THE SOFT SPHERE MODEL FOR
NUCLEAR QUADRUPOLE RESONANCE
IN RARE EARTH TRTCHLORTDES
UNDER HYDROSTATTC PRESSURE

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
DAVID HARLAN CURRENT
1971

 

 

 

vo‘fu '

LIBRARY

Michigan Lenin .
Unive; 3i (7

 

This is to certify that the

thesis entitled

Quadrupole Resonance in Rare Earth
Trichlorides Under Hydrostatic Pressure

l
The Soft Sphere Model for Nuclear 1
presented by l

T

David Harlan Current

has been accepted towards fulfillment
of the requirements for

Ph . D . Acme in Physics

 

    

 

Major professor
Edward H. Carlson
Date August 2; 1971

0-7639

 

ABSTRACT

(I

THE SOFT SPHERE MODEL FOR NUCLEAR QUADRUPOLE RESONANCE
IN RARE EARTH TRICHLORIDES UNDER HYDROSTATIC PRESSURE
by
David Harlan Current
we have measured the nuclear quadrupole resonance of 35C1
3, 3, 3 and GdCl3 at 77K as a function
of hydrostatic pressure. Pressure was generated with a helium

in CeCl PrClB, NdCl SmCl

gas piston system and measured with a Bourdon gauge. Quadrupole
resonance frequencies were measured with a simple pulsed spectro-
meter having a resolution of better than one kHz. In each

compound the quadrupole resonance frequency in a linear function

3 2

of pressure up to at least a pressure of 4.5 x 10 kg/cm . In

-1

all cases the coefficient v0 (3v/8P)T is negative; the frequencies

decrease with pressure. The coefficient decreases in magnitude

6

from -5.59:o.02 x 10' cmz/kg for CeCl to -3.86:0.02 x 10‘6 cmZ/kg

3
for GdCl . This contradicts the point charge model of electric

3
field gradients applied to these compounds. We introduce the
soft sphere model of electric field gradients. This model
includes the effects of molecular overlap on the electric field
gradient by subtracting from the point charge contribution of
near neighbor ions a term of the form q0 = F exp[(Rij-R)/pij].
This form for the overlap contribution is deduced from Hartree-
Fock, Self-Consistent-Field (HF-SCF) calculations by Matcha1
on the KCl molecule. The soft sphere parameters are assumed to

obey Gilbert's2 additivity rules R.. = R. + R. and p.. = p. + p.;
1] l J 1] l J

a Hartree-Fock-Slater calculation of ionic wavefunctions is

3 The Sternheimer antishielding

utilized to obtain their values.
factor, (1 - ya), and the overlap strength, F, are taken as
adjustable parameters to fit the zero pressure frequency and
asymmetry parameters of GdCl3. For the choices RC1 = 2.740 bohr
and pC1 = 0.429 bohr the values of the overlap strength and
antishielding parameter necessary to fit the observed quadrupole
frequency and asymmetry parameter in GdCl3 are F = 0.5695 bohr-3
and l - 7m = 30.95. The model then predicts the quadrupole
frequency in LaCl3 to within 2.5% of the measured value and the
frequency and asymmetry parameter in PrCl3 to within 0.2% and
0.5% of the measured values. Because the compressibility of
these compounds is unknown it is difficult to compare the
predictions of the soft sphere model with the observed pressure
dependence of the quadrupole frequencies. For volume decreasing

faster along the a—axis than along the c-axis the model is in

qualitative agreement with experiment.

lRobert L. Matcha, J. Chem. Phys. ii, 485 (1970).

2T. L. Gilbert, J. Chem. Phys. 49, 2640 (1968).

3F. Herman and S. Skillman, Atomic Structure Calculations,

Prentice Hall, Englewood Cliffs, New Jersey, 1963.

THE SOFT SPHERE MODEL FOR NUCLEAR QUADRUPOLE RESONANCE

IN RARE EARTH TRICHLORIDES UNDER HYDROSTATIC PRESSURE

by

David Harlan Current

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1971

DEDICATION

To my wife Diane, who has been exceedingly
patient and understanding and who has learned to
wake up late at night, and to my son Michael, who
has finally learned not to wake up late at night,

I dedicate this thesis.

ii

ACKNOWLEDGMENTS

I wish to express my appreciation to
Professor E. H. Carlson who not only suggested
this thesis topic to me and assisted in the
development of some of the ideas therein, but who
also provided constant stimulation and guidance.

I wish also to thank Professor C. L. Foiles, in
whose laboratory most of the experimental work
was performed, for valuable assistance with the
high pressure apparatus and many illuminating
discussions.

Dr. J. P. Hessler assisted with some of the
data collection and provided many valuable comments.
Nancy Hessler spent long hours typing the manuscript.
Charles Butterfield performed the HFS calculations
from which the soft sphere parameters were deduced.

Finally I wish to thank Central Michigan
University for a two year leave of absence during

which this work was completed.

iii

LIST OF

LIST OF

Chapter
I.

II.

III.

IV.

V.

TABLE OF CONTENTS

TABLE 8 O O O O O O O O O O O O O O O O O O

FIGURES O O 0 O O O O O O O O O O O O O 0

INTRODUCTION . . . . . . . . . . . . . . .
QUADRUPOLE RESONANCE IN THE HEXAGONAL RARE
EARTH TRICHLORIDES . . . . . . . . . . . .
A. Quadrupole Hamiltonian . . . . . . . .
B. Crystal Structure . . . . . . . . . .

C. Experimental Results at one atmosphere
pressure 0 C O O O O O O O I O O O O 0

EXPERIMENTAL TECHNIQUES AND RESULTS . . .
A. Sample Preparation . . . . . . . . . .
B. The Spectrometer . . . . . . . . . . .
C. High Pressure Apparatus . . . . . . .
D. Experimental Results . . . . . . . . .
THEORY . . . . . . . . . . . . . . . . . .
A. Introduction . . . . . . . . . . . . .
B. Compressibility of LaCl3 . . . . . . .

C. The Point Charge Model . . . . . . . .

D. The Soft Sphere Model of Electric Field

Gradients O O O O O O O O O O O O O O

E. Volume Dependence of the Electric Field

Gradient . . . . . . . . . . . . . . .

CONCLUSIONS 0 O O O O O O O O O O O O O 0

REFERENCES 0 O O O O O O O O O O O O O O O O O O 0

iv

Page
vi

viii

12
l4
14
14
15
19
22
22
23

27

29

42
53

56

APPENDIX A

APPENDIX B

APPENDIX C

APPENDIX D

Page

LIST

OF TABLES

Crystal Structure Data . . . . . . .

Locations of Near Ions in GdCl . .

3

Experimental NQR Frequencies and Asymmetry

Parameters

Summary of Frequency versus Pressure Data

at 77K . .

Elastic Constants, Compressibilities and
Debye Temperatures for Selected Materials

KCl Molecule after Matcha . . . . .

Ionic Parameters after Gilbert . . .

Ionic Parameters from HFS . . . . .

Parameters for GdCl :

3

Soft Sphere Model .

Results of Lattice Compression in the Soft
Sphere Model . . . .

Quadrupole
CeCl3 . .

Quadrupole
PrCl . .
3
Quadrupole
NdCl3 . .

Quadrupole
SmCl3 . .

Quadrupole
GdCl O O
3
Summary of
at 300K .

Quadrupole
ErCl3 . .

Quadrupole
YbCl 3 I 0

Pressure Coefficients at 77K for ErCl

YbCl3 . .

Frequency

Frequency

Frequency

Frequency

Frequency

Frequency

Frequency

Frequency

versus Pressure

versus Pressure

versus Pressure

versus Pressure

versus Pressure

versus Pressure

versus Pressure

versus Pressure

vi

Data:

Data:

3 and

Page

11

13

20

26
30
35
36

37

50

63

65

67

70

72

75

78

80

82

Table
D.l
D.2
D.3

D.4

Results
Results
Results

Results

of Point Charge Model
of Soft Sphere Model:
of Soft Sphere Model:

of Soft Sphere Model:

vii

Data Set A

Data Set B

Data Set C

Page
85
86
87

88

Figure
2.1
3.1
3.2
3.3

LIST OF FIGURES

GdCl3 Structure 0 O O I O O O O O O O O O O 0
Schematic Diagram of Experimental Apparatus .
High Pressure Cell . . . . . . . . . . . . .

Normalized Pressure Coefficient of Quadrupole
Frequency versus Atomic Number . . . . . . .

Electric Field Gradient at the Chloride
Nucleus in the KCl Molecule . . . . . . . . .

Quadrupole Frequency versus Compound . . . .
Asymmetry Parameter versus Compound . . . . .

Quadrupole Frequency versus Volume in the
Point Charge Model . . . . . . . . . . . . .

Quadrupole Frequency versus Volume in the
Soft Sphere Model: Data Set A . . . . . . .

Quadrupole Frequency versus Volume in the
Soft Sphere Model: Data Set B . . . . . . .

Quadrupole Frequency versus Volume in the
Soft Sphere Model: Data Set C . . . . . . .

Derivative of Frequency with respect to
Volume versus Compound in the Soft Sphere
MOdel O O O O O O O O O O O O O O O C O O O O

Lattice Constants versus Compound . . . . . .

Chloride Positional Parameters versus
commund I O C C O O O O O I O O O C O O O O

Quadrupole Frequency versus Pressure at 77K:
ceC13 O O O O O O O O C O O O O O I I O O O O

Quadrupole Frequency versus Pressure at 77K:
PrC13 O l O O O O O I I O O O I O O O O O O O

Quadrupole Frequency versus Pressure at 77K:
NdC13 O O O O O O O O 0 O O O O O O O O O O 0

viii

Page
10
l6

17

21

32
39

40

44

45

46

47

51

60

61

64

66

69

Figure

8.4

Quadrupole Frequency versus

smC1 O O I O O O O O O O O
3

Quadrupole Frequency versus

GdCl O O O O O I O O O O I
3

Quadrupole Frequency versus

ErC]. O O O O O O O O O O O
3

Quadrupole Frequency versus

YbC13 C O O O O C O C O O 0

ix

page
Pressure at 77K:

0 O O O O I I O I 71
Pressure at 77K:

0 O O O O O O O O 74
Pressure at 77K:

I O I O O O O O O 79
Pressure at 77K:

0 O O O O O O O O 81

I. INTRODUCTION

An atomic nucleus with spin greater than l/2 can have
a nonspherical distribution of nuclear charge and hence a
nuclear quadrupole moment Q. This quadrupole moment will
couple to an electric field gradient to produce a splitting
of the energy levels labeled by the z-component of the nuclear
spin. The electric field gradient is a traceless second
rank tensor characterized by a magnitude q, an asymmetry
parameter, n, which is a measure of the tensors departure
from cylindrical symmetry, and the three Euler angles which
specify the orientation of its principal axes. The nuclear
quadrupole coupling constant, qu/h, where e is the electronic
charge, h is Planck's constant and q is the magnitude of
the electric field gradient, can be measured experimentally
by inducing transitions from one energy level to another.
This technique is known as nuclear quadrupole resonance.
The theory of quadrupole resonance is reviewed in Chapter II.
If the nuclear quadrupole moment is known from other measure-
ments, then from the quadrupole coupling constant one can
determine the field gradient at the nuclear position.

