Theoretical aspects of magnetic fluctuations

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Theoretical aspects of magnetic fluctuations

Ronald E. Burgess

To cite this version:

Ronald E. Burgess. Theoretical aspects of magnetic fluctuations. Revue de Physique Appliquée,

Société française de physique / EDP, 1970, 5 (1), pp.77-82. �10.1051/rphysap:019700050107700�. �jpa-

00243378�

THEORETICAL ASPECTS OF MAGNETIC FLUCTUATIONS

By ^{RONALD E.} BURGESS,

Department ^of Physics, University of British Columbia, ^Vancouver 8, B.C., ^Canada.

Abstract.

The thermodynamic ^{treatment of} equilibrium fluctuations in magnetic systems

is discussed and applied ^to

variety ^of situations. The spectral density ^{of the} magnetic

moment ^or flux ^are derived for simply-connected ^bodies (sphere ^and cylinder) ^{and for} multiply-

connected systems (loop ^{and hollow} cylinder). ^The material is considered to be either

normal conductor,

superconductor ^or ferromagnetic. The results are valid in the presence ^of

^static

magnetic ^{field and also if the} susceptibility ^is field-dependent. ^No abrupt change ^{in fluctua-}

tions occurs as

superconductor ^{is taken} through the transition temperature. ^{The introduc-}

tion of

Josephson junction ^or

resistive link in

superconducting loop ^{is shown to} increase the flux fluctuations markedly ^with

steady ^{increase as the link is made weaker. At} high ^fre- quencies ^the fluctuation spectrum ^for

superconductor approaches that for the same system

in the normal state.

PHYSIQUE APPLIQUEZ 5, 1970,

I. Introduction.

Fluctuations of the magnetic

moment of a solid body ^of arbitrary shape ^{can be}

calculated for thermal equilibrium by ^{means of the} fluctuation-dissipation ^theorem. Similarly the flllc- tuations of flux and magnetic field within a loop ^or

other hollow body may be found.

Attention is hère devoted to various configurations

of normal conductors, ferromagnetics ^and supercon- ductors. The results are valid in the _presence of an

arbitrary ^static magnetic field and the field-dependence

of magnetization (which may be non-linear) ^enters through the differential susceptibility which is in _gene- ral a complex function of frequency.

The inapplicability ^{of the} simple form of the equi- partition principle commonly ^{used to evaluate} magne- tic fluctuations in superconductors arises from the

neglect ^of significant components ^of energy additional

to the magnetic ^{field energy.}

A superconducting ^system customarily ^exhibits appreciably ^smaller magnetic fluctuations than ^a nor-

mal metal system ^{of the same} configuration ^{and has}

a smaller relaxation time. Nevertheless the fluctua- tions are continuous through ^the transition température

as is the magnetic polarizability.

The total mean square fluctuation (integrated spec-

trum) ^{of the} magnetic ^{moment or flux is} simply ^related

to the polarizabilities ^{at 0)}

^{0 and oo and to the}

response of the system ^{to an} infinitesimal step ^function

of the external field.

In dealing ^with superconductors it proves ^to be

helpful ^to represent ^the inertia of the _{super and} normal fluid components ^{in terms} of the kinetic inductances.

Similarly the behavior of ^a Josephson junction ^{can be} represented by ^{an effective} inductance for small phase

fluctuations. These equivalences ^are adopted ^{here to} provide uniformity ^{of notation.}

In practice the fluctuations of relevance _may either be the integrated spectrum (in ^wideband systems) ^or

the low-frequency portion ^{of the} spectrum (in ^narrow

band systems). Special attention is therefore devoted

to these _limiting forms. Depending ^{on the} configura-

tion either the fluctuations of magnetic ^moment (for

a solid body) ^or of magnetic ^flux (for ^a loop ^{or hollow} body) may be of primary ^interest.

Throughout the sinusoidal response functions ^are in terms of the function exp (icôt) ^and parameters ^are expressed ^{in e.m.u.}

Spectral densities for fluctuations are defined such that the variance of a quantity ^is given by ^the integral :

II. General Relations.

We first note some general

relations connecting ^the spectrum ^of fluctuations and the integrated variance with the response function of the system ^{to an external} magnetic ^field.

