• TABLE OF CONTENTS
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 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Figures
 Abstract
 Introduction
 Theoretical treatments of solid...
 Experimental apparatus and...
 Results and discussion
 Summary of the results
 References
 Biographical sketch














Title: Nuclear exchange energy and isotopic phase separation in solid helium
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Title: Nuclear exchange energy and isotopic phase separation in solid helium
Physical Description: viii, 103 leaves : illus. ; 28 cm.
Language: English
Creator: Panczyk, Michael Francis, 1938-
Publication Date: 1968
Copyright Date: 1968
 Subjects
Subject: Nuclear engineering   ( lcsh )
Nuclear reactions   ( lcsh )
Helium -- Isotopes   ( lcsh )
Solid helium   ( lcsh )
Physics and Astronomy thesis Ph. D
Dissertations, Academic -- Physics and Astronomy -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis - University of Florida.
Bibliography: Bibliography: leaves 100-102.
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098209
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 001106685
oclc - 13586034
notis - AFK3029

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Table of Contents
    Title Page
        Page i
        Page i-a
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Figures
        Page vi
    Abstract
        Page vii
        Page viii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    Theoretical treatments of solid He3
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
    Experimental apparatus and procedure
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
    Results and discussion
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
    Summary of the results
        Page 97
        Page 98
        Page 99
    References
        Page 100
        Page 101
        Page 102
    Biographical sketch
        Page 103
        Page 104
        Page 105
Full Text















NUCLEAR EXCHANGE ENERGY AND
ISOTOPIC PHASE SEPARATION
IN SOLID HELIUM










By
MICHAEL FRANCIS PANCZYK


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
TIIE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA
1968



























69-5397

PANCZYK, Michael Francis, 1938-
NUCLEAR EXCHANGE ENERGY AND ISOTOPIC
PHASE SEPARATION IN SOLID HELIUM.

The University of Florida, Ph.D., 1968
Physics, solid state



University Microfilms, Inc., Ann Arbor, Michigan



































For My Delightful Wife,

MARY
















ACKNOWLEDGMENTS


I wish to express my sincere appreciation to those individuals

listed below for generously contributing both their time and talents

to this work.

Dr. E. D. Adams suggested this investigation and provided con-

tinual guidance throughout its entire development.

Mr. R. A. Scribner made many valuable contributions to the de-

sign and construction of the apparatus and also spent many hours help-

ing take the data. Dr. G. C. Straty contributed his vast technical

Knowledge to this work and is responsible for the particular design

of the strain gauge used in these experiments. Dr. J. R. Gonano often

participated in many helpful discussions concerning the interpretation

of the experimental results. Messers. D. C. Heberlein and J. W. Philp

were frequently called upon to assist me and always did so cheerfully.

Mr. B. McDowell often worked long hours to provide the copious quanti-

ties of liquid helium necessary to carry on this investigation.

Finally, I wish to express my sincere appreciation to my wife,

Mary, for her patience and understanding during what has been a long

and at times frustrating graduate career.
















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS. . . . . . . . . . . . . ii

LIST OF FIGURES . . . . . . . .. . . . vi

ABSTRACT . . . . . . . . . . . . . . vii

Chapter

I. INTRODUCTION . . . . . . . .... .. . .. 1

II. THEORETICAL TREATMENTS OF SOLID HE3. . . . . . 9

A. Introductory Remarks . . . . . . . . 9

B. Physical Origin of the Exchange Energy . . . 10

C. Microscopic Theory . . . . . . . .. 13

Nosanow's Theory of Quantum Crystals . . ... 15

D. Thermodynamic Theory . . .. . . . . . 27

III. EXPERIMENTAL APPARATUS AND PROCEDURE . . . . .. 33

A. Introductory Remarks . . . . . . . .. 33

B. Cryostat . . . .. . .. . . . .. . 34

Helium Refrigerators . . . . . . .. 34

Vacuum Chambers and Radiation Shields. . . ... 39

C. The Strain Gauge . . . . . . . . .. 41

D. Pressure Measurements. . . . . . . .. 48

Gas Handling and Pressure System . . . .. 48

Pressure Calibration and Measurement . . .. 51

E. Potassium Chrome Alum Salt Assembly. . . . .. 52

F. Temperature Measurements . . . . . ... 58

Temperature Calibrations and Measurements . .. 58

iv











Chapter


Temperature Regulation . . . . . .

Thermal Equilibrium Time between the Sample and


Thermometer.


Page

. 65


. . . . . . . . . 66


G. Solenoids. . . . . . . . . . . .

H. Performance of the Experiment. . . . . ... .

Sample Formation and Cooldown to 0.30K . . . .

Demagnetization Procedure. . . . . . . .

IV. RESULTS AND DISCUSSION . . . . . . . . .

A. Introductory Remarks . . . . . . . . .

B. The Thermal Expansion of the Empty Cell. . . . .

C. Nuclear Exchange Energy . . . . . . . .

Values of IJI for "Pure" He3 . . . . . .

Effects of He4 Impurities on J . . . . . .

D. Locus of the Zeros of the Thermal Expansion
Coefficient. . . . . . . . . . .

E. Isotopic Phase Separation . . . . . . .

Kinetics of the Phase Transition . . . . .

Pressure Dependence of the Energy of Solution and
Phase Separation Temperature . . . . . .

V. SUMMARY OF THE RESULTS .. . . . . . . . .


REFERENCES . . . . . . . . .

BIOGRAPHICAL SKETCH. . . . . . . .


S. . 100

. .. 103


67

68

68

69

72

72

75

77

77

85


89

97
















LIST OF FIGURES


Figure Page

1. Calculated ground state energy versus molar volume .... .22

2. Variational parameters Aand log K versus molar volume. . 23

3. Calculated nuclear exchange energy versus molar volume . 24

4. Schematic diagram of the apparatus . . . . .... 36

5. Schematic diagram of the low temperature section ..... 38

6. Capacitance strain gauge. . . . . . . . . .44

7. Pressure system. . . . . . . . ... ..... 50

8. Potassium chrome alum salt assembly. . . . .... 56

9. Simplified schematic of the de mutual inductance circuit .59

10. Schematic diagram of the CMN thermometer and the mutual
inductance system. . . . . . . . . . . 61

11. Simplified schematic of the ac resistance bridge . . .. .64

12. Characteristic isochore for the 1600 ppm He4 sample. ... 74

13. Thermal expansion of the empty strain gauge. ....... .76

14. The nuclear exchange contribution to the pressure, (AP)EX,
versus T-1 for various molar volumes . . .... ... . 79

15. Nuclear exchange energy versus molar volume. ....... 82

16. Locus of the zeros of the isobaric expansion coefficient .87

17. Pressure change, (AP)pS, due to the isotopic phase
separation in the 600 ppm He4 sample versus temperature. . 92

18. Pressure change, (AP)pS, due to the isotopic phase
separation in the 1600 ppm He4 sample versus temperature 93
dE /k
19. Plot of versus molar volume. . . . . . ... 94
dV
20. Energy of mixing versus molar volume . . . . . . 96










Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy

NUCLEAR EXCHANGE ENERGY AND ISOTOPIC
PHASE SEPARATION IN SOLID HELIUM

By

Michael Francis Panczyk

March, 1968

Chairman: Dr. E. Dwight Adams
Major Department: Physics

By making use of a sensitive capacitance type strain gauge, we

have investigated the nuclear exchange energy and isotopic phase sep-

aration in solid helium mixtures containing 20, 600, and 1600 ppm He4.

Measurements of the pressure versus temperature for constant volume

samples between.22.8 and 24.2 cm3/mole have been made down to 20 m0K,

and show the expected T-1 dependence due to nuclear spin ordering at

low temperatures. The "alues of the nuclear exchange energy were ob-

tained from the slopes of the P versus T-1 curves by means of a self-

consistent procedure. The magnitude of the exchange energy, JJl/k, is

about 0.7 mOK at a molar volume of 24 cm3/mole and decreases with in-

creasing density approximately as 1iJ = 16.4. The corresponding
dlnV

magnetic transition temperature varies from 2.4 m'K at 24.2 cm3/mole

down to 0.9 mOK at 22.8 cm3/mole. Since the data were obtained in a

temperature range well below the isotopic phase separation temperature

of the three mixtures, the values of IJI reported here are those of a

very pure He3 solid.

The isotopic phase separation in the two samples containing 600

and 1600 ppm He4 has been detected by observing.the increase in pres-

sure which occurs as the mixture separates into two enriched phases.











For the 600 ppm sample this excess pressure is = 3 x 10-3 atm, while

for the 1600 ppm sample it is = 9 x 10-3 atm. The corresponding phase

separation temperatures, at a molar volume of 24.0 cm3/mole, are ap-

proximately 0.1080K and 0.1190K respectively. A brief investigation of

the density dependence of the phase separation phenomenon indicates

that both the excess pressure and the phase separation temperature de-

crease with increasing density.

A discussion of the assumptions and approximations in the theory

of "Quantum Crystals" formulated by Nosanow has also been given. A

comparison between these theoretical calculations and our experimental

results for the exchange energy and its volume dependence indicates

that while this theory is qualitatively correct, refinements must be

made in order to obtain quantitative agreement with existing experi-

mental data.


viii
















CHAPTER I


INTRODUCTION


Helium exists in two stable isotopic forms, He3 and He4. He3

atoms have a spin of 1/2 and hence are treated theoretically by Fermi-

Dirac statistics, while He4 atoms have a spin equal to zero and thus

follow a Bose-Einstein distribution law. Since the electrostatic in-

teraction between two He4 atoms or two He3 atoms is approximately the

same, these isotopes and mixtures of various concentrations provide

excellent examples for studying the effects of quantum statistics on

the macroscopic properties of systems.

One of the most interesting properties of both the pure isotopes

and mixtures is that they remain liquids down to the absolute zero of

temperature. This is a consequence of the weak interatomic forces and

the small mass of the helium atoms. Pressures of the order of 25 at-

mospheres are required to bring about the liquid-solid transition.

Liquid He4 has been studied extensively for about fifty years. Most of

the research has centered on the properties of the superfluid phase.

F. London1 pointed out that the existence of this phase is closely

connected with the fact that the He4 atom is a boson. Liquid He3,

which is a simple fermion system, exhibits no such superfluid phase,

although the possibility of a superfluid transition similar to that

occurring in the electron gas in metals has been proposed by several

authors.2,3,4 Experimental investigations of the nuclear magnetic


_~










susceptibility5,6,7 and spin diffusion coefficient of liquid He3 in-

dicate that the magnetic properties of the system are in excellent

agreement with the theoretical predictions of Landau.9 The agreement

with respect to other properties of the liquid, however, is not quite

so good. In particular, recent specific heat measurements810 to 3 mK

do not exhibit a linear dependence on the temperature as one would ex-

pect on the basis of Landau's Fermi liquid theory. Thus it appears

that additional information about the liquid is needed in order to

determine the degree of validity of Landau's approach. In particular,

measurements of the isobaric thermal expansion coefficient to temper-

atures of a few millidegrees are very desirable.

The melting line of He3 is of particular interest because it ex-

hibits a deep minimum. Although this phenomenon is unusual, since it

implies that the solid possesses a greater degree of disorder than

liquid, its existence was predicted by Pomeranchuk11 before it was
12
actually observed in the experiments of Baum, et al. Pomeranchuk

argued that the nuclear spins in the solid should be randomly dis-

tributed down to temperatures of a few microdegrees, and hence con-

tribute Rln2 to the total entropy. Since the entropy of the liquid is

known to fall below this value at about 0.320K, there should be a change

in sign of (S -S ) at this temperature and by the Clausius-Clapeyron
i s
dP
equation Lj should be negative. Although it is now expected that the

exchange interaction in the solid near the melting curve will produce

nuclear spin alignment in the millidegree rather than the microdegree

region, this temperature range is still considerably below 0.32K,so

that Pomeranchuk's original argument for the existence of the minimum
dP
remains at least qualitatively correct. The absolute value of d cannot
dT










continue to increase at temperatures very close to absolute zero since

the Nerst Theorem implies that = 0 at T = 0. Recent strain gauge
13
measurements by Scribner, et al.13 locate the minimum at a temperature

of 0.3180K and a pressure of 28.93 atm. These measurements were car-

ried down to 0.017K and at this temperature id is still increasing.

At the present time it is expected d- will reach a maximum value at

about 7 mK and again become zero at approximately 0.5 mK.

The properties of the solid phase of helium have not received as

much attention as those of the liquid. The experiments of Grilly and

Mills14 revealed the existence of two solid phases of He3 having a

triple point with the liquid at T = 3.1480K and P = 135.9 atm. X-ray

diffraction experiments5 showed that the crystal structure of the low

pressure phase was body-centered-cubic (bcc) while the higher pressure

phase was hexagonal-close-packed (hcp). Recent pressure measurements

by Straty and Adamsl6 showed that below 10K, the bcc-hcp phase boundary

is horizontal at a pressure of about 105 atm. Further x-ray work by
17 18
Schuch and Mills and by Franck8 revealed the existence of a third

solid phase above T = 17.780K and P = 1608 atm. The crystal structure

of this phase is cubic-close-packed (ccp). Solid He4 is also found to

exist in these three crystal structures, although the details of the

PVT relations of the two solids are somewhat different. The fact that

low pressure solid He3 exists in a body-centered-cubic structure is

somewhat unusual since the stable structure for most dielectric solids
19
is cubic-close-packed. Nosanow has demonstrated that the existence

of solid helium in the bcc phase is due to the strong short range cor-

relations which arise from the large amplitude zero point motion of the

atoms, and hence is a manifestation of the quantum nature of the solid.










