Magnetic instabilities in hard superconductorsR. G. Mints and A. L Rakhmanov

Institute of High Temperature of the USSR Academy of SciencesUsp. Fiz. Nauk 121, 499-524 (March 1977)

This review treats magnetic instabilities in hard and combined Type II superconductors. We give anddiscuss in detail the criteria for stability of the critical state with respect to magnetic-flux jumps. Westudy the effect of magnetic and thermal diffusion, as well as that of the structure of a combinedsuperconductor, on the magnetic-field value for a flux jump. The theoretical results are compared with theexisting experimental data.

PACS numbers: 74.60.-w, 74.70.Lp

CONTENTS

1. Introduction 2492. Theory of Flux Jumps in Hard Superconductors 2533. Theory of Magnetic Instabilities in Combined Superconductors 2574. Experiments on Flux Jumps, Comparison of Theory with Experiment. . 260Bibliography 262

I. INTRODUCTION

Unusual physical properties and possible technicalapplications have aroused great interest in experimentaland theoretical study of hard superconductors. Thetheme is that of superconducting materials in which thecurrent density j can attain values of 105-10e A/cm2

with insignificant losses, while superconductivity per-sists in magnetic fields up to H= 10s- 10e gauss. At-taining such extremal parameters and operating of vari-ous devices under these conditions are limited in manyways by magnetic instabilities that exist in hard super-conductors. This review is concerned with presentingthe theory of the onset of these instabilities and withcomparing it with experiment.

A. Hard superconductors

In Type II superconductors, the magnetic field pene-trates in the form of quanta of magnetic flux (the valueof which is Φο = irHc/e = 2 χ 10"

7 gauss · cm2) even in an ex-ternal field He = HCi» 100-1000 gauss, which is called thefirst critical field. One can imagine the flux quantumitself (an Abrikosov vortex filament)"1 as consisting oftwo regions—the core of the vortex and its periphery.The core of the vortex consists of a practically normalmetal, its size being ξ = 100-500 A. Persistent super-conducting currents circulate in the peripheral part,which has a dimension δ£= 100-5000 A. Figure 1 showsthe current and magnetic-field distributions in the vor-tex for materials having 6L /ξ » 1 (for a more detailedacquaintance with the properties of Type Π supercon-ductors, see, e.g. t 2 ] ) .

In the equilibrium state the vortex filaments form anet (triangular or square) having a mean density n=B/Φο, where Β is the magnetic induction inside the speci-men. u ' 2 : l Yet if the superconductor contains structuraldefects, the vortices can become attached to them (thisphenomenon is called pinning) and form a metastable

configuration of the magnetic flux (for more details onthe pinning phenomenon, see, e.g., £ 3 ' 4 ] ) .

Superconductors in which the vortex filaments arestrongly bound to the metal lattice are called hard su-perconductors. Since the configuration and the energyof a vortex filament depend substantially on the tem-perature, the pinning force Fp also depends on the tem-perature. The mutual repulsion of the vortices [ 1 · ε :

causes the pinning forces to depend on the density ofvortex filaments, i .e . , on B. If we pass a transportcurrent through a Type II superconductor, then the in-teraction with it gives rise to a so-called Lorentz forcethat acts on each of the vortices1·415·1:

FL-ΙυΧΦοΙ ·

When acted on by this force, the magnetic flux goes in-to motion, energy dissipation arises, and the super-conductor transforms into the resistive state, k- 8 · 2 2 3

Yet if the superconductor is hard, then this regime canset in only when FL > Fp{T, B). We can convenientlywrite the force Fp in the form

ο ς

FIG. 1. Distribution of the current j(r) (a) and of the magneticfield H(r) (b) near the core of the vortex.

249 Sov. Phys. Usp., Vol. 20, No. 3, March 1977 Copyright © 1977 American Institute of Physics 249

Here je(T, Β) is called the critical current density. Thuspersistent currents can exist in a hard superconductoras long as j jc(T, B). A viscous-flow regime of thevortex filaments is established in the superconductor(see, e.g.1 8·1 1"1 3 3). Here,

FL=FP+i\v, (1.1)

where ην is the viscous frictional force, η is the vis-cosity, and ν is the velocity of motion of the vortexstructure. Equation (1.1) implies that

We can easily derive the relation between ν and the elec-tric field Ε that arises upon motion of the flux by usingthe equation of continuity for the flux of the vortex fila-ments

and the Maxwell equation

«tE — ± » . (1.2)

The last two equations imply that

Thus:

= j c a,E, (1.3)

where af = r\cz/B$0»anHcjB. Here ση is the conductivity

of the specimen in the normal state.

