1: NOIN
CALC * COEN (K) * DATAIINDEXIKII

TOHININ) a CALC

PRINT 30:7(N)OTOHINI1TIMCALATOHININ)oCALCaDEVaflAITIN)oNDATEIN)
SUMSQ = SUMSO o DEV*DEV*WAIT(N)
FORMAT(F10.?A 5F15.1. F15.10: 7XA8)

XNDP s
VAR? =

NDP t 1
SQRTV (SUMSQ/XNDDI

PRINT 39. VARF

PORMAT<//. 17H VARIANCE or FIT: F15.9>
PRINT 4n. SUMNATT

FDRMATIII. *SUM or UEIBNTS =t F15.9I

END

00000

94

FUNCTION TRNSPRTINN.DOT)

THIS FUNCTION CALCULATES THE TRANSPORT INTEGRAL OF ORDER NN
FOR DERYF TEMPERATURE D AND TEMPERATURE T,
NOTE THAT FUNCTIONS TINTORND AND SIMPSON ARE REQUIRED.

COMMON INN/N
EXTERNAL TINTGRNn
N=NN
n = n.00001
TRNSPRT = SIMPSON (0. D/T. 0.001. TINTGRNDI
RETURN
END

 

(1CIO

95

FUNCTION TINTGRND (2)
THIS FUNCTION DEFINES THE INTEGRAND FOR FUNCTION TRNSPRT.

COMMON INN/N
TFIZ-3OOI 2: 11 1
TINTGRND = D
RETURN
E2 = EXPF (2)
F21 = El 2 1.0
FliSn 8 E21 9 521
TINTGRND 3 El * (Z**V) / E2150
RETURN
END

 

000

99

93
97

95

94

FUNCTION SIMPSONIAA.BAERRpFCT)
THIS FUNCTION EVALUATES THE TRANSPORT INTEGRAL BY SIMPSONS RULE.

DIMENSION DXISO).EPSPISOIAX2ISOIOX3(30)2F2(30):F3(30)oF4(30):
1 FMPISR)OFBPISDIOPVALISOOSIoLVF(30).EST2(30)aESTSISO)
THIS IS A TRANSLATION OF A STANFORD SUBALGOL PROCEDURE

A=AA

EPSzFRR

PSEUOO PARAMETER SETUP. ETC.
LVL=O
ABSAREA=1.0
FST=1.0
FARFCTIAI
ABZ=IA¢8I/2.0
PM=4,00FCTIABZ)
FB=FCT¢BI
DA‘BBA

LVL=LVL+1

DXILVLI=DA/3.0

SXIDX(LVL)/6.0

ADX23A¢DX(LVL)/2.0

P134.0*FCT(AOX2)

X2(LVL)=A+DXILVLI

V2L=¥2ILVLI

FZILVLI=F6T(X2L)

Y3(LVLI=X2(LVL)*RX(LVL)

Y3L=¥3ILVLI

F3¢LVLI=PCT(¥3L)

FPSPILVLIzEPS

Y30=¥3ILVLItDXILVL)/2.0

R4¢LVL)=4.O*FCT(¥3DI

EMPILVLIBFM

ESTl=(PA¢F1*F2(LVL))tSX

FBP(IVL)=F8

ESTZILVLI=¢F2<LVL)*F3ILVL)#FMIPSX

ESTSILVLI=IFSILVL)+F4ILVL)*FB)*SX

SUN=EST1+ESTRILVL)OESTSILVL)

ABSAREAzABSAREAvABSFIESTI+ABSFIEST1)+ABSF(EST2ILVL>I
I OABSFIESTSILVLI)

IFIARSFIESTaSUMIBEPSPILVL)tARSAREA)98.98.97
IFIEST-1.0)96.97a96
IFILVLBSOI92.9S.O6
DONE ON THIS LEVEL
IFILVL930393.95.93
VALI‘ESTHSHM
VAL2=ABSAREAtePSPILVLI
PRINT O4.VAL1:VAL2
FORMAT (74HTHE RECURSION HAS DESCENDED T0 LEVEL 30: NITHO
TUT SATISFYING THE TOLERANCE./37HTHE ESTIMATE OF THE ABSOLUTE ERROR
2 IS.E16.8/4SHAND THE TOLERANCE REQUIRED AT THIS LEVEL IS.E16.8/)

