NUCLEAR EXCHANGE ENERGY AND
ISOTOPIC PHASE SEPARATION
IN SOLID HELIUM
By
MICHAEL FRANCIS PANCZYK
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
TIIE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1968
695397
PANCZYK, Michael Francis, 1938
NUCLEAR EXCHANGE ENERGY AND ISOTOPIC
PHASE SEPARATION IN SOLID HELIUM.
The University of Florida, Ph.D., 1968
Physics, solid state
University Microfilms, Inc., Ann Arbor, Michigan
For My Delightful Wife,
MARY
ACKNOWLEDGMENTS
I wish to express my sincere appreciation to those individuals
listed below for generously contributing both their time and talents
to this work.
Dr. E. D. Adams suggested this investigation and provided con
tinual guidance throughout its entire development.
Mr. R. A. Scribner made many valuable contributions to the de
sign and construction of the apparatus and also spent many hours help
ing take the data. Dr. G. C. Straty contributed his vast technical
Knowledge to this work and is responsible for the particular design
of the strain gauge used in these experiments. Dr. J. R. Gonano often
participated in many helpful discussions concerning the interpretation
of the experimental results. Messers. D. C. Heberlein and J. W. Philp
were frequently called upon to assist me and always did so cheerfully.
Mr. B. McDowell often worked long hours to provide the copious quanti
ties of liquid helium necessary to carry on this investigation.
Finally, I wish to express my sincere appreciation to my wife,
Mary, for her patience and understanding during what has been a long
and at times frustrating graduate career.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS. . . . . . . . . . . . . ii
LIST OF FIGURES . . . . . . . .. . . . vi
ABSTRACT . . . . . . . . . . . . . . vii
Chapter
I. INTRODUCTION . . . . . . . .... .. . .. 1
II. THEORETICAL TREATMENTS OF SOLID HE3. . . . . . 9
A. Introductory Remarks . . . . . . . . 9
B. Physical Origin of the Exchange Energy . . . 10
C. Microscopic Theory . . . . . . . .. 13
Nosanow's Theory of Quantum Crystals . . ... 15
D. Thermodynamic Theory . . .. . . . . . 27
III. EXPERIMENTAL APPARATUS AND PROCEDURE . . . . .. 33
A. Introductory Remarks . . . . . . . .. 33
B. Cryostat . . . .. . .. . . . .. . 34
Helium Refrigerators . . . . . . .. 34
Vacuum Chambers and Radiation Shields. . . ... 39
C. The Strain Gauge . . . . . . . . .. 41
D. Pressure Measurements. . . . . . . .. 48
Gas Handling and Pressure System . . . .. 48
Pressure Calibration and Measurement . . .. 51
E. Potassium Chrome Alum Salt Assembly. . . . .. 52
F. Temperature Measurements . . . . . ... 58
Temperature Calibrations and Measurements . .. 58
iv
Chapter
Temperature Regulation . . . . . .
Thermal Equilibrium Time between the Sample and
Thermometer.
Page
. 65
. . . . . . . . . 66
G. Solenoids. . . . . . . . . . . .
H. Performance of the Experiment. . . . . ... .
Sample Formation and Cooldown to 0.30K . . . .
Demagnetization Procedure. . . . . . . .
IV. RESULTS AND DISCUSSION . . . . . . . . .
A. Introductory Remarks . . . . . . . . .
B. The Thermal Expansion of the Empty Cell. . . . .
C. Nuclear Exchange Energy . . . . . . . .
Values of IJI for "Pure" He3 . . . . . .
Effects of He4 Impurities on J . . . . . .
D. Locus of the Zeros of the Thermal Expansion
Coefficient. . . . . . . . . . .
E. Isotopic Phase Separation . . . . . . .
Kinetics of the Phase Transition . . . . .
Pressure Dependence of the Energy of Solution and
Phase Separation Temperature . . . . . .
V. SUMMARY OF THE RESULTS .. . . . . . . . .
REFERENCES . . . . . . . . .
BIOGRAPHICAL SKETCH. . . . . . . .
S. . 100
. .. 103
67
68
68
69
72
72
75
77
77
85
89
97
LIST OF FIGURES
Figure Page
1. Calculated ground state energy versus molar volume .... .22
2. Variational parameters Aand log K versus molar volume. . 23
3. Calculated nuclear exchange energy versus molar volume . 24
4. Schematic diagram of the apparatus . . . . .... 36
5. Schematic diagram of the low temperature section ..... 38
6. Capacitance strain gauge. . . . . . . . . .44
7. Pressure system. . . . . . . . ... ..... 50
8. Potassium chrome alum salt assembly. . . . .... 56
9. Simplified schematic of the de mutual inductance circuit .59
10. Schematic diagram of the CMN thermometer and the mutual
inductance system. . . . . . . . . . . 61
11. Simplified schematic of the ac resistance bridge . . .. .64
12. Characteristic isochore for the 1600 ppm He4 sample. ... 74
13. Thermal expansion of the empty strain gauge. ....... .76
14. The nuclear exchange contribution to the pressure, (AP)EX,
versus T1 for various molar volumes . . .... ... . 79
15. Nuclear exchange energy versus molar volume. ....... 82
16. Locus of the zeros of the isobaric expansion coefficient .87
17. Pressure change, (AP)pS, due to the isotopic phase
separation in the 600 ppm He4 sample versus temperature. . 92
18. Pressure change, (AP)pS, due to the isotopic phase
separation in the 1600 ppm He4 sample versus temperature 93
dE /k
19. Plot of versus molar volume. . . . . . ... 94
dV
20. Energy of mixing versus molar volume . . . . . . 96
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
NUCLEAR EXCHANGE ENERGY AND ISOTOPIC
PHASE SEPARATION IN SOLID HELIUM
By
Michael Francis Panczyk
March, 1968
Chairman: Dr. E. Dwight Adams
Major Department: Physics
By making use of a sensitive capacitance type strain gauge, we
have investigated the nuclear exchange energy and isotopic phase sep
aration in solid helium mixtures containing 20, 600, and 1600 ppm He4.
Measurements of the pressure versus temperature for constant volume
samples between.22.8 and 24.2 cm3/mole have been made down to 20 m0K,
and show the expected T1 dependence due to nuclear spin ordering at
low temperatures. The "alues of the nuclear exchange energy were ob
tained from the slopes of the P versus T1 curves by means of a self
consistent procedure. The magnitude of the exchange energy, JJl/k, is
about 0.7 mOK at a molar volume of 24 cm3/mole and decreases with in
creasing density approximately as 1iJ = 16.4. The corresponding
dlnV
magnetic transition temperature varies from 2.4 m'K at 24.2 cm3/mole
down to 0.9 mOK at 22.8 cm3/mole. Since the data were obtained in a
temperature range well below the isotopic phase separation temperature
of the three mixtures, the values of IJI reported here are those of a
very pure He3 solid.
The isotopic phase separation in the two samples containing 600
and 1600 ppm He4 has been detected by observing.the increase in pres
sure which occurs as the mixture separates into two enriched phases.
For the 600 ppm sample this excess pressure is = 3 x 103 atm, while
for the 1600 ppm sample it is = 9 x 103 atm. The corresponding phase
separation temperatures, at a molar volume of 24.0 cm3/mole, are ap
proximately 0.1080K and 0.1190K respectively. A brief investigation of
the density dependence of the phase separation phenomenon indicates
that both the excess pressure and the phase separation temperature de
crease with increasing density.
A discussion of the assumptions and approximations in the theory
of "Quantum Crystals" formulated by Nosanow has also been given. A
comparison between these theoretical calculations and our experimental
results for the exchange energy and its volume dependence indicates
that while this theory is qualitatively correct, refinements must be
made in order to obtain quantitative agreement with existing experi
mental data.
viii
CHAPTER I
INTRODUCTION
Helium exists in two stable isotopic forms, He3 and He4. He3
atoms have a spin of 1/2 and hence are treated theoretically by Fermi
Dirac statistics, while He4 atoms have a spin equal to zero and thus
follow a BoseEinstein distribution law. Since the electrostatic in
teraction between two He4 atoms or two He3 atoms is approximately the
same, these isotopes and mixtures of various concentrations provide
excellent examples for studying the effects of quantum statistics on
the macroscopic properties of systems.
One of the most interesting properties of both the pure isotopes
and mixtures is that they remain liquids down to the absolute zero of
temperature. This is a consequence of the weak interatomic forces and
the small mass of the helium atoms. Pressures of the order of 25 at
mospheres are required to bring about the liquidsolid transition.
Liquid He4 has been studied extensively for about fifty years. Most of
the research has centered on the properties of the superfluid phase.
F. London1 pointed out that the existence of this phase is closely
connected with the fact that the He4 atom is a boson. Liquid He3,
which is a simple fermion system, exhibits no such superfluid phase,
although the possibility of a superfluid transition similar to that
occurring in the electron gas in metals has been proposed by several
authors.2,3,4 Experimental investigations of the nuclear magnetic
_~
susceptibility5,6,7 and spin diffusion coefficient of liquid He3 in
dicate that the magnetic properties of the system are in excellent
agreement with the theoretical predictions of Landau.9 The agreement
with respect to other properties of the liquid, however, is not quite
so good. In particular, recent specific heat measurements810 to 3 mK
do not exhibit a linear dependence on the temperature as one would ex
pect on the basis of Landau's Fermi liquid theory. Thus it appears
that additional information about the liquid is needed in order to
determine the degree of validity of Landau's approach. In particular,
measurements of the isobaric thermal expansion coefficient to temper
atures of a few millidegrees are very desirable.
The melting line of He3 is of particular interest because it ex
hibits a deep minimum. Although this phenomenon is unusual, since it
implies that the solid possesses a greater degree of disorder than
liquid, its existence was predicted by Pomeranchuk11 before it was
12
actually observed in the experiments of Baum, et al. Pomeranchuk
argued that the nuclear spins in the solid should be randomly dis
tributed down to temperatures of a few microdegrees, and hence con
tribute Rln2 to the total entropy. Since the entropy of the liquid is
known to fall below this value at about 0.320K, there should be a change
in sign of (S S ) at this temperature and by the ClausiusClapeyron
i s
dP
equation Lj should be negative. Although it is now expected that the
exchange interaction in the solid near the melting curve will produce
nuclear spin alignment in the millidegree rather than the microdegree
region, this temperature range is still considerably below 0.32K,so
that Pomeranchuk's original argument for the existence of the minimum
dP
remains at least qualitatively correct. The absolute value of d cannot
dT
continue to increase at temperatures very close to absolute zero since
the Nerst Theorem implies that = 0 at T = 0. Recent strain gauge
13
measurements by Scribner, et al.13 locate the minimum at a temperature
of 0.3180K and a pressure of 28.93 atm. These measurements were car
ried down to 0.017K and at this temperature id is still increasing.
At the present time it is expected d will reach a maximum value at
about 7 mK and again become zero at approximately 0.5 mK.
The properties of the solid phase of helium have not received as
much attention as those of the liquid. The experiments of Grilly and
Mills14 revealed the existence of two solid phases of He3 having a
triple point with the liquid at T = 3.1480K and P = 135.9 atm. Xray
diffraction experiments5 showed that the crystal structure of the low
pressure phase was bodycenteredcubic (bcc) while the higher pressure
phase was hexagonalclosepacked (hcp). Recent pressure measurements
by Straty and Adamsl6 showed that below 10K, the bcchcp phase boundary
is horizontal at a pressure of about 105 atm. Further xray work by
17 18
Schuch and Mills and by Franck8 revealed the existence of a third
solid phase above T = 17.780K and P = 1608 atm. The crystal structure
of this phase is cubicclosepacked (ccp). Solid He4 is also found to
exist in these three crystal structures, although the details of the
PVT relations of the two solids are somewhat different. The fact that
low pressure solid He3 exists in a bodycenteredcubic structure is
somewhat unusual since the stable structure for most dielectric solids
19
is cubicclosepacked. Nosanow has demonstrated that the existence
of solid helium in the bcc phase is due to the strong short range cor
relations which arise from the large amplitude zero point motion of the
atoms, and hence is a manifestation of the quantum nature of the solid.
20
Recent heat capacity experiments by Sdenson and coworkers2 and
by Pandorf and Edwards,21 along with the pressure measurements of
Straty and Adams,6 show that above 0.3K, these properties of the
solid are determined principally by the phonons with almost no con
tribution from the nuclear spin system. This situation cannot con
tinue to very low temperatures since the contribution to the free
energy from the phonons decreases while that from the spins increases.
As mentioned previously Pomeranchuk11 originally predicted that the
temperature range in which the spin system would determine the prop
erties of the solid should be around a few microdegrees. He based
this prediction on the idea that the He3 atoms in the solid are tightly
bound to well separated lattice sites, and hence exchange effects are
negligible. Bernardes and Primakoff22 later pointed out that the
large amplitude zero point motion of the He3 atoms produces considerable
overlap of the wave functions of neighboring atoms, and hence exchange
effects in the solid are very important. They developed a theory for
the ground state of solid He3 which predicted that antiferromagnetic
spin alignment should occur in the low density solid at temperatures
of a few hundredths of a degree. More recent calculations by Nosanow19
indicate that the exchange energy is not quite as large as originally
estimated by Bernardes and Primakoff. Nosanow predicts that the magnetic
ground state is still antiferromagnetic but that the Neel temperature
is approximately 0.2 m*K.