The relationship σ,-Β"1 is well confirmed by experi-ment (see, e.g.c l 2 ]), and it stems from the microscopictheory.c l 3 ] For hard superconductors, σ,~ (101β HCz/B)sec'1. This value is substantially smaller than the con-ductivity of pure metals, even in fields Β ~Hei. Figure

FIG. 3. Typical volt—ampere characteristicof a hard superconductor.

ο ε.

3 J ) shows a typical form of the volt-ampere charac-teristic of a hard superconductor. As we vary the elec-tric field, the volt-ampere characteristic quickly climbsonto the linear region (for E

flux is considerably faster than the redistribution ofheat. Thus, in the fundamental approximation for τ « 1,the heating of hard superconductors during a flux jumpis adiabatic. This assertion has been very convincinglyconfirmed experimentally (see, e.g., Β Ο · 2 1 · 8 " ) .

The converse limiting case of τ » 1 can be realized inthe so-called combined superconductors (see Sec. 3);and also in hard superconductors at very low tempera-tures (T

Ιιψ

FIG. 4. Fundamental experimentalscheme for studying flux jumps.He—external field, 1—specimen,2—transducers, R—recording de-vices.

FIG. 6. Qualitative relationship ofthe loss power in a hard supercon-ductor to the amplitude Hm of the ex-ternal field.l 2 4 ) H, is the field for aflux jump, and Hp is the field forcomplete penetration of the externalfield into the specimen.

Since V2r~ 0/6*, then

yl ™ \9T | < α · (1.8)

Here yt is a number of the order of unity that is deter-mined by the details of the temperature distributionthrough the specimen. We can conveniently rewritethe criterion (1.8) in the form

T

flux is considerably faster than the redistribution ofheat. Thus, in the fundamental approximation for τ «I,the heating of hard superconductors during a flux jumpis adiabatic. This assertion has been very convincinglyconfirmed experimentally (see, e.g., [20·21·28])#

The converse limiting case of τ » 1 can be realized inthe so-called combined superconductors (see Sec. 3);and also in hard superconductors at very low tempera-tures (T90 hasbeen supplied to this site. An additional amount Qt ofheat is released during the redistribution of the currentsand the field, which is equal to

FIG. 4. Fundamental experimentalscheme for studying flux jumps.He—external field, 1—specimen,2—transducers, R—recording de-vices.

FIG. 6. Qualitative relationship ofthe loss power in a hard supercon-ductor to the amplitude Hm of the ex-ternal field.t 2 4 1 Hf is the field for aflux jump, and Hp is the field forcomplete penetration of the externalfield into the specimen.

Since V2T~ θ/δ2, then

xa I dT (1.8)

Here yx is a number of the order of unity that is deter-mined by the details of the temperature distributionthrough the specimen. We can conveniently rewritethe criterion (1.8) in the form

7

2. THEORY OF FLUX JUMPS IN HARDSUPERCONDUCTORS

In this section we shall derive the equations that de-scribe the development of small perturbations of thetemperature and magnetic field in hard superconductors.One can solve these equations for a given magnetic-fieldand current distribution in a specimen and also for as-signed thermal and electrodynamic boundary conditions.Evidently a system is stable if the initial perturbationsdecay with time. We shall determine the criteria forstability by starting with this condition.

A. Stability of the critical state (Bean's equation of state)

Let us study a specimen having a plane geometry (Fig.7); the initial specimen temperature is T, while asmall deviation in the latter is θ(θ « Tc-T). In an ap-proximation linear in Θ, the heat-conduction equation hasthe form

β

FIG. 8. Qualitative λ{β) relationship for various values of T.(a) Adiabatic boundary conditions (W = 0), (b) finite heat re-moval (0

As we saw in the Introduction, the parameter β char-acterizes the release of heat per unit volume of the super-conductor, while τ is the ratio between the thermal andmagnetic diffusion coefficients. Equation (2.2) containsthe combination β- λτ. This is natural, since the nor-mal currents damp the motion of the magnetic flux anddiminish the heat release. Since τ « 1 in hard super-conductors even in weak magnetic fields (H^H^), weshall first treat the case τ = 0. In the absence of damp-ing processes, the most "dangerous" perturbations be-come the "fast" ones (perturbations with λ—00), sinceone of the stabilizing mechanisms (heat conduction) can-not operate. Actually, when τ = 0 one can show that \c= under all thermal boundary conditions (0« W Hp, sincethe left-hand side of the inequality (2.7) does not dependon He . Hence the inequality (2. 7) determines the criticalthickness bc of the specimen. When b

