THE THERMODYNA,'MIC PRESSURE IN SUPERFLUID
HELIUM AND ITS IMPLICATIOrNS FOR
THE PHONON DISPERSION CURVE
By
ALBERT ROBERT MENARD III
A DISSERTATIONS PRESENTED TO THE GRADUATE COUN CII
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE"
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVr.iITY OF' FLORIDA
1974
To Anne, who made this possible
AC KNOWLE DG:4tENTS
The author would like to express his appreciation for
help and guidance during the course of this work to the
following persons:
Dr. E. D. Adams for suggesting this problem and providing
aid in all phases of the research.
Dr. S. H. Castles for his frequent and timely advice.
Drs. L. H. Nosanow, J. S. Rosenshein and J. A. Titus for
their encouragement and support.
Bill Steeger, the late Sherman Sharp, and the late George
Harris for machining the many parts of the apparatus.
C. B. Britton, R. M. Mueller and B. Kummer for valuable
assistance and suggestions.
Pat Coleman for producing the liquid helium used in the
experiment.
Catherine Phillips for assistance in drawing the figures.
Margaret Anderson for an excellent job of typing under
difficult circumstances.
iii
TABLE OF CONTENTS
Page No.
ACKNOWLEDGMENTS . . . . . . . . .. iii
LIST OF TABLES . . . . . . . . . v
LIST OF FIGURES . . . . . . . ... vi
ABSTRACT . .. . . . . . . . . vii
CHAPTER
I INTRODUCTION
Historical Background . . . . . . 1
Neutron and Xray Scattering Measurements . 7
Measurements of the Velocity and Attenuation
of Sound . ..... . 11
Recent Theoretical Developments . . . .. 13
Recent Experimental Results . . . . .. 21
II LANDAU THEORY
Algebraic Results . . . . . . .. 25
Numerical Results . . . . . . . 34
III EXPERIMENTAL APPARATUS AND PROCEDURE
The Cryogenics . . . . . . . 40
The Sample Chamber and Pressure Measurement . 44
Temperature Measurement and Regulation . . 51
The Sample . . .. .... . . . 54
Procedure for Taking Data . . .. . 54
IV RESULTS AND CONCLUSIONS
Data Reduction . . . . . . .. 56
Data Analysis .. . . . . . . 60
Conclusions . . . . . . . 73
REFERENCES . . . . .. ...... .. ... 77
BIOGRAPHICAL SKETCH . . . . . . ... 81
LIST OF T?.BLES
Table !No. Page NI o.
I. Values of the Dispersion Parrmeter, y,
and Its D.nsityO Dcriv.Lative in c.g.s.
UniLs, Pased on Experimental Data. .. ... 35
II. Contributions of Different Terms to the
Prcssuc:e, Expressed in Atmospheres, at
Sdt,'rated Vapor Pressure and at 24 Atmos
pheres . . . . . . . . . . .36
I I
III. The Values of y and 5 at Experimental
Densities. . . . . . . . . . 67
LIST OF FIGURES
Figure No.
1. Phase diagram of 4He . . . . . .
2. Excitation spectrum at SVP . . . .
3. Excitation spectrum at 24 atm. . . .
4. Schematic drawing of the cryostat . .
5. Sample chamber . . . . . . .
6. Schematic drawing of low temperature valve
7. Pph/T4 versus T2 for all data . . .
8. y versus density . . . . . .
9. Pph /T4 versus T2 for p = 0.1474 gm/cm3
LO. Pph/T versus T for p = 0.1474 gm/cm3
Page No.
. . 14
S . 29
. . 41
S . 45
S . 50
S . 61
S . 68
S . 70
. . 72
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
THE THERMODYNAMIC PRESSURE OF SUPERFLUID HELIUM AND
ITS IMPLICATIONS FOR THE PHONON DISPERSION CURVE
By
ALBERT ROBERT MENARD III
Decemuer 1974
Chairman: E. D. Adams
Major Department: Physics
The thermodynamic pressure of superfluid 4He, for seven
densities from near saturated vapor pressure to 11 atm., has
been measured from 0.4 K to 1 K, using a capacitive strain
gauge. A comparison of these measurements with calculations
for the pressure, based on the phononroton model for super
fluid helium, led to the conclusion that anomalous phonon
dispersion exists in helium at low densities. The magnitude
of the anomalous dispersion is in good agreement with that
deduced from specific heat measurements. The conjecture of
a quadratic term in the expression for the dispersion curve
does not agree with these data. Contributions to the pres
sure not accounted for by phonons and rotons were observed.
These contributions, probably maxons, increased in magnitude
and became noticeable at a lower temperature as the density
was increased.
vii
CHAPTER I
INTRODUCTION
Historical Background
Liquid helium has been a source of fascination for phy
sicists ever since helium gas was first liquified by Kammer
lingh Onnes in 1908. The phase diagram of helium is shown
in Fig. 1. Above the transition marked the X line in Fig. 1,
helium, usually referred to as He I, despite its obvious
quantum properties, shows many of the thermodynamic and hydro
dynamic properties characteristic of classical liquids. Be
low the transition, where it is referred to as He II, its
characteristics undergo drastic qualitative changes and many
of its properties are unique to helium. The most outstanding
of these changes is that of superfluiditythe vanishing of
flow resistance for the passage of the liquid through small
channels. These properties are characteristic of natural
helium, which is nearly 100% He. The lighter isotope, He,
is present in natural helium in such small quantities (about
one part in 106) that its effects are not noticeable in most
experiments. The vast amount of theoretical and experimen
,tal work on liquid helium is well summarized by Wilks (1967),
by Woods and Cowley (1973), and by Keller (1969).
The first major step toward a theory that would explain
the unique properties of superfluid helium was the suggestion
10
I.
solid
30
X line
20 He I
He II
10
12 s
1 2 3 4 5
Temperature (K)
Fig. 1. Phase diagram of 4He [after Wiiks (1967)].
made by London (1954) that the superfluid phase was a "fourth
state of matter" representing a macroscopic manifestation of
quantum effects. He reasoned that a significant fraction of
the liquid, or perhaps the whole liquid, might exist in a
single quantum state. Because 4He has zero spin, it will
obey BoseEinstein statistics: in contrast, the rare isotope
He which has spin 1/2 will obey FermiDirac statistics. In
Bose statistics an unlimited number of particles may occupy
an energy level. This has the mathematical consequence that
an ideal gas of bosons, i.e., a large number of non
interacting particles subject to Bose statistics, will for
certain values of the density and temperature of the gas have
a significant fraction of the particles "condensed" into the
ground state. This "condensation" will occur in momentum
space. Thus particles separated by large distances in the
fluid will have their moment, hence their motion, correlated.
Fermi statistics do not permit this unless there is a pairing
of the atoms. Thus the experimental fact that 3He is not a
superfluid until less than 0.003 K, in contrast to 2.17 K for
the superfluid transition in He,is easily explained.
This suggestion by London and Tisza provided the basis
for the development of a twofluid model. London (1954)
gives a full account of the historic development of this idea
and the role played by Tisza in its development. Helium was
viewed as being composed of two interpenetrating fluidsa
superfluid with zero entropy and a normal fluid. The two
fluid model successfully predicted most of the macroscopic
4
thermal phenomena in He. It also explained the tremendous
differences in low temperature properties between 3He and
He. Ho.ever, this model had four grave weaknesses. (1) It
was a macroscopic and not a microscopic theory. (2) The
agreement between theory and experiment was qualitative, not
quantitative. (3) The theory was difficult to extend to dy
namical situations such as the propagation of sound. (4)
Finally, the theory was a model, many of whose features, such
as the existence of the condensate, could not be experimen
tally verified.
The next major advance was the Landau (1941, 1947)
theory of superfluidity. Landau viewed superfluid helium as
a perfect background fluid in which a gas of elementary exci
tations moves. These excitations, which have a definite
energy and momentum, describe the "collective motion" of the
helium atoms similarly to the use of "normal modes" to de
scribe the motion of interacting particles. The excitations
behave as quasiparticles which interact very weakly with
each other. From the energy spectrum, i.e., the relation be
tween energy and momentum for the excitations, thermodynamic
and hydrodynamic properties of liquid helium can be deduced.
The excitations were given the names "phonons" and "rotons."
Landau pictured the phonon excitations as long wave
length density fluctuations with energy, E, directly propor
tional to their momentum, p, and traveling with the velocity
of sound, c. Thus,
S= cp (1.1)
in the long wavelength limit. At shorter wavelengths (larger
p) the e versus p dispersion curve passes through a minimum,
such that
S= A + (p p) 2/2p. (1.2)
The excitations in the region of the minimum were called ro
tons. While the excitation spectrum is a smooth, continuous
curve, at low temperatures only the phonon and roton states
are significantly populated. Since the excitations were
associated with the "normal" fluid and the background was
identified with the superfluid, most of the results of the
twofluid model could be derived from Landau's theory. How
ever, there was still no physical microscopic basis, nor was
there an experimental basis, for the assumptions of the
theory, and the experimentally verified necessity that super
fluid helium be a Bose system had vanished from the theory.
If a microscopic physical basis for the energy spectrum of
the excitations could be provided, the Landau theory would
provide an adequate description for almost all phenomena in
helium.
Feynman (1953, 1954) showed that a liquid consisting of
indistinguishable particles subject to Bose statistics could
have only one kind of low energy excitations. These were
density fluctuations resembling sound waves which when quan
tized were identical with Landau's phonons. Since Feynman's
arguments were very general, based on the energy of a con
figuration of individual atoms, this provided a microscopic
explanation for the phonon part of the excitation spectrum.
In a further paper, relying again on very generalized argu
ments, Feyniian (1954b) calculated a wave function for liquid
heliuri, which included the possibility of rotational motion.
Dy applying the variational principle, the energy spectrum
was found to be
2
F(P) I S(I1) (1.3)
where () is the liquid structure factor. For the small k
lirn:ii: at ? = 0 this reduces to : (p) = cp. At higher momen
tuYm. lhec structure factor, which can be measured by Xray
scltteiing from the fluid, has a maximum which leads to a
minimum in (p). The resulting spectrum is in qualitative
agreement with Landau's proposal. An improved wave function
due to Feynman and Cohen (1956) greatly improved the agree
ment between the computed energy spectrum and the Landau type
spectrum as determined from thermodynamic measurements.
Finally, Feynman and Cohen (1957) pointed out that the exci
tations could be observed directly by neutron scattering.
Neutron scattering experiments, discussed in the next sec
tion, confirmed the excitation picture by directly measuring
the excitations and showed that their spectrum had the form
suggested by Landau. Thus the theory had a firm physical
basis and the condition that the particles obey Bose statis
tics was automatically included in the theory.
However, this was still not a "first principles" calcu
lation. There have been numerous attempts to derive the
spectrum from a consideration of the interaction potential
for two helium atoms and the application of statistical
mechanics. Despite considerable formal progress, the micro
scopic theory has not yet been developed sufficiently to give
a detailed description of liquid helium. However, consider
able progress has been made in variational calculations for
the structure factor and the ground state energy, for example,
Pokrant (1972). A comprehensive summary of these efforts and
their shortcomings can be found in Keller (1969) and in Woods
and Cowley (1972).
Neutron and Xray Scattering Measurements
After Feynman's paper, several groups performed neutron
scattering experiments. The most comprehensive of the early
works was that of Yarnell et al. (1959). This work measured
an excitation spectrum that was in good agreement with the
one proposed by Landau based on thermodynamic arguments.
Also it showed that below 1.5 K the line width of the exci
tations was quite narrow. This implied that the excitations
had a relatively long lifetime and that the interactions be
tween excitations were weak, two critical assumptions of the
theory. As the temperature approached the i line, the exci
tations became very broad and strongly interacting and
changed character for temperatures above the X line, thus
confirming that the measured spectrum was connected with
superfluidity, since the superfluid properties become less
pronounced near the line and vanish above it.