There are many reasons why the nuclear magnetic properties of
solid He3 have received so much attention. One of the most important
is the absence of any electronic contribution to the magnetic moment
of the atom. Also, since it forms a simple dielectric solid containing
localized spin 1/2 particles, He3 represents an ideal substance in
which to test the various theories of magnetism. From an experimental
point of view, the large compressibility of the solid allows one to
study the magnetic properties over a wide range of densities by ap
plication of modest pressures.
Early experimental attempts to measure the exchange energy in the
23
solid were focused mainly on the nuclear magnetic susceptibility2 and
the specific heat.24 These early experimental results were inadequate
either because of sizable amounts of He4 impurities or because the
measurements were not extended to low enough temperatures. Nuclear
susceptibility measurements are very attractive because they allow
both the sign and magnitude of the exchange energy, J, to be determined
from a single measurement. However, since the Neel temperature in the
solid is expected to be of the order of a few millidegrees, it is neces
sary to make susceptibility measurements to temperatures around 0.02K
and lower to determine J accurately. At these temperatures, especially
for the higher densities, measurements become very difficult because
of the long thermal equilibrium times between the nuclear spin system,
the lattice, and the thermometer. Nevertheless, susceptibility ex
periments have been performed by Thomson, Meyer, and Dheer25 to tem
peratures down to 0.070K. Their results indicate that the exchange
energy is negative, and the corresponding Ngel temperature, TN, is
less than 0.02*K for molar volumes greater than 22.4 cm3/mole. At
higher densities, they found a systematic increase in TN to a value
of 0.1K at a molar volume of 19.5 cm3/mole. These higher density re
sults were regarded as very tentative however, and may be due to non
equilibrium effects which arise from the He4 impurities present in the
He3. These nonequilibrium effects were more apparent in the suscepti
bility measurements made on the 1% He4 sample also studied by Thomson,
et al.,25 and are discussed in some detail by them.
Two technical problems make the determination of the exchange
energy from specific heat data difficult. The first is the large con
tribution to the specific heat from isotopic phase separation of He4
impurities present in the He3. This was first observed by Edwards,
et al.24 in experiments designed to determine the nuclear spin contri
bution tothe specific heat. These experimenters found a large anomaly
in the heat capacity of solid mixtures which suggested that at low tem
peratures the mixture separates into two phases, one rich in He3 and
the other in He4. These measurements also indicate that the phase
separation line is symmetric about a concentration of 50%, and that the
two isotopes will be completely separated at O0K. The problems pre
sented by the isotopic separation of the He4 atoms could presumably be
surmounted by either using very pure He3 gas, or lowering the tempera
ture to a region where the phase separation contribution to the specific
heat is small. This brings us to the second problem; namely, that as
the temperature is reduced, the specific heat of the solid sample be
comes less than that of the cerium magnesium nitrate thermometer. This
means that the background specific heat of the calorimeter is greater
than that of the He3 sample, and this situation greatly reduces the ac
curacy with which one can obtain the exchange energy from the experi
mental data.
Thus far, the most successful determinations of IJl and its volume
dependence have been derived from nuclear magnetic relaxation measure
ments of the spinlattice and spinspin relaxation times. Although
other experimenters have also made measurements, the most comprehensive
studies have been performed by Richardson, Hunt, and Meyer26 at Duke
and by Richards, Hatton, and Gifford27 at Oxford. These experimenters
find that in the low density bcc phase, the exchange energy JlJ/k is
approximately 1 mOK and decreases with increasing density. The NMR
data also indicate that the value of J depends very strongly on the
concentration of the He4 impurities present in the sample. Although
these measurements are by no means conclusive, it appears that the ex
change energy is much larger in impure samples than in the relatively
pure ones.
The main advantage that magnetic relaxation experiments have over
the previously mentioned thermodynamic ones, is that the exchange energy
can be determined from data obtained above 0.3"K. This temperature re
gion is within range of a helium three refrigerator and hence no para
magnetic refrigerant is necessary. Furthermore, the thermal time con
stants between the spins, lattice, and thermometer remain reasonably
short above 0.30K. The principal objection to determinations of the
exchange energy based solely on T1 and T2 data is that these relaxation
times are related to J by a rather complex formalism which has undergone
considerable numerical revision. These measurements thus constitute a
somewhat indirect determination of the exchange energy and additional
thermodynamic data are very desirable.
In this work we report the first direct determination of IJI and
its volume dependence for molar volumes between 22.8 and 24.2 cm3/mole.
The values of JI are obtained from measurements of the internal pres
sure and temperature of constant volume samples. In addition we have
also made the first pressure measurements of the isotopic phase separa
tion temperature for samples containing 600 and 1600 ppm He4.
In the following chapter we discuss the Nosanow theory19 of "Quan
tum Crystals" and also obtain an approximate equation of state for the
solid at low temperatures. This equation of state relates the exchange
energy and its volume dependence to the internal pressure and tempera
ture of the solid, and can be used to extract IJI from the P, T data.
In addition we will also obtain an expression for the increase in pres
sure due to the isotopic phase separation, based upon the assumption
that the He3 and He4 atoms mix together in a completely random fashion.
In Chapter III we describe the apparatus used in these experiments.
In particular we will discuss in detail the potassium chrome alum salt
system used to lower the temperature to a region where the effects of
the spin system become observable, and the capacitive strain gauge used
to measure the small pressure changes produced by the nuclear spin or
dering.
In Chapter IV we present our results for the exchange energy and
its volume dependence, along with the results derived from the NMR ex
periments. A comparison will be made between these experimental data
and the theoretical calculations of Nosanow and his coworkers.19'28
We also present results for the isotopic phase separation temperature,
energy of mixing, and equilibrium time constant for the samples 600
and 1600 ppm He4. The volume dependence of these quantities is also
discussed.
Finally in the last chapter we will give a brief summary of the
present situation with respect to the exchange energy and phase separa
tion, and suggest some future experiments whose results should enhance
our understanding of solid helium.
CHAPTER II
THEORETICAL TREATMENTS OF SOLID HE3
A. Introductory Remarks
In this chapter we shall discuss in detail two theoretical treat
ments of the ground state of solid He3. For want of better names, we
refer to these as the microscopic and thermodynamic approaches to the
problem. In a microscopic theory one is interested in obtaining values
for the various properties of the system such as the ground state energy
and nuclear exchange energy from a solution of the many body Schruudinger
equation, while in a thermodynamic theory one regards these quantities
as experimentally determined parameters and attempts to obtain an ap
proximate equation of state for the system. Although there have been
22,29
several microscopic theories229 proposed in recent years, we shall
discuss in detail only the most recent work of Nosanow and his cowork
ers19,28 since it appears to represent the "state of the art" as it
exists today. Goldstein30 has devoted considerable attention to the
thermodynamic properties of both liquid and solid helium, and the macro
scopic equation of state which we shall develop is similar to that ap
pearing in his most recent work. Before discussing these theoretical
attempts to calculate the ground state properties of the solid,how
ever, a few words concerning the physical origin of the exchange energy
seem appropriate.
B. Physical Origin of the Exchange Energy
When Weiss31 proposed his molecular field theory in 1907, it was
assumed that the ordinary magnetic dipoledipole interaction between
neighboring atoms was responsible for the observed spontaneous mag
netization in ferromagnets. However, it soon became evident that the
observed transition temperatures were much too high to be explained on
the basis of a simple dipolar interaction which predicts a transition
temperature Tc P2/R2, where p is the net magnetic moment, and R the
distance between neighboring atoms. For iron, the observed transition
temperature is x 1000K,while the temperature calculated from the dipolar
interaction is about 10K. Thus to account for the experimental data,
it was necessary to find an interaction which is about a thousand times
stronger than the magnetic dipole one. The discovery of this unknown
interaction had to wait about twenty years for Schroedinger and Heisen
berg to develop quantum mechanics. Shortly after the formulation of
this theory, Heisenberg32 applied it to the problem of ferromagnetism.
He showed that the interaction responsible for spontaneous magnetiza
tion was truly quantum mechanical, being a direct result of the sym
metry restrictions placed on the wave function by the Pauli exclusion
principle.
To illustrate the important role that the exclusion principle plays
in determining the ground state energy of a system of fermions, it is
useful to consider the simple example of two spin 1/2 particles inter
acting with each other through a potential V(rl2). The Hamiltonian,
omitting the dipoledipole term, is given by
H(1,2) = (V2+V2)+V(r12). (2.1)
Consider two cases:
CASE I. The particles are distinguishable so that the restrictions of
the Pauli principle need not be considered. A suitable wave function
for the two particles is then
*(rir2) = oi(rl)o (r2) (2.2)
where i and j refer to the ith and jth single particle eigenstates.
The total energy will then be
E = E +E + f(r1l) *(r2)V(rl2)i(rl)O(2)drld. (2.3)
and is independent of the relative spin orientations of the particles.
CASE II. The particles are indistinguishable so that the exclusion
principle requires the wave function to be antisymmetric. A linear
combination of the
(rl,r2) = [i(ri) (r2)i(r2) (rl)]S(s ,s2) (2.4)
where S(s1,s2) is the singlet spin function if the positive sign is
used, and the triplet spin function if the negative sign is used. The
total energy is no longer degenerate, hut instead is given by
E = E 1i(l)j 2)V(rl2)Ii(r2) (rl)drldr2 (2.5)
where the additional term is called the exchange energy Jij of the two
spins in states i and j. The total energy is now seen to depend upon
the relative spin orientations of the two particles.
The essential difference between Cases I and II lies in the dis
tinguishability of the particles, which in turn is determined by their
spacial motions. When the volumes spanned by particles 1 and 2 have a
common region, they are indistinguishable and we get J ij 0, while
if these volumes do not overlap the particles are distinguishable and
Jij = 0. The size of the nuclear exchange energy in solids is thus
seen to be a measure of how large an overlap there is between wave
packets describing the vibrations of neighboring atoms. In solid He3
the weak interatomic forces and small mass combine to produce large
amplitude zero point vibrations of the atoms about their equilibrium
positions, and hence one might expect solid He3 to exhibit sizeable
nuclear exchange effects.
In 1929, Dirac33 showed that for localized spins in orthogonal
orbitals the exchange energy can be written as
N
H =2 1 J Ii (2.6)
ex icj ij "
i
This is the famous Heisenberg Hamiltonian developed by Dirac and first
used extensively by Van Vleck.34 When written in this form, the ex
change energy appears to result from a direct two body spinspin in
teraction. Also one sees that the magnetic ground state (ferro or
antiferromagnetic) will be determined by the sign of Jij. If Jij < 0
antiparallel spin alignment will be favored and the ground state of
the system will be antiferromagnetic, while if Jij > 0 the ground
state will be ferromagnetic. Furthermore, when expressed in this form
it is immediately evident that the exchange energy represents the dif
ference between the singlet and triplet state energies. Finally, this
form is very attractive because it allows the powerful spin operator
formalism to be applied to the theory of magnetism. Perhaps because
of the clarity with which Eq.(2.6) defines the exchange energy and
magnetic ground state of a system, misconceptions about the microscopic
origin of the exchange interaction have arisen. It is important, there
fore, to realize that exchange forces result from the symmetry require
ments placed on the wave function by the Pauli exclusion principle, not
from any direct spinspin coupling. Having concluded this brief inter
lude on the physical origin of the exchange energy, we may now discuss
some of the theories of the ground state of solid He3.
C. Microscopic Theory
As previously mentioned, any attempt to calculate the ground state
energy and wave function for solids from first principles is faced with
the problem of finding a solution to the many body problem. This is a
most difficult problem and can be solved only if approximations are
made. The nature of these approximations is generally determined by
the specific system under consideration. For solids of heavy atoms,
the root mean square deviation of the particles about their equilib
rium positions is small so that the harmonic approximation for the
potential, along with uncorrelated single particle wave functions may
be used. Such calculations for solid He3 have been spectacularly un
successful. Nosanow and Shaw35 have calculated the ground state energy
of noble gas solids, using uncorrelated single particle wave functions
and a LennardJones 612 potential. For the other .heavy noble gas
solids, the theoretical value of the cohesive energy is within the ex
perimental limits, while for solid helium, the calculated value is of
the order of 30 cal/mole, while experimental value is about 4.5.5
cal/mole. From this type of calculation it is evident that uncorrelated
single particle type functions are an inadequate description of the
ground state of solid helium. The reason for this inadequacy is that
these functions do not take into account the short range correlations
which arise from the very large zero point motion of the atoms.
The first attempt to include these short range correlations into
the theory of solid He3 was made by Bernardes and Primakoff,2 who made
a variational calculation of the ground state energy and wave function.
