THERMAL CONDUCTIVITY OF SOLID HELIUM

MIXTURES IN THE PHASE SEPARATION REGION .

by

ARTHUR E. BURGESS

.Sc., Royal Military College of Canada, 196

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in the Department

of . .

Physics

We accept this thesis as conforming to the

required standard

THE UNIVERSITY OF BRITISH COLUMBIA

November, 1971

In present ing th is thes is in p a r t i a l f u l f i l m e n t of the requirements for

an advanced degree at the Un ive rs i t y of B r i t i s h Columbia, I agree that

the L ibrary sha l l make i t f r e e l y a v a i l a b l e for reference and Study.

I fu r ther agree that permission for extensive copying of th is thes is

for s c h o l a r l y purposes may be granted by the Head of my Department or

by his representat i ves . It is understood that copying or p u b l i c a t i o n

of th i s thesis for f i n a n c i a l gain sha l l not be allowed without my

wr i t ten permiss ion.

Department of

The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada

ABSTRACT

The thermal conductivities of four solid helium mixtures have

been measured in the temperature range between 0.15 K and 0.6 K. These are

the f i r s t measurements of the transport properties of solid helium mixtures

in the phase separation region. A dilution refrigerator and a novel temper­

ature controller were constructed to obtain the necessary temperatures and

temperature s t a b i l i t y .

The thermal conductivity measurements above the phase separation

were consistent with present theories of conduction in isotopic mixtures.

Just below the phase separation temperature the conductivity decreased

rapidly with decreasing temperature. Below the phase separation, the con­

ductivity was limited by boundary scattering and is interpreted in terms of

a model of phonon scattering by domains rich in one isotope embedded in a

matrix rich in the other isotope.. Phonon scattering by metastable domains

of one crystal structure embedded in a matrix of another crystal structure

has been observed above the phase separation.

- i i -

' TABLE OF CONTENTS Page

Abstract ±

Table of Contents i i

L i s t of Figures and Tables i i i

Dedication and Acknowledgements i v

Chapter

I: Introduction A. Solid Helium 1 B. Solid Helium Mixtures 4

II: Phase Separation • A. Thermodynamics of Regular Solutions 8 B. Thermodynamics of Solid Helium Mixtures 12 C. Kinetics of Phase Separation 16

III: Thermal Conductivity A. History 20 B. Theory 23 C. Previous Experimental Results in Solid Helium 30

IV: Apparatus and Experimental Procedures A. Dilution Refrigerator 33 B. Thermometry 37 C. Mixing Chamber and Temperature Controller 47 D. Sample Chamber 53 E. Crystal Growth 64 F. Data Reduction and Experimental Errors 66

V: Experimental Results 4

A. Stainless Steel and He 72 B. Phase Separation Temperature 73 C. General Results for One 10% He3 Crystal • 74 D. Above the Phase Separation 78 E. Below the Phase Separation 84 F. Time Dependent Effects 97 G. Conclusions 102

Appendix A. Tables of Experimental Data 104 B. Bibliography 114

- i i i -

FIGURES

Page 1. Liquid-solid phase diagrams of solid helium. 3

- 2. T-X phase diagram of solid helium mixtures at P % 33 atm. 6

3. Phase separation parameters for regular solution theory. 10

4. Specific heat of solid helium mixtures in the phase separation 1 3 region.

5. Schematic diagram of low temperature apparatus. 3 4

6. Tunnel diode oscillator thermometer and temperature controller c i r c u i t . 43.

7. Mixing chamber temperature controller c i r c u i t . 48

8. Sample chambers. 55

3 9. General results for 10% He (sample E). 75

10. Thermal conductivities above phase separation. 79

11. Comparison n ? results above phase separation with theory at 83 T/e'fc .03.

3 12. Conductivities below phase separation 90% He samples. 85

3 13. Conductivities below phase separation; 80% and 50%.He samples. 86

3 14. Conductivity below phase separation 10% He sample. 87

3

15. Time dependent effects in 50% He sample. 101

TABLES

1. Phase separation time constants. 17

2. Conductivity above phase separation. 81

3. Conductivity below phase separation. 93

4. Elapsed time below the phase separation for data in figure 15 100

(x 3 = .50).

- i v -

Dedicated

' to my wife

Carolyn

whose patience and understanding

were of paramount importance.

I would particularly like to thank my supervisor Dr. Michael Crooks

for suggesting this research project and for timely advice and encouragement

during the experiments. I would like to thank Dr. P.W. Matthews for suggest­

ing the self inductance thermometer and other discussions. ' I would like to

thank Dr. D.L. Johnson for taking some of the data and lending equipment as

well as a sympathetic ear.

Mr. R. Weisbach and Mr. G. Brooks helped in the construction cf the

apparatus. Mr. J. Lees and Mr. E. Williams constructed the dewars and

glassware.

Miss Rose Chabluk typed the thesis in a very accurate and speedy

manner.

I would like to thank the National Research Council for a Bursary

and Studentships and the H.R. MacMillan Family for a Fellowship.

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CHAPTER I

INTRODUCTION

In this introduction we w i l l f i r s t describe a few of the 3 \

properties of solid helium (pure He or He ) which set helium apart from

other solids. In the second part of this chapter we w i l l briefly

describe the solid isotopic mixtures of helium and state the reason for

doing the experiments described in this thesis. A. SOLID HELIUM

Helium was f i r s t l i q u i f i e d in.1908 by Kamerlingh Onnes. Early

attempts to s o l i d i f y helium by cooling the liquid at the saturated vapour

pressure were unsuccessful. In 1926 Keesom found that solid helium

exists only at pressures greater Limn 23 atmospheres. Helium Lch^vcs

in this unique manner because of i t s large zero point energy.

One usually considers a solid to be a collection of atoms which

osci l l a t e in their l a t t i c e sites like coupled harmonic oscillators. In

the quantum mechanical model a harmonic oscillator has quantized energy

levels with a minimum energy (the zero point. energy) of • -'ftco, where h

is Planck's constant. The frequency co is inversely proportional to the

square root of the atomic mass, giving helium a large zero point energy.

The van der Waals attraction between helium atoms is weak compared to

this zero point effect, therefore considerable external pressure is

necessary to sol i d i f y helium. Guyer (1969) gives the following descrip­

tion of solid helium: Because of i t s large kinetic energy the helium

atom under low external pressure can do the work necessary to escape

-2-

between i t s nearest neighbours and become n o n - l o c a l i z e d . At a s u f f i c i e n t l y

l a r g e e x t e r n a l pressure the atom can no longer escape between i t s nearest

neighbours and remains l o c a l i z e d forming a s t a b l e s o l i d .

The l a r g e zero p o i n t energy leads to a l a r g e c o m p r e s s i b i l i t y .

The molar volume of the pure s o l i d s can be decreased by a f a c t o r of two

by r a i s i n g the press u r e from 25 atmospheres to 1500 atmospheres. The

Debye temperature 8 i s a l s o very pressure dependent and i n c r e a s e s by a

f a c t o r of f i v e d u r i n g the above pressure i n c r e a s e . The l a r g e compressi­

b i l i t y thus a l l o w s one to study the p r o p e r t i e s of the s o l i d over a l a r g e

range of d e n s i t i e s and Debye temperatures.

3 4 The phase diagrams of s o l i d He and s o l i d He g i v i n g the l i q u i d -

.-.--1-.-? ~r~..J V r r ~ r > ~ 1 r*1 ^ c; ~ ^ n n l m A fVyr> *\ _"k/~»^\T r» o i - * * - - r o r ! P u K l P f ' h r r ^ f r a n Q i -* - ^ * - ~ a w - ' ~ — r \ : - i ' " ' "

t i o n l i n e s are shown i n f i g u r e 1. The phase diagram f o r the 50% mixture

i s a l s o shown f o r comparison. The hcp-bcc t r a n s i t i o n r e g i o n f o r the 50%

mixtures i s estimated by a s c a l i n g method which i s described i n Chapter V,

s e c t i o n D. The phase diagrams show the re g i o n s where the s o l i d s e x i s t and

show the minimum f r e e z i n g p r e s s u r e s .

S o l i d helium i s r e f e r r e d to as a quantum s o l i d because the

l a r g e zero p o i n t energy leads to a zero p o i n t motion of the atoms about

t h e i r e q u i l i b r i u m s i t e s which i s a l a r g e f r a c t i o n of the nearest neigh­

bour d i s t a n c e . One might ask whether i t i s reasonable to speak of phonons

i n such a s o l i d . However, the neutron s c a t t e r i n g experiments of Min-4

k i e w i c z et a l . (1968) have shown t h a t hep He has a Debye-like phonon

spectrum so we w i l l t h e r e f o r e assume throughout t h i s work that the Debye

model can be used f o r s o l i d helium.

FIGURE 1 : Liquid-Solid phase diagrams of sol id helium.

Note; In the mixed region both bcc and hep

crystal structures are present in the s o l i d .

-4-

B. SOLID HELIUM MIXTURES

Any mixture which i s not i n an ordered s t a t e has a s s o c i a t e d w i t h r

t h i s " d i s o r d e r " an entropy of mixing. In a s t a t e of complete thermal

e q u i l i b r i u m at absolute zero, the atoms of each k i n d must occupy completely

ordered p o s i t i o n s . In most m a t e r i a l however, the changes i n c r y s t a l

s t r u c t u r e are so slow at low temperatures that d i s o r d e r i s " f r o z e n " i n and

a constant r e s i d u a l entropy remains. The l a r g e zero p o i n t energy and high

m o b i l i t y of atoms i n s o l i d helium mixtures permits a r e - o r d e r i n g of the

c r y s t a l when i t becomes e n e r g e t i c a l l y p r e f e r a b l e . 3 4

S o l i d He and s o l i d He are completely m i s c i b l e i n one another

above 0.38 K. The u s u a l method of d e s c r i b i n g a thermodynamic property

A (x,V,T) of a mixture i n terms of the p r o p e r t i e s of the pure substances

A^ (V,T) and A^ (V,T) f o r a molar c o n c e n t r a t i o n of He at a temperature T i s ; A (x 3,V,T) = x 3 A 3 (V.T) + (1 - x 3 ) A 4(V,T) + A ^ x ^ V . T ) ,

where A^,(x 3 ,V,T) i s the excess property. Above about 1.4 K the p r o p e r t i e s of He

mixtures along the m e l t i n g curve can be w e l l d escribed by l i n e a r i n t e r ­

p o l a t i o n because any A , i s s m a l l . At lower temperatures the f r e e z i n g curve

f o r mixtures drops below the f r e e z i n g curve f o r e i t h e r pure component f o r

those concentrations that have been measured ( x 3 = 1% to 85%). The hcp-bcc

t r a n s i t i o n i s a l s o c o n c e n t r a t i o n dependent. The estimated hcp-bcc

t r a n s i t i o n (Le P a i r et a l . , 1965) f o r 33 atmospheres i s shown i n f i g u r e 2.

No experimental data are a v a i l a b l e below 1 K f o r t h i s t r a n s i t i o n .

3

Below 0.38 K, and pressures up to at l e a s t 60 atmospheres, He

and He^ are no longer m i s c i b l e at a l l concentrations and phase s e p a r a t i o n

-5-

i n t o two phases takes place (Edwards et a l . , 1962; Zimmerman, 1965).

Each of the phases becomes i n c r e a s i n g l y r i c h i n one of the two isotopes

as the temperature i s decreased. The T - X diagram f o r s o l i d helium mix­

tures at 33 atm. ( f i g u r e 2) showing the e q u i l i b r i u m concentrations of the

two phases i s very s i m i l a r to that of a r e g u l a r binary s o l u t i o n but

t h i s agreement i s c o i n c i d e n t a l . Volume e f f e c t s are ignored to a f i r s t

approximation i n r e g u l a r s o l u t i o n theory; whereas i n the case of s o l i d

helium mixtures, volume e f f e c t s are the dominant f a c t o r . A l a r g e d i f f e r e n c e

i n molar volumes of the two helium isotopes a r i s e s from a l a r g e d i f f e r e n c e

i n t h e i r zero p o i n t energies. The e f f e c t of the molar volume d i f f e r e n c e

w i l l be discussed l a t e r . F i g ure 2 i n c l u d e s a datum from t h i s work as w e l l

as data from work by Panczyk, S c r i b n e r , Gonanc, and Adams (Panczyk et a l . ,

1968; Adams et a l . , 1969) and Edwards, McWilliams, and Daunt (EMD) (Edwards

et a l . , 1962).

I t can be seen from the T - X diagram t h a t at pressures on the

order of 30 atm. one phase w i l l have an hep c r y s t a l s t r u c t u r e and the

other phase bec s t r u c t u r e below about 0.3 K.

The experiments described i n t h i s t h e s i s were undertaken to

determine the average s i z e s of the phase separated regions i n a s o l i d helium

mixture. We expected that phonons would be s c a t t e r e d at the boundaries 3 4 between He r i c h regions and He r i c h r e g i o n s . I f t h i s phonon s c a t t e r i n g

d i d take p l a c e , we expected to observe a change i n the temperature depen­

dence of the thermal c o n d u c t i v i t y .

We w i l l d e s c r i b e the thermodynamics and k i n e t i c s of phase separa­

t i o n i n Chapter I I . The theory of thermal c o n d u c t i v i t y and r e s u l t s of

previous c o n d u c t i v i t y measurements i n s o l i d helium w i l l be presented i n

FIGURE 2 : T - X phase diagram of solid helium mixtures at P % 33 atm.

It should be .noted that the references Panczyk et al.,1968 and Adams et al.,1969 describe results of the same, experiments.

-7-

Chapter III . The experimental apparatus and procedures w i l l be described in

Chapter IV and the experimental results w i l l be discussed in Chapter V.

-8-

' CHAPTER I I

PHASE SEPARATION

A. THERMODYNAMICS OF REGULAR SOLUTIONS

A s o l i d composed of a mixture of two d i f f e r e n t types of atoms

d i f f e r s from a pure s o l i d i n the f o l l o w i n g manner (Guggenheim, 1952):

The i n t e r n a l energy U of the s o l i d must take i n t o account the

i n t e r a c t i o n energy of u n l i k e neighbouring atoms. I f we l e t x denote the

mole f r a c t i o n of component A; and l e t the s u b s c r i p t A i n d i c a t e component

A atoms and the s u b s c r i p t B i n d i c a t e component B atoms, then ( i f k i n e t i c energy can be neglected)

U = - | N [ x E ^ + (1 - x ) E B B + x ( l - x) AE]

where N = Avogadro's Number

and E..= the i n t e r a c t i o n energy between an atom of type i and i t s nearest

neighbours, a l l of type j .

A E = 2 EAB - EAA " EBB

I f AE = 0, the mixture i s considered to be an i d e a l s o l u t i o n and

no phase s e p a r a t i o n w i l l occur. The i n t e r n a l energy i s the same f o r a l l

c o n f i g u r a t i o n s of the atoms.

I f AE > 0, a l a t t i c e c o n t a i n i n g u n l i k e neighbouring p a i r s of

atoms i s not e n e r g e t i c a l l y favoured and the s e p a r a t i o n tends to c r e a t e

regions that are i n c r e a s i n g l y pure i n one type of atom as the temperature

decreases. There i s no long range order.

-9-

If AE < Oj a l a t t i c e containing unlike neighbouring pairs is

energetically favoured. Then, in a 50% mixture for example, an alternating

( A - B - A - B - A ) long range order would tend to exist at 0 K.

3 4

Solid helium mixtures separate into regions rich in He and He

respectively and AE/k = 1.5 K (Edwards et a l . , 1962).

For a regular solution AE is independent of concentration and

temperature. In addition, the excess volume V (x) is usually small and is E

neglected in the following description of a regular solution.

The excess Gibbs free energy per mole G (x,T) of the system (ignoring E

phonon energy and entropy which are negligibly small) i s :

GE(x,T) = U E + PVE - T S M * u E - x S M

The configuration entropy S^ i s : S = _ Nk [x ln x + (1 - x) l n ( l - x)]. M

The excess internal energy is U = x ( l - x)NAE/2 and the phase separation E

temperature for x = 0.5 is = AE/4k.

Figure 3 shows the value of G (x,T) for various temperatures.

It can be seen from the figure that at temperatures below T £ there

are two minima in the free energy curve at concentrations that we w i l l

c a l l x , and (the equilibrium concentrations). If a mixture of

concentration between x , and x^ (say x = .5) is cooled rapidly to

T = .8 T c i t is energetically favourable for i t to phase separate and

the most favoured concentration for the two regions are x , and x^.

FIGURE 3 : Phase separation parameters for regular

solution theory.

- l i ­

l t should be noted for a complete description that lying

between the minima there are two inflexion points (spinodal points) where 2 2 2 2 8 G /9x = 0. Between these two points d G/dx is negative and diffusion E

w i l l take place up concentration gradients. A homogeneous mixture that is

cooled rapidly into this region w i l l i n i t i a l l y phase separate by "spinodal

decomposition" (Cahn, 1968). In the experiments described in this thesis

we have not been able to distinguish between i n i t i a l phase separation by

spinodal decomposition and by nucleation (described in section C of this

chapter).

and the locus of the spinodal points (spinodal line) are also shown in

figure 3.

The equation of the phase separation line is found by minimizing

G (x,T) and is

The locus of the equilibrium points (the phase separation line)

T 4T (1/2 -c x) ps

The equation of the spinodal curve is given by

T = 4T x ( l - x).

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B. THERMODYNAMICS OF SOLID MIXTURES

S p e c i f i c heat measurements of s o l i d He mixtures by.Edwards,

McWilliams & Daunt (1962), shown i n f i g u r e 4, i n d i c a t e d that a phase

s e p a r a t i o n occurred w i t h T^ = 0.38 K. The phase s e p a r a t i o n temperatures

T^, and s p e c i f i c heats C agreed w e l l w i t h the p r e d i c t i o n of r e g u l a r

s o l u t i o n theory w i t h the appropriate value of T . The envelope of the

s p e c i f i c heats below T p r e d i c t e d by t h i s theory i s ; ps 2

r - 'QL = ( 1 / 2 ~ X 3 ^ Nk C dT

t t x 3 ( l - x 3 ) 1 ]

where

t = kT/AE = T/4T . c

This good agreement w i t h r e g u l a r s o l u t i o n theory was r a t h e r

s u r p r i s i n g s i n c e the two regions would have d i f f e r e n t c r y s t a l s t r u c t u r e

at the pressures i n v o l v e d (30 to 40 atmospheres).