Ε" + β £ =·•= Ο,

FIG. 9. Hs is a function of T. where

(2.16)

as ι r/ruϊ = _ > £ _

The boundary conditions for E(y) are now written as:

H^HjiT) curve for a hard superconductor. In particu-lar, if we assume that

v, = v0 (TITe)\ U = h [1 - (T/Tu)},

where Tu~Te (see Fig. 2b), then we can easily findfrom (2.8) the following expression for HS(T):

(2.14)

Here H} has a maximum (see Fig. 9) at Γ=0.75 Tu.We recall that the temperature region that is too closeto Tc is not treated here. On the other hand, the ap-proximation τ « 1 usually does not hold (r~T ~3) at lowenough temperatures {T< 1 °K). Yet one can show evenin this case that Hf falls with decreasing T, and vanishesat T = 0.

B. Stability of the critical state (equation of state ofgeneral form)

As we have seen, rapidly growing (adiabatic) per-turbations are weakly damped (within limits as τ « 1)for hard superconductors, and they lead to the maxi-mum possible heating of the specimen. Evidently thisassertion depends neither on the model of the criticalstate nor on the geometry of the problem. Thus onecan find the stability criterion in the fundamental ap-proximation by assuming that τ = 0.β )

Upon neglecting the density of the normal currents andthe heat conductivity, we can easily derive an equationfor the electric field Ε"11:

Ε" + α (χ) Ε' + β (*) Ε = 0, (2.15)

where

We should impose the following electrodynamic boundaryconditions on Eq. (2.15): Ε '(±b) = £(δ&) = 0, which coin-cides with (2.3) and (2.5).

Before we proceed to solve (2.15), let us make achange of variable:

y = •

whereupon (2.15) acquires the standard form

Ε (±1) = 0, Ε" (0) = 0. (2.17)

We note that if je(H, T)=jo(T)(H), then Tt does not de-pend on H, and hence not on y. Equation (2.16) can besolved exactly, and the stability criterion has the form

where / / / = (2.18)

In the general case, one can solve (2.16) if the condi-tion (d/dy)(l/f]l)

Η

FIG. 10. Superconducting wire (geom-etry of the problem).

port current Im that a superconducting wire of radius Rcan transmit without losses (Fig. 10).

Since τ = 0, the onset of instability involves fast per-turbations (Xc>> 1), and heat conduction is not essential.By analogy to the case of plane geometry, we can showthat the stability criterion has the form β

ΡΌ,Ιί

FIG. 13. Relationship of the parameter γ to the magnetic-field gradient for different values of R (tube in an externalfield), (a) Ht

defining d'c as 3Ssk, we can easily get an estimate(/=0)M 0 ::

FIG. 15. Curves of λ(β) for various values of d. a) Adiabaticconditions at the boundary of the superconductor; b) isothermalcooling.

As we know,t 4 2 ] in a normal metal an ac electromag-netic field penetrates only to the depth of the skin layer6sk. We see from (3.1) that in our notation 6st(\)= δ(λτ')"1 / 2. Η λ = \{β)» 1, then we have 6 a k - 0, and theelectric field Ε vanishes at the superconductor-coatingboundary. With such a boundary condition, we can easilyfind from (2.2) (with τ = 0) that the λ(β) curve asymptot-ically approaches the straight line β=τ? (Fig. 15). Thusthe presence of the normal metal strongly deforms andshifts the λ= λ(β) curve for λ » 1, independently of thethickness of the coating and the thermal boundary con-ditions.

In the region λ « 1, the effect of the coating on theperturbation spectrum is less appreciable (if the coatingdoes not change the heat-removal regime from the super-conductor). In particular, under adiabatic conditionsthe position does not change of the point |3= β0 at which*(&>) = 0.C 3 2 ] since screening currents are not induced inthe normal metal when λ = 0.

Thus, with d« b and with external thermal insulation,the coating cannot raise the value of y2 beyond y* = 3.Let us see how the transition occurs from y=ir/2(T', the entire branchλ(/3)>0 lies in the region /3>3. l m

Under isothermal boundary conditions, perturbationshaving λ « 1 can appear only in the region β» 1 (see Fig.15b). Thus we can increase y2 to the value y2 = π2 by us-ing a coating. When A ' » 1 (which we know to be satis-fied for characteristic values of τ'), Eq. (2.2) and theboundary conditions imply that λ β » 1. The quantities\e and y depend on the damping properties of the nor-mal metal. The substantial increase in y with increas-ing d occurs in a range of d below some critical valued'e. Then λβ and y cease to depend on d, and they are de-termined solely by the values of τ'Μ°3:

, d>d'c. (3.2)

This effect has been detected experimentally in"*3.Evidently the critical thickness of the coating is of theorder of the depth of the skin layer 6Bt for λ = \e. Upon

3b (3.3)

Let us see how instability develops when there is goodheat removal from the volume of the superconductor.Since here λ,,» 1, a perturbation of the magnetic fieldand the temperature, grows sharply while the electricfield Ε vanishes simultaneously at the two boundariesof >the specimen. Hence the total magnetic flux in thesuperconductor is unchanged. This means that at theonset the magnetic flux rapidly (within a time of the or-der of ίχ/λ0) becomes redistributed within the super-conductor, and then the final distribution of the magneticfield and the currents is slowly established (within thediffusion time of the magnetic field through the normalmetal). I32·90·»7} Onishic43] has observed this effect ex-perimentally.

As we have seen, the stability of the critical state isin many ways determined specifically by the perturba-tions having Xe» 1. Instability with respect to them isabsolute in the sense that y cannot (within limits as Xc» 1) become elevated by any external agents (improvedheat removal, increased thickness or conductivity ofthe coating, etc.). Thus the maximum attainable valuesof y are determined by the onset of growing perturba-tions having λ » 1 .

The effect of the geometry of the current and field dis-tribution, as well as the role of the je(H) relationship,can also be treated for specimens coated with a normalmetal.

Figure 11 (curve 2) shows the Im(R/Re) relationshipfor a wire (see Fig. 10) coated with a normal metal'4 0 3

(d>d'c). In contrast to the case of a wire lacking a nor-mal coating, the parameter y does not vanish as δ-0(/ —7C), since here one of the degrees of freedom of thesystem does not disappear—the currents that compen-sate the decline in j e arise in the coating. If the radiusR of the wire is less than Λ(.«2.4Λ0, flux jumps do notarise up to /=/„. In the example that we treated earlierof Nb-25% Zr (Sec. C of Chap. 2), the characteristicvalue is x, = 4x I03 erg/cm · sec °K. If a wire made ofthis material is covered with a pure metal (Cu or Al),then T ' = 1, while we get the following estimates for Rea.ndd'e: cm; dc{fie, /c)«5xl(T

3 cm.

B. Flux jumps in combined superconductors

We shall treat in this section the problem of stabilityof the critical state of a combined superconductor (amatrix of a normal metal containing a regular structureof superconducting regions embedded in it, or fibers inthe critical state). The number Ν of superconductingfibers in the cross section of the specimen varies from2-3 to several tens and even hundreds. One can use asthe matrix either metals of good conductivity (Cu, Al),or various alloys of lesser conductivity. t*M*.™le"'.«*:i χηsuch a combined superconductor, instability can beassociated not only with losses of stability of any of thesuperconducting regions, but also withonset of collective

258 Sov. Phys. Usp., Vol. 20, No. 3, March 1977 R. G. Mints and A. L. Rakhmanov 258

effects—loss of stability of the entire distribution of cur-rent and flux as a whole. Naturally, the case of greatestinterest here is that of materials having N» 1. We shallderive the conditions below under which a flux jump de-velops in a combined superconductor made of stabilizedfibers.

For a quantitative description of collective effects, wemust derive the equation for propagation of a small per-turbation through a combined superconductor, as aver-aged over regions whose dimensions include enoughstructural elements of the combined superconductor,yet are smaller than the dimensions of the specimen it-self. Evidently such an equation is valid up to the pointwhere its solution varies over distances larger than thecharacteristic dimension of the structure, while thetime for equalizing the perturbation over the scale ofthe structure is much less than the corresponding timeof variation of the entire solution. After averaging, weget the equations for β and E:

diffusion of magnetic flux (see the Introduction). HenceXe« 1, and the thermal boundary conditions play an es-sential role.

For example, let us examine the problem of stabilityof a plate made of a combined superconductor jiaving iso-thermal conditions at its boundary. The λ=λ(/3, τ)curves have a form analogous to that illustrated in Fig.8b. The value of •y2 at which a root λ> 0 first appearsis ( f » l ) H 5 ] :

(3.7)

while the value of \e is:

(3.8)

Analogously, when τ"1« W« 1, we can easily get thecorresponding expression for Xe:

- κ ΔΘ + (3.4)

and the relationship of the current density j to the fieldE:

i = j . + «Ε. (3.5)

The quantities v, j 0 , σ, and ν that enter into (3.4) and(3.5) are the averages of the heat capacity, the criticalcurrent density (here we assume that the superconductingcurrents flow in the same direction inside the region ofaveraging), and the electric and thermal conductivities,respectively. Let us denote the relative concentrationof the superconducting metal as xs, and that of the nor-mal metal as xjjca+xn = 1). Then

The averaged value of the thermal conductivity trans-verse to the structure of the combined superconductoris determined by the details of the structure. As wecan easily verify, a good estimate is κ = (1 -

For the following treatment we must choose a modelof the critical state. Here we shall restrict the choiceto Bean's model. We can generalize to the case of anarbitrary je(H) relationship by the methods presented inSec. Β of Chap. 2.

