Marder M.P. Condensed Matter Physics

Подождите немного. Документ загружается.

Superfluid He

433

/ν„

Figure 15.7. Direct observation of two-fluid motion in liquid helium. Hydrogen crystals

trapped in vortices move downwards while others move upwards. The figure displays

the probability distribution P of different tracer particle velocities for three different flow

conditions. The scale is set by

v„

in Eq. (15.93a). [Source: Paoletti et al. (2008), p. 5.]

Experimental Observations. These traveling waves were first found experimen-

tally by Peshkov (1946). First sound has a velocity ranging from around 240 m/s

at very low temperatures to 220 m/s at the λ point. The second sound velocity

vanishes at the λ point, in accord with Eq. (15.92), has a plateau at around 20

m s^

between

and 2 K, and then rises to a value greater than 100 m s

-1

at low

temperatures, which approaches

c\/\/3

as T

—>

First sound is generated by inducing periodic variations of pressure in the he-

lium, by tapping it or other mechanical agency. Second sound is induced by tem-

perature variations, such as produced by an oscillating heater.. Pressure variations

simply will not induce it to any measurable degree, a fact that defeated the earliest

experimental attempts to measure it.

15.5.3 Direct Observation of Two Fluids

The two-fluid model of superfluids has often been regarded as a crude represen-

tation of a complicated underlying quantum mechanical state, but Paoletti et al.

(2008) found an experimental situation in which the two fluids can be observed

directly. The experiment is conducted by injecting superfluid helium with micron-

sized tracer particles of frozen hydrogen, and then injecting heat into the superfluid

from below. The heat is transported upwards by normal fluid, since the superfluid

component carries no entropy, while superfluid moves downwards to keep density

constant. Specifically, if heat Q is injected per time into a system of area

density

434 Chapter 15. Fluid Mechanics

entropy per unit mass s and temperature T the velocities of the two fluids are

That the full density p appears here is due to ( 15.93a)

the conventional definition of s.

To conserve mass density. (15.93b)

Most hydrogen crystals are carried upwards because of drag by the normal fluid.

However, some of the crystals become trapped in the cores of vortices, and the vor-

tices move with the superfluid. Thus the experiment produces images where some

particles are drifting upwards at a speed centered on the prediction of

Eq.

(15.93a),

while a smaller population drifts downwards, as shown in Figure 15.7.

15.5.4 Origin of Superfluidity

One common explanation for the superfluidity of helium follows from displaying

the dispersion relation for propagating modes. These modes, which are the ana-

log of phonons in crystalline solids, can be measured by neutron scattering, using

Eq. (13.93), and the results of such investigation are displayed in Figure 15.8. For

small values of the wave vector k, neutrons excite long waves capable of traveling

long times with little scattering. These are the sound waves described in the pre-

vious section. As the wave vector increases, the dispersion curve passes through

a maximum, called the maxon peak, and then through the roton minimum before

proceeding to wavelengths too small for measurement. In addition, there is a class

of background excitations corresponding roughly to the neutron knocking a single

helium atom into motion; this sort of excitation is heavily damped, and therefore

_ 2

5Γ

^ 1

0 12 3 4

Wave number k

(Â~

' )

Figure 15.8. Neutron scattering data giving dispersion relation for

He. Dashed line

shows Cherenkov-Landau criterion, Eq. (14.105), for exciting an excitation in the helium.

[Source: Donnelly (1991), p. 46.]

~ApsT'

-V„

—

Superfluid

435

quite broad, but does not appear to have appreciable amplitude below the solid

dispersion curve. According to the Cherenkov-Landau condition Eq. (14.105), il-

lustrated by Figure 14.19, a moving particle in superfluid helium should not be

able to transfer any energy until its velocity v

reaches the smallest value vi « 56

m s

-1

(Landau's critical velocity) for which the straight line vik is able to touch

the dispersion curve

uj(k).

In fact, negative ions injected into He II at pressures

between 21 and 25 atmospheres and below 0.5 K begin to experience drag right

around a velocity of

m s

However, explanations based upon the dispersion curve in Figure 15.8 do not

provide a persuasive account of superfluidity. The velocity

is orders of mag-

nitude larger than the characteristic velocities at which flows in narrow capillaries

lose their superfluid character. Ions injected by Allum et al. (1976) into helium

at atmospheric pressure and velocities on the order of a meter per second display

the unexpected property of traveling slower in helium the more energetic they are,

behavior that may only be explained quantitatively by supposing that the ions gen-

erate a large vortex ring that travels with them. These experimental facts can only

be appreciated by making a conceptual shift. An experimental probe such as neu-

tron scattering can confirm the existence of certain sorts of excited states. However,

explaining superfluidity requires something different. It requires making plausible

the complete absence of any excitation, whether phonon, roton, vortex, or anything

else not yet classified, capable of causing a degradation of superfluid flow. For ex-

ample, the argument needs to provide a way to understand why a superfluid flowing

m s

through a thin rough-walled vibrating quartz channel is unable to excite

any phonons in the quartz, although the quartz dispersion relation, similar to that

of Figure 14.19, will permit satisfaction of

Eq.

