27r~j

v"8 f(dr

DENSITY PROFILE OF A QUANTIZED VORTEX LINE

IN SUPERFLUID HELIUM-4

DISSERTATION

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requiremants

For the Degree of

DOCTOR OF PHILOSOPHY

By

John H. Harper, M. S.

Denton, Texas

May, 1975

Harper, John B., Densi4y Qrofil gf a Quantized Vortex

HL in. Suerfluid Helium-4. Doctor of Philosophy (Physics),

May, 1975, 174 pp., 5 tables, 20 figures, bibliography, 105

titles.

The density amplitude of an isolated quantum vortex line

in superfluid 4He is calculated using a generalized Gross-

Pitaevskii (G-P) equation. The generalized G-P equation for

the order parameter extends the usual mean-field approach by

replacing the interatomic potential in the ordinary G-P

equation by a local, static T matrix, which takes correla-

tions between the particles into account. The T matrix is a

sum of ladder diagrams appearing in a diagrammatic expansion

of the mean field term in an exact equation for the order

parameter. It is an effective interaction which is much

softer than the realistic interatomic Morse dipole-dipole

potential from which it is calculated.

A numerical solution of the generalized G-P equation is

required since it is a nonlinear integro-differential equa-

tion with infinite limits. For the energy denominator in the

T matrix equation, a free-particle spectrum and the observed

phonon-roton spectrum are each used. For the fraction of

particles in the zero-momentum state (Bose-Einstein donden-

sate) which enters the equation, both a theoretical value of

0.1 and an experimental value of 0.024 are used. The chem-

ical potential is adjusted so that the density as a function

of distance from the vortex core approaches the bulk density

asymptotically.

Solutions of the generalized G-P equation are not very

dependent on the choice of energy denominator or condensate

fraction. The density profile is a monotonically increasing

function of the distance from the vortex core. The core

radius, defined to be the distance to half the bulk density,

varies from 3.7 A to 4.7 A, which is over three times the

experimental value of 1.14 A at absolute zero.

TABLE OF CONTENTS

Page

LIST OF TABLES . . . , . , . , . . , . , . . . , .viii

LIST OF ILLUSTRATIONS . . . . . . .ix

Chapter

I. INTRODUCTION . . . ......,,, ,

II. GENERAL BACKGROUND . . . . . . . , . , . . 7

Basic Properties of Liquid HeliumPhase Dia ram

1 n 2 0Chemiical PoetiA,Heagt. capacity

Super PropertiesExperimen tal.,.-Flu,,j Model

Energy SpectrumTheories

Bose-Einstein CondensationConnection i~th SuiaerfluidityTheoreti Condensate Fract onExperimenta. Condensate Fraction

III. QUANTIZED VORTICES . . . . . . . . . . . 20Prediction of Vortices

Irrotational Velocity FieldOsborne's Experiment

Quantization of CirculationVortex LinesVortex Rings

ExperentTheoryCore Radus

V

PREVIOUS VORTEX THEORIES . . . . . . .

Hartree ModelGross-Pitaevskii EquationVortex Line.Solutions

Related TheoriesMethod of Correlated Wave Functions

Trial Wave FunctionsEnerv Variational Principle

Comparison

V. GENERALIZED GROSS-PITAEVSKII EQUATION. . 48

General Equation for Order ParameterT MatrixFactorization MethodEnergy Functional MethodLocal T MatrixVortex Line Equation

VI. NUMERICAL METHODS . . . . . . . . . . . 62

Determination of the T MatrixFrjee-Particle Energy

Observed Excitation EnergyHtlium PotentialNumerical Procedure

Determination of the Kernel K(P.PSolution of the Generalized Gross-Pitaevskii Equation

Solution frZr.o. ondensate Diensitynumerical Procedure

Boundary.ConditionsCalculations

VII. RESULTS AND DISCUSSION . . . . . . , . . . 80

T MatrixDensity Amplitude

VIII. COMPARISON WITH OTHER WORK . . . . . . . 83

Density ProfileCore RadiusChemical Potential$stgaard's T Matrix

IX. CONCLUSION . . . . . . . . . . . . . . . 89

vi

,.1IV.*

Appendix

A. SYMBOLS USED . . . . .. . . 94

B. THE ANALYSIS OF VORTEX RINGS . . . . . 99C. DERIVATION OF THE INTEGRAL EQUATION

FOR THE T MATRIX *.-.. .,. . .. 106

D. NUMERICAL SOLUTION OF T(r) . . . . . . 108

E. SERIES EVALUATION OF K(fe') FOR LARGE 1P'11'

F. NUMERICAL METHODS FOR SOLVING THEGENERALIZED GROSS-PITAEVSKII EQUATION 115

G. COMPUTER PROGRAMS -- . . . ., . . . 121

H. %STGAARD'ST MATRIX 9.9.990. .0. . . 140

vii

LIST OF TABLES

Table Page

I. Values of Some Physical Constants andHeliumr-4Data *,,. ... * , , , 143

II. Previous Values of the Condensate Fraction . . . 144

III. Previous Values of the Core Radius . , , . , , . 145IV. Parameters for the Calculated Density

Amplitude . . . - . . . . . .. . . . . . . 146

V. Symbols Used , , , . . . , , , , , , , . , . , . 94

viji

: :. .

LIST OF ILLUSTRATIONS

Figure Page

1. The Phase Diagram of He . . . . . . . . . . 147

2. The Elementary Excitation Spectrumof He II . . . . . . . . . . . . . . . . 148

3. The Relative Condensate Density Amplitudeof Kawatra and Pathria . . . . . . . . . 149

4. The Relative Condensate Density Amplitudeof Chester, Metz and Reatto . . . . . . 150

5. A Diagrammatic Representation of theMatrix Element in Eq. (5.8) . . . . . . 151

6. Factorization of the Correlation Function. . 152

7. The Graphical Form of tne T Matrix . . . . . 153

8. The T Matrix for the Free-ParticleEnergy Denominator . . . . . . . . . . . 154

9. The T Matrix for the ExperimentalEnergy Denominator . . . . . . . . . . . 155

10. The Kernel K Calculated with theRealistic Potential . . . . . . . . . . 156

11. The Kernel K Calculated with theT Matrix in Fig. 8 . . . . . . . . . . . 157

12. The Kernel K Calculated with theT Matrix in Fig. 9 . . . . . . . . . . . 158

13. The Relative Condensate Density Amplitudefor the Kernel in Fig. 11 andCondensate Density 0.1 . . . . . . . . . 159

14. The Relative Condensate Density Amplitudefor the Kernel in Fig. 11 andCondensate Density 0.024 . . . . . . . . 160

15. The Relative Condensate Density Amplitudefor the Kernel in Fig. 12 andCondensate Density 0.1 . . . . . . . . . 161

ix

LIST OF ILLUSTRATIONS-Continued

Page

The Relative Condensate Density Amplitudefor the Kernel in Fig. 12 andCondensate Density 0.024 . . . . . . . . 162

The Best Results from Figs. 13 and 14 . . . 163

The Best Results from Figs. 15 and 16 . . . 164

The T Matrix of %stgaard . . . . . . . . . . 165

A Comparison of Theoretical RelativeDensity Amplitude Profiles . . . . . . . 166

x

Figure

16.

17.

18.

19.

20.

CHAPTER I

INTRODUCTION

Only a few phenomena exhibit quantum mechanics on a

macroscopic scale: principally superconductivity, super-

fluidity, and laser action. When liquid 4He is cooled below

2.18 K at atmospheric pressure, its nature changes so

drastically that it is called He II in contrast to the

ordinary fluid, He I. Its characteristics of superfluidity,

the ability to flow with zero viscosity, and other properties

are manifestations of quantum phenomena which continue to

challenge theorists and fascinate experimentalists.

One of the earliest theories of He II was the two-fluid

model. Tisza1 assumed that He II consists of two inter-

penetrating fluids: a normal fluid with viscosity and a

superfluid with zero viscosity. Landau 2 proposed that the

normal fluid component is composed of elementary excitations.

For his postulated energy spectrum elementary excitations

can not be created below a certain critical rate of flow,

which qualitatively explained the superfluidity of He II.

The heat capacity of He II was also well fit by considering

He II to be a gas of noninteracting elementary excitations.

Experiments' using inelastic neutron scattering qualitatively

verified Landau's form of the energy spectrum. 3

1

2

A microscopic theory based on the quantum mechanics of

weakly-interacting bosons was developed by Bogoliubov,5

which gave an energy spectrum linear in the momentum for

small momentum, in agreement with Landau's energy spectrum.

Feynman6 obtained an energy spectrum similar to experiment4

by considering the elementary excitations to be density

fluctuations. By ascribing a macroscopic wave function to

superfluid helium, Feynman6 also verified Onsager's conjec-

ture7 that the circulation of superfluid about a vortex line

should be quantized.

One of the many phenomena exhibited by He II is the

existence of quantized vortex lines. They were theoretically

required to explain an inconsistency between theory and

experiment8 for rotating He II. Vinen9 first observed

quantized vortices in 1961, and two years later mobile vortex

rings were detected.10 Recent experiments 1 indicate that0

the vortices have a core radius of (1.14 * 0.05) A at

absolute zero, although the density profile has not been

obtained directly from experiment.

A mean-field theory of He II was developed independently

by Gross12 and Pitaevskii1 3 from which the density profile

of a vortex line has been calculated 14 using a s-function

approximation for the 4He interatomic potential. Their

equation, called the Gross-Pitaevskii (G-P) equation, has been

criticized 5 because, without short-range correlations, a

mean-field theory may not describe phenomena very well on the

3

size of atomic spacings. A generalization of their theory

was made by Kobe,16 who took particle correlations into

account by summing an infinite set of ladder diagrams

representing multiple scatterings.

An alternative approach to the theory of vortex lines is

to use an appropriate model many-body wave function15 in the

energy variation principle. The wave function is expressed

in terms of a parameterized radial function, from which the

system energy is calculated and minimized. In this manner

the optimum density as a function of the radial distance from

the vortex core can be obtained, but the calculations are

difficult.

In this work, the generalized G-P equation16 is solved

numerically for the density amplitude, i.e., the square root

of the radial superfluid density function, as a function of

the distance from the center of the vortex line. Since the

square of the density amplitude is the superfluid density,

this calculation gives the density profile of the vortex

line.

The realistic Morse dipole-dipole (MDD2) interatomic

helium potential is first used to calculate a T matrix,

which is an effective interaction between the particles. In

contrast to the potential, which is extremely repulsive for

small interatomic separations, the T matrix is only weakly

repulsive (about five orders of magnitude less than the

potential at zero separation distance). By using this

4

procedure, particle correlations are transferred from the

wave function to the T matrix. The T matrix is calculated

for two different energy denominators. In one case the free-

particle kinetic energy is used, and in the other case the

experimental excitation spectrum is used.

The density amplitude is calculated from the generalized

G-P equation using the T matrix as the effective interaction

between particles. In order to solve the generalized G-P

equation, the fraction of particles in the Bose-Einstein

condensate, or lowest energy state, is required. Both a

theoretical value18 of 0.1 and an experimental value19 of

0.024 are used. For each of the T matrices calculated, the

G-P equation is solved for both values of the condensate

density. In each case considered, a density amplitude, which

is a monotonically increasing function of distance from the

vortex line, is obtained which reaches 99.8 per cent of its

bulk value at a distance of 9 A. The vortex core radius is

taken to be the distance at which the computed density profile

is half the bulk density. The core radii vary between 3.7

and k.7 X, which are considerably larger than 1.14 X, the

experimental value11 extrapolated to 0.0 K. However, the

core radius is determined from experiment on the basis of

classical hydrodynamics for a hollow core. Thus the two

values can be only approximately compared.

The chemical potential A enters the generalized G-P

equation and has a strong influence on the density profile.

5

It is chosen so that the density profile approaches the bulk

density far from the core. In two of the cases considered

the chemical potential is negative, which is characteristic of

a bound system of particles. The chemical potentials

obtained previously from the G-P equation have been positive,

which indicates an unbound system.

These results show that the generalized G-P equation can

be used in conjunction with a realistic potential to yield a

physically reasonable vortex density profile, which smoothly

increases from zero at the core to a constant bulk value a

few angstroms away. Although the core radius is on the order

of the interparticle spacing, the deBroglie wavelength of the

helium atoms is significantly larger. The medium thus acts

like a continuum with a uniform density, except in the

vicinity of a vortex.

This work shows that the generalized G-P equation is a

useful method of describing vortex core structure. A real-

istic potential can be used, a negative chemical potential

can be obtained, and the resulting density profile has a

reasonable shape and size. Many possible diagrammatic contri-

butions to the interaction term in the generalized G-P

equation are neglected due to the difficulty of calculating

more complex diagrams. Inclusion of more terms could improve

the results.

Chapter II presents a general background of properties

and phenomena associated with He II, and early theories. The

6

excitation energy spectrum and Bose-Einstein condensation are

also discussed. Quantized vortices are introduced in

Chapter III, which concludes with a description of how the

core size is deduced from vortex ring experiments. Chapter IV

reviews two microscopic approaches to the theory of quantized

vortex lines. One is the ordinary G-P equation and the other

is the method of correlated wave functions. 1 5 The general-

ized G-P equation is derived in Chapter V from both an

expansion of the equation of motion and an energy functional

approach. The T matrix is defined and the generalized G-P

equation is cast into a form appropriate for investigating

vortex lines. Chapter VI contains the calculations of the

density amplitude, and Chapter VII compares the results with

the works of others. The final chapter includes a critical

summary of this investigation and suggestions for further

work. To aid the reader, Appendix A contains a list of

symbols used in this paper.

CHAPTER II

GENERAL BACKGROUND

Basic Properties of Liquid Helium

Liquid helium has been the subject of much investiga-

tion2 0-2 4 due to many unique properties related to the small

mass and closed-shell structure of the helium atom. There

are two natural isotopes of helium, 'He and 3He. Although a

composite particle, 4He acts like a boson due to the even

number of nucleons in its nucleus, which has zero spin. On

the other hand, 3He acts like a fermion due to its odd

number of nucleons. Since 3He has only a concentration of

about 1.3 ppm in natural helium, it became available in pure

form only after it could be produced artificially. Recent

experiments indicate superfluidity in 3He at 2-3 mK due to

pairing.

Ehase Dia rm

In 1908 helium was liquefied at 4.2 K by Kamerlingh

Onnes. 4He forms a unique liquid because it cannot be solid-

ified by cooling except under high pressures, j.g. 25.0

atmospheres at absolute zero. The phase diagram of 4 He is

shown in Fig. 1. Beginning at the critical point, 2.26

atmospheres at 5.20 K, the vapor pressure curve uniformly

approaches the origin with no indication of a triple point.

The nature of the liquid does, however, change across the Xk

7

W .a;.

8

line, as discussed below.

The lack of a solid phase at low temperatures and

pressures below 25 atmospheres is due to a combination of

factors. The small electric polarizability of the atom

resulting from the closed-shell electronic structure means

that the interatomic van der Waals force will provide only

a weak attraction between atoms. Due to their small mass,

helium atoms have zero-point energies26 which are large com-

pared to the attractive potential. Hence, the usually stable

configuration of a solid at sufficiently low temperatures is

not favored at pressures less than 25 atmospheres. The other

inert gases have masses which are much larger, while hydrogen

experiences a much stronger van der Waals force, so they form

stable solids at sufficiently low temperatures.

Dneity

If Avogadro's number is divided by the 'He molar volume

at absolute zero and saturated vapor pressure given in

Table 1,27 the number density is

0.3

f= 0.0213 A , (2.1)

which is approximately constant28 up to about 1.5 K. The0

average interparticle spacing is thus 3.6 A.

Within three years of its liquefaction, Kamerlingh Onnes

noticed that liquid helium passes through a state of maximum

density as it is cooled. This anomalous behavior was

strangely ignored until more careful observations30 thirteen

9

years later showed that the density changes rapidly near

2.2 K. The density rises uniformly28 along the vapor

pressure curve to a maximum of 0.0220 A"3 (0.1466 g cm"3) at

2.178 K, and then drops sharply from a cusp, being

0.0193 J-3 at 4.0 K. This observation was the first of

several indications which were soon found of an apparent

phase transition in liquid helium near 2.2 K.

Chemica otntial

According to thermodynamics, the chemical potential A

of a system of particles is the change in energy E with

respect to particle number N at constant volume Si,

DN n (2.2 )

Thus, }t >0 corresponds to an unbound system with a net

repulsion among the particles. An interparticle potential

with an attractive well which is sufficiently large should

lead to a bound system with/1,l0. Atkins2 9 obtains an

experimental chemical potential for liquid helium of

-13.2 cal mole" 1, which in energy units is -6.7 K.

gat Capacity

Of the many discontinuities23 discovered in liquid

helium, that of heat capacity was especially important since

no latent heat is associated with the transition. The shape

of the specific heat curve of liquid helium led to the

designation of the transition temperature -2.19 K as the X

>- - - - -

10

point31 or T , separating two liquid phases, He I above and

He II below the ? point.

Ehrenfest32 soon proposed a classification scheme for

phase transitions based on the behavior of the Gibbs free

energy G as a function of temperature and pressure. An

ordinary first-order transition involves a latent heat and

corresponds to discontinuous first derivatives of G, j.#.,

entropy and molar volume. But a second-order transition is

continuous in these values while having (finite) discontinu-

ities in the second derivatives s specific heat at constant

pressure, coefficient of thermal expansion, and isothermal

compressibility.

Apparently the X transition is not a true second-order

transition because the specific heat has a logarithmic dis-

continuity which is not finite at TA . The expansion

coefficient and sound velocity behave similarly. Hence,

liquid helium does not fit into Ehrenfest's scheme exactly,

although it is sometimes loosely referred to as second order.

Discontinuities have also been discovered in properties such

as latent heat of vaporization, dielectric constant, and

surface tension.

Super Properties

Experimental

In the latter part of the 1930's a series of remarkable

"super properties" of He II were discovered. Measurements33

11

showed that the heat conductivity near 2 K is three orders of

magnitude greater than for copper at room temperature. This

near elimination of temperature gradients explains the sudden

quiesence of boiling helium as it passes through the ". point

upon cooling. Also, a beaker containing He II will empty by

an anomalous flow of a thin film on the wall, which can not

be explained by ordinary capillary action.

Further, when viscosity measurements in capillary tubes

or fine powder are made, the superfluid nature of He II is

manifested. Kapitza34 found that He II has a viscosity about

10"9 times that of water. On the other hand, the damping of

oscillating disks35 in He II produces the same results as for

He I. The resulting viscosity is 106 times larger than

Kapitza's value, although for classical liquids the different

methods give the same viscosity.

Two-Fluid Model

A quite successful phenomenological explanation of super-

fluidity is the two-fluid model. Tisza' assumed that He II

consists of two interpenetrating fluids: a normal component

of density pn which behaves like an ordinary liquid, and a

superfluid component of density PS which has zero entropy ani

viscosity. The total fluid density is

Pn*ps{(2.3)

Landau2 proposed a similar theory, except that he identified

the normal component with excitations in the fluid.

..

12

The discrepancy between the two viscosity experiments

can be explained on the basis of the two-fluid model. In

capillary flow the normal fluid is clamped in place while the

superfluid passes unhindered. On the other hand, for rotating

objects only the surrounding normal fluid has any effect. In

fact, Andronikashvili36 used a stack of plates to measure the

temperature dependence of */J , which decreases from 1 to 0

between absolute zero and T) . The high thermal conductivity

of He II can be explained as superfluid convection.

At times the two-fluid model is taken too seriously, and

can lead to misconceptions. Originally based on classical

hydrodynamics, it cannot adequately describe the behavior of

He II on a quantum-mechanical basis, but the terms "normal'

component" and "auperfluid component" are firmly entrenched

in the language of superfluidity.