They reasoned that correlations in the motions of pairs of atoms could
be accounted for by replacing the true interaction potential by a
single parameter effective one. They assumed an analytical form for
this effective potential given by
V(r,A) = 442 ) e (2.7)
The parameter X was determined by fitting the known ground state energy
and root mean square deviation for crystalline He4. Their calculations
were approximately correct for the bulk properties of the solid, but
vastly overestimated the nuclear exchange energy. It is now realized
that due to the differences in crystal structure and statistics the ef
fective potential between nearest neighbor atoms of the two solids is
significantly different. The exchange integral is very sensitive to
the variations in the tail of the wave function which is determined
primarily by the repulsive part of the potential. Therefore, any ef
fective potential determined from He4data will not be accurate enough
to calculate quantitatively the exchange energy for solid He3. De
spite this shortcoming in their approach, Bernardes and Primakoff were
dJ
able to correctly predict that both J and d would be negative in the
bodycenteredcubic phase. Their value for J is about two orders of
magnitude too large and the pressure dependence of J as estimated from
variation of X with pressure is also much too large.
Nosanow and his coworkers19,28 have made a systematic study of
the ground state properties of quantum crystals by employing a varia
tional calculation of the energy based on a cluster expansion technique.
This group succeeded in calculating the ground state pressure and energy
of solid helium to a few percent, while obtaining reasonable agreement
with the available experimental values for the exchange energy. Since
the Nosanow treatment is the most comprehensive and gives the best agree
ment with all the available experimental data, a discussion of the as
sumptions and approximations of this theory will be given.
Nosanow's Theory of Quantum Crystals
If we consider a system of N particles of mass m interacting with
each other through a potential function V(rij), the Schrdedinger equa
tion is
m ~V(r icr ..N) = E (rl,r2...rN). (2.8)
i= i i< i I
For He3 the LennardJones 612 potential
V(r ) = 4( (12 )6 (2.9)
gives an adequate representation of the interatomic forces. In Eq.(2.9),
E and o are constants determined from the low temperature gas phase
data and have values of 10.220K and 2.55 X respectively.
The effects of short range correlations are included in the function
p by taking it to have the form
N N
(r...)i...r N) I i ) R f(r.), (2.10)
i=l i
in his expression is e coordinate of the i
in this expression i is the coordinate of the ith lattice site, ri is
the position vector of the ith atom and rij is the distance between the
i and jth atoms. The function i in Eq.(2.10) is not properly sym
metrized with respect to an interchange of He3 atoms and hence cannot
adequately describe exchange effects. In the actual calculation, the
two body exchange energy is accounted for by using an antisymmetrized
two particle wave function constructed from the appropriate linear com
bination of the 0i(riR ) and spin functions.
N > 4
The function iii (riRi) has the normal Hartree form. The boundary
conditions are
lim O(r) = finite (2.11a)
r 0
reO
lim O(r) = 0 (2.11b)
r +
4(r) # 0 for r > R/2, (2.11c)
condition (2.11c) permits the wave packets of neighboring He3 atoms to
overlap. Nosanow chooses #(r) to have the spherically symmetric form
4(r) = exp(Ar2/2) (2.12)
where A is a variational parameter to be determined. The function
iTjf(ri) is introduced to account for the short range correlations
in He3. By the particular functional form chosen, one can see that
only two body correlations are to be considered. The boundary condi
tions on f(r i) are
lim f(r) = 0 (2.13a)
r 0
rO
lim f(r) = constant. (2.13b)
r *+
Condition (2.13a) reflects the strong repulsion of He3 atoms at small
distances, while (2.13b) expresses the fact that at large distances the
atoms are essentially uncorrelated. One of the analytical forms used
by Nosanow for the correlation function is
f(r) = exp(K[ )12_ (611 (2.14)
where K is a variational parameter to be determined. The procedure is
to determine A and K by a variation of the energy
E = (,H (2.15)
o ( i,tI)
with respect to these parameters.
Up to this point, the main approximation in the theory is the
admission of two body correlations only. However due to the presence
of the function H f(r ), evaluation of (i,Hi) becomes difficult to
i
do without additional approximations. In order to evaluate (i,Hi),
Nosanow makes a cluster series expansion of the energy in such a way
that each successive term in the expansion makes a decreasing con
tribution to E It should then be possible to truncate the series
after a few terms, provided it converges rapidly.
To make the expansion, it is convenient to introduce the quantity
M(y) = (i,expyHi), (2.16)
so that
E = lim  InM(y). (2.17)
o y 0
In the cluster series expansion the M(y) is expressed in the form
N
M(y) = n M (y) (2.18)
n=l
where the only contribution to M arises from that volume of phase
n
space where n particles are grouped together in a cluster. Equa
tions (2.16) and (2.17) lead to an expression for the energy, Eo, of
the form
N
Eo = on (2.19)
n=l
where the Eon give the contribution to the energy of the system from
a cluster of n particles. For this series to be rapidly convergent,
it is clear that the contributions to E from clusters of four or
o
five particles must be much smaller than the single and pair particle
energies.
The derivation of the various Eon is accomplished by first cal
culating the Mn(y) and then using the expression Eon = lim lnMN().
It is quite difficult to do, especially for n > 2, and only the results
for the special forms of and f specified above will be presented here.
To conform with Nosanow's notation, we need to define the average value
of a function g(rl...ri...rN) over the weight function *(rl.. i.. r .rN
by
I IrN il)l12N... (2.20)
<8( ) l> d Il,(I i R )l(2.) )
g(ri > 2 dg(rifd, (2.21)
rJl),(rji1)g(r rZ (2. 21)
where
W> ( SJ)12d. (2.22)
The ground state energy is then given by the expression
E0 = EO1+EO2V+EO2T+E02J+E3V+E03T' (2.23)
where the various terms have the following forms.
N 21) V (Iri ij]dri
E 01= (2.24)
i=l1 (I r )(IRi d dri'
and
1
O2' Vff( ) (2.25)
i,j
with
Vff = (V(ri) m V2nf(ri))f2(ri ). (2.26)
E02T = E3T = 0 for the particular form of O(r) chosen.
E =*+ 1 )3 (2.27)
02J 4 i,j ij
where
r 2 2R2A2 2
ij 2m + J (2.28)
has been defined such that Jij is the difference in energy between
the singlet and triplet spin states. Finally
1
03V 2 i ,k (2.29)
The form of each term can be related very nicely to physically
intuitive quantities. E01 is the single particle contribution to the
energy of the solid. It has been expressed in a form closely resembling
the ground state energy of a system of N harmonic oscillators. The
second term E02V measures the contribution to the energy produced by
the atoms taken in pairs. The LennardJones 612 potential however, is
replaced by an effective potential which depends upon the form of the
short range correlation function f(rij). The term E03V is slightly
more general than EO2V. It can be considered to be the effective po
tential energy resulting from all possible interactions involving three
particles. Since this includes contributions from two particle inter
actions already counted in E02V, these must be subtracted out. E02J
is the two particle exchange energy and was calculated using a two
particle antisymmetric wave function of the form
2(,rj) = [(riji )rjW)A(Il)jRj)]f(rij)S(si,sj) (2.30)
where the plus and minus sign in Eq.(2.30) go with the singlet and tri
42R2A2
plet spin states respectively. In Eq.(2.28) the term 2m
2m
is the exchange kinetic energy and is seen to have a negative sign.
Furtherit turns out to be about three orders of magnitude larger than
2x
the second term in brackets. The expression is the contribu
tion to the exchange integral from effective potential in the overlap
region. From Eq.(2.28) one sees that the sign of J will be determined
by the relative magnitude of these two terms. If the former term is
greater, J will be negative and the magnetic ground state will be anti
ferromagnetic, while if the latter is greater, J will be positive and
the ground state will be ferromagnetic. It is interesting to note that
Ji = 0 when and = 0, that is when there is no overlap of
the various x e
the various (i.
In Nosanow's 1966 paper, which we shall refer to as N, the basic
assumption is that the three body and exchange contributions to the
ground state energy are small so that E0 is approximately given by
E0 = E01+E 02V (2.31)
The values of the parameters K and A can then be obtained by minimiz
ing E0. These values may then be used to calculate the terms, E03V
and E02J in the cluster expansion. If these terms are small, the
cluster expansion is assumed to converge rapidly.
In a subsequent paper by Hetherington, Mullin, and Nosanow,28
(HMN) the three body term E03Vi is included in EO and the parameters
A and K are chosen to minimize
EO = E01+E 2V+E03VW (2.32)
The actual variational calculation itself becomes formally identi
cal with the single particle Hartree calculation of Nosanow and Shaw35
except that the true interaction potential V(rij) is replaced by an
effective potential which is approximately given by
Veff(rj) = f2(rij)[(r) n V21nf(r)]. (2.33)
When viewed in this manner, the Nosanow approach is seen to be similar
to that used by Bernades and Primakoff. Namely, the effects of cor
relations are taken into account by replacing the true potential by an
effective one. However, in the Nosanow theory, veff is found by mini
mizing the ground state energy of He3 while in the Bernades and Prima
koff theory, the effective potential was determined from the known
ground state properties of He4. The results of these calculations are
summarized in Fig. (1), (2), and (3).
15
J
O \\
1
I \
0
1 
w 13
Ix
10
III I I I I
20 21 22 23 24
MOLAR VOLUME (CM3/MOLE)
Figure 1. Calculated ground state energy versus molar volume. The
dashed curve was obtained by minimizing E01+EO2V while the
solid curve was obtained by minimizing E01+E02v+EO3V
23
1.9 A  N .84
 HMN
LOG K
1.7 .80
AY
< 1.6 LOG K .78 0
o
1.5 .76
1.4 .74
1.3 .72
I I I I I
20 21 22 23 24
MOLAR VOLUME (CM3/MOLE)
Figure 2. Variational parameters A and log K versus molar volume.
The dashed curve gives the parameters that minimize
E01+E02V, and the solid curve gives those that minimize
E1l+Eo2V+EO3v. Larger values of A correspond to a
greater localization of the atoms about their lattice
sites.
R (A)
3.45 3.55 3.65 3.75
0.10
0.08
0.06
N
0.04
HMN
0
E
= 0.02
0.01
I I I I
18.5 20.5 22.5 24.5
MOLAR VOLUME (CM5/MOLE)
Figure 3. Calculated nuclear exchange energy versus molar
volume. Curve N was calculated using the param
eters which minimize EOI+EO2V; curve HMN was cal
culated using those which minimize Eol+E02V+EO3V.
From Fig. (1) we see immediately that the inclusion of the term
E03V has only a small effect on the total ground state energy EO, and
thus the truncation of the energy expansion seems valid. Moreover, in
HMN, a physical argument based upon the short range nature of the cor
relation function f(rij) is given which indicates that higher order
terms in the cluster expansion will be small. From Fig. (3) however,
it can also be seen that while the changes in A and K introduced by
the inclusion of E03V in E0 do not greatly affect the values of the
total energy, they do significantly alter both the value and the shape
of J(R). This should not be too surprising and can be understood by
the following argument. The main contributions to E0 come from E01 and
E02V. The former is completely independent of the choice of K and
depends only on the width of *(r), hence small changes in A will pro
duce only slight variations in E0. While EO2V does depend on both f(r)
and *(r), the dependence is such that small changes in A and K do not
greatly affect EO2V. This is because it is related to the average
values of f(r) and *(r) and not their detailed structure. On the other
hand, the quantity
t2R2A2 2x
S(R) 2+ f (2.34)
ij) 2m
depends on a knowledge of the details of both f(r) and #(r). In par
ticular, Jij is most sensitive to variations in the wings of 0(r) since
this is where the overlapping occurs. Also, it is expected that Jij
should be very sensitive to changes in the function f(r) since it is
this function which describes the short range correlations between
28
neighboring atoms. Nosanow and Mullin28 have investigated the sensi
tivity of iJ to the function f(r). They varied K from 0.14 to 0.18
and found that Jij changed by a factor of 3 while the energy changed
by only 1%. They also changed the first exponent in f(r) from 12 to
8 and found that while J varied by a factor of 4, the total energy
changed by only 10%. This sensitivity of Jij and insensitivity of E0
to slight modifications of the parameters illustrates a basic defi
ciency in all variational calculations of the exchange energy. Namely,
that, although the energy is determined to a few percent, the wave
function is not an accurate enough representation of the true ground
state function to enable one to calculate accurately the value of the
exchange energy.
Before ending this discussion of the Nosanow theory, some con
sideration should be given to the use of spherically symmetric func
tions for O(r) and f(r). It is known that the low density phase of
solid He3 has a bodycenteredcubic crystal structure. This suggests
that spherically symmetric functions will be a good representation for
the motion of atoms for small values of r. However, for large values
of r (r > ), the cubic symmetry of the lattice must be reflected in
the wave function. It is precisely in this region that the overlap
integral is large and hence the exchange energy sizeable. It is
conceivable, therefore, that the assumption of spherical symmetry in
troduces an error into the calculations of the exchange integral.
Numerical values for O(r) have been considered in the theory, and
generally speaking they tend to make the exchange energy somewhat
larger.