The microscopic theory of helium phase se p a r a t i o n was done by

M u l l i n (1968), who g e n e r a l i z e d the Nosanow (1966) theory of the pure

phases of s o l i d helium. M u l l i n found that the s o l i d helium s o l u t i o n

should be n e a r l y r e g u l a r , but f o r d i f f e r e n t reasons than the c l a s s i c a l

model.

Using a Hartree t r i a l wave f u n c t i o n made up of s i n g l e p a r t i c l e

wave f u n c t i o n s modified by a term to account f o r c o r r e l a t i o n s i n a

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.05 .1 .2 .3 .4 .5 .7 1.0 T ( !<)

FIGURE 4 : S p e c i f i c heat of s o l i d helium mixtures i n the

phase s e p a r a t i o n r e g i o n (Edwards et a l . , 1962). 4

Note that the i n d i c a t e d concentrations are He

concentrations x..

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v a r i a t i o n a l c a l c u l a t i o n , M u l l i n found that f o r helium mixtures

G E(x 3,T,P) x 3 ( l - x 3 ) [a(P) - x 3 b(P)] - TS M(x)

where a ^ 8.0 j/mole; b ^ 0.8 j/mole; at P = 36 atm.

The x 3 b ( P ) term causes an asymmetry i n the phase s e p a r a t i o n

curve. The c o e f f i c i e n t s a(P) and b(P) are c o n t r i b u t i o n s from s e v e r a l

e f f e c t s .

a(P) = c - d - e - f and b(P) = g - h

where f = -PV / x . ( l - x„) f o r excess volume V„, and e = AE(P)N/2. I f E J J • Li

t h

NE_ and are ground s t a t e energy and volume per mole of the i atoms

AE 3 = E 3 ( x 3 , P ) - E 3(1,P) % ( c ( l - x 3 ) - g x 3 ( l - x 3 ) ) / N

AE 4 = E 4 ( x 3 , P ) - E 4(0,P) % (-dx 3 - h x 32 ) / N

The only c o n t r i b u t i o n i n a r e g u l a r s o l u t i o n would be e s i n c e

one assumes V 3(0,P) = V 4(0,P) and AE 3 = AE 4 = 0. In the case of helium

mixtures, e i s very s m a l l . The important f e a t u r e of helium mixtures i s

that V 3(0,P) and V 4(0,P) are very d i f f e r e n t because of the d i f f e r e n t magni­

tudes of the zero p o i n t energy. In a mixture, the volume each atom f i n d s

i t s e l f i n i s somewhat d i f f e r e n t from the volume i n a pure substance, so

E 3 and E 4 are c o n c e n t r a t i o n dependent. M u l l i n p r e d i c t e d the value T = 0.47 K and a V_ of the form r c E

3 PV„ = - f x _ ( l - x.) where f = 0.4 P cm /mole. E J J

-15-

Panczyk et a l . (1968) measured the change in pressure of solid

samples in constant volume chambers. The excess pressure determined by

this method can be related to the excess volume. The excess volume

results and the observed asymmetry in the phase separation curve were in

good agreement with Mullin's theory.

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C. KINETICS OF PHASE SEPARATION

Measurements of properties in the phase separation region are

done after growing a crystal, annealing and then cooling below T ps

Depending on the pressure and concentration, the sample may have either

a bec or hep crystal structure or a mixture of both. When phase separa­

tion takes place the two stable phases form in some domain structure.

We w i l l assume that domains rich in the dilute isotope form in a matrix

of the more abundant isotope. At sufficiently low temperatures an hep-bec

transition w i l l take place in either the domains or the matrix. Finally

to minimize the total surface energy of the domains, the system w i l l tend

to the condition of one domain. These three processes have very different

time constants.

The experiments of Edwards et a l . and Panczyk et a l . indicated

that time constants x of the order of tens to hundreds of seconds were

associated with the phase separation ordering process. The time

constants observed by Edwards et a l . were independent of temperature and

concentration, but increased with increasing pressure. Edwards et a l .

assumed that quantum mechanical tunnelling was the diffusion mechanism. 3

Using He spin diffusion data (Reich, 1963), they estimated a diffusion

distance of 10 microns; this was the size of the pores in the sintered

copper sponge that they used for a sample chamber.

Adams et a l . (1969) observed time constants that were dependent

on temperature, concentration and pressure as well as whether cooling or

warming was taking place. These data are l i s t e d in Table 1.

-17-

Table 1 : Phase separation time constants (from Adams et a l . , 1969).

x_ P T T T 3 PS

.485 30.5 atm. .38 K .38 K 120 sec.

.35 78

.18 36

38.5 500

.084 33.5 .31 .25 140

.996 33.8 .14 .13 385

The trends observed were: increasing x as temperature increased

with longest times near the phase separation curve; increasing x as one

l e f t = .5 in either direction; increasing x as pressure increased and

larger x on cooling than on warming. The sample chamber used by Adams

et a l . had a width of 0.8 mm indicating a maximum thermal time

constant of about 1 sec.

4 Guyer and Zane (1970) predicted that in dilute mixtures of He

3 4 in He , He atoms can move through the la t t i c e by a tunnelling process i n

3

which they change places with He atoms. The estimated jump frequency f ^

is of the same magnitude as the exchange frequency in spin diffusion

0 10 Hz). A diffusion coefficient that is temperature independent would

be expected from a tunnelling process. The observed jump frequencies in

spin diffusion are pressure dependent but not sufficiently so to explain

the pressure dependence observed by Adams et a l .

-18-

Greenberg et a l . (1971) measured the spin relaxation times 3 4

and of He impurities in solid He crystals. They deduced tunnelling

frequencies that had a much stronger pressure dependence than the spin exchange

frequency. They found f = 4 x 10^ sec for P = 25 atm. and 4 -1

f^^ = 2 x 10 sec at P = 42 atmospheres.

Adams et a l . found the time constant for the crystallographic

change was much longer than the time constant for phase separation. The

measurements were made rapidly enough that the decrease in pressure due to

the transition from bec to hep or the increase due to the hep - bec

transition was small compared to the pressure changes due to phase separa­

tion.

The clustering process that takes place during phase separation

requires a net movement of matter in the crystal. The key aspect i s that,

on the average, atoms w i l l diffuse in such a way as to decrease the total

free energy of the solid. Three distinct processes are successively

involved (Fine, 1964). Stable nuclei w i l l form from fluctuations which

reach a c r i t i c a l size. The probability of such a c r i t i c a l fluctuation i s

exp(-AG /kT) where AG is the energy of formation of a c r i t i c a l nucleus, c c

Further growth w i l l lower the total free energy. The nuclei grow, at a

rate controlled by the diffusion of solute, u n t i l the concentration of the

solute in the matrix approaches i t s equilibrium value. Assuming spherical

domains, there w i l l then be a distribution of domain sizes with an average

radius Y • Since the surface free energy per unit volume of a domain

-19-

i s i n v e r s e l y p r o p o r t i o n a l to i t s r a d i u s , the l a r g e domains w i l l now

grow by the d i s s o l u t i o n of smaller g r a i n s . During t h i s t h i r d stage

(coalescence or coarsening) the average domain r a d i u s r as a f u n c t i o n of

time t f o r a d i f f u s i o n c o e f f i c i e n t D i s given by ( L i f s h i t z and Slezov, 1961):

r 3 - r 3 = 8DV x ot/9RT o m e

where a i s the su r f a c e f r e e energy per u n i t area, V i s the molar volume of m

the m a t r i x , x g i s the s u p e r s a t u r a t i o n of the m a t r i x at the s t a r t of

coarsening, and R i s the gas constant. We w i l l estimate a using the r e g u l a r

s o l u t i o n r e s u l t f o r a l i q u i d (Cahn and H i l l i a r d , 1958):

. * 1.7 . iO" 7 c i e - T ) / v 2 / 3 iiE2|££l 2 /3 m cm K

For T_ - T = 0.1 K and = 23 cm 3/mole, a & 2 x 10~ 9 J/cm 2.

We w i l l estimate r ( t ) f o r a s o l i d helium mixture u s i n g the approx-

imate parameters, D % 3 x 10 cm /sec and x^ % 0.01. Then, f o r r >> r Q

dr = 2 x 10 ^ / r 2 cm/sec. dt

— -4 'b

The time taken f o r coarsening to r = 10 cm i s t ^ 100 days. This i s only

an order of magnitude r e s u l t s i n c e the s e l e c t i o n of the value of x g was

a r b i t r a r y . In an i s o t r o p i c medium that cannot support shear the t o t a l

f r e e energy w i l l be a minimum f o r s p h e r i c a l domains. L i f s h i t z and Slezov

show that the s i z e d i s t r i b u t i o n of the domains, a f t e r coarsening has pro­

ceeded f o r some time, w i l l be sha r p l y peaked at r w i t h a c u t - o f f r a d i u s

at 1.5 r .

-20-

CHAPTER III

THERMAL CONDUCTIVITY

A. HISTORY

Heat conductivity was f i r s t defined by Fourier (1822) whose

object was "to set forth the mathematical laws which heat obeys". Fourier

defined thermal conductivity by considering a homogeneous solid enclosed

between two para l l e l and i n f i n i t e planes whose temperatures v were held

at constant values. Then,

"If F i s the quantity of heat which, during a unit of time, passes across a unit of area of the surface taken on a section parallel to the base, F = - K dv/dx. We have taken this coefficient K, which enters into the equation, to be the measure of the specific conductivity of each substance; this number has very different values for different* bodies»"

Fourier's analysis described the conduction of heat given the

boundary conditions and the various coefficients. His work did not attempt

to explain the mechanism of conductivity.

y The earliest experiments were done with long rods which were

heated at one end and cooled by radiation. In the experiments of Ingen-Hausz

(1789) the rods were coated with a thin layer of wax and therefore the

temperature was known only at the heated end and at the point where the wax

melted. This experiment was suggested by Benjamin Franklin. Despretz

(1827) determined temperatures by placing thermometers in holes equally

spaced along the rod. Using this technique, Wiedemann and Franz (1854)

determined empirically that at a given temperature the ratio of the thermal

and e l e c t r i c a l conductivities is the same for a l l metals. Lorentz (1872)

-21-

determined that this rat io i s proportional to the absolute temperature.

Very elegant experiments on the conductivity of dielectr ics

were done by de Senarmont (1847). He cut thin slabs with para l le l sides

from single crystals , coated the surfaces with a thin layer of wax and

applied heat to one point. At equilibrium, the anisotropy of the

conductivity was indicated by the shape of the edge of the molten wax.

The shape was a c i r c l e for isotropic materials such as glass or cubic

crystals , independent of the direction of the cut. However, for other

crystals the shape was e l l i p t i c a l . By. cutting the slabs in various

crystal directions de Senarmont constructed the e l l i p s o i d a l isotherms for

a crystal heated at one internal point. It was found that these e l l i p -

showed that by assuming that heat was conserved the problem of conduction

in anisotropic bodies could be.reduced to the corresponding problem

concerning isotropic bodies.

Debye (1914) attempted to explain conduction in dielectr ics by

assuming a model of heat transfer by propagation of elast ic waves i n a

continuous medium. These waves would be scattered by density variations

in the sol id that are produced by the whole set of waves.

Pauli (1925) showed that the atomic nature of the solid must be

considered when the scattering centres moved with the same order of

velocity as the waves themselves. Peierls (1929) did the f i r s t rigorous

V -22-

calculation of mutual scattering of quantized la t t i ce vibrational waves

(phonons) and showed that the thermal conductivity of dielec t r ics

increases as exp (0/bT) with decreasing temperature. Peierls showed

that two types of phonon-phonon scattering processes (normal processes

and Umklapp or flop-over processes) could occur. The normal processes

do not change the direction of energy flow whereas the Umklapp processes

change the direction of energy flow to any one of several equivalent

crystallographic directions.

-23-

B. THEORY OF THERMAL CONDUCTIVITY

If one were able to construct a perfect, i n f i n i t e , dielectric

crystal whose interatomic potentials were harmonic, then phonons created

at the heat source would travel without scattering to the heat sink.

The conductivity K of real materials i s limited by the effects of various

phonon scattering processes.

If we consider a small imaginary volume in a solid, at equi­

librium the number of phonons of frequency to is given by the equilibirum

Planck distribution N q = [exp (fito/kT)- 1] \ In the presence of a heat

current Q = -KVT, changes (-VVN) in N (the phonon population distribution

at time t) w i l l occur because of transport of phonons of group velocity v

through the walls of the volume, because of the temnp.rature gradient.

There w i l l also be population changes (8N/9t) due to phonon collisions c

with other phonons, impurities and various crystal imperfections. In the

steady state, the total time rate of change of the population must be

zero.

(3N/9t) + v" • VN = 0 c

This Boltzmann equation has not been rigorously solved.

(9N/St) c can be approximated by assuming that in the absence of a heat

current any deviation in the population from N^ damps out exponentially

with a relaxation time T. In the presence of a heat current, resistive

scattering processes tend to return the distribution to N q whereas normal

processes (which are not i n themselves resistive) lead to a displaced

Planck distribution N(£). In an isotropic medium 5 is a vector in the

direction of the heat current.

- 2 4 -

Callaway (1959) i n c l u d e d the e f f e c t of normal processes i n the

c a l c u l a t i o n by the approximation:

(m) = N ( T ) - N - N - N —o

where x i s the r e l a x a t i o n time f o r normal s c a t t e r i n g processes and x i s

the t o t a l r e l a x a t i o n time f o r the r e s i s t i v e s c a t t e r i n g processes. Callaway

c a l c u l a t e d the c o n d u c t i v i t y f o r an i s o t r o p i c Debye s o l i d ,

K = v 2

2 v-/

x c

1 - B x

N J C(k) k 2 dk

where — = — , B = - gT/nv 2 VT, X X X C N R

C(k) i s the phenen s p e c i f i c h<??t: ?r>r! k if? the ohotion wave irninher. The

constant g i s evaluated using the f a c t that the r a t e of change of the

t o t a l phonon momentum due to normal processes i s zero. The r e s u l t i n g

i n t e g r a l s can be evaluated approximately i f one r e s i s t i v e process i s

dominant. I f s e v e r a l r e s i s t i v e processes of n e a r l y equal importance are

present the i n t e g r a l s must be evaluated n u m e r i c a l l y . Some of the

s c a t t e r i n g processes are des c r i b e d below.

Normal S c a t t e r i n g

These are phonon-phonon s c a t t e r i n g processes that conserve

c r y s t a l momentum. At low temperatures we need only consider three-phonon

processes. Normal processes are not r e s i s t i v e ; however, they r e d i s t r i b u t e

energy throughout the phonon modes. I t can be seen i n Callaway's theory

-25-

that as normal processes become slower compared to defect scattering the

conductivity rises because the most strongly scattered phonon modes are

depleted of phonons.

Herring (1954) showed that elastic anisotropy has a drastic

effect on the c o l l i s i o n probabilities of phonon modes of very low frequency.

Well below the Debye temperature the relaxation for these processes was

calculated to be

where a depends on crystal symmetry and phonon polarization. At high

temperatures the relaxation rates become less temperature dependent. The

normal processes relaxation time i s rather d i f f i c u l t to determine experi­

mentally. The best estimates are from conductivity results in che pnonon

Poiseuille flow region.

Umklapp Processes

These are three-phonon processes which do cause a thermal

resistance because they change the direction of energy flow. Peierls (1929)

showed that the Umklapp relaxation time is given by

1/T u = B u T n exp [6/ b T]

where b i s a constant (b 2).

This behaviour was verified experimentally by Berman et a l .

(1951). The exponential factor i s so strong that the choice of n is not

significant. At the temperature of the experiments described in this

thesis, Umklapp processes w i l l not be important.

-26-

Bouridary S c a t t e r i n g

At low enough temperatures i n pure c r y s t a l s the mean f r e e path X

becomes comparable to the dimensions of the c r y s t a l and the thermal

c o n d u c t i v i t y becomes dependent on the.sample s i z e . This e f f e c t was f i r s t

observed by de Haas and Biermasz (1938) and explained by Casimir (1938).

The r e l a x a t i o n time i s r e l a t e d to the c h a r a c t e r i s t i c dimension L and. the

sound v e l o c i t y v by the temperature and frequency independent expression

'1_ _ v = v. T b " L " A

P o i n t Defect S c a t t e r i n g

The s t r a i n f i e l d s r e s u l t i n g from p o i n t defects give a r e l a x a t i o n

time p r o p o r t i o n a l to co 4 . Carruthers (1961) suggests that i t i s not

c o r r e c t to consider t h i s to be e s s e n t i a l l y Pv.ayleigh s c a t t e r i n g because

the s t r a i n f i e l d s are not l o c a l i z e d .

We w i l l be concerned w i t h the case i n which the p o i n t defect i s

an i s o t o p e . Pomeranchuk (1942) f i r s t p ointed out that such a defect

would cause thermal r e s i s t a n c e . Klemens (1955) derived the f o l l o w i n g

r e s u l t assuming that the s c a t t e r i n g i s due to mass d i f f e r e n c e s o n l y .

1 = A co4 = VEx. (AM./M)2co4

K . 1 1 TM

th where x^ i s the mole f r a c t i o n of the i i s o t o p e , of atomic mass M^, M = Zx.M. and AM. = M. - M. . 1 1 i i l

-27-

For the case x^ << x^ Ziman (1956) obtained a limiting conduc­

t i v i t y for mass defect scattering by a variational calculation. This

Ziman limit places a lower bound on the conductivity. The restriction

T\, < < : T., indicates that the result i s only valid for small defect N M J

3

concentrations. For helium mixtures with He concentration x^ using a

Debye model (units for V are cm /mole). K = 10" 7 6 2 ( 4 - x 0 ) 2 W/cm K. . z 3

V 1 / 3 x 3 ( l - x 3)

Berman et a l . (1959) extended the variational calculation to

cover the whole range of concentrations by including the effect of normal

processes on the thermal conductivity. In the region where only defect

and normal processes are impuj." LauL, Cue tiieo-al cuiid uC t i v i ty was found to -1/2

have a concentration dependence of approximately [x (1 - x)] instead of [x (1 - x) ] This is i n agreement with calculations using Callaway's

theory. The thermal conductivity for a Debye model of solid helium with 3

He concentration, x 3 in the intermediate concentration region is given by

(Berman, 1965 b) 2/3

K = 1.28 x 10~7 6^ ( 4 ~ X3* T~ 3 / 2 _W_ Y x 3 ( l - x3)-L/2 cmK

The Grlineisen constant y is used in the theory as an estimate

of the anharmonicity of the l a t t i c e and has been used as a single adjust­

able parameter to f i t experimental data.