With improvements in technique and technology, more work
followed, culminating in the "definitive work" by Cowley and
Woods (CW, 1971), whose work was accurate enough to permit
attempts to fit the spectrum with algebraic expressions. Re
cently Dietrich et al. (1972) undertook a comprehensive study
of the roton dip in the spectrum, including a study of its
variation with pressure. Three major problems have occurred
in all neutron scattering measurements. Neutron scattering
l
is limited to values of k greater than about 0.4 A and the
measurements for small k values have a very large intrinsic
error. Except for the roton work of Dietrich et aZ. and some
work at 24 atm. by Svensson, Woods and IMLrtel (1972), and Hen
shaw and Woods (1961), there has been no work on the pressure
variation of the excitation spectrum. Finally, all neutron
measurements have been taken above 1.1 K, so there is a ques
tion of possible variation of the curves with temperature be
low that point. Nevertheless neutron scattering is the most
direct and most accurate way of measuring the excitation spec
trum.
In addition to direct measurement of the excitation spec
trum, neutron scattering, as well as Xray scattering, can be
used to measure the structure function, S(k,u'). The structure
factor relates the differential coherent scattering cross sec
tion to the total cross section
do a k
dE S(k,w) (1.4)
dndE 4tn k
where dQ is the solid angle of acceptance of the scattered
beam, dE is the energy width of the scattered beam, ko and k
are, respectively, the incident and scattered wave numbers,
a is the total cross section and t is the energy transfer
during the scattering. Since S depends on the momentum
transfer during the scattering process, frequently S(k,w) is
written S(Q,w) where Q is a new variable defined as the
momentum transfer. Because the Fourier transform of S is the
time dependent correlation function, S is related to the spa
tial and temporal behavior of the atoms in the fluid.
As shown by Eq. (1.3), the structure factor can be re
lated to the excitation spectrum. There is reasonable agree
ment between the spectrum obtained directly from neutron
scattering and that deduced indirectly from the structure
factor as obtained in the best recent measurements of Hallock
(1972). As with the neutron scattering measurements of the
excitation spectrum, structure factor measurements are
limited to k greater than 0.33 A There has been essen
tially no work on the pressure dependence of S. Hallock did
carry the measurements down to 0.38 K, which, while it is not
T = 0 as used in most theories, does eliminate most questions
about temperature variation. Scattering can also be used to
hunt directly for the presence of the condensate, but results
so far are inconclusive because of serious experimental
problemsfor a review of the subject see Jackson (1974).
Thus scattering measurements have thoroughly demonstrated the
"reality" of Landau's excitations and have provided a quanti
tative basis for the theory '.which is in good numerical agree
ment with the values deduced from thermodynamic measurements,
but there are still gaps in the measured spectrum, particu
larly in the long wavelengthsmall p region.
By fitting the experimental scattering data with the
assumed form given by Eqns. (1.1) and (1.2), explicit values
for V, A, p, and c can be obtained. These values can then be
used in a statistical mechanical analysis to yield expres
sions for the pressure, specific heat, entropy, expansion co
efficient and the superfluid density used in the twofluid
model. This will be done explicitly in Chapter II. These
expressions can be compared directly with experimental
values. Historically, the process was reversed, with values
for p, A, p and c being deduced from thermodynamic measure
ments and these values were used to construct the excitation
spectrum. In general there is good agreement between calcu
lated and measured thermodynamic properties. This agreement
is usually expressed by comparing the values of i, p A and
c derived from scattering with those derived from thermo
dynamics. There is a major flaw in this scheme: since
thermodynamic measurements simultaneously sample the entire
excitation spectrum, any deviation of the spectrum from the
assumed forms can lead to a change in the parameters. This
is a particularly severe problem with rotons, since the
measured excitation spectrum is not parabolic except in a
very narrow region around the minimum. Most of the differ
ence between the thermodynamic values for p, A, p and c, and
0
the neutron values of p, A, p and c can be accounted for by
0
this problem. Thus the phononroton picture of liquid helium
rests on a sound phenomenological basisneutron scattering
and successfully predicts most of the thermodynamic proper
ties of helium.
Measurements of the Velocity and
Attenuation of Sound
Only one parameter in the phononroton model can be
directly measured other than by scattering. That parameter
is c, the sound velocity. Many measurements of the sound
velocity and the attenuation of sound have been made. The
most comprehensive of these was a series of measurements made
at the Argonne Laboratories, Abraham et at. (1969, 1970) and
Roach et al. (1972a, 1972b), which measured sound velocity,
attenuation of sound, and the Gruniessen constant, F pc
C dp
(where p is the density), as a function of pressure, fre
quency and temperature in the region below 0.6 K. These
measurements could be compared with the detailed predictions,
for these quantities in liquid helium, developed by Khalat
nikov and coworkersAndreev and Khalatnikov (1963), Khalat
nikov (1965), and Khalatnikov and Chernikova (1965, 1966).
Their results were based on the development of quantum hydro
dynamics for superfluid helium by Landau and Khalatnikov as
summarized by Khalatnikov (1965). For there to be attenu
ation of sound, Eq. (1.1) was modified to
E = cp(l yp2). (1.5)
Some modification was necessary, since a pure phonon spec
trum, E = cp, does not permit phonon decay because of the
impossibility of simultaneously satisfying energy and momen
tum conservation except through the quantum mechanical un
certainty of E. Experimentally the energy width is known to
be far too small to permit the observed attenuation. The
additional term containing * is known as dispersion. The
exact form of the dispersion term was a reasonable assump
tion, but was not the only possibility. Since Landau and
Khalatnikov's theory had been successful in predicting most
hydrodynamic phenomena in liquid helium, it was a major sur
prise when their predictions were qualitatively wrong about
the measured values of the velocity and attenuation of sound
as well as their variation with temperature, pressure, and
frequency at very low temperatures. The theoretical predic
tion for the attenuation, a, was that in low temperature
limit o would be proportional to T6. The experimental result
4
was that a was proportional to T Furthermore the measured
attenuation was approximately two times greater than the
calculated attenuation. As the pressure was raised, a
strange "shoulder" appeared in the data although nothing of
the sort had been predicted. The velocity of sound showed
extremely complicated behavior as a function of frequency and
temperature which had not been predicted. As Abraham et al.
(1969) remarked, "We therefore conclude that the present
theoretical formulation of sound propagation at very low
temperatures is incomplete" (p. 370). How could a model
which had so successfully predicted so many of the properties
of helium fail so miserably in the low temperature limit
where it should perform the best?
Recent Theoretical Developments
Although substantially confirmed by neutron scattering,
the phononroton picture of helium, like the twofluid model,
is a model. Keller's (1969) warning about models is appro
priate. "A model is devised to represent a physical phenome
non because the actual situation is far too complicated to be
handled directly; the mode] then incorporates the simplifying
assumptions that make the problem tractable. The first dan
ger is one of oversimplification, and this is an especially
hazardous possibility when quantum effects are involved ....
When such a [good] model exists, there is a tendency to
take it too literally, to promote it to too high a status,
and then finally to forget it is a model" (p. 17). One such
oversimplification of the phononroton model can be seen in
Fig. 2. Clearly phonons and rotons are a very poor approxi
mation to the actual excitation spectrum in the region of the
maximum of the curve and on the high k side of the minimum.
To check the possibility that this oversimplification might
affect calculations based on the model, Bendt et aZ. (1959)
12
14
I I
0.8 1.6 2.4 3.2
Momentum p/h (A)
Fig. 2. Excitation spectrum at SVP [after
Cowley and Woods].
dashed line is the phonon approximation,
dashed and dotted line is the roton
approximation.
and Singh (1968) made the laborious and complex calculations
of some of the thermodynamic properties of liquid helium
directly from the measured excitation spectrum. They found
that calculations based on the phononroton model deviate
very little from their exact calculations at low tempera
tures, but that these deviations increase sharply with in
creasing temperature, becoming quite large above 1.5 K. Be
cause the whole model collapses rapidly as one approaches the
X transition, this is not surprising. Unfortunately, no work
has ever been done on the pressure dependance of the devia
tions of the phononroton model from exact calculations,
partly because no extensive neutron scattering studies have
been done as a function of pressure. Only very high preci
sion measurements of thermodynamic properties will show de
viations from the phononroton model since the total devia
tion is of the order of 2% for temperatures near 1 K and de
creases rapidly as the temperature is lowered below 1 K.
There would appear to be little possibility that there
is any major flaw in the theory of sound propagation, since
it has worked so well for other materials. Thus attention
focused on the approximation that c(p) = cp(l yp"). If, as
Landau had assumed, y was greater than 0, then the dispersion
was normal, i.e., the type found in most materials. Normal
dispersion prohibits threephonon processes by the conserva
tion laws. Threephonon processes are those where one phonon
decays into two others, or where two phonons combine to make
a third. All theorists agree that the only way to produce
the experimentally observed T4 dependence of the attenuation
of sound at low temperatures is by allowing threephonon pro
cesses to occur.
Simon (1963) and Petick and Ter Haar (1966) showed that,
under some circumstances, threephonon processes were allowed
without dispersion. Due to the finite lifetime of the pho
nons, the phonon spectrum has a width which allows the con
servation equations to be satisfied. However, Friedlander,
Eckstein and Kuyper (1972) claimed to have shown that proper
renormalization of phonons forbids threephonon processes for
all cases of "normal" dispersion. The best fit to the CW re
rults at saturated vapor pressure was the phonon spectrum
S= cp(l yp2 6p4) with y = 0 2 x 1036 and
75 37
6 = 2.4 .3 x 10. This disagreed sharply with y = 8 x 103
derived from ultrasonic measurements by Eckstein and Varga
(1968) using Khalatnikov's approach. Havlin and Luban (1972)
were able i:o fit the ultrasonic data with a spectrum
E(p) = cp(l 6p4); but their 6  0.9 x 1075 was a factor of 3
lower than the neutron value. As will be shown in detail in
Chapter II, no thermodynamic measurement can separate the
effect of y2 from the effect of 6. So, unless y remains very
small over a considerable range of pressure, an analysis based
solely on 6 has limited usefulness. The high pressure neu
tron measurements of Svensson, Woods and Martel, which had
37
y = 6.2 0.6 x 10 argued against any assumption that y
was insignificantly small over the entire pressure range of
liquid helium.
Maris and Massey (1970) suggested that if y were less
than zerosocalled anomalous dispersionthe problems of
the theory would be solved. Anomalous dispersion unques
tionably permits threephonon processes. Further calcula
tions by Maris (1972, 1973), in which he solved the integral
equations for velocity and attenuation of sound numerically
without the customary approximations, but assumed anomalous
dispersion, were in good agreement with the experimental data
on attenuation and velocity of sound below about 14 atm.
pressure. These calculations yielded a value of 8 x 10
for y at saturated vapor pressure.
Jackle and Kehr (1971) were able to explain the shoulder
in the attenuation data by using anomalous dispersion and in
troducing a cutoff frequency. The cutoff occurred because of
an assumption that an ultrasonic phonon, i.e., one produced
by the experimenter, could only be absorbed by a thermal pho
non, i.e., one already present in the liquid due to its fi
nite temperature, if the thermal phonon had a momentum k
less than kc. This reproduced the shoulder very well if kc
was a strong function of pressure, decreasing sharply with
increasing pressure. That implied that the shape of the pho
non spectrum was very sensitive to pressure. Using available
data on the shape of the curve, they predicted that the
effect of the cutoff would be most noticeable between 14 and
19 atm., in accord with experiment.
The assumption that y is negative would radically change
the generally accepted ideas about phononphonon interactions.
While agreement with the ultrasonic data was gratifying, at
that time there w:as little other experimental evidence to
support this assumption. There were hints from the neutron
data that this might be possible, since earlier neutron work
by Henshaw and Woods (1961) had reported the possibility that
) was negative. There was also evidence from Xray scatter
ing that the small k structure factor was not inconsistent
with a negative 7. This result was given a boost with a
model calculation by lachello and Rassetti (1973), based on a
new technique of deriving the spectra from the helium poten
tial curve, yielded a model which had anomalous dispersion at
low pressures and normal dispersion at higher pressures.