96

93

92

91

90

39

38

END

97

VL91 ,
LVL;:£LVE¢LVbF)=SUNAVE
LE:L(LVL LS: i9).LE
Pv I 91, ,

O I
60 T

IVE<LVL):}
BAan(LV

FM=F1 VL) 1‘?
‘Béfiéémvw
EET=RSTé

GO TO 9

)=2

E(LVL
LXIDX<LVt§
TA:F?(LVVL,
PMsFMPTLL) 7
FB=F3(:g(LVL)/1.
”ifES;2.LVL,
::X2(LVL)

no In 99

‘ )=3

E(LVL
RX=DxtL3ti
EA=F3¢LVL)
rmzracL VL) 7
FaaFapét(LvL,/1,
FpgfggTSILVL)
§§¥3TLVL>
do ro ¢9

)
VLa3
LIL
)iPVA
)+PVAL(LVL¢2

Lol

ALILV. ’91

SU7::t!158::RB

éiMPSON=SU.

RETURN

 

OC1CJOC1CJOC7

OC?C)OC3C)O

110
130
120

530
550
560
540
510
565

98

SURROUTINF STEPREG
SIMPLE STEPUISE REGRESSION

THIS SUBROUTINE PERFORMS A LEAST SQUARES FIT TO OBTAIN

COEFFI
Y

CIENTS IN AN EXPRESSION OF THE FORM
8 A(1)*X(1) i A(2)OX(2) +

0000,000000

THE VARIABLES 1(1) AND Y ARE THE SUBSCRIPTED VARIABLE DATA(N)
DEFINED IN SUBROUTTNE DEFN.

COMMON
COMMON
COMMON
COMMON
COMMON

T(SOO).TOHCSOO).TONINTSOQ):NAT‘SOO):NAIT(500),NLO.NHI.NEX
LI~(500), NDATE(500). NOP. HTPLUS. LTH, NOVAR. NVLESS
OATA¢5). VECTOR(6:6). AVE(5). COENTS). SIGMCO<5).SIGMA<5)
NODATE. KDATE(500).S!GY. NOIN. NNNTa INDEX(5)

sv, sx, SYY. sxx. NOTPRxNTa NOTPLOT

TYPE DOUBLE COEN.919MA.SIGY.VEOTOR.OATA.RAIT

IFSTEP = 1, no NOT PRINT EAOH STEP
{PRAN : 1 DO NOT PRINT RAH SUNS AND SQUARES
IPAVE = 1 no NOT PRINT AVERAGES
[FRESH 2 1 no NOT PRIVT RESIDUAL SUMS SQUARES
[FCOEN : a DO NOT PRINT PARTIAL COEFFICIENTS
{PPRED : 1 no NOT CALC PREDICTED VALUES
IPCNST a 1 no NOT HAVE CONST TERM IN EQUATION
FORMAT(/)
EPIN=0.0000001
EFOUT=0,0n00001
TOL = .00001

IPCNSTzi
IFSTEP : 1
IFRAH = 1
IRAVE = 1
IFAVG =
IFRFSD a 1
IPCOEN a 1
CALL DEFINE (n.n. 0.0)
{NVAR = NOVAR
NOIN : O
VAR = O
K = n
FLEVEL : fl
NOENT = 0
NOMIN = 0
NOMAX = Q
NVP1 : NOVAR + 1
no 120 I s 1. NVP1
DO 320 J 8 1. NV91
VECTOR(!.J) = 0.0

no 510 N: 1.NDP

CALL DEFN(T(N). TOAIV(N))
Do 540 I 8 1. NOVAR
VECTOR(I,AOVAR t 1) = VECTOR(I, NOVAR + 1) + DATA(I) . NAIT(N)
D0 540 J 3 1: NOVAR
VECTOR(!. J) : VECTOR (I. J) + DATA (1) w DATA(J) t NAIT(N)
VECT0R(NV913 val) S VECTOR‘NVPl: NVPl) * WAITIN)
NQVMI = NOVAR . 1

