Non monotonic variation of the critical current as a function of the magnetic field in a heavily cold worked type ii superconductor

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Non monotonic variation of the critical current as a function of the magnetic field in a heavily cold worked

type ii superconductor

J. Baixeras, J. Maldy, E. Santamaria

To cite this version:

J. Baixeras, J. Maldy, E. Santamaria. Non monotonic variation of the critical current as a function

of the magnetic field in a heavily cold worked type ii superconductor. Journal de Physique, 1971, 32

(4), pp.349-355. �10.1051/jphys:01971003204034900�. �jpa-00207085�

NON MONOTONIC VARIATION OF THE CRITICAL CURRENT AS A FUNCTION

OF THE MAGNETIC FIELD

IN A HEAVILY COLD WORKED TYPE II SUPERCONDUCTOR (*)

J. BAIXERAS

Laboratoire de Génie

Electrique

^{de la Faculté} des Sciences de Paris

(**)

L. C. I.

E.,

^{B. P.}

8, 92, Fontenay-aux-Roses

J. MALDY and E. SANTAMARIA

Les Laboratoires de

Marcoussis,

^Centre de Recherches de la

Compagnie

^Générale

d’Electricité, 91,

^Marcoussis

(Reçu

^{le 24 décembre}

1970)

Résumé. 2014 Nous proposons ^un modèle de

piégeage

pouvant

expliquer l’apparition

^d’une

région

^où

dJc/dH

^est

positif.

^{Ce modèle} repose ^sur l’existence de deux différents mécanismes de

piégeage.

^{Nous avons} obtenu des résultats

expérimentaux

^{dans le cas} d’un ruban de Nb-1

%

^Zr,

que nous avons

comparé

^{avec les}

prévisions théoriques

^{en ce}

qui

^{concerne la}

dépendance

^{en tem-}

pérature

^de Hp,

champ magnétique

pour

lequel le

^courant

critique présente

^un

minimum,

^{et en}

ce

qui

^{a trait à} l’influence du paramètre

critique

^{03B1c. Nous} obtenons ainsi un bon accord

qualitatif, qui indique

que ^nos

hypothèses

^de

départ

^sont

réalistes ;

^{celles-ci sont} par ailleurs bien

étayées

par des observations au

microscope électronique.

Abstract. ²⁰¹⁴ We _propose a

pinning

^model

accounting

for the appearance of ^a

large region

where

dJc/dH

^is

positive.

This model is based on the existence of two different mechanisms of

pinning.

^The

experimental

results obtained on a

sample

^{of Nb-1}

%

^{Zr are}

compared

^{with the}

theoretical

predictions

^{of our model :} temperature

dependence

^of Hp, ^{the field} for which the critical current exhibits a

minimum,

and influence of the critical parameter ^{03B1c. The}

assumptions

involved in the calculations are well

supported by

^electron

microscopy observations,

^{and we}

obtain

qualitative

agreement ^between

theory

^and

experiments.

Classification :

Physics

^Abstracts

17.24

In this paper ^we

report

^some

experiments

^{on a cold}

rolled

sample

^{of Nb-1}

% Zr,

^{in which the movement}

of vortices is

highly anisotropic,

^{and also exhibit a}

marked increase

of Je

^{as a} function of

H,

when the current is

perpendicular

^to

the rolling

direction. In order to account for the observed

phenomenon,

^we

propose ^a

pinning

^model

by

^{which the}

pinning strength

can become an

increasing

function of the

magnetic

field. Our

experimental

^{results are in}

good qualitative agreement

^{with the}

predictions

^{of the}

model,

^{so it}

seems that the

physical assumptions

^{involved in the}

theory

^are rather realistic.

In

figure

^{1 the}

plots

^{of the}

voltage

^{across the}

sample

for two

configurations

^{of the current with}

regard

^{to the}

rolling

direction demonstrate the drastic

anisotropy

ⁱⁿ

the motion of the vortices for a current

larger

^{than the}

critical current. This

anisotropy

^{has been}

pointed

^out

previously

^{in this material}

by measuring

^{the Hall}

angle

in the mixed state

by

Williamson and one of ^us

[1].

(*) Supported ⁱⁿ part by the D. G. R. S. T. (Comité ^Electro- technique Nouvelle).

(**) Laboratoire Associé au C. N. R. S.

This is a characteristic

of many

cold rolled

samples [2].