Also Zasada and Pathria (1972) had shown a similar result;
negative y at low densities, positive y at higher densities
could be obtained for an imperfect Bose gas by using reason
able potentials. Experimental evidence discussed in the next
subsection has tended to support this hypothesis.
A recent calculation by Jackle and Kehr (1974) concluded,
on the basis of ultrasonic measurements at finite tempera
tures, that y was negative and nearly constant with pressure,
while 6 was positive and increased rapidly with pressure.
Thus the apparent normal dispersion at higher pressures would
be caused by 6 becoming so large as to dominate y rather than
by y changing signs.
All of the preceding calculations have assumed that the
phonon spectrum is given by
(p) = cp(l yp2 6p4...). (1.6)
The series is assumed to terminate with the 6 term in the
sense that higher order terms are too small to be observed.
Feenberg (1971) showed that for a simple interatomic poten
tial, such as the inverse r6, e.g., the dipoledipole poten
tial, that S(p) and E(p) must contain both even and odd
powers of p beyond the cubic term. Thus there was no a
priori reason why the expansion for E(p) should not include
all powers of p. This suggestion led Molinari and Regge (MR,
1971) to reanalyze the results of CW yielding a fiveterm fit
E(p) = cnp(l + 0.5465p 1.3529p2 + 0.2595p3 + 0.1860p4
0.0522p5)1/2 (1.7)
with both even and odd powers. Most experimental evidence
2
seems to indicate that the coefficient of the p term of the
expansion is zero. This destroys the ability of the formula
to fit the neutron data.
At about the same time, Gould and Wong (1971) showed
that for a weakly interacting Bose gas there are terms of the
form p5 log (l/p) in both e(p) and S(p). Their fairly general
arguments indicated that, to the extent that helium could be
treated as a weakly interacting Bose gas, no analytic expan
sion of E(p) was possible. However, since their complete
3 ~5
spectrum was of the form e(p) = cop + c2p + cLp log(l/p) it
would be difficult to detect the nonpolynomial terms. The
source of the singularity lay in the multiphonon interactions.
While multiphonon interactions are important in helium, it is
very questionable whether a weakly interacting Bose gas is a
good model for multiphonon interactions in helium. So far,
no one has analyzed the experimental data to look for evidence
of the predicted log (1/p) term, due to the formidable mathe
matical difficulties in doing such an analysis.
LinLiu and Woo (1974) presented a calculation based on
sum rules which the structure should obey. This yielded a
phonon spectrum at P = 0 and T = 0 of the form
c(p) = cp(l + 0.17 p 2 + 0.78 3 3.3 4). (1.8)
nimc mc mc
The positive numbers indicate anomalous dispersion. In this
calculation, the anomalous dispersion comes from both the p
termthe term with coefficient y in the usual expansionand
the p term. The p termthe term with coefficient 6 in the
usual expansionhas normal dispersion. No comments were
made about the pressure dependence of any of the coefficients.
With the exception of the work of Jackle and Kehr (1974), all
calculations of the spectra are for T = 0 at zero applied
pressure. With the exception of MR, all of the formulas for
the excitation spectrum are in agreement with experimental
data. This is hardly surprising since most experiments can
just barely detect the effect of the yp term, making posi
tive identification of the higher order terms difficult and
the assignment of accurate numerical values virtually impos
sible.
Recent Experimental Results
The principal experimental evidence has been a set of
specific heat measurements by Phillips, Waterfield and Hoffer
(PWH, 1970). They measured the specific heat of liquid
helium at four different pressures. At low pressures they
2
found evidence for anomalous dispersion. If E = cp(l yp ),
then y = 4.1 x 1037 at saturated vapor pressure. y de
creased in magnitude with increasing pressure and became
positive somewhere between 5 and 20 atm. At 20 atm., they
37
found y = + 19.6 x 10. They claimed their work could also
be fitted, though "less well" at saturated vapor pressure by
y = 0 and 6 = 4.5 x 107. No other fits were attempted.
There is some question about the validity of the analysis
performed on the specific heat data by PWH. Nevertheless,
the data are so significant that most theories are tested by
reanalyzing these data to see if they can be fitted by the
theory. No other thermodynamic data accurate enough to be
analyzed have been published.
A recent reanalysis of these specific heat data by
Zasada and Pathria (1974), using the measured neutron spec
tra instead of the phononroton approximation, yielded
y = 5.1 x 1037 at saturated vapor pressure. Similar
analysis was performed on the data at the other two "low
pressures," changing y slightly, but no fit was attempted
for the highest pressure.
Similarly the Xray measurements of S(p) of Hallock
(1972) could be analyzed to give y = 5.7 x 37 However,
(1972) could be analyzed to give y = 5.7 x 10 However,
they could alsc be analyzed to yield a quadratic term as pro
36
posed by MR or to give a = 3 x 10 i.e., = 0 to within
the quoted error of CW.
2
Since the quadratic term alp proposed by MR, was of
lower order than the usual yp3 term, there were immediate
attempts to observe it. Anderson and Sabisky (1972) measured
the acoustic thickness of helium films at 1.38 K for frequen
cies between 20 and 60 GHz. Their analysis yielded a posi
tive quadratic term in good agreement with MR. These frequen
cies were over an order of magniture greater than the usual
ultrasonic frequencies, the largest of which was 256 MHz.
The temperature was also much higher than usual, since most
measurement of phonon properties are made below 0.6 K. Re
cently Anderson and Sabisky (1974) stated that their analysis
of film properties may have been inadequate.
Roach et at. (1972c) measured the frequence dependence of
the sound velocity with an improved technique at T = 0.3 K at
frequency 30 and 90 MHz. They reported that if
E(p) = cp(l + alp), then a, = 0 0.01 A, in sharp contrast to
al = 0.275 0.030 A for Anderson and Sabisky (1972) and MR.
A reanalysis of the specific heat data by Zasada and Pathria
(1972) concluded that the inclusion of a quadratic term in
the energy spectrum led to a set of parameters whose behavior
as a function of density was too erratic to be acceptable;
whereas, setting al identically equal to zero led to a set of
parameters which, as a function of density, were in agreement
with direct measurements of the same quantities. This
experimental evidence is generally regarded as showing that
there is no quadratic term in the excitation spectrum.
The uncertainties in the experimental situation are
illustrated by two recent experiments. (1) Narayanamurti,
Andres and Dynes (1973) measured the group velocity and
10
attenuation of phonons with frequencies of 2 x 10 to 9 x
10 Hz, far in excess of any ultrasonic measurements. They
report zero dispersion over the whole range of frequencies.
As noted by Jackle and Kehr (1974), there appear to be some
inconsistencies in the analysis of Narayanamurti et al. For
example, in 1974 Dynes and Narayanamurti used threephonon
processes to analyze their experiments with heat pulses in
helium. The phonon lifetimes used in the heat pulse analysis
are inconsistent with threephonon processes occurring in the
absence of dispersion. The heat pulse experiments did show a
striking change near 15 to 17 atm. and normal dispersion at
higher pressures. (2) Mills et aZ. (1974) measured the
angular spreading of phonon beams, which they interpreted to
be evidence for negative (anomalous) dispersion below 17 atm.
and normal dispersion above that. This analysis was based on
the presence or absence of threephonon processes as shown by
the width of a phonon beam.
In summary, the weight of the evidence suggests that
there is anomalous dispersion in liquid helium at low pres
sures and normal dispersion at higher pressures. However,
more high precision experimental data are needed to decide
24
among the different theories. This thesis was undertaken to
provide high precision measurements of the thermody'namic
pressure as a function of both density and temperature below
1 K. The results 'wil] be compared with the predictions of
the different theories. The measurements were also carried
to higher temperatures to compare them with the measurements
of pressure as a function of density and temperature above
1.5 K made by Keesom and Keesom in 1933.
CHAPTER II
LANDAU THEORY
Algebraic Results
In this chapter the predictions of the thermodynamic
pressure for the different phonon spectra will be computed.
If one adopts the Landau model for helium, then the thermo
dynamic properties of liquid helium are determined by the
excitations. If there are Np excitations with momentum p
and energy c(p), the total energy of the fluid is
E {Np} = E + E Np c(p) (2.1)
where Eo is the ground state energy. This is the fundamental
assumption of Landauall of the energy of the system can be
accounted for by the excitations of the single particle exci
tation spectrum e(p). However, this is not strictly true.
As CW showed, there are other branches of the excitation
spectrum. However, these lie at such high energy, E/k greater
than 18 K, that they should have little effect on low tempera
ture (less than 1 K) properties, due to the weighting factor
exp (e/kT) in thermodynamic calculations.
By definition, the partition function for a liquid is
Z = Z E exp(BE{Np}) (2.2)
{Np }
where 6 has its usual definition, B = l/kT. Assuming that
the excitations are bosons, which would be expected for a
system of Bose particles, this equation reduces to
1
Z = Z_ I[le:.:p(6c (p) )] (2.3)
The Helmholtz free energy is defined to be F = kT log Z.
Thus, F is given by
1
F kT 7 log[lex:p(6c(p))] + Fo (2.4)
where Fo = hT log .Z i the ground state. free energy.
Assuming a continuum distribution of stOt':sC so that =
P
v 3p
SId p then
h3
kTV 1 3
F = log[lexp(RE(p))] d3p + Fo (2.5)
h3
This result assumes, in agreement with experimental results,
that E(p) is a smooth, wellbehaved function. For a fluid
one can assume isotropy, i.e., properties of the material de
pend only on the magnitude of p, not its direction. Thus one
3 2
can replace d p by 4Tp dp, yielding
4 kTV 2
F = Fo + I3 I log[lexp(B E(p))]p dp. (2.6)
h o
Integrating by parts gives
3
F = F kTV f p d[Be(p)]
S 3h3 oexp(BE(p))l (2.7
This assumes that Lim E(p) = c, which is true for all phy
sical systems.
In addition to the mathematical approximations discussed
above, there are two assumptions inherent in this approach
which must have their validity checked. (1) The excitations
are welldefined, longlived and weakly interacting so that
they can be treated as independent entities. This is con
firmed by neutron measurements. (2) The excitation spectrum
is a fixed quantity, neither varying with temperature for a
fixed density nor varying rapidly with small density changes.
To be useful, the spectrum must not vary during an experi
ment. For example, it will be assumed that the spectrum is
constant in this experiment, although the temperature varies
from 0.3 K to 0.9 K. Since neutron scattering measurements
extend only to 1.1 K, this assumption cannot be experimen
tally proven for temperatures less than 1 K. However, neu
tron measurements are almost independent of temperature be
tween 1 K and 1.5 K. Similarly, the sound velocity, which is
an explicit parameter in the excitation spectrum, varies by
less than 0.1% for temperatures between 0.1 K and 1 K and
varies slowly with density. However, theoretical calcula
tions by Ishikawa and Yamada (1972) showed that for an im
2
perfect Bose gas the higher order terms, such as yp varied
considerably with temperature and that temperature variations
in the interactions between excitations would have the same
effect as introducing new higher order terms. Because these
variations only become large for temperatures greater than 1
K, the rest of this development of the theory will assume
that the excitation spectrum is fixed in temperature.
Knowing c(p) from neutron and Xray scattering, and more
indirect sources, one calculates F from Eqns. (2.6) or (2.7).
Then, the thermodynamic functions P, S, E, and Cv are calcu
lated by the usual formulas:
S = ) E = F + TS, CV (
V ST V' V = T
T V V
In practice, the calculation of F is very difficult unless
further approximations are made. Furthermore, the calcula
tion of the thermodynamic functions is extremely difficult
unless F is known as an algebraic function of temperature,
T, and volume, V.