The electric field gradient components can be expressed

in spherical tensor notation as
qm = fr'3p(?)r§ (e.¢)d? ,

where Y? (6,¢) is a spherical harmonic, and p(;) is the electronic
charge density. This form emphasizes that the electric field

gradient gives a description of the electronic charge distribu-

l

tion in the neighborhood of the nucleus. Since only that

part of p(;) with the symmetry of Y? makes a contribution

to the field gradient, there is no contribution from the

spherically symmetric part of p(¥). Most of the contribu-

tion to the electric field gradient at the nucleus of a

given ion comes from the outer, nonspherical part of the

ion as well as from other nearby ions. Thus nuclear quadrupole

resonance provides an excellent experimental tool for investi-

gating the electronic charge density at nuclear sites in solids.
The compounds studied in this work, the trichlorides

of lanthanum through gadolinium, all have the same hexagonal

structure. An excellent x-ray study of these compounds has

recently been reported (Morosin, 1968). The geometry (ion-ion

distances and angles) varies slowly from one compound to

another. In addition whatever prOperties of the rare earth

ions that are important in determining the electric field

gradient are also expected to be a slowly varying function

35

of cation species. The C1 resonances in these compounds

have all been reported previously: in PrCl3 by Hughes,

Montgomery, Moulton and Carlson (1964); in GdCl by Carlson

3

(1966); in CeCl , NdCl and SmCl by Mangum and Utton (1967);

3 3 3

and in LaCl by Carlson and Adams (1969). No quadrupole

3
resonances in PmCl or EuCl have been measured. Although

3 3
a large amount of experimental data on quadrupole resonances
in ionic salts is available, very few of these compounds
have well determined structures. The hexagonal rare earth
trichlorides are an important exception and provide an

excellent opportunity to test theories of the electric field

gradient in ionic salts.

The earliest attempt to calculate the electric field
gradient in an ionic solid was the point charge model of
Bersohn (1958). This model assumes that the effect of each
ion in the crystal can be represented by a point charge
located at the position of each ion. The electric field
gradient at any ion is obtained by summing the contributions
of all other ions in the crystal. Bersohn calculated the

electric field gradient at Na+ in NaN03, NaClO and NaBrO

2+

3 3

as well as the electric field gradient at Cu in CuO

3+

2

in A1203. Carlson and Adams (1969) have applied

the point charge model to the rare earth trichlorides.

and at Al

Throughout its history the point charge model has enjoyed
rather limited success. It seems to do a better job of
predicting the field gradient at positive ions than at
negative ions. Burns (1959) and others (Brun and Hofer, 1962;
Raymond, 1971) have attempted to expand the point charge 1
model by including higher multipole contributions from
polarized ions. Their attempts have not always been very
successful.

In an effort to obtain additional information on the
electric field gradient in the rare earth trichlorides we
have measured the quadrupole resonance frequency as a function
of hydrostatic pressure. The application of hydrostatic
pressure changes the geometry of the unit cell, and these
geometrical changes are reflected in a change in the electric
field gradient. The details of the experiments are discussed

in Chapter III.

It is well known that molecular overlap produces a
significant contribution to the electric field gradient
(Das and Karplus, 1964). In Chapter IV we introduce the
soft sphere model of electric field gradients. This model
includes a systematic parameterization of the overlap
contribution to electric field gradients. We then use the
model to calculate the zero pressure quadrupole coupling
constants for the rare earth trichlorides. We find substantial
agreement with the experimental results. In applying the
model to the pressure dependence of the quadrupole frequencies
in these compounds, we are hampered by the lack of experimental
data on the compressibilities. However the model is in

qualitative agreement with the high pressure experiments.

II. QUADRUPOLE RESONANCE IN THE HEXAGONAL
RARE EARTH TRICHLORIDES

A. Quadrupole Hamiltonian

The Hamiltonian for the interaction of the electric
quadrupole moment of a nucleus with an electric field
gradient due to charges external to the nucleuscan be

written as (Das and Hahn, 1958)

4-.
HQ - Q

: 6E = 2 o? (va)gm . (2.1)

m
Q? are the irreducible components of the nuclear quadrupole
charge distribution operator and (VB); are the irreducible
components of the electric field gradient Operator evaluated
at the nuclear position. In terms of the nuclear quadru-

pole moment, Q, the components of gare given by Das and

Hahn (1958) as

0 _ 2eQ 2 _ 2
02 ’ 4I 21-1) [312 I 1' (2'2)
:1 _ /6eQ
QZ _ 41 21-1) [Iin + IiIzl' (2'3)
and
:2 _ /ێQ 2

The coordinates of the nuclear spin operators are taken
in some arbitrary Cartesian frame.
In the same frame the field gradient has components

2
_ a v . . _
ij ‘ axbeT (1'3 ‘ X'Y'z" (2'5)
1 3

where V is the electrostatic potential at the nucleus.

Vij is symmetric and, if V is produced by charges exterior
5

to the nucleus, traceless. If the Cartesian frame is
chosen so as to diagonalize Vij’ then the components of

the field gradient operator are given by

0 _ 1 :1 _
and
i2 1
VB = — - . .
( )2 2/5 (vXX vyy) (2 7)

< v Lilv and define the field

If we assume that IV | |
xx —’ yy

22"

gradient, q, and asymmetry parameter, n, by

eq = sz and n = (Vxx-Vyy)/sz’ (2.8)
then
(vz)g = % eq and (VE):2 = -l—-neq. (2.9)
2/5

By carrying out the sum in the Hamiltonian one

arrives at

_ 2 _ 2 1
HQ - A[3Iz I + 2 n(I+I+ + I_I_)], (2.10)
where
= __E_SQ__

A complete derivation of this result is given by
Slichter (1963).
The matrix elements of this Hamiltonian taken

between nuclear angular momentum states are

— 2—
<Im,|HQ|1m> — A[3m I(I+l)] 6m.,m

 

+ %nA/EI(1+17-m(m+l)1[1(1+1)‘(m+1)(m+2)] 6m'.m+2

 

+ %nA/[I(I+l)-m(m-l)][I(I+l)-(m“l)(m-2)] m. m_2

(2.12)

For I = 3/2 the non-zero elements are
3 3 3 3
(EIiEIHQI-f,i§> = 3A ' (2.13)
3 l 3 1
<§,i§IHQ|§,i§-> = '3A (2.14)
and
3 3 3 l
<§,i§|HQ|'2-,i-2-> = finA , (2.15)
with
A = e2q0/12 . (2.16)

The Hamiltonian is easily diagonalized and yields

two doubly degenerate energy levels with eigenvalues

E: = 3A 1 + 3 n . . (2.17)

Thus there is a single quadrupole resonance transition

with frequency given by

th = E+ - E_ (2.18)
equ l 2

B. Crystal Structure

 

The trichlorides of lanthanum through gadolinium were
shown to be isostructural by Bommer and Hohmann (1941).
Zachariasen (1948) showed that they were isomorphic to
UCl3 with space group P63/m, and measured the lattice
constants A and C. In this structure there are two
molecules per unit cell with metal ions at i(l/3, 2/3, 1/4)
and chloride ions at t(u, v, 1/4), i(-v, u-v, 1/4) and
1(v-u, -u, 1/4). Zachariasen measured the chloride

positional parameters, u and v, only for UC13.

Templeton and Dauben (1954) measured the lattice
constants for the entire series (with the exception of
LaC13). Morosin (1968) measured lattice constants and
chloride positional parameters for LaCl3, NdClB, EuCl3
and GdC13. Lattice constants and positional parameters
for CeCl3, PrCl3 and SmCl3 have been estimated from
Morosin's data. The procedure is described in Appendix A.

The volume of the unit cell varies smoothly across
the series with a minimum at GdCl3. The volume per
chloride ion (cell volume divided by six) ranges from
35.20 i3 for Lac13 to 32.16 R3 for GdCl3. The crystal
structure data is summarized in Table 2.1.

In these crystals all ions lie on mirror planes
perpendicular to the c-axis. Figure 2.1 shows two layers
of GdCl3. The large spheres are Cl- (radius 1.8 A) and
the small spheres are Gd3+ (radius 1.2 A). The radii are
taken from Pauling (1940). For the chloride labeled 0 the
three nearest metal ions are labeled a and b, and the eight
nearest chloride ions are labeled c, d and e. We have
chosen a coordinate system centered at ion 0 with the z-axis
parallel to the crystal c-axis and the y-axis bisecting
the 120° angle of the unit cell. In Figure 2.1 the z-axis
is out of the plane of the paper and the x-axis and y-axis
are horizontal and vertical respectively. The positions
of the eleven nearest ions are summarized in Table 2.2.

The existence of the mirror plane perpendicular to

the c-axis requires that one of the principal axes of

.< xfipcmdd< mom

.maMp m.cHwouoz Eouw poumswumm

.uoouuoocH ma osfio> m.chouozo

n

.Awomflv camouoz Eoum muons

 

 

eH.~m ma.~es mesom.o ammmm.o emoH.e meem.~ unseen
ee.~m me.ses eksom.o esamm.o mmme.e eeem.e ensues
AA.Nm Ne.eos Nasom.o mnmwm.o ees.e swm.e hmsoem
Nn.mm NH.HoN Aesom.o AAAmm.o m~e~.e mmmm.e ensoez
mo.em ae.eo~ eeH0m.o meemm.o mmN.e mm¢.a Ensues
me.sm eH.mo~ oesom.o mmnmm.o mmm.e mme.n assume
Hm.mm eem.HH~ mmsom.o Essen.o meem.s mAAe.A ensues
Amwvuao\> AmNVHHoo\> > a Amvo va< vasomsou

 

some musuosuum

 

.H.N wanna

10

 

Figure 2.1. GdC13 Structure

Table 2.2. Locations of Near Ions in GdC13

11

 

 

Species Labela 8(2) X(X) Y(X) 2(2)
Gd3+ a 2.9177 -2.6863 1.1388 0.0000
Gd3+ b 2.8223 1.5666 1.1388 12.0529
01' c 3.3546 3.3361 0.3514 0.0000
01‘ c 3.3546 1.3637 3.0649 0.0000
01' d 3.3169 -2.4834 -0.7873 12.0529
01' d 3.3169 1.9236 -1.7570 32.0529
01' e 3.2643 -1.1197 2.2775 32.0529

 

aLabeled in Figure 2.1.

