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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
REFERENCES AND NOTES
1. L. Tisza, Nature 141, 913 (1938).
2. L. D. Landau, J. Phys. USSR $ 71 (1941).
3. L. D. Landau, J. Phys. USSR 11, 91 (1947).
4. R. A. Cowley and A. D. B. Woods, Can. J. Phys. 4Qt,
177 (1971).
5. N. N. Bogoliubov, J. Phys. USSR 11, 23 (1947).
6. R. P. Feynman, in Progress in Low Tipereatue Phy ii,
Vol. 1, C J. Gorter, Ed. (North-Holland Publishing
Co., Amsterdam, 1955), Chapt. II,
7. L. Onsager, Nuovo Cimento 6, Suppl."2, 279 (1949).
8. D.V. Osborne, Proc. Roy. Soc. (London) Q , 909 (1950).
9. W. F. Vinen, Proc. Roy. Soc. (London) A260, 218 (1961).
10. G. W. Rayfield and F. Reif, Phys. Rev. Lett. 3J., 305
(1963); Phys. Rev. 136, A1194 (1964).
11. W. I. Glaberson and M. Steingart, Phys. Rev. Lett.
26, 1423 (1971); M. Steingart and W. I. Glaberson,
J. Low Temp. Phys. 8, 61 (1972); M. Steingart and
W. I. Glaberson, Phys. Rev. A, 985 (1972).
12. E. P. Gross, Ann. Phys. (N.Y.) 2, 292 (1960); Nuovo
Cimento 20, 454 (1961).'
13. L. P. Pitaevskii, Zh. Eksper. Teor. Fiz. 4Q, 646
(1961) [Soviet Phys.-JETP 12, 451 (1961)]
167
168
14. M. P. Kawatra and R. K. Pathria, Phys. Rev. .i1,
132 (1966).
15. G. V. Chester, R. Metz and L. Reatto, Phys. Rev. ~7,
275 (1968).
16. D. H. Kobe, Phys. Rev. A 5, 854 (1972).
17. L. W. Bruch and I. J. McGee, J. Chem. Phys. 2,
5884 (1970).
18. D. Schiff and L. Verlet, Phys. Rev. 160, 208 (1967).
19. H. A. Mook, R. Scherm and M. K. Wilkinson, Phys. Rev.
A 6, 2268 (1972),
20. K. R. Atkins, juid Helium (Cambridge University Press,
Cambridge, England, 1959), pp. 1-122.
21. W. E. Keller, Liquidc Hi- el um-4 (plenumPress, New York, 1969), pp. 1-60, 123-157, 273-341.
22. J. Wilks, hm Prorties of Liquid and Sowid Heliun
(Clarendon Press, Oxford, 1967), pp. 1-432.
23. Z. M. Galasiewicz, Helium 4 (Pergamon Press, New York,
1971). This reference includes reprints of several
important experimental and theoretical papers.
24. R. J. Donnelly, Exierimental Superfluidity (The
University of Chicago Press, Chicago, 1967), pp.
1-100, 148-210.
25. A review of the experiments and references appears
in Physics Today 27, 17 (April 1974).
26. F. London, Proc. Roy. Soc. (London) AIM, 576 (1936);
J. Phys. Chem. .4, 49 (1939),
169
27. The values in Table I are taken from Ref. 24,
28. Reference 24, p. 225.
29. Reference 20, pp. 21-23.
30. H. Kamerlingh Onnes and J.D.A. Boks, Leiden Comm.
10b (1924).
31. Since various phenomena occur at slightly different
temperatures, the ?\ point is usually defined from
the phase diagram as 2.171 K at saturated vapor
pressure, 38.05 mm Hg.
32. P. Ehrenfest, Proc. Acad. Sci. Amst. 3., 153 (1933).
33. W. H. Keesom and A. P. Keesom, Physica 2, 359 (1936).
34. P. Kapitza, Nature 141, 74 (1938).
35. W. H. Keesom and J. E. MacWood, Physica ., 737 (1938).
36. E. Andronikashvili, J. Phys. USSR 10, 201 (1946)o
37. R. P. Feynman, Phys. Rev. 4, 262 (1954).
38. R. P. Feynman, Phys. Rev. 91, 1301 (1953).
39. L. Pitaevskii, Zh. Eksper. Teor. Fiz. 31., 536 (1956)
[Soviet Phys.-JETP 4, 439 (1956)]
40. C. F. A. Beaumont and J. Reekie, Proc. Roy. Soc.
(London) A228, 363 (1955).
41. R. P. Feynman and M. Cohen, Phys. Rev. 142, 1189 (1956'
42. M. Cohen and R. P. Feynman, Rhys. REv. L.Z, 13 (1957).
43. K. Huang, Statistical Mechanics (John Wiley and Sons,
New York, 1966), Chapt. 12.
44. F. London, Nature 141, 643 (1938).
45. 0. Penrose and L. Onsager, Phys. Rev. 104, 576 (1956).
).
170
46. Reference 22, pp. 142-145.
47. G. B. Hess and W. M. Fairbank, Phys. Rev. Lett. 12,216 (1967).
48. A. L. Fetter in Lectures in Theoretical Physics,
Vol. XI-B, K. T. Mahanthappa and W. E. Brittin, Eds.
(Gordon and Breach, New York, 1969), pp. 351-353.
49. W. L. McMillan, Phys. Rev. 138, A442 (1965).
50. H. A. Gersch and J. M. Tanner, Phys. Rev. 22., 1769
(1965).
51. W. P. Francis, G. V. Chester, and L. Reatto, Phys. Rev.
A 1, 86 (1970).
52. H. A. Mook, Phys. Rev. Lett. )2, 1167 (1974).
53. 0. K. Harling, Phys. Rev. A 1, 1073 (1971); Phys. Rev.