The usual approximation for helium at saturated vapor
pressure is shown by the dotted lines in Fig. 2. Fig. 3
shows the actual excitation spectrum and the phononroton
approximation for helium at 2.1 atm. pressure. Separate free
energies are computed for the phonons and rotons. Thus, F =
F, + Fr + ph Fo, the free energy at 0 K, is independent of
temperature. The volume derivative of Fo gives Po, the pres
sure at 0 K. Po will be treated as an experimental parameter
and will not be calculated. To calculate Fr one inserts the
roton spectrum E(p) = A + (ppo) /2j into Eq. (2.6). Only a
very sra.ll error is introduced by extending the limits of
integration to 0 and o. Since the energy gap, A, is large
compared to the temperature, the logarithm can be expanded to
.. V
^~~ '
." /
* _1 i
(N) Ak62iUa
.,,,, ~rl"u`rP"~,_TM1.r ~r.~rrrur unurl ~o*nTI
r
0
r1(
co
0
iJ
0
0
0
N
>io
( 0O 0
4J 0 r
0 0
a) 0
0 E
4 0 0
0"(a
,.0 x
(U 0 0
(0 U
0 0
40 4J
Or4
r P
Loxn
0 r
1 P
H H
)r
(U (U
'OH'I
4 J ,;
4J C;( U
0 *I f0
first order, i.e., log [lexp(S.(p)) = exp(Sc(p))]. Because
of the exponential weighting factr and the relatively small
width of the minimum, the major contribution to the integral
2
will come from p'"p so that the p in the integrand of Eq.
2
(2.6) may be replaced by p This gives
41TkTV 2 e ( 2
Fr = 3 exp(A/kT)Po2 / exp((pp ) /2pkT)dp. (2.8)
Integrating gives the result
Fr = (2 (kT)1/2p 2kTV exp(A/kT))/((27r)3/213) (2.9)
This yields
P rFV = 2(kT) /2kTV exp(A/kT) (2.10)
r av
[1 + A pA 1/2 p ll 2p D]/((27)3/2133
kT A ap Il B Po
In order to compute the pressure in a useful form, the
variation of A, p and 11 with density must be known. Donnelly
(1972) calculated the following values for the Landau parame
ters as a function of density. For p in gm/cm these values
are
A/k = (16.99 57.31p) Kelvin
po/ = 3.64p/3 inverse Angstroms (2.11)
p = (0.32 1.103p)M4He grams
These simple algebraic forms, based on the neutron scattering
work of Dietrich et aZ. (1972) will be used in preference to
the graphical tables of Mills (1965) which were based on
thermodynamic data. At low pressure the two different
methods yield similar values for A, po and ji, but have sub
stantially different values for the density derivatives ,
o p
P, and At higher pressures, the two approaches dis
agree about both the values of the Landau parameters and the
values of their density derivatives. The differences between
the Mills and the Donnelly roton spectra can be seen in Fig.
3. Measurements of Sr by van den Meijdenberg et at. (1961)
yield Landau parameters in substantial agreement with
Donnelly if the corrections at low pressures suggested by van
den Meijdenberg et al. are applied to their results. Using
Donnelly's values for A, po and p and their variation with
density, one obtains
P = 2kT {[kT(0.32 1.103p) ]12 [3.64p13]2 (2.12)
r (2T) 3/21
[1/3 57.31p + .5515p [exp((57.31p 16.99)/T)]}
T 0.32 1.103p
Fph is calculated in a similar fashion. A phonon spectrum,
the most general form encompassing all proposed formulas ex
cept that of Gould and Wong,
2 3 4
S(p) = cp(l+alp + a2p + a3p + a4p ...)
is inserted in Eq. (2.7). The limits of integration are
extended to infinity. This causes complete overlap of the
integrals for Fr and Fph. However, the resulting error is
small because of the large values of r and cph in the re
gions of overlap. This yields
4
F TIV(2nk) 4
ph 180h3 c
45)1 ( 2n1k T5 10 2 2) ( 6
 5')T + (3 a2)( )
27T
15B_(7) 3 2nrk 7
+ 15B (7) (15a a 3 16 )( ) T
7 1 2 3 1 c
811
+ 1/2(541 4 + 12a22
 751 2 + 18la3 3a4)
T8]
c
(2.13)
where B(n) equals the Gamma function of n times Zeta function
of n. Hence, the pressure is
Tr (2Tk)4 4 45B(5) 2Tk 1 5
ph 8(1 + 3F)T 5 (1+ 4 F)a p ]
ph 80h c 2 c 1
+ 10/7 (2 )2 [(1 + 5)(3a 2
c 1
15 2rk
+ 15 B(7) (Z )3
87 c
8r
1 a 2 6
C) p(6a1 ) T
2 1 p P P
[(1 + 6F)(15a l2 3a3 16a 3)
1 2 3 1~c
2 +1 3 2 1 7
 p(15a + 15a 3 48 ]T
1 dp 2 dp dp 1 dp
+ 1/2 ( 2~

4 2 2
[(1 +7F)(54al + 12a2 75a1 a2 + 18cla3 34)
3 1 2 1 2 2
 p(216a + 24c 150a 2 a 75a1 9
+13 8 l 4 8
+ 18a+ 18a 3 )]T } (2.14)
+ 18 3 p 3p
where p = is the Gruniessen constant. This expression
c p
is the first four terms of the MR formula. The standard form
2 4
for most theories is (p) = cp(l yp 6p ), i.e.,
a1 = a3 = 0, = y and a4 = 6. For this case Eq. (2.13)
becomes
4 2
SnV(21rk)4 4 10 2 k T6
Fph = [T + ( ) T
ph 33 7 c
180h c
4
2 27rk 8
+ 1/2(12y2 + 36) ( 2 ) T ] (2.15)
c
and Eq. (2.14) reduces to
4 2
i 2(2nk) 4 10 2k2 y TG
p (2 { (1 + 3)T4 + () [ (1 + 5{)y p+ 3
ph 3 3 7 c p
S180h c
+ 3/2 ( 2k) [(1 + 7F)(4y2 + 6) p(8 + )T8
c [C3
(2.16)
As a check on the validity of these calculations for
pressure, E, C and S were calculated from the free energy
Eqns. (2.9) and (2.15). These expressions for E, C and S
were compared with other calculations of those quantities by
Zasada and Pathria (1972), Roach et at. (1972e) and Donnelly
(1967) and were found to be in exact agreement. These calcu
lations led to the discovery that Eq. (2) in PWH (p. 1260) is
incorrect. The correct form is
4 2 4
ITV(2 1k) 3 25 2k 5 7(4y2 + 62k 2 7
S= 315h 3 + ) T + 7(4y + 6 ) T ...I
15h c
Numerical Results
The first term in Eq. (2.16), n(2nk)4(1 + 3)T4/180h3c3,
common to all theories, will be referred'to as the phonon
pressure. Similarly, Eq. (2.12) gives the roton pressure.
To compute the magnitude of the pressure due to the disper
sion term in the phonon spectrum, one needs the values of y
and Sy/Dp. Table I gives the values of y and y/;p for three
phonon spectra derived from experimental results. The values
of y and dy/Dp for most of the theories of superfluid helium
fall within the range of values of these three spectra. The
value of Dy/;p is calculated by assuming a linear variation
of y with p, i.e., 3y/ap = (y 24 a SVP at am. SVP)
Because this thesis concerns the measurement of pres
sure, numerical estimates of the size of the various contri
butions to the pressure are necessary. Table II gives numeri
cal values of the phonon pressure, roton pressure, and the
pressure due to dispersion for each of the spectra in Table I.
These values are given at saturated vapor pressure and at 24
atn.the two extremes of pressure accessible in the liquid.
Values for both 1 K and 0.5 K are given since the upper limit
of validity for the approximation that there are only phonons
and rotons present is 1 K, and since the effects of disper
sion are clearest near 0.5 K. For all of the spectra except
MR the dispersion pressure varies as T6. For MR the disper
sion pressure varies as T which should make MR readily dis
tinguishable from the other spectra, although the absence of
any comment by IMR on the variation of their spectrum with
Table I. Values of the Dispersion Parameter, y, and Its
Density Derivative in c.g.s. Units, Based on
Experimental Data
Method
Ultrasonic
Neutron
Specific Heat
SVP
8 x 1037 a
0 c
 4.1 x 1037 c
24 Atmospheres
6 x 1037 b,d
6 x 1037 d
19.6 x 1037 e
ap
38
 7.3 X 1038
2.19 x 103
8.65 x 103
aEckstein and Varga (1968)
bMills et al. (1974), Svensson, Woods and Martel (1972)
CCowley and Woods (1971)
"Svensson, Woods and Martel (1972)
ePhillips, Waterfield and Hoffer (1970)
Table II. Contributions of Different Terms to the Pressure,
Expressed in Atmospheres, at Saturated Vapor
Pressure and at 24 Atmospheres
0.5 K
1.0 K
Phonon
SVP
24 atmn.
Roton
SVP
24 atm.
Phonon plus Roton
SVP
24 atm.
Dispersion
Ultrasonic
SVP
24 atm.
Neutron
SVP
24 atm.
Specific Heat
SVP
24 atm.
3
1.47 x 10 3
4
3.33 x 10
6
1.18 x 106
51
3.29 x 10
3
1.47 x 1.0
4
3.04 x 10
5
9.71 x 10
6
9.74 x 10
5
2.27 x 105
3.07 x 10
4
1.37 x 10
7.7 x 10 6
2
2.35 x 102
3
5.34 x 10
3
9.29 x 10
2
5.33 x 10
2
1.42 x 10
2
4.79 x 10
3
6.21 x 10
4
6.23 x 10
3
1.48 x 103
1.96 x 104
"3
8.78 x 10
4.94 x 104
Te2 rm
density makes it impossible to calculate a value for the dis
persion pressure for their phonon spectrum. The values of
pressure in Table II should be compared with Po, which ranges
from approximately 0.1 atm. for an allliquid sample just
above saturated vapor pressure to approximately 24 atm. for
an allliquid sample just below the melting curve.
It is difficult to calculate the T term in Eq. (2.16)
the next order correction termsince it depends on y
y y/Dp, 6, and 56/p. To estimate the size of the T term
at 1 K, y and Dy/Dp were taken from PWH. 6 was set equal to
2
4y and 56/%p was set equal to 8y M6/Dp, so that the contri
buLion of 6, and 56/9p to the T pressure term would be equal
to the contribution of y and Dy/8p. This gave a value of
6 = 1076 and 96/Dp = 2 x 078 at saturated vapor pressure.
76
Only Jacklc and Kehr (1974) have a 6 greater than 10 at
saturated vapor pressure. However, the large positive vari
ation of their 6 with p would tend to cancel the effect of
8
the large 6 term. Despite this maximization of the T term,
it is less than 10% of the T6 term at 1 K and decreases more
rapidly with decreasing temperature than the T term.
It is also necessary to estimate the error caused by the
inaccuracy of the phononroton approximation in the region of
the maximum of the excitation spectrum, shown in Figs. 2 and
3. If one introduces another type of excitation, called
maxons, characterized by an equation of the form
p 2
(P Po )
c(p) = A 
2p
over a restricted range, the properties of the superfluid can
be analyzed in terms of three excitations instead of two.
Unfortunately, because of the necessity of restricting the
range of integration on each of the excitations, the inte
grals in Eqns. (2.7) and (2.8) can no longer be done in
closed form. They must be done numerically. Since the spec
trum for the maxons is similar to the roton spectrum, the
equation for pressure will be similar to Eq. (2.10). Thus
the maxon pressure will vary as exp(A/kT). Substituting a
reasonable A in Eq. (2.11) gives a pressure equal to approxi
mately 5% of the roton pressure for low pressures. Since the
roton pressure is considerably smaller than the plionon pres
sure, the effect of the maxons is negligible for temperatures
below 1 K. As shown by Fig. 3, at higher pressures the situ
ation is different, the height of the maxons changes only
slightly, but the width approximately triples. Because A
increases, the pressure due to the maxons will be positive.