The curve

(a)

ⁱⁿ

figure

^{1 is}

interesting

^{in another}

respect,

^{i. e. it shows the} appearance of the ^so called

«

peak

^{effect », observed}

by

many workers on different

FIG. 1. ^- Curves E versus H are shown for two different orien- tations of J with respect ^{to the} rolling direction :

a) 0 ^{= 90°} (i. ^{e. J} perpendicular ^{to the} rolling direction).

b) 0 ^{= 0°} (J//rolling direction).

(al), (a2), (a3) correspond ^{to different values of J.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01971003204034900

350

materials

[3]. Although

many theoretical and

experi-

mental

attempts

^{have been made}

[4], [5], [6], [7], [8],

until now no

fully satisfactory explanation

^{of this}

phenomenon

^{has been}

proposed.

^{Two relevant}

aspects

of this effect appear in all the

previous

observations.

The first for pure materials results in a

sharp

^{increase of}

Yc

^{in the immediate}

vicinity

^of

H,,2.

^The

second,

^most

frequently

^{observed in}

dirty

^materials consists in a

large region

^{of H}

(for

^our

sample

from about

Hc2/2

to near

Hc2)

^{where the}

slope dJc/dH

^is

positive.

^This

last

phenomenon

^is

strongly depending

^{on the struc-}

tural state of the

sample

^{and in our}

opinion

it is related to the interaction of a

part

^{of the flux} lines with another

part

^{of vortices more}

strongly pinned

^{on suitable}

defects.

In order to find a model

accounting

^{for this}

effect,

^we

make the

following assumptions :

1)

^In the material there are classical

pinning

^{forces of}

the

type

^described

by

^Anderson and Kim and

leading

to a critical

pinning strength

per unit volume _Clc.

2)

^{On the other}

^hand,

^there exist in the material a

few

particularly strong pinning

^{centers which are}

regularly spaced.

^{The distance between} two centers is denoted

by

^« 2 d ». The vortices

pinned

^{on these defects}

can interact

by

^{means of the}

electromagnetic

^forces

with the other vortices

(which

^{we call for} convenience

pseudofree although they undergo

^a

pinning

^{force of}

the Anderson

type).

Now we consider the case when

pseudofree

^vortices

(P.

^F.

V.) attempt

^to pass between ^two

pinned

^vortices

A and B

(Fig. 2).

^The

pinning

^{force on the P. F. V.}

FIG. 2. ^- A and B represent ^{two vortices} strongly pinned, ^and

the other vortices tend to pass between A and B under the influence of the Lorentz force.

is then the sum of the classical forces and of the force

resulting

^{from the}

electromagnetic

interaction with the vortices A and B. This mecanism results in an enhanced

pinning

^{force on the «} P. F. V. » and it is obvious

that,

when there are sufhcient

pseudo

^{free vortices between}

A and

B,

^the

electromagnetic repulsive

^{forces can}

become more

important

than the classical one. If the

applied magnetic

^{field H increases so that the number}

of free vortices between the two

strong pinning centers

becomes more and more

important,

^the

pinning strength

^{can become an}

increasing

^{function of}

H,

^{i. e.}

the

« peak

^{effect » can} appear. It is

important

^to

emphasize

^{that this}

phenomenon

^{is due to} the existence.

of two different

types

^of

pinning.

The critical current is reached when the Lorentz force is

large enough

^{to overcome the sum of the}

Anderson

type

force and of the

repulsive

^{force due}

to A and B.

Thus Je

^{is defined}

by

the maximum

pinning strength acting

^{on a} P. F. V. Now the

problem

^{is to}

calculate the force on the first vortices which can move

and not the force _per unit volume

(J,,

^x

B).

^which

concerns all the vortices.

The

electromagnetic

force per unit

length

^between

two vortices is

(in

^{the limit}

K > 1) (9)

where is the London

penetration depth, 00

^{the flux}

quantum,

^and

Ir,

^-

r2 1

^the

spacing

^{between the}

vortices, Ki

the modified Bessel function.

The number of

pseudo

free vortices between A and B is of the order of d

B/n (n

⁼

B/ Wo),

^{so that the net}

force exerted

by

these vortices on the

pinning

^center,

^A,

is _1-

where rA

^and ri ^are the

position

^{vectors of the vortex A}

and the

itb

P. F. V.

respectively.

Thus a vortex

pinned

^{in A exert on the P. F. V.}

the force -

Fi.