 

566

650
735
651
652

655
670
660

680
690
780
7B

782
790
701
792

793

794
7°6
797
795
810
800
8?0
840
841
850
830
1000
1001
1002

1010
1015
1016
1017
1018

10?0
1030
1035
1040
1041
1042
1043
1045
1060
1080

n / VECTOR (NOVPL:

99

NOVEL = NOVAR * 1
CALCULATICN OF RESIDUAL SUNS 0F SQUARES AND CROSS PRODUCTS
IP‘IFCNST, 90026511735
GO TO 780

[FIVECTORCNOVpLgNOVDLII 652:652o655
PRINT 634
GO TO 910
DO 660 I 3 in NOVAR
DO 660 J = I, NOVAR
VECTOR (InJI = VECTOR (Ind) 9
NOVPLII

I = 10 NOVAR

a VECTORIIONOVPLI / VECTORINOVPLpNOVPL)
: Q1

(VECTOR(IoNOVRL) * VECTOR (J1N0VPL)

DO 690
AVEII’
NOSTEP
ASSIGN 1320 TO NUMBER

DEFR : VEOTORINOVRLpNOVPL) 9
DO 800 I = ipNOVAR
1PIVECTOR¢I.I)) 702,794,810
PRINT 793; I

GO TO 910

FORMAT (SIR ERROR RESIDJAL SQUARE VARIABLE I4.31H
LEM TERMINATED 9

PRINT 795. I
SIGMAII) 3
GO TO 800
FORMAT (1H010H VARIABLE 15,13H IS CONSTANT )
SIGMAII) ODSQRT (VECTOR (1,1))

VECTOR(I.I) : 1.0

DO 830 I laNOVMI

191 = I .
DO 830 J : IP19
VECTORIIaJ) = VECTORII:J) /¢ SIGMAII)* SIGMAIJ))
VECTOR(J.I) = VFCTOR(I:J)

MOSTER = 00$TEP * 1

IF (VECTORI NOVAR.NOVAR)) 1002.1002:1010

1.0

18 NEGATIVE:PROB

10D

1
NOVAR

NSTPni = ROSTER . 1

PRINT 1004. NSTPMi

GO TO 1331

SIGV a SIGMAINOVAR) *DSQRT (VECTORINOVARaNOVAR)/ DEFR)
DEFR =DEFR-1.0

1F (DEFR ) 1017:1017. 1020

PRINT 1019 .NORTEP

PRINT 1019. NOSTEP
GO TO 1381

VMIN I 0.0

VMAX a 0.0

NOIN = 0

Do 1050 I 3 lnNOVMI

IF (VECTOR TIJII) 10420105011060

PRINT 1044: I: NOSTFP

PRINT 1044: In NOSTEP
GO TO 1381

IF‘VECTORIlfil) m TOL) 10501108001030

VAR = VECTOR(I,NOVAR) * VECTOR(NOVARnI) / VECTORIIoI)

 

1090
1100
1120
1130
1140
1190

904

1170
1180
1190
1160
1110
1210
1220
1050
1230

903

1240
1260
1245
1246
3247
1250
1270
1290
1230
1300
1310
1311
1312
1313
1314

1315
1320
1330
1340
1345

1350
1360
1361
1370
1390
1391
1392
1393
1394
1400
3420

100

IFIVAR)1100.1050s1110
NOIN a NQIN O 1
INDEXINOIA) = I
COEN(NOIN)
SIGMCOINOIN) =
I? (VMIN) 1160at170.904

PRINT 906
PRINT 906

GO TO 910

VMIN 3 VAR

NOMIN = I

GO TO 1050

IPIVAR . VMIN)1050.1050.1170
IF (VAR . VMAX)1050.1050'1210

VMAX a VAR

NOMAX = I

CONTINUE

IF (NOIN) 903.1240,I245
PRINT 907
PRINT 90?