If the magnetic ^{moment of a} body ^{is m} exp (Í6Jt) ⁱⁿ

response ^{to a} small sinusoidal magnetic ^field H exp (Í6Jt)

their ratio is denoted by m/H

rl(6J) ^where ce(m) ^is

the complex polarizability. This function is analytic

in the lower half of the 6J-plane and it follows by

contour integration ^{that :}

where oc

a’

ia". The dissipative ^part ce"(ce) ^is

a non-negative ^odd function of frequency ^{which va-}

nishes at ce

0 and cc.

From the fluctuation-dissipation ^theorem (Landau

and Lifshitz, 1958), it may be deduced that the ma-

gnetic ^moment fluctuation spectrum ^{is :}

If the significant frequencies ^{in the} dispersive ^beha-

vior of (X( (0) ^{lie below} kT/~, ^the spectrum simplifies ^to

the classical form :

The total variance of the moment fluctuations inte- grates ^{to :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:019700050107700

78

Now the function ce(m) is related to the response function for an abrupt change ^of magnetic ^{field. If}

the applied magnetic field is increased by ^{an infini-}

tesimal step ^{8H at t}

^{0 and} 3m(t) ^{is the} resulting

time variation of the magnetic moment, the incre- mental response function is defined ^{as :}

This is in general ^{a tensor but we shall} only ^consider

cases where it is a scalar by making the coordinate

axes coincide with the principal ^{axes of the system.}

The Fourier transform relation connecting r(t) ^and et. ( (0) ^{is :}

The prompt ^and the final steady ^state response functions are related to ce(ce) by :

The variance of a Cartesian component ^{of the ma-} gnetic ^{moment is thus :}

where r(t) ^{is function} applicable ^{to the} component ^of the moment considered. This result can be further

generalized ^to give ^the autocorrelation function for the moment component :

For a normal conductor r( oo) ^{= 0 if} [J.

1 and

r( oo) > 0 if y > 1 while for a superconductor r( oo) ⁰ corresponding ^to diamagnetic ^behavior.

For all systems r(0+) ^{0 and is} simply ^{related to}

the geometry ^{of the} system.

One situation of practical ^interest is that of a long

solenoid enclosing ^the magnetic body ^{which can be}

of arbitrary shape. ^The magnetic ^{moment of the} body m ^{in the direction} of the solenoid axis ^sets _up in the solenoid a flux :

where no ^is the number of turns per unit length ^{of the}

solenoid. This flux is independent of the dimensions of the solenoid so long ^{as its} length ^is large compared

with its radius.

The observable would be the EMF induced in the solenoid by variations of the magnetic ^{moment. If}

the body is in thermal equilibrium ^at temperature ^T the spectrum of the induced EMF is :

This is just (kT/03C0) times the resistance induced in the solenoid by the presence of the body, ^{as is indeed} required by ^the general thermodynamic validity ^of Nyquist’s ^theorem.

III. Simply-Connected System.

III.1. SPHERE.

-

The sphere exemplifies ^{the case of a} simply-connec-

ted body ^{with an} isotropic demagnetization coefficient

so the polarizability ^{tensor is} particularly simple.

A sphere ^of radius a, ^normal conductivity ^{a and} permeability [L ^{= 1 has} polarizability :

where z

Ka

a(47ti(ù(j)1/2 ^{and K is the wave}

constant.

Thus oc 0

^{0 and} cc(oo) == - 1 a3 ^{gives :}

The low-frequency ^moment spectrum ^{is :}

If however the sphere ^is ferromagnetic with permea-

bility p,., these results become modified to :

where f ( z) ^{= z-1 coth z}

^{z- 2 and now :}

The limiting ^{values are :}

As expected ^{the final} steady ^state response :

is not zero but has the positive ^value appropriate ^to

a polarizable sphere ^of permeability f1. taking ^account

of the demagnetization ^{effect. The} variance and the low frequency spectrum ^{of the} magnetic ^{moment are}

greater than those for a non-ferromagnetic sphere :

We discuss the variance of the magnetic ^moment

instead of its mean square value since for ^a ferromagne-

tic system there may be ^some steady magnetization

upon which the fluctuations are superimposed. ^The permeability 03BC ^{here must be} interpreted ^{at the static}

differential permeability (dB/dH) ^{taken at the value}

of the steady magnetization.