3 Klemens and Maradudin (1961) showed that when a He isotope is

4 placed in a l a t t i c e of He atoms the la t t i c e i s distorted outward by

-28-

several percent. Klemens (1967) extended- the. calculation to include the

change in the local bulk modulus due to the changed zero point energy.

The effect of this l a t t i c e distortion is to enhance the defect scattering.

The point defect relaxation time is then calculated to be

where V,M = average molar volume and atomic mass

AV,AM = differences for the solids of pure isotopes at the same

pressure

Y i s a parameter which depends on the elastic constants of the solid.

The theory for intermediate concentrations can be corrected for

the effect of l a t t i c e strain by reducing the calculated conductivity by 1 I?

the factor (A/A^) ' .

Kinetic Theory Approximation

In order to use Callaway's analysis profitably one must have a

reasonable estimate of the normal process relaxation time. Most authors

have treated this as an adjustable parameter to obtain the best f i t to

their data. The only direct estimate of x^ can be made from thermal

conductivity or second sound measurements in which Poiseuille flow is

present. Poiseuille flow has been observed in pure, single crystals of

solid helium i n the temperature region where the only resistive process

is boundary scattering and x < x . The phonons w i l l random walk to the

-29-

crystal boundaries before a resist ive scattering takes place. The

effective mean free path is larger than the crystal diameter and is

inversely proportional to x . (See Guyer and Krumhansl (1966)). The

Poiseuil le flow measurements have a l l been made at much higher pressures

than the work presented in this thesis, leaving us with the alternative

of estimating x^ by curve f i t t i n g . Our data do not give us suff ic ient

information to do such a curve f i t t i n g . In the region below the phase

separation the phonon scattering was observed to be frequency independent

so we w i l l analyze the data quali tat ively using the kinetic approximation:

K = CvA/3 = Cv 2 x /3 .

-30-

C. PREVIOUS EXPERIMENTAL RESULTS IN SOLID HELIUM

3 4 Pure He and He

The f i r s t e x tensive measurements of the thermal c o n d u c t i v i t y 4

of s o l i d He were done by Webb, W i l k i n s o n , and Wilks (1952). They found

that the phonon mean f r e e path v a r i e d e x p o n e n t i a l l y w i t h 1/T f o r samples

grown at pressures between 50 and 150 atmospheres. This was the Umklapp

s c a t t e r i n g behaviour p r e d i c t e d by P e i e r l s . Subsequent measurements by

Walker and Fairbank (1960 b) agreed w i t h the above. Bertman et a l . (1966 b)

extended the measurements to 0.5 K and found that i n the hep phase of

both s o l i d s the thermal c o n d u c t i v i t y showed a maximum. Above the maximum

the c o n d u c t i v i t y v a r i e d e x p o n e n t i a l l y w i t h 1/T over three orders of

magnitude. Below the maximum they -found the c o n d u c t i v i t y to be c h a r a c t e r -3

i s t i c of boundary s c a t t e r i n g w i t h a T dependence and a phonon mean f r e e

path comparable to the sample diameter. A l l the above measurements were

done on c r y s t a l s grown at constant volume using the blocked c a p i l l a r y

method.

Measurements by Mezhov-Deglin (1964) and Hogan et a l . (1969) on

c r y s t a l s grown at constant pressure showed an enhancement of the thermal

c o n d u c t i v i t y peak and T 7 dependence below the peak as would be expected

from P o i s e u i l l e flow of phonons (Guyer and Krumhansl, 1966). 3

Measurements on bec He have shown an anomalously low c o n d u c t i ­

v i t y above the maximum (Walker and Fairbank, 1960 b ) . Thomlinson (1969)

measured the c o n d u c t i v i t y between 0.1 K and the m e l t i n g temperature and

-31-

found that the r e s u l t s above the maximum could not be i n t e r p r e t e d i n terms

of exp (6/bT). P l o t s of l o g K as a f u n c t i o n of 1/T gave a n o n l i n e a r

r e s u l t that could not be normalized w i t h respect to 8. At temperatures 2.5

w e l l below the maximum Thomlinson observed a T ' temperature dependence 3.5

which i n c r e a s e d to T * j u s t below the peak (suggesting that P o i s e u i l l e 2 5

flow of phonons was p r e s e n t ) . The T " dependence was c o n s i s t e n t w i t h

anomalous s p e c i f i c heat r e s u l t s obtained by Sample and Swenson (1967) and

al s o by Pandorf and Edwards (1968). Greywall and Munarin (1970) i n f e r a

very low sound v e l o c i t y i n the (101) d i r e c t i o n from measurements on o r i e n t e d 3

s i n g l e c r y s t a l s at V = 21.6 cm /mole. A l l these measurements suggest an 3

anomaly i n the bcc He phonon spectrum.

Helium Mixtures 4

Walker and Fairbank (1960) found that the c o n d u c t i v i t y of He was 3

reduced by adding He . Callaway (1961) analyzed t h e i r experimental r e s u l t s

and found that he could o b t a i n good agreement w i t h h i s theory by assuming

that the p o i n t defect s c a t t e r i n g parameter was three times stronger than

the parameter due to mass d i f f e r e n c e s c a t t e r i n g only. Klemens and Maradudin 3

(1961) showed that a 1% - 2% l a t t i c e d i s t o r t i o n about the He l a t t i c e s i t e s

would account f o r the l a r g e s c a t t e r i n g parameter. Bertman et a l . (1966) measured the c o n d u c t i v i t y of e i g h t d i f f e r -

3 ent concentrations at molar volumes of 19.5 and 20.2 cm /mole. They analyzed t h e i r r e s u l t s using the Callaway theory and a normal process

3 2 r e l a x a t i o n time 1/T m = B T W w i t h a d i f f e r e n t value of B f o r each

-32-

concentration. B^ was used as an adjustable parameter i n f i t t i n g the data.

They concluded that the point defect scattering depended on concentration

more strongly than x ( l - x) .

Berman et a l . (1965) measured the conductivity of a number of

mixtures grown at constant volume from the same starting pressure. This

gave a similar value of B^ for the group of mixtures. They analyzed their

2 2 results using 1/T . t = B„ T OJ . B„, was held fixed for each starting N N N

pressure and was determined by f i t t i n g the data using Callaway's theory.

They found that the point defect scattering could be explained using both

mass difference and la t t i ce s train contributions. The magnitude of the

l a t t i c e strain effect was smaller than the theoretical value but i t s

pressure variation was as predicted. Subsequent measurements by Berman

et a l . (1968) extended the measurements to starting pressures of 17Q0

atmospheres. The la t t i ce distort ion results could be f i t ted by

A_ = 1 + Y fAV] 2 f i l l 2

Air L v j LAMJ

with a value of Y = 3.7 as compared to the theoretical value of Y % 16.

Conductivity measurements by Berman et a l . (1965) indicated

that point defect scattering effects in bec mixtures and hep mixtures are s imilar . Values of (A/A ) % 2 were necessary to account for strain f i e l d

K enhancement in bec mixtures.

-33-

CHAPTER IV

APPARATUS

A. DILUTION REFRIGERATOR

The temperatures r e q u i r e d f o r these experiments were obtained

us i n g a d i l u t i o n r e f r i g e r a t o r (Wheatley e t a l . , 1968) w i t h two s i n t e r e d

copper heat exchangers.

The c r y o s t a t c o n s i s t e d of a g l a s s l i q u i d n i t r o g e n dewar, a

10 cm. i . d . g l a s s l i q u i d helium dewar and a metal vacuum can tha t contained 3

a 90 cm 1 Kh e l i u m pot and the d i l u t i o n r e f r i g e r a t o r . Figure 5 i l l u s t r a t e s

the c r y o s t a t and the r e f r i g e r a t o r .

Room Temperature Parts

3

The r e f r i g e r a t o r used 4 l i t e r s of helium gas (about 20% He )

which was st o r e d i n two 20 l i t e r c o n t a i n e r s when the r e f r i g e r a t o r was not

i n use. Gas was r e c i r c u l a t e d using an Edwards 2M4A mercury d i f f u s i o n 3

pump and a Welch 1402KBG mechanical pump s p e c i a l l y sealed f o r use w i t h He . The gas mixture was cleaned by a l i q u i d n i t r o g e n cooled trap c o n t a i n i n g

3 about 100 cm of Linde molecular s i e v e .

To reduce mechanical v i b r a t i o n i n the c r y o s t a t , the plumbing was

i s o l a t e d from the c r y o s t a t by bellows tees and the mechanical pump was

f u r t h e r i s o l a t e d by another set of bellows tees. The plumbing p l a t f o r m

was b o l t e d to the cement l a b o r a t o r y f l o o r and a l s o weighted by 100 pounds

of sand. >

I TD - tunnel diode circuit H - heat sink for electrical

leads

Vacuum can (8.75 cm o.d.)

1 K heat shield

C - condensor

Z^, - pressure drop capillaries

S^ - s t i l l heat exchanger

A, B - stainless steel mechanical support tubes

X^, s'intered copper heat exchangers

S - sample chamber

R - sample resistance ther­mometers

P o v a T ^ n i i v r> t - c * 5^ 11 V ^ T T V . ... \ • T 7 ...... _-_ _ _. ._ _.

copper can containing CMN crystal

high pressure.capillary

nylon thread

FIGURE 5 : Schematic diagram of low temperature apparatus.

-35-

The cryostat was supported by a w e l l braced wooden stand which

was weighted with 200 pounds of sand.

Low Temperature Parts

3

The r e c i r c u l a t i n g He was cooled i n copper heat exchangers at

80 K and 4 K, then condensed i n a sin t e r e d copper condenser at 1 K. The

pressure i n the condenser was kept above 30 mm Hg by a flow l i m i t i n g

impedance between the condenser and the s t i l l heat exchanger. The

impedance consisted of a short length of .015 cm i . d . c a p i l l a r y containing

a t i g h t f i t t i n g constantan wire. The impedance was tested at room temper­

ature and cut to the required length. The s t i l l heat exchanger consisted

of 2.9 metres of .080 cm o.d. s t a i n l e s s s t e e l c a p i l l a r y wound on a copper

spool that was bolted to the bottom of the s t i l l . A layer of copper wire

was wound between each layer of c a p i l l a r y and bonded with G.E. 7031 varnish.

Each s i n t e r e d copper heat exchanger was constructed by s i n t e r i n g

copper plugs i n t o two p a r a l l e l holes i n a s o l i d copper block. The i n t e r ­

connecting c a p i l l a r i e s were s o f t soldered to flanged end plugs that were

bolted to the copper blocks. Indium wire "0" rings provided a vacuum s e a l .

The upper h a l f of the copper mixing chamber was s i l v e r soldered

to a 0.19 cm thick copper p l a t e that served as an isothermal platform for

a t t a c h i n g experiments, thermometers and a heater. The lower h a l f was a

copper cup containing sintered copper. The two halves were flanged for an

indium "0" r i n g s e a l and bolted together.

-36-

The s t i l l was machined from a solid copper block and superfluid

film flow up the pumping line was suppressed by a .10 cm diameter electro-

polished o r i f i c e . The s t i l l had two indium "0" ring seals.

There were a total of twelve indium seals in the refrigerator.

The indium wire was li g h t l y greased before the seal was made. The only

leaks encountered took place at flanges that were slightly warped due to

silver soldering. Resurfacing the flanges solved the problem.

Mechanical support for the refrigerator was provided by three

equally spaced .635 cm o.d. stainless steel tubes soldered to the 1 K pot

and the s t i l l plate and bolted to the mixing chamber. Heat leak down these

mechanical supports limited the refrigerator to a lowest temperature of

Oft -i-.V <!*vma f" ! ? i i - ! . - . - . 1 *-.» ^ J ^ J - - ...f i i i. - „ . n J \ J *_ ^ j - ^ - • - j - i . x ^ ^ ^ w ^ j - w i i v- M ^ b * 4 ~ u_ ^ ^.I^LJ i w i o i _ u . _ L . i - t t u u b C i j j i_/Vv c i _- L / i .

0.5 mW were 8 yW at .12 K, 34 yW at 0.20 K, 70 yW at 0.30 K, and 186 yW

at 0.43 K.

-37-

B . THERMOMETRY

3 The primary thermometer was a He vapour pressure thermometer

with i t s bulb attached to the mixing chamber plate. The pressure was

measured on a butyl phthalate (C^g 0^) o i l manometer using a Cenco

cathetometer. The manometer readings were corrected for density changes

due to room temperature changes using the results of Johnson (1968). The 3

He temperatures were determined from the 1962 vapour pressure scale of Sydoriak and Sherman (1964). In a separate experiment the mixing chamber

4 plate was connected to the i K p o t by a copper rod and the 1 K pot He vapour pressure was determined by both a butyl phthalate and a mercury ~~ 4 4 manometer. The He temperatures were determined using the T^g He vapour

3 4

pressure scale (van Dijk et a l . , 1960). The He and He temperatures were

found to agree within a mil l i k e l v i n bet ween 1.1 K and 1.6 K. The primary thermometer below IK was a self inductance cerium

3

magnesium nitrate thermometer that was calibrated against the He temperature

between 0.5 K and 1.6 K. The secondary resistance thermometers were calibra­

ted using the CMN thermometer.

The vali d i t y of the low temperature scale was checked by measuring

the thermal conductivity of the stainless steel sample chamber and also by 4

measuring the conductivity of solid He .

CMN Thermometry

The primary thermometer for the temperature region below 1 K was a

cerium magnesium nitrate (CMN) magnetic susceptibility thermometer. (See

Betts et a l . , 1964).

-38-

A cylinder (1 cm diameter, 2 cm long) was cut from a single

crystal and placed in the inductance c o i l of the parallel resonant c i r c u i t

of a tunnel diode oscillator. The resonant frequency (about 1.5 MHz) was

measured by a General Radio 1191 counter.

The inductance of the c o i l depends upon the magnetic susceptibility

of the material i t contains. If L q is the inductance of the empty c o i l ,

then the inductance of the c o i l f i l l e d to a fraction n of i t s volume with a

material, whose susceptibility is x = X"* ~ ix""» is given by

L = L Q ( 1 + 4TTTJX')«

The resonant frequency f of a f i l l e d c o i l is related to the frequency of the

1 = 1 (1 + 4TTT1X').

The susceptibility of a paramagnetic material is

X'= C . T + G

c

CMN was chosen because i t s Curie temperature 8^ is very small and

Curie's Law (x = C/T) can be assumed to well below the temperature range of

this experiment (see Daniels and Robinson, 1953). It also has a very small

specific heat permitting very short equilibrium times. Because of the

anisotropy of the susceptibility the cylinder was cut from the crystal so

that the trigonal axis was perpendicular to the axis of the cylinder.

- 3 9 -

I f 2TTf T t >> 1 and 2nfx << 1 where T t i s the s p i n l a t t i c e L s L r e l a x a t i o n time and t g i s the s p i n - s p i n r e l a x a t i o n time we can assume

( f o r s m a l l e x t e r n a l f i e l d s ) that y" = X where y i s the i s o t h e r m a l o o

s u s c e p t i b i l i t y . These requirements are met by CMN at a frequency of about

1 MHz. I f the departure of the frequency of the f i l l e d c o i l from the

frequency of the empty c o i l i s s m a l l then we can use the approximate r e s u l t

1/T = A ( l - f_) f o

where

A = 1/2-TnC w i t h a t y p i c a l value of A = 391.7 K - 1

C = Curie constant

n = f i l l i n g f a c t o r £ .5.

The second order terms t h a t have been neglected i n t r o d u c e an

e r r o r i n T of about 0.2% over the temperature range of the c a l i b r a t i o n .

The values of A and f were obtained by a l e a s t squares f i t to 3

c a l i b r a t i o n s of f against the He vapour pressure temperature T. The

standard d e v i a t i o n of the f i t t e d parameter A was t y p i c a l l y 0.1%. I t

should be noted that f i s a property of the o s c i l l a t o r and places s t r i n g e n t

requirements on i t s s t a b i l i t y . The l a r g e s t s h i f t i n f observed i n a run

was 50 Hz. The o s c i l l a t o r was t y p i c a l l y s t a b l e to 10 Hz over a pe r i o d of

s e v e r a l weeks corresponding to a s t a b i l i t y of 3 ml at 1 K and .03 mK at 0.1 K.

Changes i n f during a c a l i b r a t i o n of the r e s i s t a n c e thermometers (4 hours)

were n e g l i g i b l e .

-40-

The tunnel diode o s c i l l a t o r c i r c u i t i s shown i n f i g u r e 6.

The b a t t e r y and b i a s i n g r e s i s t o r s (R^ and R^) were at room temperature.

The i s o l a t i n g c a p a c i t o r C^, the s t a b i l i z i n g c a p a c i t o r and the diode were

mounted on a c i r c u i t board mounted on the i n s i d e of the vacuum can at 4 K.

The c o i l and tank c i r c u i t c a p a c i t o r were on the mixing chamber.

The tunnel diode was a General E l e c t r i c BD-6 back diode w i t h a

t y p i c a l peak cu r r e n t of 5 uA and a peak v o l t a g e of 50 mV. When biased

between the peak (y 50 mV) and the v a l l e y 200 mV) the diode acts as a

negative r e s i s t a n c e .

Two requirements must be met i f the device i s to o s c i l l a t e .

F i r s t the r e s i s t a n c e i n the c o i l and i t s leads must be low enough so that 4-t-~ _ ~ J - - - t 4-„ * n 1 ^ ~ . TUA ~ ,•„ A T 1 ~ 4-~~ t-- J A— L - i i w A . ~~ ^ i . w ~ w i _ — . — ~ O ' • * " O " 1 - — . ' - - — . • ' — —

the f o l l o w i n g e q u i v a l e n t c i r c u i t . R i s the p a r a l l e l e q u i v a l e n t of the i-j

tank c i r c u i t r e s i s t a n c e and -R, i s the r e s i s t a n c e of the diode (which i s d

n e g a t i v e ) .