A reasonable estimate is that the contribution of the maxons
will be approximately 10% of the roton pressure at higher
density. Because the roton pressure is equal to or larger
than the phonon pressure at these densities, this will intro
duce a serious error. From Fig. 3 it would appear that the
Mills spectrum, derived from thermodyniamic measurements, is
shifted to lower p, higher A, and broader 11 in order to aver
age in the effects of the maxons. Thus the difference between
the thermod.namic values fo:r I, po and A and the neutron
values at high densities may be due to the inadequacy of the
phononroton approximation in the region of the maximum of
the spectrum. If the maxons produce a pressure large enough
to interfere with measurements at higher densities, one could
use the Mills roton spectrum to alleviate the problem.
From Table II it is evident that one needs a pressure
5
resolution of the order of 10 atm. at saturated vapor pres
sure to clearly see the dispersion pressure. At 24 atm. a
resolution of the order of 106 atm. is required, which im
plies a substantially higher instrumental resolution. If such
resolution could be obtained, pressure measurements would
differentiate between the different predictions of the dis
persion parameters.
CHAPTER III
EXPERIMENTAL APPARATUS AND PROCEDURE
The Cryogenics
The cryostat is similar to the one described by Walsh
(1963). It has been substantially modified from the one
described by Heberlein (1969). The main features of the
cryostat are a helium bath, a vacuum space, a He refrigera
tor, a 3He refrigerator and a sample chamber. Fig. 4 shows
the cryostat schematically.
The vacuum space is enclosed by a copper cylindrical
container which is attached to a flange on the cryostat. The
vacuumtight seal is made with 0.075 cm diameter pure indium
wire. The vacuum space surrounds the 4He refrigerator, the
He refrigerator and the sample chamber. When the apparatus
is first being cooled down, by filling the helium bath, the
vacuum space is filled with helium gas for purposes of ther
mal contact. It is then evacuated by pumping for several
hours with a Consolidated Vacuum Corporation PMCS2C oil dif
fusion pump backed by a Welchmodel 1400 pump. To prevent
contamination there is a nitrogen trap in the pumping line.
The 4He refrigerator has an internal volume of approxi
mately 250 cm3. It is filled from the bath via a modified
Hoke valve operated by a long shaft extending through the top
flange of the cryostat. This refrigerator can be maintained
Fig. 4. Schematic drawing of the cryostat.
A Electrical feedthroughs
B Pumping line for vacuum space
C Diffusion pump for 3He refrigerator
D Mechanical pump for 3He refrigerator
E Trap for 3He return line
F Vacuum space flange
G 4He refrigerator
H 3He refrigerator
I Low temperature valve
J Radiation shield
K Sample chamber
L Vacuum can
M Pumping line for 4He refrigerator
B
'3
at a temperature of approximately 1.2 K for over 24 hours by
pumping on the enclosed helium with a Kinney model KC46
pump.
A radiation shield, attached to the underside of the 4He
refrigerator, surrounds the 3He refrigerator and the sample
chamber. The shield was constructed in the following manner.
A mat of #44 bare copper wire was wound on a drum. A coat of
General Electric 7031 varnish was applied to hold the mat
together. The mat was then fitted around and epoxied onto a
thin phenolic plastic form. The upper ends of the copper
wires were stripped of varnish and soldered onto a copper
ring which screwed into another copper ring bolted to the
bottom of the 4He refrigerator. This arrangement provided
the largest possible crosssectional area for experimentation
below the 4He refrigerator.
The 3He refrigerator was designed for continuous opera
tion. The refrigerator holds approximately 1 cm3 of liquid
3 3
3He. To increase thermal contact between the He and a cop
per flange, which is the base of the 3He refrigerator,
approximately 1 meter of thin (0.025 cm) copper foil was
placed in the refrigerator and hard soldered to the base.
The returning 3He gas is forced to pass through a trap im
mersed in liquid nitrogen. This trap consisted of several
layers of copper wire mesh followed by Linde molecular sieve
#13 X. To provide the pressure drop needed to liquefy the
returning 3He gas, an impedance was placed in the return line
below the 4He refrigerator. The impedance was made by
inserting approximately 18 cm of 9 mil copper wire into a
10mili.d. coppernickel capillary. A National Research
Corporation model B2 oil diffusion pump backed by a Welch
model ]402 pump with an oil shaft seal for closed system
operation was capable of reducing the temperature of the
sample chamber to about 0.45 K. By closing a value on the
He return line and operating for periods up to two hours in
the "single shot" mode, colder temperaturesabout 0.39 K
for the sample chamberwere achieved.
The sample chamber is supported by a 15 cm long, pitch
bonded graphite rod approximately 1 cm in diameter. To pro
vide thermal contact between the 3He refrigerator and the
sample chamber, a copper rod with a diameter of 0.5 cm was
screwed into the base of the 3He refrigerator and a length
of copper braid with a diameter of 0.4 cm was hard soldered
into a copper rod which screwed into the sample chamber. The
two copper pieces were connected by a lead heat switch. The
heat switch was 0.30 cm wide by 1 cm long by 0.03 cm thick.
It was joined to the copper pieces with a thin coat of soft
solder. Around the switch was placed a small, 1 cm bore,
superconducting solenoid capable of generating a field of
1000 gauss with a current of 3 amps. The solenoid was
powered by a 6 volt battery which also provided the current
for the heater on the persistent switch.
The Sample Chamber and Pressure Measurement
The sample chamber shown in Fig. 5 was a modification of
the capacitive strain gauge described by Straty and Adams
G .\ H
F///
//,*.*.'* _  
.... ... I tli t
Fig. 5. Sample chamber
A Body of the chamber
B Diaphragm support
C Guard ring
D Moveable capacitor plate
E Fixed capacitor plate
F Diaphragm
G Sample
H Inlet Capillary
The copper brush is not shown
(SA, 1969). The cell had a volume of approximately 5 cm and
a height of 0.42 cm. To increase thermal contact to the sam
ple, the interior of the cell was filled with fine copper
wires using the technique of Kirk, Castles and Adams (1971).
These wires had a surface area of approximately 250 cm and
filled about 30% of the volume of the cell. The main body of
the cell was made of copper, while the diaphragm and the
capacitor plates were made of berylium copper. The diaphragm
was 0.2 cm thick. The upper capacitor plate was epoxied onto
a post machined into the center of the diaphragm, so that as
the diaphragm flexed the upper plate moved. The lower capa
citor plate was held in a fixed position by a guard ring
bolted to the piece containing the diaphragm.
Changes in the sample pressure cause a deflection of the
diaphragm, which produces a change in the capacitance by
changing the spacing between the capacitor plates. Expanding
Eq. (7) in SA, one can show that
P P = A(C C ) + B(C C )
0 0 0
+ D(C C 3 + E(C C) +... (3.1)
where P is pressure and C is capacitance. In practice, the
coefficients D and E are so small that the contribution of
those terms is negligible except for very large pressures.
Because of the very small spacing of the plates, 0.0012 to
0.0025 cm, any imperfections in the plates will cause short
ing at low pressure. The plates were reasonably free of
imperfections, since a pressure of about 15 atm. was reached
before shorting. In order to reach higher pressure, the
plate spacing was increased to 0.0075 cm. With this spacing,
the plates remained unshorted to a pressure of over 30 atm.
The capacitance was measured with a General Radio type
1620 A capacitance bridge. An Ithaco model 391 A lockin was
used to detect the balance of the bridge. A General Radio
1321 A audio oscillator provided a 10 v, 500 Hz signal to
drive the bridge. With this arrangement, changes in capaci
tance as small as 10 pF could easily be detected. This
6
corresponds to a pressure change of from 1 x 106 atm. to
5 x 106 atm., depending on the pressure. As the pressure
increases, the sensitivity increases because the plates are
closer together. Thus this apparatus could easily detect the
dispersion pressure.
The capacitance was measured by a threeterminal tech
nique which uses a separate coaxial line for each plate with
neither plate grounded. This prevented shifts in capacitance
due to small movements of the leads. To reduce any effect of
changing temperature on the bridge, it was placed in a large,
insulated box. The temperature of the laboratory was regu
lated so that the largest temperature drift over the course
of a run was 10 F. The measured drift rate of the bridge
never exceeded 2 x 10 pF for 8 hours.
The calibration of the strain gauge consisted of observ
ing the capacitance of the gauge versus the pressure read on
a gauge in the external system. The external gauge was a
Heise model #7770a dial type Bourdon Gauge. Calibration of
the strain gauge extended over a wide range of pressures
from 0 to 10 or 14 atm. The calibration points, consisting
of a capacitance and a corresponding pressure, were used to
find the coefficients of Eq. (3.1) by means of a least
squares program. For this calibration P was 0 and C was
the capacitance of the cell at 4 K when evacuated. The re
sulting fit had an RMS deviation of less than 0.01 atm. The
absolute accuracy of the calibration measurements was no bet
ter than 0.05 atm., the absolute accuracy of the gauge. How
ever, the relative accuracy, i.e., the accuracy in the measure
ment of small changes in pressure at a given pressure, was
much higher being limited by the resolution of the strain
gauge and the accuracy of the derivative dP/dC. The values
of dP/dC must be fairly accurate to obtain a good fit over a
wide range. The thermodynamic pressure is the pressure at
constant volume. Obviously, the volume of the cell is not
strictly constant. SA, in Eq. (8), show that
,P dP 1 dV
( dT (1 + kV d) (3.2)
3T V dT kTV dP
where kT is the isothermal compressibility of the sample.
For this apparatus the correction term is less than 0.5%.
Furthermore, Boghosian and Meyer (1966) showed that kT is
virtually independent of temperature below 1 K, varying by
less than 0.01%, and that kT varies very slowly with pres
1 dV
sure. The geometrical factor V dP is independent of tempera
ture and pressure below 1 K for small pressure changes. As
shown in Table II, the total variation in pressure between 0
2
and 1 K is approximately 102 atm. Such pressure changes
1 dV
have negligible effect on k and V dP. Thus the correction
for the change in volume of the cell is constant with tempera
ture below 1 K. The analysis of the data will be concerned
mainly with the temperature dependence of the pressure, so a
smallless than 0.5%constant correction should not affect
the analysis of the data. For that reason the correction
will be ignored in subsequent analysis.
Since the thermodynamic pressure is the pressure at con
stant volume and constant density, it is necessary to isolate
the cell from the external pressure system. At first a valve
on top of the cryostat was used. This proved unsatisfactory
because of variations in the pressure in the filling capil
lary. These variations were caused by the normal changes in
the level of the liquid helium bath which changed thermal
gradients in the filling capillary even though it was inside
an evacuated tube. To alleviate this problem a low tempera
ture valve, shown in Fig. 6, based on the design of Roach et
al. (1972d) was built. The valve was mounted immediately be
low the sample chamber so that the valve was at the same
temperature as the sample cell. The valve seal was made by
a Teflontipped brass stem sealing against a brass seat. The
end of the brass stem was threaded and then machined to a
conical shape. The inside of the Teflon tip was machined to
the same geometry and screwed onto the brass stem. In this
way, the Teflon tip was fully supported by metal. The valve
Z77zziLLL/_ '
_7 
........ : :..=..=...
1D
4E
Fig. 6. Schematic diagram of low temperature valve.
A Valve body
B Accuator bellows
C Valve bellows
D Accuator piston
E Valve stem
F Valve seat
was an hydraulic one, operated by changing the gas pressure in
a capillary leading to the valve. At the start of an experi
ment, helium gas was let into the capillary and liquefied by
contact with the bath until the valve actuator and the lower
section of the capillary were full of liquid helium. There
after, the valve could be opened and closed easily by changing
the gas pressure in the filling line. A pressure of about
6 atm. was required to fully seal the valve. A small SA type
strain gauge monitored the pressure in the actuator.
Unfortunately, the valve had a small superfluid leak.