^{We can calculate a mean value of the}

force exerted

by

^{A on each P. F. V.}

lying

^{between A}

and B

(the right

calculation would be the determination of the

pinning

^{force on the less}

pinned

^{P. F.}

V. ;

nevertheless we think that the mean value is a

good approximation) :

Putting

y ⁼

2/ À Jn,

^we get :

But we have interest

only

^{in the}

component parallel

^to

the direction of

possible

^{motion of the P. F.}

V. ;

^{so we}

introduce a

coefficient f3

^{to evaluate the effective}

pinning

force directed

along

this direction and result-

ing

^{from the described mechanism.}

In order to take into account the Anderson

type pinning

^{forces we make the}

following approximation :

the mean value of this force on each P. F. V. is

fp

⁼

^oc,,In.

Therefore the total mean

pinning

^force

acting

^{on one P. F. V.}

lying

between A and B is

Let us call

Hp

^the

magnetic

^{field for which} the critical current exhibits a minimum as a function of H. This

particular

^value of H is of no

physical signification,

but appears ^{as a} convenient

parameter

for the compa- rison between the theoretical

previsions

^{and the}

experi-

mental results. In order to determine

Hp

^{we calculate}

the value

yp

^{for which}

dF/dy

^vanishes.

For the calculation of

df /dy,

^{we take into account}

only

the first three terms of the sum involved in the eq.

(4).

^{That is a}

sufficiently

^accurate

method,

^{as the} function

Kl

^decreases

rapidly :

We have also

and in order to determine

yp,

^{we have to solve}

equation

The term between brackets is an universal function and this leads to an easy

graphical

^{solution. In}

prin- ciple

^the

problem

^of the calculation of

Hp

^{is solved but}

the value

of f3/d is

unknown. With the

object of testing

our

model,

^we circumvent this

difnculty by calculating P/d

^{from the}

experimental

^{value of}

_yp

^at 4.2 OK. Then

we calculate the values of the function

yp

⁼

yp(T)

^for

several

temperatures

^{and we} compare them with the

experimental

^results

(In

^{fact we calculate}

Hp

^from

yp).

From the values of

He2

^and

He at

^4.2

OK,

^{we deter-} mine À ⁼ 630

À,

^{and the}

plot of Jc

⁼

f(H)

^{for low}

values of H

gives

^us oc,,, ⁼ 6.3 x

10’

M. K.

S.,

^the

experimental

^value

of yp being yp

^{= 2.45. So we obtain}

from

equation (7)

To determine the function

yp(T),

^{we make use of the}

following

relations :

the later is an

experimental

result that ^we have obtained for T

varying

from 2 "K to 5.18 OK.

We take

so that

and,

^{for different}

temperatures,

^{we have to solve the}

equation (7),

^{which becomes :}

Figure

^{3 is a}

plot

^{of the}

experimental

^{values of}

Hp

and of the curve calculated from the

present model, showing

^a very

good agreement.

FIG. 3. ^- The solid curve represents ^the variation of Hp ^{vs T}

as determined theoretically ^{from our} model. + : experimental

data taken for decreasing ^values of T. d : experimental ^data

taken for increasing ^{values of T. The} experiments ^{for T} > 4,2,OK

were carried out in a bath of liquid ^{helium under} pressure.

Our calculation seems to take into account

only

^the

free vortices

lying approximately

^{on the same line as}

the vortices A and

B ;

^{in fact} the influence of all the other vortices is included in the classical

pinning strength

ac. However we

neglect

^the

rigidity

^{of the flux}

line

lattice,

^{but this effect is}

important only

^near

Hc2

and

Hp

^{is about}

Hc2/2,

^{so our}

assumption

^{seems rather}

valid.

It is _very

interesting

^{to examine the structure of our}

samples

^{in order to check the}

validity

^{of our} assump- tions and to

specify

^{the nature of}

imperfections

respon- sible for the observed

phenomena.

Electron

microscopy

examinations reveal the exis- tence of two main kinds of structural

imperfections (Fig. 4) :

- cold work cells with diameter in the range 0.2 gm- 0.6 _tim formed

by tangles

^of

dislocations,

- thin dislocation

walls, nearly perpendicular

^{to the}

sample.surface,

about 0.15 _gm

apart

^from each other.

352

Because of a strong

(100) [ 110]

texture, these walls ^are

nearly parallel

^{to the}

rolling

^direction

(the misalign-

ment is lower than 20

degrees).

This second kind of defects is

directly responsible

^for

the observed

anisotropic

^behaviour.