-41-

r

! 1.5 V

General Radio 1191

frequency counter

9 stage Tschebyscheff

f i l t e r

Shasta 854A

amplifier

•

1 K Hz , high pass

f i i t e r

<

Shasta 854A

amplifier

r

L

Bias control Room temperature

H-P Moseley 7100 B

chart recorder

Hewlett-Packard 5 62A

digital/analog convertor

mixing mixing chamber

temperature controller

VTVM

r "

10 pF

H h

103 pF

at A K

i

heater

CMK crystal

" ^ l o 3 ^ 1 P F L = 11 yH JL

on mixing chamber

J

FIGURE 6 : Tunnel diode osc i l la tor thermometer and temperature controller

c i r c u i t .

-42-

The second requirement f o r o s c i l l a t i o n places r e s t r i c t i o n s on the r e s i s t o r s

i n the b i a s i n g c i r c u i t . The source r e s i s t a n c e must be l e s s than the

absolute value of the negative diode r e s i s t a n c e to prevent s w i t c h i n g . For

d e t a i l s of d e s i g n , see Reybey (1962) or Boghosian et a l . (1966).

The f i r s t requirement se t s a lower l i m i t on the Q of the c o i l .

The c o i l c o n s i s t e d of 70 turns of #36 AWG w i r e wound on the CMN c y l i n d e r so

that i t s ends were e q u i d i s t a n t from the ends of the c y l i n d e r . The w i r e was

bonded w i t h GE 7031 v a r n i s h . The c y l i n d e r was placed i n a vacuum can i n

such a way that about .5 cm of the lower end of the c r y s t a l was i n a t i g h t

f i t t i n g h o l e i n the copper base that had been p r e v i o u s l y f i l l e d w i t h GE 7031

v a r n i s h . The i n s i d e of the s t a i n l e s s s t e e l w a l l s of the can were coated

w i t h s o f t s o l d e r . The superconducting s o l d e r reduced w a l l l o s s e s that

would degrade the Q of the c o i l . A brass end cap c o n t a i n i n g a two lea d

e l e c t r i c a l s e a l and a copper f i l l i n g c a p i l l a r y was s o f t soldered to the 4

s t a i n l e s s s t e e l tube. The can was f i l l e d w i t h one atmosphere of He gas

at room temperature, the f i l l i n g tube was pinched o f f and sealed w i t h s o f t

s o l d e r . The helium f i l m (about 5 microns t h i c k ) would h o p e f u l l y provide

good thermal contact at low temperatures. The tank c i r c u i t c a p a c i t o r was .

attached to the R.F. leads at the top of the can.

I t was found that the temperature i n d i c a t e d by the CMN thermometer

(which would be the s p i n temperature averaged over the volume of the c o i l )

was dependent on the s i g n a l amplitude V g of the o s c i l l a t o r (measured across

Cy)' Taking the mixing chamber temperature to be the CMN temperature

e x t r a p o l a t e d to V = 0,the d i f f e r e n c e between the CMN temperature and the

-43-

mixing chamber temperature was found to be

Because the Q of the c i r c u i t i s unknown i t i s not p o s s i b l e to

give an absolute value f o r the constant B. For R.F. heating one would 2

expect a heat i n p u t r a t e Q = - cox^'h^ (Casimir, 1940) where h Q i s the

peak to peak amplitude of the magnetic f i e l d of frequency co. AT could be

kept below 1 mK by s e t t i n g the b i a s very near the peak v o l t a g e . The s i g n a l

across was then about 0.2 mV.

In the i n i t i a l CMN thermometer that we constructed the tank

c i r c u i t c a p a c i t o r was mounted on the 4K p l a t f o r m w i t h the tunnel diode and

"as contact?'?, to tb? c o i l hv a superconductor — c^at-pA r ? p i 11 gry. I t was

found that R.F. h e a t i n g of the sample r e s i s t o r s took place because of the

l a r g e c u r r e n t i n the l e a d . P l a c i n g the c a p a c i t o r on the CMN c o i l can

reduce these c u r r e n t s by the f a c t o r Q and no measurable h e a t i n g of the

r e s i s t o r s occurred.

The CMN thermometer w i l l not give r e p r o d u c i b l e r e s u l t s i f

f a c t o r s other than temperature a f f e c t the measured resonant frequency. At

1 K a change of 1 mK causes a change of 3.7 Hz i n the resonant frequency.

The c i r c u i t had a s t a b i l i t y of 1 Hz over an hour i n t e r v a l (except f o r

o c c a s i o n a l noise pulses which w i l l be discussed i n the s e c t i o n on the

temperature c o n t r o l l e r ) and a long term s t a b i l i t y of 10 Hz over a p e r i o d of

weeks.

- 4 4 -

The resonant c i r c u i t was i s o l a t e d from the b i a s i n g c i r c u i t and

the frequency measuring c i r c u i t by the c a p a c i t o r C^. The s i g n a l across 4

(about 0.2 mV) was a m p l i f i e d by 10 by two Shasta 854A wide band a m p l i f i e r s .

The r e s u l t a n t s i g n a l was f i l t e r e d by a 9 stage h i g h pass Tschebyscheff

f i l t e r and the frequency was measured by a General Radio 1191 counter. The

f i l t e r ( H a n s e l l , 1969) had a cut o f f frequency of 1 MHz and an a t t e n u a t i o n

of 1,000 at 500 KHz.

Resistance Thermometry

Speer 470 carbon r e s i s t o r s were used as r e s i s t a n c e thermom­

e t e r s i n these experiments. The r e s i s t a n c e s were measured by a 318 Hz

Wheatstone b r i d g e u s i n g a P r i n c e t o n JB-6 l o c k - i n a m p l i f i e r as a s i g n a l

source and d e t e c t o r . Using the two phase c a p a b i l i t y of the JB-6, the b r i d g e

was balanced r e s i s t i v e l y and r e a c t i v e l y during each measurement.

The output reference s i g n a l was reduced by a v o l t a g e d i v i d e r and

i s o l a t e d from the bri d g e by a Hammond 585 D transformer. This combination

reduced the reference v o l t a g e to a maximum of 13 mV across the b r i d g e . In

f a c t , the l a r g e s t reference v o l t a g e used was 2.5 mV. The e r r o r s i g n a l

passed through a T r i a d G 10 i s o l a t i o n transformer, was a m p l i f i e d by a f a c t o r 3

of 10 by a P r i n c e t o n CR4 a m p l i f i e r and was detected by the JB-6. General

Radio decade r e s i s t a n c e s and capacitances were used f o r b a l a n c i n g the b r i d g e .

The bridge noise l e v e l was reduced to a low l e v e l (0.1 uV peak to

peak r e f e r r e d to the inp u t of the CR4) by s h i e l d i n g i n such a way that the

ground w i r e of the bridge was connected to the s h i e l d at only one p o i n t and

the s h i e l d was grounded at only one p o i n t . The r e s i s t o r s were not grounded

to the c r y o s t a t . Severe he a t i n g of the r e s i s t o r s by t h e r m o e l e c t r i c e f f e c t s was

observed when one end of the r e s i s t o r had been grounded. This had a l s o

-45-

caused d r i f t s in the cal ibrat ion as the room temperature varied.

The R.F. noise heating of the resistors was reduced by placing

.022 uF polystyrene capacitors i n p a r a l l e l with the resis tors . These

capacitors were mounted on the mixing chamber plate. The use of these

large capacitors prevented us from doing AR measurements.

When resistance measurements were made, the reference voltage

was reduced u n t i l no self heating of the resistors was observed. At 0.2 K

this gave a power dissipation of 10 ^ watts. Under these condition, the 4

bridge resolution was about 1 part i n 2 x 10 .

A l l resistors except those on the sample chamber were greased into

snug holes d r i l l e d i n the element whose temperature was to be measured.

The sample resistors were prepared by grinding off about 1/3 of the

insulation down to the graphite core. A //38 AWG copper wire was soldered

to each p i g t a i l , coiled in a layer over the resistor body and bonded with

GE 7031 varnish. A third wire was coiled and bonded and i t s ends were

soft soldered to the element making thermal contact with the sample chamber.

The copper coi ls provided thermal contact, fixed stray capacitance and pro- .

vided shielding.

The sample resistors were calibrated i n the following manner. 4

The sample chamber was f i l l e d with sol id He to provide good thermal

contact between the resis tors . The temperature of the mixing chamber was

held constant u n t i l no resistance change was evident, then the resistances

and the CMN frequency were measured. The measured resistances were f i t t e d

- 4 6 -

to a polynomial by a l e a s t squares f i t (LQF) program.

1/2 + P R = A Cl - f_) = 1/ T

o where R =.resistance

f = CMN frequency

c a l i b r a t i o n constants of CMN thermometer

P^, P^* P^ = constants that give the best f i t to the c a l i b r a t i o n data.

About 30 c a l i b r a t i o n p o i n t s were taken between 0.1 K and 0.8 K

at equal i n t e r v a l s i n 1/T. Tables of R(T) were then c a l c u l a t e d f o r each

thermometer using the above f i t t e d polynomial. The computer program gave

the d i f f e r e n c e , f o r each c a l i b r a t i o n p o i n t , between the measured temperature

and the temperature c a l c u l a t e d using the above equation. Because of the

r e s t r i c t e d number of parameters the f i t t e d curve d i d not e x a c t l y f o l l o w the

measured R(T) curve, however the d i f f e r e n c e was u s u a l l y l e s s than 2 mK.

For a given measured temperature the c a l c u l a t e d temperatures of the sample

r e s i s t o r s u s u a l l y agreed to 0.1 mK. Measurements between the c a l i b r a t i o n

points- were c o r r e c t e d by l i n e a r i n t e r p o l a t i o n between the d i f f e r e n c e s which

v a r i e d s y s t e m a t i c a l l y over the c a l i b r a t i o n range w i t h a t y p i c a l change of

0.3 mK between c a l i b r a t i o n p o i n t s . During the longest run (5 weeks) the

c a l i b r a t i o n s changed by l e s s than 0.1 mK.

-47-

C. MIXING CHAMBER TEMPERATURE CONTROLLER

The large specific heat and low thermal conductivity of solid

helium mixtures below the phase separation temperature caused long thermal

time constants i n the conductivity sample. To do accurate conductivity

measurements we had to keep the mixing chamber temperature constant for

periods up to six hours. The refrigeration capacity of the dilution

refrigerator varies with temperature so we could control the temperature

by varying the power input to an el e c t r i c a l heater bolted to the mixing

chamber platform. We designed and constructed an analog device to provide

this control automatically.

The voltage analog of the mixing chamber temperature was compared

to the voltage analog of the desired temperature and the airrerence signal

was used to increase or decrease the power dissipated in the heater. The

ci r c u i t diagram of the controller is shown in figure 7.

The. resonant frequency of the CMN thermometer was measured by a

General Radio 1191 Frequency Counter. The binary coded decimal output of

the counter was converted to a DC voltage (0 to 100 mV) by a Hewlett-Packard

562A d i g i t a l to analog convertor. The output voltage of the convertor was

V = 100 A mV B

where A is a two or three digit number

B = 100 or 1000.

Bias Spare Temp. set

-o + 9 V T

to V T V M °

to D/A convertor

i co i

to heater

FIGURE 7: Mixing chamber temperature controller c i r c u i t .

-49-

We could select any three digits or the last two digits from

the counter display for conversion to a DC voltage. The maximum available

sensitivity was 1 mV/cycle. The 0.1 mV/cycle sensitivity was normally

used.

Three additional potentiometer controlled voltages were provided;

a temperature setting reference voltage (0 to 100 mV), a heater bias

voltage (two ranges, 100 mV and 1 V f u l l scale) and a spare (two ranges).

To maintain a desired temperature we set the empirically determined

voltage on the bias potentiometer and a voltage on the temperature reference

equal to the desired output voltage of the D/A convertor. The spare poten­

tiometer was used for controlling the heater power manually or to replace

one of the other potentiometers while changing temperatures.

The above voltages were applied to the input resistors of the

Philbrick 1001 Operational Amplifier which was chosen for i t s low offset

d r i f t . If voltages and are applied to the inverting input through

resistors R/k^ and R/k^; and and are applied through R/k^ and R/k^ to

the non-inverting input then the output voltage V q i s

V = k. V. + k, V. - k. V. - k. V. o 4 4 3 3 2 2 1 1

i f k^ + k^ = k^ + k^.

The f i r s t input was grounded to satisfy the second equation. For

control at low temperatures a l l the input resistors had the same value. At

high temperatures increased gain was necessary tc compensate for the

decrease in thermometer sensitivity so the temperature and reference

-50-

temperature input r e s i s t a n c e s were reduced by a f a c t o r of 10.

The method of changing temperatures i s s t r a i g h t forward. To

reduce temperature one turns o f f the c o n t r o l l e r and allows the mixing

chamber to c o o l . The b i a s v o l t a g e i s adjusted to provide heater power

approximately equal to the r e f r i g e r a t i o n c a p a c i t y at the new temperature.

The c o n t r o l l e r i s turned on when the CMN frequency i s i n the d e s i r e d range

and the temperature becomes steady w i t h i n 2 minutes. To go to a higher

temperature one grounds input r e s i s t o r and increases the b i a s v o l t a g e .

When the d e s i r e d frequency range i s reached R^ i s switched back to the D/A

output. The c o n t r o l l e r reduces the time f o r going from one steady temper­

ature to another by an order of magnitude compared to the time taken using

manual heater v o l t a g e c o n t r o l .

Below 0.4 K the temperature could be c o n t r o l l e d to ± 0.1 mK f o r

per i o d s of many hours. When the c o n t r o l l e r was d i s t u r b e d by a noise pulse

i t returned to steady s t a t e w i t h a time constant of about 1 second. This

corresponded to the sampling p e r i o d of the frequency counter. I f good

s t a b i l i t y was to be obtained, such noise pulses (which are due to e x t e r n a l

sources) had to be removed from the s i g n a l to the counter. Normal s h i e l d i n g

and grounding p r a c t i c e s were f o l l o w e d , w i t h i n the c o n s t r a i n t s of the

equipment a v a i l a b l e . Most noise pulses were found to have F o u r i e r

components up to about 500 KHz so i t was necessary to construct a f i l t e r

w i t h a very sharp r o l l o f f i f 1 MHz s i g n a l s were to pass unattenuated. A

nine stage Tschebyscheff f i l t e r ( H a n s e l l 1969), was s e l e c t e d as the best

cho i c e . I t had a c u t - o f f frequency of 1 MHz and an a t t e n u a t i o n of 1000 at

-51-

500 KHz. The cir c u i t diagram is shown below.

R , R = 1 K

C 1 S C 9 = 106 pF

C 3, C 5, C ? = 66 pF

To eliminate modulation of the signal by noise pulses in the

transformer.

We w i l l summarize the differences between this temperature con­

t r o l l e r and two previous types of controllers. The simplest method of

temperature control i s to manually adjust the electrical power input to a

heater mounted on the mixing chamber to maintain a steady temperature. The

experimenter is a feed back element in the system and he must make frequent

adjustments in power input to correct for variations in refrigeration power.

The disadvantages of such a controller in a long experiment are obvious.

The second type of controller uses a resistance thermometer as the sensing

element. The measured resistance is compared to the desired resistance in

an AC Wheatstone bridge and the out-off balance signal i s used in a feed­

back c i r c u i t to change the power input to the heater. To do a thermal

L 2, L g = 117 pH

L 4, L 6 = 107 pH

- 5 2 -

c o n d u c t i v i t y experiment at these low temperatures with the above c o n t r o l l e r

one would need two AC resistance bridges i n add i t i o n to a magnetic s u s c e p t i ­

b i l i t y thermometer. The novelty of the method described i n t h i s thesis

l i e s i n the use of the magnetic thermometer as the temperature sensing

element. The main advantage i s the e l i m i n a t i o n of one AC resi s t a n c e bridge

which saves money and space.

-53-

D. SAMPLE CHAMBER

In order to do accurate and reasonably rapid thermal conductivity

measurements we needed to construct a sample chamber which met the

following requirements.

- a b i l i t y to withstand internal pressures of up to 100 atmospheres.

- low thermal conductivity in parallel with the sample.

- heat flow parallel to the sample axis only, in the v i c i n i t y of the

thermometers.

- a thermal time constant as short as possible.

- good thermal contact between the sample ends and the heater and the

heat sink.

- thermometer separation and.temperature gradients large enough to permit

temperature differences large compared to temperature resolution.

- a b i l i t y to measure sample pressure.

These requirements were not a l l compatible so a compromise was

necessary. Two sample chambers were constructed as shown in the figure

8. The second was constructed after the f i r s t one was accidentally

stressed beyond the tensile strength of the pressure c e l l . The sample

chambers consisted of two parts, a conductivity c e l l and a pressure c e l l .

These w i l l be discussed separately below.

Thermal Conductivity Cell

This c e l l consisted of a stainless steel tube (.318 cm o.d. x

.015 cm wall), an e l e c t r i c a l heater and several thermometer connections.

The most important requirements of this c e l l were low thermal conductivity

-54-

i n the c e l l walls and isotherms in the sol id sample which were plane and

perpendicular to the cylinder axis. The . f i rs t requirement was met by the

stainless steel tube. The largest correction for conduction in the walls

was 25% of the total conductivity.

The overall length of the conductivity c e l l , L, was restricted

by the thermal time constant, x, for temperature changes

2 A T % A T q exp (-t/x) x = 4 L PC

° IT K

where p = density of helium sample

C = specif ic heat of sample

K = thermal conductivity of sample.

At temperatures below the phase separation the specif ic heat is

very large. The resulting time constant i s as high as 30 minutes for

L = 2 cm. Since waiting times of 8 to 9 time constants were necessary to

establish "steady state" conditions a small value of L was desired.

The requirement for plane perpendicular isotherms meant that i n

the v i c i n i t y of the thermometers the c e l l wall and adjacent sol id helium

had to be at the same temperature. Near the heat sink and the heater, the

walls are at a higher temperature than the adjacent helium because of the

Kapitza resistance. This temperature difference 0 decreases exponentially,

as one goes away from the ends, with a characteristic length a.

6 = 6 exp (-x/a) where x = distance to end.