20
Recent heat capacity experiments by Sdenson and co-workers2 and

by Pandorf and Edwards,21 along with the pressure measurements of

Straty and Adams,6 show that above 0.3K, these properties of the

solid are determined principally by the phonons with almost no con-

tribution from the nuclear spin system. This situation cannot con-

tinue to very low temperatures since the contribution to the free

energy from the phonons decreases while that from the spins increases.

As mentioned previously Pomeranchuk11 originally predicted that the

temperature range in which the spin system would determine the prop-

erties of the solid should be around a few microdegrees. He based

this prediction on the idea that the He3 atoms in the solid are tightly

bound to well separated lattice sites, and hence exchange effects are

negligible. Bernardes and Primakoff22 later pointed out that the

large amplitude zero point motion of the He3 atoms produces considerable

overlap of the wave functions of neighboring atoms, and hence exchange

effects in the solid are very important. They developed a theory for

the ground state of solid He3 which predicted that antiferromagnetic

spin alignment should occur in the low density solid at temperatures

of a few hundredths of a degree. More recent calculations by Nosanow19

indicate that the exchange energy is not quite as large as originally

estimated by Bernardes and Primakoff. Nosanow predicts that the magnetic

ground state is still antiferromagnetic but that the Neel temperature

is approximately 0.2 m*K.

There are many reasons why the nuclear magnetic properties of

solid He3 have received so much attention. One of the most important

is the absence of any electronic contribution to the magnetic moment

of the atom. Also, since it forms a simple dielectric solid containing










localized spin 1/2 particles, He3 represents an ideal substance in

which to test the various theories of magnetism. From an experimental

point of view, the large compressibility of the solid allows one to

study the magnetic properties over a wide range of densities by ap-

plication of modest pressures.

Early experimental attempts to measure the exchange energy in the
23
solid were focused mainly on the nuclear magnetic susceptibility2 and

the specific heat.24 These early experimental results were inadequate

either because of sizable amounts of He4 impurities or because the

measurements were not extended to low enough temperatures. Nuclear

susceptibility measurements are very attractive because they allow

both the sign and magnitude of the exchange energy, J, to be determined

from a single measurement. However, since the Neel temperature in the

solid is expected to be of the order of a few millidegrees, it is neces-

sary to make susceptibility measurements to temperatures around 0.02K

and lower to determine J accurately. At these temperatures, especially

for the higher densities, measurements become very difficult because

of the long thermal equilibrium times between the nuclear spin system,

the lattice, and the thermometer. Nevertheless, susceptibility ex-

periments have been performed by Thomson, Meyer, and Dheer25 to tem-

peratures down to 0.070K. Their results indicate that the exchange

energy is negative, and the corresponding Ngel temperature, TN, is

less than 0.02*K for molar volumes greater than 22.4 cm3/mole. At

higher densities, they found a systematic increase in TN to a value

of 0.1K at a molar volume of 19.5 cm3/mole. These higher density re-

sults were regarded as very tentative however, and may be due to non-

equilibrium effects which arise from the He4 impurities present in the










He3. These nonequilibrium effects were more apparent in the suscepti-

bility measurements made on the 1% He4 sample also studied by Thomson,

et al.,25 and are discussed in some detail by them.

Two technical problems make the determination of the exchange

energy from specific heat data difficult. The first is the large con-

tribution to the specific heat from isotopic phase separation of He4

impurities present in the He3. This was first observed by Edwards,

et al.24 in experiments designed to determine the nuclear spin contri-

bution to-the specific heat. These experimenters found a large anomaly

in the heat capacity of solid mixtures which suggested that at low tem-

peratures the mixture separates into two phases, one rich in He3 and

the other in He4. These measurements also indicate that the phase

separation line is symmetric about a concentration of 50%, and that the

two isotopes will be completely separated at O0K. The problems pre-

sented by the isotopic separation of the He4 atoms could presumably be

surmounted by either using very pure He3 gas, or lowering the tempera-

ture to a region where the phase separation contribution to the specific

heat is small. This brings us to the second problem; namely, that as

the temperature is reduced, the specific heat of the solid sample be-

comes less than that of the cerium magnesium nitrate thermometer. This

means that the background specific heat of the calorimeter is greater

than that of the He3 sample, and this situation greatly reduces the ac-

curacy with which one can obtain the exchange energy from the experi-

mental data.

Thus far, the most successful determinations of IJl and its volume

dependence have been derived from nuclear magnetic relaxation measure-

ments of the spin-lattice and spin-spin relaxation times. Although










other experimenters have also made measurements, the most comprehensive

studies have been performed by Richardson, Hunt, and Meyer26 at Duke

and by Richards, Hatton, and Gifford27 at Oxford. These experimenters

find that in the low density bcc phase, the exchange energy JlJ/k is

approximately 1 mOK and decreases with increasing density. The NMR

data also indicate that the value of J depends very strongly on the

concentration of the He4 impurities present in the sample. Although

these measurements are by no means conclusive, it appears that the ex-

change energy is much larger in impure samples than in the relatively

pure ones.

The main advantage that magnetic relaxation experiments have over

the previously mentioned thermodynamic ones, is that the exchange energy

can be determined from data obtained above 0.3"K. This temperature re-

gion is within range of a helium three refrigerator and hence no para-

magnetic refrigerant is necessary. Furthermore, the thermal time con-

stants between the spins, lattice, and thermometer remain reasonably

short above 0.30K. The principal objection to determinations of the

exchange energy based solely on T1 and T2 data is that these relaxation

times are related to J by a rather complex formalism which has undergone

considerable numerical revision. These measurements thus constitute a

somewhat indirect determination of the exchange energy and additional

thermodynamic data are very desirable.

In this work we report the first direct determination of IJI and

its volume dependence for molar volumes between 22.8 and 24.2 cm3/mole.

The values of |JI are obtained from measurements of the internal pres-

sure and temperature of constant volume samples. In addition we have

also made the first pressure measurements of the isotopic phase separa-

tion temperature for samples containing 600 and 1600 ppm He4.










In the following chapter we discuss the Nosanow theory19 of "Quan-

tum Crystals" and also obtain an approximate equation of state for the

solid at low temperatures. This equation of state relates the exchange

energy and its volume dependence to the internal pressure and tempera-

ture of the solid, and can be used to extract IJI from the P, T data.

In addition we will also obtain an expression for the increase in pres-

sure due to the isotopic phase separation, based upon the assumption

that the He3 and He4 atoms mix together in a completely random fashion.

In Chapter III we describe the apparatus used in these experiments.

In particular we will discuss in detail the potassium chrome alum salt

system used to lower the temperature to a region where the effects of

the spin system become observable, and the capacitive strain gauge used

to measure the small pressure changes produced by the nuclear spin or-

dering.

In Chapter IV we present our results for the exchange energy and

its volume dependence, along with the results derived from the NMR ex-

periments. A comparison will be made between these experimental data

and the theoretical calculations of Nosanow and his co-workers.19'28

We also present results for the isotopic phase separation temperature,

energy of mixing, and equilibrium time constant for the samples 600

and 1600 ppm He4. The volume dependence of these quantities is also

discussed.

Finally in the last chapter we will give a brief summary of the

present situation with respect to the exchange energy and phase separa-

tion, and suggest some future experiments whose results should enhance

our understanding of solid helium.















CHAPTER II


THEORETICAL TREATMENTS OF SOLID HE3


A. Introductory Remarks


In this chapter we shall discuss in detail two theoretical treat-

ments of the ground state of solid He3. For want of better names, we

refer to these as the microscopic and thermodynamic approaches to the

problem. In a microscopic theory one is interested in obtaining values

for the various properties of the system such as the ground state energy

and nuclear exchange energy from a solution of the many body Schruudinger

equation, while in a thermodynamic theory one regards these quantities

as experimentally determined parameters and attempts to obtain an ap-

proximate equation of state for the system. Although there have been
22,29
several microscopic theories229 proposed in recent years, we shall

discuss in detail only the most recent work of Nosanow and his co-work-

ers19,28 since it appears to represent the "state of the art" as it

exists today. Goldstein30 has devoted considerable attention to the

thermodynamic properties of both liquid and solid helium, and the macro-

scopic equation of state which we shall develop is similar to that ap-

pearing in his most recent work. Before discussing these theoretical

attempts to calculate the ground state properties of the solid,how-

ever, a few words concerning the physical origin of the exchange energy

seem appropriate.










B. Physical Origin of the Exchange Energy


When Weiss31 proposed his molecular field theory in 1907, it was

assumed that the ordinary magnetic dipole-dipole interaction between

neighboring atoms was responsible for the observed spontaneous mag-

netization in ferromagnets. However, it soon became evident that the

observed transition temperatures were much too high to be explained on

the basis of a simple dipolar interaction which predicts a transition

temperature Tc P2/R2, where p is the net magnetic moment, and R the

distance between neighboring atoms. For iron, the observed transition

temperature is x 1000K,while the temperature calculated from the dipolar

interaction is about 10K. Thus to account for the experimental data,

it was necessary -to find an interaction which is about a thousand times

stronger than the magnetic dipole one. The discovery of this unknown

interaction had to wait about twenty years for Schroedinger and Heisen-

berg to develop quantum mechanics. Shortly after the formulation of

this theory, Heisenberg32 applied it to the problem of ferromagnetism.

He showed that the interaction responsible for spontaneous magnetiza-

tion was truly quantum mechanical, being a direct result of the sym-

metry restrictions placed on the wave function by the Pauli exclusion

principle.

To illustrate the important role that the exclusion principle plays

in determining the ground state energy of a system of fermions, it is

useful to consider the simple example of two spin 1/2 particles inter-

acting with each other through a potential V(rl2). The Hamiltonian,

omitting the dipole-dipole term, is given by


H(1,2) = (V2+V2)+V(r12). (2.1)









Consider two cases:

CASE I. The particles are distinguishable so that the restrictions of

the Pauli principle need not be considered. A suitable wave function

for the two particles is then


*(rir2) = oi(rl)o (r2) (2.2)

where i and j refer to the ith and jth single particle eigenstates.

The total energy will then be


E = E +E + f(r1l) *(r2)V(rl2)i(rl)O(2)drld. (2.3)


and is independent of the relative spin orientations of the particles.

CASE II. The particles are indistinguishable so that the exclusion

principle requires the wave function to be antisymmetric. A linear

combination of the

(rl,r2) = [i(ri) (r2)i(r2) (rl)]S(s ,s2) (2.4)

where S(s1,s2) is the singlet spin function if the positive sign is

used, and the triplet spin function if the negative sign is used. The

total energy is no longer degenerate, hut instead is given by


E = E 1i(l)j 2)V(rl2)Ii(r2) (rl)drldr2 (2.5)

where the additional term is called the exchange energy Jij of the two

spins in states i and j. The total energy is now seen to depend upon

the relative spin orientations of the two particles.

The essential difference between Cases I and II lies in the dis-

tinguishability of the particles, which in turn is determined by their

spacial motions. When the volumes spanned by particles 1 and 2 have a










common region, they are indistinguishable and we get J ij 0, while

if these volumes do not overlap the particles are distinguishable and

Jij = 0. The size of the nuclear exchange energy in solids is thus

seen to be a measure of how large an overlap there is between wave

packets describing the vibrations of neighboring atoms. In solid He3

the weak interatomic forces and small mass combine to produce large

amplitude zero point vibrations of the atoms about their equilibrium

positions, and hence one might expect solid He3 to exhibit sizeable

nuclear exchange effects.

In 1929, Dirac33 showed that for localized spins in orthogonal

orbitals the exchange energy can be written as

N
H =-2 1 J Ii (2.6)
ex icj ij "
i
This is the famous Heisenberg Hamiltonian developed by Dirac and first

used extensively by Van Vleck.34 When written in this form, the ex-

change energy appears to result from a direct two body spin-spin in-

teraction. Also one sees that the magnetic ground state (ferro or

antiferromagnetic) will be determined by the sign of Jij. If Jij < 0

antiparallel spin alignment will be favored and the ground state of

the system will be antiferromagnetic, while if Jij > 0 the ground

state will be ferromagnetic. Furthermore, when expressed in this form

it is immediately evident that the exchange energy represents the dif-

ference between the singlet and triplet state energies. Finally, this

form is very attractive because it allows the powerful spin operator

formalism to be applied to the theory of magnetism. Perhaps because

of the clarity with which Eq.(2.6) defines the exchange energy and

magnetic ground state of a system, misconceptions about the microscopic










origin of the exchange interaction have arisen. It is important, there-

fore, to realize that exchange forces result from the symmetry require-

ments placed on the wave function by the Pauli exclusion principle, not

from any direct spin-spin coupling. Having concluded this brief inter-

lude on the physical origin of the exchange energy, we may now discuss

some of the theories of the ground state of solid He3.