The above remarks are intended only to illustrate some of the
problems associated with any theoretical attempt to calculate the
exchange energy accurately. In point of fact, the clarity and physical
basis for the assumptions and approximations, coupled with the good
agreement between the theoretical calculations of Nosanow and experi
mental data make this work a significant contribution to the understand
ing of solid He3.
D. Thermodynamic Theory
In this section we will obtain an approximate equation of state
for a solid composed mainly of He3 atoms but containing a small amount
of He4 impurities. To do this it is necessary to construct suitable
models for the various degrees of freedom of the system. The choice
of these models will be based upon both experimental information and
theoretical intuition.
The experiments of Edwards, et al.24 show that below 0.50K the
specific heat of the lattice in the bcc phase can be represented by a
Debye term plus a term arising from the phase separation of a regular
mixture. We may thus write this contribution to the free energy in
the form
FL(x,V,T) = FD(x,V,T)+FpS(x,V,T)+UO(x,V). (2.35)
In Eq.(2.35) U0(x,V) is the zero point energy, FD(x,V,T) and FpS(x,V,T)
are the Debye and phase separation contributions to the free energy,
and x is the concentration of the He4 atoms.
Since the exchange energy arises from localized spin 1/2 particles,
one should be able to treat the magnetic interactions on the basis of
a Heisenberg Hamiltonian of the form
H = 2 Jij(xV)I'j (2.36)
Si exhane i b n ni ,
where Jij(xV) is the exchange integral between nearest neighbor atoms,
and the summation extends over nearest neighbor atoms only. Using this
Hamiltonian, one can then calculate a partition function Qex(x,V,T)
from which the magnetic free energy Fex(x,V,T) may be obtained. The
total free energy will then be the sum of Fex and F and is given by
F(x,V,T) = U0(x,V)+FD(x,V,T)+FPS(x,V,T)+Fex(x,V,T). (2.37)
The equation of state of the system can then be obtained from the re
lation
P(x,V,T) = F(x T) (2.38)
L DV ,T
In order to determine the functions FD, FPS, and Fex it is useful
to divide the temperature scale into three regions as shown below.
III II I
STwo Phase Region Phase Separation Region Single Phase Region
W F = Fex(o,V,T) F = Fps(x,V,T) F FD(Xo,V,T)
0.06 0.30
Temperature
In region I, where FD(x,V,T) is the main contributor to the energy,
the solid exists as a single homogeneous phase having a temperature inde
WT(xo,V)
pendent concentration x. We may, therefore, write FD(x,V,T) = FDI (xT
where 6(xo,V) is the characteristic Debye temperature for the solid.
Region II is the isotopic phase separation region. In this temperature
range the solid is transformed from a single homogeneous phase of con
centration x into two separated phases. Initially, the He4 concentration
of these separated phases varies rapidly with the temperature; however,
by the time one reaches 0.060K, the separation into pure phases is
practically completed. Below about 0.060K, therefore, the solid exists
in the form of a large nearly pure He3 phase in equilibrium with a small
He4 phase. The energy in this range will thus be essentially that of a
pure He3 system. We may then write the magnetic contribution to the
energy in the form F (x,V,T) = F (o,V,T). The total free energy then
becomes
rB(x ,V)i
F(x,V,T) = UO(V)+F D[ T +FS(x,V,T)+Fex(o,V,T). (2.39)
We will now obtain expressions for the various terms in Eq.(2.39).
The spin partition function Q ex(v,T) is given formally by the expression
Hex/kT +k i I
Qex(V,T) = trace e = trace e i< (2.40)
Rushbrooke and Wood36 have made a series expansion of this function in
powers of . They find that at high temperature (T >> J/k) Eq.(2.40)
reduces to
nQ Nln(21+1)+i NzI2 (+1)2( )2. (2.41)
in9ex 3 Nln(21+)+
For He3 in the bcc phase, I = 1/2 and z = 8 so that the above equation
reduces to
lnQe N[ln2+ ( )2]. (2.42)
Fex is related to the partition function Qex by the relation
Fex(v,T) = kTlnQex, (2.43)
so that the magnetic contribution to the pressures becomes
2 dJ 1
P (VT) 3Nk()2 d (2.44)
which may finally be put in the form
3R j 2 1
Pex(V,T) = yex() 2, (2.45)
where
x dlnVJ (2.46)
Yex dlnV
We can see from Eq.(2.45) that in Region III the pressure is pro
portional to T1 with the constant of proportionalty being directly re
lated to the strength of the exchange interaction J.
To obtain an expression for the free energy in the phase separation
region we recall that the specific heat measurements of Edwards, et al.24
indicated that the He3 and He4 atoms mix together to form a regular
solution. As a consequence of this, the internal energy and entropy of
the system may be written as
U(x,V,T) = [xE44+(1x)E33+2x(lx)E ], (2.47)
S(x,T) = Nk[xlnx+(lx)ln(lx)], (2.48)
E33 E44
where EM = E34 2 and Eij is the energy of interaction between
an atom of isotope i and its nearest neighbors of isotope j. For the
case x << 1, Eqs.(2.47) and (2.48) reduce to
U (E33+2xE), (2.49)
S = Nk(xlnxx). (2.50)
When written in this form, one sees that the stable configuration of
the system depends upon the sign of EM. If EM is positive (2E34> E33+E44),
then the energy of the mixed phase is greater than that of the separated
phases and the system will separate into pure isotopes at absolute zero.
To find the temperature at which this separation begins, we minimize the
free energy F = UTS. For the present case, we have
FpS(x,V,T) = i(E33+2xEM)+NkT(xlnxx). (2.51)
The equation for phase separation curve is derived from the stability
condition
S= 0. (2.52)
V,T
This yields an expression for the phase separation curve in the Tx
plane given by
x = eEM/kT (2.53)
where x is the concentration of the He4 enriched phase at a temperature
T and volume V. The free energy in this two phase region is obtained
by substituting the expression for x given by Eq.(2.53) into Eq.(2.51).
The result is the simple expression
FpS(x,V,T) = E33NkTeEM/kT. (2.54)
The internal pressure arising from the isotopic phase separation is then
given by
R dEM MIT (2.55)
PS k dV (2
Equation (2.55) shows that the size of PPS is linearly related to the
rate of change of the energy of mixing with density.
Finally the contribution to the pressure in the high temperature
re(xo,V)
region can be obtained from the Debye free energy FD [ and the
expression
xF 3F
PD(xo,V,T) = ,D ) D (2.56)
x,T
or alternatively
1 dO
PD(XoV,T) = UD(T,O) dV. (2.57)
For solid helium T << 8D and this becomes
3.4R T 3
P ) Ty, (2.58)
D 5V 1
dine
where y dlnV is the Gruneisen parameter.
dlnV
The approximate equation of state for the system is then
3R 1 R dE 3
oV) V1 + EMkT R Ty. (2.59)
P(x,V,T)Po 01 ex Y k dV 5V
This equation will be used in Chapter IV to obtain the values of IJI,
'ex, and EM.
CHAPTER III
EXPERIMENTAL APPARATUS AND PROCEDURE
A. Introductory Remarks
The apparatus used to obtain the experimental results reported
in this work incorporates many of the standard techniques of low tem
perature physics with enough new ideas to make it somewhat unique. It
seems natural that we should describe these distinguishing features
with considerable detail and present only a brief description of those
sections which are conventionally used in low temperature research
throughout the world. Also, as is usually true, the unique sections
of the apparatus are also the most important in determining the suc
cess of these particular experiments. For these reasons we shall de
scribe in detail the design and construction of both the potassium
chrome alum salt assembly and the capacitance strain gauge, while pre
senting only a brief description of the helium refrigerators, vacuum
systems, pressure system,and superconducting solenoids. For more de
tails on these latter sections of the apparatus,the reader is referred
to the earlier works of P. J. Walsh3 and G. C. Straty39 and also to
the standard books40 on the techniques used in low temperature physics.
We have also tried to follow the same philosophy in describing the
manner in which the experiments were performed. Long discourses on
experimental procedure from initial cooldown to final shutdown tend to
become somewhat boring, and hence we have included only those aspects
of the experimental procedure which indicate the precautions taken to
insure the validity of the final results.
B. Cryostat
The cryostat in which the experiments were performed is a modified
version of that described by Walsh38 and is shown schematically in
Fig. (4). A more detailed drawing of the low temperature section is
shown in Fig. (5). Three stages of refrigeration are required to re
duce the temperature of the He3 sample from 4.2 to 0.020K. The tem
perature io lowered initially to 1.0K by reducing the vapor pressure
above a liquid He4 bath. A further reduction to 0.30K is obtained using
a continuously operating He3 refrigerator system. The final stage of
cooling is accomplished by adiabatic demagnetization of a potassium
chrome alum [CrK(SO4)212H20] salt pill.
Helium Refrigerators
In this apparatus, the 1K He4 bath was contained in a cylindrical
container which could be filled with liquid from the outer bath by
means of a modified Hoke valve (not shown). This valve has a stem
which extends through the top flange of the cryostat so that it may
be operated from outside the helium dewar. To increase the thermal
contact between the boiling liquid and its container, a copper spiral
wound from 0.013 inch copper sheet was soldered to the bottom of the
inside surface. The volume of the container is about 250 cm3 and one
filling provided 10K operation for a period of about forty hours. A
model KC46 Kinney pump provided the necessary pumping speed to maintain
Figure 4. Schematic diagram of the apparatus.
A. Sample filling capillary
B. Exchange gas pumping line
C. He4 bath pumping line
D. Manostat
E. He3 refrigerator diffusion pump
F. N2 cold trap
G. He3 refrigerator pump
H. He4 bath
I. He3 refrigerator
J. Vacuum jacket
K. He4 recovery line
L. Connection to oil and mercury manometers
M. Vacuum flange
N. Electrical Connections
0. Potassium chrome alum salt
36
N
L
G
Figure 5. Schematic diagram of the low temperature section.
A. Various pumping lines
B. Main support and vacuum flange
C. Sample filling capillary
D. He3 refrigerator return line
E. Electrical connections
F. Indium gaskets
G. He4 bath
H. Vacuum chambers
I. Evaporator section of He3 refrigerator
J. Lead heat switch
K. Nylon support tube
L. Potassium chrome alum salt
M. Demagnetization and zinc heat switch solenoids
N. Zinc heat switch
0. Sample chamber
P. 2000 #44 copper wires
Q. Vacuum jackets and radiation shields
R. Cerium magnesium nitrate thermometer
S. Primary of the mutual inductance system
T. Measuring and compensating secondaries
U. Teflon spacers
38
E
L
M
P
S
I1
the temperature at 1"K,even in the presence of substantial heat loads.
Temperature control in the region between 4.20and 1K was accomplished
by pumping through a diaphragm type manostat which allowed the pressure
to be regulated to 1%. Vapor pressures were measured by mercury and
oil manometers connected into the pumping line. The bottom of the con
tainer was used as a support flange for the inner vacuum jacket which
also served as a 10K radiation shield.
The He3 refrigerator system was designed for cyclic operation. It
consisted of a cylindrical container having a volume of 1 cm3 which
served as the evaporator, an NRC type B2 oil diffusion pump in series
with a Welsh DuoSeal pump modified for closed system operation, and a
constricted capillary used to produce the pressure necessary to re
liquify the returning He3 gas. A liquid nitrogen cold trap, located in
the He3 return line, was used to prevent oil vapors from entering the
cryostat and possibly plugging the pressure dropping capillary. The
dead volume inside the Welsh pump above the oil was used to store He3
gas during shutdown periods. A network of mercury and oil manometers
along with a CVC type GM 100 McLeod gauge was connected into the pump
ing line. These could be used to measure the vapor pressure of the
liquid He3 under static conditions during temperature calibrations.
A carbon resistor which served as both heater and thermometer was at
tached to the evaporator section. By closing the return line before
demagnetization it was possible to maintain a temperature of about
0.280K at the evaporator section of the refrigerator.
Vacuum Chambers and Radiation Shields
Two vacuum chambers and four radiation shields were used to reduce
the heat leak into the sample and cerium magnesium nitrate (CMN)
thermometer. An outer jacket, used to provide thermal isolation from
the main He4 bath, was attached to a flange located in the cryostat
stem. An indium gasket made from 0.075 cm diameter wire was used for
the vacuum tight seal. A second vacuum chamber, surrounding the He3
refrigerator and salt system, was attached to the bottom of the 1K
bath again using an indium gasket for the vacuum tight seal. During
temperature calibrations, this chamber was filled with He4 exchange
gas used to provide thermal equilibrium between the helium baths and
the thermometers. The walls of the chamber were thermally grounded.
to the inner He4 bath and hence provided a 1K radiation shield for the
He3 refrigerator. A third shield, in the form of a copper plated
brass cage, was screwed onto a support flange thermally grounded to
the He3 refrigerator. This cage, which contained eight small windows
used for viewing the potassium chrome alum salt, completely surrounded
the pill thereby providing it with 0.30K ambient. The construction
and.use of the final radiation shield will be discussed in the salt
assembly section of this chapter.