12

the electric field gradient at each chloride ion be along
the c-axis. Experimentally this has been shown to be Vxx
(Hughes, et al., 1964). All three chloride ions in the
molecule are chemically equivalent. That is, the diagonal
elements of the electric field gradient tensor are the

same at each ion, and there is a single pure quadrupole
frequency, but the orientation of the tensors differs by
120°. In addition, since the crystals lack a two fold

axis in the mirror plane, it is not easily possible to
determine one end of the c—axis from the other (Carlson and
Adams, 1969). Thus the orientation of the electric field
gradient principal axes can only be experimentally

measured to within a multiple of 60°. For these reasons

we have not attempted to calculate the orientation of the

principal axes, except to be sure that Vxx was always along

the c-axis.

C. Experimental Results at One Atmosphere Pressure
35

 

Cl nuclear quadrupole resonance frequencies have
been reported for all of the hexagonal rare earth
trichlorides with the exception of PmCl3 and EuCl3.
Asymmetry parameters have been accurately measured only

for PrCl and GdCl . The experimental data is summarized

3 3
in Table 2.3.

Table 2030

Experimental NQR Frequencies and Asymmetry Parameters

 

 

Compound 41 7J(MHz)4K '1)(MHz) 77K V(Wiz)3OOK
LaCl3 --- 4.167 --- ---
CeCI3 --- 4.387 4.377 4.341
PrC13 .4937 4.567 4.562 ---
NdC13 --- 4.729 4.722 4.676
SmC13 --- 5.033 5.027 4.976
GdC13 .4265 5.315 5.308 5.248

 

Uncertainties are 1 in the last figure.
are taken from Hessler (1971).
and Adams (1969).

'S are measured at T<4K and
Frequencies at 4K are taken from Carlson

III. EXPERIMENTAL TECHNIQUES AND RESULTS

A. Sample Preparation

 

Anhydrous rare earth trichloride powder purchased
from Lindsay was used in the preparation of single crystal
samples. Our laboratory method is similar to that
described by Garton, et al. (1964). The powder was
melted and distilled under vacuum into a Vycor growing
tube, which was then slowly lowered through a gradient

furnace. The only exception was SmCl the powder was

3.
not distilled but merely melted in the presence of HCl gas
and allowed to flow into the growing tube. As the
crystals are hydroscopic, they are stored under mineral
oil when removed from the growing tube.

Small samples were prepared for resonance measurements
by cutting single crystals perpendicular to the c-axis on
a diamond saw to a length of about 1/2 inch. The resulting
pieces are then cleaved parallel to the c-axis to yield
hexagonal samples about 1/8 inch in diameter. The samples

were coated with a thin layer of GE7031 cement for

protection against moisture.

B. The Spectrometer

 

The spectrometer used in this work was a simple pulsed
instrument developed by Parks (1967), and referred to as
"the minipulser". A pulse of radio frequency power derived
from an external oscillator is applied to a coil wrapped
directly on the sample. The pulse amplitude is about

200 volts, the pulse length about 25 microseconds, and the
14

15

repetition rate a few milliseconds. In this work the

coil axis was parallel to the c-axis of the sample. The
same coil acts as a receiver to pick up the induced

signal from the sample. This signal is amplified,
multiplied by the signal from the external oscillator,

and displayed on an oscilloscope. The quadrupole

resonance frequency is determined by adjusting the
oscillator frequency until a zero beat condition is
observed. The oscillator frequency is monitored continuous-
ly with an electronic counter.

The minipulser has several advantages for this work.
Signals are strong enough so that direct oscillosc0pe
display is possible without the need for signal averaging
techniques. Frequency measurements can be made rapidly
and with high precision (frequency differences of 100Hz
can be resolved). In addition, if one is willing to
sacrifice some loss of signal strength, a relatively
large input capacitance (of the order of 500 p.f.) can be

tolerated.

C. High Pressure Apparatus

A schematic diagram of the experimental apparatus is
shown in Figure 3.1. Figure 3.2 shows details of the BeCu
pressure cell. The high pressure gas seal between the two
halves of the cell is made with a series of stainless
steel rings which are coated with a thin layer of indium
(Goree, et al., 1965). These rings provided an excellent

seal and could be reused at least six times without

16

see. 5 a:

new c_m\_
cu.» nieom

ooeaom

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OssauOgQ t
can or

28823

 

 

 

 

 

 

ana 2:32“.

:32.

.32-:
3:: :o

 

 

3369:0on

 

 

Schematic Diagram of Experimental Apparatus

Figure 3.1.

Pressure Cell 3

5e .
I \

/////\\\\\\\\\\\

/////,////////////// ///////\4..

//////s

\t

18
recoating. The upper half of the cell is connected to

5/16 o.d. by 1/16 i.d. stainless steel, high pressure
tubing by a standard cone seal.

The high pressure tubing is electrically grounded
and serves as a shield for the Spectrometer lead. The
lead is extracted from the high pressure region through
a U-shaped tube, approximately 20 inches deep, filled
with Dow Corning 704 silicone diffusion pump oil. When
frozen in liquid nitrogen, the oil expands and provides
a seal which easily withstands pressures up to 90,000
pounds per square inch. However, some care must be taken
to be sure that there are no trapped bubbles of air in
the oil before it is frozen.

Hydrostatic pressure is generated in helium gas by
an oil driven piston and a 10:1 intensifier. The gas is
slowly admitted to the system through a liquid nitrogen
cooled trap to remove impurities. Pressure was measured
with a Bourdon gauge. The manufacturer claims 2% accuracy
for this gauge. Although no absolute calibration was
attempted, the gauge has been checked against manganin
resistance data to verify the manufacturer's claim.

Our experimental procedure was to wrap the sample
with a suitable coil (generally about 20 turns of #36
Cu wire) and mount it in the pressure cell. The cell and
oil seal were then attached to the rest of the pressure
apparatus and the oil seal slowly frozen. With the cell
at room temperature the Spectrometer was adjusted and the

pressure was raised to approximately 80,000 p.s.i. When

19

a signal could be seen, room temperature data was taken.
The pressure was then reduced to about 15,000 p.s.i. and
the cell immersed in liquid nitrogen. We allowed at

least 30 minutes for thermal equilibrium to be established
before taking data at 77K. In all cases, except YbCl3

(see Appendix C) the pressure was cycled at least once

to check for hysteresis. None was ever observed.

D. Experimental Results

Raw data for the 35Cl resonance in the hexagonal

 

trichlorides as a function of pressure is presented in
Appendix B. The data was analyzed by least squares methods

(Barford, 1967) according to the equation

v(P) = v0 + mP (3.1)

where m is (av/8P)T. In all cases m is negative and
independent of pressure over the range of pressures used.

The results at 77K are summarized in Table 3.1. v0 is

the pure quadrupole frequency measured at one atmosphere.

The uncertainty in 00 is the standard deviation of at

least five separate measurements. v and (3v/3P)T are

0

obtained from the least squares fit to the raw data.
The uncertainties here are statistical best estimates

of error. The normalized pressure coefficient

031 (av/8P)T is calculated from v and (av/3P)T. It

0
is plotted with respect to compound in Figure 3.3.

20

 

 

000.0 a 000.0- 000.0 14. 000.7 0.0 0 0.0000 0.0 0 0.0000 0800
000.0 a 000.0- 20.0 0 000.7 0.0 .1. 0.0000 0.0 0 0.0000 0880
000.0 a 02.0- 000.0 0 000.7 0.0 a 0.0000 0.0 H 0.003 0802
000.0 0. 02.0- 000.0 a 000.7 0.0 0 0.0000 0.0 H 0.000 0800
000.0 a 000.0- 000.0 a 02.7 0.0 0 0.0000 0.0 0 0.200 0800
0.55 0-0: .08 03.3%? 32:05 08 393 3:00? 330? 0509.60

 

x00 um sumo ousmmopm msmuo> hocosvoum mo sumsssm

 

.H.m canoe

 

fi
-—1
—
u.-
—-1
—
d

5.6 -

*4

5“” XCI

TE»
“x 500 b
e
0
O
.9 4.8- 1:
406 _

- v." (aw/3P)Y
J)
45
1

4.2~

4.0 -

3.8 -

 

l l l l l L 1

 

 

58 59 60 6| 62 63 64

Atomic number

Figure 3.3. Normalized Pressure Coefficient of Quadrupole Frequency
versus Atomic Number

IV . THEORY

A. Introduction

 

The calculation of electric field gradients in ionic
solids is a complicated quantum mechanical problem. Since
the necessary many-ion wave functions are not known, the
usual procedure is to assume an appropriate two-ion model
for the field gradient and then sum the contribution from
each ion in the crystal. Once the two-ion model has been
chosen, the calculation of the electric field gradient
becomes a geometrical problem.

we define the normalized pressure coefficient of the

quadrupole frequency as
-1 _-
v0 (av/3P)T - B[8(v/vo)/3(V/Vo)]T (4.1)

where v0 and V0 are the frequency and unit cell volume at

zero pressure. The bulk compressibility, 8, is given by

._'l _
8 - VO (8V/8P)T — a + a + a (4.2)

l 2 3 '
The three linear compressibilities along the crystallographic
axes are al, a2 and d3. For hexagonal crystals, symmetry
requires that a1 = a2. In order to properly calculate the
dependence of the quadrupole frequency on cell volume it is
necessary to know the compressibility ratio r = 03/01 as a
funcwion of pressure. We consider this point in Section B
alcnng with the bulk compressibility. In addition it is

GSSential to have information concerning the variation of

tbs? <2hloride positional parameters with pressure. However

22

23

no experimental data is available and we assume that they
are independent of pressure.

In Section C we briefly discuss previous attempts to
calculate electric field gradients in solids in the two-ion
approximation. In Section D we introduce the soft sphere
model used in this work, and present the results of calcula-
tions of quadrupole frequencies and asymmetry parameters for
all of the hexagonal rare earth trichlorides. Finally in
Section E we combine the results of Sections B and D to
calculate normalized pressure coefficients from the soft
sphere model and compare these calculations with the

experimental results.

B. Compressibility of LaCl

 

3

The best and most complete information on the compressibility
in these compounds would be a full X-ray measurement of the
lattice constants and positional parameters as a function of
pressure and temperature. Such experiments are difficult and
have not been performed. Indeed even the bulk compressibility
has not been measured. The only piece of evidence on the
elastic behavior in these compounds appears to be a recent
paper by Stedman and Newman (1971) in which the elastic
constants, Cij' of LaCl3 are calculated from optical data.

Stedman and Newman give three sets of elastic constants
and label them A, B and C. They are reproduced here as part
of Table 4.1. The different sets result from different
assumptions concerning the nature of the bonds in LaCl3, but

all are derived by assuming Hooke's Law forces between ions.