C. Microscopic Theory


As previously mentioned, any attempt to calculate the ground state

energy and wave function for solids from first principles is faced with

the problem of finding a solution to the many body problem. This is a

most difficult problem and can be solved only if approximations are

made. The nature of these approximations is generally determined by

the specific system under consideration. For solids of heavy atoms,

the root mean square deviation of the particles about their equilib-

rium positions is small so that the harmonic approximation for the

potential, along with uncorrelated single particle wave functions may

be used. Such calculations for solid He3 have been spectacularly un-

successful. Nosanow and Shaw35 have calculated the ground state energy

of noble gas solids, using uncorrelated single particle wave functions

and a Lennard-Jones 6-12 potential. For the other .heavy noble gas

solids, the theoretical value of the cohesive energy is within the ex-

perimental limits, while for solid helium, the calculated value is of

the order of 30 cal/mole, while experimental value is about -4.5.5

cal/mole. From this type of calculation it is evident that uncorrelated

single particle type functions are an inadequate description of the

ground state of solid helium. The reason for this inadequacy is that









these functions do not take into account the short range correlations

which arise from the very large zero point motion of the atoms.

The first attempt to include these short range correlations into

the theory of solid He3 was made by Bernardes and Primakoff,2 who made

a variational calculation of the ground state energy and wave function.

They reasoned that correlations in the motions of pairs of atoms could

be accounted for by replacing the true interaction potential by a

single parameter effective one. They assumed an analytical form for

this effective potential given by


V(r,A) = 442 )- e (2.7)


The parameter X was determined by fitting the known ground state energy

and root mean square deviation for crystalline He4. Their calculations

were approximately correct for the bulk properties of the solid, but

vastly overestimated the nuclear exchange energy. It is now realized

that due to the differences in crystal structure and statistics the ef-

fective potential between nearest neighbor atoms of the two solids is

significantly different. The exchange integral is very sensitive to

the variations in the tail of the wave function which is determined

primarily by the repulsive part of the potential. Therefore, any ef-

fective potential determined from He4data will not be accurate enough

to calculate quantitatively the exchange energy for solid He3. De-

spite this shortcoming in their approach, Bernardes and Primakoff were
dJ
able to correctly predict that both J and d- would be negative in the

body-centered-cubic phase. Their value for J is about two orders of

magnitude too large and the pressure dependence of J as estimated from

variation of X with pressure is also much too large.










Nosanow and his co-workers19,28 have made a systematic study of

the ground state properties of quantum crystals by employing a varia-

tional calculation of the energy based on a cluster expansion technique.

This group succeeded in calculating the ground state pressure and energy

of solid helium to a few percent, while obtaining reasonable agreement

with the available experimental values for the exchange energy. Since

the Nosanow treatment is the most comprehensive and gives the best agree-

ment with all the available experimental data, a discussion of the as-

sumptions and approximations of this theory will be given.


Nosanow's Theory of Quantum Crystals


If we consider a system of N particles of mass m interacting with

each other through a potential function V(rij), the Schrdedinger equa-

tion is


-m ~V(r icr ..N) = E (rl,r2...rN). (2.8)
i= i i< i I

For He3 the Lennard-Jones 6-12 potential


V(r ) = 4( (12 )6 (2.9)


gives an adequate representation of the interatomic forces. In Eq.(2.9),

E and o are constants determined from the low temperature gas phase

data and have values of 10.220K and 2.55 X respectively.

The effects of short range correlations are included in the function

p by taking it to have the form

N N
(r...)i...r N) I i -) R f(r.), (2.10)
i=l i in his expression is e coordinate of the i
in this expression i is the coordinate of the ith lattice site, ri is










the position vector of the ith atom and rij is the distance between the

i and jth atoms. The function i in Eq.(2.10) is not properly sym-

metrized with respect to an interchange of He3 atoms and hence cannot

adequately describe exchange effects. In the actual calculation, the

two body exchange energy is accounted for by using an antisymmetrized

two particle wave function constructed from the appropriate linear com-

bination of the 0i(ri-R ) and spin functions.
N -> 4
The function iii (ri-Ri) has the normal Hartree form. The boundary

conditions are


lim O(r) = finite (2.11a)
r 0
reO


lim O(r) = 0 (2.11b)
r +-


4(r) # 0 for r > R/2, (2.11c)


condition (2.11c) permits the wave packets of neighboring He3 atoms to

overlap. Nosanow chooses #(r) to have the spherically symmetric form


4(r) = exp(-Ar2/2) (2.12)


where A is a variational parameter to be determined. The function

iTjf(ri) is introduced to account for the short range correlations

in He3. By the particular functional form chosen, one can see that

only two body correlations are to be considered. The boundary condi-

tions on f(r i) are

lim f(r) = 0 (2.13a)
r 0
rO

lim f(r) = constant. (2.13b)
r *+









Condition (2.13a) reflects the strong repulsion of He3 atoms at small

distances, while (2.13b) expresses the fact that at large distances the

atoms are essentially uncorrelated. One of the analytical forms used

by Nosanow for the correlation function is


f(r) = exp(-K[ )12_ (611 (2.14)


where K is a variational parameter to be determined. The procedure is

to determine A and K by a variation of the energy


E = (,H (2.15)
o ( i,tI)

with respect to these parameters.

Up to this point, the main approximation in the theory is the

admission of two body correlations only. However due to the presence

of the function H f(r ), evaluation of (i,Hi) becomes difficult to
i do without additional approximations. In order to evaluate (i,Hi),

Nosanow makes a cluster series expansion of the energy in such a way

that each successive term in the expansion makes a decreasing con-

tribution to E It should then be possible to truncate the series

after a few terms, provided it converges rapidly.

To make the expansion, it is convenient to introduce the quantity


M(y) = (i,expyHi), (2.16)


so that


E = lim -- InM(y). (2.17)
o -y 0

In the cluster series expansion the M(y) is expressed in the form

N
M(y) = n M (y) (2.18)
n=l









where the only contribution to M arises from that volume of phase
n
space where n particles are grouped together in a cluster. Equa-

tions (2.16) and (2.17) lead to an expression for the energy, Eo, of

the form
N
Eo = on (2.19)
n=l

where the Eon give the contribution to the energy of the system from

a cluster of n particles. For this series to be rapidly convergent,

it is clear that the contributions to E from clusters of four or
o
five particles must be much smaller than the single and pair particle

energies.

The derivation of the various Eon is accomplished by first cal-

culating the Mn(y) and then using the expression Eon = lim lnMN().

It is quite difficult to do, especially for n > 2, and only the results

for the special forms of and f specified above will be presented here.

To conform with Nosanow's notation, we need to define the average value

of a function g(rl...ri...rN) over the weight function *(rl.. i.. r .rN

by




I IrN il)l12N... (2.20)



<8( ) l> d Il,(I i -R )l(2.) )
g(ri > 2 dg(rifd, (2.21)


r-Jl),(rj-i1)g(r rZ (2. 21)


where








W> ( -SJ)12d. (2.22)

The ground state energy is then given by the expression


E0 = EO1+EO2V+EO2T+E02J+E3V+E03T' (2.23)

where the various terms have the following forms.

N 21) V (Iri- ij]dri
E 01= (2.24)
i=l1 (I r- )(I-Ri d dri'

and
1
O2' Vff( ) (2.25)
i,j
with

Vff = (V(ri)- m V2nf(ri))f2(ri ). (2.26)

E02T = E3T = 0 for the particular form of O(r) chosen.

E =*+ 1 )3 (2.27)
02J 4 i,j ij

where

r 2 -2R2A2 2
ij 2m + J (2.28)

has been defined such that Jij is the difference in energy between
the singlet and triplet spin states. Finally

1
03V 2 i ,k (2.29)

The form of each term can be related very nicely to physically
intuitive quantities. E01 is the single particle contribution to the










energy of the solid. It has been expressed in a form closely resembling

the ground state energy of a system of N harmonic oscillators. The

second term E02V measures the contribution to the energy produced by

the atoms taken in pairs. The Lennard-Jones 6-12 potential however, is

replaced by an effective potential which depends upon the form of the

short range correlation function f(rij). The term E03V is slightly

more general than EO2V. It can be considered to be the effective po-

tential energy resulting from all possible interactions involving three

particles. Since this includes contributions from two particle inter-

actions already counted in E02V, these must be subtracted out. E02J

is the two particle exchange energy and was calculated using a two

particle antisymmetric wave function of the form


2(,rj) = [(|ri-ji )|rj-W|)A(I|-|l)|j-Rj)]f(rij)S(si,sj) (2.30)


where the plus and minus sign in Eq.(2.30) go with the singlet and tri-
42R2A2
plet spin states respectively. In Eq.(2.28) the term 2m
2m
is the exchange kinetic energy and is seen to have a negative sign.

Furtherit turns out to be about three orders of magnitude larger than
2x
the second term in brackets. The expression is the contribu-

tion to the exchange integral from effective potential in the overlap

region. From Eq.(2.28) one sees that the sign of J will be determined

by the relative magnitude of these two terms. If the former term is

greater, J will be negative and the magnetic ground state will be anti-

ferromagnetic, while if the latter is greater, J will be positive and

the ground state will be ferromagnetic. It is interesting to note that

Ji = 0 when and = 0, that is when there is no overlap of
the various x e
the various (i.









In Nosanow's 1966 paper, which we shall refer to as N, the basic

assumption is that the three body and exchange contributions to the

ground state energy are small so that E0 is approximately given by


E0 = E01+E 02V (2.31)

The values of the parameters K and A can then be obtained by minimiz-

ing E0. These values may then be used to calculate the terms, E03V

and E02J in the cluster expansion. If these terms are small, the

cluster expansion is assumed to converge rapidly.

In a subsequent paper by Hetherington, Mullin, and Nosanow,28

(HMN) the three body term E03Vi is included in EO and the parameters

A and K are chosen to minimize


EO = E01+E 2V+E03VW (2.32)

The actual variational calculation itself becomes formally identi-

cal with the single particle Hartree calculation of Nosanow and Shaw35

except that the true interaction potential V(rij) is replaced by an

effective potential which is approximately given by


Veff(rj) = f2(rij)[(r) n V21nf(r)]. (2.33)

When viewed in this manner, the Nosanow approach is seen to be similar

to that used by Bernades and Primakoff. Namely, the effects of cor-

relations are taken into account by replacing the true potential by an

effective one. However, in the Nosanow theory, veff is found by mini-

mizing the ground state energy of He3 while in the Bernades and Prima-

koff theory, the effective potential was determined from the known

ground state properties of He4. The results of these calculations are

summarized in Fig. (1), (2), and (3).

















15

-J
O \\
1







I \

0
1 -
w 13



Ix




10-





III I I I I





20 21 22 23 24

MOLAR VOLUME (CM3/MOLE)
Figure 1. Calculated ground state energy versus molar volume. The
dashed curve was obtained by minimizing E01+EO2V while the
solid curve was obtained by minimizing E01+E02v+EO3V-






23




1.9- A --- N --.84
-- HMN


LOG K




1.7 --.80

AY

< 1.6 LOG K -.78 0
o



1.5 .76




1.4 -.74




1.3 -.72



I I I I I
20 21 22 23 24
MOLAR VOLUME (CM3/MOLE)
Figure 2. Variational parameters A and log K versus molar volume.
The dashed curve gives the parameters that minimize
E01+E02V, and the solid curve gives those that minimize
E1l+Eo2V+EO3v. Larger values of A correspond to a
greater localization of the atoms about their lattice
sites.








R (A)

3.45 3.55 3.65 3.75
0.10

0.08


0.06


N
0.04

HMN

0
E
= 0.02






0.01






I I I I
18.5 20.5 22.5 24.5

MOLAR VOLUME (CM5/MOLE)

Figure 3. Calculated nuclear exchange energy versus molar
volume. Curve N was calculated using the param-
eters which minimize EOI+EO2V; curve HMN was cal-
culated using those which minimize Eol+E02V+EO3V.









From Fig. (1) we see immediately that the inclusion of the term

E03V has only a small effect on the total ground state energy EO, and

thus the truncation of the energy expansion seems valid. Moreover, in

HMN, a physical argument based upon the short range nature of the cor-

relation function f(rij) is given which indicates that higher order

terms in the cluster expansion will be small. From Fig. (3) however,

it can also be seen that while the changes in A and K introduced by

the inclusion of E03V in E0 do not greatly affect the values of the

total energy, they do significantly alter both the value and the shape

of J(R). This should not be too surprising and can be understood by

the following argument. The main contributions to E0 come from E01 and

E02V. The former is completely independent of the choice of K and

depends only on the width of *(r), hence small changes in A will pro-

duce only slight variations in E0. While EO2V does depend on both f(r)

and *(r), the dependence is such that small changes in A and K do not

greatly affect EO2V. This is because it is related to the average

values of f(r) and *(r) and not their detailed structure. On the other

hand, the quantity

t2R2A2 2x
S(R) 2+ f (2.34)
ij) 2m

depends on a knowledge of the details of both f(r) and #(r). In par-

ticular, Jij is most sensitive to variations in the wings of 0(r) since

this is where the overlapping occurs. Also, it is expected that Jij

should be very sensitive to changes in the function f(r) since it is

this function which describes the short range correlations between
28
neighboring atoms. Nosanow and Mullin28 have investigated the sensi-

tivity of iJ to the function f(r). They varied K from 0.14 to 0.18









and found that Jij changed by a factor of 3 while the energy changed

by only 1%. They also changed the first exponent in f(r) from 12 to

8 and found that while J varied by a factor of 4, the total energy

changed by only 10%. This sensitivity of Jij and insensitivity of E0

to slight modifications of the parameters illustrates a basic defi-

ciency in all variational calculations of the exchange energy. Namely,

that, although the energy is determined to a few percent, the wave

function is not an accurate enough representation of the true ground

state function to enable one to calculate accurately the value of the

exchange energy.