Some ambiguous ^{results can} arise if the sphere

with 03BC > 1 be assumed to have zero conductivity

i.e. an ideal ferromagnet. The value of x(0) ^is unchanged ^{but the} evaluation of z(oo) ^{and hence}

of tl( (0) ^{for zero} conductivity ^is indeterminate. In this situation and others of similar character it must

be recognized ^{that the} conductivity ^{of a real ferro-} magnet may be _very small but positive ^{for eu} > 0.

Thus in evaluatingf( (0) ^{to obtain} ce (0) ^{we let m - aJ}

and 03C3 ~ 0+ which again ^{leads to} the value found above for oc ( oo ) .

A further factor is that the permeability ^{itself will}

in general ^{be a} complex ^{function of} frequency ^{due to}

the inevitable inertia of the spin ^{motion in a varying} magnetic ^{field. Thus in} evaluating ce oo) ^it is relevant that ^not only ^is f (z) frequency dependent ^{but so also}

is _[.L. These considerations are without effect if f ( 00 )

is taken as 0, yielding again ^{the above} expression ^for

var m even when there is dispersion ^of 03BC.

It is important ^{to remember that the} conductivity 03C3(03C9) ^is complex and bounded at high frequency. ^For

a normal conductor in which the electrons can be

considered to have a momentum relaxation time _{Tn :}

where nn ^{is the} (normal) ^electron density. ^{Thus as}

(ù -+ oo the wave constant :

where 11n

m/nne2.

Thus K and z do not tend to infinity ^{in this} limit,

and the corresponding characteristic length :

for a typical metal is about 2 X 10-6 cm. Thus except ^{for very small} particles ^we may take z(co) > ¹

as m - oo and thus f( (0)

^{0 as} assumed above.

This limit is equally valid for a ferromagnetic ^con-

ductor.

If however our concern is with _very small spheres

the earlier expressions for the variance of the magnetic

moment must be modified to take account of the finite value of K( oo) and becomes :

In discussing superconducting ^{bodies we shall use}

the approximation ^{of London} (London, 1961).

The results for ^a superconducting sphere ^{can then be}

derived from those of a normal conductor of the same

configuration by ^a simple procedure using ^{the two-}

fluid approach.

The wave constant K used above for steady-state periodic excitation at frequency ^{m in a} normal conduc-

tor with unity permeability ^{is :}

For a superconductor ^we modify ^{this to} include the contributions of both the super and normal fluids :

where 11s

m/ns ^{e2 with} ns

superfluid number den-

sityand À is the London penetration depth

(As/41t)1/2.

The limiting values of the wave constant K(to) ^are

then :

where n _{= ns} --+- _nn ^and A -1

As’ ⁺ An’.

The essential difference between the normal and

superconducting spheres is that for the latter z(O) ^is

not zero but 2/À.

Thus in calculating ^{the low} frequency spectrum

care is needed in approximating ^since z(w) ^{is not} necessarily ^small (as for the normal sphère). ^Indeed

if the radius of the sphere ^is large compared ^{with the}

London penetration depth :

so the low-frequency ^moment spectrum ^{is :}

which is smaller by the factor (45/2) (03BB/a)3 ^{than the}

low frequency ^moment fluctuations in the corres-

ponding ^normal sphere.

If a/03BB ^{is small} compared ^with unity ^{we recover the}

result found for the normal sphere ^{at low} frequency :

The presence ^of the superfluid ^{does not therefore}

manifest itselfin the magnetic fluctuations for particles

small compared ^{with h.}

Since the penetration depth ^{03BB increases as T in-}

creases to Tc, it is ^{seen that the} fluctuation spectrum increases rapidly ^and steadily ^with temperature ^and joins smoothly ^to the value for a normal conductor

at T

T,. ^{This is a} general inference for an arbi- trary shape of the conductor since K((ù) ^{varies conti-} nuously ^with temperature through T,.

The integrated spectrum ^{of the moment} fluctuations in a superconducting sphere ^{also has two} limiting

forms depending ^on whether the radius is large ^or

small compared ^{with À.}

For a » h ^{the moment} variance is smaller by ^the

factor 3(h - h’) la compared ^{with the same} sphere

in the normal state :

where 03BB’

(A/4n) . For a ~ 03BB’ :

which is determined by the normal fluid and corres-

ponds ^{to the} result for the normal sphere ^with p.

1.