(14.105) at all velocities.

The only simple explanation for such phenomena is that helium undergoes a

transition to a radically new state below the λ transition. A model for this state

is the weakly interacting Bose gas, which is studied as a model for superfluidity

in Problem 7. For a highly interacting system such as helium, one cannot hope

that the ground state of the ideal Bose gas, in which all particles occupy the same

single-particle ground state, provides a quantitative description. However, one can

abstract one of the features of this ground state and can guess that the ground-state

density of

He is almost completely uniform. That is, unlike a crystal in which

atoms adopt specific locations relative to one another, the ground state density of

He is symmetrical and featureless, except at system boundaries where it rapidly

drops to zero. The helium is able to flow through complicated constricted geome-

tries because its wave function is truly smooth, and it does not exert periodic forces

upon the containing walls.

This state can be characterized by a wave function

Ψ(Γ)

that gives the quantum

mechanical amplitude for finding condensed helium atoms at position r. One way

to define this wave function is to suppose that one has found the complete wave

function ΨΝ(7Ι

■ ■ ■

Ov) for N helium atoms. This wave function might be the

ground state eigenfunction or else a low-lying excited state. Suppose that one also

knows ΨΝ

$?\ . . . 7N+$, which is supposed to be essentially the same as ψκ, but

436 Chapter 15. Fluid Mechanics

with one extra atom added at

?N+\.

Define

Φ(Γ) = J d

■ ■ ■

7Ν)ΨΝ+Ι (n . . . 7

, 7) (15.94)

Far away from r, the two wave functions coincide, but in its neighborhood, they

differ, because

ΨΝ+Ì must accommodate the extra particle. The time evolution of

Φ may be determined approximately in closed form. Let

N pi

= V~ + U(n...r

) (15.95)

f^2m

be the Hamiltonian for the helium. Applying Schrödinger's equation to Eq.

(

15.94),

almost all kinetic energy terms cancel out and

■

r—

^φ

Ν+λ

'φ*

-'φ*

'Κ

Ν+

\'φ

Ν+

(15.96)

7—ψ

I —-—

- +

N+l

(n . . . r

, r) - U

{r\ . . . r

) \ φ

Ν+ι

(15.97)

Carrying out exactly the integrals associated with the potential energies U is a task

lying somewhere between the difficult and the impossible. However, one can make

a guess about the nature of the result based upon the picture of low-lying states of

helium as corresponding to a featureless liquid. Suppose that the potential energies

and t/#+i have broad minima of nearly uniform energy corresponding to many

equivalent locations of the atoms r\ . . . r^. Suppose further that for any atomic

configuration not lying within this minimum, the amplitudes ψη and

ΨΝ+Ì drop

very rapidly to zero. Thus, the low energy wave functions act like projection oper-

ators picking out a large number of atomic configurations of low potential energy,

and the functions

and ί/jv+i can be replaced by their minimum values. Given

this hypothesis and recalling that the chemical potential μ is defined as £^+i

—

8-N,

one has

-h ΟΦ -/i

— -κ- = ^ * + μΦ· (15-98)

ι at 2m

The wave function evolves in an effective potential given by the chemical potential

for the whole fluid.

The only difference between the wave function for a stationary system and one

moving at a uniform velocity v is that the latter has a phase of

imv

Therefore

it is valuable to rewrite Φ in terms of amplitude and phase, as

*(r) =

y/ne®.

(15.99)

Placing Eq. (15.99) into Eq. (15.98) and taking both n and φ to be real gives two

equations, when (15.98) is separated into real and imaginary parts. The first is

^ = -V--V<M, (15.100)

at m

Superfluid

437

which

is an

equation

continuity

for

density

n if

= —V0

(15.101)

is tentatively identified

as the

superfluid velocity.

The

imaginary part

(15.98)

= -(μ + ^

/2) + ^^·

(15.102)

2m ^Jn

The last term

Eq. (15.102)

only appreciable near system boundaries, where

the density

helium

dropping

zero,

and

because helium

is a

nearly

in-

compressible fluid, this last term

negligible

bulk flow. Once this term

dropped,

Eq.