Before ending this discussion of the Nosanow theory, some con-

sideration should be given to the use of spherically symmetric func-

tions for O(r) and f(r). It is known that the low density phase of

solid He3 has a body-centered-cubic crystal structure. This suggests

that spherically symmetric functions will be a good representation for

the motion of atoms for small values of r. However, for large values

of r (r > ), the cubic symmetry of the lattice must be reflected in

the wave function. It is precisely in this region that the overlap

integral is large and hence the exchange energy sizeable. It is

conceivable, therefore, that the assumption of spherical symmetry in-

troduces an error into the calculations of the exchange integral.

Numerical values for O(r) have been considered in the theory, and

generally speaking they tend to make the exchange energy somewhat

larger.

The above remarks are intended only to illustrate some of the

problems associated with any theoretical attempt to calculate the

exchange energy accurately. In point of fact, the clarity and physical

basis for the assumptions and approximations, coupled with the good










agreement between the theoretical calculations of Nosanow and experi-

mental data make this work a significant contribution to the understand-

ing of solid He3.



D. Thermodynamic Theory


In this section we will obtain an approximate equation of state

for a solid composed mainly of He3 atoms but containing a small amount

of He4 impurities. To do this it is necessary to construct suitable

models for the various degrees of freedom of the system. The choice

of these models will be based upon both experimental information and

theoretical intuition.

The experiments of Edwards, et al.24 show that below 0.50K the

specific heat of the lattice in the bcc phase can be represented by a

Debye term plus a term arising from the phase separation of a regular

mixture. We may thus write this contribution to the free energy in

the form


FL(x,V,T) = FD(x,V,T)+FpS(x,V,T)+UO(x,V). (2.35)


In Eq.(2.35) U0(x,V) is the zero point energy, FD(x,V,T) and FpS(x,V,T)

are the Debye and phase separation contributions to the free energy,

and x is the concentration of the He4 atoms.

Since the exchange energy arises from localized spin 1/2 particles,

one should be able to treat the magnetic interactions on the basis of

a Heisenberg Hamiltonian of the form


H = -2 Jij(xV)I'j (2.36)
Si exhane i b n ni ,

where Jij(xV) is the exchange integral between nearest neighbor atoms,










and the summation extends over nearest neighbor atoms only. Using this

Hamiltonian, one can then calculate a partition function Qex(x,V,T)

from which the magnetic free energy Fex(x,V,T) may be obtained. The

total free energy will then be the sum of Fex and F and is given by


F(x,V,T) = U0(x,V)+FD(x,V,T)+FPS(x,V,T)+Fex(x,V,T). (2.37)


The equation of state of the system can then be obtained from the re-

lation


P(x,V,T) = F(x T) (2.38)
L DV ,T

In order to determine the functions FD, FPS, and Fex it is useful

to divide the temperature scale into three regions as shown below.



III II I
STwo Phase Region Phase Separation Region Single Phase Region
W F = Fex(o,V,T) F = Fps(x,V,T) F FD(Xo,V,T)






0.06 0.30

Temperature

In region I, where FD(x,V,T) is the main contributor to the energy,

the solid exists as a single homogeneous phase having a temperature inde-
WT(xo,V)
pendent concentration x. We may, therefore, write FD(x,V,T) = FDI (xT

where 6(xo,V) is the characteristic Debye temperature for the solid.

Region II is the isotopic phase separation region. In this temperature

range the solid is transformed from a single homogeneous phase of con-

centration x into two separated phases. Initially, the He4 concentration









of these separated phases varies rapidly with the temperature; however,

by the time one reaches 0.060K, the separation into pure phases is

practically completed. Below about 0.060K, therefore, the solid exists

in the form of a large nearly pure He3 phase in equilibrium with a small

He4 phase. The energy in this range will thus be essentially that of a

pure He3 system. We may then write the magnetic contribution to the

energy in the form F (x,V,T) = F (o,V,T). The total free energy then

becomes

rB(x ,V)i
F(x,V,T) = UO(V)+F D[ T- +FS(x,V,T)+Fex(o,V,T). (2.39)


We will now obtain expressions for the various terms in Eq.(2.39).

The spin partition function Q ex(v,T) is given formally by the expression

-Hex/kT +k i I
Qex(V,T) = trace e = trace e i< (2.40)


Rushbrooke and Wood36 have made a series expansion of this function in

powers of -. They find that at high temperature (T >> J/k) Eq.(2.40)

reduces to


nQ Nln(21+1)+-i NzI2 (+1)2( )2. (2.41)
in9ex 3 Nln(21+)+

For He3 in the bcc phase, I = 1/2 and z = 8 so that the above equation

reduces to


lnQe N[ln2+ ( )2]. (2.42)


Fex is related to the partition function Qex by the relation


Fex(v,T) = -kTlnQex, (2.43)


so that the magnetic contribution to the pressures becomes









2 dJ 1
P (VT) 3Nk()2 d (2.44)


which may finally be put in the form

3R j 2 1
Pex(V,T) =- yex() 2, (2.45)


where


x dlnVJ (2.46)
Yex dlnV


We can see from Eq.(2.45) that in Region III the pressure is pro-

portional to T-1 with the constant of proportionalty being directly re-

lated to the strength of the exchange interaction J.

To obtain an expression for the free energy in the phase separation

region we recall that the specific heat measurements of Edwards, et al.24

indicated that the He3 and He4 atoms mix together to form a regular

solution. As a consequence of this, the internal energy and entropy of

the system may be written as


U(x,V,T) = -[xE44+(1-x)E33+2x(l-x)E ], (2.47)



S(x,T) = -Nk[xlnx+(l-x)ln(l-x)], (2.48)

E33 E44
where EM = E34- 2 and Eij is the energy of interaction between

an atom of isotope i and its nearest neighbors of isotope j. For the

case x << 1, Eqs.(2.47) and (2.48) reduce to


U (E33+2xE), (2.49)



S = -Nk(xlnx-x). (2.50)


When written in this form, one sees that the stable configuration of










the system depends upon the sign of EM. If EM is positive (2E34> E33+E44),

then the energy of the mixed phase is greater than that of the separated

phases and the system will separate into pure isotopes at absolute zero.

To find the temperature at which this separation begins, we minimize the

free energy F = U-TS. For the present case, we have


FpS(x,V,T) = i(E33+2xEM)+NkT(xlnx-x). (2.51)


The equation for phase separation curve is derived from the stability

condition


S= 0. (2.52)
V,T

This yields an expression for the phase separation curve in the T-x

plane given by


x = e-EM/kT (2.53)


where x is the concentration of the He4 enriched phase at a temperature

T and volume V. The free energy in this two phase region is obtained

by substituting the expression for x given by Eq.(2.53) into Eq.(2.51).

The result is the simple expression


FpS(x,V,T) = E33-NkTe-EM/kT. (2.54)


The internal pressure arising from the isotopic phase separation is then

given by

R dEM MIT (2.55)
PS k dV (2

Equation (2.55) shows that the size of PPS is linearly related to the

rate of change of the energy of mixing with density.









Finally the contribution to the pressure in the high temperature
re(xo,V)
region can be obtained from the Debye free energy FD [---- and the

expression

xF 3F
PD(xo,V,T) -= ,D ) D (2.56)
x,T

or alternatively

1 dO
PD(XoV,T) = UD(T,O) dV. (2.57)


For solid helium T << 8D and this becomes

3.4R T 3
P ) Ty, (2.58)
D 5V 1

dine
where y dlnV is the Gruneisen parameter.
dlnV
The approximate equation of state for the system is then

3R 1 R dE 3
oV) V1 + EMkT -R Ty. (2.59)
P(x,V,T)-Po 01 ex Y k dV 5V


This equation will be used in Chapter IV to obtain the values of IJI,

'ex, and EM.
















CHAPTER III


EXPERIMENTAL APPARATUS AND PROCEDURE


A. Introductory Remarks


The apparatus used to obtain the experimental results reported

in this work incorporates many of the standard techniques of low tem-

perature physics with enough new ideas to make it somewhat unique. It

seems natural that we should describe these distinguishing features

with considerable detail and present only a brief description of those

sections which are conventionally used in low temperature research

throughout the world. Also, as is usually true, the unique sections

of the apparatus are also the most important in determining the suc-

cess of these particular experiments. For these reasons we shall de-

scribe in detail the design and construction of both the potassium

chrome alum salt assembly and the capacitance strain gauge, while pre-

senting only a brief description of the helium refrigerators, vacuum

systems, pressure system,and superconducting solenoids. For more de-

tails on these latter sections of the apparatus,the reader is referred

to the earlier works of P. J. Walsh3 and G. C. Straty39 and also to

the standard books40 on the techniques used in low temperature physics.

We have also tried to follow the same philosophy in describing the

manner in which the experiments were performed. Long discourses on

experimental procedure from initial cooldown to final shutdown tend to










become somewhat boring, and hence we have included only those aspects

of the experimental procedure which indicate the precautions taken to

insure the validity of the final results.



B. Cryostat


The cryostat in which the experiments were performed is a modified

version of that described by Walsh38 and is shown schematically in

Fig. (4). A more detailed drawing of the low temperature section is

shown in Fig. (5). Three stages of refrigeration are required to re-

duce the temperature of the He3 sample from 4.2 to 0.020K. The tem-

perature io lowered initially to 1.0K by reducing the vapor pressure

above a liquid He4 bath. A further reduction to 0.30K is obtained using

a continuously operating He3 refrigerator system. The final stage of

cooling is accomplished by adiabatic demagnetization of a potassium

chrome alum [CrK(SO4)2-12H20] salt pill.


Helium Refrigerators

In this apparatus, the 1K He4 bath was contained in a cylindrical

container which could be filled with liquid from the outer bath by

means of a modified Hoke valve (not shown). This valve has a stem

which extends through the top flange of the cryostat so that it may

be operated from outside the helium dewar. To increase the thermal

contact between the boiling liquid and its container, a copper spiral

wound from 0.013 inch copper sheet was soldered to the bottom of the

inside surface. The volume of the container is about 250 cm3 and one

filling provided 10K operation for a period of about forty hours. A

model KC-46 Kinney pump provided the necessary pumping speed to maintain
























Figure 4. Schematic diagram of the apparatus.

A. Sample filling capillary

B. Exchange gas pumping line

C. He4 bath pumping line

D. Manostat

E. He3 refrigerator diffusion pump

F. N2 cold trap

G. He3 refrigerator pump

H. He4 bath

I. He3 refrigerator

J. Vacuum jacket

K. He4 recovery line

L. Connection to oil and mercury manometers

M. Vacuum flange

N. Electrical Connections

0. Potassium chrome alum salt






36












N



L


















G















Figure 5. Schematic diagram of the low temperature section.

A. Various pumping lines

B. Main support and vacuum flange

C. Sample filling capillary

D. He3 refrigerator return line

E. Electrical connections

F. Indium gaskets

G. He4 bath

H. Vacuum chambers

I. Evaporator section of He3 refrigerator

J. Lead heat switch

K. Nylon support tube

L. Potassium chrome alum salt

M. Demagnetization and zinc heat switch solenoids

N. Zinc heat switch

0. Sample chamber

P. 2000 #44 copper wires

Q. Vacuum jackets and radiation shields

R. Cerium magnesium nitrate thermometer

S. Primary of the mutual inductance system

T. Measuring and compensating secondaries

U. Teflon spacers







38






E




















L

M









P




S




I1










the temperature at 1"K,even in the presence of substantial heat loads.

Temperature control in the region between 4.20and 1K was accomplished

by pumping through a diaphragm type manostat which allowed the pressure

to be regulated to 1%. Vapor pressures were measured by mercury and

oil manometers connected into the pumping line. The bottom of the con-

tainer was used as a support flange for the inner vacuum jacket which

also served as a 10K radiation shield.

The He3 refrigerator system was designed for cyclic operation. It

consisted of a cylindrical container having a volume of 1 cm3 which

served as the evaporator, an NRC type B-2 oil diffusion pump in series

with a Welsh Duo-Seal pump modified for closed system operation, and a

constricted capillary used to produce the pressure necessary to re-

liquify the returning He3 gas. A liquid nitrogen cold trap, located in

the He3 return line, was used to prevent oil vapors from entering the

cryostat and possibly plugging the pressure dropping capillary. The

dead volume inside the Welsh pump above the oil was used to store He3

gas during shutdown periods. A network of mercury and oil manometers

along with a CVC type GM 100 McLeod gauge was connected into the pump-

ing line. These could be used to measure the vapor pressure of the

liquid He3 under static conditions during temperature calibrations.

A carbon resistor which served as both heater and thermometer was at-

tached to the evaporator section. By closing the return line before

demagnetization it was possible to maintain a temperature of about

0.280K at the evaporator section of the refrigerator.