GO TO 910

PRINT 65: SIGV
GO TO 1350

IF (IFCNST) 900.12SO,1246
CNST = 0.0

GO TO 1300

CNST B AVE(NOVAR)

DO 1280 I = inNOIN

J 3 INDEXIII

CNST a ONST e (COENIII r AVEIJII

IFIIFSTEP) 900'131001320

IF INCENT) 1311,131I:1313

PRINT 91,NOSTEP. K

00 TO 1314

PRINT 92.NOSTEP. K

PRINT 70aFLEVELc SIGY.CNST.
IINDEXIJ,ICOENTJIOSIGMCO(J,I J = 1: NOIN I

00 T0 NUMBER. (1320,1580)

FLEVEL e VMIN * DEFR I VECTOR INOVARINOVAR)
IFIEFOUT t FLEVEL) 1350. 1360; 1340

K = NOMIA

NOENT = 0

GO TO 1391

FLEVEL = VMAX * DEFR I (VECTORINOVARaNOVARI-
IF (EFIN n FLEVEL) 1370.1361.1380

1: (5010) 1380:1380-1370

K = NOMAX

NOENT = K

IFIK) 1392.1392,1400

PRINT

PRINT 1395:
GO T0 910
00 3410 I = 1.NOVIR
IF II-K) 1430.1410o1430

VMAX)

1395: NOSTEP

NOSTEP

a VECTOR(I:NOVAR) . SIGMAINOVAR) / SIGMA (I)
(SIGV / SIGMAII)) *DSQRT (VEOTORII.I))

 

101

1430 DO 1440 J = 1, NOVAR
1450 IF (JQK) 1460-144nn1460

1450 VECTOR(I:J) = VFCTOR(I-J) 9 (VECTORIIaK) ' VECFOR (KIJ) / VECTOR

EIK.k)I
1440 anTINUF
1410 CONTINUF
1470 D0 1480 I
1490 IF IIaKI 1
1500 VECTOR (I.
1480 CONTINUE
1510 DO 1520 J = 1. NOVAR
1530 IF IJ-K) 1540.1520.1540
1540 VECTOR(KpJ) = VECTOR IK.JI / VECTOR (K.K>
16?0 CONTINUE
1550 VFCTOR(K.K) : 1.0 / VECTORIKaK)

1550 GO TO 1000
1380 PRINT 75. NOSTEP
1381 IF (IFSTEP) 900, 1500:1570
900 PRINT 905
GO To 910
1570 ASSIGN 1580 T0 NUMBER
1571 00 10 1310
1500 PRINT 1585,(L.VECTOR(LoL):L=1.NOVMI )
910 CONTINUE
65 FORMAT (25H0 STANDARD ERROR OF Y = F12.6 I

1: NUVAR

500-1480:1500
K) a - VECTOR (1.x) / VECTOR (K.KI

70 FnRMAT I11H F LEVEL F12.4/25H STANDARD ERROR 0F Y = F12.4/12,
1 CONSTAAT F13.5/55H VARIABLE COEFFICIENT STD ER

20R nr coer // c150 x-13.515.5.215.5))
75 FORMAT (10H COMPLETED 15.200 STEPS 0F REGRESSION)
o1 FORMAT I9RUSTEP No.75 /19H VARIABLE REMOVED 18)
02 FORMAT «9009159 No.75 /2oa VARIABLE ENTERING 18)
654 FORMAT c310 zeno NuMBER or DATA. so LONG.)