FIG. 4. ^- Typical aspect ^{of the} samp!e ^{structure observed} by transmission electron microscopy. ^{In this} picture, ^{the contrast is} adjusted ^{to see the two main} kinds of defects : thin dislocations walls (W) ^oriented along ^the rolling ^direction (R. D.) ^{and cold}

work cells formed by networks of dislocations (C).

We think that the

peak

^{effect is} connected with both kinds of defects : the dislocation walls are efhcient in

guiding

^{the vortices} and then force them to pass between the

strong pinning

centers ; both ^{kinds of} defects

(tangles

^and dislocation

walls)

contribute to Y,,,.

The

changes

ⁱⁿ orientation and

spacement

^{of these} walls are

responsible

for the existence of the

strongest pinning

^centers.

According

^{to the estimated value of}

d/fl

^{and to the}

microscopic examinations,

^{it seems that}

f3

^{must have a} very small value

(in appendix

^{we have}

calculated

fi

^for

only

^{one P. F. V. between A and}

B).

It is

noteworthy

^{that we} have also noticed this

phenomenon

^{in the case of} pure niobium

strip showing

after cold work the same structure as our

samples

^of

Nb-1 %

^{Zr. In our}

opinion,

^{the zirconium content}

plays

^no

prominent part

ⁱⁿ the appearance of the

peak

effect. It

just

^{induces an increase of the K}

parameter.

Besides the action of

temperature

^{on the value of}

Hp,

our model allows us to

interpret

^{at least}

qualitatively

several

experimental

observations. From

equation (7)

^it

is _easy to see that

decreasing

values of the critical _para- meter oc

correspond

^to

decreasing

^{values of}

HP

^{and to}

an increase of the relative

amplitude

^{of the}

peak effect ;

the calculation is

straightforward.

To act on ac ^we have modified the structural state of the

samples by

^{means of two different} treatments :

1)

Modification of the surface state : chemical or

mechanical

polishing

^{of the}

sample

surface results in a

strong

^{decrease of} ac which is

apparent

^{from the}

curves of the critical current

density Jc

^{vs the}

applied magnetic

^field

(Fig. 5).

^In

agreement

^{with the}

predic-

tions of the

model,

these surface treatments induce an

increase of the

peak

^effect

amplitude

^{and a decrease}

of

Hp.

^The

^agreement

^{between measured and theore-}

tical values of

HP

^{is not so}

^good

^{as obtained}

^previously

but is

quite satisfactory (deviation

^{of 20}

%)

^{if we take}

into account the

rough

determination of _oCc’

FIG. 5. ^- Influence of chemical and mechanical polishing ^of

the sample ^{surface on the critical current} density Je [Jc ^{is defined}

as the current density ^{for which a} voltage ^of 10-7 V appears

across the sample] ^{and the} peak effect. Ae is the total decrease of sample thickness after the polishing. ^{The curve for Ae = 3} pm is obtained by mechanical polishing. ^{This treatment} provides

very smali stretches at the surface and induces an increase of the surface current. This phenomenon explains ^{the observed} anomaly

in the low current critical field.

2)

^{Influence of}

metallurgical

treatments : cold-

working

^{of the}

samples

^{induces an increase of} ac and thus a decrease of the

peak

^effect

amplitude.

^The

influence of vacuum heat treatments is more

complicat-

ed : for

annealing temperatures

^lower than 570 °C we

observe a decrease of etc connected with the

expected

variation of the

peak

^effect

amplitude.

^For

higher temperatures

^{we observe the}

opposed

^behaviour

(Fig. 6) (In

^this

Fig. Io

^{is the current for which the}

peak

effect appears in the curves

E(H) -

^see

Fig. 1.)

Electron

microscopic

examinations show that

these

anneals have

only

^a noticeable action on the

tangles

^of

dislocations and have

practically

^{no action on the}

dislocation walls

parallel

^{to the}

rolling

^direction

(if

^the

annealing temperature

^{does not} exceed 1 000

OC).

^This

observation corroborates the fact that the increase of

Je

is connected with these

tangles.

FIG. 6. ^- Influence of the annealing temperature ^{on the current} 10 ^{for which the} peak ^effect appears in the curves E(H), ^see figure ^1. Here, lo is ^{the current carried} by ^a sample ^{one centi-}

meter wide. The variation of lo is connected with the same

modification of the critical current for a given magnetic ^field.