Vacuum Chambers and Radiation Shields

Two vacuum chambers and four radiation shields were used to reduce

the heat leak into the sample and cerium magnesium nitrate (CMN)










thermometer. An outer jacket, used to provide thermal isolation from

the main He4 bath, was attached to a flange located in the cryostat

stem. An indium gasket made from 0.075 cm diameter wire was used for

the vacuum tight seal. A second vacuum chamber, surrounding the He3

refrigerator and salt system, was attached to the bottom of the 1K

bath again using an indium gasket for the vacuum tight seal. During

temperature calibrations, this chamber was filled with He4 exchange

gas used to provide thermal equilibrium between the helium baths and

the thermometers. The walls of the chamber were thermally grounded.

to the inner He4 bath and hence provided a 1K radiation shield for the

He3 refrigerator. A third shield, in the form of a copper plated

brass cage, was screwed onto a support flange thermally grounded to

the He3 refrigerator. This cage, which contained eight small windows

used for viewing the potassium chrome alum salt, completely surrounded

the pill thereby providing it with 0.30K ambient. The construction

and.use of the final radiation shield will be discussed in the salt

assembly section of this chapter.

Electrical leads, with the exception of the coaxial lines used

for leads to the capacitor plates, were brought into the outer vacuum

chamber through a small stainless steel tube. A vacuum tight glass-

to-metal seal was used to bring the leads through the 10K bath into

the inner vacuum chamber. Thermal grounding to 0.3K was accomplished

by soldering the leads to small glass-to-metal seals mounted on the

He3 refrigerator. The leads were made from #36 Advance wire and a

sufficient length was allowed between the refrigerators and the salt

pill to provide the necessary thermal isolation. Coaxial lines were

used as leads to the capacitor plates of the strain gauge wherever











possible. Where unshielded leads were used, care was taken to separate

them as far as possible in order to reduce distribution capacitance.

This distributed capacitance was measured with the leads disconnected

at the capacitor plates, and found to be about 0.1 pf. At the working

pressures encountered in these experiments, the strain gauge capacitance

had a value of about 12 pf. The quantity C = C where AC is the
Cg+%
g d
capacitance change produced by a pressure change AP, and C and Cd are

the strain gauge and distributed capacitance respectively, was affected

by less than 0.8% by Cd. More importantly, we observed no evidence

which indicated that the value of Cd changed during the course of

these experiments. In particular, it was possible to refill the li-

quid nitrogen bath, as well as the outer and inner He4 baths, without

affecting the value of the distributed capacitance. We are quite cer-

tain therefore that the observed capacitance changes resulted entirely

from changes in the value of the strain gauge capacitance, C .

A discussion of the final stage of cooling and thermometry will

be given after the description of the sample cell and pressure measure-

ments.



C. The Strain Gauge


In order to make pressure measurements on samples of solid He3,

it is necessary to incorporate onto the sample cell some pressure

sensing device. This is because when the sample freezes in the cell a

solid plug also forms in the filling tube thereby isolating the sample

chamber from any external pressure sensing devices. In these experi-

ments a capacitance strain gauge which relates the internal pressure of










the sample to the capacitance of a parallel plate capacitor was used.

The details of the strain gauge and sample cell are shown in Fig. (6).

The main section of the cell was machined from a 5/8 inch long,

7/8 inch diameter stainless steel (type 304) cylinder. A hole, 1/2 inch

in diameter and 3/8 inch deep was bored into one end of the cylinder,

while the opposite end was machined, as indicated in Fig. (6), to a

depth of 0.205 inches. The resulting diaphragm, having a diameter of

1/2 inch and a thickness of 0.045 inches, constitutes the active ele-

ment of the strain gauge. A stainless steel plug, containing a copper

piece used to increase the thermal contact between the He3 sample and

the chamber walls, was designed to fit tightly into one end of the

cylinder to a depth of 5/16 inch. The plug, which also contained a

0.025 inch i.d. copper capillary, was silver soldered into the main

body of the chamber through two access holes drilled into the walls.

Because of the snugness of the fit, no solder or flux flowed into the

sample volume. The He3 sample on which the pressure measurements were

made, was located in a 1/2 inch diameter, 1/16 inch long cylindrical

volume. This pancake geometry,with its large surface area to volume

ratio, helps relieve any internal pressure gradients which may occur

during the formation of the solid, and also helps to decrease the

thermal time constant between the He3 sample and the chamber walls.

This point will be discussed in greater detail later in this chapter.

A copper cylinder, containing four wells used to hold the resistance

thermometers and the heater, was soldered to the steel section. This

cylinder also contained a #8 copper wire which served as a thermal

grounding post for attaching a zinc heat switch used to provide thermal

contact between the salt pill and the sample chamber. The switch was
















Figure 6. Capacitance strain gauge.

A. 2000 #39 copper wires to potassium chrome alum salt

B. Sample filling capillary

C. Zinc heat switch

D. Bakelite support

E. A #8 copper wire

F. Resistance thermometer

G. Access holes for soldering

H. Copper cylinder

I. Stainless steel chamber and plug
1 1"
J. x diameter sample volume

K. 0.045" diaphragm

L. Capacitor plates

M. 2000 #44 copper wires to CMN thermometer

N. Bakelite support for the CMN thermometer





44
A


B
C
D

E

F












M
*k fr^










in the form of a thin foil having an area to length ratio of 0.25 mm.

The cell was supported by a bakelite rod which extended from the salt

pill.

The sample cell was filled with liquid through a stainless steel

capillary having an o.d. of 0.033 cm and a 0.006 cm wall. A ff36 Ad-

vance wire was inserted into the capillary to reduce its volume still

further. A length of about 25 cm was thermally anchored to the He3

refrigerator thereby providing a sufficiently long solid He3 plug to

prevent any slippage of material into the pressure cell. In addition

to this, a length of about 20 cm was thermally grounded to the He4 bath.

No problems with plug slippage were encountered during the course of

these experiments.

After the components of the chamber were soldered together, the

two capacitor plates were mounted onto the strain gauge. The active

plate, in the form of a circular disk having an area of 0.625 cm2, was

fastened with epoxy to the diaphragm. Tissue paper and epoxy serve to

electrically insulate the plate from the chamber walls. The fixed plate

was made in two sections as indicated in Fig. (6). The inner disk,

having an area of 0.625 cm2, was pressed into a tapered hole in the

outer guard ring. Two layers of 0.001 inch mylar were used to elec-

trically insulate the central plate from the outer guard ring. The

plate was attached to the main section of the chamber by means of four

#4-36 steel screws.

The sensitivity of the gauge to a pressure change AP is propor-

tional to A/d2 where A is the plate area and d is the plate spacing.

In order to obtain a high sensitivity, it is therefore advantageous

to use as small plate spacing and as large a plate area as possible.











In this experiment, the plate area was 0.625 cm2 and the plate spacing

at one atmosphere was chosen to be 0.0025 cm. To insure that the plates

would not short together at the working pressure of approximately forty

atmospheres, a method for setting the plate spacing devised by Straty39

was used. After the active plate was fastened to the diaphragm, the

sample chamber assembly was chucked in a lathe and light cuts were taken

across both the chamber and the plate. Facing both the chamber and the

plate in a single cut insured that both surfaces would be co-planar.

The desired plate spacing was obtained by inserting a brass shim be-

tween the main body of the cell and the fixed plate. This particular

geometry allowed both the active and fixed plates to be mounted on the

same piece and thereby reduced undesirable changes in the plate spac-

ing which arise from a differential thermal expansion of materials in

the gauge. The choice of steel rather than copper as the material to

be used for the main body of the strain gauge was motivated by the re-

quirement that the diaphragm exhibit no pressure hysteresis. An earlier

chamber constructed entirely from copper possessed enough hysteresis to

prevent accurate determinations of the exchange energy. The present

chamber has no detectable hysteresis.

The theoretical sensitivity of the gauge can be calculated from

the equation for the deflection of a circular membrane fixed around

its circumference. If a pressure change AP is distributed uniformly

over its surface, the diaphragm will deflect an amount given by41

0.054R4AP (
6 ,-- t3 (3.1)
wherEt

where


R = radius of the diaphragm in inches










t = thickness of the diaphragm in inches

E = modulus of elasticity in psi

and

AF = pressure change in psi.
AC
The fractional change in capacitance C- is approximately given by

AC 6
AC d (3.2)
C d

where d is the plate spacing. Solving equations (3.1) and (3.2) for

AP gives

EtsdAC
6P = 0.54RC (3.3)


The strain gauge capacitance has a value of about 12 pf and is measured

by means of a General Radio type 1615A capacitance bridge used in con-

junction with a type 1404B standard capacitor. This arrangement per-

C
mits a ~- = 10-7 to be measured. Using the values of E, t, R and d ap-

propriate to this chamber, one obtains a minimum detectable pressure

change of 3 x 10-5 atm. The measured sensitivity at forty atmospheres

was also 3 x 10-5 atm.

The data obtained in these experiments consist of a series of

values of pressure as a function of temperature for a given solid sample

contained within the volume of the sample cell. Although the determina-

tion of the sample pressure depends upon the deflection of the chamber

diaphragm, the following considerations show that for all practical

purposes the measurements are performed on constant volume samples. If

we consider the.pressure as a function of temperature and volume we ob-

tain

dP = a dT+ ~ dV, (3.4)
V T










or equivalently

SdP 3 dV (3.5)
S - T (3.dT5)

By making use of the definition of the isothermal compressibility

S ~ Eq.(3.5) can be written as


V d 1+ -b d (3.6)


where 8 is the compressibility of the helium sample.
The factor V-I dV
The factor V-1 V- is determined by the mechanical properties of

the chamber and for this chamber is about 2 x 10-5 atm-l. In the

volume range covered by these experiments the value of B varies from

about 3 x 10-3 atm-1 to 5 x 10-3 atm-1, and hence the second term in

brackets will never be greater than 10-2. Thus = '( to within

1% or better.



D. Pressure Measurements


Gas Handling and Pressure System

The He3 pressure and gas handling system is shown schematically

in Fig. (7). A system of three glass bulbs and two Toepler pumps was

used for both storing and moving the gas. The Toepler pumps were con-

nected through a liquid helium cold trap into one leg of a mercury

U-tube pressure system. The U-tube has sufficient length so that the

He3 side could be evacuated with a pressure of one atmosphere on the

opposite side. This side of the U-tube is filled with oil and con-

nected to a dead weight gauge which served as both calibrating de-

vice and pressure manostat. A small correction was made to the pressure






















Figure 7. Pressure system.

A. Vacuum line

B. Sample filling capillary

C. Oil reservoir

D. Pressure gauge

E. Dead weight gauge

F. He4 cold trap

G. Toepler pumps

H. Mercury U-tube

I. Hydraulic pump

J. Glass storage bulbs

K. Connection to cryostat

L. N2 cold trap

M. Nylon insulator and pressure seal
























































-LL











to compensate for the difference in the mercury levels in the U-tube.

The Toepler pump connection to the gas side of the U-tube was made a-

bout 20 cm below the top so that the sample gas could be trapped in

the U-tube at a low pressure. The sample filling capillary connection

at the top of the tube was made through a Nylon seal. Since the U-

tube was electrically insulated from its supporting structure, the

mercury in the tube could then be used as a switching device to actuate

an alarm which signaled the entrance of mercury into the capillary

system. A small section of the external filling capillary was im-

mersed in a nitrogen cold trap so that if a pressure leak developed in

the capillary system no mercury would enter the cryostat.


Pressure Calibration and Measurement

The calibration of the strain gauge as a function of pressure was

performed using the AMICO #47-2161 dead weight tester. The accuracy

of this gauge is 0.05%, however, the uncertainty in the relative posi-

tions of the mercury columns limits the absolute accuracy at all pres-

sures to about 0.03 atm. Calibration points were taken for both in-

creasing and decreasing pressures with no detectable hysteresis.

Since the temperature at which the calibrations were made was held

just above the freezing temperature of the He3 sample corresponding to

the particular density to be studied, the calibration conditions were

not identical to those under which the experimental data were obtained.

This was necessary to prevent a solid plug from forming in the capillary

system and isolating the strain gauge from the external pressure system.

To determine the effect of temperature on the capacitance of the strain

gauge, a separate demagnetization was performed with the sample cell











evacuated. Aside from a small anomaly occurring at about 0.15K, the

temperature variation of the capacitance was completely negligible.

This anomaly will be discussed in more detail in the following chapter

since it has some effect on the quantitative accuracy of the phase

separation data.

As was stated previously, the capacitance was measured by means

of a General Radio type 1615A capacitance bridge used in conjunction with

a General Radio 1404B standard capacitor. The GR 1615A capacitance

bridge was located inside a Styrofoam container and its temperature

regulated electronically to within 0.20C. This was done in order to

reduce the drift of the capacitance bridge reading which resulted from

variations in the room temperature. It was determined empirically that

the drift rate of the bridge reading with room temperature was about

10 af/oC.

The strain gauge capacitance was about 12 pf and could be measured

to *1 af, giving a relative sensitivity, --, of about 10-7. The ca-

pacitance and pressure are linearly related over the range of the ex-

perimental data taken on a particular density, and hence the conver-

sion of capacitance readings to pressure value was accomplished in

a straightforward manner.



E. Potassium Chrome Alum Salt Assembly


Potassium chrome alum was chosen as the paramagnetic refrigerant

because it possesses a Schottky type specific heat anomaly at about

15 mOK, and hence may be used to lower the temperature to this region.

Furthermore, it has a large specific heat in the temperature range











between 0.015K and 0.10K, thereby permitting one to make measurements

over a period of many hours, providing the residual heat leak is kept

low. The size and geometry of the salt pill were chosen to utilize a

previously constructed niobium zirconium solenoid.