(15.102) turns into Euler's equation: Taking

its

gradient,

and

using

Eq. (15.101)

one has

m—-+mV^f

= -V/x

(15.103)

„_

^» Use the

identity

-V)v

-—

^7h =

TjsX

,î

xT)

(15.104)

ßl

■> ' jjj the

tact that because

is the

gradient

ot a

scalar,

V x v

vanishes. Thus

one

obtains

Eq. (15.73).

Vortices.

In the

course

deriving

Eq.

(15.104),

the

observation that

is the

gradient

the scalar function

φ, and

that therefore

V x v

= 0 was

crucial.

this

description

liquid helium

correct,

should therefore

impossible

to set the

superfluid into rotational motion. This conclusion

experimentally incorrect.

is true that

if a

bucket

helium

cooled below

the λ

point

and the

bucket then

slowly made

rotate,

the

superfluid component

of the

helium remains station-

ary. However,

instead

the

bucket

is set

into rotation above

the λ

point,

and

then

cooled down below

the

superfluid transition,

the

fluid continues

rotate,

as can be

deduced from

the

bowing

the liquid free surface. Nevertheless,

the

rotating

su-

perfluid continues

exhibit

the

fountain effect,

superfluidity

is not

completely

destroyed.

The explanation

this apparent paradox

that superfluid

capable

sus-

taining vortices, rotating states

whose cores superfluidity

destroyed

and the

phase

φ is not

defined. They

are

perfectly analogous

dislocations

solids.

one assumes that

the

phase

φ is

destroyed only along isolated lines,

but

that other-

wise

the

wave function

defined

and

single-valued,

one

arrives

at the

prediction

that flow around superfluid vortices must

quantized

units

of h/m. To see why,

suppose

for

example that

the

phase

φ is not

defined along

line passing

in the z

direction through

the

origin,

and

integrate

the

superfluid velocity

along

closed

contour

C in the x-y

plane around

the

vortex core.

The

condition that

the

wave

function

Φ be

single-valued

ds-V

2πΙΗ/ίΠ

The

hase

can

change

by an

integer multi-

(

15.105

)

pie / of 2π

going around

loop

and

leave

the

wave function single-valued.

? h

/ d r Z· V X V

= K = I —.

Using Stoke's theorem

and

defining

K to be

(15.106)

J j{

' m the

total circulation enclosed

by the

contour

438

Chapter 15. Fluid Mechanics

The quantization of vorticity has been observed directly, as shown in Figure 15.9.

}-<

-1

-2

-3

-4

-5

-4

-2-101234

Angular velocity

)

Figure 15.9. Circulation κ of

He vortex in units of h/m

as a

function of

the

rotation rate

container. The experiment is conducted by passing a wire of about 25μπι in diameter

through the superfluid and observing its vibrational modes. If a vortex is present, its core

can preferentially attach to the wire. The additional angular momentum lying along the

wire changes its vibrational frequency and allows a direct measurement of

the

circulation

the

vortex. [The method is due to Vinen (1961), while the source of

these

data is Karn

al.

(1980), p. 1799.]

The discussion of vortices in He II also implies a new explanation of the phe-

nomenon of superfluidity. Suppose that superfluid is flowing through a narrow

channel from one bath of helium to another. In the two baths the fluid is essentially

at rest, so each of them is characterized by a constant phase of the wave function,

say φ\ in the first and

Φ2

in the second. Within the narrow channel, the phase

changes linearly (up to multiples of 2π from φ\ to 02). In order for the flow in the

channel to degrade, the gradient of φ must decrease. However, it must at all times

be compatible with the boundary conditions, which means that the change in φ

must in some respect be discontinuous and occur in discrete

jumps.

The superflow

can only decay by spawning a vortex, and the difficulty of accomplishing such a

transition accounts for the persistence of superfluid flow.