Energy Spectrum

In order to calcukte the thermodynamic properties of

He II, its energy eigenvalues must be known. The strongly

interacting system of particles is considered to be a weakly

interacting system of elementary excitations (quasiparticles)

whose energy spectrum approximates the actual low-lying

states of the system. The energy spectrum is a dispersion

relation for elementary excitations in the fluid, which

expresses the energy E(k) as a function of wavenumber k.

The momentum of an excitation is given by ib, where i is

Planck's constant divided by 2.ir

..

13

Theories

Landau2 proposed that He II can be described in terms of

elementary excitations of the fluid, analogous to phonons in

a crystal lattice. In fact, to explain the observed T3

dependence of the specific heat at low temperatures, low-

lying phonon excitations were postulated. In order to

explain the observed specific heat at higher temperatures,

Landau added excitations called "rotons" in an upper branch,

which require a minimum energy for formation. Rotons were

presumably related to some type of collective rotational

motion. According to this model there are few excitations at

low temperatures, so the liquid is essentially all superfluid,

which accounts for frictionless flow.

Landau2 developed a set of linearized hydrodynamical

equations which predicted temperature waves called second

sound, in which P and P oscillate out of phase. Since his

predicted value of the speed of second sound was too high,

Landau revised his energy spectrum without theoretical

justification to a single continuous curve with a roton dip,

given by the dashed curve in Fig. 2. His energy spectrum has

three parameters which are fit to experimental data.

A quantum-mechanical variational approach to the energy

spectrum of an interacting boson system was proposed by

Feynman.7 Since the repulsion at small atomic separations

makes a uniform density highly probable, 8 the most reason-

able low-energy, non-phonon excitation would be a "stirring"

: -

14

of the atoms. But due to the orthogonality of such states

to the ground state and the indistinguishability of atoms,

only a small translation of each atom is required to bring the

wave function from its maximum to its minimum value, so the

gradient of the wave function - and hence the energy - must

be large. Therefore phonons constitute the only low-energy

states, but the stirring mode at higher energy might corre-

spond to rotons. A variational calculation for such states

shows that the energy is

E(k) = zk2/2M 5(k), (2.4)

where M is the atomic mass and S(k) is the liquid structure

factor. This energy spectrum is shown as the curve F in

Fig. 2. Pitaevskii3 9 derived the same expression from

Landau's quantum hydrodynamical viewpoint. Experimental

values40 of S(k) show a maximum at about 2 X1, where the

roton dip of the energy spectrum falls. Although the energy

at the roton minimum proved to be over twice the experimental

value, a later modification which included backflow around

a moving atom greatly improved the agreement with experiment.

This spectrum is shown by curve FC in Fig. 2.

Experiment

The excitation spectrum of He II has been determined

experimentally by inelastic neutron scattering. Cohen and

Feynman42 showed that, at low temperatures, a neutron

scattering from a helium atom creates a single excitation

15

(quasiparticle) in the liquid. The energy and momentum of

the resulting quasiparticle can be determined from the

energy and momentum of the neutron before and after

scattering.

The latest results of Cowley and Woods' are shown by

the experimental points in Fig. 2. The resulting curve is

qualitatively similar to Landau's,3 except that beyond the

roton minimum the experimental curve begins to level off and

data points are difficult to obtain. The slope of the linear

region near k = 0 is consistent with the experimental speed

of sound, as expected from the phonon interpretation of

excitations in this region.

Bose-Einstein Condensation

It is tempting to associate the superfluid component of

He II with the theoretical condensation of an ideal boson

system at low temperature into the ground state. But there

are difficulties in understanding how real boson systems

behave, and the fraction of condensed particles as determined

from experiment and theory differs significantly from the

relative superfluid density.

An ideal Bose gas exhibits a condensation,13 or macro-

scopic occupation of the ground state, below a certain

critical temperature. It is classified as a first-order

phase transition,' 3 and is referred to as Bose-Einstein

condensation. If helium were an ideal Bose gas it would

begin condensation at 3.13 K, and the condensate would have

16

zero entropy.

Connection ith Superfluidity

London 4 explained Tisza's two-fluid model1 by analogy

with the -X transition to Bose-Einstein condensation. The

comparison of liquid He II with a Bose gas is not unreason-

able because the fluid density is much smaller than for most

liquids, and the viscosity is gas-like above 1.5 K. When

interactions between particles are introduced, quantum

theory45 shows that condensation should still occur, but to

a lesser extent. The interactions may, however, be strong

enough to influence profoundly other properties of the

fluid. Indeed, the calculated ratio of density in the con-

densate to total density does not agree with the experimental

ratio of P5/o , and the;\ transition is not first order.

A good microscopic theory of He II should predict the

observed energy spectrum. The energy of an ideal Bose gas

is the free-particle form

E(k= Z2M ,(2.5)

which does not agree with the observed linear behavior for

small k. Bogoliubov5 considered the case of a weakly-

interacting Bose gas, which should be a better approximation

to He II than the ideal gas. Using the formalism of second

quantization, he treated Bose-Einstein condensation by

replacing the creation and annihilation operators for the

macroscopically occupied zero-momentum state with c-numbers in

:

17

the Hamiltonian. The system Hamiltonian is then partially

diagonalized by a canonical transformation to give the

energy spectrum of the elementary excitations. Bogoliubov

neglected the number of particles excited out of the zero-

momentum state compared to the number of particles in that

state, and obtained the spectrum

ELk) (i~k1/2 M2 + 2nCVkk2)]>(2.b)

where Vk is the Fourier component of the interatomic

potential and n is the density. If the number of particles

excited out of the zero-momentum state is not negligible, n

is replaced by no, the density of particles in the zero-

momentum state. Equation (2.6) is linear for small momentum,

characteristic of phonons, and quadratic for large momentum,

typical of free particles. By choosing nok suitably, the

roton minimum can be reproduced, but the potential is then

only a pseudopotential.

The relationship between superfluidity and Bose-Einstein

condensation is not well understood. Superfluidity has not

been shown to be a consequence of condensation. In fact,

Landau2 emphasized the independence of his theory from Bose-

Einstein condensation and thought that superfluidity had

nothing to do with it.

Theoretical Condensate Fraction

Attempts to reconcile the superfluid density ratio

which can be defined operationally from the two-fluid model,

18

with the condensate fraction no/n, where no is the condensate

number density, have not been successful.

When the normal mass density Pn is defined in terms of

the total momentum of excitations created with a drift

velocity in helium moving at absolute zero, it and the

related superfluid density are statistical quantities which

can not be identified with individual atoms or groups of

excitations. Experiments3 6 ,47 indicate that at

absolute zero.

In microscopic theories based on condensation phenomena,

the factor n0M arises for mass density instead of the pheno-

menological PS. The distinction has been emphasized by

Fetter,48 although many investigators do not differentiate

between them. Both theoretical and experimental values of

the condensate fraction near absolute zero are much less than

unity, in contrast to P.Penrose and Onsager45 used a crude wave function to

describe a system of hard spheres interacting via two-body

forces. For this system the condensate fraction is

no/n = 0.08 at absolute zero, which means that 92 per cent

of the atoms are excited out of the zero-momentum state.

McMillan49 considered the single-particle density matrix ofbosons interacting through a Lennard-Jones potential. His

numerical computations, based on a variational treatment of

a trial wave function, gave no/n = 0.11. A similar calcula-

tion by Schiff and Verlet18 gave n0/n = 0.10. A study of the

19

phase transition from a solid to a superfluid using a cell

model led Gersch and Tanner50 to a value of 0.06. Francis,

Chester, and Reatto5 1 used a parameterized pair wave

function to calculate no/n = 0.10, but inclusion of zero-

point motion lowers it to 0.08. The theoretical values of

the condensate fraction are summarized in Table II.

Experimental Condenaate rcin

Mook, Scherm, and Wilkinson'9 experimentally determined

no/n = (2.4 1)% by neutron inelastic scattering between

1 and 2 K. They obtained corrections to the impulse approx-

imation for the scattering cross section as a series of

inverse powers of the momentum transfer k, Scattering by the

condensate produces a peak of width proportional to k*,

which is distinguishable from the noncondensate contribution

to the width proportional to k. Recently Mook52 revised the

value to (1.8 t 1)%, which is still compatible with the

earlier result. Previously, Harling53 had experimentally

determined the ratio to be (8.8 t 1.3)% at 1.27 K. Recently

Jackson54 examined the line-shape broadening and concluded

that the relative condensate density is less than 0.04,

and may even be vanishingly small.

The experimental values are summarized in Table II.

The next chapter introduces another peculiar phenomenon in

He II, which is the existence of quantized vortex lines and

rings. Determination of their structure is the subject of

the following chapters.

CHAPTER III

QUANTIZED VORTICES

One of the macroscopic quantum phenomena associated with

He II is the existence of quantized vortex lines and rings.

The circulation, or line integral of velocity around a

closed loop, is an integral multiple of h/M, where h is

Planck's constant and M is the mass of a helium atom. This

chapter describes the prediction and confirmation of quantized

vortices in He II, along with a theoretical analysis and

experiments concerning their structure.

Prediction of Vortices

Irrotatinal Yelocity Fie

Landau,2 in his phenomenological two-fluid model of

superfluid helium, obtained hydrodynamic equations for the

velocity of the superfluid v$ and of the normal component Vn

by requiring that the fluid satisfy the relevant conservation

laws. An important condition that he imposed on vs is that

superfluid flow be irrotational, j.l.,

1 7 *5 = 0 .( 3 . i )

This conservation of vorticity is a requirement of non-

viscous flow in classical hydrodynamics, implying that no

turbulence or vortex formation exist. The lack of viscosity

20

21

in He II has been firmly demonstrated by persistent current

experiments in rotating systems. However, dissipation

sets in above a certain critical rate of rotation, but for

slow rotation Eq. (3.1) should apply rigorously.

Qsborne's Experiment

In a cylindrical bucket of rotating He II the only

solution of Eq. (3.1) is v = 0 if vortex lines do not exist.

The vanishing curl implies by Stoke's law that the line

integral of vs around any closed loop is zero. But any such

loop can be shrunk to a point in a simply-connected region,

so that v$ must vanish everywhere. The absence of superfluid

velocity has been verified experimentally by observing the

transfer of angular momentum to the walls of the vessel as

rotating He I is cooled through the A point.

Since only the normal fluid should be in rotation, the

height of the free surface above the minimum is

z = hor)P/p ,(3.2)

where h(r) is the classical parabolic result

( c"(3.3)

for a point a distance r from the axis of the container

moving with angular velocity w , and g is the acceleration

due to gravity. Careful experiments by Osborne8 showed,

however, that z = h(r) at low temperatures where PIP 0.1,

even for small c> to avoid critical rotation.

22

The superfluid apparently undergoes solid-body rotation,

which casts doubt on the validity of Eq. (3.1). One possible

explanation is that even the small values of wo which were

used greatly exceeded the critical rate above which viscosity

sets in. An alternative explanation is the existence of

linear singularities parallel to the axis, so that the con-

tainer becomes a multiply-connected region. Physically, the

singularities could be vortex lines. For a single vortex

line the irrotational condition in Eq. (3.1) has the non-

trivial solution

s rK.(3o4)which represents circular flow inversely proportional to the

distance from the singularity. Although rs-+oo as r-0,'a

model can be adopted which has special properties for small

r, such as solid-body rotation or zero density at r < a for

some radius a. Classical vortices have the flow pattern of

Eq. (3.4), but their core structure can not be determined by

hydrodynamics.

There is strong theoretical evidence for the existence

of vortex lines in He II. In this manner, experiments with

rotating He II can be explained while maintaining the vanish-

ing curl in Eq. (3.1).

Quantization of Circulation

In a footnote to a paper on classical hydrodynamics

theory, Onsager7 wrote, "Vortices in a suprafluid (fja) are

23

presumably quantized; the quantum of circulation is h/m,

where m is the mass of a single molecule." Later Feynman6

obtained this result by considering the superfluid at the

point r to be described by a macroscopic wave function

l1Ji= y(r) , which can be written in the form

Ifs (3.5)

The superfluid density is then

s f' (3.6)

The real phase factor S can be identified as the velocity

potential,

(3.7)

where (z) is the superfluid current. density function

(3.8)

The irrotational condition of Eq. (3.1) is an immediate

consequence of taking the curl of Eq. (3.7).

The circulation K of the superfluid is defined to be

the line integral of velocity around a closed contour, which

in this case is

/M Vs.(3.9)

24

The single-valuedness56 of the wave function implies that

QS, the change in S, must be an integral multiple n of 21r.

The circulation is then

i = nh/M(3.10)

so it is quantized in units of h/M = 0.997 x 10-13 cm2 sec'i

for OHe. The superfluid velocity away from the vortex line

is then

\V; = 1c/Zrrr(3.11)

by doing the integral in Eq. (3.9) in cylindrical coordinates.

which is in agreement with the velocity in Eq. (3.4).

Vortex Lines

Quantized circulation was first detected in 1961 by

Vinen.9 Due to the Magnus force, a wire vibrating in a

fluid with circulation K has its plane of vibration rotated

at a rate proportional to K . A thin wire stretched along

the axis of a slowly rotating cylinder of He II was set into

vibration, and the electromotive force induced in it due to a

external magnetic field was monitored. The recorded ampli-

tude is a function of the orientation of the plane of vibra-

tion, and the circulation K is deduced from the frequency.

Values of K which were stable against repeated large

vibration of the wire were either zero or values close to h/M,

but the accuracy was not good.

25

The experiment was improved by Whitmore and Zimmermann,57

who increased the detector sensitivity, determined the

direction of circulation, and made measurements after the

vessel stopped rotating. They found the circulation to

persist for hours, with transitions between stable values.

The circulation made spontaneous changes in the number of

quanta between +3 and -3.

Further evidence has been provided by the observation5 8

of single vortex lines in a narrow rotating cylinder. Since

electrons form microscopic bubbles59 which can be trapped by

vortices, the vortices can be charged, and the electrons

extracted and detected. The amount of charge collected is

proportional to the number of lines present. The number of

vortices increases with angular velocity, which is consistent

with the result60 obtained by the minimization of the free

energy.

Since kinetic energy is proportional to 1(2, and hence

n2 through Eq. (3.11), a collection of vortices with one

quantum of circulation has a lower energy than fewer

multiply-quantized lines. The creation of more vortices asthe kinetic energy of rotation is increased helps explain

the observations of Osborne8 that He II rotates like a solidbody, in violation of Eq. (3.1). Since v = ox r for solid-

body rotation, the vorticity is

\7x> =2w.

(3.12)

26

The corresponding circulation from Eq. (3.9) is the same as

for a vortex line density of

(Y= ZMw/h.(3.i3)

For c>= 1 rad/sec, 6" 2000 lines/cm2 and solid-body rotation

is a good approximation. A recent experimental produced

photographs of an array of vortices. A phosphor screen was

photographed when electrons trapped on the cores of vortex

lines were extracted by means of an electric field. They

were distributed irregularly rather than in a triangular

lattice configuration. The triangular lattice was calcula-

ted to have a slightly lower energy than a square lattice,62

although the energy difference was too small to be experi-

mentally significant.

Vortex Rings

Experiment

Although the existence of vortex lines is well estab-

lished, it has not been possible to determine their

structure or core redius directly. Such information can,

however, be obtained in part from experiments on the more

mobile vortex ring, which is a vortex line with ends joined

rather than terminating on the vessel or free surface.

In their brilliant experiment, Rayfield and Reif10

studied the speed v of ions in liquid helium by a time-of-

flight method. Alpha particles provided by a 210Po source

27

were accelerated in the He II by an electric potential V

applied to a grid and later stopped by a retarding poten-

tial VR applied to a second grid. They moved with

mobilities about 10"5 times that of free ions in vacuum

with very little energy loss. The ion speed was deter-

mined by a time-of-flight velocity spectrometer. By

applying a small square-wave modulation voltage of frequency

V to a third grid halfway between the other two grids, at

a distance L from either, the ions whose time of flight

over L is half the period, .,..

L/v = 1/2v,(3.i )

are slightly accelerated over the whole path 2L. Ions with

the speed given by Eq. (3.14) are detected at the collector

plate, while ions with other speeds are not. For a given

potential V, the ion speed given by Eq. (3.14) is obtained

by sweeping through values of the frequency V to find the

maximum current.

Rayfield and Reif10 found two remarkable results. The

ion speed was about 10-5 times that expected for free helium

ions, indicating that many atoms were moving as a unit with

the ion. Even more remarkable was the observation that,

contrary to the behavior of free ions, the more energy

given to the ion, the slo a it moved. The experimental

result that the speed is approximately inversely proportional

to the energy can be explained by assuming that the ions

a8

create vortex rings and are then trapped on their cores.

Theory

Classical hydrodynamics63 gives the energy of a large

vortex ring with radius R much larger than the core size a as

E- (3.15)

and its velocity is

\V %/(4TrR) (D9>-&) (3.16)

where

In (8R/a)

(3.17)

p is the fluid density, and oL and /3 depend on the model of

the core used. These expresssions are derived in the first

section of Appendix B.

A model for the vortex ring is necessary since classical

theory does not provide any information about the core

structure. For a circular filament with a core undergoing

solid-body rotation, Lamb63 determined o(= 7/4 and , = 1/4,

whereas for a hollow core Hicks64 found o(. = 2 and = 1/2.

Neither model is strictly physical since the expected

relation for group velocity,

(3.18)

v

29

where P is the classical impulse63

(3.19)

does not hold. This situation was examined by Roberts and

Donnelly65 who showed that Eq. (3.18) is true for o( = 3/2

and P = 1/2, assuming a hollow core. In this case vortex

rings behave like classical "quasiparticles."

Qor~e.Radius

Rayfield and Reif10 analyzed their data using Lamb's

values 6 3 for a solid core and, in a footnote, for a hollow

core. Application of Eqs. (3.15)-(3.17) to the circulation

and core radius is described in the last section of Appendix

B. They found that Eq. (B29) for (Ev) vs. In E produced a

linear graph. If a slope given by Eq. (B27) and a density of

0.1454 g/cm 3 is used, the data give a circulation of

' = (1.00 t 0.03)x 10-13 cm2/sec, (3.20)

which is equal to the theoretical value of h/M = 0.997x 10 -3cm2/sec to within experimental error. The intercept of the

line gives the core radius

a = (1.28 t 0.13) A. (3.21)

Roberts and Donnelly 6 5 found the same value for a by using

their relations for E and v with a hollow core.

Recent experiments have shown a slight pressure and

temperature dependence of the core radius, which is

>.

30

(1.28 0.05) A at 0.35 K. If the values are extrapolated

to 0.0 K, the radius becomes (1.14 * 0.05) . The temper-

ature dependence agrees with a semi-phenomenological model6 6

in which the core is filled with normal fluid and has a

polarized tail of rotons outside the core. A much cruder

determination of the core radius by Gamota and Sanders6 7

gave a = (0.90 * 0.50) X. These values are summarized in

Table III.

Classical hydrodynamics has been used here to explain the

experimental results, but a more difficult problem is the

development of a theory of quantized vortices based on

quantum theory. The following chapter introduces two

approaches to this problem.

CHAPTER IV

PREVIOUS VORTEX THEORIES

The previous chapter has shown how quantized vortices

make the mathematical requirements of a superfluid compat-

ible with observed rotational properties. Classical hydro-

dynamics adequately describes the behavior of vortex rings,

such as the relationship between energy and velocity, but

is incapable of providing information about the core radius.

Hence, some model of the core structure must be adopted10 ,68

to interpret experimental results for the core radius,

although the resulting values are not very model-dependent.

It is challenging to understand vortex properties on a

microscopic basis. Quantum theory also enables the core

density profile to be calculated from first principles.