Lett. 24, 1046 (1970).
54. H. W. Jackson, Phys. Rev. A LO, 278 (1974).
55. Reference 20, pp. 96, 120.
56. E. Merzbacher, Amer. J. Phys. )Q, 237 (1962).
57. S. C. Whitmore and W. Zimmermann, Phys. Rev. 166,
181 (1968); Phys. Rev. Lett. , 389 (1964).
58. R. E. Packard and T. M. Sanders, Phys. Rev. A 6,
799 (1972).
59. Reference 24, Chapt. 6.
60. G. B. Hess, Phys. Rev. 161, 189 (1967).
61. G. A. Williams and R. E. Packard, Phys. Rev. Lett. 2,280 (1974).
171
62. D. Stauffer and A. L. Fetter, Phys. Rev. x68,
156 (1967).
63. H. Lamb, Hydrodynamics (Dover Publications, New York,
1945), pp. 236-241.
64. W. M. Hicks, Phil. Trans. Roy. Soc. 175A, 183, 190
(1884).
65. P. H. Roberts and R. J. Donnelly, Phys. Lett. fA,
137 (1970).
66. W. I. Glaberson, D. M. Strayer, and R. J. Donnelly,
Phys, Rev. Lett. 3.2, 1740 (1968).
67. G. Gamota and T. M. Sanders, Jr., Phys. Rev. A 4,
1092 (1971).
68. D. Amit and E. P. Gross, Phys. Rev. 145, 130 (1966).
69. D. H. Kobe, Amer. J. Phys. 4, 1150 (1966).
70. C. N. Yang, Rev. Mod. Phys. ,4, 694 (1962).
71. E. P. Gross, J. Math. Phys. 4, 195 (1963).
72. A. L. Fetter and J. D. Walecka, Quan u iTheory _ f_May.-rticle Systems (Mc Graw-Hill, New York, 1971),
pp. 488-492.
73. V. L. Ginzburg and L. P. Pitaevskii, Sov. Phys.-JETP
Z, 858 (1958). They obtained and showed a solution
for an equation having the form of the G-P equation,
but it represented the superfluid wave function near T .
74. A. L. Fetter, Phys. Rev. 138, A429 (1965).
75. R. Jastrow, Phys. Rev. _8, 1479 (1955).
-^- -
172
76. H. L. Frisch and J. L. Lebowitz, The, Euilibirim Theory
1 Classical Fluids (W. A. Benjamin, New York, 1964),
Section II.
77. Amit and Gross (Ref. 68) use a &-function potential
because "...the potential-energy integral is hopelessly
complicated even for the simplest V(x-y)."
78. B. A. Lippmann and J. Schwinger, Phys. Rev. 2,469 (1950).
79. N. H. March, W. H. Young,and S. Sampanthat, TheJMany-
.Bod yjProblem , uQuantum Mechanics (Cambridge University
Press, Cambridge,England, 1967), pp. 192-201.
80. K. A. Brueckner and K. Sawada, Phys. Rev. 106, 1117,
1128 (1957).
81. D. F. Goble and L. E. H. Trainor, Can. J. Phys. 46,
839 (1968).
82. K. A. Brueckner and D. T. Goldman, Phys. Rev. 116,
424 (1959).
83. E. $stgaard, J. Low Temp. Phys. 4, 239, 585 (1971).
84. D. H. Kobe, Ann. Phys. (N.Y.) _,Z, 15 (1968).
85. D. B. deHaan, ouvelles Tables d'Interales Definies
(G. E. Strechert, New York, 1939), 160.5.
86. Reference 85, j161.5.87. V. H. Smith, Jr. and A. J. Thakkar, J. Low Temp. Phys.
1', 331 (1973).
88. I.B.M. Corporation, System/360 Scientific Subroutine
Package (I.B,.M.Corp.,White Plains, N.Y., 1968), p. 301.
173
89. L. P. Smith, Mathematical Methods for Scientists and
Engineers (Prentice-Hall, New York, 1953), pp. 380-385.
90. P. J. Davis and P. Rabinowitz, Numerical Integration
(Blaisdell Publishing Co., Waltham, Mass., 1967),
pp. 164-165, 198.
91. P. A. Stark, Introduction to Numerical Methods (Macmillan
Co., New York, 1970), pp. 284-288.
92. F. W. J. Oliver, in Ha0dbook o'Mathematical Functions,
M. Abramowitz and I. A. Stegun, Eds. (National Bureau
of Standatds, Washington, D. C., 1964), pp. 374-379.
93. Reference 88, p. 62.
94. F. Freudenstein and B. Roth, A.C.M.J. 18, 550 (1963).
95. Reference 22, pp. 542-545.
96. G. W. Goble and D. H. Kobe, Phys. Rev. A 1o, 851 (1974).
97. A. L. Fetter, Phys, Rev. Lett. , 986 (1971).
98. A. L. Fetter, Ann. Phys. (N.Y.) ZQ, 67 (1972).
99. D. H. Kobe, J. Math. Phys. 1, 1507 (1969).
100. B. Carnahan, H. A. Luther, and J. 0. Wilkes, Applied
Numerical Methods (John Wiley & Sons, New York, 1969),
pp. 272-296.
101. Reference 85, 8.13.
102. Reference 85, 117.13.103. Reference 100, p. 74.
104. Reference 100, pp. 308-309.
105. B. H. Brandow, Ann. Phys. (N.Y.) 2, 214 (1970).
174
106. F. Byckling, Phys. Rev. 15., 71 (1966).
107. G. V. Brown and M. H. Coopersmith, Phys, Rev. 18,
327 (1969).
"A404MMAs-