Group Title: thermodynamic pressure in superfluid helium and its implications for the phonon dispersion curve
Title: The Thermodynamic pressure in superfluid helium and its implications for the phonon dispersion curve
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Title: The Thermodynamic pressure in superfluid helium and its implications for the phonon dispersion curve
Physical Description: vii, 81 leaves. : illus. ; 28 cm.
Language: English
Creator: Menard, Albert Robert, 1943-
Publication Date: 1974
Copyright Date: 1974
 Subjects
Subject: Liquid helium   ( lcsh )
Phonons   ( lcsh )
Superfluidity   ( lcsh )
Physics thesis Ph. D
Dissertations, Academic -- Physics -- UF
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 77-80.
Additional Physical Form: Also available on World Wide Web
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General Note: Vita.
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Bibliographic ID: UF00097554
Volume ID: VID00001
Source Institution: University of Florida
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Resource Identifier: alephbibnum - 000582351
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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 X-ray 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 phonon-roton 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 superfluidity--the 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 Bose-Einstein statistics: in contrast, the rare isotope

He which has spin 1/2 will obey Fermi-Dirac 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 two-fluid 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 fluids--a

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 quasi-particles 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

two-fluid 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 X-ray

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 X-ray 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 X-ray 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







problems--for 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 wavelength-small 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 two-fluid

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 phonon-roton picture of liquid helium

rests on a sound phenomenological basis--neutron 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 phonon-roton 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 co-workers--Andreev 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 phonon-roton picture of helium, like the two-fluid 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 phonon-roton 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 phonon-roton 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 phonon-roton 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 phonon-roton 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 three-phonon processes by the conserva-

tion laws. Three-phonon 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 three-phonon pro-

cesses to occur.

Simon (1963) and Petick and Ter Haar (1966) showed that,

under some circumstances, three-phonon 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 three-phonon 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 zero--so-called anomalous dispersion--the problems of

the theory would be solved. Anomalous dispersion unques-

tionably permits three-phonon 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 phonon-phonon 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 X-ray 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 dipole-dipole 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 five-term 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 non-polynomial 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.

Lin-Liu 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

term--the term with coefficient y in the usual expansion--and

the p term. The p term--the term with coefficient 6 in the

usual expansion--has 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 phonon-roton 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 X-ray 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 three-phonon

processes to analyze their experiments with heat pulses in

helium. The phonon lifetimes used in the heat pulse analysis

are inconsistent with three-phonon 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 three-phonon 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 Landau--all 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[l-e:.:p(-6c (p) )] (2.3)



The Helmholtz free energy is defined to be F = kT log Z.

Thus, F is given by

-1
F --k-T 7 log[l-ex: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[l-exp(-RE(p))]- d3p + Fo (2.5)
h3


This result assumes, in agreement with experimental results,

that E(p) is a smooth, well-behaved 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[l-exp(-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 well-defined, long-lived 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 X-ray 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 phonon-roton

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 + (p-po) /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- u-nurl ~-o*nTI
r


0







r-1(
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 [l-exp(-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(-(p-p ) /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 10-4


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 all-liquid sample just

above saturated vapor pressure to approximately 24 atm. for

an all-liquid sample just below the melting curve.

It is difficult to calculate the T term in Eq. (2.16)--

the next order correction term--since 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 phonon-roton 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

phonon-roton 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

vacuum-tight 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 PMCS-2C 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 feed-throughs

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 KC-46

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 cross-sectional 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

10-mil-i.d. copper-nickel capillary. A National Research

Corporation model B-2 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 temperatures--about 0.39 K

for the sample chamber--were 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 lock-in 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 three-terminal 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 #7770--a 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 10-2 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

small--less 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 Teflon-tipped 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 10-6 pF/min to 10- pF/min. Since

the leak rate was constant in time, a simple correction fac-
dC
tor--the 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 lock-in 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 i-1
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 out-of-balance signal of the lock-in 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-

of-balance signal from the lock-in 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 10-4 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 error--no more than 0.0003 gm/cm3 for densities between

0.1451 and 0.1725 gm/cm3--because 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 non-linear, 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,

non-horizontal 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 T-4 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 phonon-roton

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 phonon-roton 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 = 4-2 + 6-7 (r + ) from Eq. (2.16)
1-+7 3p 3p
are about equal to 4-2 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

9-84 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 _I-L--







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 self-consistent 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


















































































(N o C
C4N N r-A
0 0 0
0 0 0


qI/


o

X
r)


II
-1


C




I --

0) > o





o E
U) -H ,- I
I) ,- U ,
0 .


n E>
04 0
mn -H -H
r Lin 4-4


00 *>
0 c0 >



O -o

4-) r Ori
4 0
C) (J *
-4 0 4-)
4-4 i;
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-


4-4


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 phonon-roton 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 phonon-roton 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 phonon-roton 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.












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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 Dc-tor
of Philosophy.

December, 1974


Dean, Graduate School




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