24

This model seems questionable to us for two reasons. First
it requires attractive forces between chloride ions, and
second it attempts to derive information about the long
wavelength acoustic phonons from the behavior of the optical
phonons.

These objections not withstanding, the lack of experimental
data on the elastic behavior of these compounds forces us to
examine the results of Stedman and Newman in some detail. It
is possible to calculate the compressibility of a crystal
from a knowledge of the elastic constants. For a general
review of this tOpic see Hearmon (1946). The linear compress-
ibilities are usually written in terms of the elastic compliance

constants, $1.,
3
a. = E s.. , 1 = 1, 3 . (4.3)

The compliance constants are related to the elastic constants
by (Cady, 1964)
6

6
Z s. c . = Z
k=l 3k k3' k=l

C . . (4.4)

. S . = 6..
3k kJ' 33
For hexagonal systems it is a straightforward matter to solve
Eq. (4.4) for Sij in terms of Cij and substitute the result

into Eq. (4.3). In terms of the elastic constants the linear

compressibilities are given by

a = dz = (C33 - C13)/D (4.5)

and

(C + C12 - 2C13)/D (4.6)

3 = 11

25

where

) - 2c 2 (4.7)

D = 13

+ C

C33(C11 12

The compressibilities calculated in this way from the data
of Stedman and Newman are presented as part of Table 4.1.
They seemed to us to be rather small.
In order to examine the validity of these elastic constants
we have calculated the Debye temperature that results from them

(Carlson, Current and Foiles, 1971). In LaCl the Debye

3

temperature, 0 has been calculated from low temperature

D'
specific heat data to be l49.5:l.5 K (Varsanyi and Maita,
1965). Betts, Bhatia and Horton (1956) derive three approximate
formulae for calculating GD as a function of elastic constants
for hexagonal systems. Wolcott (1959) gives a set of numerical
tables. Using these four methods, we have calculated OD from
each of the sets of Cij of Stedman and Newman. The results
for the second method of Betts, Bhatia and Horton appear in
Table 4.1 along with some data on other compounds for comparison.
There is good agreement among the four methods. However the
calculated Debye temperatures are from three and one-half to
four times larger than the measured value. We are therefore
forced to conclude that the elastic constants of Stedman and
Newman are in error.

The following argument is used to estimate the size of
this error. The Debye temperature is proportional to the

mean sound velocity, which for hexagonal systems is given by

(wo1cott, equation 6)

f/C447p (4.8)

V
m

.Hmwumume 002:0 .Aommfiv Houufix Scum sump Hmucmeaummxmw

.00000 . 00000 . 0000 .000000 000.0000.0
.000000 000302 000 00000000
.o:Hw> pousmmch

.N.0 000 0000: AommHV couuom 0:0 mwumsm .muumm 5000 00000500000

.000000 00000000
0 .00

.00000008 00050 .AoooHV 00000 8000 00mw
can Aommav Houuax 800m museumcoo ofiumwam

o Eoum 6000030000 .wx\NEo 0-00 mo 000::b

.NEo\oc>v N000 mo 000:20

 

26

 

n
000 000 --- 00.0 --- --- --- 000.0 --- --- 000.0 000 0 00000
000 000 0.0 00.0 --- --- --- 000.0 --- --- 000.0 000.0 0000
000 000 0.0 00.0 --- --- --- 000.0 --- --- 000.0 000.0 0000z
--- --- 0000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000000
0.000 0.000 --- 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000000000
0.000 0.000 --- 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000000000
0.000 0.000 --- 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000.0 000000000
00000 08 000000 ® 0 0\ 000 refine 0000 00 00 0000 0000 0000 0000 0000

 

macfiumumz pouooamm you mousuwuodEmH canon paw 0000000300000daoo .mucmumcoo uuummam

 

.0.0 00000

27

where p is the density and f is a numerical function of

ratios of the other elastic constants. By examining Wolcott's
tables, we conclude the f is slowly varying. If we assume
that the relative sizes of the Cij's of Stedman and Newman

are approximately correct, then we are forced to conclude

that their magnitudes are too large by a factor of from 12 to
16. Based on this argument we estimate that B for LaCl3 is

6 cmz/kg. If the simple

somewhere between 6 and 8 x 10-
scaling argument is not valid, this estimate could be off by
as much as a factor of two or three. Based on the values of
8 for other chloride salts, our personal feeling is that the
estimate is probably too large rather than too small. As far
as the ratio of linear compressibilities is concerned, we have
assumed a value somewhere between 0.4 and 0.7, but stress the

fact that this choice is almost completely arbitrary. Certainly

a good experimental determination of these numbers is needed.

C. The Point Charge Model

 

In the point charge model of electric field gradients
one assumes that each ion in the crystal can be represented
by a spherical charge distribution having negligible overlap
with other ions. One then replaces this spherical distribution
by an appropriate point charge located at the nuclear position.
The contribution to the field gradient from each ion is summed
over the entire crystal:

_ 3
qsum " g zza/Ra I (4.9)

where Rd is the distance to the ion a and za is its charge.

28

In performing this sum the contribution of each ion must be
rotated into a common coordinate system. Details are
presented in Appendix D.

A closed shell ion such as Cl- has a spherical electronic
charge distribution which produces no field gradient at the
nucleus. However if such an ion is placed in an external
field gradient the electronic charge distribution is distorted.
This distortion produces an additional contribution to the
field gradient at the nucleus which is proportional to the
externally applied gradient. The ratio between the gradient
produced by distortion and the external gradient is called
the Sternheimer antishielding factor (Y0): It has been
calculated by Sternheimer (1956) and others (Lucken, 1969,

p. 90). Sternheimer's result is Y0 = ~56.6. The net field

gradient at the nucleus is then

Burns and Wikner (1961) have calculated Y0 = -27 for C1_ using
contracted wavefunctions which they claim are representative
of C1- in solids. Attempts to fit the point charge model to
the observed quadrupole frequencies in various solids usually
result in antishielding factors of from -10 to -40.

Point charge calculations for some of the hexagonal rare
earth trichlorides have been reported by Carlson and Adams
(1969). They found (1 - Y0) = 15.4 necessary to fit the

observed frequency in LaCl With this same value of the

3.
antishielding parameter the calculated frequency in GdCl3 is
too low. In addition the asymmetry parameters are poorly

predicted.

29

We have carried out point charge calculations for all
of the hexagonal rare earth trichlorides using the structure
data discussed previously. We have chosen (1 - Y0) = 17.59

so as to fit the observed frequency in GdCl The results

3.
for all compounds are given in Appendix D in Table D.1.
The frequencies and asymmetry parameters are plotted with

respect to compound in Figures 4.2 and 4.3 in the next section.

D. The Soft Sphere Model of Electric Field Gradients

 

Consider two closed shell ions, e.g. K+ and Cl-, separated
by a large internuclear distance. It is assumed that the
field gradient at the chloride nucleus is correctly described
by the point charge model together with the theoretical
antishielding parameter. As the ions are moved together
it is clear that the point charge model must eventually fail.
The ions no longer see each other as point charges, but as
distributions of charge which become nonspherical as the
outer parts of ions come into contact and distort each other.

Several authors have stressed the need for including an
overlap contribution to the gradient. Das and Karplus (1959)
suggested that a contribution to the gradient proportional to
the overlap integral between ions in the KCl molecule was
necessary to fit the experimental data. In a later paper
(Das and Karplus, 1965) they performed such a calculation
and showed that the overlap effect was important. Several
papers (Taylor, 1968; Sawatsky and Hupkes, 1970; Sharma, 1970;
Sharma, 1971) have dealt with the same effects in Al O

2 3'

Cr203 and Fe203.

30

In the paragraphs that follow we will discuss a systematic
method of introducing such overlap effects into field gradient
calculations. We first discuss a Hartree-Fock, self-consistent
field (HF-SCF) calculation by Matcha for the KCl molecule.

From this calculation we deduce a functional form for the
overlap contribution to the chloride field gradient. We then
discuss a paper by Gilbert on the Born-Meyer repulsive potential
between closed shell ions and use the results of a Hartree-Fock-
Slater (HFS) calculation of ionic wavefunctions to extend
Gilbert's work to other ions. Finally we introduce the soft
sphere model of electric field gradients and discuss its
application to the hexagonal rare earth trichlorides. This
section concludes with some comments on the limitations of

the model and its application to other crystal structures.

Matcha has recently calculated various molecular properties
of alkali halide molecules using HF-SCF wavefunctions. In one
paper (Matcha, 1970) he considers KCl and presents results
for the gradient at the chloride nucleus for several inter-

nuclear separations. His results are summarized in Table 4.2.

Table 4.2. KCl Molecule after Matcha

 

 

R(b0hr) 4.300 4.700 5.039 5.300 5.650

(bohr-3) -0.76931 -0.37517 -0.10043 0.06069 0.18729

qC1

 

At the equilibrium separation of the molecule (R = 5.039 bohr)

3

the measured field gradient is 00.003 bohr- (Ramsey, 1956).

The point charge model alone (with Y0 = -56.6) gives q = 0.89 bohr-

3

31

Das and Karplus (1965) obtain values for q between 0.33 and
0.13 bohr-3 when they include some effects of molecular
overlap. Matcha's result of q = -0.10 bohr-3 represents
substantial improvement over the above. In addition Matcha's
calculations allow us to examine the dependence of g on
internuclear separation.

We decompose Matcha's results into two parts, one due
to the point charge contribution and the other due to the
wavefunction distortion caused by orbital overlap. The overlap

contribution is then

qo = q - 2(1 - rml/R3 = q - q (4.11)

p I
where q is Matcha's calculated value and Y0 is -56.6. For

the three largest separations calculated by Matcha we find

go = B expl-R/pl. (4.12)

with B = -752.0 hohr"3 and p = 0.7594 bohr. In Figure 4.1,

q, q and q0 are plotted with respect to internuclear

P
separation. The exponential dependence of the overlap
contribution to the electric field gradient seems reasonable
to us. Since the ion distortion is a function of the
molecular overlap integral, it should fall off rapidly with
increasing separation. We would expect that the parameter p
is in some way characteristic of the "softness" on an ion
Pair.

In attempting to parameterize these ideas we were lead

to 51 paper by Gilbert (1968) which proposed an extension of

the: liorn-Meyer repulsive interaction between closed shell

32

 

I136; r I l ' I1----l

  
   
  

Molecule

0.50

 

(bohr‘3)
l

0.00)

Grodlent

-O.50*

 

 

~I.oo— 1

 

l L l
5.0 6.0 7.0

 

Separation (bohr)

Figure 4.1. Electric Field Gradient at the Chloride Nucleus in the KCl
Molecule

33
ions. Gilbert writes the interaction energy between ions
i and j as

Uij = f pij epoRij - R)/pij], (4.13)
where Rij and pij are characteristic radii and softness*
parameters for the ion pair, and f is a standard force.