-55-

- Stycast 2850 GT epoxy

nylon spacer heater

.033 cm d i a . copper wires

\

heater on brass plug

copper brus

T . 5 cm

i 3 fins 0.5 cm

spacing

N

;,=. dummy strain gauge on copper base

K strain gauge on thin wall brass tube

l 1 •

Itr #1 #2

FIGURE 8 : Sample chambers

f

-56-

For the geometry and samples i n t h i s work a 4 1 mm. To minimize

6 i n the v i c i n i t y of the thermometers the. temperature d i f f e r e n c e at the

ends, 6 , should be s m a l l and the d i s t a n c e between the thermometers and o

the ends should be l a r g e . This d i s t a n c e was about 5 mm i n both sample

chambers. The geometry at the ends of the..conductivity c e l l prevented plane

isotherms i n those r e g i o n s . A c a l c u l a t i o n u s i n g a s u i t a b l e model i n d i c a t e d

that d i s t o r t i o n s due to the end geometry would. decrease as exp (-Trr/x)

where r i s the r a d i u s of the tube and x i s the d i s t a n c e from the end. E r r o r

due to t h i s e f f e c t would be n e g l i g i b l e at the thermometers.

The temperature d i f f e r e n c e at the ends was minimized by p r o v i d i n g

l a r g e contact areas between the sample and sample chamber. The K a p i t z a -3 A 2

boundary r e s i s t a n c e between s o l i d helium and copper i s % 80 T K cm /Watt

( P o l l a c k , 1969).

The f i r s t heater was placed i n s i d e the s t a i n l e s s s t e e l tube. I t

c o n s i s t e d of 320 cm of constantan w i r e (.005 cm d i a . ) b i f i l a r l y wound on two

s t a i n l e s s s t e e l wires that a l s o served as e l e c t r i c a l leads through the

pressure s e a l at the end of the s t a i n l e s s s t e e l . t u b e . The heater was non-2

i n d u c t i v e , had a r e s i s t a n c e of 230 ft, and a surface area of 0.5 cm . The

s t a i n l e s s s t e e l . w i r e s passed through holes d r i l l e d i n a nylon spacer that

was a t i g h t f i t to the i n s i d e of the tube. The end of the s t a i n l e s s s t e e l

tube was degreased, cleaned w i t h a h y d r o c h l o r i c - n i t r i c a c i d s o l u t i o n and

then sealed w i t h Stycast 2850 GT epoxy (Anderson, 1968). The S t y c a s t epoxy

i s loaded w i t h quartz to match the thermal c o n t r a c t i o n of copper. The s e a l

was i n i t i a l l y t e s t e d by r a p i d c y c l i n g between room temperature and 80 K w h i l e

-57-

pressurized to 50 atmospheres with helium gas. No leaks were detected by

a Veeco MS90 Mass spectrometer leak detector when the sample chamber was 4

f i l l e d with superfluid He at 25 atmospheres. Assuming the above value of

Kapitza resistance the temperature difference between the heater wire and

the sample would be 20 mK/yW at 0.2 K.

The second heater was constructed by s i lver soldering a brush of

several hundred bare copper wires.(.012 cm dia . ) into a hole d r i l l e d i n a

brass plug. To prevent solder from entering the part of the brush that

extended into the conductivity c e l l , the solder was fed into the opposite

end of the hole. The plug was then soft soldered into the end of the

stainless steel tube. A b i f i l a r heater (240 Q) of constantan wire (.005 cm

dia . ) was wound on the plug and bonded with GE 7031 varnish. Uncertainty

i n the position of the s i lver solder boundary l imits us to determining a

2

lower l i m i t (2 cm ) for the surface area of the brush and an upper l imit

(5 mK/yW) for the temperature difference between the heater and the sample

at 0.2 K.

The two thermometers on the f i r s t conductivity c e l l were attached

to two copper wires (.033 cm dia . ) that passed through two sets of holes

d r i l l e d 1 cm apart i n the stainless steel tube. The wires passed through

the centre of the tube and were perpendicular to the tube axis to within 2 ° .

The solder f i l l e t s at the holes were reduced by f i l i n g away excess solder.

The maximum temperature.difference between the wire and the sample due to

e l e c t r i c a l heating of the thermometers was estimated to be 30 yK at 0.2 K.

This arrangement was not satisfactory because of poor mechanical support for

-58-

the resistors . Severe vibrational heating resulted and i t was necessary

to construct a frame and grid of nylon threads to support the resis tors .

The second sample chamber had three brass fins (.025 cm thick)

s i l v e r soldered to the stainless steel tube with 5 mm separations. The

f i l l e t s were reduced to .040 cm wide by f i l i n g away the excess solder. The

third wire of each resistance thermometer was soft soldered to a f i n and

the resistor was bonded to the f i n by vacuum grease.

The lower end of the conductivity c e l l was soft soldered to the

copper base of the sample chamber. The e l e c t r i c a l leads from the heater

were. 4 cm long constantan wires (.01 cm dia . ) and the thermometer leads

were 4 cm long constantan (.005 cm d i a . ) . A l l leads were thermally anchored

to the copper base as well as at the lead heat sink on the mixing chamber

platform. The thermometer leads were also anchored to the f i n s .

Pressure C e l l

Thephase separation time constants observed.by Edwards et a l .

(1962) and Adams et a l . (1962) were pressure dependent. This indicated that

the domain size might be pressure dependent. The phase diagram of helium

mixtures i s not well enough known to permit accurate prediction of the sample

pressure from the freezing pressure. It was therefore desirable to measure

the pressure of the sol id sample.

A thin wall (.005 cm) brass sleeve (.635 cm o.d.) was soft soldered

to the base of the sample chamber. The annulus between the sleeve and the

base was connected to the central hole by six equally spaced radial holes of

-59-

0.1 cm diameter. A constantan f i l m , s t r a i n gauge.. (Budd, type C6-141 ~ 1000)

was glued to the brass -using Eastman 910 contact cement. The s t r a i n gauge

had a gauge f a c t o r , g = > of 2.1 and a nominal room temperature

r e s i s t a n c e of 1000 fi. An I d e n t i c a l gauge was glued to a copper surface i n

contact w i t h the mixing chamber. The d i f f e r e n c e between the two r e s i s t a n c e s

was measured by a Wheatstone br i d g e . The s t r a i n gauge s e n s i t i v i t y was

.05 ft/atm. p e r m i t t i n g a r e s o l u t i o n of .1 atmospheres. The gauge was c a l i ­

b rated against the e x t e r n a l Heise pressure gauge, when helium gas was

condensed i n t o the sample chamber i n p r e p a r a t i o n f o r c r y s t a l growing.

C o r r e c t i o n s were made f o r the.temperature dependence of AR, which changed

by .08 9, between 2.5 K and 0.1 K. The pressure measurements, however, were

not as accurate as the above r e s o l u t i o n would i n d i c a t e . The s t r a i n gauge on

the f i r s t ores sure c e l l fwhif-h had ? . 05 cm arm..1 us". Inst a f a c t o r of 5 i n

s e n s i t i v i t y between the 10th and 11th c y c l e from room temperature. The

pressures measured by t h i s gauge ( a l l samples reported except x^ = 0.1) are

t h e r e f o r e r e p o r t e d to ± 0.5 atm. The pressure c a l i b r a t i o n s f o r these samples

were done at 0.5 K immediately a f t e r the samples were melted by reducing the

e x t e r n a l pressure. The u n c e r t a i n t y i n the pressure of the samples i n the

second chamber were a l s o 0.5 atm. This pressure c e l l had a .02 cm annulus

and i t was found that pressure d i f f e r e n c e s of up to 8 atmospheres could

e x i s t between the pressure c e l l and the c o n d u c t i v i t y c e l l . The pressure

i n d i c a t e d by the s t r a i n gauge would drop on the f i r s t anneal a f t e r c r y s t a l

growth. The c r y s t a l was a l t e r n a t e l y cooled and annealed u n t i l the i n d i c a t e d

pressure became constant. The r e s u l t i n g m e l t i n g p o i n t s agreed w i t h the data

of Le P a i r et a l . (1965). I t i s r a t h e r s u r p r i s i n g t h a t such l a r g e inhomogen-

i t i e s can be maintained. Other authors have used a n n u l i of s i m i l a r

-60-

dimensions (Tedrow and Lee, 1969) or smaller (Edwards et a l . , 1962), and

have not reported inhomogenities. The interconnecting holes were observed

to be free of obstruction before the heater plug was soldered onto the

stainless steel tube.

The annulus and radial holes served as a heat sink of variable

^effective contact area between.the solid helium and the copper base.

Consider a model D f the heat sink in which a copper rod of diameter r^

extends into a cylinder of radius containing the helium sample.

assume 8T 8x

= 0 Jx = L

8T 8x x = 0

- 5 _ KA

K = sample conductivity

R = Kapitza resistance . 2 A = irr„

Assume that in an element of sample between x and x + dx

Q J = Q , (to next helium element) + Q (to copper) In h c

If we define t = ~ and t is small compared to x^, then

82T - ( T " Tc) 3x a

= 0 a = KRt

T c = temperature of copper base

-61-

3 -3

Since K a T and R a T approximately, then a is nearly constant

with respect to temperature. Then 9 = T - i s : • 1 - -

0 - Q a cosh L/g ( L) 2 rt K sinh L/g

If the temperature gradient in the sample i s to be 10 mK/cm then

6 (x = 0) % 3.5 mK independent of temperature for the f i r s t sample chamber.

Some care was required with the pressure c e l l because the calculated

collapsing pressure was 1.1 atmospheres and the calculated room temperature

bursting pressure was 50 atm.

The base of the sample chamber was threaded.and bolted to the

mixing chamber plate. A thin layer of Apieson "M" grease was used for thermal

Sample Heater Circuit

The heat dissipated i n the sample heater was determined by a four

lead measurement. The current was determined by measuring the voltage drop

across a General Radio 100 ft.. (± .025%) resis tor . The voltage across the

heater was measured by potential leads connected to the current carrying leads

at the mixing chamber. Half the power dissipated in the leads between the

mixing chamber and the heater (1.5% of the heater power) was assumed to go

to the heater. The leads were not grounded in the cryostat. No detectable

thermal EMF's were present. A l l voltages were measured by a Leeds and

Northrup 8662 potentiometer whose calibration had been checked against another

precision potentiometer.

-62-

The heater was shunted by a 0.1 yF polystyrene c a p a c i t o r mounted

on the mixing chamber p l a t f o r m .

Heater powers i n these experiments, on helium ranged from 10 7 watts

to 5 x 10 ^ watts. The background heat leak to the heater was determined by

measurements on the empty sample chamber and was found to vary between -9 -9

2 x 10 watts and 3 x 10 watts. The l a r g e s t u n c e r t a i n t y i n the heater power was t h e r e f o r e 1%.

Sample Gas Handling System

I t was necessary to prepare helium mixtures of known concentrations

and apply pressures up to 60 atmospheres to the mixture i n the sample

chamber.

3 A system w i t h two g l a s s t o e p l e r pumps of 400 cm volume, a s t a i n -

3

l e s s s t e e l t o e p l e r pump of 500 cm volume and ass o c i a t e d vacuum equipment

and storage f l a s k s was constructed (Walker and Fairbank, 1960). The volumes

and pressures i n the g l a s s t o e p l e r pumps could be determined to 0.1% of f u l l

s c a l e .

A f t e r the mixtures were prepared they were passed through a Linde

molecular s i e v e trap at 80 K to remove i m p u r i t i e s before being introduced

i n t o the s t e e l t o e p l e r pump. This was u s u a l l y done when the sample chamber

was below 2 K so that a minimal amount of gas was l e f t i n the g l a s s part of

the system.

The s t e e l t o e p l e r pump was connected to the sample chamber by a

c a p i l l a r y which passed through a l i q u i d n i t r o g e n trap before e n t e r i n g the

-63-

c r y o s t a t . The l a r g e bore c a p i l l a r y C.081 cm o.d. x .015 cm w a l l ) was

connected to a s m a l l bore c a p i l l a r y (.030 cm o.d. x .010 cm w a l l ) at 4 K.

This c a p i l l a r y was th e r m a l l y anchored to the I K pot, to the s t i l l , to each

heat exchanger and the mixing chamber before e n t e r i n g the bottom of the

sample chamber.

The s t e e l t o e p l e r pump was p r e s s u r i z e d by n i t r o g e n from a commer­

c i a l h i g h pressure c y l i n d e r . The pressure on the n i t r o g e n s i d e of the pump

was measured to an accuracy of ± 0.1 atm. by a Heise Bourdon gauge. The

mercury l e v e l on the helium s i d e was determined by d e t e c t i n g the p o s i t i o n

of an i r o n b a l l f l o a t i n g on the mercury, using a magnetic s u s c e p t i b i l i t y

d e v i c e . This device (Vignos and Fairbank, 1966) permitted the mercury l e v e l

to be determined to w i t h i n 1 mm.

4 The He gas was obtained from the l a b o r a t o r y 25 l i t r e l i q u i d

3 4

helium dewars. The He gas (0.1% He ) was purchased from Monsanto Chemical

Cor p o r a t i o n .

-64-

E. -CRYSTAL'GROWTH

C r y s t a l s were grown by the blocked c a p i l l a r y method. The 1 K pot

and the s t i l l were kept s e v e r a l hundred m i l l i k e l v i n s warmer than the f r e e z i n g

temperature f o r the d e s i r e d sample pressure w h i l e the pressure on the sample

was being i n c r e a s e d . The temperature of the mixing chamber was hel d 50 to

100 mK above the f r e e z i n g pressure once the d e s i r e d pressure was reached,

and was kept constant u n t i l the upper sample.thermometer had cooled to w i t h i n

50 mK of the mixing chamber temperatures. The 1 K pot was then cooled

r a p i d l y causing a plug of s o l i d to form, b l o c k i n g the c a p i l l a r y .

The r e f r i g e r a t o r was s t a r t e d and the mixing chamber was cooled

r a p i d l y to w e l l below the me l t i n g temperature of the f i n a l mixture to f a c i l -

- — • — — A- — — j — — o - _. . i_, • -'

c r y s t a l s were then annealed by warming u n t i l the sample j u s t s t a r t e d to

melt. A sudden increase i n the sample pressure i n d i c a t e d the s t a r t of

m e l t i n g .

Rapid c r y s t a l growth was necessary to prevent c o n c e n t r a t i o n g r a d i ­

ents from forming i n the s o l i d i n the r e g i o n where the thermal c o n d u c t i v i t y

i s measured. Growth had to be r a p i d enough to prevent d i f f u s i v e r e d i s t r i b u ­

t i o n of the two isotopes i n the l i q u i d . In t h i s l i m i t there w i l l be

l a r g e c o n c e n t r a t i o n gradients near the ends of the sample chamber and a

c e n t r a l r e g i o n w i t h a s m a l l g r a d i e n t . We w i l l d e f i n e the end regions as

those regions where the co n c e n t r a t i o n x d i f f e r s from the

-65-

i n i t i a l average concentration x Q by more than 2.5%. The approximate

solution of a rather involyed expression (Pfann, 1966) gives the following

estimate of the length L of the end regions.

L = 4D/f

where f i s the freezing rate and D is the diffusion coefficient of the

liquid.

-5 2 Using the value D = 3 x 10 cm /sec (Garwin and Reich, 1959) and

_3

a typical freezing rate of 4 x 10 cm/sec, L = .03 cm. The concentration

can therefore be considered to be constant over the length of the conducti­

vity c e l l .

The poly crystalline structure resulting from this ranld o.rowth can

then be corrected by annealing. In fact, no difference in conductivity

between annealed and unannealed samples was observed, except for a rather

special case which w i l l be discussed in the experimental results. The 10% He

samples in the second sample chamber were checked for concentration gradients

by comparing the conductivities of the two sections of the crystal in the

region above the phase separation. The results agreed within experimental

error (5%). It i s unlikely that the average concentration in the sample

chamber was different from the average gas concentration. The sample chamber

was f i l l e d rapidly through a capillary and the capillary was always blocked

in a time that was short compared to the time required for fractionation to

establish the equilibrium concentrations in the liquid and gas. Mass spec­

trometry analysis of gas samples from three stages of a similar experiment

(Berman et a l . , 1965) showed no significant difference in the three samples.

-66-

F. DATA REDUCTION AND EXPERIMENTAL ERRORS

A thermal c o n d u c t i v i t y measurement was done by supplying a measured

e l e c t r i c a l power input Q to the sample heater and measuring the e l e c t r i c a l

r e s i s t a n c e of each Speer r e s i s t o r mounted on the sample chamber. The

measured r e s i s t a n c e s were then converted i n t o temperatures u s i n g the R(T)

t a b l e s c a l c u l a t e d from the c a l i b r a t i o n polynomial of that run and c o r r e c t e d

f o r the m i s f i t of the c a l i b r a t i o n curve by l i n e a r i n t e r p o l a t i o n between c a l i ­

b r a t i o n p o i n t s . The temperature d i f f e r e n c e AT between two r e s i s t a n c e

thermometers was determined by s u b t r a c t i o n .

The average thermal c o n d u c t i v i t y K at the average temperature T

between two thermometers was c a l c u l a t e d using the equation

• K = Q L

A AT

where L i s the measured di s t a n c e between the thermometers and A i s the cross

s e c t i o n a l area of the sample c a l c u l a t e d from the measured i n s i d e diameter of

the sample tube.

The above equation i s the steady s t a t e s o l u t i o n of the heat flow

equation

V 2T = _1 9T where K = K_ K 8 t pC

f o r the flow of heat i n a homogeneous i s o t r o p i c medium w i t h d e n s i t y p,

s p e c i f i c heat C and constant thermal c o n d u c t i v i t y which i s f r e e of

sources and s i n k s . I f the thermal c o n d u c t i v i t y i s temperature dependent the

-67-

solution to the above equation for the experimental conditions with K = BT n

and temperatures and at the thermometers is

Q -=' BA CT * + 1 - T . n + X ) . (ri + 1)L 1 1

If AT s - T 2 is small compared to T E 0^ + T )/2, then using a binomial

expansion we find that

Q = KAAT [" 1 + n (n - 1) Mj_V L L 24 \ T/

In this work AT/T *; 0.1 and n -S 3, so by ignoring the second order term, we

introduce a maximum error in the calculated average conductivity K of 0.25%.