A cross section of the salt pill and its support assembly is shown

in Fig. (8). Thermal contact with the He3 refrigerator was made by

bolting the copper support flange to the bottom surface of the evapora-

tor. Apiezon N grease was used as a thermal bonding agent. The flange

was threaded so that it could be used to support the 0.30K radiation

shield which completely surrounded the salt. One end of a Nylon support

tube as screwed nto the bottom of the copper flange while the other end

was clamped to copper wires in contact with the salt crystals. The

Nylon support tube, which also served to center the salt pill inside

its 0.30K cage, had an o.d. of 0.625 cm, a wall thickness of 0.5 mm,

and a length of 5 cm. A lead heat switch in thie form of a thin foil

having an area to length ratio of 0.1 mm thermally linked the mounting

flange with the salt pill. The switch was positioned so as to be

closed (normal) when the salt was fully magnetized and open (super-

conducting) after the initial step in the demagnetization had taken

place. Before being installed, the lead was etched in warm nitric

acid in an attempt to improve its on-off ratio.

The salt pill, which consisted of a mixture of 170 grams of pow-

dered potassium chrome alum and Apiezon J oil sandwiched between sheets

of #39 insulated copper wire, was assembled in a specially constructed

press. The mixture was tightly compressed into a cylinder 16.0 cm

long and 2.92 cm in diameter. The copper wire sheets were arranged so

that those coming out the top of the salt were thermally separated by










about a 1/16-inch layer of salt from those extending out the bottom.

This was done so that after demagnetization, the thermal impedence be-

tween the He3 refrigerator and the sample would be as large as pos-

sible. The total contact area between the copper wires and the salt-

J oil mixture is about 2000 cm2. Using the specific heat and Kapitza
42
boundary resistance data of Vilches and Wheatley, we calculated a

thermal time constant at 0.02'K for the salt-J oil-copper wire system

of 15 minutes. A Speer grade 1002, 1/2 watt resistor having a nominal

value of 220 ohms was embedded in the body of the salt and could be

used to monitor its temperature during the experiment.

The salt pill itself was contained inside a phenolic tube having

an inside diameter of 3.0 cm and a wall thickness of 0.8 mm. A cylin-

drical sheath, made from two layers of perpendicularly wound #39 in-

sulated copper wire lightly coated with epoxy resin, was glued to the

inside wall of the phenolic tube. This sheath was used to provide

thermal contact between the salt crystals and a radiation shield which

completely surrounded the He3 sample and cerium magnesium nitrate (CMN)

thermometer. The radiation shield, which was made by glueing two layers

of #39 insulated copper wire onto a phenolic tube, was supported by a

Bakelite flange as indicated in Fig. (8). The o.d. of the shield is

the same as the i.d. of the copper sheath and hence they fit together

very snugly. Apiezon N grease was used as the thermal bonding agent

between the sheath and shield. This arrangement provided an ambient

temperature of about 0.0250K for the sample and CMN thermometer system.

Small cotton balls cemented to the top of the salt pill, and a Nylon

spacer screwed into the bottom of the 0.0250K shield, were used to

provide the final alignment of the salt in its 0.30K cage.



















Figure 8. Potassium chrome alum salt assembly.

A. Copper support flange

B. Electrical terminals

C. Lead heat switch

D. Nylon support tube

E. Sample filling capillary

F. Nylon clamp and terminal strip

G. Sheets of copper wires

H. Phenolic tube

I. Salt and J-oil mixture

J. 220 ohm Speer resistor

K. Bakelite support flange

L. Copper wire sheath

M. Brass support screw

N. 0.0250K radiation shield


_ __ ~_~~~






56
A
B

D
E
F




G






















N











Before ending this discussion of the salt system, a few remarks

concerning the residual heat leak should be made. In any of the stand-
40
ard books40 on low temperature techniques, one can find formulas for

calculating heat leaks due to conduction down solid supports, and ra-

diation from surrounding walls. In most cases these calculations yield

results which are correct to within about an order of magnitude only.

This is due in part to the difficulty of estimating the effects of im-

purities and strains on the low temperature thermal conductivity of

materials. Also it is difficult in many cases to include quantitatively

the effects of the thermal boundary resistance between solids at low

temperatures. Finally, estimates of the vibrational input from mechan-

ical pumps and other sources are at best educated guesses. For these

reasons, the author believes that the choice and dimensions of materials

to be used in the design and construction of a paramagnetic salt system

should be governed primarily by their successful use in similar systems.

In this respect the exhaustive study of the properties of materials at
8 42 43
low temperatures by the cryogenic group at the University of Illinois8,243

has been extremely helpful.

In these experiments the sample warm up rate at 21 moK was less

than 0.1 mK/hr. The corresponding residual heat leak was approximately

15 ergs/min. Because of the rather elaborate precautions taken to iso-

late the sample from vibrations and high temperature radiation, we be-

lieve the major portion of this residual heat input comes from conduc-

tion down the lead heat switch, Nylon support tube,and cotton spacers.











F. Temperature Measurements


Temperature Calibrations and Measurements

The magnetic susceptibility of powdered cerium magnesium nitrate

(CMN) is known44 to obey Curie's law, X = C/T, to temperatures as low

as 6 mK and perhaps lower. Since the constant C can be determined by

measuring X in a known temperature region, the substance is an excellent

one to use for very low temperature thermometry. However because of

the relatively small size of the constant C (about 1/10 as large as

that of potassium chrome alum), considerable care must be taken to avoid

spurious contributions to the measured susceptibility for other weakly

magnetic materials present in the cryostat.

In these experiments the relative susceptibility of ten grams of

powdered cerium magnesium nitrate (CMN) was used as the primary thermom-

eter. The CMN was in the form of a right circular cylinder with the

diameter equal to its height. The average dimension of the CMN crystals

was about fifty microns. Two thousand #44 Formvar insulated copper

wires having an area of 150 cm2 were embedded in the CMN crystals with

Apiezon N grease, and were used to establish thermal equilibrium be-

tween the CMN and the He3 sample. This point will be discussed in

greater detail in the following section.

The relative susceptibility was measured using a ballistic-circuit

shown schematically in Fig. (9). In tWis arrangement a measuring cur-

rent, supplied by the battery, is reversed through the primary coil of

the mutual inductance system MI which surrounds the salt pill. This

induces a current pulse through the secondary which, is measured by

the deflection of the ballistic galvanometer G.


~










59























--- -0
u

I S











U4

010
44









u




rz
I0




LL j






"-I
01J.







-4-






cJ



CD I..c




4a











Under these conditions the size of the ballistic deflection is

proportional to the susceptibility of the salt, so that we nay write


6 = ax+6 = m +6. (3.7)
o T o


The constants m and 6o depend upon the coil geometry and the Curie

constant of the CMN,and are determined by plotting 6 versus T-1 in

the 40 to 1IK calibration region.

The deflections were measured with a Leeds and Northrup type 2284D

galvanometer critically damped by means of a 390 ohm shunt resistor.

When used in this manner the Coulombic sensitivity and period are

5 mm/nc and 5.3 seconds respectively. Galvanometer readings were

taken visually and after some experience could be estimated to tenths

of a millimeter.

The mutual inductance coils are shown schematically in Fig. (10).

The vertical distance between the bottom of the potassium chrome alum

cooling salt and the top of the CMN thermometer salt was 9 1/2 inches.

At this distance the contribution to the measured susceptibility from

the potassium chrome alum is less than 0.2%.43 The primary of the

mutual inductance system is a 5 inch long solenoid located on the 1K

shield. It consisted of three tightly wound layers of #30 Formvar in-

sulated copper wire separated from each other by Mylar sheets having a

thickness of .001 inches. To avoid eddy current heating, the measur-

ing field was varied from 5 gauss in the calibration region down to

1/2 gauss at low temperatures. The maximum power dissipated in the

primary was about 1 mW and caused no measurable heating of the 10K

refrigerator. The secondary was wound on the 0.30K shield and con-

sisted of two nearly identical coils each 1 1/4 inches long with a













THERMAL CONNECTION
TO SAMPLE CHAMBER


2000 INSULATED
COPPER WIRES


S10K THERMAL SHIELD


0.30K THERMAL SHIELD


SALT SHIELD






NYLON CONTAINER


POWDERED CMN
PLUS N GREASE





SECONDARY



PRIMARY














Figure 10. Schematic diagram of the CMN thermometer and the mutual
inductance system. The drawing is approximately to
scale.











center separation of 1/14 inches. The coils were made from 20 layers

of #40 Formvar insulated copper wire. Each layer contained approxi-

mately 250 turns and successive layers were separated by one mil

Mylar sheets. The wire was wound as tightly as possible in order to

keep the relative positions of the turns fixed. These two coils are

sometimes referred to as the measuring and compensating coils since

they are connected together in opposition so that the induced EMF's

will approximately compensate in the region where the salt's suscepti-

bility is small. This is very desirable since the accuracy with which

one can read the galvanometer deflections depends greatly on their

size. To obtain maximum sensitivity over the entire temperature range,

and also to avoid the occurraice of nonballistic deflections, two Gen-

eral Radio 107L mutual inductors were also used in the external cir-

cuit.

Since it generally took about thirty hours to investigate a single

density and since several densities were studied, it was necessary that

the mutual inductance system possess good stability over a period of

several days. Temperature calibrations, taken at various times during

the course of this and other experiments, indicated that this was in-

deed the case. It has been observed that over a period of a few weeks

neither the slope nor the intercept of the deflection versus T-1 curve

changed by more than 3%.

The CMN susceptibility was calibrated against the He3 and He4 vapor

pressures in the temperature range from 1.1 to 3.20K. Both He3 and He4

vapor pressures were used in the calibration between 1.1K and 1.50K,

with the two calibrations agreeing to within 1%. The calibration was

accomplished by first calibrating the germanium and carbon resistors











against the helium vapor pressures with the inner vacuum chamber cun-

taining 100 microns of He4 exchange gas. The gas was thun pumped away

and both helium baths lowered to their working temperatures of 1.0K

and 0.3"K. The temperature of the CMN and the resistors was then con-

trolled by a heater, and the CMN calibrated against the resistors.

This was done so that the shields upon which the primary and secondary

coils were mounted would be at the same temperatures during the calibra-

tion as they were during the experiment itself. Also by maintaining

the shields at a constant temperature, any temperature dependent dia-

magnetic contributions to the susceptibility arising from the brass

shields were eliminated. It is believed that the absolute temperatures

are accurate to within about 1%, while below 0.08*K relative temperature

changes as small as 0.1 mOK could be measured.

The carbon and germanium resistors were calibrated against the CMN

and served as secondary thermometers down to 0.04K. Both resistors

exhibited good temperature reproducibility upon cycling; in particular,

no measurable change in the calibration of the germanium resistor was

observed even after the apparatus had been allowed to warm to room

temperature and then recooled. The thermometers fit snugly in wells

drilled into the copper section of the sample chamber. Thermal con-

tact between the resistors and the copper walls was insured by melting

Apiezon N grease and causing it to flow over the entire surface of the

resistor.

Resistances were measured by means of a 21 cps ac resistance

bridge shown schematically in Fig. (11). The phase-sensitive detector

is a Princeton Applied Research type JB-4 lock-in containing a variable

frequency oscillator which was used to drive the bridge circuit. The


___~

































































CI-
101
C-

a.!
A



t o
Cto











power dissipated in the resistors was reduced from 10-9 W above 1.0K

down to less than 10-12 W at 0.040K. Resistances could be measured to

at least 0.5%, corresponding to a temperature sensitivity over the en-

tire range of the resistors of about 0.2 m'K.


Temperature Regulation

In this apparatus a zinc foil was utilized as a thermal switch be-

tween the cooling salt and the sample chamber. This was done so that

the temperature of the sample could be varied over a wide range while

maintaining the temperature of the salt at a fairly constant value.

With the zinc in its superconducting state, it was possible to raise the

sample temperature, over a period of four to five hours, to about 0.07K

while the temperature of the salt remained below 0.030K. This is a very

desirable arrangement since it allows one to obtain several sets of pres-

sure and temperature measurements on a given density.

The temperature of the sample was regulated by manually adjusting

the current through a 33 Kf metal film resistor located in one of the

chamber wells. Regulation at the lowest temperature was accomplished

with the zinc heat switch normal since only small power inputs were

necessary to raise the temperature of the sample above that of the potas-

sium chrome alum heat sink. At higher temperature, however, where the

thermal gradient between the sample and salt was large, regulation was

accomplished with the switch superconducting. It was also desirable

to have the switch open when passing through the phase transition re-

gion, since here it was sometimes necessary to maintain a large thermal

gradient for a long period of time while the mixing of the two phases

took place.


__











Thermal Equilibrium Time between the Sample and Thermometer

In order to insure that the measured thermal expansion be pro-

duced by the helium sample alone, it is necessary to exclude all other

materials from the sample volume. This means that the CMN thermometer

used to measure the temperature of the He3 spin system must be located

outside the chamber. This requirement poses little problem at tem-

peratures greater than about 0.10K since in this region the Kapitza

boundary resistance is small and hence thermal equilibrium between

sample and thermometer occurs rather rapidly. However, at lower tem-

peratures, the Kapitza resistance increases rapidly and some care must

be taken to avoid producing a long thermal time constant between the

sample and thermometer. A schematic diagram of the thermal path be-

tween the nuclear spins and CNN thermometer is shown below.