It is a general result that for superconducting ^bodies

with dimensions small compared ^{with À} the relevant electron density is that of the normal fluid in deter-

mining ^the magnetic ^moment fluctuations.

III.2. LoVG CYLINDER.

A long cylinder ^{of a}

normal conductor with unity permeability ^has pola- rizability in the axial direction given by :

where ^a is the radius and 1 the length ^{of the} cylinder

and z

a( 4ni(j(ù) 1/2 ^{as for the} sphere.

Thus (x(0)=0 ^and oc ( oo) _ - a2 lj4 ^{for the} cylinder ^radius large compared with the skin depth :

The low frequency spectrum ^{of the moment fluc-} tuations is :

If the cylinder ^is ferromagnetic ^the polarizability

becomes :

where the first term represents ^{the induced} magnetiza-

tion and the second term the diamagnetic effect of the

eddy ^currents.

The variance and low frequency spectrum ^{are now :}

where _{fL ===} fL’ - ifL".

If the cylinder ^is superconducting ^{we make the same}

modification to the wave constant K as we did for the sphere. ^{For a » h} the variance of the magnetic

moment is :

80

which is 2 (h

À’) la times the variance for the cylinder

in the normal ^{state whereas} for ^a radius small compared

with the London penetration depth :

A generalization ^{of the moment} variance for ^a solid

superconducting body ^of arbitrary shape ^and having

all dimensions large compared with X is :

where S. is the surface area of the body ^{and D is the} demagnetization coefficient in the direction of the observed component ^{of the moment} (e. g. for ^a cylin-

der D

0, ^{for a} sphere ^D

1/3).

III .3. MEDIUM COMPOSED ^{OF A LATTICE OF MAGNE-}

TIC PARTICLES.

Here we consider a medium with

macroscopic polarizable particles distributed in ^a cubic array and for simplicity ^of isotropy ^we consider the

particles ^{to be} spherical. ^The magnitude ^{and orien-}

tation of the particle ^{moment is} determined by ^the

local field.

The relation between the local field H1 ^and magne- tization M assumed to be given by the Lorentz relation :

where Ho is the external field.

If 03B11(03C9) ^{is the} polarizability ^{of each} particle ^and

n is the particle density :

A long cylindrical sample of the medium (of ^vo-

lume V) ^{thus has} polarizability :

Hence if the particles ^have static differential per-

meability lL :

where F

4na3/3 ^{is the} particle filling ^factor.

The variance of the axial magnetic ^{moment of the} cylinder ^{is : -.}

For a given filling ^{factor F} the variance increases

by the factor 3/ (1 - F) as p. goes from 1 to cc. It may be inferred that the filling factor is of more

significance ^{than the} particle permeability ^{in deter-} mining the fluctuations.

IV. Ring ^{or Hollow} Cylinder.

^{IV.1. NORMAL}

CONDUCTOR. - The elementary multiply-connected

systems ^{of interest are the} planar loop ^{and the} long

hollow cylinder. ^{In both cases the} magnetic pro-

perties ^are simple if the thickness of the conductor is small compared with the other dimensions and the field penetration depth.

We first consider a planar loop of normal conductor with impedance Z(co) ^{and area} A. Then the pola- rizability in the direction normal to the plane ^{of the} loop ^{is :}

The fluctuation spectra ^{of the} loop flux (D is :

where L is the inductance of the loop.

We shall express the results for hollow systems ⁱⁿ

terms of the flux spectrum since this would usually ^be

the observable of most direct interest.

Now the impedance ^is composed ^{of the} magnetic

and the electron contributions :

where Ln

(mlnn e2) (piS) ^and Rn = Ln/Tn while p

and S are the perimeter and cross-sectional ^area of the loop conductor. Provided that this representation

is valid _up to frequencies of the order of Rn/L ^{so that}

skin effect and radiation impedance ^{are not} important

the fluctuation of the flux is :

and the-mean total _energy is :

’

,

which signifies ^{that the} magnetic energy and the electron kinetic _energy (which ^both depend ^on 12) together correspond ^{to one} degree ^{of freedom in} formulating ^the equipartition ^result.

In practice Ln ^will customarily be very small com-

pared ^{with L. Thus for a circular} loop ^{of radius a}

and circular cross-section radius b :

which will be negligible ^{for a metallic} loop ^{unless b}

is less than about 10-6 cm.