There are two approaches to the calculation of the

vortex density profile. The Hartree model describes the

behavior of a particle in terms of the average field of

all the others in the condensate. A product wave function

is used. The method of correlated wave functions is based

on the energy variational principle with a trial wave function

which takes particle correlations into account.

31

32

Hartree Model

Gross-Ptaevs ii auction

Gross1 2 and Pitaevskii13 developed a quantum-mechanical

vortex theory for a Bose gas with a two-body potential V.

The system Hamiltonian in second quantization 9 is

(1 Xf (X) X +(1 x x')) Xx'O )Cp) c' d. (,

The commutation relations for the field creation and

annihilation operators, Y(r) and N'F) respectively, are

[]P(i-)) 'y(Fjj = ( -,.

CPfi(r), PVf')] = 0)

The Heisenberg equation of motion for the field operator

l3 /(rI) is

When the Hamiltonian in Eq. (4.1) and the commutation

relations in Eq. (4.2) are used in Eq. (4.3), the operator

equation of motion

IA ?x)S xI xi (xQ'()dx ' (4.4)

is obtained.

33

A system of N particles is characterized by the

(normalized) wave function 4TN(I,2,...,N) in which the

argument i (i=1,2,...,N) represents the space-time

coordinate (ri, t) of particle i. The order parameter

(x) is defined as the off-diagonal matrix element of the

field annihilation operator,

As a first approximation, the wave function N is assumed to

be a Hartree product,

a

N71, 2,..., N).(4.6)

Taking the non-vanishing matrix elements of Eq. (4.4), the

operator equation is transformed to a c-number equation for

the order parameter. With the product wave function of

Eq. (4.6), the result is the Gross-Pitaevskii (G-P) equation

where q*00 is the complex conjugate of (e(x) and e J .The mean-field nature of the G-P equation occurs in the

integral, which gives the average field at x due to all the

other particles.

The meaning of the order parameter ( can be deduced

from the density matrix approach to homogeneous boson

systems proposed by Penrose and Onsager.4 5 Ordinary fluids

34

have no long-range configurational order, but if Bose-

Einstein condensation occurs, long-range correlations are

induced. For superfluidity to occur it is usually assumed

that the single-particle reduced density matrix

P=r <W'-t)_ Iv t)1 l fir':~)) trw(t> (4.8)

has off-diagonal long-range order.45970 This means that thedensity matrix can be written as

(4.9)

where the factorized term, defined by Eq. (4.5), is finite

in the limit of large separations i-') :> , but

vanishes in the limit. Hence, < is the condensate wave

function, the magnitude of which is the square root of the

condensate density. When the Hartree product wave function in

Eq. (4.6) is used in Eq. (4.8), ,= 0 in Eq. (4.9).

In his original theory, Gross1 2 replaced the field

operators in Eq. (4.4) with c-number functions to obtain

Eq. (4.7) by considering the system to be semiclassical.

He later71 pointed out that for free particles (V = 0),

Eq. (4.7) leads to a particle density proportional to

cos2('Irx/L) in a one-dimensional geometry where -L/2 x L/2.

This produces an unphysical hump in the density at the

center. A small repulsive potential should flatten out the

curve to a uniform density except near the walls. This is

35

accomplished by adding a Hartree self-consistent potential

o VI12 to the Hamiltonian.

Pitaevskii13 obtained the G-P equation from the

equation of motion in Eq. (4.4). The field operator

is expressed as a c-number condensate wave function ' plus

a small operator correction term )c for the weakly inter-

acting system. When

(4.10)

is substituted into the operator equation of motion in

Eq. (4.4), the c-number terms vanish if72

aP)-t r V (\1Ox'I) 9x')I dx' 9 (x) 4, )

which is the G-P equation in Eq.(4.?).

ortx u SoleionsIn order to describe vortex lines, the condensate wave

function in cylindrical coordinates ( p. 9 , z) is assumed to

have the form

(4.12)

where f is a real function, I is an integer, a is the

chemical potential, and the core lies along the z axis.

The number density of the condensate is then

I (X) I= hC f ( ) (4.13)

36

For large P , f(P ) -+ 1 so that lxtZ approaches the bulk

value no.

The simplest approximation is to use a &-function

for the two-body potential,

1 x-x'O --= /, Six-x'}. (4.14 )

The coefficient V0 is an adjustable parameter which is not

often calculated explicitly, but is absorbed in the "healing

length" which characterizes the core size and is used to

make the spatial variable dimensionless. The speed of

sound is sometimes used to determine V0, as described later.

If Eq. (4.12) is substituted into the G-P equation in

Eq. (4.7) with the &-function potential, the result is

d2+ I + ( -y)\/ O$5 . (4.15)

The boundary condition at the origin is f(O) = 0, so the

solution of Eq. (4.15) for small p is the Bessel function

of order JI,

RP4~A. J.([2 N /X2fK ] ) or - (4.16)

where A is a constant. Since the asymptotic behavior of f is

*)41,P -;o, (4.17)

the chemical potential is given by

Y)oVO >0 ((4.18)

37

In this theory the chemical potential must be positive for

a repulsive potential, and the system is unbound.

Gross12 also shows that Eq. (4.42) leads to a z

component of angular momentum of h per particle, and anazimuthal velocity 1 /M p , in agreement with Eq. (3.11).

Equation (4.16) introduces a characteristic length a, the

de Broglie wavelength or healing length

CA = .(21n (4.19)

which characterizes the rise in f from zero to unity.

The equation for the density amplitude in Eq. (4.15)

was solved in dimensionless form by Ginzburg and Pitaevskii,? 3

although in a different context. They presented only a

small sketch with no table of values nor any indication of

their method.

A numerical solution of the G-P equation, Eq. (4.15) in

dimensionless form, was performed by Kawatra and Pathrial4

(K-P) for the cases -1 1,2,3. Near the axis the solution is

&j=A,J1 ( /Ky 1 so, (4.20)

where the coefficient A, adjusts the slope. For large pthe solution to Eq. (4.15) is the power series expansion

~ 1 ~(~j2 t 4~ (Cdr) 4 p-oo. (4.21)

For a solution obtained by a Chebyshev method, K-P showed

that the previous two forms match only for particular values

of A. . When IX= 1, the value is Al = 1.1665, corresponding

38

to an initial slope of 0.5832. The value of a was not

given, but can be deduced from Eq. (4.20) for f(0.1) to be0

a = 1.001257 A. This value also satisfies f(25) in

Eq. (4.21). The chemical potential is then j._ +6.0443 K

from Eqs. (4.18) and (4.19), which indicates an unbound

system. The solution is a monotonically increasing function

as shown in Fig. 3.

Related Theories

Since the G-P equation in Eq. (4.15) can not be solved

in closed form, Fetter assumed an approximate form for f,

(4.22)

which is proportional to for small arguments and

approaches unity for large p0 . If X = a as given in

Eq. (4.19), so that fF(a) = 2*, the vortex energy per unit

length is

6F x(roo M 0 (:.5 P/a)(4.23 )

in a cylindrical container of radius R. This is somewhat

larger than the numerical solution 1 3 '7 3

J ()L/ R/a)) (4.24)

but is qualitatively the same as found by the K-P method.14

The energy per unit length of a vortex ring determined from

Eq. (3.15) with oc= 7/4 would equal the above results if

39

0.17 and 0.05 were subtracted from the logarithm terms of

Eqs. (4.23) and (4.24), respectively.

Fetter improved his theory by letting > be a

parameter, and minimizing the energy with respect to it,

which gives

(4.25)

The calculation of energy per unit length for this case

replaces the argument of the logarithm term in Eq. (4.23)

by 1.50 R/a, very close to the solution E,. Hence,

Eq. (4.25) is superior to the nonvariational relation -"= a.A quantum-mechanical approach to vortex rings has

been presented by Amit and Gross.68 They proposed a

phenomenological model of He II in which the density near

a vortex line has the form

= 4n 2 ( +ce, (' C, (4.26)

with the bulk value n beyond a. Gross71 used Eq. (4.4) for

the field operator equation of motion with the &-function

potential of Eq. (4.14) to obtain

Ct =+ //2. (4.27)

The annihilation operator 4 is replaced by the c-number

expression in Eq. (3.5). V0 is the strength of a two-body

&-function potential in Eq. (4.14). When a stationary state

is assumed for the equation of motion, the energy can be

40

expressed as

E zVM\h)PP( kf "I.(4.28)

The optimum value of a is determined by substituting

Eq. (4.26) into Eq. (4.28) and minimizing the energy.

The Bogoliubov relation for the speed of sound, u = E/k,

is obtained from the long wavelength form of Eq. (2.6).

Using the relation

uj =n\//M (4.29)

they obtain

C = o z(0.12M nV$~ ! A. (4.30)

This is four times the deBroglie wavelength 4

O,)-MniZ0.7 A. (4.31)

For a vortex ring, Eq. (4.26) becomes the density

measured from the vortex core. The velocity field is

determined in the same manner as in Appendix B. Variation

of the energy per unit length with respect to a gives the

energy in Eq. (3.15) for large rings with d = 1.67. The

value of a in the density profile of Eq. (4.26) is then

1.0 J.

41

Method of Correlated Wave Functions

Tral Wave Functions

An alternative approach to the investigation of He II

is based on the construction of a trial wave function

containing interparticle correlations, and the energy

variational principle. Some model for the form of the wave

function is adopted which contains parameters that can be

varied to produce a minimum energy. Such procedures18'50

have met with some success in determining the binding energy

of He II.

Determination of a suitable trial wave function is

based on a combination of physical arguments and mathematical

tractability. A variety of rather simple functions with

physically desirable features can be assumed. For ground-

state calculations one of the most popular forms is the

Jastrow?5 wave function ( , which is a product of expo-

nential terms:

(4.32)

The function J depends on the relative coordiantes r .

Similar forms are used ftv the distribution function in

classical fluid theories, and quantum-mechanical calculations

can be carried out by analogy with the classical procedures.

Excited states are assumed to be described by one or more

density fluctuation operators acting on 43

42

For application to quantized vortices, the model wave

function should take into account the strong interatomic

potential which induced two- and several-body correlations

in the wave function, and should lead to the velocity field

in Eq. (3.11) far from the vortex core. A parameterized

radial function is introduced into the trial wave function,

from which the vortex density is determined.

A theory of quantized vortices based on the procedure

just outlined has been developed by Chester, Metz, and

Reatto15 (CMR). They adopted as a model wave function

~cMR -1 ^ fi@r) F 4.33)

where 10 is the exact ground-state wave function. In

cylindrical coordinates (ff 9 , z), S is the velocity

potential

(4.34)

as in Eq. (4.12). Since the resulting velocity in Eq. (3.11)

diverges for small p , the real function R(r) = R(p) is

chosen to be a function of p only such that

R(P) - , 0- (4.35)

sufficiently rapid to ensure convergence, and approaches unity

for large p . The form of R determines the density profile.

Energy Yaiational Principle

In order to make a calculation, an expression for the

energy must be found in terms of the radial function R,

which is written in parameterized form.

The Hamiltonian

T -V (4.36)

minus the exact ground-state energy E0 is

K= H -u.(4.37)

Since the eigenvalue of H' in the ground state vanishes,

H'1 o>-CTtv-E0 ) > 0) =(4.38)

we then find

$PCMRI CMR> 'q (TIV- U)) F~u>

CMR LTI I +l t , R (T t-E)%

S< TF4[>) TV9T

where we define

-T F o - o) - FCT O). (4.40)

The expectation value of H' is

= l. 'vN nr~r (4.41)

44

where the number density is defined as

fl NiT' r d~r f. TdrA,' rN(4

The energy variational principle is used to determine

R from Eq. (4.41). If R is chosen to have the form

R( (4.43)

for some real function g, Eq. (4.42) is the same as the

density of a classical system having an equilibrium

probability distribution 1 under the influence of an

external potential U given by

where kB is Boltzmann's constant and Teff is an effective

temperature. The density can then be calculated as in the

classical case, using the Percus-Yevick (P-Y) or

convoluted-hypernetted-chain approximation methods.76

These seem fairly accurate at liquid heliium densities18

and are estimated to give results accurate to 10-15 per cent.

The P-Y equation is obtained from the N-particle

probability density function I)CM t12 through the grand

partition function,15 From the variation of a general

function which has been expanded in terms of the density,

one obtains the integral equation

rr e rx L'fiS CI(-r') 1r)'- r ] I)(4.45)

45

where C is the direct correlation function, whose Fourier

transform is related to the liquid structure factor S0 ,by

S )=(Tnh .(4.46 )

S0(k) can be obtained50 from x-ray and neutron scattering

experiments at finite temperatures except for small k, where

Feynman6 showed that at zero temperature it should be pro-

portional to k. Hence, C (k) can be obtained from experiment

by Eq. (4.46) and then Fourier transformed so that Eq. (4.45)

can be solved iteratively for a given function g.

The form of Eq. (4.43) is still general, since for any

R the function g can be written as -2 In R. The most

successful radial functions found by CMR are

(4.47)

and

cz( ( a)(4.48)

in terms of a variational parameter a. For both cases the

minimum energy occurs at a x 1.2 A, assuming a cylindrical

container of radius 6 X. The function R2 is Fetter's

approximation in Eq. (4.22).

The density amplitude corresponding to a solution of

Eq. (4.42), in which n(r) is normalized to the bulk density

i, is shown in Fig. 4. The curve is obtained by using R2

in Eq. (4.48), but is characteristic of both trial functions.

46

The oscillatory nature of the particle density is qualita-

tively similar to S0(k), which was put into the theory.

Comparison

Solutions for the density amplitude using the Hartree

product wave function have the general form shown in

Fig. 3, while the use of the correlated wave function

leads to a density amplitude shown in Fig. 4. In the former

case, the density is \CI = n f2. In the latter case, due

to the presence of the true ground-state wave function <o ,in Eq. (4.33), the density is given by Eq. (4.42). Since

the behavior of Figs. 3 and 4 is quite different, the ques-

tion arises as to whether the actual density profile

increases monotonically towards its bulk value or whether it

has maxima and minima reminescent of the pair distribution

function, which is the Fourier transform of the liquid

structure factor.

An experiment which would determine the density profile

directly is not available at present. The core radius,

chosen to be the distance at which the density is half its

bulk value, is found to be about 0.5 A in the CMR theory,which is much smaller than the K-P value of about 1.4 A in

Fig. 3. The value of the healing length can be chosen

arbitrarily in the K-P calculation. The core radius which

is inferred from experiments on the basis of classical hydro-

dynamics with a hollow core is 1.14 A. One way of deciding

which density profile is the best is to use the calculated

47

density profile in a classical calculation of the energy

of the vortex ring as a function of its velocity.

The Hartree theory is criticized15 because it does not

include short-range correlations. An improvement presented

in the following chapter uses a realistic interatomic

potential and takes multiple scatterings into account.

. . - - - .

CHAPTER V

GENERALIZED GROSS-PITAEVSKII EQUATION

In the previous chapter the Gross-Pitaevskii (G-P)

equation was discussed as an important method of studying

the properties of a weakly-interacting Bose gas, in which

the depletion of the condensate is small. For He II the

atomic interactions are sufficiently strong to cause a

depletion of the zero-momentum state of over 90 per cent.19,54

The object of this chapter is to derive a generalization

of the G-P equation for the order parameter in which a real-

istic interatomic potential can be used, and which coritsins

no arbitrary adjustable parameters. In spite of mathema-

tical difficulties,77 the generalized G-P equation obtained

here is solved for vortex line solutions as discussed in the

following chapters.

The generalized G-P equation is derived for the order

parameter or condensate wave function, in which the average

field due to all the other particles is determined by

considering multiple scatterings. These processes are taken

into account by a summation of ladder diagrams to give a

T matrix. In lowest order it reduces to the ordinary G-P

equation. This approach permits a realistic potential with a

strongly repulsive core and an attractive tail to be used in

the calculation of the average effective field due to all the

other condensate particles.

48

49

General Equation for the Order Parameter

A generalization of the G-P equation based on rigorous

application of time-dependent perturbation theory has been

accomplished by Kobe.1 6 In that approach the Hamiltonian

for a system of interacting bosons is

H = kt\. (5.1)

It has a form similar to Eq. (4.1), but the kinetic energy

term K is

K J2 (5.2)

where ) is the chemical potential, and the two-body potent-

tial is

Y (r'') V( i (DY-r) dr' d r (5.3)

The interatomic potential depends on the relative coordinate

of two atoms, ,.,.,

(5.4)

The Schrbdinger equation for H is

NNt (5.5)

where (() is the time-dependent N-particle wave function.

For superfluidity to occur it is usually assumed that the

single-particle reduced density matrix

:

50

((J )', t -= N Ct) ) 47(g) ~'))l pg))J(5.6)

has off-diagonal long-range order,45'70 expressed by

Eq. (4.9). The non-vanishing term of Eq. (4.9) involves the

condensate wave function or order parameter 4(r(t) , which

is postulated to be

When the Hartree product in Eq. (4.6) is used for 4 , the

density is given by (C as in Eq. (4.9).

The equation of motion for is

- vCyQZ ) (i~t)

+ M '/( ) < 1P,, (t) 4 t( )' P( ) 4r) J ZE& > clr'(5 .8 )

when Eq. (5.5) is used in the time derivative of Eq. (5.?),

and the commutation relations in Eq. (4.2) are used. The

integral term represents the mean field due to the particles

at r', and includes correlations through the true wave

functions TN and ??4

It is at this point that a technique must be decided

upon for dealing with the matrix element in the last term

of Eq. (5.8) before a calculation can proceed. A simple

factorization into three off-diagonal elements > ,

----------

51

<l4 W ') I t ') Pa t ) ~~C ( f t E ) q (rl(5 .9)

would be exact if Y were a Hartree product. If Eq. (5.9)

is substituted into Eq. (5.8), the ordinary time-dependent

G-P equation

i (it- 2t V,~j)C~t ' (V- I W itTIdiC(50(5.10)

is obtained.

T Matrix

The approximation in Eq. (5.9) is the simplest form,

but does not take correlations into account since lYN is

given by a Hartree product. More general approximations

which do consider correlations involve summing terms invol-

ving multiple scattering processes. In this section the

correlations are taken into account by using a T matrix,

which is the sum of all ladder diagrams.

If G0 is the propagator

o E_____ (5.11)

where H0 is the unperturbed Hamiltonian, ) is the unper-

turbed ground-state wave function, and E0 is the true ground-

state energy, the T matrix is defined78 by

7 \1+\/GET(5.12)

...

52

which is an operator equation known as the Lippmann-

Schwinger (L-S) equation. Upon iteration it becomes

T= v I (G&v

(5.13)

where W is defined as the wave operator. If < and 14 are

the unperturbed and true wave functions of a system of

particles, respectively, an alternate form of the L-S

equation?8 can be written

(5.14)

By applying the potential V and using Eq. (5.13), we obtain

/W T}. (5.15)

Thus, the particle correlations in the true wave function \l

are transferred to the T matrix, which is an effective

potential acting on the unperturbed system.

Originally developed for scattering theory, the Tmatrix has been used to overcome divergences in the pertur-

bation expansions for nuclear matter. 7 9 For the problem ofa high-density hard-core boson system, Brueckner and Sawada80

replaced V by T in Bogoliubov's theory to obtain an energy

spectrum with an apparent roton dip, but they neglected the

depletion. Goble and Trainor8 1 repeated the calculations

using pseudopotentials and treating the depletion

53

self-consistently to obtain a reasonable phonon-roton

spectrum. The appearance of the T matrix in the generalization

of the G-P equation is therefore not unexpected, 1 2 as it is

a common procedure in dealing with realistic hard-core

potentials for systems such as finite nuclei8 2 and liquid

helium.83 But previously the T matrix has been introduced

in an g, IA ,. manner8 0 rather than developed as a natural

extension16 of the Hartree-Fock equations.