905 FORMAT (42H ERROR IN CONTROL CARD. PROBLEM TERMINATED)

906 FORMAT (25H ERROR. VMIN PLUS. SOLONG)

907 FORMAT czaH ERROR-NOIN MINUS. SOLOVG )

1004 FORMAT (1H037HY SQUARE VON—POSITIVE.TERHINATE STEP
1019 FnRMAT (1H099H NO MORE nEGREES FREEDOM STEP I 5 )

I 5)

1044 FORMAT I1“010H ROUARE XnI5ai7H NEGATIVE. SOLONG 15:6H STEPS)

1395 FORMAT (12H K=0. STEP 16a 7H SOLONG)

1536 FORMAT (24H0 DIAGONAL FLEMFNTS //20H VAR.N0.
1(1H I 7. F16,6))
RETURN
END

VALUE/I

 

OCDC30

600

605

606

610

620

102

sUBROUTINE earrcw

T0 PLOT EXPERIMENTAL AND CALCULATED T1 VERSUS TEMPERATURE
ON A L00 L00 SCALE.

COMMON T(%00).TON(500).TONIN(500).HAT(500).HAIT(500).NLO.NHI.NEX
COMMON LIN(500). NDATEISOO). NDP. HIPLUS. LTH. NOVAR. NVLESS
COMMON DATAIS). VECTOR(6.6). AVEIS). COENIS). SIGNCOIS).SIGMA(S)
COMMON NOBAYE. KDATE(500).SIGY. NOIN. NNNT. INDEXIS)
COMMON SY. sx. SYY. sxx, NOTPRINT. NOIPLOT
DIMENSION TCISDR). VKURVISoo)
TYPE DOUBLE CDENISIGM‘DSIGYQVECTORIDATA'NAIT
TYPF ROUBLE YKURV
TYPE DOUBLE BSCHG
BSCHG = 0.4342944819
PLOT T1ICALC) VS TEMP
TCflI a 1.0
no 600 K a 2990
TCIK) = TCIKa1I ¢ 0.1
70(91) = 10.0
00 605 K = 92. 370
TCIK) a TCIKal) + 0.5

no 610 K 3 1:370

YKURV(K) S 0.0 3 CALL DEFNITC(K)A 0.0)

no 606 J g 11 NOIN

YKURVCK) 3 YKURV(K) * COENIJ) t DAIAIINDEXIJII
YKURV(K) 3 DLOGF (YKURV‘K)) * BSCHG

TCIKI = LOGF (TCIK)) t BSCHG
ESTABLISH ROUNDS FOR PLOTTER
LORNU = XFIXFI YKURV(370)) w 1 $ XLOBD = LOBND
KIRND a YFIXF (YKURVI1I) + 1 0 XHIBO = KIBND
INITIALIZE PLOTTER
CALL PL0T(0.OI 0.0. 0! 1000! 1000)
CALL PLOTI0.0. 30.. 2. 100.. 100.)
CALL anT (XLOBHC 2.5: 0: 8Y0 8X)
SET UPPER ROUNn 0N LENGTH OF PLOTTPR PAPER
LTH = 6 t 4 t XABSFIKIBND , LOBND) * LTH
CALL PLOT (LTH. 0.0. 3)
CALIBRATE PLOTTER PAPER AND PLOT CURVE
CALL PLOT (XLOBD: 0.00 1: SY: 8X)
CALL pLOT (YHIBna 0.0. 1: SY. SX)
CALL PLOT (YKURVI1). TCIl). 2)
D0 620 K = 2.370
CALL PLOT (YKURVIK) , ICIK). 1)
CALL PLOT (XLOB0. 2.0. 2)
CALL PLOT (YHIBH: 2.0: I)
cALL PLOT (XHIBn. 0.0. 1)
LHIP a XABSFIKIRND a 1) $ LLOM = XABSFILDRND + 1)
00 630 L 3 LHIP: LLONa 2
YL = -L
CALL PLOTIYLn 0.0. 2)
CALL PL0T(YL) 9.5: 1)
LH = ~L'1
IF (LN " LOBND) 635. 635. 625

625
630
635

650

103

YL = L" $ CALL PLOT (YL: 2.5:
CALL PLOT (YL: 0.0: 1)
CALL PLOTIXLOBD. 1.0: 2)
CALL PLOTIXHIRD, 1,0. 1)
PLOT EXPERIMENTAL POINTS
DO 650 K = 1. N09
KAR : KDATEIK’
EX = LGGP(TIKII v BSCHG $ NY
CALL PLOT‘NYD Ex: 2' SY’ SX)
CALL CHARAC (NYaPXI KAR.SY0 SX:
HIPLUS é XHIBD * 4.0
CALL PLOT (HIPLUS. 7.5: 2: SY: SX)
END

2)

= LOGFITONIK)) * BSCHG

SYY.