He3 He3 Chamber CMN
Spins 1 Lattice TLC Walls TCT Ther-
mometer





The rate at which the nuclear spins come into equilibrium with

the lattice is determined by the spin-lattice relaxation time. Meas-

urements of T1 in the bcc phase have been performed by the Duke and
27
Oxford27 groups down to temperatures of 0.040K in magnetic fields of

a few hundred gauss. From these results one can reasonably assume that

at 0.020K in zero magnetic field T1 should be no longer than a few

minutes for any of the densities studied in this work. Since an es-

sentially fixed lattice temperature can be maintained for hours, the

spins and lattice will have sufficient time to come into thermal equi-

librium.











The CMN salt crystals and He3lattice come into thermal equilib-

rium with one another through contact with the walls of the sample

chamber. The rate at which this process occurs is determined by the

boundary resistance between the He3 and the chamber walls, and between

the CMN crystals and copper wires embedded in them. This Kapitza re-

sistance between two surfaces can usually be expressed in the form

AT a (3.8)
RB ATn

where a and n are empirical constants, and A is the area of contact

between the two surfaces. An accurate calculation of the thermal

time constant for the He3-CMN system cannot be made, chiefly because

of the lack of experimental data on the Kapitza resistance of solid

He3. The measured value at 0.020K was about five minutes. Since

the drift rate at the lowest temperature is less than about 0.1 mOK

per hour, we can expect the He3 and CMN to be in excellent thermal

equilibrium at all times during the experiment.



G. Solenoids


A niobium zirconium solenoid previously described by Lichti45 was

used to produce the necessary magnetic field for the demagnetization.

The solenoid has a 2 inch inside diameter and is 6.88 inches long. It

is equipped with a persistent switch so that it may be operated for

many hours without excessive boiling of the liquid helium in the outer

bath. A small resistive short was soldered across the terminals of the

solenoid to give a time constant for the parallel R, L circuit of about

five minutes. A Magnion type CFC 100 power supply was used to provide

the current for the solenoid. A current of 17 amps in the solenoid











produced a field of 13.2 kG at the center. At 0.30K this corresponds

to an H/T ratio of 4.4 x 104 gOK-1, a value at which the magnetization

of the potassium chrome alum is essentially complete. Several attempts

were made to determine the residual field of the solenoid after de-

magnetization. These were only partially successful so that at the

present time we can only say that the residual field appears to be

less than 50 G. This probably caused the final temperature to be

slightly higher than might otherwise have been attained.

A small niobium solenoid also located in the outer bath was used

to provide the 60 gauss necessary to actuate the zinc heat switch. The

vertical distance between this solenoid and the CMN thermometer was 6

inches, and it produced no detectable field at the position of the CMN.



H. Performance of the Experiment


Sample Formation and Cooldown to 0.3K

After the completion of the He4 transfer, the field in the main

solenoid was turned up to 13.2 kG and the current was made to persist.

The He4 exchange gas was pumped for several hours until the reading on

the leakage meter of an MS9A Veeco leak detector went below an em-

pirically determined value which indicated that the exchange gas had

been essentially removed. At this point, the inner He4 bath was filled

with liquid and pumped to 1K, He3 gas was condensed into the evaporator

section of the refrigerator and its temperature lowered to the vicinity

of 0.30K. Within about two hours, the temperature of the cell was be-

low 2K and the sample gas was condensed into the sample system. When

the temperature of the sample was within 0.1K of the freezing temperature










corresponding to the particular density to be studied, the refrigerators

were warmed and the strain gauge calibrated against the dead weight

gauge. After calibration, the pressure on the liquid was held at the

desired value by the dead weight gauge and the He3 refrigerator quickly

cooled below the freezing temperature of the sample. A solid plug then

formed in that section of the filling capillary thermally grounded to

the He3 refrigerator so that the desired density was obtained in the

sample cell.

As the sample cooled, the melting curve was reached and the pressure

in the cell dropped rapidly until the chamber became completely filled

with solid. Ihe location of the melting point was determined by the

drastic change in slope which occurs when the sample leaves the melting

curve and enters into the all solid region of the phase diagram. The

molar volume was determined from the point of intersection of the solid

isochore with the melting curve using the data of Grilly and Millsl4

and Mills, et al.46 The temperature was then held within 0.01K of the

melting point and the solid annealed for thirty minutes. After anneal-

ing, the process of cooling the sample was allowed to continue. When

the temperature reached about 0.8K, the return valve on the He3 refrig-

erator was shut. The apparatus could then be left unattended for eight

to ten hours while the temperature of the salt and sample cooled to 0.31

at which point the inner bath was refilled and the demagnetization begun.


Demagnetization Procedure

The magnetic field was decreased exponentially, by letting it de-

cay with the L/R time constant of about 5 min, from 13.2 kG down to

about 9 kG. During this initial step in the demagnetization process











the lead heat switch became superconducting thereby isolating the chrome

alum salt from the He3 refrigerator. The field was decreased from 9 kG

to 3 kG in three steps over a period of about two hours. At this point

the temperature of the sample was approximately 0.080K, which is well

below the phase separation temperature of the 600 and 1600 ppm mixtures.

It was then necessary to wait for the isotopic phase separation to be-

come essentially complete. This waiting period ranged from about

thirty minutes at a molar volume of 24.0 cm3/mole to about four hours

at molar volume of 23.0 cm3/mole. During this time, the temperature of

the sample remained stationary and the onset and completion of the phase

separation could be determined by watching the rate of change of the

pressure with time. After determining that the phase separation was

essentially complete, the demagnetization was continued. The sweep

time on the power supply was adjusted so that the field would be

turned down to 2000 gauss in about one hour. During this time, values

of the capacitance, resistance, and ballistic deflection were taken.

The sweep time of the power supply was then readjusted so that the

final 2000 gauss would be turned off in approximately three hours. A-

gain, readings of the capacitance and deflection were taken as the

sample cooled. Several checks of the thermal equilibrium between the

CMN thermometer and the He3 sample were made by stopping the demagneti-

zation process and watching the rate of change of the temperature and

pressure. At all temperatures, it was found that the demagnetization

proceeded slowly enough for the thermometer and sample to attain good

thermal equilibrium. The lowest temperature, generally about 21 mOK,

was obtained about one half hour after the final field was turned off.

No heating of the sample was done for at least another hour; during

this period no detectable change in temperature occurred. heat was then


____







71


applied to the sample and its temperature was raised to about 0.08K

over a period of about five hours. The heat was then removed and the

sample allowed to recool, generally reaching a temperature in the

vicinity 25 m"K. This procedure was repeated at least once for every

density so that at least three sets of deflections versus capacitance

readings were obtained below 0.08K. The sample was then warmed through

the phase transition region and up to the melting curve. The He3 re-

frigerator was allowed to warm above the freezing temperature of the

sample and the pressure calibration checked. In all cases this calibra-

tion agreed with that taken previously to within 0.01 atm.
















CHAPTER IV


RESULTS AND DISCUSSION


A. Introductory Remarks


In this chapter we present values for the nuclear exchange energy

and isotopic phase separation temperature obtained from three samples

containing 20, 600, and 1600 ppm He4 impurities. The data are limited

to large molar volumes by the long equilibrium time for the isotopic

phase separation in the 600, and 1600 ppm samples, and by the pressure

sensitivity for the 20 ppm sample. The results for the volume depend-

ence of the nuclear exchange energy, phase separation temperature, and

energy of mixing, will be discussed in terms of the approximate equa-

tion of state developed in Chapter II. To facilitate this discussion,

and also to indicate the relative size of the pressure changes produced

by the various degrees of freedom in the solid, we present in Fig. (12)

a typical isochore obtained using the 1600 ppm sample. As was done in

Chapter II, the temperature scale has been divided into three regions.

At temperatures greater than 0.30K, the phonons make the largest con-

tribution to the free energy and we find the pressure to be proportional

to T4. In the phase transition region there is a sharp increase in the

pressure as the mixture separates into two phases. Finally at tempera-

tures below 0.06K, the phase separation is essentially complete,

and the pressure change, (AP)ex, arising from the nuclear spin system

is proportional to T-1.


































a) (U E-4 3 p h *
-1 000 0. 0N

m 1 A 0 H
C rq o 0 M .C t 00


04 M4 C () U) 00
a -Hic "d ;4 0 1.4


0. 0. *rl C 0.
C O) $4 o M
o < l amC
0 O.-H $O -
1 0 to 00 O a)
a) $4 m0 I- U
= 0 C3 L. 3 E0
41 4 : o U ra d
a w 10 0 0 10
o ac c E *r a .c ,C
44 U ( 0 4( H, 4
O11 r1W-H ) U) r-I >,
O 0 1 0 4-10 0 -
Xm bo 4J ) 4 3 rl U)

o0 4) 1 m) im rd
S 0 m a) '0 *
m-H p "t 0 ( .0
41 0 41 -1 0 4H 4 r-1
0o00H04- o


Um t14 41 CD U r n)
*H 0 H 0 k 44 a 14 H r

u0 E4 0 4H Om 1
U El 4J 0o (1) Ed (V 0) 4
mC a) .0i= i w bo a 0
0 a 4 0 m 9 0
to 0 1 4 0 1
S4- 4 uN 4 CC -
u 4M -a Ed a 0 U 4J 6





CM

-H










So
















C O






0 -0
.. .

w








/ 00










SI I I


(wuo .01) dV











Before presenting the results for the exchange energy and the

phase separation temperature however, it is necessary to discuss a

small anomaly in the thermal expansion of the strain gauge itself.



B. The Thermal Expansion of the Empty Cell


As previously mentioned, the pressure calibration fo the strain

gauge was performed at a temperature just above the sample freezing

temperature corresponding to the particular density to be studied.

The calibration conditions were thus not identical to those under

which the experimental data were obtained. For this reason a separate

demagnetization was performed, with the sample chamber evacuated, to

determine the effect of the thermal expansion of the sample chamber

itself on the capacitance. The result of this demagnetization is shown

in Fig. (13). In this graph the capacitance change due to the thermal

expansion of the chamber has been converted into an equivalent pres-

sure change so that its effect may be more easily compared with the

pressure changes produced by the thermal expansion of the solid helium

samples. It can be seen that a strange anomaly exists in the thermal

expansion of the sample cell in the temperature range from 0.080K to

0.300K. This is also the temperature region in which the phase separa-

tion occurs, and hence the contribution to the capacitance change re-

sulting from the thermal expansion of the chamber must be included in

the analysis of the phase separation data. For the 600 ppm sample, the

chamber contribution to the total capacitance change is about 10%,

while for the 1600 ppm sample it amounts to approximately 4%. Below

0.060K, the thermal expansion of the chamber is completely negligible



























00






0-
*4



a
hie
o h)

W 44





01

0




In CLU 0
/ (


/ ^ j
A Q (





^ ^w^ o ,


(WjD -t01) dV











and hence does not affect the values obtained for the nuclear exchange

energy.

It is interesting to speculate about the possible source of this

anomaly in the thermal expansion of the sample chamber. If the thermal

expansion coefficient, which is proportional to the derivative of the

P versus T curve shown in Fig. (12), is plotted against temperature,

the result suggests that some type of cooperative transition occurs

in the chamber. A possible explanation for this behavior is that a

magnetic transition takes place in the stainless steel section of the

sample cell.



C. Nuclear Exchange Energy


Values of IJI for "Pure" He3

The values given below for the nuclear exchange energy and its

volume dependence are obtained from PVT data taken in the temperature

region from 0.020K to 0.060K. In this range the He4 impurity in the

He3 rich phase is less than 0.2 ppm, and hence the values of IJI are

those for very pure He3. The molar volumes studied in this work range

from 22.8 cm3/mole to 24.2 cm3/mole. At smaller molar volumes, the

absolute value of J becomes too small to be measured with the present

pressure sensitivity.

The equation of state for dilute mixtures developed in Chapter II

indicates that at low temperatures the contribution to the pressure

from the nuclear spin system is given by


) R e j 2 1
(AP)ex ex (4.1)

where (AP)ex is the increase in pressure relative to some arbitrary
ax



































) 0 0) 0
- H Cd .ri C
: Xc rI H w
0 0 ( o 0 *Hr
C 0 H 0 )
1 C4-1 4J m
1 o *rI W 4
o 1e
j4I O 3 i a a
m .0
o o ca c

O-wP0 0o0.0
'I r-i a) 4 a C .
0- C4-i 0 W C m0o 14
0 -i H0 4 3C 41
rl CR O J .J CX
1. 44 0 0 -H x8 to
*r *to oM d40u C 8 a
o W CROL) a 0.
w t0) 4 4 U
0 um 1 d 0 m* OC
0 *r-0 -1 CL0 0
0( 4 H Ur 00 -4 0
H60 *l r-( r-l
SA 0 m CD
1i I 0 .0 1 bo
) O e- 0 4 30 0
U O0 0> .0 0" 3
XCR mt4. W I x
om U u
pH w"
C0 W 0 r0 4)
0 4 4U 4 100 C 4 3.