Factorization Method

The generalized G-P equation can be obtained by a

diagrammatic factorization of the last term in Eq. (5.8),

which leads naturally to the T matrix approximation. This

heuristic derivation is much simpler than the perturbation

expansion.16 A diagrammatic expression for the potential

energy term in Eq. (5.8) is given in Fig. 5. Dashed lines

represent the potential V(i,r') between points r and r', and

the large box represents all possible processes which can

take place consistent with the Hamiltonian. The integration.

over r' is indicated by a line looping back to the box.

As an initial approximation to Fig. 5, the order para-

meters, represented by the small boxes, are factored out on

the left side of Fig. 6. This approximation represents two

particles excited out of the single-particle condensate which

interact in all allowed ways. After a final interaction,

one particle returns to the condensate. If only successive

: -

54

scatterings and free propagation occur, the process is

represented by the right side of Fig. 6.

The T matrix on the right side of Fig. 6 is an

effective interaction, which is a sum of ladder diagrams

defined by Fig. 7. This figure can be iterated to display

the entire sum of terms. If the lines joining the potential

and T matrix are interpreted as free propagators G, this

equation is the same as Eq. (5.12). The full equation,1 6

obtained by taking appropriate matrix elements of Eq. (5.12),

is

7~(1234)-= \/ (13i) -G c' G(2'c)T(3'' 34])(5.16)

where (1) = (r 1 ,t 1 ), (2) = (r2 t2). etc., and T(1234) =b2iT)34>.

Since the two-body potential is local in space and instan-

taneous in time, we have

/ 012 ))_IV(,2.) 1-3) (-)+ {- (2 -3)] 5.17)

where the space-time c-function means

-3--r3) 90,t3) (5.18)}

and the two-body potential is

Ti i(5.19)

The single-particle propagator is given by16

.. , _

55

GAD= ua unge E^t,-3) N(5.20)

where the step function O is unity for a positive argument

and zero otherwise. The eigenfunctions un and eigenvalues

E, satisfy the single-particle Schrodinger equation

2V1UhUM) = E Cur),(5.21)

where U(s) is the external field, which may be zero.

From the diagram in Fig. 6, the generalized G-P

equation in Eqa (5.8) becomes

+ 5d'fCX( (r)~ 3 rtt.) ( rt'pit')Ar2c 3 cr .-0o (5.22)

Only two time arguments t and t' occur, where t =t1 = t2 and

t'= t3 =t4 . Due to time translational invariance they are

written as t - t'. Equation (5.22) can also be obtained

from perturbation theory by using the time-evolution

operator in the interaction picture and the coherent-state

representation. 16

Energy Functional Method

Equation (5.22) is the generalized G-P equation, but it

is possible to examine another derivation and determine which

approximations lead to the same result. For most applications

a time-independent equation, which is obtained here by using

i i Noon

56

an energy functional method, may be sufficient.

The total energy of the system can be expressed

approximately as a functional of 1(?) and then minimized.

The resulting equation for c gives the best function under

the given approximations. The Schrdinger equation is

eE lY(5.23)

where W is the exact ground-state wave function and E is

the ground-state energy. Intermediate normalization with

respect to the unperturbed state 4 , the eigenfunction of K

in Eq. (5.2), is

1 (5.24)

so that the exact energy E is given by

EN= IP . (5.25)

Equation (5.25) can also be written as

E=<' (K+v>)

~ll, <CK + I( (5.26)

where Eq. (5.15) is used for the T matrix and the kinetic

energy term is approximated. Equation (5.26) can be

written more fully as

57

where the static limit of the T matrix has been assumed.

If it is assumed that 4 is a Hartree product, E becomes

the functional

F 1i~ S~'~ ~ P( r) sr

(5.28)

The true ground-state energy E is not a guaranteed lower

bound on E3 N in this case because the Rayleigh-Ritz

variational principle is not used. Equation (5.27) is not

exact due to the approximations made in Eq. (5.26), and

can not be written as the expectation value of the Hamil-

tonian with respect to some trial wave function.

An equation for T which gives the lowest energy cannow be obtained by minimizing Eq. (5.28) with respect to *.

The functional minimization gives

Ej= (p)T(r 4(r) ck *) dl

=o. (5.29)

For a time-independent problem, Eq. (5.22) reduces to

Eq. (5.29)

Local T Matrix

In general the T matrix in Eq. (5.16) is non-local,

i.j., it does not depend only on the relative coordinates of

the two particles. Nonlocality makes calculations extremely

58

difficult, so a local approximation is used to provide a

more tractable equation. The approximation is most clearly

made in momentum space. First, matrix elements of Eq. (5.16)

for T(1234) are taken with respect to the set of single-

particle functions defined in Eq. (5.21). Then a Fourier

transform in time produces the usual79 L-S equation

"'J"-(E~,,Eh) io ' (5.30)

where C is the frequency and the vanishingly small

imaginary term is introduced to assure convergence.

Plane-wave states will be used for the static case,

o = 0. For the excitation of particle pairs from the zero-

momentum condensate, c3 = 4=0 and k1 - 2 =k, which lead to

T4-~rloQ> <1 7J )V~u>+ ~<,1 IV) ><W1T)VNP_2p . (5.31)

The matrix elements of V are

(5.32)

where is the volume of the system and the Kronecker delta

9 , which is unity for zero argument and zero otherwise,

insures overall momentum conservation. The function V(k) is

the Fourier transform of the potential V(r). Then Eq. (5.31)

59

can be written as

~T(k) = \q(k) - Q V((apTcp /2E)(53P (5.33)

where T(k) is defined similarly to V(k).

Equation (5.33) is also obtained from the Bogoliubov

canonical transformation, but the energy denominator in

that case is the quasiparticle energy Ep instead of the bare

energy in Eq. (5.33). This similarity lends support to

dressing the energy denominator p, corresponding to

dressed propagators in the intermediate states in Fig. 7.

In configuration space the potential V(r) is the Fourier

transform of V(k),

\J(r'7Y (2mr) 3 3 V (k ) k) 54(5.34)

while the local T matrix in configuration space is the

Fourier transform of T(k),

~T~r) = (2TY)-3 Se- T(ke),(5.35)

Equation (5.33) is then transformed into

T~r: \(r) -C(i) ~T(r') d r' (5.36)

where the kernel is

C(,ir) = Jc5k rSnkrc5jnkr'/EOe), (5.37)

Derivations of Eqs. (5.36) and (5.37) are given in Appendix C.

60

As r-eoo the kernel in Eq. (5.37) vanishes, so the

asymptotic behavior of the T matrix is

T(r) -+ V(r) a r-4,.(5.38)

However, at short distances T differs significantly from V

due to the effect of strong short-range correlations.

Finally, in the local, static, central T-matrix approx-

imation, the generalized G-P equation in Eq. (5.28) becomes

~~V 9( (' ( r')(fir' fi{r) = ,(5.39)

using the properties in Eq. (5.17), where T(r,r')= T( \r-ir'\ )

is calculated from Eq. (5.36).

Vortex Line Equation

The density profile of an isolated vortex line in

rotating superfluid helium can be determined from the conden-

sate wave function.. Particles not in the condensate are

assumed to be decoupled from the condensate. For simplicity,

surface and boundary effects are neglected for the line,

which lies along the symmetry axis of a cylindrical container

of large radius. The condensate wave function in cylin-

drical coordinates (g>, 9, z) is then independent of z, and

is assumed12,13 to have the separable form

e(5.40)

with boundary conditions f(0)= 0 and f() - 1 as p4o.

61

The normalization is chosen so that far from the vortex core

, ,',(5.41)

where no is the bulk condensate density. The circulation

quantum number P2 is a positive integer, usually chosen to

be unity, as discussed in Chapter III.

When Eq. (5.40) for the stationary nonuniform conden-

sate is substituted into Eq. (5.39), the generalized G-P

equation becomes

+ of --?2-J (S ) '')o' 4Zo , (5.42)

where o(= h2 /2M. The kernel K( , ') is an integral over

the relative coordinates 9 and z of the spherically

symmetric T matrix, T( r -ir'I ). Since the vortex line has

cylindrical symmetry and the T matrix has spherical symmetry,

the kernel K is given by

The next chapter describes the numerical solution of

Eq. (5.42) to determine the density profile.

CHAPTER VI

NUMERICAL METHODS

Three basic calculations are discussed in this chapters

the determination of the T matrix from Eq. (5.36), the

kernel K in Eq. (5.43), and solution of the generalized G-P

equation in Eq. (5.42) for vortex lines. The latter is a

nonlinear integro-differential equation, for which there areno general solution techniques. Nonlinear equations

typically require more sophisticated techniques than theirlinear counterparts, and a unique solution is not guaranteed.

The various equations are written in discrete form for

the finite procedures appropriate for computation on a

digital computer. These procedures were programmed in

FORTRAN IV language and run on an IBM Model 360/50 computer.

Determination of the T Matrix

The integral equation for T(r) given in Eq. (5.36) isderived in Appendix C. Since V(r) is large for small r,T(r) may be quite sensitive to the integral term. There is acalculational advantage to define the ratio

=(r T(r)/V(r) (6.1)

The desired equation can then be written as

Cr= 1 - odr' C ((rr'D (rV(6.2)

62

63

where the kernel C is given by Eq. (5.37).

In order to proceed in the evaluation of g, the energy

denominator in the kernel C in Eq. (5.37) must be chosen.

Generally the integration over k must be done numerically,

but in certain cases it can be performed exactly.

reZu-PLrti&LU Enerz

The integral in Eq. (6.2) can be evaluated in closed

form in the case of a free-particle-like energy denominator

for C,

E(k ck2p ,(6.3)

where the constant }Lt could be taken as the chemical poten-

tial, and off= h2/2M. This type of energy denominator is of

the form originally obtained in Eq. (5.33), and also occurs

in the reference spectrum procedure of $stgaard.83

If pA is negative and expressed as

JU 0/'Y(6.4)

the kernel integration in Eq. (5.37) is85

S5nh'r 5 inJhr' / ol ( k2(y.)

(24 1 s r-r) - cs rt)/t

4dY (6.5)

The sign of (r- r')/ depends on the relative size of r and

r'. Denoting the larger (smaller) of r and r' by r> (r).

the kernel is

2 ej'rd.(6.6)

For positive values of a similar analysis yields 8 6

C~r r,) r CosC(r>\ Sin(-<Q. (6.7 )2at'C

If p= 0, both Eqs. (6.6) and (6.7) reduce to

C(') r (6.8)

Calculations for the )A< 0 case are therefore based on the

integral equation

9 n Srr (6.9)

Observed Eerg Spectrum

If the observed phonon-roton excitation energy spectrum

in Fig. 2 is used for E(k), the kernel in Eq. (5.37) must

be approximated numerically. This energy denominator is

obtained in the equation for the Fourier-transformed pair

potential when the Bogoliubov canonical transformation is

made on the full Hamiltonian. The Fourier-transformed pair

65

potential can be simply related to the local T matrix.84

An empirical fit84 to the experimental energy spectrum,

in terms of a dimensionless momentum

X= k c,(6.10)

is

E3k () {_tMx)+2 (tX) 6.1

The dressed free-particle energy is

t x) = /2Mc2 +tA 6(x)-11(6.12)

where j0(x) is the zero-order spherical Bessel function. The

set of parameters giving the best experimental fit are8

a=2.011

0= 43.3 K

A = 3.22 K. (6.13)

Helium Potential

Bruch and McGee17 have discussed several semiempirical

potential functions for helium. The long-range tail of each

has an r-6 and an r-8 term to account for dipole-dipole and

dipole-quadrupole interactions, respectively. It joins

smoothly to a short-range Morse or Frost-Muslin potential.

In each case the potential and its first derivative are

continuous. They17 recommend their second Morse dipole-

dipole (MDD2) potential for condensed state calculations

66

since it fits the second virial coefficient data of dilute

helium gas better than others. The second virial coefficient

is fit by the MDD2 potential to within 6 per cent accuracy

over a wide range of temperatures, from 1 - 2000 K. The

parameters were also chosen to fit data for the viscosity

and mutual diffusion coefficients. This potential has been

adopted for the subsequent calculations.

The MDD2 potential is given by'7

I D.75 [.2cC:-x) 2c o _ k (r< ;)

VWr) I t %. 2.77 K (rrd

S(6.14)

where x= r/rm. Values of the constants are

c = 6.12777

r0= 3.6828 Xrm= 3.0238 A. (6.15)

The MDD2 potential also has the calculational advantage of

being finite at the origin ( V(0)= 2.25 x 10 6 K), so it is

Fourier transformable for calculations in momentum space.8 7

Numerical Procedures

In general the kernel C(r,r') in Eq. (5.37) must be

approximated numerically. The g function is evaluated at

equally-spaced points ri separated by an appropriate step

size h, where

u

67

r =0. (6.16)

If 0 ri :10 is chosen with h-=0.1 A, 5151 evaluations

of the k-integral in Eq. (5.35) would be required, taking

into account the symmetric nature of C with respect to r and

r'. Much computation can be saved by expressing the

integral in Eq. (5.35) as

ZW c ' = Sdk 5i kr 5i kr / E Oc)

Sk cs k(r--r')/Ek) cb Cc r' (6.17)

so that only 201 integrations corresponding to 0 5)r* r'1 .20 X are required.

The integrals in Eq. (6.17) are evaluated by applying

the IBM 8-point Gaussian quadrature subprogram QG888to

each quarter period of the cosine function. The integration

terminates when the contribution of one complete period is

less than 10-5 of the accumulated sum of terms. This typi-

cally occurs when the upper limit reaches from about 20 A_1

to 600 I 1.Unfortunately the evaluation of Eq. (6.17) fails if

E(0)= 0, which is the usual case. But this problem can be

avoided by calculating the integral for small k separately:

Ak 5iikr/E(k)

- kc <r-rr/) c-dk cs )<(rr')/ Ek) . (6.18)

68

The first integral in Eq. (6.18) is finite at k= 0, and is

evaluated by subroutine QG8 for a-2 X.A, below which most

of the structure in E(k) occurs. The accuracy of this

method was checked by evaluating the exact result in

Eq. (6.9) for the free-particle energy given by Eq. (6.3).

The error varies from 0.3% to 10-4%.

A procedure similar to the Fredholm method89 for inte-

gral equations of the second kind, in which the unknown

function appears both inside and outside of the integral, is

used to solve Eq. (6.2) once the kernel C(r,r') has been

tabulated. By writing the integral as a finite sum of dis-

crete terms, the equation for g is converted into a set of

n linear algebraic equations, one for each r1, which is

solved simultaneously. Convergence is assured by increasing

the number of equations (decreasing h) until successive

solutions agree. Details of the method of calculation

appear in Appendix D.

In order to insure accuracy and guard against spurious

or divergent solutions, the set of equations in Eq. (D6) was

solved successively for n= 20, 34, 50, and 100. The latter

two values of n produced almost identical results for the

energy denominator in Eq. (6.3) with )4 equal to the experi-

mental chemical potential, -6.7 K. As a further check, the

solution was substituted back into the set of equations,

which were satisfied. The solution T= gV is shown in Fig. 8

with the points calculated for n= 20 given for comparison.

69

A dashed curve shows the MDD2 potential. As expected, the

asymptotic behavior of T in Eq. (5.38) is confirmed, with

T(r) V(r) for r>4 A.

Solution of the T matrix for the observed energy spec-

trum in Eq. (6.11) is found in the same manner as above for

n= 100. Figure 9 shows T(r) for this case. Again the MDD2

potential V(r) is given by a dashed curve for comparison,

which shows that T(r) AV(r) for r>4.5 X.

Determination of the Kernel K(C, )

The kernel K(P, P') defined by Eq. (5.43) can now be

evaluated for a given interaction. Three forms of the inter-

action are examined: (1) the T matrix in Fig. 8 with the

energy denominator in Eq. (6.3), (2) the T matrix in Fig. 9

with the observed excitation energy in Eq. (6.11), and (3)

the MDD2 potential.17 Results of the double integration are

stored in matrix form, the rows (columns) corresponding to

values of (5; ((g{). The symmetric nature of K(g,(o') cuts

computing time in about half, and if some elements of K are

quite large, many others can be neglected in comparison

without significantly affecting the value of K.

Evaluation of the kernel

4S t c T2( (6.19)

is based on an adaptive Newton-Cotes integration subprogram

SIMP.90 It is adaptive in that a given integral is divided

70

into three subintervals, over which each is integrated by

a three-point rule. If their sum differs from the same rule

applied to the entire integral by more than a given tolerance,

the subintervals are further subdivided until agreement is

achieved. For given values of e and P', the 9 integration

in Eq. (6.19) is performed with a tolerance of 10-. Inter-

polation of the tabular array for T is made when necessary by

a third-degree Lagrange polynomial method.9 '

Even though the functional form of the MDD2 potential

is given, a direct integration is prohibitive due to

difficult functions arising from the complicated limits.

The MDD2 potential in Eq. (6.14) involves two different

expressions in spherical coordinates, whereas the integra-

tion in Eq. (6.19) is in cylindrical coordinates. Thus a

numerical integration is required. For the T matrix in

Eq. (6.19) the integrand is not known in functional form,

so a numerical integration is also required there. But some

of the matrix elements for either case can, however, be

evaluated by a series solution for \ f') t 3.7 X since r

will then lie in the tail region of V ( x T) in Eq. (6.14) for

any and z. This series solution is obtained in Appendix E.

The infinite z integration is also performed by the

subprogram SIMP between the limits 2n-1 and 2 n successively

for n= 1,2,3,... until the last integral is less than 10'

times the accumulated integral from 0 to 2n. This procedure

gives a better indication of convergence than equally-spaced

71

limits when the integral approaches zero for large f'.

When T in Eq. (6.19) was replaced by the MDD2 potential,

the integrations were performed from p= 0 to e= 10.2 A at0intervals of 0.1 A. Typical results for K with eP1.0 X

are shown in Fig. 10. For larger values of (0the curve

becomes bell-shaped with a slowly diminishing maximum at

and half-width of about 2 A.A more complicated K( ,e') is obtained when the T matrix

with the free-particle energy denominator (h 2k2 /2M -Aq) is

used. Figure 11 shows K corresponding to pA= -6.7 K for

several values of 9. In this case K was determined from

P 0 to 13 A at intervals of 0.1 A. Note that the maximum

values here are about four orders of magnitude smaller than

for the bare potential in Fig. 10. Since values of ( through

9.0 A were desired, Eq. (E17) was used for P'?13 A.Even more structure is found in the curves corresponding

to the observed energy spectrum. Results for several values

of are shown in Fig. 12. The integrals were calculated0

from (0 = 0 to 9.0 A at intervals of 0.2 X. A smaller inter-

val would increase accuracy somewhat, but would require

excessive computer time, and interpolation can be used when

more accuracy is required. Values of K for (>9 A are

essentially the same as for the free-particle spectrum.

72

Solution of the Generalized G-P Equation

Now that K( ) has been evaluated, the generalized

G-P equation

) ) offo PP (6.20)

can be solved. First a related solution is obtained by

ignoring the nonlinear term. The linear solution is then

used as a starting point for the nonlinear problem. The

numerical procedures are discussed and the results are shown.

Solution f Z r Condensate Density

The difficulty in solving Eq. (6.20) is that it is a

nonlinear integro-differential equation with infinite limits.