The long thin hollow cylinder ^is very similar to the

loop ^for equilibrium fluctuation properties. ^{The flux}

fluctuation spectrum ^{is still} (kTL2j1t) ^Re (11Z) ^where

Z is now the impedance ^{for a current} sheet with peri- meter p = 2a ^{and S = 1 d :}

where ^a is the radius of the cylinder, ^{l its} length ^and

d the wall thickness which is assumed to be small.

Accordingly ^{the mean} square flux fluctuation is

k TL2/ (L ⁺ L.) ^{as for the} loop. ^{The ratio} LJL ^of

kinetic to magnetic inductance is :

If this is small compared ^with unity ^{the mean} square axial magnetic ^{field at} any point within the hollow

region ^is given by : ^-.

which is the value corresponding ^to equating ^{the ave-}

rage field _energy ^to (1/2)kT.

IV. 2. SUPERCONDUCTOR.

For a superconducting loop ^{or hollow} cylinder ^the impedance Z(cù) ^{can be}

written in the first approximation for the two-fluid model :

where Lg

(mlns e2) (piS) is the effective inductance associated with the kinetic energy of the superfluid component ^while Rn

L.1-r. ^{is the} resistance asso-

ciated with the normal fluid. _{Here Tn} is the momen-

tum relaxation time of the normal electrons.

The spectrum and variance of the flux fluctuations

are :

where Q

L(LS ⁺ Ln) ⁺ Ls Ln.

It is seen that the flux fluctuation for a given loop

or hollow cylinder is much smaller in the superconduc- ting ^state than in the normal state if L is large compared

with L, ^and Ln. Then for the superconducting loop :

Nevertheless it must be stressed that the fluctuations do not undergo ^a discontinuous change ^{at the tran-}

sition temperature, ^{since as this} temperature ^is approa- ched from below _n, tends to zero, Ls ^{tends to} infinity

and (D2 uniformly ^{tends to the value} kTL2/(L ⁺ Ln)

for the normal ^state.

The reason for the large ^{decrease of D2 below} Tc ^is

that the kinetic _energy terms (1/2)Ln I2n ^and (1/2)Ls I2s

for the normal and superfluids considerably ^exceed

the magnetic ^term ( 1 /2) L12

03A62/2L. ^{From the two-}

fluid model it is readily found that the _energy ratios

are :

and :

The field fluctuations inside a hollow superconduc- ting cylinder thus decrease rapidly ^{as the} temperature is reduced; ^{for instance in} going ^{from t}

^{0.9 to}

t

0.5 the RMS field is reduced approximately

sevenfold (t

T/Tc).

It should be remarked that the equations ^derived

here incorporate ^the conservation and quantization

of fluxoid in the form (D + L, I,

^const.

Nhl2e.

Quantization ^{is not however} relevant in the discussion of fluctuations unless one is concerned with transitions from one fluxoid ^{state to} another but this topic ^lies

outside the scope of the present paper.

IV. 3. SUPERCONDUCTING LOOP WITH JOSEPHSON JUNCTION.

^{Since this} problem has been discussed earlier (Burgess, 1967, 1968), it will be outlined here and the essential results presented. ^The Josephson junction ^{can be} conveniently represented ^{as a} parallel

combination of its capacitance C, the conductance G due to dielectric loss and quasi-particle tunneling ^and

thé effective inductance Li

n/2eI1 representing ^the Josephson relations between voltage, phase cp and the

junction ^current 7j whose maximum or critical value is 11.

The conservation and quantization conditions are :

where IJ ^{is the} Josephson tunneling ^{current in the} junction. ^{If the} system is in thermal equilibrium ^the

fluctuation sources are given by ^the Nyquist ^relation applied ^{to the} Rn ^of the normal fluid in the super-

conductor and to the conductance G of the junction.

The spectral density of the flux fluctuations is then :

The mean square flux fluctuation is thence :

which is always greater ^{than the flux} fluctuation in the superconducting loop ^{without a junction.}

If L is large compared ^with Zg ^and Ln ^this simplifies

to :

If the junction conductance G is not too great ^the system ^{can be} approximated by ^the loop inductance L connected to the junction ^reactance composed ^{of C}

and L J ⁱⁿ parallel. ^{Then the} fluctuation spectrum of the current has its maximum ^at the frequency :

and has its minimum at the (slightly) ^lower frequency equal ^to Josephson’s "plasma frequency" :

Thus the spectrum ^{has its most} conspicuous ^features

of a minimum at 0153j and a maximum at 03C9J(1 -p LJIL) 1/2

in a relatively ^narrow range of frequency.