0U 400 4xJ 40
41I4 H a A3
0 0 44 W I W O 0<
< r'd 0. 0 .c J-'10


r


1:4
r


































00
4 I -(






a. 0.
I =\ dII



0 0


0 (
I' CI
(D


> fi ri 4ir
0i W W N

ca 0
coqo





(W4D 2,01) (dV)











reference point, and


x = dInIJI (4.2)
ex dlnV

In Fig (14) (AP)ex is plotted versus T-1 for various molar volumes and

He4 concentrations. In the experimental procedure section of this

work, we explained how it was possible to obtain values of P versus T

for several runs on a given density. As can be seen, the data obtained

on these various runs exhibit extremely good reproducibility. In view

of the small sizes of the pressure changes measured, this fact is indeed

comforting. The rapid decrease in (AP)ex at a value of T-1 = 12 is due

to the mixing of the separated phases in the 600 ppm sample.

It can be seen from Eq.(4.1) that the slopes of the (AP)ex versus

T-1 curves are equal to y ex( Since this expression is propor-

tional to J2, these measurements cannot be used to determine the sign

of J. Also, since it contains both J and its logarithmic derivative,

a self-consistent procedure must be used to extract J from the data.

The first step in this process is to assume an initial value of yex

based on the NMR data.26,27 Equation (4.1) is then solved for IJ(V)I

at each molar volume studied. These initial values, IJ(V)I can be

used to compute a new, self-consistent value of yex. The correct values

of the exchange energy are then computed using Eq.(4.1) and this self-

consistent value of yex'

The values of IJl/k obtained using this self-consistent procedure

are presented in Table I and Fig. (15). Also shown for comparison are

the results derived from NMR data, and those calculated by

Hetherington, et al.28 The solid line fitted to the data indicates

that yex = -16.4 is valid over the density range covered by this work.



























Figure 15. Nuclear exchange energy versus molar volume. The
various symbols represent values of IJI/k obtained
using different initial He4 concentrations. Also
shown for comparison are the values of IJI/k de-
rived from the relaxation measurements of Richard-
son, Hunt, and Meyer (RHM); Richards, Hutton, and
Giffard (RHG); and those calculated by Hethering-
ton, Mullin, and Nosanow (HMN).


























I~


































HMN
-


.,,

0.06
0.05

0.04


23 23.5 24

MOLAR VOLUME (CM3/MOLE) (Log Scale)










TABLE I


Smoothed Values of the Nuclear Exchange
Energy in Solid He3


V IJl/k TN = 31Jl/k
(cm3/mole) (mK) (mK)


24.4 0.90 2.70
24.0 0.68 2.04
23.6 0.52 1.56
23.2 0.39 1.17
22.8 0.30 0.90




From these results it is seen that in the bcc phase IJl/k is about

0.9 mK at a molar volume of 24.4 cm3/mole, and decreases with in-

creasing density approximately as dln- = 16.4. The corresponding
dlnV
Ndel temperature, obtained from the expression TN = 3k varies from

about 2.7 mOK at 24.4 cm3/mole down to 0.9 mK at 22.8 cm3/mole.

The agreement between our values of IJI/k and those derived from

the NMR data of Richardson, et al.26 is excellent. Since these pres-

sure measurements constitute a direct method for determining IJ| this

agreement may be interpreted as additional confirmation of the validity

of the NMR formalism. Richardson47 has suggested that the discrepancy

between the Duke and Oxford values for IJI arises from the latters'use

of the block capillary technique to determine the molar volume of the

solid. This variance in IJI can be resolved if one assumes that the

plug in the Oxford filling capillary slipped enough while forming the

low density solid to decrease the molar volume by about 0.8 cm3/mole.

At higher densities, the values of IJI determined by the Duke and Ox-

ford workers are in better agreement, and hence Richardon's suggestion

seems reasonable.










A comparison of the theoretical calculation of HMN and the experi-

mental data indicates that while the theoretical slope agrees well with

that found experimentally, the magnitude of J predicted by HMN is about

an order of magnitude too small. One can understand how such a situa-

tion arises by recalling the theoretical expression for J obtained

earlier,

2R2A2 f2r i x 2eff r )> x
J; = + 2 (4.3)
S 2m f(r ij)> f (r)>

The first term in this equation is large in magnitude and negative in

sign, while the second term is of the same order of magnitude but

positive in sign. One sees, therefore, that the theoretical value of

J is obtained from the difference of two large numbers, each of which

is accurate to about 10%. It is evident that such a situation could

easily produce a relatively poor absolute value for the exchange energy,

yet still yield a good value for its volume dependence.

The observed decrease in IJI with increasing density is interest-

ing and can be understood at least qualitatively on the basis of the

following discussion. As the density is increased the average separation

between atoms becomes smaller and intuitively one might expect the over-

lap region between nearest neighbor atoms to increase. This would of

course increase the absolute value of the exchange energy. This in fact

does not occur because, in addition to reducing the interatomic spacing,

the increase in density also produces an increase in the kinetic energy

of the atoms. This increase in kinetic energy is accomplished by a

greater localization of there He3 atoms to the vicinity of their equilib-

rium positions. In more formal language we could say that the increase

in density produces an increase in the curvature and a decrease in the









extent of the wave functions describing the motions of the atoms. Since

in solid He3 the exchange energy arises as a consequence of the over-

lapping which occurs in the tails of these functions, any decrease in

their physical extent produces a smaller absolute value for the ex-

change energy.


Effects of He4 Impurities on J

Before terminating this discussion of the exchange interaction

in solid He3, a brief examination of the effects of small amounts of

He4 impurities on J will be made. The NMR data of RHG taken on a

sample containing 5000 ppm He4 indicate that the presence of a He4

atom causes a distortion of the He3 lattice, which in turn produces an

increase in the value of J in the vicinity of the impurity site. These

workers found that it was possible to express J in the form


j2 = J2 +J2 (4.4)
bulk imp'

where (Jimpl/k = 0.1'K is independent of density over the molar volume

range from 18.3 cm3/mole to 20.0 cm3/mole. If the data of Garwin and

Reich,48 obtained with a sample containing 1% He4 impurities, am also

analyzed according to Eq.(4.4) it is found that the value of JiJ p/k

is = 0.5 mK. On the basis of these two results it appears that the

parameter Jimpl increases rapidly as the He4 concentration is increased.

The recent magnetic susceptibility experiments of Cohen and Fair-

bank49 performed on samples which contained 0.5, 100, and 3000 ppm He4

impurities also indicate that the exchange energy increases with in-

creasing He4 concentration. Originally the susceptibility measurements

taken on the 3000 ppm sample also indicated that the exchange energy









increased rapidly with increasing density between molar volumes of

22.6 cm3/mole to 21.0 cm3/mole. Subsequent experiments by the same in-

vestigators, however, have not confirmed this result, so that at the pres-

ent time the question of how a small amount of He4 impurities affects

the magnetic interactions in solid He3 remains unanswered.


D. Locus of the Zeros of the Thermal Expansion Coefficient

In Chapter II we obtained an approximate equation of state for

solid He3 given by
3R(V ) R ex 1 3n
P(V,T) = Po(V) x T T, (4.5)

where we have omitted the contribution from the isotopic phase separa-

tion. Differentiation of Eq.(4.5) with respect to temperature at con-

stant volume gives

(ap) 3R J2 1 + 12r4 T 3
ST) V Yex Ty Y ") (4.6)
S+ (4

A straightforward thermodynamic derivation using the definitions of

the isobaric thermal expansion coefficient a -) and isothermal

compressibility 8 V (-) leads to the expression (-) = .
T V
Goldstein30 has pointed out that the expansion coefficient be-

comes zero when the negative contribution for the spin system becomes

equal in magnitude to the positive contribution from the lattice.

Thus, there will be a line in the P-T plane which is the locus of the

zeros of the isobaric thermal expansion coefficient. By rewriting

Eq.(4.6) in the form


= ( .) = Ai(V) + A(V)T3, (4.7)
B =aP =AI V + A2(V)T3, (4.7)





























Exchange Lattice
aw















Liquid


I I I I

.16 .18 .20 .22

TEMPERATURE (OK)

Figure 16. Locus of the zeros of the isobaric expansion coef-
ficient.










one can see that the locus of the zeros of a is given by the relation

AI(V). 1/5
T a= A2(V) (4.8)


We have obtained A1 and A2 from measurements of the slopes of the iso-

chores in the low and high temperature limits. The result for T =0 is

shown in Fig. (16).

Roughly speaking, one can say that this locus divides the P-T

plane such that at temperatures and pressures to the left of the line,

the properties of the solid are determined primarily by the nuclear

spins, while to the right of the T =0 line they are determined primarily

by the lattice.



E. Isotopic Phase Separation


Kinetics of the Phase Transition

One of the most striking features of the isotopic phase separation

in solid He3-He4 mixtures is the rapid increase in the equilibrium time

constant, T, with increasing pressure. Although we have not made a de-

tailed study of the dependence of on density we have noted that it in-

creases from a value of few minutes at a molar volume of 24.2 cm3/mole

to about one hour at a molar volume of 23.0 cm3/mole. This rapid in-

crease in T with density suggests that diffusion of the atoms by quantum

mechanical tunnelling is the mechanism by which the separation into pure

phases is accomplished. It also appears that T is considerably longer

in the cooling direction than in the warming direction, which is an

indication that the atoms can mix more readily than they can separate.

The actual spacial distribution of the isotopes in the two phase

region is presently Uncertain. In liquid mixtures, a visually observable










bulk stratification of the two phases occurs due to the differences in

mass of the two isotopes. It is highly unlikely that such a bulk sep-

aration also takes place in the solid. It seems more reasonable to

picture the phase separated solid to be composed of a number of locally

enriched regions, whose dimensions are very large compared to inter-

atomic distances. In dilute solutions the number of these He4 en-

riched regions is quite small, and hence the bulk properties are es-

sentially those of a very pure He3 system.


Pressure Dependence of the Energy of Solution and Phase Separation Tem-
perature

The theoretical development of the isotopic phase separation

phenomenon given in Chapter II was based on the assumption that the

He3 and He4 atoms mix together randomly to form a regular solution.

This model was proposed by Edwards, et al.24 in an effort to explain the

observed discontinuity in the specific heat of He3-He4 mixtures. The

shape of the specific heat anomaly is similar to that associated with

an order-disorder transition, and one interprets this as evidence for

the separation of'the mixture into two phases. The measurements of

Edwards. et al., at a pressure of 35.8 atm., were performed on seven

different He4 concentrations ranging from 0.03 to 80%. The data in-

dicate that the phase separation curve is symmetric about a concentra-

tion of 50% in the T-x plane.

Mullin50 has recently developed a theory of the phase separation

in solid He3-He4 mixtures using techniques similar to those employed

by Nosanow in his study of the pure isotopes. Mullin's analysis leads

to the conclusion that solid helium solutions should be nearly regular

but for different reasons than those originally suggested by Edwards,










et al.24 Two important predictions of Mullin's work are that the

phase separation curve should be unsymmetrical, and that the phase

separation temperature should decrease with increasing density.

In this section we present some preliminary results for the pres-

sure dependence of the phase separation temperature, and the energy of

mixing. We wish to emphasize that these results are to be regarded

as tentative until confirmed by future experiments performed with

mixtures containing greater amounts of He4 impurities.

The pressure change at constant volume due to the phase separa-

tion of a dilute, regular mixture was obtained in Chapter II and is

given by

d(Ek) E kT
(P) R d(E k) e-ElkT, (4.9)
(AP)S= R dV (9)
E33 E44
where El = E34- -2 -- is the energy of mixing. The equation for

this phase separation line has, for small He4 concentrations, the

simple form


x = e-EkT. (4.10)


Substitution of Eq.(4.10) into Eq.(4.9) yields the following expres-

sion for (AP)PS,

d(E1k)
(AP)pS = Rx d--V- (4.11)
dE k
One can see from this equation that if (AP)pS > 0, then dV > 0,

and EM should decrease with increasing density. Further by rewriting

Eq.(4.10) in the form

EM I 4
T Mk- (4.12)
PS k lnx










it can be seen immediately that the pressure dependence of TPS is

similar to that of EM.

We have measured (AP)pS as a function of volume over the range from

23.0 cm3/mole to 24.2 cm3/mole for initial He4 concentrations of 600

and 1600 ppm. The results are shown in Figs. (17) and (18). The curves

shown have been obtained by a graphical smoothing procedure as follows.

The capacitance values were first plotted against temperature and a

smooth curve placed through the data. The contributions to the ob-

served capacitance change from the exchange energy and sample chamber

anomaly were then subtracted from this smooth curve. The resulting

values for (AP)PS versus T are those which appear in Figs. (17) and

(18). We wish to emphasize thatwhile tis process is necessary to obtain

the shapes of the (AP)PS versus T curves it is not needed to establish

the essential fact that (&P)PS decreases as the density is increased.
dEM/k
The values of dV computed from Eq.(4.11), and the measured values

of (AP)PS' are shown as a function of molar volume in Fig. (19).

While there is some scatter in the data, it can nevertheless be seen
dEM/k
that -- decreases monotonically from u 0.05"K mole/cm3 at 24 cm3/mole

to 0.030K mole/cm3 at 23 cm3/mole. Using an average value of 0.04K

mole/cm3, and Eq.(4.12), one finds that the phase separation tempera-

ture of the 1600 ppm sample should decrease by 6 mOK between molar

volumes of 24 and 23 cm3/mole. This is in qualitative agreement with

the calculation of Mullin50 which indicates that over a similar pres-

sure range the phase separation temperature of a 50% mixture should

decrease by 0.020K.

Finally we have determined the phase separation temperatures and

energy of mixing as a function of pressure from the inflection points




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