Experimentation showed that in a given run the superfluid leak
dC
rate, i.e., d, was constant in time regardless of pressure
differential across the valve. No dependence of the leak rate
on temperature was observed for temperatures well below the X
line. The leakage rate changed with each opening and closing
of the valve, varying from 106 pF/min to 10 pF/min. Since
the leak rate was constant in time, a simple correction fac
dC
torthe e3apsed time multiplied by could be added to
each capacitance reading, thus removing the effects of the
leak on the capacitance readings. Upon disassembling the
valve, the problem appeared to be small flecks of solder on
the valve stem. Two subsequent attempts to clean the valve
to prevent leakage failed, resulting in much larger leaks.
Temperature Measurement and Regulation
The temperature of the sample chamber was measured by a
calibrated Cryocal Gerxmanium thermometer whose electrical
resistance varied from 1000 ohms at the ). line to 400,000
ohms at 0.3 K. The thermometer was tightly mounted in a
copper block bolted to the sample chamber. A light coat of
Apezion ';' grease was used to provide thermal contact. The
leads to the thermometer were Midohm wire whose resistance
is independent of temperature. The leads were thermally
anchored to the He refrigerator, the He refrigerator and
the sample chamber by epoxying them to copper posts bolted to
these objects. The resistance of the thermometer was mea
sured or an AC resistance bride described by Castles (1973).
A Princeton Applied Research model HR S lockin served as
both the source of the 20 Hz driving signal and as the de
tector of the balance of the bridge. A G.R. standard decade
resistor with a minimum resolution of 0.] ohms was used as
the known resistance in the bridge. Using this arrangement
it was easy to resolve 0.02% changes in the resistance. This
corresponds to a temperature resolution of better than 0.1 mK
over the entire range of this experiment.
This thermometer had been calibrated against the vapor
pressure of He and He between 0.4 K and 4.2 K by Philp
(1969). He found that a form
N 8 i1
T = Z Ai (R ) (3.3)
i=l1
where T is temperature and R is resistance with N = 5 fitted
the temperature "with an RMS deviation of less than 1 mK
above 0.6 K." (p. 39) A further calibration of the thermometer
against the susceptibility of CMN, extending to 0.29 K was
done. These results were fitted by the same form using the
program RESFIT written by Philp. These fits gave a set of
coefficients, Ai, which could be substituted in Eq. (3.3) to
give the temperature for any given resistance. This yielded
a fit with an RMS deviation of less than 2 mK over the whole
range, with a slight increase in the deviations near 0.3 K.
The outofbalance signal of the lockin was used to
modulate the power supplied to a heater on the sample cham
ber. The heater was a 2000 ohm metal film resistor attached
to the copper braid from the heat switch, just above the sam
pie chamber. A metal film resistor was used because its
resistance was almost independent of temperature. By suit
ably adjusting the bias current and the magnitude of the out
ofbalance signal from the lockin with an external circuit
built for that purpose, the temperature of the sample chamber
could be stabilized to within the resolution of the bridge at
any temperature between 0.4 K and 1.2 K. Above 1.2 K it was
necessary to reduce the cooling power of the refrigerator be
cause the heater had insufficient power to stabilize the
temperature. This could be done by pumping on the 3He with
only the mechanical pump, or by turning off the 3He refrigera
3 4
tor entirely, or by turning off both the 3He and He refrig
erators, depending on the temperatures required. For the
very lowest temperatures, all heating and regulation were
turned off and the system was allowed to slowly drift to
colder temperatures. Thus regulated, the temperature was
quite stable. The largest observed drift was 4 mK over
several hours.
The Samole
The sample was commercial grade helium from Air Products
and Chemicals Incorporated (Airco). It was passed through a
nitrogen trap with the same copper wire, molecular sieve com
bination as the 3He return line trap previously described.
The gas then passed through a trap filled with Linde #13x
molecular sieve immersed in liquid helium. Testing with a
Veeco MS 9 leak detector, modified to scan both the 3He and
4He peaks, revealed that the He concentration was much less
than 0.01%.
Procedure for Taking Data
Data were taken by recording the pressure at a fixed
temperature after the system came into equilibrium. The sam
ple was then warmed or cooled to a new temperature and ano
ther reading was taken. In most runs, the apparatus was
first allowed to cool overnight. Starting from this tempera
ture, the sample was systematically warmed in approximately
25 mK steps to some predetermined temperature greater than
1 K. Then the sample was cooled in 25 mK steps back to the
original temperature. To assure consistency between runs,
the same set of temperatures was used for each data pass. In
the early runs that extended to the X line, it was found that
after warming to the X line the apparatus could be cooled
well below the starting temperature, but would suddenly and
irreversibly warm up to above the starting temperature. This
was traced to inadequate thermal grounding of the fill capil
lary and valve actuator capillary. After correcting this
problem, it was possible to repeatedly cover the whole P ver
sus T curve both warming and cooling. The leakage rate
through the valve was measured by holding the cell at a con
stant temperature for about an hour and observing the chance
in capacitance. Since the P versus T curve has a maximum
near 1.1 K at saturated vapor pressure and near 0.7 at 24
atm., the leakage rate was measured at this point to minimize
the effect of any inaccuracies in the thermal regulation.
CHAPTER IV
RESULTS AND CONCLUSIONS
Data Reduction
The raw data consist of capacitance values versus resis
tance bridge readings which can be converted to pressure ver
sus temperature. Before converting the capacitance values to
pressures, two corrections to the capacitance were made. As
described in Chapter III, the correction for the leakage
through the valve was done by adding a constant times the
elapsed time to each capacitance value. The correction is al
ways additive because the fill capillary is evacuated to re
duce the heat leak, so that the mass flow is out of the cell.
By filling the capillary to a pressure greater than that of
the cell, the mass flow is into the cell, the capacitance
rises and the correction is subtracted from each value. In
the worst case, the total leakage during a run changed the
4
density of the liquid by less than one part in 10 Thus, the
approximation of constant density remains valid. After apply
ing this correction, the values of capacitance for both warm
ing and cooling generally coincided. However, in some of the
runs there were obvious discontinuities in the capacitance
values, e.g., in one set of data a group of points taken
while cooling the sample was exactly 4 x 104 pF larger than
the same points taken while warnring the sainple. Philp had
reported "Physical shocks . caused the capacitance to
shift discontinuously" (p. 56). The problem was eventually
traced to excessively tight vibrational coupling of the
cryostat to the main support frame. Small contacts with the
support frame, e.g., closing the 3He return line valve or
bumping an exposed corner of the frame would repeatedly cause
discontinuous shifts in capacitance, although moving heavy
objects in the laboratory or banging the apparatus violently
other than on the main frame or dewars produced no shifts in
capacitance. After discovering the cause of the problem,
contact with the frame was avoided. No further discontinuous
shifts occurred.
Corrections for the shifts were applied in a direction
which made capacitance versus temperature a smooth function.
After these corrections were applied to the capacitance
values, the capacitances taken when warming the sample agreed
very closely with the capacitances taken when cooling the
sample. In all cases the differences between warming and
cooling were within three of the smallest resolved units of
capacitance over the entire range from 0.4 K to 1.0 K and
were frequently zero. This agreement, as well as the sta
bility of the capacitance at a given temperature indicated
that the sample was in thermal equilibrium.
The corrected capacitances were converted to pressure by
Eq. (3.1), using the coefficients developed in the calibra
tion procedure described in Chapter. III. The resistances
were converted to temperatures by Eq. (3.3) using the
coefficients developed in the calibration procedure described
in Chapter III. From the pressure values, the density was
obtained by comparing the pressure at the lowest temperature,
0.4 K, with the pressure versus density data at 0.1 K found
in Table III of Abraham et al. (Ab, 1970). A linear interpo
lation was used between their data points which introduced a
slight errorno more than 0.0003 gm/cm3 for densities between
0.1451 and 0.1725 gm/cm3because the pressure versus density
function is not strictly linear. To within the accuracy of
the data of Ab (1970) 0.01 atm., the measured pressure is
constant below about 0.8 K so that no significant error is
introduced by using the pressure value at 0.4 K. To check
this determination of density, measurements were extended to
the A line and the pressure at the X line was compared with
the measurements of pressure versus density along the A line
by Keesom and Keesom (KK, 1933). Good agreement, i 0.0005
gm/cm3, was found between densities determined by comparison
with KK and those determined by comparison with Ab (1970),
despite differences of several atmospheres in pressure be
tween the N line and 0.4 K for a given run. Having estab
lished this correspondence at several pressures, subsequent
runs were not extended to the A line because of experimental
difficulties in doing this. For this reason, the values of
density obtained from Ab (1970) were used.
Knowing the density of the sample and the temp er.ture at
each point, the roton pressure can be calculated from Eq.
(2.12). Alternatively, one could have determined the roton
parameters from the data and compared these values with
values determined from other experiments. This was not done
for three reasons. (1) Determining the roton parameters
from the data would introduce six new parameters, A, A/p,
p, a/a/p, p and Po/9p. Four of these parameters, p,p//p,
p and Po/p, are difficult to measure; hence they are
poorly known. (2) Such an analysis would increase the num
ber of variables to 10 for the simplest spectrum. This is
too many to be well determined from just the 20 data points
in a run. (3) This would have required nonlinear, least
squares fitting routines which are difficult to develop and
use, and can lead to spurious results.
Since P = P + + P P where Ph includes both the
o r ph ph
Pph' the term proportional to T4, and the dispersion pres
sure, the only remaining parameter to be determined is Po'
the pressure at 0 K. Ideally, this would be determined by
lowering the temperature until the pressure reaches a con
stant value. Since this requires a temperature of less than
0.25 K, this was impossible in this experiment. Another,
less accurate, possibility is to extrapolate the lowest tem
perature pressures by assuming that temperature is suf
ficiently low that the only contribution to the pressure is
the Pph which varies as T4. This requires a substantial
number of points below 0.4 K. Since heat leaks limited the
experiment to T greater than 0.4 K, such a method was useless.
P was determined from a least squares fit of
o4
P P= P + AT + BT + CT8. As a check on the fit, the
r o
sound velocity was calculated from Ab (1970). The calculated
value agreed closely, : 1 to 3 m/sec out of 250 to 300 m/sec,
with the experimental values given in Ab (1970). Other in
ternal evidence, discussed later, suggests that this gave a
reasonable value for P The reduced data are a set of values
of Pph = P (P + P ) versus temperature.
Because the least squares routine used will not accept
more than one value of pressure for each temperature, the
points taken cooling and warming were averaged together to
produce a single pressure at a given temperature. The co
efficients determined by the least squares fits were converted
to meaningful quantities by using Eq. (2.16) with the values
of C, F and p from Ab (1970) and the fundamental constants from
Appendix I of Donnelly (1967).
Data Analysis
In order to display the T4 and T6 terms in the expres
sion for Pph as given by Eq. (2.16), a plot of Pph/T4 versus
T2 was made. Fig. 7 shows this plot for the experimental
densities. If there were no dispersion, i.e., if only the
phonon pressure were present, the graph would be a horizontal
line. On this type of plot a T6 dependence is a straight,
nonhorizontal line. The lines along the left edge of the
figure indicate the position of this horizontal line for each
run. The lines are labeled with the density of the sample for
that run. This is a plot of the "raw" data, i.e., every ob
served point in this temperature range has been reduced and
plotted. At higher temperatures the difference between points
is smaller than the size of the symbol used to plot them. The
0,022
0. /1 64
v
0v v
V
V
V
30
7
" C v
O
o o
0.1474
0.020 ..
P .0i
0 o o
0.1 '36
,. 151.9
C. 014.
0.012
03. O0 l1 576
0o0
00 0
0 0 0
a O
S a0
0
FAJ 1
a
0.1549
A
a
a
0.1
0 0
0.24
0.2
0o
S9D 0 9
0.3
0. 4
0.5
0.6
T2 (K2)
Fig. 7. Pph/T4 versus T2. Lines indicate position of line of
zero dispersion for density given on the line. Solid tri
angle 0.1464, solid circle 0.1474, open circle 0.1496,
solid square 0.1519, open triangle 0.154' open square
0.1576.