Errors in the Conductivity

Tnjp o f f o m T i f - p / " f*n c ^ f T c f v the f^teP.flv - ? f a t p requirement of the abo-i7e

equation during each measurement by keeping the mixing chamber temperature

as constant as possible and waiting u n t i l the temperature difference between

the upper and lower resistance thermometers became "constant". The temper­

ature difference was considered to be "constant" when the relative time rate

of change of AT was 0.1%/hour. In fact, i t was not possible to keep the

mixing temperature constant, but d r i f t s were always less than 0.1 mK/hr.

Errors in AT due to this d r i f t were determined experimentally to be

negligible. During measurements of time dependent effects i n the region

50 mK below the phase separation and in the region above T^g where metastable

crystallographic domains were observed we did not wait unt i l AT was "con­

stant". Data taken under these conditions w i l l be clearly identified in the

discussion of results. The error due to temperature d r i f t s in data that is

not identified as time dependent w i l l always be less than 0.2%.

- 6 8 -

The relative accuracy of AT was limited by the relat ive accuracy

of the calculated resistor temperatures. The relative uncertainty i n T for

each resistor was 0.05 mK with a resulting uncertainty in AT of ± 0.1 mK.

AT was always larger than 10 mK except i n the temperature region just above

the phase separation where AT was always greater than 5 mK. Therefore, i n

general the relat ive error i n AT due to the method of calculation was less

than 1%. The resistance thermometers were calibrated after each cycle to

room temperature and the calibrations were frequently checked during each

run by measuring AT with the heatex* turned o f f . This also permitted a stat­

i s t i c a l check on the error i n the calculated AT and the results agreed with

the above estimate of ± 0.1 mK. The error i n AT due to the Kapitza resistance

end effects i s estimated to be less than 0.1 mK in the worst case.

The power input to the heater was determined by measuring the

voltage drop across a resistor in series with the heater as well as the

voltage across the heater and the heater leads to the mixing chamber. The

—6 uncertainty in these measurements was 2 x 10 v o l t s . The largest relative

error in Q due to these measurements was 0.25%. This relative error was

-9

always less than the uncertainty in Q of 10 watts due to a time varying

background heat input. In the worst case the relat ive error due to this

uncertainty in the background was 1%.

The total relative error in K due to the combination of the above

errors i s 4.5% in the worst case, and 1.5% in the typical case. This

estimate i s substantiated by the scatter i n the data.

-69-

The sample and sample chamber walls proyide parallel paths for

heat flow. The maximum contribution of the walls was 25% of the total

conductivity. However, this contribution was less than 5% in a l l measure­

ments except for three, data points of one sample.. The conductivity measure­

ments were corrected for this effect with an uncertainty in the correction

of 5% and a typical uncertainty in K of less than 0.25%. The f i n and wire

separations on the two sample chambers were measured using a travelling

microscope and the resulting uncertainty in L was .05 mm. The lower limit

on the inside diameter of the sample chamber (nominally .287 cm) was

determined using a #33 d r i l l (.287 cm o.d.) and the uncertainty in the

diameter was estimated to be .025 mm. The error in L/A due to thermal con­

traction between room temperature and 1 K (0.3%) w i l l be neglected. The

resulting absolute error in L/A is approximately 2.5% for conductivity

measurements using the outer thermometers. The absolute error in Q is the

same as the relative error. The total estimated absolute error in K is 10%

in the worst case and 5% in a typical case. The disagreement between the

two sets of stainless steel conductivity measurements in two temperature

ranges and their disagreement with the measurements of Seward et a l . (1969)

were within the above estimated errors in K.

Errors in the Temperature Scale

A

The absolute error in the temperature scale T of this work i s

mostly due to two approximations made i n obtaining the equation

^ = A (1 - f _ ) . T f

o

-70-

Curie's law has been found to hold f o r CMN down to a temperature of 6 mK

(Daniels and Robinson, 1953) so error.due to departure from Curie's law can

be neglected. An estimate of the demagnetization co r r e c t i o n A (which we

have neglected) i s given by A. =' (4TT - D) C, where C i s the Curie constant and 3

D i s the demagnetizing f a c t o r . For a c y l i n d e r with a length/diameter r a t i o

of 2 magnetized l o n g i t u d i n a l l y D = 2.3 (Brown, 1962) and A =1.9 C = 1.5 mK. ft

I f our temperature scale i s defined as T , then the absolute temperature

T = T + A. The error due to the approximation used i n c a l c u l a t i n g the ft

d e f i n i n g equation (see Chapter IV) i s e = T - T = 0.2/T mK. These two

corrections to T are p a r t i a l l y c a n c e l l i n g , leading to an uncertainty ft

(T - T) of less than 1% i n the absolute temperature. The error i n the

temperature scale due to the previously discussed heating e f f e c t was kept

below 0.5 mK during c a l i b r a t i o n by operating the tunnel diode at a very low

s i g n a l l e v e l . The absolute error i n T due to uncertainty i n the c a l i b r a t i o n

parameters of the CMN thermometer i s 0.2%, independent of temperature. The

t o t a l absolute er r o r i n the temperature scale i s le s s than 2% i n the worst

case (T = 100 mK) and i s p r o p o r t i o n a l l y l e s s at higher temperatures (0.5% at

.5 K). The temperature dependences of the conductivity measurements of the

s t a i n l e s s s t e e l tube and the s o l i d He sample place an upper l i m i t of 3% on

err o r i n T which i s consistent with the above estimate.

-71-

CHAPTER V

EXPERIMENTAL RESULTS

Thermal conductivity measurements were done on the stainless 4

steel conductivity tube, a He crystal and on the following mixture crystals

over the temperature range 0.6 K to 0.15 K.

A x 3 = 0.900 ± .003 P = 35.5 ± .5 atm.

B 0.900 ± .003 30.0 ± .5

C 0.800 ± .006 33.5 ± .5

D 0.50 ± .015 30.0 ± 1.

E 0.103 ± .003 33.3 ± .5

F 0.103 ± .003 28.0 ± .5

The pressures are measured above the phase separation temper­

ature.

We w i l l describe the experimental results in the following order.

4 Thermal conductivities of stainless steel and pure He .

Determination of the phase separation temperatures. 3

General results for one 10% He sample.

Conductivities above the phase separation temperatures.

Conductivities below the phase separation temperatures.

Time dependent effects.

-72-

A. STAINLESS STEEL AND He

The thermal c o n d u c t i v i t y of the sample tube (304 s t a i n l e s s s t e e l )

was measured to determine the c o r r e c t i o n to the s o l i d mixture c o n d u c t i v i ­

t i e s , to determine the background heat flow along the tube and to check f o r

systematic e r r o r s at low heat i n p u t s . The r e s u l t s were given by

1 K to 2.5 K K = (5.7 ± .3) T x 10~ 4 W/cmK.

0.1 K to 0.4 K K = (6.0 ± .3) T x 10~ 4 W/cmK.

The s c a t t e r at low temperatures was due mainly to a background heat l e a k -9

(2 to 3 x 10 watts) which v a r i e d w i t h time. These measurements are i n

good agreement w i t h those of Seward et a l . (1969) who found ( i n the temper­

a t u r e range 1 K - 7 K) K = (5.5 T + .018 T ) x 10 W/cmK.

4 The thermal c o n d u c t i v i t y of He (obtained from a l a b o r a t o r y l i q u i d

4

He storage dewar) at 36 atmospheres was measured to check f o r systematic

e r r o r s at high heat inputs and low values of AT. The r e s u l t s were given by

K = (3.2 ± .1) T 3 W/cmK.

I f the c o e f f i c i e n t (3.2) i s taken to be exact, the temperature dependence

exponent i s 3.0 ± .1. The c a l c u l a t e d phonon mean f r e e path was .071 cm,

which i s much smaller than the tube diameter (.29 cm). The e r r o r i n t h i s

measurement was mainly from the determination of AT. The c a l c u l a t i o n

procedure f o r l i m i t e d us to an u n c e r t a i n t y of ± 0.1 mK and a t y p i c a l AT

was 3 mK. The temperature dependence i s i n good agreement w i t h measurements

of Bertman et a l . (1966 b ) . P o l y c r y s t a l l i n e samples were d e l i b e r a t e l y

grown to give as low a c o n d u c t i v i t y as p o s s i b l e .

-73-

B. PHASE SEPARATION TEMPERATURE

The very long time constants for changes in conductivity made i t

d i f f i c u l t to do accurate determinations of the phase separation temperature

T . We therefore limited ourselves to a measurement in the region of low ps 3

He concentration where the results of Panczyk et a l . (1968) and Edwards

et a l . (1962) were not in complete agreement. . This datum, plotted in 3

figure 2, agreed with Panczyk's results within experimental error. The He concentration was 10.3 ± .3% and the phase separation temperature was

.320 ± .010 K. Panczyk's data w i l l be used to define T in this work. ps

A lower l imit for T could be determined in a short time (10 min.) ps

by rapidly lowering the mixing chamber temperature. The rate of temperature

change measured by the B c t m p x t res is to., s i . c c i c a o c < I obrupcly ct c particular

temperature. This temperature was reproducible for each crystal to a few

millidegrees and was 30 to 40 mK below T • It is probable that this temper­

ature indicated the onset of rapid nucleation.

The determination of T for = 0.1 was done by measuring the

conductivity and the change in conductivity with time. The f i r s t measurements

were made 10 mK above T where the conductivity remained constant to 0.5% ps

over a period of 20 minutes. The average temperature was then reduced i n

5 mK steps u n t i l the conductivity was observed to change with time.

C. GENERAL RESULTS FOR ONE 10% He 'SAMPLE

A t y p i c a l r e s u l t f o r the thermal c o n d u c t i v i t y of a mixture i s 3

shown i n f i g u r e 9. This sample (E) was 10% He at P = 33,3 atmospheres, When

c r y s t a l growth was complete c o n d u c t i v i t y measurements were made between 0.6 K and

0.3 K (a -> b ) . T n e r e s u l t s i n t h i s temperature range i n d i c a t e t h a t the

thermal r e s i s t a n c e i s due to i s o t o p i c s c a t t e r i n g . -The datum at 1.0 K i s

from measurements by Berman et a l . (1965) on an 11% mixture. Berman's

sample had a f r e e z i n g pressure of 60 atmospheres whereas t h i s c r y s t a l began

f r e e z i n g at 54 atmospheres. The agreement between the two c o n d u c t i v i t y

r e s u l t s i s s a t i s f a c t o r y .

The v e r t i c a l l i n e at .31 K (b -»• c) shows the r e d u c t i o n i n c o n d u c t i v i t y 3

r h i o t o T>bg.«?*» P.p.oprstxor> and the bcc—hen t r a n s i t i o n i n the He r i c h component.

The mixing chamber temperature was adjusted o c c a s i o n a l l y to keep the average

temperature i n the c o n d u c t i v i t y c e l l at approximately .31 K. The time

constant a s s o c i a t e d w i t h the c o n d u c t i v i t y change was 70 minutes. This i s

much longer than e i t h e r the 10 minute thermal time constant of the sample chamber or the 140 second phase s e p a r a t i o n time constant (see Table 1), and i s taken as an i n d i c a t i o n of a t r a n s i t i o n to bcc s t r u c t u r e i n the phase

3

separated He r i c h component. The time constant f o r c o n d u c t i v i t y change was

longer c l o s e to the phase s e p a r a t i o n temperature. Using t h i s e f f e c t we were

able to vary the c o n d u c t i v i t y of the lower h a l f of the c r y s t a l more q u i c k l y

than the upper h a l f so that at the end of 5.5 hours the c o n d u c t i v i t y of the

two d i f f e r e d by a f a c t o r of 1.6.

-76-

The data below .3 K (d -> e) can be fitted' using the empirical equation -2 3

K = 2.7 x 10 T W/cmK and are indicative of boundary scattering with a

constant mean free path. These data were obtained by cooling the annealled

crystal rapidly from above the phase separation to .25 K and then turning on

the sample heater. The conductivity at .27 K reached a minimum value after

2.5 hours and then slowly increased so that 5 hours later the conductivity

was 5% higher than the minimum value. At .22 K and .17 K the conductivities

became constant (to within 0.1%/hr.) after 6 and 4 hours respectively. The

measurements made going from d to e lay on the same line as measurements

from e to d. The sample was then warmed to .3 K and measurements

(d -> f) were made while passing through the phase separation from below.

We waited, for about 3 hours at each measurement and the conductivities K n p a f f f pnr.*-!i~ r~~*tt" *~r. t.-.T*? f V 1 . r > 0 1 °L/r>r».i"r* rl'iT"! T>rr r h - * 0 . r - - 1 ' T r * p . TV«p /->QT*H**r* t - v * t v VOP.a

rapidly with temperature between .31 K and .320 K to a value 15% higher than 3

the extrapolated T value for .32 K. A subsequent experiment on the same

crystal gave T = .320 ± .005 K on our temperature scale, ps

The data above .335 K with one exception involved waiting times

less than 30 min. which was long compared to the thermal time constant and'

short compared to the time constant for the crystallographic change. The three

measurements at .43 K were made over a period of 2.5 hours, the conductivity

increased by 10% during this time. The sample was again cooled after reaching

(g ->- h) 0.65 K. The resulting conductivities are indicated by the dot-dash

line. Measurements made while going to 0.7 K (h -> i) lay on the same line. 3

These data suggest that boundary scattering by domains remains after the He

and He^ have gone back into uniform solution. An extremely long time

-77-

constant f o r the bcc-hcp t r a n s i t i o n i s expected above T (see sec. F of t h i s ps

chapter) and we hypothesize bcc "ghosts" of the domains i n the hep mat­r i x above T . The reduced c o n d u c t i v i t y due to s c a t t e r i n g by these "ghosts"

ps

was observed f o r a l l the concentrations and. pressures used i n t h i s work

except x^= .5. Given the observed phase s e p a r a t i o n time constants (a

maximum of 10 minutes was observed by Adams et a l . ) i t i s u n l i k e l y that the

mixture i s s t i l l phase separated at these h i g h temperatures.

This hypothesis of s c a t t e r i n g by c r y s t a l l o g r a p h i c "ghosts of the

phase s e p a r a t i o n " was te s t e d by a subsequent experiment. The c r y s t a l was

annealed and i s o t o p i c s c a t t e r i n g behaviour was observed between 0.6 K and

0.32 K. The c r y s t a l was cooled to j u s t below the phase s e p a r a t i o n tempera­

t u r e f o r a p e r i o d of time and then measurements were made between .32 K and

.6 K i n a time short compared to the c r y s t a l l o g r a p h i c r e l a x a t i o n time. By

va r y i n g the time below phase s e p a r a t i o n the boundary s c a t t e r i n g c o n t r i b u t i o n

to the thermal r e s i s t a n c e was v a r i e d over a wide range. The r e s u l t i n g

c o n d u c t i v i t i e s i n d i c a t e d that two s c a t t e r i n g mechanisms were i n v o l v e d ,

i s o t o p i c s c a t t e r i n g and boundary s c a t t e r i n g . The dotted l i n e s i n the

f i g u r e (c* *>• j , c" k) are smooth f i t s to these data.

D. ABOVE THE'PHASE SEPARATION

The results of thermal conductivity measurements above the phase

separation are shown i n figure 10. The approximate phase separation

temperature is indicated by the v e r t i c a l bar at the low temperature end of

each curve. The uncertainty in T is typical ly ± 10 mK. The results of

rapid (5 min.) measurements below T ^ g lay on extrapolations of the high

temperature curves. Smooth curves to indicate the l imit ing results below

T are shown for several samples. PS

The results w i l l be compared to the theory for point defect

scattering at intermediate concentrations. It w i l l be assumed that the

specif ic heat of mixtures is a l inear combination of the specific heats of

the two isotopical ly pure sol ids .a t the "sample pressure in the c rys ta l lo -

3 graphic phase of interest . The 0 of bcc He at a fixed molar volume is

3

nearly constant in this temperature range (Pandorf and Edwards, 1968) so T

behaviour for the specif ic heat w i l l be assumed. The ratio of 0 's for the

pure He solids which was observed at lower molar volumes w i l l be assumed

to hold at the higher molar volumes of this work. This ratio,63/6^ = a> is

equal to 1.23 for bcc and 1.17 for hep He solids (Edwards and Pandorf, 1966)

Then ( i / 0 m ) 3 = x 3 d / e 3 ) 3 + (1 - X 3 ) ( i / 8 4 ) 3 .

The samples with x 3 > 0.1 have a bcc crystal structure. The crystal

structure is uncertain i n the case of x 3 = .1 and P = 28 atm. The bec-mixed

l i q u i d t r i p l e point for x ^ = .1 is at approximately T f c = 1.85 K and Pfc = 36

atmospheres. There is no conclusive experimental data for the bcc-hcp

-80-

transition at lower temperatures at this concentration. We found that

the lower lines of the mixed bcc-hcp regions that were determined 3

by Vignos and Fairbank for several He concentrations lay on a common line,

within experimental error, when plotted on a reduced p - t diagram. The

reduced parameters were p = P/Pfc and t = T/T , where Pfc and Tfc are the

pressure and temperature for the bcc-mixed-liquid tr i p l e point at a given

concentration. For example; at t = 0.6, p = 0.83. This suggests that,

for x 3 = 0.1, at 1.1 K the mixture is mixed above and bcc below 30 atmospheres.

We conclude that the 10% crystal i s a mixture of hep and bcc at 33 atm. (see

also Miyoshi" et al„, 1970). We w i l l assume an bcc structure for the purpose

of calculation but the result w i l l be of l i t t l e significance.

The conductivities, calculated using the theory of isotopic

scattering at intermediate concentrations (see Chapter III), -7 1 -V? 1.28 x 10 ' 6 J (4 - x „ r / Z T T 7 .

1 ( A / V 1 / 2 Y Ix 3 (1 - x 3 ) ] 1 / 2 C m K

are shown in table 2. In the bcc phase, y = 2.2 (Sample and Swenson, 3

1967) , and in the hep phase at V = 20.9 cm /mole y = 2.9 (Jarvis et a l . , 1/2

1968) . The correction for l a t t i c e strain, (A/A ) is calculated using a

value of Y = 3.7. The calculated conductivity, K , i s compared to the

experimental conductivity, K , at T = 0.6 K (T/9 % .03).