From spectroscopic data for alkali halide monomers Gilbert
deduces the additivity rules Rij = R1 + Rj and pij = pi + pj,
where Ri and pi are parameters for individual ions. From

the same spectroscopic data he has assigned values to individual
ion parameters. These are listed as part of Table 4.3.

In trying to extend to other ions the meaning of Gilbert's
parameters we have carried out a series of calculations of
ionic wavefunctions using the HFS method. We used a computer
program originally written by Herman and Skillman (1963).

A working copy of the program was kindly provided by Dr. D.Y.
Smith of Argonne National Laboratory. We have observed a
correlation for alkali ions between the radii quoted by
Gilbert and the natural logarithm of electronic charge density
calculated by HFS. In addition the variation of the charge

density is nearly exponential near Gilbert's value of radius.

we define an HFS softness parameter by

p(HFS) = - AR/A(£nps) , (4.14)

where pS is the electronic charge density from HFS and the

slope is evaluated at Gilbert's value of R. For the alkali

 

*Earlier workers have called p a hardness parameter. We
prefer Gilbert's characterization since a larger value of
0 implies a softer repulsive force.

34

ions lnpS seems to be consistently near -4.3. The results
are more obscure for the halide ions. Gilbert's parameters
along with those deduced from the HFS calculation are
presented in Table 4.3.

we define radii and softness parameters for metal ions
from the HFS calculation. The parameters are evaluated where
the natural logarithm of the electronic charge density is
-4.3000. The values of Ri and pi for various ions of interest
in this work are presented in Table 4.4. The residual charge
<3utside a sphere of radius Ri is of the order of one electron.
Note that the halide radii are considerably smaller than
those of Gilbert.

We define the electric field gradient at ion i in the
soft sphere model as

q- = Z (qj - qj) . (4.15)
1 j¢i p 0

‘Mhere the sum runs over all ions in the crystal. The point

Cfliarge contribution is given by

q; = 2 zj(l - yi)/R3 , (4.16)

“fliere R is the distance from ion i to ion j with charge zj.
Tums antishielding factor for ion i is treated as a free
Parameter. For Cl- we expect its value to be somewhat less
titan the theoretical free ion value, since a chloride ion
cOnstrained in a potential well will certainly polarize
1£385 in response to an external gradient than a free ion.

Tale overlap contribution is given by

q2 = F epoRij - R)/oij] . (4.17)

35

Table 4.3. Ionic Parameters after Gilbert

 

 

 

Ion R(Gilbert)a ,O(Gilbert)a 16,6s (R = RC) ,o(HFs)b
Li 1.31 0.131 -4.34 0.205
Na+ 1.80 0.150 -4.33 0.163
x+ 2.35 0.200 -4.28 0.338
Rb+ 2.59 0.217 -4.35 0.369
05+ 2.87 0.245 -4.28 0.417
F' 2.59 0.338 -5.10 0.470
01' 3.59 0.449 -5.89 0.585
Br‘ 3.90 0.488 -6.08 0.619
I“ 4.37 0.546 -6.31 0.660
8Values in bohrs taken from Gilbert (1968).

bValues

in bohrs from HFS calculations.

36

 

 

Table 4.4. Ionic Parameters from HFS
Ion R(bohr) P(bohr) Residual Chargea
Li+ 1.306 0.2047 0.08
Na+ 1.801 0.2481 0.17
K+ 2.361 0.3304 0.36
Rb+ 2.590 0.3690 0.52
0s+ 2.880 0.4176 0.73
F' 2.230 0.4172 0.64
01' 2.740 0.4947 1.01
Br' 2.896 0.5277 1.20
1‘ 3.140 0.5803 1.50
La3+ 2.636 0.3267 0.42
Ce3+ 2.608 0.3204 0.42
Pr3+ 2.581 0.3148 0.40
N63+ 2.556 0.3096 0.37
963+ 2.531 0.3048 0.38
Sm3+ 2.507 0.3004 0.36
Eu3+ 2.484 0.2962 0.34
Gd3+ 2.462 0.2923 0.32

¥

aCharge in units of e outside characteristic radius.

37

The additivity rules for the soft sphere parameters are
assumed to hold. We treat F as a free parameter and restrict
the overlap contribution to near neighbors (R i 4.0 i).
The soft sphere parameters for metal ions are taken from
HFS calculations. We discuss three possible choices of
parameters for C1. below.

In what we call data set A we Choose the chloride
parameters directly from the HFS calculation. This gives
RC1 = 2.740 bohr and pc1 = 0.4947 bohr. Since F and RCl
are not independent parameters (the quantity F exp(RC1/pij)
enters for all ions) we fix gclat this value. For data set
B we return to the Matcha calculation which indicates a
combined softness for K+ plus C1. of 0.7594 bohr. By sub-
tracting the HFS value for K+ we obtain a softness for C1-
of 0.4290 bohr. For data set C we arbitrarily choose
pCl = 0.4000 bohr. For each set we then adjust the free
parameters F and (l - 7”) so as to fit the observed quadrupole
frequency and asymmetry parameter in GdCl3. The parameters
for the three data sets are listed in Table 4.5. With these

parameters we then calculate the electric field gradient for

each of the other hexagonal rare earth trichlorides.

Table 4.5. Parameters for GdCl3: Soft Sphere Model

 

 

 

-3
Set RC1(bohr) pC1(bohr) F(bohr ) (l Ym)
A 2.740 0.4947 0.4346 28.25
B 2.740 0.4290 0.5695 30.95

C 2.740 0.4000 0.6710 32.99

 

38

The detailed results are tabulated in Appendix D, and the
values of quadrupole frequency and asymmetry parameter are
plotted with respect to compound in Figures 4.2 and 4.3.
The agreement with the experimental values is quite
good. For set B the computed values of v are within 2.5%
of the experimental values for all compounds, while n for

PrCl is calculated to within 0.5% of the measured value.

3
The substantial improvement over the point charge model
should be noted. There is still room for improvement however,
since the variation of frequency with compound is not quite
right from LaCl3 to NdCl3. None of the three sets gives a
fast enough increase in frequency as ones goes from LaCl3 to
NdClB.

Two other attempts to fit the experimental data within
the context of this model were made. While not resulting
in a good fit they serve to illustrate some useful points.
First, if the Gilbert radius for C1. is used the gradient
is completely dominated by the near neighbor chlorides. The
amount of overlap required to reduce the frequency and
asymmetry parameter to their experimental values in GdCl3
is sufficient to cause the gradient tensor to be in the wrong
direction. That is, in its principal axis system, the
smallest element (Vxx) is not along the c-axis as it should
be. The second point involves the use of the full anti-
shielding parameter for the free Cl- ion. We assumed the

HFS values for the Gd3+ soft sphere parameters and set F = 1.0

and (l - yw) = 57.6. We then varied the chloride parameters

Frequency (MHz )

39

 

 

 

I I I I I I I I
525*-
///
//
//
I
/
5.00- ,1,
I I,
I ”
.l,”
I, ’II
I
///:/I
P I [”1
415? ,/
I
I
I
450-
A
4.25- 3
C
(Q 1 1 1 11 I 12 .1
L0 CO Pr Nd Pm Sm Eu Gd

Figure 4.2.

Quadrupole Frequency versus Compound

 

0.65

0.60

0.55

0.50

40

 

 

 

 

 

 

’1 I I I 1 I r I
Paint Charge \
_
P
h
A
h— E!§;::::::::::::;;:::::::
c -——— \:::\‘
1 I 1 1 1 1 1 1
L0 Ce Pr Nd Pm Sm Eu Gd

Figure 4.3. Asymmetry Parameter versus Compound

41

to fit the experimental data for GdCl Using these chloride

3.
parameters and the same F and antishielding factor, we

3+

varied the Pr soft Sphere parameters to obtain a fit to

the experimental data for PrCl The resulting soft sphere

3.
parameters were both significantly less than those assumed
for Gd3+. This is contrary to both the HFS calculations and

our intuition.

There are a number of effects we have neglected in this
model. Some of them we expect to be partially included in
the parameterization of our model and the others we expect
to be small. Those partially included are the effects of
polarization of the chloride ion at whose nucleus the calcu-
lation is carried out, and the contributions due to three-body
forces. In addition to an external field gradient at a
chloride nucleus there is also an electric field. This field
makes some contribution to the polarization of the chloride
electronic wavefunction which in turn produces a contribution
to the gradient at the nucleus. The dipole polarizability
of Cl- is not well known, but we expect that at least part of
this effect is included in the antishielding parameter. Many-
body effects are certainly important, but we feel that they
are at least partially included in our parameterization of
the overlap contribution. In other chlorides with different
geometries they may manifest themselves through changes in
the overlap strength F or the antishielding parameter. We
have completely neglected any dipole or higher moment contri-

bution to the lattice sums. The largest of these effects

42

would be from the more easily polarizable Cl“ ions. But
since they are symmetrically placed, their effect would
largely cancel. In addition most of the gradient appears
to originate from the three nearest metal ions which have
very small polarizability.

A variation of this model has been applied by Carlson

(l97l) to CstCl In this compound the overlap contribution

3.
to the electric field gradient at the chloride nuclei appears
to originate entirely from two nearby Pb2+ ions. The crystal
exists in two phases, one cubic with a single frequency and
the other tetragonal with two frequencies. Carlson has

used the Pb-Cl separation of 2.802 A in the cubic phase as
the sum of the characteristic radii and deduces F = 0.8098
bohr-3 and pPb + 0C1 = 0.546 bohr from the two frequencies

in the tetragonal phase. His relatively large value for F

and small value for 0 partially result from his use of the

free ion value for the Cl- antishielding parameter.

E. Volume Dependence of the Electric Field Gradient

 

The calculation of the volume dependence of the electric
field gradient is a straightforward procedure. The lattice
constants of the unit cell are reduced in a systematic way and
the field gradient is calculated for each set of lattice
constants. Since the linear compressibility ratio r = 03/01
is so poorly known we reduced the values of the lattice
constants A and C in such a manner as to correspond to a
wide range of values for r. We made calculations in GdCl3
for the extreme values r = 0.0 and r = w as well as for

several intermediate values. The choice r = 0.0 corresponds

43

to holding the value of C fixed and allowing all of the
volume reduction to occur by shrinkage of the crystal along
the a-axis. The other extreme, r = w, corresponds to
holding the value of A constant and allowing all of the
volume reduction to occur by shrinkage along the c-axis.
The choice r = 1.0 corresponds to an isotropic compression
in which A and C both decrease at the same rate (constant
C/A ratio).