If it were made linear by setting n equal to zero it would

become, for negative , ,

+ P (6.21)

With the change of variable

\g (6.22)

Eq. (6.21) can be written as

ZV J Wjt) - (\?-).1 ) ,(6.23)

The solution of Eq. (6.23) is the modified Bessel function 92

6 of order I , which is related to the ordinary Bessel

function J- by

73

(w =- J( w) .(6.24)

At the origin ((o = 0 for t 1, but is concave upward, and

approaches the asymptotic form9 2 ew/(2 Tr w)* as w--o. This

behavior is radically different from the expected behavior

f(P) -+p1for the solution of Eq. (6.20). Hence, simple linear-

ization or perturbation techniques are difficult to apply.

Boundary Conditions

Two boundary conditions must be specified for the

solution of a second-order differential equation which de-

scribes a physical situation. Usually the function value

and its derivative at the origin are specified, but for

Eq. (6.20) the latter is not known.

COe condition for vortex lines is the assumption that

f(0)= 0, j.e., a vortex core is devoid of condensate along

its axis, thereby producing a multiply-connected region.

The derivative f'(0) is not known. Hence, the problem is

approached on the basis of two-point boundary conditions, the

second point being chosen in the asymptotic region where

f(p) e 1. There is some uncertainty in the best value of f

and ( in the asymptotic region, but the results are not

very sensitive to the choice. Since f(p) should level off

and approach unity for large , the boundary condition

f(9 X) = 0.98 was adopted. This is very close to Fetter's10v02value 0.9803? fo a 1.28 A in Eqs. (4.22) and (4.25).

74

Dependence of the solution on this boundary condition is

discussed later in this chapter. An iterative procedure for

solution is now applied with fixed end points.

NumMricaL ,Procedures

In order to solve Eq. (6.20) it is first written in

discrete form using finite differences and numerical quad-

rature. Appendix F shows how this procedure produces a set

of simultaneous nonlinear algebraic equations to be solved

by iteration.

The function f is evaluated at N equally-speced points,

requiring N-2 equations and the boundary conditions f ((,) and

f (P .).This means that the upper limit of the integral is

also p, and the remainder of the integral, I( Cj) in Eq. (F?),

must be approximated for each equation. Computing time

increases rapidly with N, so PH is chosen to be 9 A with0

step size 0.2 A, making N = 46. For each , the remainder

integral is calculated as

00

)SFI~s({'I ae' (6.25)

In the first integral on the right-hand side, fF is Fetter's

approximation in Eq. (4.22) with ~h determined by10 a= 1.28 Xin Eq. (4.25). In the last integral, f ti1 and K8 is the

kernel in Eq. (5.43) calculated by the series expansion in

Eq. (E1?). Both integrals are evaluated by the IBM

75

subroutine QUADR, a five-point quadrature for tabulated

functions. The infinite upper limit is approximated by Pl,/

the value for which the integral from 9 - A to is

less than 10-3 times the integral from 13 2 to . The

values ranged from 33 A for = 8.8 A to 55 A

for = 0.2 A.For a given value of condensate density no, the G-P

equation in Eq. (F?) is solved with several different values

of because a consistent determination of through

Y) /(6,p26)

the generalization of Eq. (4.18), is not practicable in most

cases due to slow convergence of the integral as a function

of p . Equation (6.26) is obtained by letting (9'O in

Eq. (6.20). Since J affects the results substantially,

the value which produces the correct asymptotic behavior,

f ()-v1 as P-eo, is used. A trial-and-error procedure deter-

mines the optimum ' .

The solution of Eq. (G7) by the method of perturbed

parameters94 proceeds according to the following outline.

First obtain a similar equation with a known solution, repre-

sented by Eq. (F11). By choosing no0 = 0, Eq, (F7) is the

finite difference form of the linear differential equation

in Eq. (6.21), whose solution is the modified Bessel function

(w). The values of are obtained by a polynomial

approximation.92 Then n0 is the perturbed parameter and is

76

successively increased at each step until its chosen value

is reached, as described in the next subsection.

Calculations

Certain numerical values must be assigned to Eq. (F7)

before a solution can proceed. First, two values of the

condensate density are used: a theoretical 8 estimate of

no = 0.1 n and an experimental19 value no = 0.024 n, discussed

at the end of Chapter II. The azimuthal quantum number is

J= 1, the energetically favored value.

The last coordinate should be as large as practicable

to assure that the solution reaches the asymptotic region

f x1, yet not cause excessive computing time and memory0

requirements. As a result, 6N = 9.0 A was chosen with a

step size h = 0.2 X so that N = 46, which is a manageable

number of simultaneous equations.

The overall procedure was tested on a similar equation

with a known solution, namely the ordinary G-P equation

r~ - (6.27)

Using the value y= 6.0443 K deduced from the numerical eval-

uation of K-P, 14 was regarded as the perturbed parameter.

The solution differed from the K-P result at most by three

parts in 104. This gives a high degree of confidence in the

basic numerical analysis and demonstrates that the step size

0.2 X does not cause substantial error.

77

In the solution of Eq. (F?), no is the perturbed

parameter in the form o\no for \= 0 to1 in steps of

0.2 or 0.25, depending on the rate of convergence. For each

value of ?\, about four or five iterations are required for

convergence of the simultaneous equations, i.j., when

successive solutions differ by less than one part in 105.

The norm in Eq. (F19) helps ensure that the resulting values

are indeed a solution of the set of equations. If the solu-

tions diverge so that the norm exceeds 5 '10-4, the increment

in '\ is halved to remain in the domain of convergence.

Figure 13 shows the solution of Eq. (6.20) for the case

of no = 0.1 n, using the kernel calculated from the T(r) in

Fig. 8 with a free-particle energy denominator. Curves

corresponding to several values of )A are displayed. It is

difficult to determine exactly which value of M gives the

best asymptotic behavior of f, but there is a slight in-

flection point for m = -0.175 K at (0 = 8.3 A, and a maximum

for = 0.0 K at = 8.4 A. The best solution seems to be

-0.1 K with an uncertainty of about *0.05 K.

For the case no= 0.024 n with the single-particle energy,

Fig. 14 shows curves corresponding to four values of ,A. A

maximum occurs at ( = 8.7 X for = 0.2 K, and an inflec-

tion point is located at = 8.2 X for f{= 0.0 K. Since

the curve for = 0.1 K (not shown) appears too steep near0

= 9 A, the best chemical potential is close to ), 0.15 K.

78

Figure 15 shows the results for the case no= 0.1 n,

using the kernel calculated from the T(r) in Fig. 9 with the

observed energy spectrum in its denominator. There is an

inflection point for J -0.52 K at P = 7.7 2, and amaximum for -0.45 K at P(= 7.4 X. The best solution

seems to be A = -0.48 K with an uncertainty of about

*0.03 K.

For the case no= 0.024 n with the experimental energy,

Fig. 16 displays curves corresponding to five values of )1.L.

A maximum occurs at = 8.8 X for = 0.1 K, and the curve

is too steep for ) = 0.0 K near (0 = 9 X to produce thecorrect asymptotic behavior. The best chemical potential

lies close to = 0.08 K.

These solutions are not very sensitive to the particular

choice of fN~0. 9 8 . For example, a value of fN= 0.99 raises

all the solution values in Fig. 16 past 5 2 by about 0.01,so . = 0.08 K still produces a reasonable solution,

although a slightly smaller J. may be preferable. A smaller

fN such as 0.95 would require a slightly larger value of }u.

Although there are no fundamental restrictions on how small

fN can be chosen, in practice the approximations for I((O) in

Eq. (6.25) are no longer valid if the curve does not approach

unity near 9 X, and either larger step sizes or more equations

would be required.

Unfortunately, no physical solution was obtained using

the kernel in Fig. 10 calculated from the bare potential due

79

to the large numbers involved. There was a solution for

small JA in which the curve is practically zero until a

sharp rise occurs near = 8 X, rapidly increasing to 'But in this case the integral of f K((, t') is approximately

constant for fO7 X and y can therefore be evaluated from

Eq. (6.26). Fetter's f(p) in Eq. (4.22) gives a value of

about 8 x 106 KA for the integral, or A 2 x 10 K, an

unphysically large value. Many of the kernel values are of

this order of magnitude due to the large contribution from

the potential near the origin, so the numbers involved in the

matrix algebra for solving simultaneous equations become too

large, causing overflow in the computer.

In the next chapter the significance and interpretation

of these results are presented, and in Chapter VIII they

are compared with previous calculations. A listing and

description of the computer programs used in the calculations

is given in Appendix G.

CHAPTER VII

RESULTS AND DISCUSSION

Chapter VI described in some detail the method of

solving the generalized G-P equation in Eq. (6.20). This

chapter summarizes and discusses the results of the T matrix

and the density amplitude f.

T Matrix

Equation (5.12) formally defines the T matrix. A local

approximation of this effective potential is necessary to

produce a tractable form of the generalized G-P equation for

numerical solution, and is given by Eq. (5.36). The integral

equation is solved by iteration, and the solution depends on

the chosen interatomic potential V(r) and the energy denomi-

nator E(k) of the kernel C(r,r') in Eq. (5.37). Each

calculation used the MDD2 potential given by Eq. (6.14).

Two energy denominators were used. One was the bare

single-particle energy minus the experimental chemical

potential,

EN)K_ ( 2k ,,%t G. K.(7.1)

The resulting T matrix in Fig. 8 is approximately constant

at 13.4 K from = 0 to 1 A, rises to a maximum of 43.35 K0

at 2.1 A, and falls rapidly to a minimum of -10.6 K at0

3.1 A. Beyond about 4 , T and V are practically identical,

80

81

as expected from Eq. (5.38).

The other energy denominator used was the experimental

energy spectrum approximated by Eq. (6.11). This choice is

perhaps more realistic than the bare energy, since particle

correlations are included. The energy denominator is auto-

matically dressed when the Bogoliubov transformation is

used84 to derive Eq. (5.33) for the local T matrix. Figure 9

shows the resulting curve, which is similar to Fig. 8 except

that for rg1.6 X it now turns upward to a maximum at the

origin of 60.2 K. The maximum near 2.2 X is 5 K lower andthe minimum at 3.1 A is 2.3 K lower than for the free-

particle denominator.

According to Eq. (5.15), T(r) is an effective potential

which takes into account short-range correlations between

particles. Beyond about 4 X, T(r)VV(r) and their zeros

coincide near 2.7 X. But for small r, T is much softer than

the bare potential, which climbs to the value V(0)= 2.25x 106

K. The ratio of T to V in Eq. (6.1) is g(0)= 5.95 10- for

the free-particle denominator and g(0) =2.67 x10-5 for the

phonon-roton spectrum.

Density Amplitude

The numerical results of the solution of the generalized

G-P equation are summarized in Table IV and the corresponding

curves are given in Figs. 17 and 18. In each case these

results are the solution for the best value -of the chemical

potential which gives the best asymptotic behavior. They all

82

have the same qualitative form characteristic of other mean-

field calculations,12,13,14 which rise linearly near the

origin and approach unity at large distances.

When the vortex density increases monotonically from

zero to its bulk value there is no clear-cut core radius.

The effective core radius is chosen here to be

(Q) =(7.2)

the distance at which the density is half the bulk value.

Figure 17 shows the results for the free-particle

energy denominator in Eq. (7.1). The solution for no= 0.1 n

corresponds to a chemical potential 4= -0.1 K, and a

core radius a = 4.1 A. This curve is slightly higher than the

other case, no= 0.024 n, where y = 0.15 K and a = 4.7 X.

These latter two values are the largest of the four cases

considered.

Similar results occurred for the phonon-roton energy

denominator. The larger condensate density no= 0.1 n pro-

duced the smallest core radius, a = 3.7 A, and the most

negative chemical potential, }A = -0.48 K.0 0

In summary, the core radii vary between 3.7 A and 4.7 A.

The chemical potentials are small in magnitude, but two are

negative, which is characteristic of a bound system of parti-

cles. The next chapter compares these results with previous

calculations.

CHAPTER VIII

COMPARISON WITH OTHER WORK

Results of the calculations discussed in the previous

chapter can perhaps be best analyzed when compared with

previous work in the literature. There has been an earlier

calculation by a different method of the spatial variation

of the T matrix. Several calculations of the vottex

density profile have previously been made. These calculatiorB

are discussed in this chapter.

Density Profile

Previous calculations of the density profile of an

isolated vortex line have been based on the approaches

described in Chapter IV. Mean-field theories have used the

G-P equation, and calculations1 4 '74 based on it have been

limited to an approximation of the two-body potential by a

'-function, V(r) = V S(r). V0 is an adjustable parameter

used to fit the core radius. The solutions obtained are

qualitatively similar to the solution of the generalized G-P

equation obtained here. The most detailed evaluation using

the &-function approximation is the K-P solution,14 shown

in Fig. 20 as the dashed curve for comparison with the solid

curve, which is the upper solution in Fig. 18.

A second approach, used by Chester, Metz, and Reatto,15

is the method of correlated wave functions. Their model

83

84

wave functions have built-in short-range correlations and

vanish at the vortex core. An approximate expression in

Eq. (4.42) for the number density involves a parameterized

function g determined by energy minimization, and the

direct correlation function Co obtained from the experimental

liquid structure factor. The resulting relative density

amplitude f= n* is shown by the dotted line in Fig. 20.

It is qualitatively different from the mean-field results,

since it has density oscillations similar to the pair

correlation function, which is the Fourier transform of the

liquid structure factor S0 (k).

The condensate density, however, need not resemble the

pair correlation function; the single-particle density does

not directly describe interparticle correlations. The ther-

mal deBroglie wavelength of helium atoms at temperature T is

T ZkT6 (8.1)

and represents the quantum-mechanical uncertainty in

position. When the temperature is e0.7 K the wavelength

is Z10 1, which greatly exceeds the average interparticlespacing of 3.7 X. In this case the condensate is a continuum

and the density would be constant in the absence of a vortex

line. Thus, a core radius ^-1 Ais plausible near absolute

zero. It is also difficult to understand what would sustain

density maxima and minima near the core under these circum-

stances.

85

Core Radius

No experiment has directly determined the core radius

of an isolated vortex line. Experimental values are obtained

from the classical hydrodynamical analysis of the experimen-

tal energy y!. speed of vortex rings with some specific

model of the core structure. Rayfield and Reif1 0 found the

core radius a = 1.28 * 0.13 A for solid-body rotation withinthe core and a = 1.00 * 0.10 A for a hollow core. When

Roberts and Donnelly65 adjusted the hydrodynamical equations

to describe classical quasiparticle behavior as discussed

in Chapter IV, they found a = 1.28 * 0.13 A for a hollowvortex core. Later experiments analyzed from the quasi-

particle point of view give a = 1.29 * 0.05 A at 0.35 K, but

the results can be extrapolated to a = 1.14 t 0.05 A at0.0 K.

Theoretical estimates of the core radius give various

values. Amit and Gross68 used a semiphenomenological

quantum variational approach to find the bulk density

established at a = 1.9 A from Eq. (3.26) for vortex lines.

This implies that the half-density point occurs at 0.78 A.Fetter's density profile in Eq. (4.22) is not very useful

for determining the core radius since ?\ is related to thedeBroglie wavelength in Eq. (4.19), which in turn depends on

the adjustable parameter V . Indeed, an experimental value

of a is often used in Eq. (4.22) to obtain the density

profile, and to calculate V0 from Eq. (4.19). The density

86

profile is often shown as a dimensionless ratio r/a. The

half-density point of Fig. 3 for the K-P solution of the

ordinary G-P equation gives the core radius a = 1.6 X.When the same criterion is applied to the density profile

obtained by CMR15 in Fig. 4, the core radius is about

0.5 A. For another trial function the core radius is about

0.8 1.

Core radii calculated from the generalized G-P equation

in Eq. (6.20) and listed in Table IV are larger than those0

obtained by other methods. In particular, the values 3.7 A

to 4.7 A are over three times the zero-temperature value11

1.14 * 0.05 A.

Chemical Potential

An incidental result from calculating the density

amplitude is the determination of the chemical potential .

The asymptotic expression in Eq. (6.26) should give. A, but

in practice it does not converge sufficiently rapidly. The

kernel in Eq. (6.26) is known only through ( = 9 A, and

when the integral is plotted as a function of p , the curve

does not level off quickly enough to extrapolate it to an

asymptotic value.

Table IV shows that two values of p which lead to a

physical profile are negative, characteristic of a bound

system, and each corresponds to a condensate fraction of 0.1.

The chemical potential obtained from the G-P equation by

87

Eq. (4.18) are always positive, which is characteristic of an

unbound syatem of particles.

Equation (4.18) for > is a direct consequence of the

G-P equation in Eq. (4.15). Since Vo is an adjustable param-

eter, a criterion must be found for its determination. One

possibility is to use the long-wavelength limit of the Bogo-

liubov energy spectrum in Eq. (2.6). In this case the speed

of sound u is given by95

{= (YVV/j' _ /(8.2)

Using the values for u and M in Table I, Eq. (8.2) gives

JA= 27.3 K. This gives a core radius of 0.47 A from

Eq. (4.19). Another possibility is to solve Eq. (4.19) for

JA= n0V0 and substitute values for the healing length which

have been obtained elsewhere. For example, if a is equal to

1.28 A (Rayfield and Reif10), 1.13 A (Hess and Fairbank ),

or 1.00 X (Kawatra nad Pathria 4), the corresponding values

of p' are 3.7 K, 4.8 K, and 6.04 K, respectively.

The chemical potentials determined in this work are

small in magnitude, but in some cases are negative. On the

other hand, the ordinary G-P equation always gives positive

values, indicative of an unbound system. However, a direct

calculation of the binding energy per particle d by

%stgaard gives -4.0 to -5.0 K, which is comparable to the

experimental value 2 9 >,= -6.7 K.

88

$stgaard's T Matrix

The only previous calculation of the T matrix in

configuration space for He is that of %stgaard.83 He used

a nonlocal, momentum-dependent T matrix based on a modified

Brueckner theory80 and the reference spectrum method. This

theory is discussed in more detail in Appendix H. Figure 19

shows the results of his calculations for two values of the

final-state relative momentum q. Curve A corresponds to

q 0.4 X and curve B corresponds to q 1.0 X . This

momentum dependence results from a partial wave expansion of

the L-S equation and the nonlocality of the T matrix.

Although different approaches and approximations were

used, the agreement between the T matrices of this work and

%stgaard83 is remarkable. The main qualitative differences

occur near the origin, where $stgaard extrapolated his curve,

assuming that it continues decreasing linearly towards the

origin. Beyond about 1.5 X, Figs. 8 and 19 are quantitatively

similar, indicative perhaps of their similar energy denomi-

nators, given in Eqs. (7.1) and (H2), respectively.

CHAPTER IX

CONCLUSION

The density profile of an isolated quantum vortex line

in He II has been determined by a numerical solution of the

generalized Oross-Pitaevskii equation in Eq. (5.42).

Results of the calculations for the density amplitude f(p)

are shown in Figs. 17 and 18, and numerical values of the

core radius and chemical potential are listed in Table IV.

Previous calculations of He II vortex properties based

on a mean-field approach 4 have been limited to the simple

matrix element factorization in Eq. (5.9) and approximation

of the potential by a I-function. A more general equation

is obtained in Chapter V which takes multiple particle

scattering into account by summing the class of diagrams in

Fig. 6. The result is to replace the interatomic potential

V by a T matrix, which is an effective potential to be used

with an uncorrelated Hartree product wave function.

Both a free-particle energy and the experimental energy

spectrum of He II are used in conjunction with the MDD2

potential to calculate local T matrices which agree quali-

tatively with a previous nonlocal calculation.

The generalized G-P equation is then solved with two-

point boundary conditions. The conditions are f(O) = 0 since

the core must be hollow at the origin, and f(9 X) = 0.98,

89

90

where the density should be close to its bulk value. A value

for the condensate density must also be chosen. Both an

experimental and theoretical value were used, 0.024 n and

0.10 n, respectively, where n is the bulk He II number density.