A quantitative theory for the decay of superfluid flow may be obtained by com-

bining a basic fact of statistical mechanics with some results from classical hy-

drodynamics. The basic fact of statistical mechanics is that the probability of any

configuration of energy δ is exp[—t/kßT]. The results from hydrodynamics, dis-

cussed by Donnelly (1991) p. 23, are that a vortex ring of vorticity κ, radius R,

and hollow core at constant pressure of size a in a stationary fluid of density p has

energy

dr\ pv

{r) = \ρκ

ΙΙ(η - §), η = ln(SR/a)

(15.107)

Superfluid He

439

and momentum

and moves at velocity

<9£

κ(η—\

[ dr pv{7) = pnnR

(15.108)

This is the rate at which the whole vortex drifts, (15.109)

dP\/

ATÏR

'ike a smoke ring

Consider now superfluid flowing at velocity

in a pipe. If a vortex forms in this

moving superfluid, the change in energy due to the vortex will be

tot

— Ρ

ν$ If the velocity field of the vortex has

sub- (15.1 10)

tracted from it and substituted intoEq. (15.107),

this form results immediately, using Eq. (15.108)

and omitting the constant energy of flow v

For small radii R, the energy of a system with a vortex increases with R, but even-

tually the vortex is sufficiently successful in reducing the kinetic energy of fluid

motion that it becomes energetically favorable to make it even larger. The saddle

point in energy beyond which this change occurs is found from

-dR-

= 0

^R--jR-

= 0

= V

(15

Π1)

That is, the energy of the vortex reaches the saddle point when it moves against the

prevailing flow

just the speed needed to sit stationary in the laboratory frame, as

shown by Langer and Fisher (1967).

This theory predicts that superflows will decay through formation of vortices

of radius R

= κ(η

—

^)/4πν

, whose energy is £

^pn

—

|) and whose

probability of formation is proportional to exp[—/3£

]. Superflows should last ex-

ponentially longer as either the temperature or their velocity decreases. A com-

parison of these predictions with experiment was carried out by Langer and Reppy

(1970).

15.5.5 Lagrangian Theory of Wave Function

Because the the wave function of liquid helium is rather a mysterious object, it is

interesting to obtain equations for it in a more systematic way, one that provides a

general procedure for finding effective equations of motion in complicated many-

body systems. That is, there is a definite set of equations governing the motion of

all the helium atoms, but the vast number of atoms and the strength of their inter-

action makes a direct solution impossible. On the other hand, there is every reason

to suspect that most important features of the helium are encoded in a single wave

function, a collective

coordinate,

that is much simpler to understand. The question

is how to pass from the microscopic laws for the helium atoms to dynamical rules

for the wave function in a controlled and consistent way.

Demircan et al. (1996) showed that one way to proceed is to use the formalism

of Appendix B.3, which shows that the time-dependent Schrödinger equation de-

rives from a Lagrangian. If one can think of a good approximate wave function for

440

Chapter 15. Fluid Mechanics

helium, the formalism describes how it will evolve in time. Feynman (1953) sug-

gested a way of building a wave function. Imagine that the ground-state wave func-

tion of helium in some experimental geometry is known, and call it φ{τ\ . . . r#).

This ground state wave function may be tremendously complicated, and cannot be

found in practice, but in the end one will need to know surprisingly little about it.

Only three things need to be assumed. First, it is a true ground state, so

ih.

dt*

V/ means -â-.

(15.112)

The interactions described by the potential Ü may be quite strong. Second, because

He is a boson, φ is symmetric under interchange of any of its arguments. Third,

because φ is a ground state, there are no currents flowing anywhere in it, which

means that

-0*V,<;

Φν

(15.113)

Next imagine that all the low-lying excited states of the wave function are of the

form

V>(n . . . r

) = exp[^ Φ(?/, ή] Φ(λ · ·

)

„ΣΦ,

(15.114)

The essential point in this single-mode approximation is that there is only a single

function *(r, t) shared by all the particles; this is the wave function of

He. The

wave functions in Eq. (15.114) are not guaranteed to be normalized; one can divide

through by a normalization factor at this point, but it is simpler to use a Lagrange

multiplier later that ensures \ψ\

integrates to one.

Adopt abbreviated notation in which Ψ/ is shorthand for *(r/, t), and the ar-

guments of φ are not indicated explicitly. The effective Lagrangian defined by

Eq. (B.16) corresponding to Eq. (15.114) is

= J ά

7φ*β^^^)

■ a ^

vì

■U

,Σ,» *«»),

(15.115)

In evaluating L, the only term that is difficult to evaluate is the one involving the

spatial gradient:

ΣΦ;

e i'

ΣΦ;

■V?

*'"<

2„Σφ„

φνΐβ^

+2(ν

/<?

Σφ

'")

(V/0) +*

Σφ

'" V^

Expand derivative.

|0|2

ΣΨ,„

(ν

/ί?

ΣΨ,„ ). (ν

;

|0|

) + β

Σφ

/"0*V?<î

Using Eq. (15.113). Only middle term changes.