Table 2 : Conductivity above phase separation.

Sample A B C D E # F

x 3 0.90 0.90 0.80 0.50 0.103 0.103

V cm3 23.6 24.2 23.5 22.5 20.8 21.1 m

ft

e

mole G_ K 18.5 17.0 18.0 17.0 6, = 24.0 22.0 3 4

6 v 18.8 17.3 18.7 18.8 23.0 20.9

A/AK 1.36 1.36 1.38 1.48 1.70 1.70

D (x) — - | T - 5 - 1 7 4.05 3.90 3.44 12.9 7.92 cmK

KT <T = -6> JEJl 1 1 • 1 8 - 7 0 8 - 3 8 7 - A 0 27.1 17.1

K (T = .6) mW 3.85 2.85 3.15 2.55 6.9# 4.40 cmK

K /K_ .347 .328 .377 .344 .250# .265 e 1

* 4 Samples E and F are nearer He so the estimated 0. i s l i s t e d . 4 Sample E has mixed hcp-bcc crystal structure, bec structure has been assumed for the calculation.

In figure 11 the ratio ^/K^ as a function of x ^ (1 - x^) at 4

T/0 = .03 for this work is compared to similar results in hep He rich 3

mixtures and hep He rich mixtures (Berman et a l . , 1965) and LiF isotopic

mixtures (Berman et a l . , 1965 b) at the same reduced temperature. The LiF

data were also analyzed by the Callaway method and at this reduced

-82-

temperature about 10% of the thermal r e s i s t a n c e i n the i n t e r m e d i a t e

c o n c e n t r a t i o n samples was found to be due to boundary s c a t t e r i n g . The

r a t i o K^/Kj was reduced f o r the d i l u t e samples by r e l a t i v e i n c r e a s e i n

boundary and Umklapp s c a t t e r i n g . The disagreement between the L i F measure­

ments and the theory was a t t r i b u t e d to the f a i l u r e of y as an estimate of

the e f f e c t of anharmonicity on the normal processes r e l a x a t i o n time. This

theory has been found t o overestimate the c o n d u c t i v i t y of a number of

d i f f e r e n t i s o t o p i c mixtures by a f a c t o r of order 2 (Berman et a l . , 1959).

We conclude t h a t , w i t h i n the l i m i t s imposed by the parameters; 9 m

Y and A/Aj,, these r e s u l t s f o r bcc helium mixtures are i n agreement w i t h

measurements on other s o l i d mixtures at the same reduced temperature.

The t h e o r y nredi'cts that i f i s o t o n i c s c a t t e r i n g i s the s o l e r e s i s ­

t i v e s c a t t e r i n g mechanism the r a t i o K /K should be temperature independent -3/2 • ' and one should observe K a T . The observed c o n d u c t i v i t i e s do not f o l l o w

t h i s simple r e l a t i o n s h i p , suggesting that other s c a t t e r i n g mechanisms are

present. Edwards et a l . , (1962) observed a " t a i l " on the s p e c i f i c heat -

measurements that extended to about 1.5 T and i n t e r p r e t e d t h i s as an ps

e f f e c t due to s h o r t range order. Such an e f f e c t would enhance the i s o t o p e 2

s c a t t e r i n g s i n c e a c l u s t e r of n s c a t t e r i n g s i t e s s c a t t e r s w i t h n times the

s t r e n g t h of n random s i t e s (Morse and Feshbach, 1953). There may a l s o be a

boundary s c a t t e r i n g c o n t r i b u t i o n . The mean f r e e path f o r boundary s c a t t e r ­

i n g i n a p o l y c r y s t a l l i n e sample i s approximately the average g r a i n s i z e . We

observed no s i g n i f i c a n t change i n the c o n d u c t i v i t y a f t e r annealing the

r a p i d l y grown samples which suggests that boundary s c a t t e r i n g i s not an

important c o n t r i b u t i o n .

-83-

A

At + •f-B

A

+ THIS W O R K

k H e 4 R I C H , h e p

A H e 3 R I C H , h e p

O Lf F

. 0 . 0 5 . 1 0 . 1 5 . 2 0 . 2 5

' X-(i-x)

FIGURE 11 : Comparison of results above the phase separation

with tbeory at T/0 % .03. The He rich and He

rich measurements in the hep phase are due to

Berman et a l . (1965). The LiF results are from

Berman et a l . (1965 b).

-84-

E. BELOW THE PHASE SEPARATION

The measurements of the sample conductivities are shown in 3 figures 12, 13 and 14. These results (for a l l samples except 50% He ) were

time independent. The results forx^= .90, P =» 35.5 atm., were reproduced

to within 5% on three different crystals. Measurements in the 50 mK region

below T that were obtained-while cooling from above T were not conclusive ps ps because of the very long time constants.

The temperature dependences of the conductivity for the 10% and

90% samples are in good agreement with the temperature dependences of con­

ductivity and specific heat measurements in the pure isotopes. The calcu­

lated mean free paths are temperature independent.

The apparent temperature dependence of the mean free path in the

80% sample (A % 6.8 T - 0 * 2 microns) and the 50% sample (A % 2.8 T ~ 0 , 3 microns)

is similar to the effect seen in highly polycrystalline solids (Berman,

1952). Klemens (1958) has suggested that a grain boundary can act both as a

surface (A = est) and as a row of dislocations (A a T ^ ) . A combination of

these scattering processes w i l l then give an intermediate temperature

dependence.

Dilute Mixtures

The conductivities of the 10% and 90% mixtures were observed to

reach time independent limits (which we shall refer to as the limiting conduc­

t i v i t i e s ) . The limiting conductivity of each sample could be described by the

empirical equation K = BT n. The conductivities of the dilute mixtures and

specific heats of the matrix solids had the same temperature dependence in d i ­

cating that the phonon mean free paths (A = 3K/Cv) were temperature independent.

-85-

' FIGURE 12 Conductivities below the phase separation A: x 3 = .9 . P = 35.5

B: .9 30.0 .

-87-

T (K) FIGURE 14 : Conductivity below the phase separation

temperature. E: x, = .103 P = 33 atm.

-88-

The limits could be reached most quickly when the average temperature of

the crystal was 30 to 50 mK below the phase separation, temperature. The .

limiting conductivity was reproduced for several 90% crystals at the same

pressure.

A complete theoretical model would .be extremely complex. We w i l l

analyze the results of the 10% and 90% mixtures using the following simple

model.

We assume that domains of the least abundant isotope are

distributed randomly in a matrix of the more abundant isotope. The existence

of domains is supported by evidence of a constant mean free path which is

indicative of scattering by surfaces. The temperature independence of the

mean free path suggests.that the domain dimensions are larger than the o

dominant phonon wave lengths ( 700 A at 0.15 K). These domains are assumed

to be spheres of constant size and composition. The composition of these

domains can be considered constant below 0.2 K since phase separation is 3

nearly complete at that temperature. At 0.2 K the equilibrium He concentra­

tions of the phases are 97% and 1%. The limiting conductivity indicates

that the domain sizes became time independent.

The assumption of spherical shape is not easily j u s t i f i e d . The

equilibrium shape w i l l be that which minimizes the total free energy. Since

the molar volumes of the two phases are different there is a strain energy

per unit volume, a , which depends on the modulus of r i g i d i t y of the matrix

and the shape of the domain (see Burke, 1965). The value of a^, is largest

for a spherical domain and smallest for a disc shaped domain. The surface

-89-

free energy of a domain per -unit volume, cr^, w i l l be minimized for a spher­

i c a l domain. A small modulus of r i g i d i t y favours spherical shape; however, 3

the anisotropy of bec He (Greywall and Munarin, 1970) suggests that

spherical domains are unlikely. The assumption w i l l be used since no better

estimate of shape is available.

In the region just below T^g the compositions of the two components

(domains and matrix) are temperature dependent. A complete theoretical cal­

culation would have to include isotopic scattering within these components.

In addition, conservation of matter requires that the ratio of domain to

matrix volume change as the compositions change. This w i l l alter the domain

scattering cross section. The hep-bec transition region probably occurs

above 0.2 K and this also w i l l alter the domain scattering cross section.

In the limit of complete phase separation of a mixture of molar

concentration x of the least abundant isotope A; the domains w i l l

occupy a fraction, p, of the total volume of the solid. If the molar

volumes of the two components A and B at the sample pressure are and

respectively, then

p = x A VA/V where V - x A V A + xfi V B

We assume for dilute mixtures that the domains are small i n

comparison to the phonon mean free path. In the limit of dilute mixtures

we can use the model of plane wave scattering from n spheres of average

radius, a, per unit volume. The details of the nature of the domain walls

and phonon propagation inside the domains w i l l be accounted for by assuming

-90-

a scattering cross-section, a = sA where A is the geometric cross-section

and s i s a proportionality parameter. For. a hard sphere that does not

distort the matrix, s = 2. The mean free path for phonon scattering in this

model i s then A = 1/no" and the ratio a/s i s given by

a/s = "j;

The mean free path i s calculated from the experimental data (table 3) by

means of the kinetic theory of gases approximation, using the specific heat

and sound velocity of the matrix isotope, and the measured thermal conducti­

vity. The mean free path i s then A = 3K/Cv and

9 K x A V A

hence a/ s = — — AC v V

C!once^t.ter! M i - V t ' 1 1 * 0 . ? ?

The thermal conductivities of the 80% and 50% mixtures did not have

the same temperature dependences as the associated specific heats. This

suggests a temperature dependent mean free path. In addition, the 50% mix­

ture conductivity did not become time independent at a lower limiting value.

The conductivity rose with time and the value reached after 82 hours below

T cannot be considered to be an upper limit, ps

The model used in the dilute cases is unlikely to be valid

when the domain volume i s not a small proportion of the total volume. We

w i l l make no attempt to develop a r e a l i s t i c model for this situation but w i l l

define an average mean free path \. We w i l l use the kinetic theory approxi­

mation with arithmetic averages of the specific heats and sound velocities of

the pure isotopes for the molar volumes that correspond to the sample pressure,

then X = 3K/Cv.

-91-

The mean free path, defined in this way should give an order of

magnitude estimate of the domain size. The average, domain size for 80% w i l l

be determined using X = 4a/3ps.. . The average domain size for 50% w i l l be

calculated using X = 2a/s.

Specific Heats and Sound Velocities of the Mixtures

Once phase separation has occurred the molar volumes of the two

separated phases are different from the average molar volume. Well below

T we can assume that each component i s one of the pure isotopes.

The two separated phases have different crystal structures, molar

volumes and Debye temperatures. We w i l l determine the molar volumes of the

pure isotopes at a given pressure from data of Edwards and Pandorf (1965,

1968) without correcting for the excess pressure due to phase separation and

the resultant change in molar volume (AV/V = 0.5%).

4 The specific heat of hep He has been found to exhibit Debye

4 behaviour below T/0 = .03. The Debye temperature of hep He for a given

molar volume w i l l be obtained from the data of Edwards and Pandorf (1965). 3

The specific heat of bec He does not exhibit Debye behaviour (Pandorf and

Edwards, 1968; Sample and Swenson, 1967). When i t is interpreted using a

temperature dependent 6, the resulting 0(T) reaches a maximum at T/0 = .02 3

and decreases at lower temperatures. The specific heat of bec He can be 2 5

expressed by the expirical equation C = AT to within experimental error.

We w i l l determine the value of A by f i t t i n g the above equation to the

specific heat data of Sample and Swenson (1967) at T = 0.4 K.

-92-

We w i l l assume that the velocity of sound i s constant and given

by the Debye model

2 -1/3 v = (k0/n) (6TT N/V) , where N i s Avogadro's number,

4 3 with 0 (T = 0) for He and 6 for bcc He . Sound velocity measurements by max

3 Greywall and Munarin (1970) in bcc He at one molar volume indicated a 8 smaller than 0 and were interpreted as 0 (T = 0). However Trickey and Adams max ,

(1970) suggest that the 8 determined in this way is the value for the measure­

ment temperature (1.7 K). Since there are no measurements at low temperatures

we w i l l use the above assumptions.

Experimental Results

i'ne empirical expressions for the experiment-any decermiueti conduc­t i v i t y K and the values of the mean free path A and the ratio of domain radius

e

to scattering proportionality parameter a/s calculated using our model are

li s t e d in Table 3. The concentrations and pressures were determined experi­

mentally. The methods of obtaining the molar volumes, specific heats and

sound velocities have been described in the preceding section. The results

for sample F (x^ = 0.103, P = 28 atm) are not included because a crystallo­

graphic transformation, with a very long time constant, was taking place in

the matrix.

Table 3 : Conductivity below phase separation

Sample

X3

V

atm.

0.90 0.90 0.80 0.50 0.103

35.5 30.0 33.3 30 33.5

V 3 cm3 23.9 24.5 24.1 24.5 24.1 mole

V 4 cm3 20.3 20.6 20.4 20.6 20.4 mole

C mJ 8.15 T 2' 5 10.2 T 2 , 5 8.3 T 2 , 5 7.8 T 2 , 6 4.34 T 3

cm K M, 208 193 217 > 238 299 / sec

K J2W 113 T 2* 5 244 T 2* 5 41.0 T 2 , 3 15.5 T 2* 3 27.0 T 3

6 cmK

X (T = .2 K) microns 20.0 37.2 9.4 4.5 6.25

a/s microns 1.26 2.3' 1.2 2.2 0.56

- 9 4 -

Scattering by the Domains

The average radius of the spheres is not determined uniquely by

these experiments because the value of the scattering cross section

proportionality parameter is unknown. We w i l l estimate the average radius

by assuming a hexagonal array of domains in which each domain grows by

mutual diffusion of the two isotopes within a sphere of radius R surround­

ing i t . The time constant x of such a growth process is approximately

(Crank, 1956)

x % r 2

I T 2 D

The mutual diffusion coefficient D w i l l be estimated using the tunnelling

atoms do a three dimensional random walk in steps of a Q (the nearest

neighbour distance) with a jump frequency £34' then the diffusion coefficient 2

is given by D % a f-./6. The average radius R of the spheres for a given o 34 x is then R - IT a ( x f . / 6 ) 1 / 2

o 34

4 4 For the 10% He sample we use the values x = 140 sec, f ^ = 7 x 10

sec and a = 3.7 x 10 ^ cm. o

Then R % 1.5 microns, the average separation between domain centres

is 2R % 3 microns, and the average radius of the domains i s a % R/2 % 0,75

microns. The value of the proportionality parameter is then s % 1*5. Given

the crudeness of the model, we only take the result as an indication that s

is of order unity.

-95-

The domains w i l l scatter phonons because strain f i e l d s w i l l exist

around them and also because the density and elast ic constants of the sample

w i l l change at their boundaries.

We w i l l estimate the second contribution s,, . = a /A .to the scatter-

ing proportionality parameter by assuming that i t i s equal to the reflect ion

coefficient that is associated with a plane boundary between two f luids of

different density and sound veloci ty . We w i l l use the solid helium sound

veloci t ies given by the Debye model. Below the crystal structures and

isotopic concentrations are different and s B % 0.2. When crystallographic

domains exist above T ^ g only the sound veloci t ies are s ignif icant ly d i f f e r ­

ent and s % 0.01. This large change in s (and therefore i n X and K) on

passing upward through T was not seen experimentally (see figure 9). ps

The strain f i e l d created when an oversized sphere of radius a i s

placed i n a spherical hole in an elast ic medium is given by (Carruthers,

1961)

V(r) = A r / r 2 r > a

= 0 r < a

where r i s a unit vector i n the radial direction and A is a constant which

measures the strength of the dis tor t ion . The probability of scattering of

phonons by such a f i e l d has not been calculated for the case in which the

phonon wavelengths are small compared to the radius of the sphere. We

w i l l describe the scattering cross-section in the following manner. We

assume that the scattering takes place on the surface of a sphere of radius

-96-

= a (1 + 6) where <S i s p r o p o r t i o n a l to the s t r e n g t h of the s t r a i n f i e l d .

The s c a t t e r i n g cross s e c t i o n of s t r a i n f i e l d i s then

0 = B-rra2 = BTra 2(l + <5)2

1

where B i s a p r o p o r t i o n a l i t y constant. The p r o p o r t i o n a l i t y parameter i s 2

then given by s g = B (1 + S) . I f 6 i s sm a l l compared to 1, then the s c a t t e r i n g cross s e c t i o n a w i l l not change s i g n i f i c a n t l y when the composi-

s

t i o n of the domains change and the phonon mean f r e e path above T w i l l then

be n e a r l y equal to X below T • This r e s u l t agrees w i t h the experimental

r e s u l t f o r x^ = 0.1 and suggests s t r a i n f i e l d s c a t t e r i n g i s stronger than

s c a t t e r i n g due to changes i n d e n s i t y and e l a s t i c constants.

The above r e s u l t s l e a d us to conclude that the s t r a i n f i e l d

s c a t t e r i n g at the domains i s so strong that the domains s c a t t e r i i i c e nard

spheres.

F. TIME DEPENDENT EFFECTS

A l a r g e amount of superheating..and...supercooling of the v a r i o u s -

c r y s t a l l o g r a p h i c phases was observed.. These, metastable e f f e c t s have been 3

observed i n the bcc-hcp t r a n s i t i o n measurements., i n . pure He .. S t r a t y and

Adams (1966) were able to supercool bec samples, which i n e q u i l i b r i u m would

enter the mixed phase r e g i o n at 1.7 K, to 0.33 K.

Because of the very long time constants of these phase changes, we

observed the unusual e f f e c t of i s o t o p i c s c a t t e r i n g combined w i t h v a r i a b l e

boundary s c a t t e r i n g .

The n u c l e a t i o n p r o b a b i l i t y of a c r y s t a l l o g r a p h i c phase which i s

thermodynamically favoured i s p r o p o r t i o n a l to exp(-AG^/KT) where the energy

b a r r i e r , AG^, i s i n v e r s e l y p r o p o r t i o n a l to the d i f f e r e n c e (AG^) between the

volume f r e e energy of the e x i s t i n g phase and the favoured phase.