The volume dependence of the quadrupole frequency in
GdCl3 for the point charge model is illustrated in Figure 4.4.
The frequency and volume have both been normalized to their
zero pressure values. For all positive values of r the
frequency increases as the volume decreases. This is exactly
the opposite of the observed behavior. For this reason we
will restrict ourselves to a discussion of the volume
dependence in the soft sphere model only.

In the soft sphere model the exponential dependence of
the overlap contribution to the field gradient can result in
a smaller gradient as ions are moved closer together. The
volume dependence of the quadrupole frequency in GdCl3 for
the three data sets of the soft Sphere model is illustrated
in Figures 4.5, 4.6 and 4.7. Again we have performed the
calculations for a wide range of values of r. The scale on
each of these figures is the same so that comparisons among
them can be made easily. Several comments are in order.

First, the volume dependence for each of the data sets
of the soft sphere model changes drastically as a function of

r. Compression along the c-axis alone (r = w) yields

44

 

 

 

 

l l I l
l. Point Charge Model _
GdCl3
|.03'- f8” —
>~ l— ..l
0
c
O
3
3
I: 1021- "L0 _.
"0.5
3 "0.0
o: b -
'6
E
O
z l.Ol- _
LOO-
r- -
l J l l
.980 .985 .990 .995 LO

Narmallzed Volume

Figure 4.4. Quadrupole Frequency versus Volume in the Point Charge Model

45

 

 

 

 

 

l I F l
1.02 .. Overlap Model A .4.
r 3 00

_ Gd Cl 3 ..

l.0| l— .....l
)0
o
I:

:3 _. f3‘LC) -‘
or
2
IL

1: |.00 l— "05

o
.2.‘
'3

E "' .—
3
:2

0.99 l- r80.0 _.

b .4

0.98 - _.

I J l l
.980 .985 .990 .995 LC

Narmallzed Volume

Figure 4.5. Quadrupole Frequency versus Volume in the Soft Sphere Model:
Data Set A

46

 

 

 

 

 

l l l l
— r. w d
Overlap Model B
LO". —.
GdCl3
>0
0
5 1.00—
3
g r= Lo/
u.
‘U
0
2 r805
'6 0.99— .-
E
5’
L 3
"0.0
0.98— .4
l l l l
.980 .985 .990 .995 LG

Narmallzed Volume

Figure 4.6. Quadrupole Frequency versus Volume in the Soft Sphere Model:
Data Set B

47

 

 

 

 

 

l l l l
"5.0 r800
|.O|l- .1,
F- -!
“2.0 ~
LOO—
>5
0
C
O
a b q
0’
O
l: r-|.O
3
= "0.6
E
3 "' r-O.5 "‘
z
0.98— --
*- r80.0 -
Overlap Model C
0.97... GdC|3 _‘
l l J l
.980 .985 .990 .995 LG

Normalized Volume

Figure 4.7. Quadrupole Frequency versus Volume in the Soft Sphere Model:
Data Set C

48

frequencies which increase as the volume is decreased, while
compression along the a-axis alone yields decreasing
frequencies. Second, for a fixed value of r, av/BV decreases
as one goes from data set A to data set C. This is easily
understood, since data set C contains a larger overlap
contribution to the field gradient than data set A. In a
qualitative way one can consider the point charge component
of the field gradient to contribute positive 3v/3V and the
overlap component to contribute negative av/BV. Finally,
all of the curves are slightly non-linear; for av/av
negative they curve upward and for av/BV positive they
curve downward. For r = 0.5 in data set B we tried to
estimate whether or not this effect was observable. If

6 cmz/kg, then at a pressure of

we assume 8 = 7 x 10-
5 x 103 kg/cm2 (about 75,000 p.s.i.) the effect would
produce a deviation from linearity in the frequency versus
pressure curve (Figure 8.5) of a few kHz. Since the
uncertainty in frequency measurements is of the order of
one kHz, we would not expect this possible non-linearity
to be observable at the pressures achieved in this work.
It is interesting to note that a distinct non-linearity
was observed in YbCl3 (Figure C.2). Unfortunately, as is
pointed out in Appendix C, analysis of the field gradient in
this compound must await better structural information.

we have calculated the volume dependence of the electric

field gradient in the soft sphere model for all of the hexa-

gonal rare earth trichlorides. We have arbitrarily limited

49

these calculations to values of r between 0.7 and 0.4.
The elastic constants of Stedman and Newman (1971) seem
to indicate a choice within this range, but our decision
was based more on convenience than on belief in their
numbers. The volume derivatives of the quadrupole frequency
and asymmetry parameter of each compound for the three data
sets of the soft sphere model are given in Table 4.6.
All derivatives are evaluated at the zero pressure limit.
The normalized volume dependence of the quadrupole
frequency in the soft sphere model is plotted with respect
to compound in Figure 4.8. The three shaded areas in the
figure correspond to the three different data sets of the
soft Sphere model. Each Shaded area corresponds to choices
of r between 0.7 and 0.4. For values of r outside this
range the curves are Similar. In order to compare these
calculations with the experimentally measured normalized
pressure coefficients, it is necessary to assume a value of
the bulk compressibility for each compound. We have assumed

6 cmZ/kg for each compound and calculated an

8 = 7 i l x 10-
"experimental" value of 3(v/vo)/3(V/Vo). These values are
also plotted in Figure 4.8. The error bars attached to
these values are a result of the assumed uncertainty in B
and are somewhat arbitrary as discussed in Section B. A
smaller value of B will result in an increase in the
"experimental" value of 3(v/vo)/3(V/Vo). In addition any

variation of B from one compound to another will also

shift these values.

50

Table 4.6. Results of Lattice Compression in the Soft Sphere Model

 

r = 0(3AK1 = 0.7 Data Set A Data Set B Data Set C

 

Compound 9 g 71/742 4) ’1- 9111/1102 9 ’L J (21/ Va) A 7L
A (v/vo) )(v/vo) 3(v/vo) 9 (v/vo) a (v/vo) J (v/vo)

 

LaC13 0.063 0.42 0.566 0.70 0.949 0.93
CeC13 0.047 0.45 0.533 0.71 0.902 0.93
PrCl3 0.028 0.46 0.499 0.72 0.854 0.92
NdC13 0.015 0.48 0.473 0.73 0.817 0.93
SmC13 -0.043 0.51 0.366 0.76 0.672 0.92
EuC13 -0.070 0.54 0.320 0.78 0.610 0.95
Gd013 -0.095 0.56 0.278 0.80 0.553 0.96

1’ = ’(3/‘(1 = 0.4

LaCl3 0.201 0.50 0.731 0.90 1.135 1.23
CeCl3 0.196 0.52 0.710 0.89 1.099 1.20
PrC13 0.188 0.53 0.687 0.89 1.063 1.17
NdCl3 0.185 0.56 0.670 0.90 1.035 1.17
SmCl3 0.148 0.60 0.581 0.92 0.904 1.14
EuC13 0.128 0.64 0.539 0.95 0.844 1.17

GdCl3 0.107 0.66 0.500 0.98 0.790 1.18

 

 

.l

I 7r l’ T ’1 Ti

Assume ,6 a - 'I‘ilmlO'6 cm 2llIa

.0
‘8
(3
I
7.
‘4

\l
8.
\ l ,\
.19

.°
N
on

0.00 A —

 

I—
l-
)—
l—
l—
—
-
p-

 

 

Figure 4.8. Derivative of Frequency with respect to Volume versus Com-
pound in the Soft Sphere Model

52

There are several important comments that can be made
about Figure 4.8. All of the calculated volume derivatives
for each of the soft sphere data sets decrease as one goes
from LaCl3 to GdCl3. This behavior is consistent with
experiment. We might point out that in the point charge
model the volume derivatives, in addition to being negative
for all compounds and all positive values of r, increase
slightly as one goes from LaCl3 to GdC13. The "experimental"
values fall on a smooth curve, but the calculated derivatives
seem to have a kink at NdC13. We are unsure of the origin
of this anomaly. We made a series of calculations in which
we attempted to discover the effects of the experimental
uncertainties in the lattice constants of NdCl3 on the
derivative of frequency with respect to volume. The effects
of such structural uncertainties are totally invisible on
the scale of Figure 4.8.

Because of our lack of information about 8 and r we
are unwilling at this point to make a choice between data
sets B and C. Data set A probably can be rejected as having
too small an overlap contribution to the field gradient.
we stress again that accurate values of bulk compressibilities
and linear compressibility ratios in these compounds are

essential to interpret the experimentally measured pressure

dependence of the quadrupole frequency.

V. CONCLUSIONS

In introducing the soft Sphere model of electric field
gradients we have attempted to include in a systematic
fashion the effects of wavefunction overlap. we have deduced
a Simple exponential dependence for the overlap contribution
to the field gradient from HF-SCF calculations for the KCl
molecule. we acknowledge the fact that in extending this
idea from a diatomic molecule to a closely packed ionic solid,
we are undoubtedly omitting important many-ion effects. We
feel confident that future theoretical work will indicate a
more comprehensive scheme for representing the overlap
contribution.

We feel that representation of ions by a characteristic
radius and softness parameter, which is an integral part of
the soft sphere model, is an extremely useful idea. Using
HFS calculations of ionic wavefunctions, we have made a
connection between Gilbert's empirical parameters for alkali
and halide ions, and parameters for other ions.

For the hexagonal rare earth trichlorides we have
demonstrated that the soft Sphere model of electric field
gradients gives far better agreement with experiment than
the earlier point charge model. The soft sphere model seems
to indicate that the value for the Sternheimer antishielding
parameter for C1. in these solids is about -30 as opposed to
the theoretical free ion value of -56.6.

The pressure dependence of the quadrupole frequency in
3, SmCl3 and GdCl3 has been measured. It

53

CeCl PrC13, NdCl

3'

54

is found that the frequency decreases linearly with increasing
pressure in all compounds. For any physically reasonable
value of the linear compressibility ratio, the point charge
model when applied to these compounds predicts that the
frequency Should increase with pressure. This constitutes
further evidence of the unsuitability of the point charge
model. On the other hand the soft sphere model predicts
that the frequency should decrease almost linearly with pressure,
provided that the linear compression along the c-axis is
Slower than that along the a-axis. Future work Should include
a measurement of the geometrical changes in these compounds
under pressure. The best method would be an x-ray measurement
of the lattice constants and chloride positional parameters
as a function of pressure. A direct measurement of the elastic
constants would also be useful.

AS soon as good crystal structure data is available for
TmCl

orthorhombic TbCl3 and monoclinic DyCl HoCl3, ErCl

3' 3’ 3’
YbCl3, and LuC13, the model Should be applied to these com-
pounds. The monoclinic salts represent a severe change in
geometry from the hexagonal salts, and a different value of
the overlap strength may be necessary. However, we would
expect the antishielding parameter to be roughly the same.
In TbCl3 one of the two chloride sites has an environment
quite similar to that in the hexagonal salts. we expect
that the soft Sphere model will predict the frequency and

asymmetry parameter well, using the same overlap strength

and antishielding factor as in GdC13.