Electrical leads, with the exception of the coaxial lines used
for leads to the capacitor plates, were brought into the outer vacuum
chamber through a small stainless steel tube. A vacuum tight glass
tometal seal was used to bring the leads through the 10K bath into
the inner vacuum chamber. Thermal grounding to 0.3K was accomplished
by soldering the leads to small glasstometal seals mounted on the
He3 refrigerator. The leads were made from #36 Advance wire and a
sufficient length was allowed between the refrigerators and the salt
pill to provide the necessary thermal isolation. Coaxial lines were
used as leads to the capacitor plates of the strain gauge wherever
possible. Where unshielded leads were used, care was taken to separate
them as far as possible in order to reduce distribution capacitance.
This distributed capacitance was measured with the leads disconnected
at the capacitor plates, and found to be about 0.1 pf. At the working
pressures encountered in these experiments, the strain gauge capacitance
had a value of about 12 pf. The quantity C = C where AC is the
Cg+%
g d
capacitance change produced by a pressure change AP, and C and Cd are
the strain gauge and distributed capacitance respectively, was affected
by less than 0.8% by Cd. More importantly, we observed no evidence
which indicated that the value of Cd changed during the course of
these experiments. In particular, it was possible to refill the li
quid nitrogen bath, as well as the outer and inner He4 baths, without
affecting the value of the distributed capacitance. We are quite cer
tain therefore that the observed capacitance changes resulted entirely
from changes in the value of the strain gauge capacitance, C .
A discussion of the final stage of cooling and thermometry will
be given after the description of the sample cell and pressure measure
ments.
C. The Strain Gauge
In order to make pressure measurements on samples of solid He3,
it is necessary to incorporate onto the sample cell some pressure
sensing device. This is because when the sample freezes in the cell a
solid plug also forms in the filling tube thereby isolating the sample
chamber from any external pressure sensing devices. In these experi
ments a capacitance strain gauge which relates the internal pressure of
the sample to the capacitance of a parallel plate capacitor was used.
The details of the strain gauge and sample cell are shown in Fig. (6).
The main section of the cell was machined from a 5/8 inch long,
7/8 inch diameter stainless steel (type 304) cylinder. A hole, 1/2 inch
in diameter and 3/8 inch deep was bored into one end of the cylinder,
while the opposite end was machined, as indicated in Fig. (6), to a
depth of 0.205 inches. The resulting diaphragm, having a diameter of
1/2 inch and a thickness of 0.045 inches, constitutes the active ele
ment of the strain gauge. A stainless steel plug, containing a copper
piece used to increase the thermal contact between the He3 sample and
the chamber walls, was designed to fit tightly into one end of the
cylinder to a depth of 5/16 inch. The plug, which also contained a
0.025 inch i.d. copper capillary, was silver soldered into the main
body of the chamber through two access holes drilled into the walls.
Because of the snugness of the fit, no solder or flux flowed into the
sample volume. The He3 sample on which the pressure measurements were
made, was located in a 1/2 inch diameter, 1/16 inch long cylindrical
volume. This pancake geometry,with its large surface area to volume
ratio, helps relieve any internal pressure gradients which may occur
during the formation of the solid, and also helps to decrease the
thermal time constant between the He3 sample and the chamber walls.
This point will be discussed in greater detail later in this chapter.
A copper cylinder, containing four wells used to hold the resistance
thermometers and the heater, was soldered to the steel section. This
cylinder also contained a #8 copper wire which served as a thermal
grounding post for attaching a zinc heat switch used to provide thermal
contact between the salt pill and the sample chamber. The switch was
Figure 6. Capacitance strain gauge.
A. 2000 #39 copper wires to potassium chrome alum salt
B. Sample filling capillary
C. Zinc heat switch
D. Bakelite support
E. A #8 copper wire
F. Resistance thermometer
G. Access holes for soldering
H. Copper cylinder
I. Stainless steel chamber and plug
1 1"
J. x diameter sample volume
K. 0.045" diaphragm
L. Capacitor plates
M. 2000 #44 copper wires to CMN thermometer
N. Bakelite support for the CMN thermometer
44
A
B
C
D
E
F
M
*k fr^
in the form of a thin foil having an area to length ratio of 0.25 mm.
The cell was supported by a bakelite rod which extended from the salt
pill.
The sample cell was filled with liquid through a stainless steel
capillary having an o.d. of 0.033 cm and a 0.006 cm wall. A ff36 Ad
vance wire was inserted into the capillary to reduce its volume still
further. A length of about 25 cm was thermally anchored to the He3
refrigerator thereby providing a sufficiently long solid He3 plug to
prevent any slippage of material into the pressure cell. In addition
to this, a length of about 20 cm was thermally grounded to the He4 bath.
No problems with plug slippage were encountered during the course of
these experiments.
After the components of the chamber were soldered together, the
two capacitor plates were mounted onto the strain gauge. The active
plate, in the form of a circular disk having an area of 0.625 cm2, was
fastened with epoxy to the diaphragm. Tissue paper and epoxy serve to
electrically insulate the plate from the chamber walls. The fixed plate
was made in two sections as indicated in Fig. (6). The inner disk,
having an area of 0.625 cm2, was pressed into a tapered hole in the
outer guard ring. Two layers of 0.001 inch mylar were used to elec
trically insulate the central plate from the outer guard ring. The
plate was attached to the main section of the chamber by means of four
#436 steel screws.
The sensitivity of the gauge to a pressure change AP is propor
tional to A/d2 where A is the plate area and d is the plate spacing.
In order to obtain a high sensitivity, it is therefore advantageous
to use as small plate spacing and as large a plate area as possible.
In this experiment, the plate area was 0.625 cm2 and the plate spacing
at one atmosphere was chosen to be 0.0025 cm. To insure that the plates
would not short together at the working pressure of approximately forty
atmospheres, a method for setting the plate spacing devised by Straty39
was used. After the active plate was fastened to the diaphragm, the
sample chamber assembly was chucked in a lathe and light cuts were taken
across both the chamber and the plate. Facing both the chamber and the
plate in a single cut insured that both surfaces would be coplanar.
The desired plate spacing was obtained by inserting a brass shim be
tween the main body of the cell and the fixed plate. This particular
geometry allowed both the active and fixed plates to be mounted on the
same piece and thereby reduced undesirable changes in the plate spac
ing which arise from a differential thermal expansion of materials in
the gauge. The choice of steel rather than copper as the material to
be used for the main body of the strain gauge was motivated by the re
quirement that the diaphragm exhibit no pressure hysteresis. An earlier
chamber constructed entirely from copper possessed enough hysteresis to
prevent accurate determinations of the exchange energy. The present
chamber has no detectable hysteresis.
The theoretical sensitivity of the gauge can be calculated from
the equation for the deflection of a circular membrane fixed around
its circumference. If a pressure change AP is distributed uniformly
over its surface, the diaphragm will deflect an amount given by41
0.054R4AP (
6 , t3 (3.1)
wherEt
where
R = radius of the diaphragm in inches
t = thickness of the diaphragm in inches
E = modulus of elasticity in psi
and
AF = pressure change in psi.
AC
The fractional change in capacitance C is approximately given by
AC 6
AC d (3.2)
C d
where d is the plate spacing. Solving equations (3.1) and (3.2) for
AP gives
EtsdAC
6P = 0.54RC (3.3)
The strain gauge capacitance has a value of about 12 pf and is measured
by means of a General Radio type 1615A capacitance bridge used in con
junction with a type 1404B standard capacitor. This arrangement per
C
mits a ~ = 107 to be measured. Using the values of E, t, R and d ap
propriate to this chamber, one obtains a minimum detectable pressure
change of 3 x 105 atm. The measured sensitivity at forty atmospheres
was also 3 x 105 atm.
The data obtained in these experiments consist of a series of
values of pressure as a function of temperature for a given solid sample
contained within the volume of the sample cell. Although the determina
tion of the sample pressure depends upon the deflection of the chamber
diaphragm, the following considerations show that for all practical
purposes the measurements are performed on constant volume samples. If
we consider the.pressure as a function of temperature and volume we ob
tain
dP = a dT+ ~ dV, (3.4)
V T
or equivalently
SdP 3 dV (3.5)
S  T (3.dT5)
By making use of the definition of the isothermal compressibility
S ~ Eq.(3.5) can be written as
V d 1+ b d (3.6)
where 8 is the compressibility of the helium sample.
The factor VI dV
The factor V1 V is determined by the mechanical properties of
the chamber and for this chamber is about 2 x 105 atml. In the
volume range covered by these experiments the value of B varies from
about 3 x 103 atm1 to 5 x 103 atm1, and hence the second term in
brackets will never be greater than 102. Thus = '( to within
1% or better.
D. Pressure Measurements
Gas Handling and Pressure System
The He3 pressure and gas handling system is shown schematically
in Fig. (7). A system of three glass bulbs and two Toepler pumps was
used for both storing and moving the gas. The Toepler pumps were con
nected through a liquid helium cold trap into one leg of a mercury
Utube pressure system. The Utube has sufficient length so that the
He3 side could be evacuated with a pressure of one atmosphere on the
opposite side. This side of the Utube is filled with oil and con
nected to a dead weight gauge which served as both calibrating de
vice and pressure manostat. A small correction was made to the pressure
Figure 7. Pressure system.
A. Vacuum line
B. Sample filling capillary
C. Oil reservoir
D. Pressure gauge
E. Dead weight gauge
F. He4 cold trap
G. Toepler pumps
H. Mercury Utube
I. Hydraulic pump
J. Glass storage bulbs
K. Connection to cryostat
L. N2 cold trap
M. Nylon insulator and pressure seal
LL
to compensate for the difference in the mercury levels in the Utube.
The Toepler pump connection to the gas side of the Utube was made a
bout 20 cm below the top so that the sample gas could be trapped in
the Utube at a low pressure. The sample filling capillary connection
at the top of the tube was made through a Nylon seal. Since the U
tube was electrically insulated from its supporting structure, the
mercury in the tube could then be used as a switching device to actuate
an alarm which signaled the entrance of mercury into the capillary
system. A small section of the external filling capillary was im
mersed in a nitrogen cold trap so that if a pressure leak developed in
the capillary system no mercury would enter the cryostat.
Pressure Calibration and Measurement
The calibration of the strain gauge as a function of pressure was
performed using the AMICO #472161 dead weight tester. The accuracy
of this gauge is 0.05%, however, the uncertainty in the relative posi
tions of the mercury columns limits the absolute accuracy at all pres
sures to about 0.03 atm. Calibration points were taken for both in
creasing and decreasing pressures with no detectable hysteresis.
Since the temperature at which the calibrations were made was held
just above the freezing temperature of the He3 sample corresponding to
the particular density to be studied, the calibration conditions were
not identical to those under which the experimental data were obtained.
This was necessary to prevent a solid plug from forming in the capillary
system and isolating the strain gauge from the external pressure system.
To determine the effect of temperature on the capacitance of the strain
gauge, a separate demagnetization was performed with the sample cell
evacuated. Aside from a small anomaly occurring at about 0.15K, the
temperature variation of the capacitance was completely negligible.
This anomaly will be discussed in more detail in the following chapter
since it has some effect on the quantitative accuracy of the phase
separation data.
As was stated previously, the capacitance was measured by means
of a General Radio type 1615A capacitance bridge used in conjunction with
a General Radio 1404B standard capacitor. The GR 1615A capacitance
bridge was located inside a Styrofoam container and its temperature
regulated electronically to within 0.20C. This was done in order to
reduce the drift of the capacitance bridge reading which resulted from
variations in the room temperature. It was determined empirically that
the drift rate of the bridge reading with room temperature was about
10 af/oC.
The strain gauge capacitance was about 12 pf and could be measured
to *1 af, giving a relative sensitivity, , of about 107. The ca
pacitance and pressure are linearly related over the range of the ex
perimental data taken on a particular density, and hence the conver
sion of capacitance readings to pressure value was accomplished in
a straightforward manner.
E. Potassium Chrome Alum Salt Assembly
Potassium chrome alum was chosen as the paramagnetic refrigerant
because it possesses a Schottky type specific heat anomaly at about
15 mOK, and hence may be used to lower the temperature to this region.
Furthermore, it has a large specific heat in the temperature range
between 0.015K and 0.10K, thereby permitting one to make measurements
over a period of many hours, providing the residual heat leak is kept
low. The size and geometry of the salt pill were chosen to utilize a
previously constructed niobium zirconium solenoid.
A cross section of the salt pill and its support assembly is shown
in Fig. (8). Thermal contact with the He3 refrigerator was made by
bolting the copper support flange to the bottom surface of the evapora
tor. Apiezon N grease was used as a thermal bonding agent. The flange
was threaded so that it could be used to support the 0.30K radiation
shield which completely surrounded the salt. One end of a Nylon support
tube as screwed nto the bottom of the copper flange while the other end
was clamped to copper wires in contact with the salt crystals. The
Nylon support tube, which also served to center the salt pill inside
its 0.30K cage, had an o.d. of 0.625 cm, a wall thickness of 0.5 mm,
and a length of 5 cm. A lead heat switch in thie form of a thin foil
having an area to length ratio of 0.1 mm thermally linked the mounting
flange with the salt pill. The switch was positioned so as to be
closed (normal) when the salt was fully magnetized and open (super
conducting) after the initial step in the demagnetization had taken
place. Before being installed, the lead was etched in warm nitric
acid in an attempt to improve its onoff ratio.