The l a r g e amount of superheating i n the 10% samples can be explained

i n the f o l l o w i n g way. The bcc-hcp t r a n s i t i o n i n the uniform 10% mixture i s

i n the temperature and pressure region of t h i s work so AG^ i s s m a l l . When the 3

sample cools through the phase s e p a r a t i o n , domains o f . n e a r l y pure He form,

the value of AG^ f o r the domains inc r e a s e s c o n s i d e r a b l y , and the c r y s t a l l o ­

graphic phase change to bec s t r u c t u r e i n the domains and/or hep s t r u c t u r e i n

the m a t r i x takes place w i t h a r e l a t i v e l y short time constant. 3

When the phase separated c r y s t a l i s warmed above T^g the He and

He^ go back i n t o a uniform 10% s o l u t i o n r a p i d l y (Tf\.10 minutes). The bec

domains and the hep ma t r i x w i l l now have l i t t l e tendency to r e t u r n to the

-98-

equilibrium crystal structure because of the small value of A.G and a long

time constant is expected.

The inverse effect would be expected for the 90% samples. On 4

cooling, the He rich domains that form have concentrations.close to 10% and.

the small AG^ leads to a long time constant for crystallographic change in

the domains. When the sample is warmed above T the concentration of the ps

3

domains returns to 90% He and the large value of AG^ leads to a rapid

change in crystal structure. Both these effects were observed. For the 10% samples, the hcp-bcc

time constants were an hour or less on cooling through T but were longer ps

than an hour (calculated values of up to 7 hours) above T . The time con-ps

stants for the 90% samples were a l l longer than 30 minutes when the domains

were forming. The "ghosts" of the domains sample B were gone after 20 to

30 minutes at T = .35 K.

One of the supercooling effects was not understood. Sample A was

cooled below T many times. If the heater was turned on at .25 K the ps limiting conductivities were obtained. On three occasions the sample was

cooled to .2 K before the heater was turned on. Care was taken to ensure

that the experimental conditions were reproduced as closely as possible. On

two of these occasions the conductivity was 30% higher than the limiting

'value and constant within the resolution of the thermometers for 30 minutes. 2.15

Measurements at lower temperatures gave the result Ka T " . When the

sample was warmed to .25 K the conductivity went to the limiting value. On

the third occasion the sample was annealed and then cooled to .15 K before

the heater was turned on. The conductivity approached the limiting value

at .17 K with a time constant of about an hour.

- 9 9 -

The time dependent e f f e c t s i n the 50% sample were very d i f f e r e n t

from the ones discussed above. The r e s u l t s of the f i r s t experiment are

shown i n the f o l l o w i n g f i g u r e . The data are numbered and the time of each

measurement i s recorded on the page opposite, the graph. The sample was cooled 2

through T at t = 0. The c o n d u c t i v i t y was p r o p o r t i o n a l to T f o r the

f i r s t three measurements. In a second experiment the c r y s t a l was held at ml'T .3 K f o r 10 hours, the c o n d u c t i v i t y was .57 -~? a f t e r 4 hours and increased ' cmK

by 10% i n the next s i x hours. The r e s u l t s of these two experiments i n d i c a t e

that the domain s i z e doubles i n 50 ± 10.hours. We i n t e r p r e t t h i s as an

e f f e c t due to coarsening (see Chapter 2). The r e l a t i v e l y short time span of

the second experiment and the changing temperatures i n the f i r s t experiment

do not permit us to a c c u r a t e l y determine the time dependence of t h i s e f f e c t .

-100-

Table 4 : Elapsed time below the phase.separation for data

in figure 15. QK ~ .50)

Datum Time

1 5.5 hrs.

2 8.5

3 14.5

4 21.0

5 r 26.5, 28.5

6 33.5

7 38.5

8 46.0, 47.5, 49.0

9 55.5

10 61.0

11 64.0

12 " 6 7 . 0

13 70.0

14 73.5

15 r 77.0, 80.0

16 • 82.5

-101-

3 I 1 1 T i—r

2

0.7

0.5

! ) l A cmK

0.3

0.2

0.1

1 6 + +8

0. 03" " T (K)

as L

0.7 .0

FIGURE 15 : Thermal conductivity of the 50% sample. The data

are numbered in the time order of measurement.

-102-

G. CONCLUSIONS

The thermal c o n d u c t i v i t y measurements of s o l i d helium mixtures described

i n t h i s t h e s i s are the f i r s t such measurements done below 0.5 K. The meas­

urements above 0.5 K (Bertman et al.,1966 and Berman et al.,1965;1968) were

done to t e s t t h e o r i e s of i s o t o p i c s c a t t e r i n g . Our measurements were a l s o the

f i r s t measurements of t r a n s p o r t p r o p e r t i e s of phase separated s o l i d helium

mixtures. The two previous measurements were measurements of s p e c i f i c heat

(Edwards et al.,1962) and excess pressure at constant volume (Panzcyk et a l . ,

1968). We know of no other thermal c o n d u c t i v i t y measurements done on hetero­

geneous s o l i d s i n the low temperature r e g i o n where only phonon s c a t t e r i n g at

the boundaries betx^een the two components i s important.

The thermal c o n d u c t i v i t y measurements above the phase se p a r a t i o n

temperature i n d i c a t e t h a t the l a t t i c e s t r a i n enhancement of i s o t o p i c

s c a t t e r i n g i n bcc s o l i d helium mixtures i s s i m i l a r to the enhancement i n

hep s o l i d helium mixtures.

The thermal c o n d u c t i v i t y of o r i g i n a l l y homogeneous mixtures decreases

a b r u p t l y at the phase s e p a r a t i o n temperature. For d i l u t e mixtures below

T , the phonon mean f r e e path i s constant. The phonons are s c a t t e r e d by ps'

surfaces whose s p a t i a l dimensions are l a r g e compared to the dominant phonon

wave l e n g t h s . Our model suggests domains w i t h t y p i c a l dimensions of one

micron. There i s no l a r g e change i n mean f r e e path when phase separated

mixtures remix l e a v i n g c r y s t a l l o g r a p h i c domains. This i n d i c a t e s that the

s c a t t e r i n g cross s e c t i o n s of domains are not s t r o n g l y dependent on the

d e n s i t i e s and e l a s t i c constants of the domains.

3 The temperature dependences of the c o n d u c t i v i t i e s below T^g (T

A 2 5 3 f o r a He r i c h m a t r i x and T * f o r a He r i c h m a trix) agree w i t h previous

measurements on the pure s o l i d s . ,'

-103-

The measured phase separation for the sample with = .103 and

P = 33.5 atmospheres agrees with the measurements of Panczyk et a l . (1968)

with i n experimental e r r o r .

Suggestions f o r Further Research

Thermal c o n d u c t i v i t y measurements are not well suited f o r precise

determination of the nature of the phase separated domains because both

l o n g i t u d i n a l and transverse phonons of a range of frequencies are present,

and a theory of s c a t t e r i n g of phonons by large scale s t r a i n f i e l d s i s not

a v a i l a b l e . A more s u i t a b l e method would be the measurement of the a t t e n ­

u a t i o n of sound. D e t a i l e d theories of attenuation i n the Raleigh l i m i t are

a v a i l a b l e and measurements could be made with f i x e d frequencies and p o l a r -

i nations.

The a b i l i t y to vary the properties of the domains by melting the 3

He r i c h domains or warming the phase separated mixture above T would

permit the t e s t i n g of theories of s t r a i n f i e l d s c a t t e r i n g .

One would expect the f o l l o w i n g r e s u l t f o r thermal c o n d u c t i v i t y

measurements at lower temperatures. As the temperature i s reduced the

mean free path w i l l become temperature dependent. When the dominant phonon -2

wavelength i s about 1 micron (T ^ 5 mK) one would expect X a T . Below • -A

1 mK X w i l l increase as T u n t i l X equals the diameter of the sample. The

mean free path w i l l then become constant. The thermal c o n d u c t i v i t y w i l l

reach a minimum at a temperature on the order of 1 mK, then increase to a

maximum and f i n a l l y decrease with a constant mean free path.

-104-

' APPENDIX

A. ' TABLES'OF EXPERIMENTAL DATA

The data i n this appendix are l i s t e d in the time sequence of the

measurements. The elapsed time associated with each measurement is given.

The elapsed time is measured from the end of the f i r s t anneal.

Some measurements were taken while AT was not "constant". These

measurements are indicated by an asterisk. In most of these cases the con­

ductivity was slowly changing because a crystallographic phase transforma­

tion was taking place.

-105-

Sample A: x 3 = .90 P = 35.5 atm.

t

hours

0.8

K

mW/cmK

4.92

5.14

4.52 .

4.33

5.63

5.76

2.8 5.80

6.0 3.28

8.8 2.24

10.5 1.36

12.5 .81

grew new crystal

1.0 4.83

4.96

5.47

5.60

5.68

1.8 5.93

4.5 2.95

5.7 .2.04

7.8 1.43

11.0 3.22

14.2 1.71

T

K

.478

.460

.523

.541

.417

.394

.375

.240

.206

.169

.140

.496

.468

.418

.394

.377

.350

.203

.167

.138

.241

.184

t

hours

K

mW/cmK

16.7 1.01

19.1 2.30

21.5 3.31

23.5 1.07

25.7 5.47

5.53

4.96

4.86

26.4 5.82

6.00

26.8 5.60

30.0 3.00

crystal annealed

41.8 4.60

4.60

5.90

5.72

6.34

6.63

6.66

- 7.0

•6.8

5.15

T

K

.150

.204

.239

.149

.417

.395

.465

.484

.353

.324

.306

.202

.526

. 5 2 1

.425

.443

.391

.367

.358

.330

.300

.484

-106-

t K T

hours mW/cmK K

5.34 .465

5.14 - .486

4.94 .504

44.0 4.50 .537

45.7 2.54 * .207

46.0 2.50 * .206

46.3 2.46 * .206

46.7 2.43 * .206

49.5 2.19 .208

52.3 1.32 .172

52.8 5.05 .483

5.05 .464

5.94 .414

6.40 .369

53.9 7.05 .341

annealed for 6 hours

66.5 6.06 .414.

6.26 .392

7.04 .347

6.48 .372

6.93 .345

7.2 .332

7.5 .307

7.6 .298

7.4 * .294

t K T

hours mW/cmK K

7.5 * .290

68.5 7.5 * .285

71.4 1.68 .170

75.0 1.52 * .170

77.9 1.43 * .170

78.9 3.87 .599

4.02 .585

4.78 .516

4.48 .534

4.21 .562

7.30 .317

7.30 * .299

7.7 * .293

7.5 * .290

7.3 * .305

7.7 .278

7.5 * .269

7.4 * .275

7.4 * .264

7.7 * .257

7.7 * .265

7.5 * .281

82.5 7.7 * .266

-107-

Sample B: x 3 = .90 P = 29.8 atm.

t K T t K T

hours mW/cmK K hours mW/cmK K

45.0 4.20 * .231 4.75 .393

46.5 4.05 .233 4.65 .416

50.3 3.85 .196 5.35 .333

53.5 2.08 .160 - 4.80 .394

56.0 3.75 .196 94.5 4.65 .416

58.0 4.58 .427

59.3 1.58 .135 4.46 .439

62.2 1.75 .144 4.13 .461

65.5 3.00 .175 3.61 .511

70.6 3.96 .195 3.85 .491

73.9 4.02 .212 ' 3.86 .480

77.2 4.44 .231 3.26 .540

80.2 4.38 .243 3.19 .551

82.5 4.45 .248 3.08 .570

84.3 4.65 .260 5.45 .333

warming very slowly ' 5.05 .353

87.5 4.78 .333 4.95 .368

91.3 4.90 .329. 97.5 4.70 .387

91.8 5.10 .353 102.0 2.20 .158

• 5.05 .368 102.7 4.55 .417

-108-

Sample C: x 3 = .80 P =

t K T

hours mW/cmK K

2.0 3.11 .601

3.22 .582

3.59 .517

3.52 .537

2.7 3.77 .508

4.00 .460

3.93 .472

3.88 .492

4.12 .454

4.33 .432

4.42 .419

4.31 .422

4.55 .397

4.0 4.42 .384

7.2 3.03 .294

10.7 1.97 .268

14.2 1.56 .245

17.0 1.02 .203

21.0 .65 .163

23.5 .45 .142

25.5 .82 .185

27.5 1.34 .226

t K T

hours mW/cmK K

30.3 1.58 .242

annealed for 25 min.

3.73 .490

3.75 .466

4.00 .421

4.08 .397

4.10 .383

4.12 .380

4.22 .352

31.5 4.20 .335

annealed for 1 hour

4.68 .349

4.50 .377

33.0 4.75 .332

annealed for 1 hour

4.59 .377

4.75 .348

4.88 .331

4.82 .321

5.00 .314

35.0 3.37 .565

-109-

Sample D: x 3 = .50 P = 30.0 atm.

t

hours

K

mU/cmK

2.16

2.26

2.31

2.52

2.53

2.55

2.80

2.64 O K C ^ - • "O —«"

2.76

2.84

2.83

2.80

2.75

2.92

2.86

2.88

2.87

T

K

.680

.656

.655

.600

.589

.583

.534

.545

.519

.506

.483

.469

.461

.435

.420

.410

.405

t

hours

3.5

9.0

12.0

18.0

24.5

30.0,32.0

37.0

42.0

59.0

64.5

67.5

70.5

73.5

77.0

80.5,83.5

86.0

K

mW/cmK

2.82

.406

.327

.157

.798

.249

.802

.515

A O s S i 0 S.? ^ 1 O A

.941

.631

.348

.232

.187

.450

.786

1.03

T

K

.466

.279

.249

.174

.343

.187

.314

.244

.305

.247

.186

.159

.141

.208

.267

.308

-110-

Sample E: x 3 = .103 P = 33.3

t K T

hours mW/cmK K

2.5 7.17 .511

7.10 .505

6.85 .545

6.85 .550

6.60 .589

• 6.65 .583

3.0 6.30 .635

6.24 .629

7.75 .451

7.9 .446

8.0 .401

8.15 .407

8.70 .371

8.9 .365

8.9 .341

8.9 .335'

5.0 9.2 .316

5.5 9.1 .310

12.5 .87 .310

19.0 .510 .267

23.0 .280 .217

25.5 .135 .170

27.5 .088 .148

t K T

hours mW/cmK K

30.0 .062 .129

32.4 .78 .305

36.2 .90 * .318

39.5. 1.04 * .320

42.5 1.00 * .325

45.0 1.05 * .329

46.0 1.05 * .329

48.6 1.08 * .331

49.2 1.11 * .337

50.0 1.20 * .367

50.5 1.19 * .386

51.0 1.18 * .385

51.4 1.33 * .407

51.6 1.56 * .435

52.2 1.60 * .433

53.6 1.68 * .431

54.1 1.98 * .470

54.3 2.29 * .522

54.5 2.98 * .588

54.8 3.27 * .602

55.1 . 3.45 * .601

55.4 3.03 * .530

2.78 * .478

-111-

t

hours

55.7

56.2

57.0

K

mW/cmK

2.43 *

1.99 *

2.29 *

2.64 *

2.97 *

3.43 *

2.31 *

3.53 *

4.01 *

T

K

.436

.387

.424

.465

.518

.587

.420

.599

.697

annealed for 20 min.

7.70 .471

8.40 .357

sample below 0.3 K for 2 hours

then 8 hours at 0.6 K

71.1 5.62 * .597

5.25 * .474.

4.72 * .402

4.40 * .351

71.9 5.55 * .710

annealed for 10 min.

73.7 6.90 .594

7.85 .466

74.4 8.45 .340

75.5 8.30 .330

t

hours

75.8

76.2

76.5

76.8

77.2

77.5

77.8

78.2

78.5

79.4 O r\ r\

80.5

81.0

81.25

81.5

81.75

82.0

82.3

82.6

83.1

83.3

annealed

l e f t crystal

92.6

K

mW/cmK

7.8 *

7.0 *

5.90 *

5.15 *

4.65 *

4.30 *

3.85 *

3.58 *

2.86 *

2.38 *

2.20 *

2.53 *

2.82 *

3.08 *

2.65 *

3.87 *

4.10 *

4.90 *

5.30 *

T

K

.306

.307

.307

.308

.309

.310

.311

.312

.313

• Jw >

.309

.308

.333

.371

.434

.374

.524

.520

.585

.681

for 30 minutes then

at .6 K for 8.5 hours

6.78 .589

7.60 .464

-112-

t K T

hours mW/cmK K

93.0 8.05 .356

7.8 .328

7.8 .325

93.9 7.0 * .319

94.5 6.8 * .319

6.5 * .318

95.2 6.1 * .317

95.3 6.1 * .314

t K T

hours mW/cmK K

95.8 5.7 * .314

96.6 6.5 * .345

97.0 6.65 * .362

97.25 6.50 * .392

97.5 6.55 * .427

97.8 6.70 * .471

98.1 6.60 * .526

98.3 6.40 * .598

-113-

Sample F : : x 3 = 0.103 P = 28.0 atm.

t K T t K T

hours mW/cmK K hours mW/cmK K

0.3 4.42 .595 17.8 .23 * .166

5.00 .519 18.8 .76 * .244

5.62 .460 19.7 1.17 * .286

6.25 .412 20.5 1.88 * .352

7.05 .362 20.9 2.77 * .413

1.0 7.18 .337 21.1 3.25 * .455

7.31 .320 21.2 3.71 * .509

1.6 7.0 * .285 21.3 4.15 * .590

2.1 6.9 * .285 4.3 * .670

2.6 6.7 * .285 21.7 4.25 * .710

3.1 4.0 * .278 Sample at 0.4 K f o r 4 houi

7.3 2.0 * .261 26.0 4.1 * .419

1.6 * .255 4.3 * .519

10.8 1.32 * .252 4.25 * .596

11.7 1.27 * .245 4.12 * .674

13.7 .61 *' .219 3.90 * .357

15.5 .31 * .189 27.0 3.20 * .321

15.9 .32 * .188

-114-

B. BIBLIOGRAPHY

E.D. Adams, M.F. Panczyk, R.A. S c r i b n e r , and J.R. Gonano Q-969) PltOCde.dln.g6 o$ tkz EZzvzntk JiiteAiicutlGnat Confiznzncz on Low Tzmp&iatuAz Pkyi>ic6, St. AndAZUS, 1968 Ed. J.F. A l l e n , D.M. F i n l a y s o n , and D.M. M c C a l l , Univ. St. Andrews, p. 461.

A.C. Anderson (1968) Rzv. Sci. InAt, 39., 605

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