7 V
V
S7
7 V V
0 0
0
0
0 0
0 0 0
0 0
scatter at the lower temperatures is largely due to instru
4
mental resolution, magnified by the T4 weighting factor.
The error bars represent instrumental resolution. The smooth
ness with which the plots approach the zero of temperature is
indicative of the correctness of the method of choosing Po.
Experimentation showed that small changes in P produced pro
nounced curvature in the low temperature end of the plot
upward when Po was too low, downward when Po was too high.
In the low temperature limit, the dispersion pressure becomes
negligible so that Pp is equal to P Thus these plots
must join the horizontal lines smoothly in the low tempera
ture limit. The two highest densities are smooth in the low
temperature limit but do not extrapolate to the phonon pres
sure. The reason for this is unknown.
The most noticeable and unexpected feature of these plots
is the sharp upward curvature of each plot as the temperature
increases. Because the curvature is upward, this represents
an "excess pressure." Since the excess pressure contribution
has a noticeable convex curvature, its temperature dependence
is stronger than T6. For higher densities, the magnitude of
the excess pressure is larger relative to Pph. Also at
higher densities, the excess pressure becomes noticeable at
lower temperatures. The curvature of the plots due to the
excess pressure contribution is greater at higher densities
than it is at lower densities.
To test whether this excess pressure arose from the T
term in Eq. (2.16), the points up to T = 0.85 K were fitted
with a least squares routine to the functional form
P = P + AT4 + BT6 + CT8 + Pr. Then, systematically the
high temperature points were removed. This sharply lowered
the RMS deviations, typically by a factor of five when
changing from 20 to 14 points, and substantially changed, by
a factor of two, the value of the coefficients B and C.
Furthermore, the values of B and C for points up to 0.85 K
were well outside the range of predicted values. In contrast,
the values for B and C for points up to 0.7 K were in reason
able accord with predicted values. The magnitude of the ex
cess pressure contribution substantially exceeds the value
estimated for the T8 term in Chapter II. Thus it was con
cluded that excess pressure was not due to the T term in Eq.
(2.16).
The higher order terms in Eq. (2.16) and in the expan
sions of other thermodynamic functions are small since these
expansions are rapidly convergent with increasing powers of
T, so it seems unlikely that the excess pressure arises from
omitted higher order terms. While there is some evidence
that the shape of the spectrum, hence y and 6, may vary with
temperature, a variation with temperature of the magnitude
required to produce excess pressure seen here would totally
destroy the Landau approach which is well supported by other
measurements. The roton pressure could be in error, but
there is no reason to believe that P is incorrect.
The likeliest explanation for the excess pressure is the
maxons, i.e., contributions from the maximum of the excitation
spectrum which are poorly approximated by the phononroton
model. As was argued in Chapter II, the maxon contribution
to the pressure would be positive, increase with increasing
density and would vary as exp (1/T). This is consistent
with the observations, since T + T + T would be a poor
approximation to exp (1/T). In addition, there is evidence
that a similar phenomenon was observed in the specific heat
measurements. In their analysis, PW'H had to limit themselves
to T less than 0.7 K to achieve good fits. Furthermore, the
PWH value of y at 20 atm. is a factor of 3 higher than y
determined by neutron scattering and ultrasonic measurements,
which are in good agreement with each other. PWH let the
data determine the roton parameter, A, in contrast to this
work which calculates the roton contribution to the pressure.
The PWH values of A agree much more closely with Mills (1965)
values than with DonLnelly's (1972) values. As noted ir, Chap
ter II the Mills spectrum disagrees with the neutron scatter
ing values in a fashion that would suggest an attempt to
average over the maxons. The Mills spcctrLum is based on
thermodynamic measurements. This suggests that all thermo
dynamic data see the effect of maxons but that these contri
butions are obscured by allowing the roton parameters to vary.
Even with this variation of the roton parameters, the Mills
spectrum remains a poor approximation to the maxons. Such an
error in approximation might explain the discrepancy between
y of PWH at 20 atm. and f'o from other measurements. Neutron
scattering determines y by fitting the spectrum in the small
p region. This fit is cut off below the maximum of the spec
trum. Thus the maxons have no effect on the determination of
y. Ultrasonic measurements are usually taken at 0.1 K, well
below the temperatures at which maxons contribute. Both the
specific heat and these pressure measurements extended to
0.9 K. At that temperature, the calculation of the thermo
dynamic properties directly from the excitation spectrum at
saturated vapor pressure by Singh (1968) showed that there
were detectable discrepancies between calculations based on
the phononroton model and direct calculations. It appears
from these data that these discrepancies are larger and be
come detectable at lower temperatures as the density is in
creased.
Turning to the low temperature portion of the data, it
is evident that all of the curves have an initial downward
slope. This slope appears to decrease with increasing density.
The coefficient of the T6 term in Eq. (2.16) is proportional
to 1/c2, where c is the sound velocity. Since, as the density
increases, c increases, some decrease in slope of the plots in
Fig. 7 would be expected. To calculate the value of
Y= (Y Y) from Eq. (2.16) at different densities,
1+5F 5p
the following procedure was adopted. Values of c and r, taken
from Ab (1970) were used to calculate P ph and the constants
in the coefficients of the T6 and T8 terms in Eq. (2.16). The
quantity P (P + P + Ph ) was least square fitted to both
6 6 8
a form BT and a form B T + CT Data points were systema
tically removed frcm the high temperature end until the effects
of the excess pressure were negligible. Two criteria were
used to determine the absence of the excess pressure. (1)
The quality of the fit from the data for the form BT6 must
not depend systematically on T. If the data used in the fit
extend into the excess pressure region, the deviations in
crease sharply at the high temperature end. (2) The value
of y derived from B must agree reasonably well with the value
derived from B
The results of this procedure are shown in Table III for
the four lowest densities for both the T6 fit and the T6 + T8
fit. The three higher densities fail to join the horizontal
AT4 line smoothly at T = 0, probably due to the increasing
magnitude of the excess pressure, so that fits of this type
are useless. The actual numbers are accurate to only 10 to
25% because of the relatively small number of points fitted
(10 to 15) and the sizeable changes that occur when the
points are removed. Nevertheless, it is clear that y is
negative and decreases in magnitude as the density increases.
The numbers for = 42 + 67 (r + ) from Eq. (2.16)
1+7 3p 3p
are about equal to 42 0 (8y ), the contribution of
1are about equal to +7.1 'ap
higher powers of y. There is very little improvement in the
RMS deviations when the T term is included. These values of
yf should be compared with values of y based on the spectra
in Table I. At saturated vapor pressure, these are 8.5 x 107
for the ultrasonic case, 2.5 x 1037 for the neutron case,
38
and 1.26 x 108 for the specific heat case. A plot of y
versus density is shown in Fig. 8.
Table III. The Values of y and 6 at Experimental Densities.
Density
in gr/cm3
0.1464
0.1472
0.1496
0.1519
,a
Y
38
1.06 x 103
1.19 x 1038
9.95 x 1037
5.80 x 1037
,b
Y
1.75 x 1038
1.56 x 1038
1.95 x 1038
6.64 x 1037
,b
8.75 x 1076
5.10 x 1076
984 x 1076
1.94 x 1076
derived from the T6 fit
derived from the T6 + T8 fit




0.5 i
a 0
n
0
0 "
*
1.5 [
0.1450
0.1500 0.1550
density (gm/cm3
0.1600
Fig. 8. y'
square
closed
square
dashed
versus density. Open triangle this work, open
PWH assuming y of CW at 24 atm., open circle PWH,
triangle ZP assuming y of CW at 24 atm., closed
ZP, dashed line is based on normal dispersion,
and dotted line is based on CW.
0
0.5
1 
1 _IL
To illustrate the agreement of the pressure data with
other data, Fig. 9 shows a comparison of the best data with
a calculated plot for the same density, assuming y has the
PWH value as corrected by Zasada and Pathria (1974) and cal
culating ( )/(p p ), assuming
up (24 atm. SVP) 24 atm. SVP a mi
24 atm. has the CW value and ySVP has the PWH value. The
data clearly follow the trend of the calculation up to the
temperatures where the excess pressure becomes detectable.
Further intercomparison of theories is shown in Fig. 9 by
plotting y' derived from the BT6 fit versus density. For
comparison, the values of y versus density for the specific
heat data are shown. These are calculated using values of y
of PWH and those of PWH as corrected by Zasada and Pathria.
The value of Yy/9p is calculated by assuming a linear fit be
tween the lowest and highest densities of PWH. Also shown
are the values of y based on the ultrasonic and neutron spec
tra given in Table I. It is evident that there is good agree
ment between values of y' derived from pressure measurements
and those derived from specific heat measurements; and there
is poor agreement with the neutron and ultrasonic values.
The ultimate goal of these measurements is to derive
values of y as a function of density. To do this one takes
the values of y as a function of density and calculates y in
a selfconsistent way. This is done by choosing an initial
i
value of y and 3y/9p which yield the correct y at that den
sity. These are used to calculate y at another density, which
in turn determines 9y/9p at that density from y These
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en, a
0 U) 0 U
4 rd 1 I
cn uJ
0  4J
en \ e
*a E in
DO a
0 C)
ul II kH
rd Or
4) 0 4
C)
011
O
u
4
0
a
44
E
ON
cn
II
0
O;
(
values are used to calculate y and 3y/9p at the next density.
The process is repeated until a consistent set of y and 9y/9p
are calculated, i.e., 9y/p and y are smooth functions of den
sity. For the present data, this is a very imprecise calcula
tion because there are too few values of y the values of y
are too uncertain and the range of densities covered is too
restricted.
If the excess pressure contributions were removed, there
would be a much greater range of P versus T from which to
i i
extract y This would greatly improve the accuracy of y.
Also, y could have been determined over a wider range of den
sities. Nevertheless, the fact that y is negative and in
creases with increasing density leads to the conclusion that
at low pressure y is negative and that 9y/9p is positive.
The possibility of a linear dispersion term was conjec
tured by MR. To check this possibility, a plot for the best
data of Pph/T4 versus T was made. The results are shown in
Fig. 10. If Pph varies at T5 at low temperatures, the plot
would be a straight line. The curvature of the plot at low
temperatures is obvious. Furthermore, the curve does not
extrapolate properly to T = 0. This contrasts with the linear
appearance of the plot of Pph/T4 versus T2 in Fig. 8. Thus
6 5
the dispersion pressure appears to vary as T rather than T
at low temperatures. From Eq. (2.14), this implies that
al
[(1+4r)(al )p~ ] = 0. Since all of the plots in Fig. 7
appear to be similar to the best data at low temperatures,
one can conclude that Tal/8p = 0 and thus al = 0. This con
tradicts the MR form of the dispersion curve.
9
0
00
Ie
I 0
C;
d o
J  _~ __ ____c
co
CO
r/
o
o
(M/ ) /ld
*)
o
nC)
o
0
>1
r:
c,
ro
c~I
0)
4 J
0
E
0
0
Cr
o 'd
p
cci
tC o
$4 fl
04rS
a4 rj
Conclusions
Three conclusions can be drawn from this work. (1) The
MR form of the excitation spectrum is probably incorrect.
(2) There is anomalous dispersion in the excitation spectrum
of superfluid helium at low densities. The magnitude of the
dispersion parameter and density derivative have values simi
lar to those derived from the specific heat measurements of
PWH. (3) There are contributions to the pressure other than
phonons and rotons, probably maxons. These contributions in
crease in magnitude as the density is increased. These con
tributions must be removed before accurate conclusions can be
made about the values of y and oy/3p as a function of density,
particularly at higher densities.