The

sharp

decrease of the critical current near

He2

is not

explained by

^our model. It could be caused

by

the fact that the Lorentz force overcomes the

pinning

force of the

stronger

defects but we

prefer

^another

explanation.

^Indeed several observations in other

experiments (10) suggest

^{that this}

phenomenon

^is

related to the

disappearance

^{of the vortex}

concept

^when the distance between the cores becomes

sufficiently

small.

The

experimental

^{results we}

reported

^{here are in}

good qualitative agreement

with the theoretical _pre- dictions of the

proposed

^{model of}

supplementary pinning.

^{In our}

opinion

^{this fact means that the mecha-}

nism of interaction between a few

strongly pinned

vortices and the other vortices is

quite realistic,

^{so we}

intend _{to carry} out a more

sophisticated

calculation which should

permit

^a

quantitative comparison

^with

the

experimental

results. The

physical

^{basis of our}

calculation are the

following :

- We suppose the existence in the material of a

square lattice of defects on which vortices are

strongly pinned.

The distance between two

neighbours

^is

«2d».

- We consider 4 P. F. V. in the interior of an

elementary

cell of the defect lattice

figure

^7.

As we have

already pointed

^out, the forces

acting

^on

the P. F. V. ^are of three kinds :

- the classical Anderson

type pinning forces ;

^we

have taken

equal

^to

a,,In

^{the mean value on each P. F.}

V.,

- the

electromagnetic

interaction with all the other

vortices,

- the Lorentz force due ^to the presence of the

transport

current, ^{whose mean value is} taken

equal

^to

J4PO

per unit

length.

FIG. 7. ^- The strongly pinned vortices ABCD determine the

elementary cell in which lie 4 P. F. V (1, 2, 3, 4). The P.F.V. l’and 2’ lie in an adjacent cell. The calculation of the net force acting

on 1 or 2 take all the other vortices into account.

For different values of the Lorentz force

(Fel)

^we

try

to determine the

equilibrium position

^{of the 4 P. F. V.}

considered ;

the value of this force for which the

equilibrium

^{is no more}

possible corresponds

^{to the}

critical Lorentz

force,

^{i. e. to the mean value of the} critical current taken across the material. In the

figure

⁷ one can see the vortices which are involved in the calculation.

Using

^the

symetry

^{in the}

cell,

^we

consider two P. F. V.

only and

^{for each of them we take}

into account 9

neighbours.

^The calculation has been carried out with the aid of a

computer :

^{first of all the}

net force on the two P. F. V. under consideration is calculated and then the P. F. V. are

displaced by

^a

certain amount in the direction of this force. This

displacement

^is

proportional

^to the absolute value of the force. The calculation continues in this way until the net force reaches ^a _very small

(near zero) value, considering

^{that the}

equilibrium position

^{is then}

attained. Next the Lorentz force is

changed

^{and a}

certain number of

equilibrium positions

^{are obtained}

until the Lorentz force exceeds the critical

value,

^for which no more solution of the

problem

^{exists. An}

important

feature of the calculation is that the

equili-

brium

position

^reached

by

the vortices are the _same, for a

given

^value of Lorentz

force,

^{whatever are the} initial

positions

^{chosen for} the calculation. This fact

means that the obtained solution is the

right

^one.

We have determined the

equilibrium positions

^and

the critical currents for various cases. In the

figure

⁸

one can see the evolution of the

equilibrium positions

when the Lorentz force is increased. The Table 1

gives

for different values of « 2 d » the

corresponding

354

TABLE I

Comparison

^between

experimental

and theoretical values

of

^{the critical current}

density Jc, for

^two

particular

choices

of

^{the distance 2} d between the

strongly pinned

vortices. The values

of

^{B are obtained}

_assuming

^that

4 P. F. V. lie in an

elementary

^cell.

magnetic induction,

^{as well as} the calculated and the

experimental

value of the critical current

Jc.

It is

interesting

^{to note} that the increase of

Fel

^from

zero to the critical value induces

only

^{a small}

change

in the vortex

positions.

^{This is related to} the fact that the

electromagnetic

interactions between vortices _may lead to much

greater

^{forces than} the Lorentz

force ;

FIG. 8. ^- Equilibrium positions ^{of the vortices 1 and 2 for}

different values of the Lorentz force Fei. In this case

2 d = 6 À = 3 780 A .

for Xc/n ^- Fet = - 70 x 10-7 no more equilibrium position exists.

thus a small

displacement

^{from the}

equilibrium posi-

tions of the vortices is sufficient to balance the influence of

Fel, provided

^{that an uniform current} distribution is

assumed across the material. The

disagreement

^between

the calculated and the

experiniental Je

^{values seems to}

prove,

consequently,

that the last

assumption

^{is not}

valid.