For an extremely ^weak junction ^with very small Il

such that Li ^{exceeded L the flux} fluctuations approach

the value for a superconducting loop ^{with a resistive} junction discussed in the next section. If the Josephson junction ^{is not too weak so that L is large} compared

with Li ^we must revert to the unapproximated ^(D2

where we see that Lj, Ls ^and Ln ^are competing ^for

control of the current fluctuations which are then _very much smaller than kT/L.

IV.4. SUPERCONDUCTING LOOP ^WITH RESISTIVE

JUNCTION.

^{If a} superconducting loop ^is interrupted by ^a junction ^(e. g. of dielectric or normal metal)

which cannot support ^a supercurrent ^{the current} fluctuations are markedly ^enhanced.

First we consider a narrow resistive region ^{of resis-}

tance R which is too wide however to permit tunneling

of a supercurrent. ^In this case we do ^not invoke the

Josephson conditions and no quantization ^condition

for flux and junction phase ^is applicable. ^The system behaves like a simply-connected superconductor ^to

which the potential difference

IR is applied. ^The

two-fluid model is used for the superconductor ^{to treat}

time varying ^{currents. There are two} fluctuation

sources

the resistance Rn associated with the normal

82

fluid in the superconductor ^{and the R of} the resistive

junction.

In terms of the notation used previously :

This spectrum ^{as well as} the response of the system

to an abrupt change of external magnetic ^{field corres-} ponds ^{to two} relaxation times _03C41 and _T2 whose recipro-

cals are the ^roots 81 and S2 ^{of the} quadratic equation : S2 Q - 8[Rn(L + LS) ⁺ R(Ln + Ls)] + RRn

^0.

These relaxation times will be approximately (L ⁺ Ls) IR ^and (Ln + Ls) I Rn if Ls ^{is small} compared

with L or Ln that is for temperatures ^{not too close to} Tc.

On the other hand the superconducting loop ^without

a junction ^{has a} single relaxation time :

which is much smaller than the value (Ln ⁺ L)/Rn

for the normal loop ^but always larger ^than in

Ln/Rn.

The relaxation time is continuous at T,.

The flux fluctuation is :

which is _very close to that for the normal loop. ^Thus

when only ^a very small segment ^{of a} superconducting loop ^{is resistive the flux} fluctuations are greatly ^enhan-

ced over the value for a completely superconducting loop ^and approach ^the limiting ^value kTL2J(L ⁺ Ln) .

No discontinuity ^{occurs in the flux} fluctuations when the temperature ^{is taken} through Tc ^{because :}

is a constant and is equal ^to the value of Ln ^{when the} loop is in the normal state.

The results for the loop presented above show that there is a steady ^increase in the flux fluctuations _pro-

ceeding ^{in the} sequence : superconducting loop

~

superconducting loop ^with Josephson junction

~

superconducting loop ^with resistive junction

^nor- mal loop. ^The popularly cited value of k T L for the flux variance is a good approximation ^{for the last two}

of these situations but a gross overestimate for the first two situations. This results from the inhibiting

influence of fluxoid conservation on flux fluctuations in the superconducting loop ^{with or without a} Joseph-

son junction. ^{With a resistive} junction ^{there is no}

fluxoid conservation or quantization. ^{One criterion}

for the existence of fluxoid conservation in a loop ^is

that the limit of iú) 1 z (ú)) ^{as 03C9 ~ 0 is non-zero.}

REFERENCES BURGESS (R. E.), Quantization ^and Fluctuations in Su-

perconductors. ^Proc. Symposium ^on Physics ^of Super- conducting Devices, Charlottesville, 1967, pp. H1-H17.

BURGESS (R. E.), Thermodynamic Fluctuations in Super-

conductors. Proc. Conference

Fluctuations in Su-

perconductors, Asilomar, 1968, _{pp. 47-79.}

LANDAU (L. D.) and LIFSHITZ (E. M.), Statistical Physics (Pergamon, 1958), sections 119-122.

LONDON (F.), Superfluids, ^1, Macroscopic Theory ^of

Superconductivity (Dover, 1961), chapter ^B.