55

In conclusion we feel that the soft Sphere model of
electric field gradients in solids, by including the effects
of molecular overlap, provides an important step forward

in the theory of nuclear quadrupole resonance.

REFERENCES

REFERENCES

Barford, N.C., 1967, Experimental Measurements: Precision,

 

 

Error and Truth, Addison-wesley, London, p. 62.

 

Bersohn, Richard, 1958, J. Chem. Phys., 29, 326.

Bersohn, R. and Shulman, R.G., 1966, J. Chem. Phys., 45, 2298.

Betts, D.D., Bhatia, A.B. and Horton, G.K., 1956, Phys. Rev.,
l_4, 43.

Bommer, H. and Hohmann, E., 1941, A. Anorg. Allgem. Chem.,
248, 373.

Bridgman, P.W., 1928, Am. Jour. Sci., Fifth Series, 15, 287.

Brun, E. and Hofer, 8.5., 1962, Z. Krist., 1_l, 37, 63.

Burns, Gerald and Wikner, E.G., 1961, Phys. Rev., 121, 155.

Cady, w.G., 1964, Piezoelectricity, Vol 1, Dover Publications,

 

New York.

Carlson, E.H., 1966, Bull. Am. Phys. Soc., _1, 377.

Carlson, E.H., 1969, Physics Letters, 22A, 696.

Carlson, E.H., 1971, J. Chem. Phys., to be published.

Carlson, E.H. and Adams, H.S., 1969, J. Chem. Phys., 51, 388.

Carlson, E.H., Current, D.H., and Foiles, C.L., 1971,
submitted to J. Chem. Phys.

Das, T.P. and Hahn, E.L., 1958, Solid State Physics, Supplement
I, edited by F. Seitz and D. Turnbull, Academic Press,
New York.

Das, T.P. and M. Karplus, 1959, J. Chem. Phys., 30, 848.

Das, T.P. and Karplus, M., 1965, J. Chem. Phys., 42, 2885.

Garton, G., Hutchings, M.T., Shore, R. and Wolf, W.P., 1964,

J. Chem. Phys., 41, 1970.
56

57

Gilbert, T.L., 1968, J. Chem. Phys., 49, 2640.

Gopal, E.S.R., 1966, Specific Heats a; Low Temperatures,

 

 

Plenum Press, New York.

Goree, W.S., McDowell, B., and Scott, T.A., 1965, Rev. Sci.
Inst., 16, 99.

Hearmon, R.F.S., 1946, Revs. Modern. Phys., 18, 409.

Herman, F. and Skillman, S., 1963, Atomic Structure Calculations,

 

Prentice-Hall, Englewood Cliffs, New Jersey.

Hessler, J.P., 1971, dissertation, The Low Temperature

 

COOperative Behavior 91 the Rare Earth Salts GdCl

 

3

 

and PrC13, Michigan State University.
Huges, W.E., Montgomery, C.G., Moulton, W.G. and Carlson,
E.H., 1964, J. Chem. Phys., 11, 3470.

Kittel, C., 1956, Introduction 19 Solid State Physics, Second

 

 

Edition, John Wiley and Sons, New York.

Lucken, E.A.C., 1969, Nuclear Qpadrupple Coupling Constants,

 

Academic Press, London.

Mangum, B.w. and Utton, D.B., 1967, Bull. Am. Phys. Soc., 12,
1043.

Matcha, Robert L., 1970, J. Chem. Phys., 53, 485.

Morosin, B., 1968, J. Chem. Phys., 19, 3007.

Parks, 8.1., 1967, dissertation, N.M.R. 1p Paramagnetic NdBr3

 

and U13 and 1p,Antiferromagnetic Q13, Florida State

 

University.
Ramsey, N.F., 1956, Molecular Beams, Oxford University Press,
London, Table XI.

Raymond, Michael, 1971, Phys. Rev. B., 1, 3692.

58

Sawatzky, G.A. and Hupkes, Julieks, 1970, Phys. Rev. Letters,
11, 100.

Sharma, R.R. and Das, T.P., 1964, J. Chem. Phys., 11, 3581.

Sharma, R.R., 1970, Phys. Rev. Letters, 15, 1622.

Sharma, R.R., 1971, Phys. Rev. Letters, 16, 563.

Slichter, C.P., 1963, Principles g£_Magnetic Resonance,

 

Harper & Row, New York.
Stedman, G.E. and Newman, D.J., 1971, J. Chem. Phys., 1:, 152.
Sternheimer, R. and Foley, H., 1956, Phys. Rev., 193, 731.
Taylor, D.R., 1968, J. Chem. Phys. 18, 536.
Templeton, D.H. and Dauben, C.H., 1954, J. Am. Chem. Soc.,

16, 5237.
Varsanyi, F. and Maita, J.P., 1965, Bull. Am. Phys. Soc.,

10, 609.
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Zachariasen, W.H., 1948, J. Chem. Phys., 16, 254.

APPENDICES

APPENDIX A
LATTICE CONSTANTS AND POSITIONAL PARAMETERS

Morosin (1968) measured the lattice constants of
LaC13, NdC13, EuCl3 and GdCl3 with an uncertainty of less
than 0.001 A. He also measured the chloride positional
parameters to within 0.1%. He suggests the following
procedure, based on the earlier work of Templeton and
Dauben (1954), for estimating the lattice constants and
3, PrCl3 and SmCl3. When
plotted against atomic number, the value for C obeys a

positional parameters of CeCl

smooth relationship while the value for A obeys two

linear relationships, one for LaCl3 through NdCl and

3’

a second for SmCl through GdC13. These graphs are shown

3
in Figure A.1.
Values of u for CeCl3 and PrCl3 were obtained by

linear interpolation of Morosin's data between LaCl3 and
NdC13, while the value for SmCl3 was obtained by
extrapolating the values for EuCl3 and GdCl3. Morosin's
values for v obey a smooth curve from LaCl3 through
EuC13. Values of v for CeC13, PrCl3 and SmCl3 were
interpolated from this curve. The curves for u and v

are shown in Figure A.2. The interpolated values

together with Morosin's data are listed in Table 2.1.

59

A (Angelrame)

C (AnaelramO)

60

 

I41!

'h46

14MB

'h42

14!)

'L38

‘136

d

 

4440

4u35

4030

I4J25

4020

44J5

4n|0

l l l l l I L

 

 

Figure A.1.

SMCI 3 EUC's GdCl 3

Lattice Constants versus Compound

 

61

 

 

 

 

 

T I l T T l r T
P
0.389 -
F q
0.386 P -
P I!
00387 - -
0.3020 l- ..
)- .1
“F
>
OJKMSI- ..
<5
" 'l
1.
0030.2 _ q
1 1 1 l 1 l J l
LaCl3 CeCI3 PrCl3 NdCI3 SmCl3 EIICI3 (MCI3

Figure A.2. Chloride Positional Parameters versus Compound

 

APPENDIX B

HEXAGONAL RARE EARTH TRICHLORIDE DATA

Raw data for the 35Cl quadrupole resonance as a

function of hydrostatic pressure at 77K and 300K is
listed in Tables 3.1 through B.5. Missing data at 300K
indicates that signals were too weak to measure. In
each table data is listed in the order in which it was
recorded. All frequency values are the average of five
readings with standard deviation of less than 1 kHz.

The uncertainty in pressure values is judged to be

0.5 x 103 p.s.i. Data at 77K are plotted in Figures B.l
through B.5. On all graphs the Size of the circle
representing a datum is greater than the error associated
with it. The room temperature data is Similar to that

at 77K. It is not plotted, but the results of a least

squares analysis are presented in Table B.6.

62

63

Table 8.1. Quadrupole Frequency versus Pressure Data: CeCl3

 

 

 

T = 77K

Date P(103 p.s.i.) ‘U(kHz)
10/21/70 13.0 4356.
10/21/70 19.6 4346.
10/21/70 25.0 4335.
10/21/70 30.0 4327.
10/21/70 35.0 4317.
10/21/70 39.9 4309.
10/21/70 45.9 4299.
10/21/70 50.2 4291.
10/21/70 55.3 4282.
10/21/70 61.2 4273.
10/21/70 65.4 4266.
10/21/70 70.3 4257.
10/21/70 75.4 4248.
10/21/70 81.2 4239.
10/21/70 38.3 4312.
10/21/70 52.7 4288.
10/21/70 22.9 4339.
10/21/70 8.1 4364.
10/21/70 0.015 4377.

 

64

 

 

 

 

 

I T r I
4.38— --1
CeCl3 T = 77 K
4.36 l-
403‘ -
4.32 l-
“J
I
I
" 4.30 -
>6
0
c
O
:I
6'
b 4.28 '-
h.
4.26 '-
4.24 r-
1 l l 1
20 40 60 80
Pressure ( I03 pei)
Figure 8.1. Quadrupole Frequency versus Pressure at 77K: CeCl3

65

Tablg 8.2. Quadrupole Frequency versus Pressure Data: PrCl3

 

 

 

T = 77K

Date 2(103 p.s.i.) 1)(kHz)
10/1/70 17.7 4533.3
10/1/70 40.7 4495.2
10/1/70 60.2 4462.3
10/8/70 23.1 4524.7
10/8/70 30.0 4513.5
10/8/70 39.9 4497.2
10/8/70 50.2 4479.0
10/8/70 60.0 4464.4
10/8/70 65.0 4455.9
10/8/70 70.7 4445.5
10/8/70 80.3 4430.4
10/8/70 75.7 4437.5
10/8/70 58.9 4464.9

10/8/70 0.015 4561.9

 

(MHz)

Frequency

66

 

 

 

 

 

I l f’ l
4.56 —l
l'-"rCl3 T = 77K
4.54 l—
4L521-
4.50 l-
4.48 1-
4.46 l—
4.44 l-
L I l l
20 4O 60 80
Pressure ( IO3 psi)
Figure 8.2. Quadrupole Frequency versus Pressure at 77K: PrC13

67

 

 

 

Table 8.3. Quadrupole Frequency versus Pressure Data: NdC13
T = 77K
Date P(103 p.s.i.) 'V(kHz)
9/10/70 17.1 4696.1
9/10/70 40.9 4657.2
9/10/70 65.3 4620.0
9/10/70 53.0 4638.4
9/10/70 29.5 4675.8
9/10/70 17.1 4695.9
9/10/70 9.0 4708.6
9/10/70 5.0 4715.9
9/10/70 13.5 4702.0
9/10/70 19.9 4691.6
9/10/70 32.2 4671.3
9/10/70 43.8 4653.0
9/10/70 57.8 4631.1
9/10/70 62.0 4624.2
9/10/70 48.5 4646.0
9/10/70 37.2 4663.6
9/10/70 34.9 4667.7
9/10/70 24.9 4683.0
9/10/70 0.015 4722.3

68

Table 8.3. (cont'd.)