SXX)

100

SUBROUTINE CHARACIYP: XP: KT: 3Y0 SK: SYYp SXX)
TO PLOT SYMBOLS AT THE EXPERIMENTAL POINTS.
DIMENSION YI6). XI6)
GO TO (100.200.300.400.500.600.700.800.900.1000.1100.1200)o KT
To PLOT A DIAMOND AROUT POINT (X.Y)
100 ASSIGN 150 To NUMBER
00 T0 405
150 CALL PLOT (VP. XIi). 2. SYY. SXX)
CALL PLOTIYf4). XP. 1)
CALL pLOTIYP. XIZI. 1)
CALL RLOTIYIS). XP. 1)
CALL pLMIYP. XI1>. 1)
CALL PLDTIYP, xv. 2)

RETURN

To PLOT A RIGHT POINTING TRIANGLE ABOUT POINT (X. Y)
200 A = .1
205 xt1I= xv + A l1.739

Y¢1I= YP

 

XI25= XP 9 A l3,464
207 YIZI= YP-e A /2.0

XI35= Xf2)
VISA: YP t A /2.0
YI4I= YI1)
XI4I=X(1)

208 CALL PL0TIY<1). XIII. 2. SYY. SXX)
210 CALL PLOIIYIJ). XIJ). 1. svv. sxx>
CALL PLRTIYP, X». 2, SYY. SXX)

RETURN
TO PLOT AA X AT POUNT (x.Y)

300 ASSIGN 350 TO NHMRER

305 A= .1
X‘1,= XP * 9354 . A
YI1I=YP a .354 o A
XISI: XP 5 .354 *A
YI3I= YR A .354 t A
XI4I= XIS)

7(4): Y¢1I
XI2)= XI1)
VIZ): YIS)
90 T0 NUMBER. (350.550)

350 cALL PLoTcYI1). XI1I. 2. SYY. SXX)
CALL pLfiTIY‘S’: XI35: 19 SYY: SXX)
CALL PLOTIYIZ’: X‘2,! 2: SYVQ SXX)
CALL PLOTIYt4). X(4). 1. SYY. Sxx)
CALL PLOTIYP. xp. 2, SYV. Sxx>
RETURN
TO PLOT A CROSS AT ROINT (X.Y)

400 ASSIGN 450 TO NUMRER

405 A = .1
XI1I= XP * 95 * A
XI293 XP 9 .5 * A
YI313 Y? a ,5 t A
YI4I= VP 6 ,5 t A

C

450

500
550

550

600

700

707

800

900

905

910

915

105

GO TO NUMEERI (4500 150)

CALL PLOTIYP. XI1). 1. svv. sxx>
CALL PLDTIYPO XCZ): 10 SYY» SXX)
CALL PLOTIYI3): XP: 20 SYY. SXX)
CALL pLOTIYI4’: XP. 10 SYY: SXX)
CALL anT(YP, XP: 2, SYY: SXX)
RETURN

TO PLOT A SQUARE AT POINT (XOY)
ASSIGN 550 T0 NUMBER

90 TO 305
CALL PLOTIYII): XII): 2. SYYO SXX)
XI5)= XI1)
YI55= YIi)

DO 560 J=1.5
CALL PLOTIYIJ). XIJ). 1. SYY. SXX)
CALL PLnTIYP, XP. 2, SYY. SXX)
RETURN
TO PLOT A LEFT POINTING TRIANGLE ABOUT POINT (x, Y)
A = .1
YI1) = YP
XIZ) 3 XP + A/3.464
GO TO 207
TO PLOT AA UprRD PRINTING TRIANGLE ABOUT POINT (X. Y)
A = .1
XI1) = x9
YI1) = VP w A/1,732
YI2) t YP ~ A/3.464
XIZI 3X9 * A’ZQO
XIBI I xP - A12.0
YIS) I VI?)
XIAI I XII)
YI4) 8 VII)
GO TO 208
TO PLOT A DONNNARO ROINTIVG TRIANGLE ABOUT POINT (X. Y)
A = .1
XIII = YR
YI1) = YP - A/1.732
Y¢2) = YP . A/3.464
GO TO 707