The salt pill, which consisted of a mixture of 170 grams of pow
dered potassium chrome alum and Apiezon J oil sandwiched between sheets
of #39 insulated copper wire, was assembled in a specially constructed
press. The mixture was tightly compressed into a cylinder 16.0 cm
long and 2.92 cm in diameter. The copper wire sheets were arranged so
that those coming out the top of the salt were thermally separated by
about a 1/16inch layer of salt from those extending out the bottom.
This was done so that after demagnetization, the thermal impedence be
tween the He3 refrigerator and the sample would be as large as pos
sible. The total contact area between the copper wires and the salt
J oil mixture is about 2000 cm2. Using the specific heat and Kapitza
42
boundary resistance data of Vilches and Wheatley, we calculated a
thermal time constant at 0.02'K for the saltJ oilcopper wire system
of 15 minutes. A Speer grade 1002, 1/2 watt resistor having a nominal
value of 220 ohms was embedded in the body of the salt and could be
used to monitor its temperature during the experiment.
The salt pill itself was contained inside a phenolic tube having
an inside diameter of 3.0 cm and a wall thickness of 0.8 mm. A cylin
drical sheath, made from two layers of perpendicularly wound #39 in
sulated copper wire lightly coated with epoxy resin, was glued to the
inside wall of the phenolic tube. This sheath was used to provide
thermal contact between the salt crystals and a radiation shield which
completely surrounded the He3 sample and cerium magnesium nitrate (CMN)
thermometer. The radiation shield, which was made by glueing two layers
of #39 insulated copper wire onto a phenolic tube, was supported by a
Bakelite flange as indicated in Fig. (8). The o.d. of the shield is
the same as the i.d. of the copper sheath and hence they fit together
very snugly. Apiezon N grease was used as the thermal bonding agent
between the sheath and shield. This arrangement provided an ambient
temperature of about 0.0250K for the sample and CMN thermometer system.
Small cotton balls cemented to the top of the salt pill, and a Nylon
spacer screwed into the bottom of the 0.0250K shield, were used to
provide the final alignment of the salt in its 0.30K cage.
Figure 8. Potassium chrome alum salt assembly.
A. Copper support flange
B. Electrical terminals
C. Lead heat switch
D. Nylon support tube
E. Sample filling capillary
F. Nylon clamp and terminal strip
G. Sheets of copper wires
H. Phenolic tube
I. Salt and Joil mixture
J. 220 ohm Speer resistor
K. Bakelite support flange
L. Copper wire sheath
M. Brass support screw
N. 0.0250K radiation shield
_ __ ~_~~~
56
A
B
D
E
F
G
N
Before ending this discussion of the salt system, a few remarks
concerning the residual heat leak should be made. In any of the stand
40
ard books40 on low temperature techniques, one can find formulas for
calculating heat leaks due to conduction down solid supports, and ra
diation from surrounding walls. In most cases these calculations yield
results which are correct to within about an order of magnitude only.
This is due in part to the difficulty of estimating the effects of im
purities and strains on the low temperature thermal conductivity of
materials. Also it is difficult in many cases to include quantitatively
the effects of the thermal boundary resistance between solids at low
temperatures. Finally, estimates of the vibrational input from mechan
ical pumps and other sources are at best educated guesses. For these
reasons, the author believes that the choice and dimensions of materials
to be used in the design and construction of a paramagnetic salt system
should be governed primarily by their successful use in similar systems.
In this respect the exhaustive study of the properties of materials at
8 42 43
low temperatures by the cryogenic group at the University of Illinois8,243
has been extremely helpful.
In these experiments the sample warm up rate at 21 moK was less
than 0.1 mK/hr. The corresponding residual heat leak was approximately
15 ergs/min. Because of the rather elaborate precautions taken to iso
late the sample from vibrations and high temperature radiation, we be
lieve the major portion of this residual heat input comes from conduc
tion down the lead heat switch, Nylon support tube,and cotton spacers.
F. Temperature Measurements
Temperature Calibrations and Measurements
The magnetic susceptibility of powdered cerium magnesium nitrate
(CMN) is known44 to obey Curie's law, X = C/T, to temperatures as low
as 6 mK and perhaps lower. Since the constant C can be determined by
measuring X in a known temperature region, the substance is an excellent
one to use for very low temperature thermometry. However because of
the relatively small size of the constant C (about 1/10 as large as
that of potassium chrome alum), considerable care must be taken to avoid
spurious contributions to the measured susceptibility for other weakly
magnetic materials present in the cryostat.
In these experiments the relative susceptibility of ten grams of
powdered cerium magnesium nitrate (CMN) was used as the primary thermom
eter. The CMN was in the form of a right circular cylinder with the
diameter equal to its height. The average dimension of the CMN crystals
was about fifty microns. Two thousand #44 Formvar insulated copper
wires having an area of 150 cm2 were embedded in the CMN crystals with
Apiezon N grease, and were used to establish thermal equilibrium be
tween the CMN and the He3 sample. This point will be discussed in
greater detail in the following section.
The relative susceptibility was measured using a ballisticcircuit
shown schematically in Fig. (9). In tWis arrangement a measuring cur
rent, supplied by the battery, is reversed through the primary coil of
the mutual inductance system MI which surrounds the salt pill. This
induces a current pulse through the secondary which, is measured by
the deflection of the ballistic galvanometer G.
~
59
 0
u
I S
U4
010
44
u
rz
I0
LL j
"I
01J.
4
cJ
CD I..c
4a
Under these conditions the size of the ballistic deflection is
proportional to the susceptibility of the salt, so that we nay write
6 = ax+6 = m +6. (3.7)
o T o
The constants m and 6o depend upon the coil geometry and the Curie
constant of the CMN,and are determined by plotting 6 versus T1 in
the 40 to 1IK calibration region.
The deflections were measured with a Leeds and Northrup type 2284D
galvanometer critically damped by means of a 390 ohm shunt resistor.
When used in this manner the Coulombic sensitivity and period are
5 mm/nc and 5.3 seconds respectively. Galvanometer readings were
taken visually and after some experience could be estimated to tenths
of a millimeter.
The mutual inductance coils are shown schematically in Fig. (10).
The vertical distance between the bottom of the potassium chrome alum
cooling salt and the top of the CMN thermometer salt was 9 1/2 inches.
At this distance the contribution to the measured susceptibility from
the potassium chrome alum is less than 0.2%.43 The primary of the
mutual inductance system is a 5 inch long solenoid located on the 1K
shield. It consisted of three tightly wound layers of #30 Formvar in
sulated copper wire separated from each other by Mylar sheets having a
thickness of .001 inches. To avoid eddy current heating, the measur
ing field was varied from 5 gauss in the calibration region down to
1/2 gauss at low temperatures. The maximum power dissipated in the
primary was about 1 mW and caused no measurable heating of the 10K
refrigerator. The secondary was wound on the 0.30K shield and con
sisted of two nearly identical coils each 1 1/4 inches long with a
THERMAL CONNECTION
TO SAMPLE CHAMBER
2000 INSULATED
COPPER WIRES
S10K THERMAL SHIELD
0.30K THERMAL SHIELD
SALT SHIELD
NYLON CONTAINER
POWDERED CMN
PLUS N GREASE
SECONDARY
PRIMARY
Figure 10. Schematic diagram of the CMN thermometer and the mutual
inductance system. The drawing is approximately to
scale.
center separation of 1/14 inches. The coils were made from 20 layers
of #40 Formvar insulated copper wire. Each layer contained approxi
mately 250 turns and successive layers were separated by one mil
Mylar sheets. The wire was wound as tightly as possible in order to
keep the relative positions of the turns fixed. These two coils are
sometimes referred to as the measuring and compensating coils since
they are connected together in opposition so that the induced EMF's
will approximately compensate in the region where the salt's suscepti
bility is small. This is very desirable since the accuracy with which
one can read the galvanometer deflections depends greatly on their
size. To obtain maximum sensitivity over the entire temperature range,
and also to avoid the occurraice of nonballistic deflections, two Gen
eral Radio 107L mutual inductors were also used in the external cir
cuit.
Since it generally took about thirty hours to investigate a single
density and since several densities were studied, it was necessary that
the mutual inductance system possess good stability over a period of
several days. Temperature calibrations, taken at various times during
the course of this and other experiments, indicated that this was in
deed the case. It has been observed that over a period of a few weeks
neither the slope nor the intercept of the deflection versus T1 curve
changed by more than 3%.
The CMN susceptibility was calibrated against the He3 and He4 vapor
pressures in the temperature range from 1.1 to 3.20K. Both He3 and He4
vapor pressures were used in the calibration between 1.1K and 1.50K,
with the two calibrations agreeing to within 1%. The calibration was
accomplished by first calibrating the germanium and carbon resistors
against the helium vapor pressures with the inner vacuum chamber cun
taining 100 microns of He4 exchange gas. The gas was thun pumped away
and both helium baths lowered to their working temperatures of 1.0K
and 0.3"K. The temperature of the CMN and the resistors was then con
trolled by a heater, and the CMN calibrated against the resistors.
This was done so that the shields upon which the primary and secondary
coils were mounted would be at the same temperatures during the calibra
tion as they were during the experiment itself. Also by maintaining
the shields at a constant temperature, any temperature dependent dia
magnetic contributions to the susceptibility arising from the brass
shields were eliminated. It is believed that the absolute temperatures
are accurate to within about 1%, while below 0.08*K relative temperature
changes as small as 0.1 mOK could be measured.
The carbon and germanium resistors were calibrated against the CMN
and served as secondary thermometers down to 0.04K. Both resistors
exhibited good temperature reproducibility upon cycling; in particular,
no measurable change in the calibration of the germanium resistor was
observed even after the apparatus had been allowed to warm to room
temperature and then recooled. The thermometers fit snugly in wells
drilled into the copper section of the sample chamber. Thermal con
tact between the resistors and the copper walls was insured by melting
Apiezon N grease and causing it to flow over the entire surface of the
resistor.
Resistances were measured by means of a 21 cps ac resistance
bridge shown schematically in Fig. (11). The phasesensitive detector
is a Princeton Applied Research type JB4 lockin containing a variable
frequency oscillator which was used to drive the bridge circuit. The
___~
CI
101
C
a.!
A
t o
Cto
power dissipated in the resistors was reduced from 109 W above 1.0K
down to less than 1012 W at 0.040K. Resistances could be measured to
at least 0.5%, corresponding to a temperature sensitivity over the en
tire range of the resistors of about 0.2 m'K.
Temperature Regulation
In this apparatus a zinc foil was utilized as a thermal switch be
tween the cooling salt and the sample chamber. This was done so that
the temperature of the sample could be varied over a wide range while
maintaining the temperature of the salt at a fairly constant value.
With the zinc in its superconducting state, it was possible to raise the
sample temperature, over a period of four to five hours, to about 0.07K
while the temperature of the salt remained below 0.030K. This is a very
desirable arrangement since it allows one to obtain several sets of pres
sure and temperature measurements on a given density.
The temperature of the sample was regulated by manually adjusting
the current through a 33 Kf metal film resistor located in one of the
chamber wells. Regulation at the lowest temperature was accomplished
with the zinc heat switch normal since only small power inputs were
necessary to raise the temperature of the sample above that of the potas
sium chrome alum heat sink. At higher temperature, however, where the
thermal gradient between the sample and salt was large, regulation was
accomplished with the switch superconducting. It was also desirable
to have the switch open when passing through the phase transition re
gion, since here it was sometimes necessary to maintain a large thermal
gradient for a long period of time while the mixing of the two phases
took place.
__
Thermal Equilibrium Time between the Sample and Thermometer
In order to insure that the measured thermal expansion be pro
duced by the helium sample alone, it is necessary to exclude all other
materials from the sample volume. This means that the CMN thermometer
used to measure the temperature of the He3 spin system must be located
outside the chamber. This requirement poses little problem at tem
peratures greater than about 0.10K since in this region the Kapitza
boundary resistance is small and hence thermal equilibrium between
sample and thermometer occurs rather rapidly. However, at lower tem
peratures, the Kapitza resistance increases rapidly and some care must
be taken to avoid producing a long thermal time constant between the
sample and thermometer. A schematic diagram of the thermal path be
tween the nuclear spins and CNN thermometer is shown below.
He3 He3 Chamber CMN
Spins 1 Lattice TLC Walls TCT Ther
mometer
The rate at which the nuclear spins come into equilibrium with
the lattice is determined by the spinlattice relaxation time. Meas
urements of T1 in the bcc phase have been performed by the Duke and
27
Oxford27 groups down to temperatures of 0.040K in magnetic fields of
a few hundred gauss. From these results one can reasonably assume that
at 0.020K in zero magnetic field T1 should be no longer than a few
minutes for any of the densities studied in this work. Since an es
sentially fixed lattice temperature can be maintained for hours, the
spins and lattice will have sufficient time to come into thermal equi
librium.