ΣΦΓ/

e i'

(15.116)

(15.117)

(15.118)

Superfluid

441

Ί β

^'\φ\

ν^^>'-\φ\

^Ην

β^''')

^>'φ*Ψ

φ (15.119)

Integrate

middle

term

by parts.

*V72J.

|j.|2|

ΣΦ;,+Φ,,

'vfa-ΗΊν,Φ,

Two

terms

cancel.

(15.120)

With the use of Eqs. (15.112) and (15.120) it is simple to evaluate L from

Eq. (15.115) and find

L = j ά

Τ\φ\

>'*ϊ

*>'Σ

' 0Φ; h

.-.

.,'

(15.121)

An approximation to Schrödinger's equation is obtained by taking the variation of

L with respect to

Φ(Γ,

t). In order to ensure that φ remain normalized, the variation

will be carried out with a constraint enforced by a Lagrange multiplier μ, so the

equation of motion for Φ is

5Φ*

L-μ

/rl^l^'*?^

Enforce

constraints

taking

the

thing

that

_ n

supposed

remain

fixed,

multiplying

it by a

constant,

and

adding

it to the

Lagrangian.

(15.122)

Before completing the equation of motion for Φ, it is helpful to define two quanti-

ties.

The density n\ (r) is

„,(?) = f d

r |</>|

ί>

+φ

'' Σ δ(?-7ι) (15.123)

·* ι

and the structure function S(r, ?') is

S(r, r') = ^ f d

f \φ\

>'^>

*' Σ

(7-

7ι

)

ρ-

7ιι

). (15.124)

The functional derivative with respect to Φ* in (15.122) has two pieces. First,

there is a contribution from (V/Φ/Ι

, and second there is one from the factors of

εχ

ρ[Σ/'

Φ*']- Performing the functional derivative with the methods of Appendix

B gives

ν·«,νΦ(?) = ^ j d

r\i)\

8{r

-r)

. ΟΦ, H

|πτ |2

ίΤζ^τ

-—|νΦ,|

-μ

dt 2m

(15.125)

ih- '

IV7,T,«'M2

dt 2m

\νπη-μ

S{r,Y).

(15.126)

To proceed further, assume that Φ is small, and that therefore anything involving

more than one power of Φ may be neglected. In particular, any spatial variation of

can be neglected, and the term involving |νΦ(Γ')|

can be thrown out. S(r, r')

becomes the static structure factor of

the

ground-state wave function φ, and because

442

Chapter 15. Fluid Mechanics

the ground state should be translationally invariant, S should depend only upon

—

r'. Taking the Fourier transform of

Eq.

(15.126) gives

2 ;,2

■Ψ(& α;) = [hw*{q, ω) -

ö{q)ö{u)]S{q)

(15.127)

Hijj(q)

■

6.02k

Â]

2mS(q)

S(q)

K So long as one stays away from (

5 128)

= o, q = 0.

where S(q) is precisely the static structure factor measured in neutron scattering.

Equation (15.128) is due to Feynman (1953) and permits comparison of excitation

energies (obtained with inelastic neutron scattering) with the static structure factor,

as shown in Figure 15.10. The agreement shown in the figure is good only within

a factor of two and has been improved substantially through elaboration of the

theory, as discussed by Feynman (1972), Chapter 11, and Glyde (1994).

□ From S(q)

- Neutron scattering

1 2

Wave number

(Â~

' )

Figure 15.10. Comparison of Feynman's expression (15.128) for excitations in helium

with direct measurement using neutron scattering. The neutron scattering data are from

Donnelly (1991) p. 46, and data for

S(q)

are from Svensson et

al.

(1980).

15.5.6 Superfluid

Because

He particles are fermions, they cannot drop into a Bose condensed state

with the same ease as

He. They nevertheless form a superfluid. The Pomeranchuk

effect makes it possible to reach very low temperatures in

He. Pomeranchuk ob-

served that at temperatures below 100 mK the entropy of liquid

He falls below

that of the solid; under pressure,

He nucleates solid crystals that cool the cell in

which they grow. Building such cells,

Osheroff,

Richardson, and Lee (1972) ob-

served slight changes in the slope of pressure versus temperature at temperatures

around 2.5 mK, and pressures around 30 atmospheres. They first attributed these

anomalies to phase changes in the solid, but subsequently determined that the liquid

was changing phase and becoming superfluid. Essentially, pairs of

He nuclei join

together forming effective Bose particles, which can then condense, in a manner

reminiscent of superconductivity. Wheatley (1975) and Leggett (1975) reviewed

the experimental and theoretical situation not long after the original discovery of