Certainly

^{the current flows}

preferably

^{near the}

strong pinning

^{centers and} therefore the vortices near A and B

experience

^a

greater

^force

th an Je Wo (J,, being

the mean value of the critical

current).

From these conclusions it _appears that the first

problem

^to solve is how the current flows in ^a

dirty

material.

Specially

the calculation of the current distri- bution should take into ^account the

position dependent

curvature of the vortices.

We ^are indebted to Professor G. Fournet forinterest-

ing

^and

helpful discussions,

^{and we} thank Professor J. Friedel for his remarks on the

manuscript.

^Discus-

sions with Dr.

Cyrot

^{are also}

acknowledged.

Appendix.

^{- We are}

capable

^of

calculating

^the

value

of fl

^{for the case of}

only

^{one free vortex between}

the two

point

^defects A and B. For this _purpose we

determine the

position

of the free vortex,

correspond- ing

^{to the} maximum of the

electromagnetic pinning

force

acting

^{on this vortex. This force is}

r, d

^{and 0 are} defined in

figure 2,

^and

so fi

^{= cos 0. The}

maximum value of F is

given by dFpldO

^{= 0 :}

From the estimated value

of d,

^{we can take} the appro- ximation

which leads to

and we have to solve an

equation

^with

only trigono-

metric functions :

A

graphical

solution of this

equation

^leads

to fl rr

^0.3

Références

[1]

^BAIXERAS

(J.)

and WILLIAMSON

(S. J.),

Sol. State Comm., 1967,

5,

^599.

[2]

^HAKE

(R. R.),

^LESLIE

(D. H.)

and RHODES

(C. G.),

VIII International Conference on Low Tempe-

rature

Physics,

p.

342,

Butterworths,

London,

1963. STAAS

(F. A.),

^NIESSEN

(A. K.),

^DRUYVES-

TEYN

(W. F.)

and VAN SUCHTELEN

(J.), Phys.

Letters, 1964, 13, ^293.

[3] ^{See for}

example,

BERLINCOURT

(T. G.), Phys.

Rev.,

1959,

114, ^{969. LE BLANC}

(M.

^A.

R.)

^and

LITTLE

(W. A.),

Proc. of Seventh International Conference on Low Temperature

Physics,

p. 362,

University

of Toronto Press, Toronto, 1960.

BERLINCOURT (T.

G.)

^{and HAKE}

(R. R.), Phys.

Rev. Letters, 1962, 9, ^293.

[4] ^ANDERSON (P.

W.)

^{and KIM} (Y.

B.),

^{Rev. Mod.}

Phys.,

1964, 36, ^39.

[5]

^MAXWELL

(E.),

^Proc. of tenth International Confe-

rence on Low Temperature

Physics,

paper n° 5103, Moscow, 1966.

[6]

^MAXWELL

(E.),

^SCHWARTZ

(B. B.)

and WIZGALL

(H.).

Phys.

Letters, 1967, ²⁵ A, ^139.

[7] ^MAXWELL

(E.),

^SCHWARTZ

(B. B.),

^WIZGALL

(H.),

^and

HECHLER

(K.),

^Journ.

App. Physics, 1968, 39, 2568.

[8] See forex

ample

and further

bibliography,

^SUTTON

(J.)

and BAKER

(C.), Phys.

Letters, 1966, 21, ^601.

KELLER

(E. L.),

^COFFEY

(H. T.),

^PATTERSON

(A.)

and AUTLER

(S. H.), Applied Phys.

Letters,

1966,

9, 270. JONES

(K. A.)

^{and ROSE}

(R. M.), Phys.

Letters, 1968, ²⁷ A, ^{412. CHANG}

(C. C.),

^MC

KINNON

(J. B.)

and ROSE-INNES

(A. C.),

^Sol.

State Comm., 1968, 6, 639. COFFEY

(H. T.), Phys.

Rev., 1968, 166, ^447.

[9]

^{DE GENNES}

(P. G.), Superconductivity

^{of metals and}

alloys,

^{W. A.}

Benjamin,

^{Inc. New} York, ^Amster-

dam,

^1966.

[10]

^MALDY

(J.),

^{to be}

published.