The CMN salt crystals and He3lattice come into thermal equilib
rium with one another through contact with the walls of the sample
chamber. The rate at which this process occurs is determined by the
boundary resistance between the He3 and the chamber walls, and between
the CMN crystals and copper wires embedded in them. This Kapitza re
sistance between two surfaces can usually be expressed in the form
AT a (3.8)
RB ATn
where a and n are empirical constants, and A is the area of contact
between the two surfaces. An accurate calculation of the thermal
time constant for the He3CMN system cannot be made, chiefly because
of the lack of experimental data on the Kapitza resistance of solid
He3. The measured value at 0.020K was about five minutes. Since
the drift rate at the lowest temperature is less than about 0.1 mOK
per hour, we can expect the He3 and CMN to be in excellent thermal
equilibrium at all times during the experiment.
G. Solenoids
A niobium zirconium solenoid previously described by Lichti45 was
used to produce the necessary magnetic field for the demagnetization.
The solenoid has a 2 inch inside diameter and is 6.88 inches long. It
is equipped with a persistent switch so that it may be operated for
many hours without excessive boiling of the liquid helium in the outer
bath. A small resistive short was soldered across the terminals of the
solenoid to give a time constant for the parallel R, L circuit of about
five minutes. A Magnion type CFC 100 power supply was used to provide
the current for the solenoid. A current of 17 amps in the solenoid
produced a field of 13.2 kG at the center. At 0.30K this corresponds
to an H/T ratio of 4.4 x 104 gOK1, a value at which the magnetization
of the potassium chrome alum is essentially complete. Several attempts
were made to determine the residual field of the solenoid after de
magnetization. These were only partially successful so that at the
present time we can only say that the residual field appears to be
less than 50 G. This probably caused the final temperature to be
slightly higher than might otherwise have been attained.
A small niobium solenoid also located in the outer bath was used
to provide the 60 gauss necessary to actuate the zinc heat switch. The
vertical distance between this solenoid and the CMN thermometer was 6
inches, and it produced no detectable field at the position of the CMN.
H. Performance of the Experiment
Sample Formation and Cooldown to 0.3K
After the completion of the He4 transfer, the field in the main
solenoid was turned up to 13.2 kG and the current was made to persist.
The He4 exchange gas was pumped for several hours until the reading on
the leakage meter of an MS9A Veeco leak detector went below an em
pirically determined value which indicated that the exchange gas had
been essentially removed. At this point, the inner He4 bath was filled
with liquid and pumped to 1K, He3 gas was condensed into the evaporator
section of the refrigerator and its temperature lowered to the vicinity
of 0.30K. Within about two hours, the temperature of the cell was be
low 2K and the sample gas was condensed into the sample system. When
the temperature of the sample was within 0.1K of the freezing temperature
corresponding to the particular density to be studied, the refrigerators
were warmed and the strain gauge calibrated against the dead weight
gauge. After calibration, the pressure on the liquid was held at the
desired value by the dead weight gauge and the He3 refrigerator quickly
cooled below the freezing temperature of the sample. A solid plug then
formed in that section of the filling capillary thermally grounded to
the He3 refrigerator so that the desired density was obtained in the
sample cell.
As the sample cooled, the melting curve was reached and the pressure
in the cell dropped rapidly until the chamber became completely filled
with solid. Ihe location of the melting point was determined by the
drastic change in slope which occurs when the sample leaves the melting
curve and enters into the all solid region of the phase diagram. The
molar volume was determined from the point of intersection of the solid
isochore with the melting curve using the data of Grilly and Millsl4
and Mills, et al.46 The temperature was then held within 0.01K of the
melting point and the solid annealed for thirty minutes. After anneal
ing, the process of cooling the sample was allowed to continue. When
the temperature reached about 0.8K, the return valve on the He3 refrig
erator was shut. The apparatus could then be left unattended for eight
to ten hours while the temperature of the salt and sample cooled to 0.31
at which point the inner bath was refilled and the demagnetization begun.
Demagnetization Procedure
The magnetic field was decreased exponentially, by letting it de
cay with the L/R time constant of about 5 min, from 13.2 kG down to
about 9 kG. During this initial step in the demagnetization process
the lead heat switch became superconducting thereby isolating the chrome
alum salt from the He3 refrigerator. The field was decreased from 9 kG
to 3 kG in three steps over a period of about two hours. At this point
the temperature of the sample was approximately 0.080K, which is well
below the phase separation temperature of the 600 and 1600 ppm mixtures.
It was then necessary to wait for the isotopic phase separation to be
come essentially complete. This waiting period ranged from about
thirty minutes at a molar volume of 24.0 cm3/mole to about four hours
at molar volume of 23.0 cm3/mole. During this time, the temperature of
the sample remained stationary and the onset and completion of the phase
separation could be determined by watching the rate of change of the
pressure with time. After determining that the phase separation was
essentially complete, the demagnetization was continued. The sweep
time on the power supply was adjusted so that the field would be
turned down to 2000 gauss in about one hour. During this time, values
of the capacitance, resistance, and ballistic deflection were taken.
The sweep time of the power supply was then readjusted so that the
final 2000 gauss would be turned off in approximately three hours. A
gain, readings of the capacitance and deflection were taken as the
sample cooled. Several checks of the thermal equilibrium between the
CMN thermometer and the He3 sample were made by stopping the demagneti
zation process and watching the rate of change of the temperature and
pressure. At all temperatures, it was found that the demagnetization
proceeded slowly enough for the thermometer and sample to attain good
thermal equilibrium. The lowest temperature, generally about 21 mOK,
was obtained about one half hour after the final field was turned off.
No heating of the sample was done for at least another hour; during
this period no detectable change in temperature occurred. heat was then
____
71
applied to the sample and its temperature was raised to about 0.08K
over a period of about five hours. The heat was then removed and the
sample allowed to recool, generally reaching a temperature in the
vicinity 25 m"K. This procedure was repeated at least once for every
density so that at least three sets of deflections versus capacitance
readings were obtained below 0.08K. The sample was then warmed through
the phase transition region and up to the melting curve. The He3 re
frigerator was allowed to warm above the freezing temperature of the
sample and the pressure calibration checked. In all cases this calibra
tion agreed with that taken previously to within 0.01 atm.
CHAPTER IV
RESULTS AND DISCUSSION
A. Introductory Remarks
In this chapter we present values for the nuclear exchange energy
and isotopic phase separation temperature obtained from three samples
containing 20, 600, and 1600 ppm He4 impurities. The data are limited
to large molar volumes by the long equilibrium time for the isotopic
phase separation in the 600, and 1600 ppm samples, and by the pressure
sensitivity for the 20 ppm sample. The results for the volume depend
ence of the nuclear exchange energy, phase separation temperature, and
energy of mixing, will be discussed in terms of the approximate equa
tion of state developed in Chapter II. To facilitate this discussion,
and also to indicate the relative size of the pressure changes produced
by the various degrees of freedom in the solid, we present in Fig. (12)
a typical isochore obtained using the 1600 ppm sample. As was done in
Chapter II, the temperature scale has been divided into three regions.
At temperatures greater than 0.30K, the phonons make the largest con
tribution to the free energy and we find the pressure to be proportional
to T4. In the phase transition region there is a sharp increase in the
pressure as the mixture separates into two phases. Finally at tempera
tures below 0.06K, the phase separation is essentially complete,
and the pressure change, (AP)ex, arising from the nuclear spin system
is proportional to T1.
a) (U E4 3 p h *
1 000 0. 0N
m 1 A 0 H
C rq o 0 M .C t 00
04 M4 C () U) 00
a Hic "d ;4 0 1.4
0. 0. *rl C 0.
C O) $4 o M
o < l amC
0 O.H $O 
1 0 to 00 O a)
a) $4 m0 I U
= 0 C3 L. 3 E0
41 4 : o U ra d
a w 10 0 0 10
o ac c E *r a .c ,C
44 U ( 0 4( H, 4
O11 r1WH ) U) rI >,
O 0 1 0 410 0 
Xm bo 4J ) 4 3 rl U)
o0 4) 1 m) im rd
S 0 m a) '0 *
mH p "t 0 ( .0
41 0 41 1 0 4H 4 r1
0o00H04 o
Um t14 41 CD U r n)
*H 0 H 0 k 44 a 14 H r
u0 E4 0 4H Om 1
U El 4J 0o (1) Ed (V 0) 4
mC a) .0i= i w bo a 0
0 a 4 0 m 9 0
to 0 1 4 0 1
S4 4 uN 4 CC 
u 4M a Ed a 0 U 4J 6
CM
H
So
C O
0 0
.. .
w
/ 00
SI I I
(wuo .01) dV
Before presenting the results for the exchange energy and the
phase separation temperature however, it is necessary to discuss a
small anomaly in the thermal expansion of the strain gauge itself.
B. The Thermal Expansion of the Empty Cell
As previously mentioned, the pressure calibration fo the strain
gauge was performed at a temperature just above the sample freezing
temperature corresponding to the particular density to be studied.
The calibration conditions were thus not identical to those under
which the experimental data were obtained. For this reason a separate
demagnetization was performed, with the sample chamber evacuated, to
determine the effect of the thermal expansion of the sample chamber
itself on the capacitance. The result of this demagnetization is shown
in Fig. (13). In this graph the capacitance change due to the thermal
expansion of the chamber has been converted into an equivalent pres
sure change so that its effect may be more easily compared with the
pressure changes produced by the thermal expansion of the solid helium
samples. It can be seen that a strange anomaly exists in the thermal
expansion of the sample cell in the temperature range from 0.080K to
0.300K. This is also the temperature region in which the phase separa
tion occurs, and hence the contribution to the capacitance change re
sulting from the thermal expansion of the chamber must be included in
the analysis of the phase separation data. For the 600 ppm sample, the
chamber contribution to the total capacitance change is about 10%,
while for the 1600 ppm sample it amounts to approximately 4%. Below
0.060K, the thermal expansion of the chamber is completely negligible
00
0
*4
a
hie
o h)
W 44
01
0
In CLU 0
/ (
/ ^ j
A Q (
^ ^w^ o ,
(WjD t01) dV
and hence does not affect the values obtained for the nuclear exchange
energy.
It is interesting to speculate about the possible source of this
anomaly in the thermal expansion of the sample chamber. If the thermal
expansion coefficient, which is proportional to the derivative of the
P versus T curve shown in Fig. (12), is plotted against temperature,
the result suggests that some type of cooperative transition occurs
in the chamber. A possible explanation for this behavior is that a
magnetic transition takes place in the stainless steel section of the
sample cell.
C. Nuclear Exchange Energy
Values of IJI for "Pure" He3
The values given below for the nuclear exchange energy and its
volume dependence are obtained from PVT data taken in the temperature
region from 0.020K to 0.060K. In this range the He4 impurity in the
He3 rich phase is less than 0.2 ppm, and hence the values of IJI are
those for very pure He3. The molar volumes studied in this work range
from 22.8 cm3/mole to 24.2 cm3/mole. At smaller molar volumes, the
absolute value of J becomes too small to be measured with the present
pressure sensitivity.
The equation of state for dilute mixtures developed in Chapter II
indicates that at low temperatures the contribution to the pressure
from the nuclear spin system is given by
) R e j 2 1
(AP)ex ex (4.1)
where (AP)ex is the increase in pressure relative to some arbitrary
ax
) 0 0) 0
 H Cd .ri C
: Xc rI H w
0 0 ( o 0 *Hr
C 0 H 0 )
1 C41 4J m
1 o *rI W 4
o 1e
j4I O 3 i a a
m .0
o o ca c
OwP0 0o0.0
'I ri a) 4 a C .
0 C4i 0 W C m0o 14
0 i H0 4 3C 41
rl CR O J .J CX
1. 44 0 0 H x8 to
*r *to oM d40u C 8 a
o W CROL) a 0.
w t0) 4 4 U
0 um 1 d 0 m* OC
0 *r0 1 CL0 0
0( 4 H Ur 00 4 0
H60 *l r( rl
SA 0 m CD
1i I 0 .0 1 bo
) O e 0 4 30 0
U O0 0> .0 0" 3
XCR mt4. W I x
om U u
pH w"
C0 W 0 r0 4)
0 4 4U 4 100 C 4 3.
0U 400 4xJ 40
41I4 H a A3
0 0 44 W I W O 0<
< r'd 0. 0 .c J'10
r
1:4
r
00
4 I (
a. 0.
I =\ dII
0 0
0 (
I' CI
(D
> fi ri 4ir
0i W W N
ca 0
coqo
(W4D 2,01) (dV)
reference point, and
x = dInIJI (4.2)
ex dlnV
In Fig (14) (AP)ex is plotted versus T1 for various molar volumes and
He4 concentrations. In the experimental procedure section of this
work, we explained how it was possible to obtain values of P versus T
for several runs on a given density. As can be seen, the data obtained
on these various runs exhibit extremely good reproducibility. In view
of the small sizes of the pressure changes measured, this fact is indeed
comforting. The rapid decrease in (AP)ex at a value of T1 = 12 is due
to the mixing of the separated phases in the 600 ppm sample.