TO PLOT A DONNNARD POIVTING Y ABOUT POINT (x.Y)
A = .1 T A0 a .394 t A
XI1) = xP + An S VIS) 3 XP 9 A0 T XI4)
YI1) = YIS) = YP 4 A0 0 YI4) = Y? 0 A0
XI2) = XI!) : XR 5 YI2) = YIS) 8 YR
CALL PLOTIYI1). xcl). 2. SYY. SXX)
D0 910 J 3 103
CALL PLOT (YIJ). X(J)a 1)
CALL PLOT (VIA). VIA). 2)
D0 915 ..I 3 4.5
CALL PLOTIYIJ). XIJ). 1)
CALL PLOT (YP. KP. P)
RETURN

TO PLOT AN UPHARO POINTIVG Y ABOUT POINT (Y.Y)

XP

 

1
i

106

A = 01 $ A0 3 9394 * A
XI1) = XP 9 A0 $ XIS) = XP + A0 3 X(4) = XP
Y(1) = V(3) = Y9 - A0 S Y(4) = YP * A9
GO TO 905
T0 PLOT A LEFT POINTING Y ABOUT POINT (X'Y)
A = .1 T AG 3 .354 * A
XI1) ' XIS) = XP * A0 S XC4) : XP 9 AQ
Y¢15 3 VP * A0 $ Y(3) = YP ' A0 1 Y(4) = YP
GO TO 905
TO PLOT A RIGHT POTNTING Y ABOUT POINT (XoY)
A = 01 fi AG 3 .354 * A
XI1) I XIS) = X9 . A0 S XI4) = X? + AU
YI1) = VP - A0 3 vIS) = Y? A AG 1 YI4) = YP
GO TO 905
END

 

APPENDIX IV.

ASSIGNMENT OF WEIGHTS IN CURVE FITTING

 

107

 

108

Use of the least squares analysis for fitting
theoretical functions to the experimental data requires

minimizing the weighted sum

2
1g % -% wi , (ALLl)
1obs. lcalc.
1 l
where N is the number of data points. The method of as—
signing values to the weights wi will now be discussed.
Let Yi and yi be the ith observed and calculated

values, respectively, of a dependent variable. For the

set of such values, the least squares condition is that

(Y1 — yi)2wi (Au.2)

IIMZ

i 1

be minimum. Let AYi be the error in the measured value

Yi' Then from the theory of statistics,30
w my >2 = 0 (Au 3)
i i ’ '
where n is a constant. Hence
_ n
wi — —2. (MA)
(AYi)

In our experiments, we estimate that the percentage error
in the measured Tl values is constant over the entire

range of values, i.e.,

—= (:3 (AMoS)

 

109

where C is constant. Combining (A4.M) and (AM.5) results
in

w. =D——1§. (101.6)

The implication of (Au.6) is that the weights used in (A4.l)

should be of the form

2 (ALL?)

(I
wi (T10

bsi)
Least squares curve fitting was performed by the
regression analysis subroutine STEPREG in the computer
program TOWPLOT (Appendix III). This subroutine requires
that the sum of the weights for a set of data points be

equal to the number of points, i.e.,

N
Z w. = N. (Au.8)

Wi = W . (AU.9)
T1
i=1 Obsi

In our measurements, the fractional error in measured

temperatures was much less than that for T Therefore,

1‘
the error in measured T values was not considered in de-

riving (AM.9).

 

 

 

 

 

I‘l‘slllal‘ll