More work needs to be done in several areas before pres
sure measurements can be used to discriminate among the dif
ferent theoretical forms for the anomalous dispersion. At
present the uncertainties in this experiment are too great to
permit this to be done with any confidence. The data need to
be improved with more runs of the quality of the best one.
The data should be extended to lower temperatures if possible
to permit an unequivocal assignment of Po based on the data.
There is a need for a greater number of data points. The
25 mK spacing was used so that the change in capacitance for
each change in temperature was clearly resolvable. However,
an additional set of data could be taken 25 mK apart with
each point being the middle of one of the present intervals.
This would double the number of data points and improve the
accuracy of the fits.
A method of removing the excess pressure should be de
vised so that the full range of temperature can be used in
fitting the temperature dependence of the dispersion term.
This will require an extensive neutron scattering study of
the region of the maximum of the excitation curve as a func
tion of density. Such work is in progress at the Brookhaven
National Laboratory. Using those data, one could calculate
the contribution due to the maxons. Such calculations could
be used to reanalyze both these pressure data and the speci
fic heat data of PWH. Such a reanalysis would improve the
accuracy and the reliability of the values of y, 6 and their
density derivatives.
Studies of the dispersion curve have widespread appli
cability, and are not just limited to studies of helium. For
example, the differences among the theories largely arise
from differing treatments of the interactions among excita
tions. Excitation models are common throughout physics, par
ticularly in solid state physics where the concept of phonons
originated. A common problem in all of these models is the
treatment of the interactions of excitations. In a solid the
problem is particularly complicated because of the existence
of three branches of the excitation curve. Helium, a simple
liquid, is a good substance for testing different theories.
Thus, studies on helium could yield theories applicable to
many materials.
Dispersion curves appear not only in solids but also in
liquids and amorphous solids. Goda (1972) has noted that
"longitudinal phonon dispersion of some amorphous solids and
simple liquids are of the phononroton type as observed in
liquid 4He. Such behavior seems to exist also in liquid
metals" (p. 1064). Is the observed anomalous dispersion
unique to helium, or is it a common feature of all topologi
cally disordered substances? Perhaps precision pressure
measurements of liquid neon or liquid argon could answer
that question.
The universality of phononroton type dispersion curves
suggests another reason for making detailed studies of the
dispersion curve of superfluid helium. Khalatnikov (1965)
gave a simple argument that claims to show that any liquid
with a roton minimum will be superfluid at T = 0. That im
plied that the form of the dispersion curve "caused" super
fluidity. The Khalatnikov argument does not predict any
clearcut transition from superfluid to normal fluid as the
temperature is raised unless the spectrum changes. But
helium has a phononroton spectrum above the X line, and is
not a superfluid at that point. There are qualitative dif
ferences in the excitation spectrum of helium above and be
low the X line. The line width is much broader and the
temperature dependence of quantities such as the roton mini
mum are different. Takeno and Goda (1972) present a model
which successfully predicts the spectrum of He I as well as
other simple liquids but fails for He II. Accurate knowledge
of the excitation spectrum for He II could lead to a better
understanding of the fundamental nature of superfluidity.
76
This work has shown that the excitation spectrum of super
fluid heliLum has anomalous dispersion at low densities and
that for precision measurements the contributions of the maxons
must be included in the theory; but it does not provide the
final answer as to which theory is correct.
REFERENCES
Abraham, B. M., Y. Eckstein, J. B. Ketterson, M. Kuchnir, and
J. Vignos, 1969, Phys. Rev. 181, 347.
Y. Eckstein, J. B. Ketterson, M. Kuchnir, and
P. R. Roach, 1970, Phys. Rev. Al, 250.
Anderson. C. H., and E. S. Sabisky, 1972, Phys. Rev. Letters
28, 80.
and E. S. Sabisky, 1974, Bull. Am. Phys. Soc. 19,
436.
Andreev, A., and I. M. Khalatnikov, 1963, Zh. Eksperim. i
Teor. Fiz. 44, 2058, [J.E.T.P. 17 (1963), 384].
Bendt, P. J., R. D. Cowan, and J. L. Yarnell, 1959, Phys. Rev.
113, 1386.
Boghosian, C., and H. Meyer, 1966, Phys. Rev. 152, 200.
Castles, S. H., 1973, Ph.D. Dissertation, University of
Florida.
Cowley, R. A., and A. D. B. Woods, 1971, Can. J. Phys. 49,
177.
Dietrich, O. W., E. H. Graf, C. H. Huang, and L. Passell,
1972, Phys. Rev. AS, 1377.
Donnelly, R. J., 1967, Experimental Superfluidity (University
of Chicago Press, Chicago).
__ 1972, Phys. Letters 39A, 221.
Dynes, R. C., and V. Narayanamurti, 1974, Bull. Am. Phys. Soc.
19, 435.
Eckstein, S., and B. B. Varga, 1968, Phys. Rev. Letters 21,
1311.
Feenberg, E., 1971, Phys. Rev. Letters 26, 301.
Feynman, R. P., 1953, Phys. Rev. 91, 1301.
1954, in Progress in Lo' Temperature Physics,
Volume I, edited by C. J. Gorter (Interscience Publishers,
New York).
1954b, Phys. Rev. 94, 262.
and M. Cohen, 1956, Phys. Rev. 102, 1189.
and M. Cohen, 1957, Phys. Rev. 107, 13.
Friedlander, D. R., S. G. Eckstein, and C. G. Kuyper, 1972,
Phys. Rev. Letters 30, 78.
Goda, M., 1972, Prog. Theor. Phys. 47, 1064.
Gould, iH., and V. K. Wong, 1971, Phys. Rev. Letters 27, 301.
Hallock, R. B., 1972, Phys. Rev. A5, 320.
Havlin, S., and M. Luban, 1972, Phys. Letters 42A, 133.
Heberlein, D. C., 1969, Ph.D. Dissertation, University of
Florida.
Henshaw, D. G., and A. D. B. Woods, 1961, Phys. Rev. 121,
1266.
Iachello, F., and M. Rassetti, 1973, Lettere al Nuovo Cim. 7,
295.
Ishikawa, K., and K. Yamada, 1972, Prog. Theor. Phys. 47,
1455.
Jackle, J., and K. W. Kehr, 1971, Phys. Rev. Letters 27, 654.
and K. W. Kehr, 1974, Phys. Rev. A9, 1757.
Jackson, H. W., 1974, Phys. Rev. A10, 278.
Kammerlingh Onnes, H., 1908, Leiden Communication 108.
Keesom, W. H., and A. H. Keesom, 1933, Leiden Communication
224d.
Keller, W. E., 1969, Helium3 and Helium4 (Plenum Press, New
York).
Khalatnikov, I. M., 1965, Introduction to the Theory of Super
fluidity (Benjamin, New York).
and D. M. Chernikova, 1965, Zh. Eksperim. i Teor.
Fiz. 49, 1957, [J.E.T.P. 22 (1966), 1336].
and D. M. Chernikova, 1966, Zh. Eksperim. i Teor.
Fiz. 50, 411, [J.E.T.P. 23 (1966), 274].
Kirk, W. P., S. H. Castles, and E. D. Adams, 1971, Rev. Sci.
Instr. 41, 1007.
Landau, L. D., 1941, Zh. Eksperim. i Teor. Fiz. 5, 71, trans
lated in I. M. Khalatnikov, 1965, Introduction to Super
fluidity (Benjamin, New York).
1947, Zh. Eksperim. i Teor. Fiz. 11, 91, trans
lated in I. M. Khalatnikov, 1965, Introduction to Super
fluidity (Benjamin, New York).
London, F., 1954, Superfluids Volume II: Macroscopic Theory
of Superfluid Helium (Dover Publications Inc., New York).
LinLiu, YuReh, and ChiaWei Woo, 1974, J. Low Temp. Phys.
14, 317.
Maris, H. J., 1972, Phys. Rev. Letters 28, 277.
1973, Phys. Rev. A8, 2629.
and W. F. Massey, 1970, Phys. Rev. Letters 25,
220.
Mills, R. L., 1965, Ann. Phys. (New York) 35, 410.
Mills, N. G., R. A. Sherlock, and A. F. G. Wyatt, 1974, Phys.
Rev. Letters 32, 978.
Molinari, A., and T. Regge, 1971, Phys. Rev. Letters 26, 1531.
Narayanamurti, V., K. Andres, and R. C. Dynes, 1973, Phys.
Rev. Letters 31, 687.
Petick, C. J., and D. Ter Haar, 1966, Physica 32, 1905.
Phillips, N. E., C. G. Waterfield and J. K. Hoffer, 1970,
Phys. Rev. Letters 25, 1260.
Philp, J. W., 1969, Ph.D. Dissertation, University of Florida.
Pokrant, M. A., 1972, Phys. Rev. A6, 1588.
Roach, P. R., J. B. Ketterson, and M. Kuchnir, 1972a, Phys.
Rev. A5, 2205.
J. B. Ketterson, M. Kuchnir, and B. M. Abraham,
1972b, J. Low Temp. Phys. 9, 105.
80
B. M. Abraham, J. B. Ketterson, and M. Kuchnir,
1972c, Phys. Rev. Letters ?_, 32.
J. B. Ketterson, and M. Kuchnir, 1972d, Rev. Sci.
Instr. 43, 898.
J. B. Ketterson, B. M. Abraham, and M. Kuchnir,
1972e, Phys. Letters 39A, 251.
Simon, S., 1963, Proc. Phys. Soc. (London) 82, 401.
Singh, A. D., 1968, Can. J. Phys. 46, 1801.
Straty, G. C., and E. D. Adams, 1969, Rev. Sci. Instr. 40,
1393.
Svensson, E. C., A. D. B. Woods, and P. Martel, 1972, Phys.
Rev. Letters 29, 1148.
Takeno, S., and M. Goda, 1972, Prog. Theo. Phys. 48, 724.
van den Meijdenberg, C. J., K. W. Taconis, and R. de Bruyn
Ouboter, 1961, Physica 27, 197.
Walsh, P. J., 1963, M.S. Thesis, University of Florida.
Wilks, J., 1967, Properties of Liquid and Solid Helium
(Clarendon Press, Oxford, England).
Woods, A. D. B., and R. A. Cowley, 1973, Reports Prog. Phys.
36, 1135.
Yarnell, J. L., G. P. Arnold, P. J. Bendt, and E. C. Kerr,
1959, Phys. Rev. 113, 1379.
Zasada, C., and R. K. Pathria, 1972, Phys. Rev. Letters 29,
988.
and R. K. Pathria, 1974, Phys. Rev. A9, 560.
BIOGRAPHICAL SKETCH
Albert Robert Menard III was born July 17, 1943 at Boston,
Massachusetts. A National Merit Scholarship finalist, he
graduated from Boulder High School in Boulder, Colorado in
June, 1961. In June, 1965, he received the Bachelor of Arts
degree from Amherst College, Amherst, Massachusetts. He then
enrolled in the Graduate School of the University of Minnesota,
where he received the degree of Master of Science in March,
1969.
In September, 1969, he enrolled in the Graduate School
of the University of Florida to pursue the degree of Doctor
of Philosophy. From January through August, 1973, he was a
National Science Foundation trainee.
Mr. Menard is a member of Sigma Pi Sigma and the American
Physical Society.
He is married to the former Anne Elaine Dozer of Green
castle, Indiana. They have one daughter, Laura Elizabeth,
born in June, 1972.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
E. Dwight Vidams, Chairman
Professor of Physics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Athur A. Broyles
Professor of Physics and Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and qualiLy,
as a dissertation for the degree of Doctor of Philosophy.
Thomas L. Bailey
Professor of Physics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Wi'ley P. irk
Assistafit Professor of Physics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Arun K. Varma
Associate Professor of Mathematics
This thesis was submitted to the Gradutate Faculty of the
Department of Physics in the College of A.rts and Scirences
and to the Graduate Council, and was accepted .s parti.Ll
fulfillment of the requirements for the degree, of Dctor
of Philosophy.
December, 1974
Dean, Graduate School