The solution has the proper physical behavior of

approaching unity asymptotically only for certain values of

the chemical potential. Table IV shows that the chemical

potentials are small, and in two cases are negative, which is

characteristic of bound-particle systems. This result is an

improvement over the larger positive values due to the

repulsive g-function approximation of the ordinary G-P

equation.

The resulting curves are physically reasonable in that

they increase monotonically from zero to a constant bulk

value far from the core. The core radius is chosen to be

the distance from the origin at which the density f2 is

half of its bulk value. The results given in Table IV are

over three times larger than the experimental value of

1.14 A at 0.0 K. It should be noted, however, that the

latter value is based on classical theory, assuming a

hollow core, and is not obtained directly by actual

measurement.

The calculations in this work have demonstrated the

feasibility of using a generalized G-P equation with a

realsitic interatomic potential to study quantum vortices

in He II. Amit and Gross68 claimed that the calculation

91

would be "hopelessly complicated" even for the simplest

two-body potential with finite extension. The generalized

G-P equation includes particle correlations in a mean-field

theory. Chester, Metz, and Reatto15 criticized the ordinary

G-P equation for not taking these particle correlations

into account. The density profile was obtained without

assuming , priori a core structure as is often done in

both theory and experiment.10,68

Determination of the local T matrix is valuable for

problems such as a microscopic theory9 6 of the He II

excitation spectrum. Quite good agreement is obtained

between the T matrix calculated here and that of %stgaard,8

who calculated a nonlocal T matrix by a different method.

Since the T matrix is an effective interaction which

allows particle correlations to be taken into account for

use with an unperturbed wave function, T(r) should be much

softer than the bare potential V(r) for small r, and

approach V(r) for large r. This behavior is shown in Figs.

8 and 9.

Although the core radii found here are over three times

larger than experimental values, it is difficult to make a

valid comparison between them, since interpretation of

experimental results is made through a classical vortex

theory, which is incapable of giving the core structure. Somemodel for the core must be assumed, so the resulting radius

is the value for that particular model, not necessarily for

92

the actual vortex. A possible extension of this work is to

explore the relation between theory and experiment more care-

fully. In particular, if the density profile calculated

here were assumed for the core structure instead of the more

artificial models previously used, would the energy vs.

velocity curve for vortex rings1 0 be reproduced? If not,

an approximate form such as Eq. (4.22) could be used with an

adjustable core radius to fit experiment and perhaps obtain

a more meaningful core radius value.

Inclusion of higher-order diagrams other than the

ladder approximation in Fig. 6 for the correlation function

in Eq. (5.8) could lead to more accurate results. If only

the bare potential, the first term in Fig. 7, is used, no

physical solution of the generalized G-P equation was

obtained. Additional terms besides the T matrix approxi-

mation might provide further improvement.

Only the condensate is taken into account here,

although Fetter97 has demonstrated that for a weakly-

interacting inhomogeneous boson system, the core contains

pairs of noncondensate particles with zero total angular

momentum. The noncondensate density in the core is 1.4

times the corresponding bulk density.98 Fetter48 believes

the same behavior should exist in He II. Since experiment"

indicates that the noncondensate fraction is 0.976, the

total core density would then exceed the bulk value.

93

Fetter98 proposes a sophisticated trial wave function to

account for this, but an alternative is to use the pair

and single-particle condensate Hartree-Fock-Bogoliubov

theory99 with a realistic potential or a T matrix similar

to this work.

Besides trying to improve the results presented here,

a possible extension of this work is the calculation of

the dependence of the vortex core radius on temperature

and pressure, which has been measured experimentally.11

These factors also influence the condensate density, but the

dependence has not been measured. The local T matrix could

be calculated with the temperature-dependent phonon-roton

spectrum as the energy denominator and the factor

coth(Ep/2kBT) in the integrand of Eq. (5.37). But due to

difficulties in correctly calculating the temperature depen-

dence96 of the excitation spectrum and the large core radii

obtained here, this extension does not appear to be fruitful

at this time.

The calculations presented here make a significant

contribution to the study of vortex lines by demonstrating

that a first-principles calculation of vortex core structure

using the generalized G-P equation is feasible.

APPENDIX A

SYMBOLS USED

The symbols used in this paper are presented below in

order of-appearance to aid the reader. Each symbol is

defined and the equation in which it first appears is given.

TABLE V

SYMBOLS USED

Symbol Definition EquationNumber

f Bulk number density (2.1)

Chemical potential (2.2)

E Energy of a boson gas (2.2)

N Particle number (2.2)

Volume (2.2)

Density of He II (2.3)

Density of normal component (2.3)P Density of superfluid component (2.3)

k Wave number (2.4)

Mass of a helium atom (2.4)

5(k Liquid structure factor (2.4)

k Fourier component of V (2.6)

94

95

TABLE V -- Continugd

Symbol Definition EquationNumber

\s Superfluid velocity (3.1)

Z Height of free surface (3.2)

h(r) Classical free surface shape (3.2)

L1 Angular velocity (3.3)

r Radial distance (3.3)Acceleration due to gravity (3.3)

C Proportionality constant (3.4)

Unit vector in 9 direction (3.4)

Macroscopic wave function (3.5)

S Velocity potential (3.5)

Particle current density (3.7)

K Circulation (3.9)

h An integer (3.10)

Unit vector in z direction (3.12)

Vortex line density (3.13)

L Distance between grids (3.14)

V Speed of ions (3.14)

Modulation frequency (3.14)

R Radius of a vortex ring (3.15)

Y) Symbol for compound term (3.15)

d, Model parameter for E (3.15)

(3 Model parameter for v (3.16)

Core size (3.17)

96

TABLE V -- Continued

Symbol Definition EquationNumber

Classical impulse (3.18)

Po Bulk density of He II (3.2)

V0 Strength of S-function potential (3.23)

1 Hamiltonian (4.1)

X Coordinate and time (1,t) (4.1)

(4 Field annihilation operator (4.1)

Field creation operator (4.1)

Dirac <-function (4.2)

((r1-) Order parameter (4.5)

14N N-particle wave function (4.5)

6(it ) Reduced density matrix (4.8)

7 Operator correction term (4.10)

Yo Condensate number density (4.12)

J Azimuthal quantum number (4.12)

5 Radial density amplitude (4.12)

Q(x Bessel function of order.,J (4.16)

Numerical coefficient (4.20)

Variational parameter (4.22)

& Energy per unit length (4.23)

u. Speed of sound in He II (4.29)

LPCMR Model wave function of CMR (4.33)

Exact ground-state wave function (4.33)

F Ratio of 4g to o (4.33)

97

TABLE V -- Continued

Symbol Definition EquationNumber

I Kinetic energy operator (4.36)

H' Excitation energy operator (4.37)

() External potential (4.44)

Boltzmann's constant (4.44)

- Effective temperature (4.44)

CO Direct correlation function (4.45)

Bulk number density (4.45)

(K Energy operator T -AN (5.1)

T T matrix (5.12)

GQ Free propagator (5.11)

V Wave operator (5.13)

Unperturbed wave function (5.14)

UY, Eigenfunction (5.20)

O Step function (5.20)

C, Eigenvalue (5.20)

, Angular frequency (5.30)

C(T) Kernel for T (5.36)

E, Quasiparticle energy (5.37)4() Radial relative density amplitude (5.40)

Bulk condensate density (5.40)

h2/2M (5.42)

U General effective potential (5.43)

Kernel for G-P equation (5.44)

98

TABLE V --Continued

Symbol Definition EquationNumber

Ratio of T to V (6.1)

Wavenumber parameter (6.4)

Greater (lesser) of r,r' (6.6)

x Dimensionless momentum term (6.10)

q Adjustable parameter (6.10)

Dressed free-particle energy (6.21)

b Adjustable parameter (6.11)

(x) Zero-order spherical Besselfunction (6.11)

A Adjustable parameter (6.12)

Step size (6.16)

T(rg) An integral over k (6.17)Dimensionless variable (6.22)

W) Modified Bessel function (6.24)

APPENDIX B

THE ANALYSIS OF VORTEX RINGS

This appendix considers two aspects of classical vortex

ring theory. First, the energy of a ring with solid-body

rotation in the core is obtained in a manner similar to

Lamb.63 The ring velocity is briefly discussed. Then the

applications of these results to determine the core radius

and circulation is presented.

Energy and Velocity of Vortex Rings

Lamb's classical hydrodynamical theory63 of vortices

is used here to find expressions for the energy and velocity

of vortex rings. The kinetic energy of a vortex ring is

based on a calculation of the stream function

where the flux is the rate of flow across an open surface

S, defined by

d? (B2)

where c is an element of area. By considering the flux

through an annular region centered on the z axis of a

cylindrical coordinate system, the axial velocity is

A P ' (B3)

99

100

A band encircling the axis gives the radial velocity

~(B4)

and there is no azimuthal Velocity component for a ring

centered on the axis.

Vorticity (3 is defined as the curl of the velocity,

( ) \T (B5)

so that the circulation 1K can be expressed by

(B6)

using Stoke's theorem with the contour bounding the

surface I.

The velocity can also be expressed as the curl of a

vector S,

V XS (B7)

for which the divergence vanishes by choice of gauge. In this

case the vorticity becomes

(B8)

which has the solution

101

V IJ6 3

(B9)

The stream function can now be found using Stoke's theorem,

C7 c -6)W -

(310)

where is the radial distance from the axis.

Using

Z. (2-&fl(B1l)

and

1, _ ( '+ P ,(B12)

we can perform the angular integration in Eq. (B9) in

cylindrical coordiantes to obtain

I0( (F CosV d

(B13)

where F(C\) and G( \) are elliptic integrals and

(B14)

S~r = 4n)

ZP . i2Tf

00e 0 (0r3V- ("a) - (,60

~ > *09.

102

For a large vortex ring of radius R, r, --2R >>r , so that

^) 1. In this case the elliptic functions are approximately

FzI((B15)

and

-tT(B16)

Assuming that the vorticity 0 is approximately constant in a

core centered at CZR, Eq. (B13) becomes

EL[)l ' L Ji(fR/rJ>IL (B17)

This integral can be evaluated as

(B18)

where

(B1 9)

The energy can now be found from

J \ + T >

= 1 - S d L(B20)

using Eqs. (B3) and (B4), where is the fluid density.

Integrating Eq. (B20) by parts, we obtain

103

(B21)

If cN is approximately constant, Eq. (B6) is

k L a, A(B22)

so the kinetic energy can be written as

T = Jg /2, 2aR (S-//2s

(B23)

The velocity of a vortex ring is obtained by defining

the mean position as

(B24)

and radius

(B25)

for a collection of circular filaments into which the ring

is divided. The time derivative of zo can be expressed in

terms of T, and the result is

(B26)

104

Determination of the Core Radius and Circulation

Rayfield and Reiff0 first used Eqs. (B23) and (B26) to

analyze experimental data on vortex rings. If the energy E

and the velocity v are measured, the core radius a and circu-

lation y, can be solved for graphically. For simplicity,

define

(B27)

and

(B28)

Then the product of E and v is

B ~ (B29)

since g9~10 for large rings.10 Equation (B29) is now solved

for 9 and substituted into its definition in Eq. (B27), with R

written in terms of E through Eq. (B23) to obtain

(B3L)

C is approximately a constant compared with the other two

terms.

105

A graph of (Ev)e _v$. In E should be a straight line

if the approximations are valid, with slope Bi and

y-intercept B (C - ln a). Hence, the circulation ( can be

computed from the slope through Eq. (B28) and the core

radius a is obtained from the intercept.

APPENDIX C

DERIVATION OF THE INTEGRAL EQUATION FOR THE T MATRIX

The equation for the T matrix in momentum space is

~k) it\v(k) - 4 N/(7g) T4)/ 2 E~p>(Cl)

where the energy denominator in Eq. (5.33) may be dressed.

An equation for T(r) is obtained via the Fourier transform

in Eq. (5.35). The result is

E

Letting

(03)

Eq. (C2) can be expressed as

The transform of V in Eq. (5.34) is now employed to give

T1r)?r 7(r) {1I,?Q T~p/ZE]CS* (C5)

Using the inverse of Eq. (5.35), the expression for

T becomes

106

107

(06)

The angular integration can be performed if we recall that

3ae Scb 2eiPt' in0 co

Zm fSl f, dA.r(07)

~-sTmcoeeIf the sum is converted into an integral according to

Eq(2rr3 (0) ecme

Eq. (C6) becomes

T(r) = V)

where the kernel is

CF 5 Yr Inp

which are given in Eqs.

(C8)

(C9)

(do)

e, LT(rl) d'r) ) 2E,

r #'o

U C(rr'YT(r') cr]0

(5.36) and (5.37).

APPENDIX D

NUMERICAL SOLUTION OF T(r)

An integral equation for the T matrix is derived in

Chapter V, and Eq. (5.36) can be expressed as

CYO; - C(r ( r)\/((') dr (D1)

where C(rr') is the kernel in Eq. (5.37), which is tabulated

for various values of r and r'. The potential V(r) is

given in Eq. (6.14), and g(r) is the ratio T(r)/V(r). The

solution of Eq. (D1) is based on the Fredholm procedure,89

in which the integral is approximated by a finite sum. A

system of simultaneous linear algebraic equations is obtained

which can be solved by matrix methods.

The integral in Eq. (D1) is approximated by a sum of

N terms, which are the integrand values at equally-spaced

intervals between zero and an appropriate upper limit L.

The step size for both r and r' is then

K-LN. (D2)

If the integrals I(r,r') in the kernel have been evaluated

according to Eq. (6.18), the equation for g can be written as

108

109

which is further approximated by

(D4)

The function F is

F~rY> -I Cr r') Vr / r. (D5)

The set of N simultaneous equations in Eq. (D4) is

linear in the set of unknowns ig(ri)} , and becomes an exact

relationship in the limit N-" . A more suggestive form of

Eq. (D4) is

(D6)

where the primed summation denotes i' J. In order to avoid

a zero denominator (r1 = 0), the terms with i= 1 have been

removed. Hence, the value g(0) must be extrapolated from

the solution for nonzero arguments.

The set of equations in Eq. (D6) is solved by a sub-

program100 SIMUL which solves simultaneous linear equationsby using a maximum pivot strategy in Gauss-Jordan complete

elimination. Numerical solutions of systems of linear

equations for large N may not be accurate due to increased

round-off and truncation errors. Therefore Eq. (D6) is

110

solved for the modest value N= 20. The calculation is thenrepeated for successively larger values of N= 33, 50, and

100. The solutions for all cases are well-defined forN = 100 and does not change appreciably with increasing N.

APPENDIX E

SERIES EVALUATION OF K(f,C' ) FOR LARGE I r--)Evaluation of the kernel K in Eq. (6.19) is possible

for the case T(r)~V(r) for large r by a series expansion.

The relative coordiante r is, in cylindrical coordinates,

(E1)

When 1 ']l ? 3.7 A, no combination of 9 and z will produce

r C3.7 A, so V(r) consists only of the 6-8 attractive tail

region in Eq. (6.14) throughout the integration.

The potential for 1-') > 3.7 A has the form

(E2)

so the kernel can be expressed as

(E3)

111

112

where

a(= ct+ 2 z

S 2('(E4)

Now consider the integral

(E5)

where c is an integer. Using the substitution

X = Cos( (E6)

in Eq. (E5), we obtain

=5' dx(c.HXZ)~ i (4' (t,4x ct

(E7)

If

c(E8)

the integral becomes

Ic = 1 of l ( d -. 6>(05<8)~C-

=ca/d<

113

Ct)

Q _x2

I I/ xl ~'7

(E9)

where the binomial coefficient is

(ElO)

Since odd values of n in Eq. (E9) will cause cancellation in

Eq. (E7), the remaining integrals have the value 10 1

cSoX X2 fl ~(E11)

-T (2Z~)

Finally, Eq. (E7) becomes

Q (C-p L(2i P. (E12)

The z integration involves only the last factor ofEq. (E12) in the form

C c) -> (E13)

which can be evaluated from the relationship102

- - ) (23 -'! (2)Vat5(E14)

Cti)' E',)

Cn)

where, for the present case, j =«2n + c - 1. Then

sd i (ne 2rat3)'! / )7i 2nc ZI .'-t-C/)1

so the double integration is

1o 2c-3

;Ito 1(2y, 2c

[ 'f"

Finally, Eq. (E3) becomes, for c=3,4 above,

1<5 (c )

which is used in Eq. (6.25) for f'>13 X.

114

(E15)

(E16)

(E17)

Fcd

[[2t>!0 2)2msA (1+3)9- e

APPENDIX F

NUMERICAL METHODS FOR SOLVING THE GENERALIZED

GROSS-PITAEVSKII EQUATION

The generalized G-P equation derived in Chapter V is

z

c! ~ 'P a K(P'f) r9d 'l fr-= O. (Fi )

In order to obtain a computer solution of this equation, itmust be written in discrete form. Then a useful algorithm

must be adopted for the iterative solution procedure. Eachis discussed in this appendix.

Discrete Form

A solution of Eq. (Fl) is to be found between the points0and at discrete intervals of size

h N e(N-1). (F2)

The function f evaluated at Pi is denoted by

i= (e) >(F3)

its first derivative is approximated by

115

116

and its second derivative is approximately

P +--/i (P5)

The integral in Eq. (Fl) is computed by Simpson's

six-point rule,103 so N should be of the form

N =5n +1 , n 1,2,3,... . (F6)

One difficulty with the integral is the infinite upper limit,

which means that f(P) should be known for all p 0. However,

in practice some cutoff value (J, must be used, assuming that

the integrand makes a negligible or calculable contribution

beyond e .Equation (Fl) can be written in the discrete form

(of.,-l+ +/hz ((1.3/-z + _ -

N-1, (F7)

where I( P,) is the approximate value of the integral beyond

N , and the remaining terms in the large parentheses comefrom the integration formula.1 03 The function g is defined by

(F8)

117

Since a two-point boundary condition is used for Eq. (Fl),

f1 and fN must be given. Equation (F7) is a set of N- 2

nonlinear algebraic equations to be solved simultaneously

for the set of values {fi

Method of Perturbed Parameters

Solutions of systems of nonlinear equations usually

require iterative techniques. An extension of the Jacobi

or Gauss-Seidel methods104 for linear equations is useful

because each step is regarded as a new approximation which

does not inherit round-off errors from previous steps. The

relative simplicity of this approach is often paid for by its

slow convergence. Indeed, it may even diverge if the initial

approximation is not sufficiently close to the actual solu-

tion. In the present problem these methods were unstable.

There was also no apparent advantage in eliminating the first

derivative by transformation or expressing the problem as

two simultaneous first-order equations.

One procedure which minimizes the dependence on the

initial trial function is the method of perturbed parameters.5

This approach begins with the known solution of a similar set

of equations and slowly "deforms" them into the desired

system of equations. In general, consider the set of

equations

(F9)

118

The domain of convergence by traditional methods is inversely

proportional to N and the degree of the equations. 94 It is

assumed that each function Fi contains j terms in the form

L ~Ckqy(S 0 (oo

where each C>ik is a product of the f's: f ,.fifj,

f ifjfk'...for ij,k,,...m 1,2, ... ,N, and cik is the

corresponding coefficient.

Now a different set of coefficients ci 5is chosen by

inspection such that the set of equations

k=)(Fii)

has a known solution f . Then f is the initial trialsolution of {F1j in Eq. (F10). Since there is no guarantee

that f(O lies in the domain of convergence of JFi} , the

coefficients ci* are changed slightly to form a new set of

equations F" whose convergence domain includes * If

the solution of F diverges, the new coefficients cc"ik

must be chosen closer to c *. If the iterative solution

converges, a new set of coefficients is chosen and the process

is repeated. The coefficients are successively chosen closer

and closer to {cik} . For example, at step p one solves

(F12)

119

where

( (F13)

and ), is a fraction which increases at each step, beginning

with zero. As )-+1 the coefficients cS-cik and the

solutions f -+ f if each step keeps the previous solution

within the domain of convergence.