It can be seen from Eq.(4.1) that the slopes of the (AP)ex versus
T1 curves are equal to y ex( Since this expression is propor
tional to J2, these measurements cannot be used to determine the sign
of J. Also, since it contains both J and its logarithmic derivative,
a selfconsistent procedure must be used to extract J from the data.
The first step in this process is to assume an initial value of yex
based on the NMR data.26,27 Equation (4.1) is then solved for IJ(V)I
at each molar volume studied. These initial values, IJ(V)I can be
used to compute a new, selfconsistent value of yex. The correct values
of the exchange energy are then computed using Eq.(4.1) and this self
consistent value of yex'
The values of IJl/k obtained using this selfconsistent procedure
are presented in Table I and Fig. (15). Also shown for comparison are
the results derived from NMR data, and those calculated by
Hetherington, et al.28 The solid line fitted to the data indicates
that yex = 16.4 is valid over the density range covered by this work.
Figure 15. Nuclear exchange energy versus molar volume. The
various symbols represent values of IJI/k obtained
using different initial He4 concentrations. Also
shown for comparison are the values of IJI/k de
rived from the relaxation measurements of Richard
son, Hunt, and Meyer (RHM); Richards, Hutton, and
Giffard (RHG); and those calculated by Hethering
ton, Mullin, and Nosanow (HMN).
I~
HMN

.,,
0.06
0.05
0.04
23 23.5 24
MOLAR VOLUME (CM3/MOLE) (Log Scale)
TABLE I
Smoothed Values of the Nuclear Exchange
Energy in Solid He3
V IJl/k TN = 31Jl/k
(cm3/mole) (mK) (mK)
24.4 0.90 2.70
24.0 0.68 2.04
23.6 0.52 1.56
23.2 0.39 1.17
22.8 0.30 0.90
From these results it is seen that in the bcc phase IJl/k is about
0.9 mK at a molar volume of 24.4 cm3/mole, and decreases with in
creasing density approximately as dln = 16.4. The corresponding
dlnV
Ndel temperature, obtained from the expression TN = 3k varies from
about 2.7 mOK at 24.4 cm3/mole down to 0.9 mK at 22.8 cm3/mole.
The agreement between our values of IJI/k and those derived from
the NMR data of Richardson, et al.26 is excellent. Since these pres
sure measurements constitute a direct method for determining IJ this
agreement may be interpreted as additional confirmation of the validity
of the NMR formalism. Richardson47 has suggested that the discrepancy
between the Duke and Oxford values for IJI arises from the latters'use
of the block capillary technique to determine the molar volume of the
solid. This variance in IJI can be resolved if one assumes that the
plug in the Oxford filling capillary slipped enough while forming the
low density solid to decrease the molar volume by about 0.8 cm3/mole.
At higher densities, the values of IJI determined by the Duke and Ox
ford workers are in better agreement, and hence Richardon's suggestion
seems reasonable.
A comparison of the theoretical calculation of HMN and the experi
mental data indicates that while the theoretical slope agrees well with
that found experimentally, the magnitude of J predicted by HMN is about
an order of magnitude too small. One can understand how such a situa
tion arises by recalling the theoretical expression for J obtained
earlier,
2R2A2 f2r i x 2eff r )> x
J; = + 2 (4.3)
S 2m f(r ij)> f (r)>
The first term in this equation is large in magnitude and negative in
sign, while the second term is of the same order of magnitude but
positive in sign. One sees, therefore, that the theoretical value of
J is obtained from the difference of two large numbers, each of which
is accurate to about 10%. It is evident that such a situation could
easily produce a relatively poor absolute value for the exchange energy,
yet still yield a good value for its volume dependence.
The observed decrease in IJI with increasing density is interest
ing and can be understood at least qualitatively on the basis of the
following discussion. As the density is increased the average separation
between atoms becomes smaller and intuitively one might expect the over
lap region between nearest neighbor atoms to increase. This would of
course increase the absolute value of the exchange energy. This in fact
does not occur because, in addition to reducing the interatomic spacing,
the increase in density also produces an increase in the kinetic energy
of the atoms. This increase in kinetic energy is accomplished by a
greater localization of there He3 atoms to the vicinity of their equilib
rium positions. In more formal language we could say that the increase
in density produces an increase in the curvature and a decrease in the
extent of the wave functions describing the motions of the atoms. Since
in solid He3 the exchange energy arises as a consequence of the over
lapping which occurs in the tails of these functions, any decrease in
their physical extent produces a smaller absolute value for the ex
change energy.
Effects of He4 Impurities on J
Before terminating this discussion of the exchange interaction
in solid He3, a brief examination of the effects of small amounts of
He4 impurities on J will be made. The NMR data of RHG taken on a
sample containing 5000 ppm He4 indicate that the presence of a He4
atom causes a distortion of the He3 lattice, which in turn produces an
increase in the value of J in the vicinity of the impurity site. These
workers found that it was possible to express J in the form
j2 = J2 +J2 (4.4)
bulk imp'
where (Jimpl/k = 0.1'K is independent of density over the molar volume
range from 18.3 cm3/mole to 20.0 cm3/mole. If the data of Garwin and
Reich,48 obtained with a sample containing 1% He4 impurities, am also
analyzed according to Eq.(4.4) it is found that the value of JiJ p/k
is = 0.5 mK. On the basis of these two results it appears that the
parameter Jimpl increases rapidly as the He4 concentration is increased.
The recent magnetic susceptibility experiments of Cohen and Fair
bank49 performed on samples which contained 0.5, 100, and 3000 ppm He4
impurities also indicate that the exchange energy increases with in
creasing He4 concentration. Originally the susceptibility measurements
taken on the 3000 ppm sample also indicated that the exchange energy
increased rapidly with increasing density between molar volumes of
22.6 cm3/mole to 21.0 cm3/mole. Subsequent experiments by the same in
vestigators, however, have not confirmed this result, so that at the pres
ent time the question of how a small amount of He4 impurities affects
the magnetic interactions in solid He3 remains unanswered.
D. Locus of the Zeros of the Thermal Expansion Coefficient
In Chapter II we obtained an approximate equation of state for
solid He3 given by
3R(V ) R ex 1 3n
P(V,T) = Po(V) x T T, (4.5)
where we have omitted the contribution from the isotopic phase separa
tion. Differentiation of Eq.(4.5) with respect to temperature at con
stant volume gives
(ap) 3R J2 1 + 12r4 T 3
ST) V Yex Ty Y ") (4.6)
S+ (4
A straightforward thermodynamic derivation using the definitions of
the isobaric thermal expansion coefficient a ) and isothermal
compressibility 8 V () leads to the expression () = .
T V
Goldstein30 has pointed out that the expansion coefficient be
comes zero when the negative contribution for the spin system becomes
equal in magnitude to the positive contribution from the lattice.
Thus, there will be a line in the PT plane which is the locus of the
zeros of the isobaric thermal expansion coefficient. By rewriting
Eq.(4.6) in the form
= ( .) = Ai(V) + A(V)T3, (4.7)
B =aP =AI V + A2(V)T3, (4.7)
Exchange Lattice
aw
Liquid
I I I I
.16 .18 .20 .22
TEMPERATURE (OK)
Figure 16. Locus of the zeros of the isobaric expansion coef
ficient.
one can see that the locus of the zeros of a is given by the relation
AI(V). 1/5
T a= A2(V) (4.8)
We have obtained A1 and A2 from measurements of the slopes of the iso
chores in the low and high temperature limits. The result for T =0 is
shown in Fig. (16).
Roughly speaking, one can say that this locus divides the PT
plane such that at temperatures and pressures to the left of the line,
the properties of the solid are determined primarily by the nuclear
spins, while to the right of the T =0 line they are determined primarily
by the lattice.
E. Isotopic Phase Separation
Kinetics of the Phase Transition
One of the most striking features of the isotopic phase separation
in solid He3He4 mixtures is the rapid increase in the equilibrium time
constant, T, with increasing pressure. Although we have not made a de
tailed study of the dependence of on density we have noted that it in
creases from a value of few minutes at a molar volume of 24.2 cm3/mole
to about one hour at a molar volume of 23.0 cm3/mole. This rapid in
crease in T with density suggests that diffusion of the atoms by quantum
mechanical tunnelling is the mechanism by which the separation into pure
phases is accomplished. It also appears that T is considerably longer
in the cooling direction than in the warming direction, which is an
indication that the atoms can mix more readily than they can separate.
The actual spacial distribution of the isotopes in the two phase
region is presently Uncertain. In liquid mixtures, a visually observable
bulk stratification of the two phases occurs due to the differences in
mass of the two isotopes. It is highly unlikely that such a bulk sep
aration also takes place in the solid. It seems more reasonable to
picture the phase separated solid to be composed of a number of locally
enriched regions, whose dimensions are very large compared to inter
atomic distances. In dilute solutions the number of these He4 en
riched regions is quite small, and hence the bulk properties are es
sentially those of a very pure He3 system.
Pressure Dependence of the Energy of Solution and Phase Separation Tem
perature
The theoretical development of the isotopic phase separation
phenomenon given in Chapter II was based on the assumption that the
He3 and He4 atoms mix together randomly to form a regular solution.
This model was proposed by Edwards, et al.24 in an effort to explain the
observed discontinuity in the specific heat of He3He4 mixtures. The
shape of the specific heat anomaly is similar to that associated with
an orderdisorder transition, and one interprets this as evidence for
the separation of'the mixture into two phases. The measurements of
Edwards. et al., at a pressure of 35.8 atm., were performed on seven
different He4 concentrations ranging from 0.03 to 80%. The data in
dicate that the phase separation curve is symmetric about a concentra
tion of 50% in the Tx plane.
Mullin50 has recently developed a theory of the phase separation
in solid He3He4 mixtures using techniques similar to those employed
by Nosanow in his study of the pure isotopes. Mullin's analysis leads
to the conclusion that solid helium solutions should be nearly regular
but for different reasons than those originally suggested by Edwards,
et al.24 Two important predictions of Mullin's work are that the
phase separation curve should be unsymmetrical, and that the phase
separation temperature should decrease with increasing density.
In this section we present some preliminary results for the pres
sure dependence of the phase separation temperature, and the energy of
mixing. We wish to emphasize that these results are to be regarded
as tentative until confirmed by future experiments performed with
mixtures containing greater amounts of He4 impurities.
The pressure change at constant volume due to the phase separa
tion of a dilute, regular mixture was obtained in Chapter II and is
given by
d(Ek) E kT
(P) R d(E k) eElkT, (4.9)
(AP)S= R dV (9)
E33 E44
where El = E34 2  is the energy of mixing. The equation for
this phase separation line has, for small He4 concentrations, the
simple form
x = eEkT. (4.10)
Substitution of Eq.(4.10) into Eq.(4.9) yields the following expres
sion for (AP)PS,
d(E1k)
(AP)pS = Rx dV (4.11)
dE k
One can see from this equation that if (AP)pS > 0, then dV > 0,
and EM should decrease with increasing density. Further by rewriting
Eq.(4.10) in the form
EM I 4
T Mk (4.12)
PS k lnx
it can be seen immediately that the pressure dependence of TPS is
similar to that of EM.
We have measured (AP)pS as a function of volume over the range from
23.0 cm3/mole to 24.2 cm3/mole for initial He4 concentrations of 600
and 1600 ppm. The results are shown in Figs. (17) and (18). The curves
shown have been obtained by a graphical smoothing procedure as follows.
The capacitance values were first plotted against temperature and a
smooth curve placed through the data. The contributions to the ob
served capacitance change from the exchange energy and sample chamber
anomaly were then subtracted from this smooth curve. The resulting
values for (AP)PS versus T are those which appear in Figs. (17) and
(18). We wish to emphasize thatwhile tis process is necessary to obtain
the shapes of the (AP)PS versus T curves it is not needed to establish
the essential fact that (&P)PS decreases as the density is increased.
dEM/k
The values of dV computed from Eq.(4.11), and the measured values
of (AP)PS' are shown as a function of molar volume in Fig. (19).
While there is some scatter in the data, it can nevertheless be seen
dEM/k
that  decreases monotonically from u 0.05"K mole/cm3 at 24 cm3/mole
to 0.030K mole/cm3 at 23 cm3/mole. Using an average value of 0.04K
mole/cm3, and Eq.(4.12), one finds that the phase separation tempera
ture of the 1600 ppm sample should decrease by 6 mOK between molar
volumes of 24 and 23 cm3/mole. This is in qualitative agreement with
the calculation of Mullin50 which indicates that over a similar pres
sure range the phase separation temperature of a 50% mixture should
decrease by 0.020K.
Finally we have determined the phase separation temperatures and
energy of mixing as a function of pressure from the inflection points