 

 

 

T = 300K

Date 9(103 p.s.i.) 1J(kuz)
9/10/70 17.8 4649.6
9/10/70 40.9 4612.5
9/10/70 64.7 4574.8
9/10/70 51.5 4595.1
9/10/70 30.2 4628.9
9/10/70 17.8 4649.3

9/10/70 0.015 4676.4

 

69

 

 

I I I 71
4'72 Mac:3 T=77K _.
4L70!-’ -1
’3 4.681— —.
3:
1!
)5
(a
:i‘9lflip- -d
1:
U’
0
5
IL
4.64- .—
4.62- --l
l 11 l l

 

 

 

I5 30 45 60

Pressure (IO3 psi)

Figure 8.3. Quadrupole Frequency versus Pressure at 77K: NdCl3

70

Table 8.4. Quadrupole Frequency versus Pressure Data: SmC13

 

 

 

 

 

 

T = 77K
Date P(1O3 p.s.i.) 1)(kHz)
9/9/70 19.0 4999.4
9/9/70 40.9 4965.6
9/9/70 61.0 4937.2
9/9/70 67.0 4926.2
9/9/70 50.9 4950.1
9/9/70 30.9 4981.0
9/9/70 17.2 5001.6
919/70 25.0 4989.3
9/9/70 55.7 4943.0
9/9/70 45.2 4957.9
9/9/70 36.1 4972.3
9/9/70 8.6 5014.1
9/9/70 0.015 5026.8
T = 300K

P(103 p.s.i.) 19(kHz)

9/9/70 18.2 4950.7
9/9/70 41.0 4914.4
9/9/70 60.3 4887.2
9/9/70 0.015 4975.6

 

(MHz)

Frequency

71

 

 

 

 

 

1 17’ I I

SmCl 'T=77K
502 3 .1
5.00 - a
4%98-- -
4&961- -
4&94l- '-
4.92 '- 'H

l l l, I

I5 30 45 60

Pressure ( lO3 psi )
Figure 8.4. Quadrupole Frequency versus Pressure at 77K: SmC13

72

 

 

 

Table 8.5. Quadrupole Frequency versus Pressure Data: GdCl3
Date P(103 p.s.i.) 1)(kHz)

7/14/70 14.8 5286.9
7/14/70 22.2 5276.7
7/14/70 24.0 5274.4
7/15/70 10.5 5293.1
7/15/70 23.9 5274.0
7/15/70 45.3 5242.3
1/13/71 18.8 5281.9
1/13/71 30.0 5264.9
1/13/71 40.0 5251.5
1/13/71 49.8 5236.9
1/13/71 60.2 5221.4
1/13/71 69.2 5208.9
1/13/71 55.3 5229.6
1/13/71 45.7 5242.7
1/13/71 35.3 5257.4
1/13/71 25.0 5272.8
1/13/71 14.3 5288.5
1/13/71 7.8 5297.8
1/13/71 0.015 5307.6

73

Table 8.5. (cont'd.)

 

 

 

T = 300x

Date 11(103 p.s.i.) wkuz)
7/23/70 5.0 5241.5
7/23/70 10.0 5233.4
7/23/70 21.0 5220.4
7/23/70 30.0 5207.3
7/23/70 40.2 5190.7
7/23/70 50.2 5177.4
7/23/70 34.9 5199.5
7/23/70 25.1 5212.4
7/23/70 0.015 5248.4
1/13/71 59.6 5165.8
1/13/71 75.3 ‘ 5144.1
1/13/71 45.7 5185.6
1/13/71 54.9 5174.0
1/13/71 64.9 5159.8
1/13/71 70.2 5152.0

1/13/71 19.6 5222.6

 

(MHz)

Frequency

74

 

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£128

£126

£124

SJNZT-

 

l l

GdCl

l

T=77K

 

 

 

Figure 8.5.

I5 30

45

Pressure (l03 psi)

Quadrupole Frequency versus Pressure at 77K:

GdC13

75

 

 

 

 

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APPENDIX C

DATA FOR ErCl3 AND YbCl3

ErCl3 and YbCl3

There are two chemically inequivalent chloride ion sites

are isomorphic to monoclinic AlClB.

in the molecule resulting in two quadrupole frequencies
separated by about 50 kHz. The lower frequency line is
twice as intense as the upper line. The former is

and the latter as v .

2 l
The crystal structure parameters for these compounds

referred to as v

are poorly known. For a summary of the locations of
available data see Morosin (1968), reference 12. For
this reason field gradient calculations in these compounds
have not been carried out. During the course of this work,
however, the pressure dependence of the quadrupole
frequencies in ErCl3 and YbCl3 was measured.

In both compounds the quadrupole frequencies increase
with pressure, each of the two lines in a given compound

35Cl resonances

having a different slope. Raw data for the
is given in Tables C.l and C.2. All data is for T = 77K,
with the exception of the stronger line in ErCl3 where some
data at 300K was also taken. The data at 77K is plotted
in Figures C.l and C.2.

The resonance lines in ErCl3 at 77K disappeared above
a pressure of about 55 x 103 p.s.i., and did not reappear
when the pressure was lowered. This effect was observed

in two different samples, one of which was kept free of

mineral oil. As the critical pressure was approached the
76 ‘

77

resonance signals lost strength. This probably means that
the lines broadened. The effect was not observed at 300K
at pressures up to 65 x 103 p.s.i. We suspect that what
we observed was a crystallographic transition to the

orthorhombic TbCl structure. Monoclinic DyCl3 is known

3
to undergo such a transition when cooled to 77K (Carlson,
1969). No further investigation of the transition in
ErCl3 was attempted.

In YbCl3 the lower frequency line has a distinct
non-linear increase with pressure, while the upper line
appears to be strictly linear. Since the data for ErCl3
is of relatively poor quality such behavior there can

not be ruled out. The results of a least squares analysis

of the data for ErCl3 and YbCl3 are presented in Table C.3.

78

Table C.1. Quadrupole Frequency versus Pressure Data: ErCl3

 

 

 

 

 

 

T = 300K
Date P(103 p.s.i.) 1)2(kHz) 'Z)1(kHz)
9/8/70 19.7 4429.4 --- a
9/8/70 50.5 4939.3 ---
9/8/70 65.8 4444.8 ---
9/8/70 46.3 4438.0 ---
9/8/70 30.3 4433.2 ---
9/8/70 0.015 4424.5 4476.8
T - 77x

P(103 p.s.i.) 112(kuz) 1J1(knz)
9/8/70 14.1 4454.9 4518.0
9/8/70 30.0 4460.3 4528.0
9/8/70 45.0 4466.5 4537.2
10/20/70 20.2 4456.5 4519.7
10/20/70 26.0 4458.8 4525.0
10/20/70 32.0 4461.2 4529.5
10/20/70 38.0 4464.5 4532.1
10/20/70 44.0 4466.5 4537.0
10/20/70 50.2 4468.2 4539.6
10/20/70 20.1 4455.5 4520.2
10/20/70 54.5 4472.0 too weak
8/9/70 0.015 4449.2 4508.5

 

a
Line not measurable.

0' (MHZ)

79

 

4£¥3

4453

4n52

4.5I

 

 

l 21, l

 

 

Figure 0.1.

IS 30 45
Pressure (lospsi)

Quadrupole Frequencies versus Pressure at 77K:

4.47

&

§

0
”2(MH2)

4&45

ErCl3

80

Table C.2. Quadrupole Frequency versus Pressure Data: YbC13

 

 

 

T = 77K
Date 2(103 p.s.i.) 'Z/2(kHz) 7p’l(knz)
1/15/71 21.8 4787.1 4847.8
1/15/71 24.9 4787.7 4849.6
1/15/71 30.0 4789.3 4852.6
1/15/71 35.0 4790.9 4854.9
1/15/71 40.1 4792.3 4857.4
1/15/71 45.0 4793.8 4860.6
1/15/71 50.4 4795.7 4863.3
1/15/71 55.0 4796.9 4865.9
1/15/71 60.2 4798.7 4868.9
1/15/71 65.2 4800.3 4871.3
1/15/71 70.0 4801.9 4874.1
1/15/71 75.3 4804.1 4877.3
1/15/71 80.5 4806.2 4879.0
8/9/70 0.015 4781.1 4836.0

 

81

 

4.88 r—

4.87

”I (MHz)

 

 

 

 

4.88
4.85
-4t8l
4.84
q 4.80
73
I
2
- 4.79 N
3
- 4.78
l l l 1
20 40 80 80

Pressure (Iospei)

Figure C.2. Quadrupole Frequencies versus Pressure at 77K: YbC13

82

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APPENDIX D
LATTICE SUMS

All lattice sums were computed from FORTRAN programs
written for the CDC 6500 computer at Michigan State
University. Sums were done over neutrally charged
clusters of eight ions in a roughly spherical volume
of radius 50 A. Approximately 2500 such clusters were
included in the sums. The components of the electric

field gradient tensor were computed in a Cartesian frame

defined by
x = /3 a(A-B)/2 , (D.1a)
y = a(A+B)/2 and z = cC, (D.1b)

where a and c are the lattice constants, and A, B and C
are distances along the (right-handed) crystallographic
axes. The convergence of these sums was tested by
comparing the results of a 50 A sum with one of 100 A.
The results differed by less than 1 part in 104 for each
component of the tensor.

In these calculations the contribution to the field

gradient from an ion located at (xl,x2,x3) is given by

_ _ _ 2 2 . . _
Vij - (Qp QO)(3xixj R 6ij)/R , (1,3 — 1,2,3).
(D.2)
R is the distance to the ion with charge 2 and Qp and

Q0 are the point charge and overlap contribution given by

2(1 - Ym)/R3 (D.3)

Qp

and
83

84

Q0 = F exp[(RaB - R)/pa81 . (0.4)

R and pas are the soft-sphere parameters for the Cl-

08
ion at the origin and the particular ion at distance R.
Qo is included in the calcuation only for ions with

R‘: 4.0 A. F and (l - Y0) are treated as adjustable
parameters.

In the tables that follow the total electric field

gradient is tabulated for the chloride ion at (u, v, 1/4).

V , V , V

and V are the components of the gradient
xx yy 22 xy

in the Cartesian frame defined above. sz and Vyz are

zero from symmetry. V Vy and V2 are the diagonal elements

x!
of the field gradient tensor in its principal axis system.
The relationship between the diagonal elements of the

field gradient in atomic units (bohr-B) and the quadrupole

frequency is

H

_ _ 2
v0 — 9.3614 V2 1 + 3 n MHz . (D.5)

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