Newton - Raphson Iteration

Each set of equations in Eq. (F12) is solved by the

Newton - Raphson iterative procedure. If the temporary

solution after the rth iteration is denoted by f'r , the

solution at step p is

(F14)

where 8 is small. Then each function can be expanded as

(00 j * O ,(F15)

If the partial derivatives are denoted by

Sa (PA ) (F16)

Eq. (F15) can be expressed in the matrix form

(F17)

120

Equation (Fl?) is solved for by the same subprogram

SIMUL described in Appendix D for the T matrix. The solution

is used to determine

(F18)

which is the new approximate solution for the next iteration.

A solution is found when this procedure is performed R times

until f C fJR,) within a preset tolerance, such as fivesignificant figures in the present calculations.

Convergence is monitored throughout the calculations by

checking whether the norm

o ' (F19)

is less than a given tolerance level, such as 0.0005.If F increases, the solution is diverging, but if it decreasesto a small value and then increases, numerical round-off

errors are becoming important and the iteration should beterminated.

A listing and description of the computer programs usedto perform the calculations discussed here are found inAppendix G.

APPENDIX G

COMPUTER PROGRAMS

This appendix contains the listings of the three

principal computer programs which were used in solution of

the generalized G-P equation. The first program is TMAT12,

which is the calculation of the T matrix. Next is MATINT,

the evaluation of the kernel in Eq. (5.43). The actual

solution of the generalized G-P equation is performed by

GPIT14. Each of these programs is discussed, along with any

functions or subroutines required.

Each program was written in FORTRAN IV language and

run on an IBM Model 360/50 computer. A copy of the punched

card decks is available on request.

Program TMAT12

TMATI2 is used to evaluate the T matrix from the

integral equation in Eq. (5.36). The integral in Eq. (5.37)has been previously computed as a matrix of values, and is

read into memory as external data. The values are stored

in the single-subscripted variable C according to the pre-

scription C(ri,r ).+C([i(i-1)/2] +j).The preliminary values include the upper limit U of the

integral and the step size H. Also read in as data are thevalues of the MDD2 potential V at the positions R(I)= rg.After the array C has been read, the values corresponding to

121

122

r1 = 0 must be eliminated because of the denominator in

Eq. (5.37), as indicated in Eq. (D6).

A coefficient matrix D is then built up, which

contains the coefficients of the set of simultaneous linear

equations in Eq. (D6). The set of equations is solved by

the function SIMUL, and the resulting values are transmitted

back to the main program through the COMMON statement. It

should be noted that what is obtained by this procedure is

not T(r), but rather g(r)= T(r)/v(r).

Li Pr&oram

A listing of the main program follows:

C 4 a* 4 * *PROGRAM TMAT1? 4 +C

IMPLICIT REAL48(A-HO-Z)

CoPMON Q(1CO,11)G(100)-~t DESIGN C(5151),V(100)1R( 100),Z(300)

CC + 4 44 PRELIMINARY VALUES 4C

ToL=1.OE-19Pi3.1415926536

1 KEAD(5,100) U H100 Fo MAT(2F1o.5)

t U/HF,0N+

RIT(E,106) UHN106 F0oR dAT(' U='F10.3,i10Xj H= ,RF1O 3,s1Xat NJI3///)

CC 4* 4 4 4* VALUES OF R AND V 44C

00 2 I=1,N2 K AD(5,103) (I)V(I)

103 oRMAT (F10. 3, E1.8)CC 4 4 * EVALUATION OF K-INTEGRAL * 4 4 4 4 4CC *SPECIAL CAf3E OF E(K)EXPERIMENTAL : y

123

00 4 I=11,101J=T*( I 1 )/2+1

K=I4(I+1)/24 FE,((5,104; (C(M),M=JK)

104 FORMAT(5FE15.8)C 4 * * * * * ELIMINATE R=O INTEGRALS 4 4 4 4 4 4

00 3 I=1,10000 3 J=11I,

3 C(I*(I-1)/2+dj)=C(14(I+1)/2+J+1)

CC 4 4 4 4 * COMPUTE ELEMENTS OF COEFFICIENT MATRIXC

20 DO 9 I1=1,N00 12 J=1,NIF(I.Et'>J) GO TO 7IF(J.GT*I) GC TO 8K=J+I*(I-1 )/2

0(I ,J )=H*R(J) V(J)*C(K)/(PI4R(I))G0 TO 12

7 K=4( I+1)/2D(I ,J )=I.+h4V(I)4C(K)/PIGO TO 12

8 K=I+J4(J-1)/20(I J )=H*R(J)*V(J)*C(K)/(PIR(I))

12 CONTINUE

9 CortTINUE

CC 4 * 4 4 4 4 SOLUTION AND OUTPUT 4 4 4 4 4 4C

DET=3ItUL (N TOL, I, NP1)WRITE(6,101) HN 'UiTOL

101 F0pMAT( 5XF4210X,I3, 0XF10.3,1c0XF10*7)DO 11 NI=1N

13 P(I)=R(I )TM(I)=f.(I)*V(I)

11 wKTL(&1 102) P( I )1G( I ),TM( I)V(I)102 FPM AT(F7.2,3( 0XE15.8) )

VRITE(7,105) (T;1(I)I=i;N)105 FORMAT(5E15.8)

NP1= N+1Do 2 I=1,N

J=N P1 "IP(.,+1)=P(J)

25 TM(J+1)=TM(J)P( 1)= 0 + 0TM(1)=TM(2)CALL PPLOT ( P, T1 10110, 6 )

99 STOPEND

124

_Funtion SIMUL

The subprogram SIMUL95 solves a set of simultaneous

linear equations by using a maximum pivot strategy in Gauss-

Jordan complete elimination. The array A contains the

coefficients of the equations, and the solution values

are stored in X, both of which are placed in COMMON with

the main program.

FUNCTION SIMUL(NEPS, INDNRC)LI ENSION IRO(1Q1)*JCOL(1o1),JO r)(101),Y(O1 )

C COMMON A(100,101),X(100)COrMON D ((131 1311311135)>A()3, R1) (R 131 ),APP (131 ),X( 131

CG(a.

M A'

IF( IND.GE.O) MAX=N+1

CC # * 4 EGIN ELIMINATION PROCEDURE 4*

C5 DETER=1

DO 18 K=1,N

K MI=K -1.

PIVOT=O000 11 I=1,N00 11 d= 1N

CC 4 4 4 LOCATE PIVOT ELEMENT # 4 4 4 4 4C

IFHK.E+1) GO TO 66o 8 JSCAN=1, KM1

IF(J.EQJCUL(JSCAN)) GO TO 118 CON'JTINUE

L0 6 I SCAN=1,KM1TF (I.E .IRW (ISCAN)) GO TO 11

6 CO'T1'NUE66 CO[TINLE

9 IF( AL-(A(IJ)).LE. AE3S(PIVOT)) GO TO11>IVO T=A (Ii J)I1F08(K)=IJCCL ( K )=J

11 CO TINUEIF,( A S(PIV0T).cGT.EPS) GO TO 13

SI "UL=oe0

RETURN

125

CC * UPDATE DETERMINANT VALUE AND NORMALIZEC

13 IRCWK=IROW(K)JCOLK=-COL(K)IF(DETER.LT.1.QE30) GO TO32

G=1.GC TO 33

32 DETERPLE TER*PIVOT33 DO 14 J=1, MAX14 A(IROWKJ)=A(IROWK:J)/PIVOT

CC * CARRY OUT ELIMINATION AND DEVELOP INVC

A(IROWKJCCLK)=1./PIVOT00 18 I=1,N

AIJCK=A( IJCOLK)IF(I.E> .IRO K)-GO TO 18

A ( IJCCLK )=-AIJCK/PIVOTr0 17 J=1, MAX

17 IF(JNE. JCOLK) A(I;J)=A(IJ)-AI JCK*A(IROWK, J)1 CONTINUE

CC -* + ORDER SOLUTION VALUES AND ADJUST SIGNC

DO 20 I=11NIqWI=IRO ( I)JCE LI=JCOL ( I)J r D( IRO I) =JCOLI

20 IF(UINDGEpc) X (JCOLI)=A( TROWIs MAX)IN.TCH:=0

NM =N 1Do 22 I=1,NM1IP1t=I+1CR 22 J=IP1.NI JOR (J). GE jUORD(I)) CO TOP2?

JT&P= FjOf-0 ( J)IJC! ; ( J) =JOKE (I)

I =jTEMPT \ (CHi=;NTCH+1

22 CONTINtuEIF IN'4CH/ 2.NE * INTCH ) ETER-DETEP

CC - UNSCRAMBLE INVERSE MATRIX IF CALLED FOC

24 IF(INDLE.c) O TO 26Sri3UL=f)ETERRET U RN

126

26 DO 28 J=1,NDO 27 1=1sNIpn! I=IROW( I)JCOLI=JCOL( I)

27 Y(JCCLI)=A(IR0WIiJ)DO 28 I=1,N

23 A(IsJ)=Y(I)

Do 30 I=1,NDo 29 J=1,NIRCW j=IROW (J)J CCL J=JCOL ( J)

29 Y(IROJ)=A(I,JCOLJ)Do 30 J=1,N

30 A(IJ)=Y(J)SIr-UL=METERIF(G.E~C1.) PRINT 100

100 FOfMAT ('ODETER GREATER THAN 10**30. STEP 45 SKIPP1ERFLOW. SItlUL INCORRECT.'//)

RETURNEND

Program MATINT

MATINT performs a double integration of the T matrix

over the cylindrical variables z and. - , given in Eq. (5.43).

Values of the T matrix obtained from program TMAT12 are read

in as data. The integrations are performed by the function

DUBINT, and stored in the array VAL; in terms of Eq. (5.43),

VAL(IR,IRP) = K(P, p ).

m.in rr Pr l

The main program initializes values and reads in

relevant data. It also starts the double integration

process, for f0 and ' at intervals of 0.2 A.

127

C PROGRAM MATINTCOrMON T( 155)X(151 ),TSPRD IMENSION V A L( 151,151 ), I TItE (3)

A0=0.EPS=.001PRINT 10000 1 N=1,151

1 X(:)(N"1)/1o.DO 2 N=1,101,5

2 RE AD(5,101) T(N),T(N+),T (N+2),T( N+3),T(N+4)101 FORMAT( E15.7)

00 20 I=1,100K=102*I

20 T(k)=T (K-1)T (1)=T (102)

CC COMPUTE INTEGRAL FOR VARIOUS VALUES 0C

L=1

H=77Do 3 IR=91,91,2R=x (IR )W2ITE(7>104) LR

104+ FOPMAT(IRF1Q,2)DO 3 IRP=MIR,2RP=X(IRP )T S=RR+-RP4RP

PR=20 *,<4RP

VAL(IR>,IRP )=4. DUBINT(A0,EPS)CC 4 OUTPUT 4C

WRI TE ( 6,i06) X (IRP) sV AL ( IR, IPP)

106 FOVMAT(5X),FbE2, )?,E15.7)3 WRIITE(7,105) VAL(I~,IRP)

105 FoRM-AT ( E15,7)STCPEND

128

Function UBINT

This is the first step in the double integration

process. It evaluates the integral of the function from

A to B. If this is more than 10 times the total

accumulated integral from AO to B, another longer interval

is evaluated and compared with the total. This continues

until the tolerance is satisfied. Hence, this is performing

the infinite z integration; it stops when new contributions

are negligible to the integral.

FUNCTION DUBINT(A0,EPS)C

AppN=1T=(.DIF=1 .E-4

= S=P IN(A,8,EpS)

T=S+T

IF( AB (S).LE.(DIF- ABS(T))) GO TO 10N N + 1A =5 2.-*N

Go TO 510 UrfINT=T

RETURN

END

Fincticn EjAPIN

SIMPIN is an integration algorithm which uses an

adaptive, nonrecursive procedure.86 This means that theintervals are dividen into subintervals, to which the

procedure i3 reapplied until an internal chick shows thatsufficient accuracy has been attained.

129

FUNCTION SIMPI4(A1,BTOL)C NOLRECURSIVE ADAPTIVE INTEGRATION PROCEDURE

DIMENSION DX(30),EPSP(30),Y2(3n ),X3(30),F2'(30), F3($F Mp(30),FBp (30),EST2(30),PVAL (30,3),NRTR (30 ),EST3(

A=A1ABSAR=0LVL=0EST=0.K=(EPS=TOLDA=B-ADIF=DAFA=FUN(A)FM=4"*FUN( (A+B) .5)FB=FUN(B)

1 LVL=LVL+110 DX(LVL)=DA/3.

SX=X(LVL)/6.F1=4.FUN(.5 DX(LVL)+A)X2(LVL)=A+DX(LVL)F ?( LVL )=FUN (X2(LVL) )X3(LVL)=X2(LVL)+UX(LVL)F3(LVL)=F.UN (X3(LVL)EPSP (LVL )=EPSF4(LVL)=4.4FUN(DX (LVL) *t5+X3(LVL) )FMP (LVL )=FMET1=SX4(FA+F1+F2(LVL))Ff3'(LVL) =F3EST2(LVL)=SX*(F2(LVL)+F3(LVL)+FM)E3T3(LVL)=sX4(F3(LVL)+F4(LVL)+FB)SU'=EST1+EST2(LVL)+FST3(LVL)A?-A 3;=AAR" AfS(EST)+ ABS(FEST1)+ AF3S(EST2(LVL))+$EST3(LVL))

K= :+1

IF( ABS(EST-SUM).EPSP(LVL)*APSAR) ?,?233 IF(LVL.LT 30) GO TO 42 IF(K.FE.t1) GO TO 3

LVL=LVL-I73 L=NRTR(LVL)

PVAL(LVL L) =SUMGO TO (11,12,13), L

4 NRTR(L\VL)=1

E sTi.EST IF({=F1

F 3A~2(LVL )7 EP=EPSP(LVL)/1'7

DA=DX(LVL)GO TO I

11 PT R(LVL)=2FA=F2(LVL)FM:FMP( LVL )F3zF3(LVL)EST=EST2(LVL)

130

A=) ? ( LVL)Go TO 7

12 NRTR(LVL)=3FA=F3 ( LVL)FMF4(LVL)FB=FhP (LVL)EST=EST3(LVL)

A=X3 (LVL)Go TO 7

13 SU I=PVAL (LVL, 1)+PVAL (LVL, 2) +PVAL (LVL, 3)IF(LVL-1) 5,5,2

5 SIBPIN=SUM17 RETURN

END

Function FUN

FUN is simply the integrand for the integration over z.

Hence, it is the integral of T over the azimuthal variable,

from 0 to 'IT . This integral is evaluated by the function

SIMP, which is identical to SIMPIN except that its integrand

is the function FTN.

FUN CT ICN FUN(X)Z=X*x

FU = IMP (O. O 3. 141593,.O01, Z)

RETURNEN IC"

Function FTN

FTN is the value of T(r) corresponding to the

variables , [', 9 , and z given in Eq. (5.43). If r :10,

the value of the potential V(r) is calculated. Otherwise,

the desired value of T is obtained by interpolation of the

di crete set of T values stored in memory.

131

FUNCTION FTN(X,Z)C

COMMON Y( 155),B( 151 )sTSPRARG=TS-PR*COS(X)+ZIF(ARG. GE..) GO TO 2C=COS (X)IF(AFS(ARG)fLT.(1.OE-4)) ARGeO.

2 D=SoRT(ARG)CALL INTERP(DS,101)IF(S.NE.O) GO TO 1FTN=-(1*47+3.98/(D*D))*6.934E-3/(D/10. )**6RETURN

1 FTN=SRETURN

END

Subroutine INTERP

This subprogram interpolates values from a table of

increasing positive arguments X(t),...,X(N) and the

corresponding function values Y(1),...,Y(N). In the interior

it uses a cubic approximation with two points on either side

of the chosen value. At the ends, a quadratic approximation

uses the extreme value and the two neighboring interior values.

SUpRxUTINE INTERP(XPYPN)C

COVMON Y(155)sX(151),TSRPIF(XPGGTTX(N)) GO TO 1IF(XP(E.X(1)) GO TO 2

1 YP=O"RETURN

2 IF(XP.GE"X(3)) GO TO 3iF(XPEQL.X(1)) GO TO 9IF(XP.EO.X(2)) GO TO 6

132

J=4GO TO 11

6 YP=Y(?)RETURN

9 YP=Y (1)RETURN

3 K=N-2DO 4 J=31KIF(XP.LE.X(J)) GO TO 10

4 CONTINUEIF(X EQeX(N)) GO TO 8IF(XP2EQ.X(N-1)) GO TO 7J=KGO TO 11

7 YP=Y(Na1)RETURN

8 YP=Y(N)RETURN

CC INTERPOLATION IN INTERIOR 4*

C10 IF(XP.NE.X(J)) GO TO11

YP=Y (J)RETURN

CX11 IF(X(J-2).E .O".ANO.X(J-1)"E('"0.) GO TO 13

IF(X(J-1).EQ.Q.) J=J+1

Go TO 1213 J=J+21? D14=X(J-2)-X(J+1 )

X13=-X(Jg2)-X(J)

O1a:=X(C-?2)-X(J3-1)OO5=X(J-2))X(J+2)003=X(J-3)-X(J+2)G04=X (J-3 ),cX ( J+1)

-3=X(J-3r)1X(J)CJQZ_= '(J-3)"X (J 1 )

9?4=x ( J-3 ) -X (J2),2E=X(J-1)-X(J+2)

u24=X (J-1) "X (J+1)

U23=X J-1) -X (J)D3!=X(J) -X(J+2)U34=X(J)-X(J+1)

D45=X(J+1)-X(J+2)Xo0=XP-X (J-3)X1=XPX ( J-2)X2=XP-X (J-l)X 3=XP-X (J )-X4=XP-X (J+1)X5=XP X(J+2)

133

PNiUM =X1X X2X3*X 4X0*XiYP=PtWU-4 ( Y (J-3) / ( D05*D403D0.?241+X0 ) -Y ( J-2) /

1 (n 01 5*(> x 13*U12*DOI*X1) +y ( J-1 ) /Ni<D2 6*D2 4*23*D12

2D24X? ).-Y (J) / (D23*D3 D23*D m3DD3*X3)+Y( J+1) / (304 *D)3440;14*4*X 4) -Y ( J+2 )/ (0 45*D:39*P25 4D15+DO5

4X5))20 RETURN

END

Program GPIT14

The program GPIT14 solves the generalized G--P equation

in Eq. (5.42) by expressing it in finite difference form

as in Eq. (F?).

air ?PrograM

The initial input data include the chemical potential

CP, relative condensate density DEN, quantum number for

circulation XL, step size H, and the second boundary condition

BC2.

Next the kernel values are read into the array D, which

are the results of program MATINT. Also input are the

integrals B, which are the fixed values obtained elsewhere

from Eq. (6.25). The first approximate solution is used

the first time through the program. The number of equations

can be reduced by eliminating some of them if the step size

is chosen larger than 0.1 10

Then the equation is evaluated: first the integral and

then the first and second derivatives. Since the trial

solution is not exact, the value will be nonzero, and it is

stored in the array CAL. These values are used to check

134

for convergence through the norm. If the solution set is

not diverging, the coefficients of the simultaneous set of

nonlinear equations are determined and stored in the array

DET. These equations are then solved by iteration through

the function SIMUL. The perturbed parameter in this instance

is T, which goes from zero initially to unity in steps of

0.2 if no convergence problems arise. Therefore T plays

the role of \ in Eq. (F13), by multiplying the chemical

potential and the integral term.

CD IMENSI ON T IN (131 );8(131 ) F (131 );CA L(131), SOL (131 )

CC 4 * INITIAL VALUES 4 4 4 4 4 4

CAPP ( I )=0 . OCAL (1)=0.0F(1 )=0.0SQL (1)=O.OK=t'

N=91NNI1=90N 1M 2

N M()

NM3=8EPS=1 GE-1oAL=6t)595

I RE'D( V100) CPDLNXLH, BC2100 FnRMAT(2L15.7,2F5.2,E15.7)

IF( XL*Ec.Q) GO TO 62CHK=50.T=c.G= . 10H =M HH

DENA-QEN/AL

)FN2 2. *DEN ACPA=(cP/ALIF(K.NL. 0) GO TO 65

CC 4 * + 4 4 READ KERNEL VALUES * * * 4 4 +C

135

DO 41 1=1,103R(I)= (I-1)/10"

DO 41 J=1,10341 D(IJ)=0,0

2 READ(5b101) MRHO101 FORMAT(I3,F10.2)

J=PH0'1 O. +1'.1Do 3 I=MJ,5

3 RFAD(5,102) D (J,I )D.(JI+1),D(JI+2),D(JI+3),D(J,102 FORMAT(5E15.8)

R(J)="RHOIH(RHO.LT.1O.2) GO TO 2L)F 1A) K=1s1j5

10 READ(5, 110) IsJsD(IsJ)

110 F0RMAT(2I5,E15.7)CC I(TEGRATI0N VALUESC 9 TO INFINITY

DO 4 1=1,704 (T)=+0sO

00 6 I=71,916 SEAD(5,108) ?(I)

108 FOPMA T ( E15.e8 )

CALL PPLOT(RB, 1,NO,6)PRINT 112

112 FOKMAT(///50X, 'APPROXIMATE VALUE OF INTEGRAL FROMCC APPROXIMATION FUNCTIONC

DO 30 .=2,NC VAF=;" T(ABS(CPA ))R(I)C 30 F(I)= FI(VAR)

30 F(I)=IDO 31 I=2,;

3. F(I )=F(I)/F (N)*EC2CALL PFLOT ( ,F,1,91,0,6)PRINT 121

121 FORMAT (///50X,' APPRXIMArT ON FUNCTION')65 00 87 I=2,N

SOL (I)=F (I)

87 APP(I)=F(I)89 K=0

CC * REDUtCE NUMBER OF EQUATIONSC

NK-1G C*H+e001NTIV=90/NK+1IF(NDIV.fe cQ.N) ( O TO 72N=NDIV

136

NMI=N-,1

NM? =Nri2NM3=N-3NPi=N+1

00 71 1=2,NJ=I*NK-1

L (I)=E3(J)

R (I)=R(J)F (I)=F( J)SOL (I) =SOL ( J)APP(I)=APP(J)Do 71 M=1,IL=M4NK-1D( IM)=0( JL)

71 CONTINUEDO 5 I=1,101IP=I+100 5 J=IP,102

5 D(IJ)=D(J,I)CCC * VALUE OF DeE. USING SOL * 4C

72 00 94 I=2,NM1CALL S.IMP6(S,1,N, I,H)

66 D2=(APP(I+1)+APP(I-1)-2e*APP(I))/HSW=( APP( 1+1)-APP I-1) )/(2,*H)Ss( I)=S+3( I )

D(NP1, I)=S

94 CAL(I)=D2+O/R(I)+(CPA4T-1,0/(R(I)4R(I))-DENAMSS(I)95 CAL(N)=0.0

K=K+1

CC CHECK FOR DECREASING NORM 4 4C

SSC=O.0DO 67 I=2,NMI

67 SSC=SSG+CAL(I)4CAL(I)ANCRS=QR T ( SSG )IF(K.LF.3) GO TO 68IF(ANOPRI.LTrCHK) GO TO 681R ITE_(6106) CHK, ANORMK

106 FORMAT(' NORM INCREASED FRPM',FI.7, ' TO',F1O.7,1BER .I4//)

IF (AkifRM. LEe(.0005)) GO TO 59G= /2.IF(GLE.(.001)) GO TO1T =T-G0

IF(APP (5) .EQ. OOL(5)) GO TO033DO 32 1=1,N

32 APP(i)=SOL(I)

137

33 GO TO 7268 CHK=ANCRM

CC i is DETERMINATION OF JACOBIAN i i * is

CDo 53 I=1,NM2IP=I+1DO 52 J=1,NM2JP=J+1

52 DET(I>J)=T4(-DEN2*SIMCO(JP)*R(JP)*D(IPJP)*APP(JP)53 UET (I, NM1) =-CAL (IP)

Do 54 I=1,NM2IP=I+1CALL SIMP6(S,1,NIPH)

54 DET(,I>I)=DET(II)*2"/HS+C:PA*T-1"/(Ri(IP).R(IP)) *DEN

DO 55 I=1,NM3IP=I+1

55 DEFT(IIP )=DET(I, IP )+1+/Hs+i./(P.*H*R(IP) )

DO 56 I=2,NM2IP I+1

56 DET(II=I)=DET(II-1)+1./HS1/(2.*H*R(IP))CC i i * CALCULATE NEW APPROXIMATION *C

i)ETVAL=SIMUL(NM2,EPSa,,NM1)DEL (NM)=O.0

UVAL=AS ( [ETVAL )IF( UVALGE.EPS) GO TO 57WPI TE (6,1O7) UETVAL

107 FOFMAT ( '1DETERMINANT OF JACOfIAN NEAR-SINGULAR WIT60 TO 1

57 S =258 Ap(I)=APP)(I)+DEL(I-i)90 DO 8 I=f>NM13

IF(ArS(APP(I)mSOL(I))9GT.(.00OQ1)) GO TO 8185 CONTINUE

Go TO 981 WRTTE(6801) K, TANORM

801 FORMAT (t'1ITFRAAT IDN' , I4, 1OXF' PARAMFTEP=',F5.2//12X,1'OLD VALUE',5X, 'DIFFERENCE',10x, 'D.E. VALUE',10X, '

DO 84 I=?,N84 *PTTE ( 6, cO:-) R( I)> APP (I), SQL (I) iOEL (I-1)' CAL( I)

802 FOCMA T (F: .d,3(bXs F1O. 7) >1oXF10#7 )

DO 8C I=2,N80 SOL(I)=APP(I)

IF(K.GE.7) GO TO 60GO TO 72

138

59 WRITE(6806) KANORM806 FOPMAT('ISOLUTION FOR K=',I4,' WITH NORM=',E14.7/

Do 88 I N88 WRITE(6,8O5) IR(I),APP(I )

805 FO MAT(' R('JI3J')=',F6.2,5XF10.7)60 CALL PPLOT(RAPP i1,NOA6)

WR TTE (& 683) K, CPDEN, Ks T8O3 FOR MAT (///20X,' SLUT ION FUNCTION AT K=',I12,10X>,'CH

1IX, 'OEN=',F9.6, 1OX, 'STEP', 13, OXi 'PARAMETER=',F5.2T=T+G

K=0C HK =50

IF(T.LEi.1) GO TO 72K=1

(O TO 162 STOP

END

Subroutine SIMP6

This subroutine is an integration algorithm using

Simpson's six-point rule from (M-1)/1O to (j-1)/10 at

intervals of 1H, where J- M must be a multiple of five,

and J and N are integers. The integrand is the function

FUN.

SUHROUTIr E SIMP6( S,MJKH)CC SIX-POINT SIMPSON'S RULE FRONT (hM-1)/10 TO (J"1)/10C J-M MULTIPLE OF 5C

rP4=M+MP5=M+5

JMU=J-5J MI = J -JM I =J-i,(001 N=MP'4, JM 1,>

1 S=+37 .. (FUN (KN N-3)+FUN(K,N) )+2?O,i(FUN(KNw2)+FUDo 2 N='M P 1 JM' hj

2 S=+19?. FUN(KN)(S+( FUN (KsM)+FUN(K,, J) ) 95 ) H/283

rETUP N

E ND

139

Function FUN

FUN is the integrand of the integral term in Eq. (5.112).

FUNCTION FUN( I, J)Crj ,MCN D( 131,13b), A(131#131),R(131)sAPP(131)s xFU=P(J)4D( IsJ) APP(J)*APP(J)RETURNEND

Function SIMCO

SICO furnishes the appropriate coefficient in

Simpson's six-point rule, which is required in the evaluation

of the Jacobian DET.

FUNCTION SIMCO( I)Cc SxvCO FURNISHES THE APPROPRIATE COEFFICIENT IN SIM

C

IF(I(I .F 1) *N "(I.NE." ) G3O TO I

1 C=IJ=,.#(C/ I/5)+1*1GO TO (2,'+,2 3,3)sJ

2 S1MFCO=l ' 32O8 3

RETURNN

3 SIN C0="e868 b5

k E.j-UP N4 3I 'CO=.b59722

RETURN

EN

APPENDIX H

$STGAARD'S T MATRIX

Although the T matrix approach of dealing with hard-

core repulsive potentials has been applied to many problems,

. solid helium105 and the excitation spectra of Bose.. 1o6 107liquids and gases, the formalism is almost entirely

applied in momentum space to obtain a nonlocal T metrix.

BycklinglO 6 wrote an integral equation for a T(r), but did

not attempt to solve it. A somewhat similar expression was

derived and solved by stgaard, 8 3 which is outlined below.

The relation between the unperturbed free-particle

two-body wave function c and the true wave function 1j

in Eq. (5.14) takes the explicit form

(HI)

based on a modified Brueckner theory where the energy

operator e represents the excitation energies of two

particles,

e(k) -g/M (ky'z). (H2)

The energy spectrum parameter Y is proportional to the

average self-consistent potential.

140

141

When Eq. (H1) is expanded into partial waves, it becomes

= S G(rv') V(( U1 (g') dr', (H3)

where JL is qr times the spherical Bessel function of order

L and

For L= 0 the last two equations bear strong resemblance to

Eqs. (5.36) and (5.37) for T and its kernel. In the

expansion, q represents what $stgaard calls the intermediate-

state relative momentum between interacting particles, but

since T is nonlocal it seems more appropriate that q be the

relative momentum after scattering, in the final state.

A partial-wave interaction operator or T matrix is

defined by

(H5)

For a given potential, Eq. (H5) is evaluated by considering

only s-wave states (L=O) and solving Eq. (H3) numerically

for u0 . Some averaging and interpolation is necessary due to

the zeros of J0(qr). A rather elaborate iteration scheme

including three-body correlations provides values of

between -4.0 and -5.0 K for the binding energy. The depletion

of the zero-momentum state is calculated to be about 40 per

cent. Qualitative agreement with the excitation spectrum in

142

Fig. 2 is obtained.

The results for Ta(r) = tq(r) using the Frost-MusulinO.q

potential are shown in Fig. 19 for the cases q = 1.0 A and

0.4 140. Note that extrapolation is necessary for r 0.4 Xand that the results are sensitive to q. Results for the

Yntema-Schneider potential are practically identical.

TABLE I

VALUES OF SOME PHYSICAL CONSTANTS

AND HELIUM-4 DATA #

Symbol Definition Value

h Planck's constant 6.6256 X10-2?erg sec

kB Boltzmann's constant 1.38054 x1016erg K"

N0 Avogadro's number 6.02252 x 1023mole 1

M Mass of 4He atom 6.648 x 1024g

Vm Molar volume of 4He 27.5793 cm3 mole '(0.0 K)

u Speed of sound in He II 238 m seco(0.0 K)

These values are taken from the appendices of Ref. 24.

143

TABLE II

PREVIOUS VALUES OF THE CONDENSATE FRACTION n /n

n0/n Source Reference

0.08 Penrose and Onsager 45

0.11 McMillan 49

- 0.105 Schiff and Verlet 184

o 0.06 Gersch and Tanner 50

0.101 Francis, Chester, and 51

Reatto

X0.04

0.088

0.024*0.01

0.0180.01

Jackson

Harling

Mook, Scherm, and

Wilkinson

Mook

54

53

19

52

1144

H

)

x

1'_ _ _ _J_ __ _ _ _ _ _ _ __ _ _ __I_ __ _ _ _ _ _ _

TABLE III

PREVIOUS VALUES OF THE CORE RADIUS

0Value (A) Source Reference

H 1.6 Kawatra and Pathria 14

+ 0.78 Amit and Gross 68)

0o 0.5 Chester, Metz, and 15

Reatto

1.28 Rayfield and Reif 10CI

1.28 Roberts and Donnelly 65

1.14 Glaberson and Steingart 11p4

145

TABLE IV

PARAMETERS FOR THE CALCULATED DENSITY AMPLITUDE

Energy denominatora Condensateb Core Radiuso ChemicalFraction Potential

(K)

Free-particled

Phonon-Rotone

0.1

0.024

4.1 I01

0.15

Ht

0.1

0.024

3.7

4.5

-0.48

0.08

a Form of E(k) chosen for Eq. (5.3?)

b Ratio of condensate to bulk density, n/n

c Distance from origin to point of half bulk density.

d Eq. (7.1)

e Approximated by Eq. (6.11)

146

147

40

0

T

SolidHe

Melting

Hel

r

Critical Point -

_CEy0vELow

10

TEMPERATURE in K

Fig. 1--Tho pThase diagrn of He (Ref. 22)

30

c

w

.)

<U)

w

o..

20

10

-U--

*

w'IT

i

.AnIL. AID%, dwaMIlk

148

40

30 FC

FC

2 0 /L

10~L

k

0 I 2Momentum Ik in

Fig, 2--The elementary excitation spectrum of He II.The experimental data points are those of Cowley andWoods (Ref. 4) obtained from neutron scattering. Thedashed curve L is the empirical spectrum of Landau(Ref. 3). Curves F and FC are due to Feynman (Ref. 37)and F'eynman and Cohen (Ref. 41), respectively.

l.G

f(p)0 O102

0.C0 I 2 3

p

4 5

inA

6 7

Fig. 3-The relative condensate density amplitudef(p) as a function of radial diatance from the corecalculated from the Gross-Pitaevskii euation in Eq(4.15) by Kawatra and Pathria (Ref. 14),

149

1

U I I I I I I I

8 9

150

1.2-I0

Q.8

0.6

0.4

Q2

041( 1 2 3 4 5 6 7 8

pin A

Fig. 4--The relative density amplitude calculatedP rm Emo. (4.42) by CR (Ref. 15). The curve is deter-mined from the trial function in Eq. (4.48).

f(p)

3

151

1

I

Fig. 5--A dia ra mmatic representation of thematrix element in Eq. (5,8), The dashed line repre-seats the two-body potential V and the box representsthe correlation function.

152

.sT

Fig. 6-Factorization of the correlation functionin Fig. 5. The small boxes represent the orderpa rareter ce. The T matrix results when only freepropagation and scattering occurs in the large box,

153

T +

Fig. ?--The graphical form of the T-matrix equationsin Eq. (5.16). When iterated the T matrix is a sum ofladder diagrams.

1

i

.L

-"- "

I

' T

154

5. 2 350

p inlA4 5 6

Fig. 8--The T matrix calculated from Eq. (5.36)with the free-particle energy denominator E (k)= =i"kZ/2M

+ 6.7 K. The dashed curve is the tI'DD2 potential(Ref. 17). Triangular points give the solution ofEq. (5.36) for N=20. The solid curve is the solutionfor N=1OO.

t

I

ii- I

V(r) -

i

1

1

I _

t

i

T

I I I 1 i

6 0

~40

20

0

-20 C 7. l

w

I

155

60

V(r)

40

20

0

-20 - I----I'- _ __ __ _ __ __ _ __ __ _ __ __ _

L 2 30 4 5 6 7rinA

Fig. 9--The T matrix calculated from Eq. (5.36)with the experimental energy spectrum of Eq. (6.11) inthe denominator. The dashed curve is the MDD2 potential(Ref. 17).

156

\p=o.0 A

x 5

4- 4-0.2

S04

.- -- - 10.I60*

00.4 081.2

p' in A

Fig. 10--The k rnel K(r, ') calculated from Eq.(6.19) using the realistic MDD2 potential (Ref, 17).

,Aol - I- - ... " - -

157

500

400

300

200

-- 100

0

-100-200

-3000123 4 5 6 7 8 9 10

O

p' in A

Fig. 11--The kernel K(p,p') calculated from Eq.(6.19) using the T matrix in Fig. 8.

0K~ ~\ p0.O A

2.00

2 0\

4.0

,

158

1000

800

600

400

200

0

~200

-4004 5

0p'ifl A

6 7 8

Fig. 12--The kernel K(Pep' ) calculated from Eq(6.19) using the T matrix in Fig. 9.

P=0. A

1.0

.4.-

2.0

\ -

c

Q

9I 2 3J

159

I.0

Q8

f (p)0.Q4

02

0.0

no=0.l n

"I-

I -

n0 =O.I-

) 2 30

p in A

5 6 789

1%) u3Re1tativehcondensate density amplitude(O) calculated from the generalized Gross -pitaevskernain Fig. q (s for various values of,,. Theeno lin.Fig. 11 is used with a condensate fractionof' 0.1 +nst

_- --, .r

I 4

160

1.0

0.8

0.6

0.4

02

0.0( 2 3 4 5 p

pinlA

7 8 9

Fig 14 -eRelative condensate density amplitudef(e) calculated from the generalized Gross-Pitaevskiiequation in Eq. (6.20) for various values of,). Thekernel in Fig.I1 is used with a condensate fraction

n 0 .024 n

.0"

Al-

T

-- __

r-

) I

161

1.0

0.8

0.6f(p) 0 .6

p0.4

0.2

.0() I 2 3 4 50p inA

6 7 8 9

Fig l R'aivccaddstedensity amplitudef) calculated from the gnrlzdGosptesiequat in -Rgoeneralized ro s P ~aevs 11equation ~ in E~q. (6.20) for various value fA hkernel in Fi. . 12 .ivros values of )4. Thekerne, iF is used with a condensate fractionof0.1

no=0.lIn

.06

i-

, . -

F'

162

no= 0 .024 n1.0

0.8

f(p)0 *6

0.4

0.2

0.00 1 2 34 560

p in A

Fig. 16*--Relative condensate density amplitudef(') calculated from the generalized Gross-Pitaevskiiequation in Eq. (6.20) for various values of fi. Thekernel in Fig. 12 is used with a condensate fractionof 0,024.

163

1.0

0.8

0.6

0.4

0.2

0.0) 2 40O

p mA6 8

Fig. 17--Thebbest relativehcondensate densitycondition from the generaljzed G eatontc boundaryto the T matrix in Fig. 8.urresponding

f(p)

no= 0 .10 n

AL=-O.IO Kn=0.024 n-L =0.15 K

w I

164

1.0

no=0.10 nQ8 -/-=-0.48 K

no=0.024 n

0.6. =0.08 Kf(p)

04

0.2

0 2 p 6 8

Fig. 18 -The best relative condensate densityamplitudes which best satisfy the asymptotic boundarycondition from the generalized G-P equation correspondingto the T matrix in Fig. 9.

165

50

40 B

30 A

20C

10

0

-10-

0 012 3 4 5rinA

Fig, 19--The T matrix of ,stgaard (Ref. 83), fortwo values of q in Eq. (H5). The dashed lines areextrapolations. Curve A corrsponds to q=0.4/A andcurve B corresponds to q=1.Q/X,

166

1.2

1.0 -..

0.8-

f(p) 0.6

0.4

I I

00

p z 4 f6 8

Fig. 20--A comparison of theoretical relativedensity amplitude profiles. The dotted curve is thatof Ci4R (ref. 15), the dashed curve is the K-P s alutjo(Ref. 14) of Eq. (4,16), and the solidtcurve is theinupper solution in Fig. 17.

.vim

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"A404MMAs-