Upon eliminating Ε from Eqs. (3.4) and (3.5), we caneasily derive the equation for 0[ 4 5 ] :

θ Ι ν - λ (1 + τ) θ" + λ (λτ - β) θ = 0. (3.6)

In (3.6) the spatial derivatives are taken with respectto the variable τ/b, where b is the characteristic dimen-sion of the specimen, while the 9(t) andE(t) relationshipsare taken in the_form Θ, E-expiXfy/vb2}. In order todetermine λ = λ(/3, f), we must impose the usual thermaland electrodynamic boundary conditions on (3.6).

As a rule, τ for combined superconductors is greaterthan unity, and it can attain values up to 102-104. Iff » 1, then heat redistribution runs much faster than the

In the fundamental approximation with f » 1, the crite-rion (3. 7) coincides with the known expressiont 4 e·4 7 1 thathas been found from qualitative arguments.

Let us now derive the criterion for applicability of(3.6). The characteristic structural scale of the com-bined superconductor is b/-JN, while the minimal scaleof variations in the solutions is the depth i r t of the skinlayer in the normal metal. In the frequency region λ~XC that is of interest to us, we evidently have 5>k~ 6/VXCT. Hence the following condition must be satisfied:

-V (3.9)

In temperature equalization, the thermal diffusion inthe superconducting fiber occurs slowest of all, and thecorresponding time ts proves to be of the order of

while the characteristic time ts of a flux jump is:

We get from the condition 11»ts the second condition forapplicability of (3.6):

&. (3.10)

In particular, when W « l , Eqs. (3.9) and (3.10) implythat

κ,ν

The condition τ » 1 permits us in many cases to sim-plify substantially the problem of studying stability.: 4 5 ]

In fact, if we substitute the explicit current-field re-lationship (3.5) into the Maxwell equation (1.5), we get

259 Sov. Phys. Usp., Vol. 20, No. 3, March 1977 R. G. Mints and A. L. Rakhmanov 259

FIG. 16. Relationship of yVr tothe heat removal W for a combinedsuperconductor ( W » T ^ , T » 1 ) .

40 W

ized. Hence, in the fundamental approximation withf » 1 , stability does not depend on the transport current,and is determined solely by the dimension 26 of the re-gion through which the current is flowing (i .e. , by theamount of heat release). " 5 ]

By using (3.11) and the thermal boundary conditions,we can easily obtain y (see (1.7)) for an arbitrary qualityof heat removal. We have the following relationship fordetermining y{W) for a flat plate (see Fig. 7):

(3.13)

If the heat removal is not too small (W>> f 1 ) , then, aswe can easily show by using Eq. (3.6) with the appro-priate boundary conditions, the characteristic time ofthe flux jump is much smaller than the magnetic-dif-fusion time in the specimen (see the Introduction). Hencewe find that XeT» 1 (in particular, with ideal refrigera-tion, \ef ~ f

2 / 3 ) . Since ΔΕ is a finite quantity, we havein the fundamental approximation:

This gives the relation between θ and E. Upon substitut-ing the found relationship into the heat-conduction equa-tion, we get an equation for θ in the stated approxima-tion:

Λθ + ί-ϊ- — λ)θ = Ο. (3.11)

Hart" 6 · 4 7 3 has derived this equation from qualitativearguments.

We should evidently impose on Eq. (3.11) the usualthermal boundary conditions, whereupon we can easilyfind the corresponding solution for specimens havingvarious geometries and current and magnetic-field dis-tributions; existence of solutions having λ> Ο impliesloss of stability. Evidently a solution exists in all casesif

γι-,.. (3.12)Here yx is some eigenvalue of the problem to be solved.Eq. (3.12) implies that λ = 0/τ - y? , and hence,

The condition dj/&t = O that we have used implies thatinstability sets in at a frozen magnetic flux in the funda-mental approximation with f » 1. One can also derive(3.11) directly from (3.6) in the limit as f, λ ? - °°. Thegiven derivation merely explains the nature of the courseof the process.

For a hard superconductor, stability breaks down in-dependently in different regions of the specimen that dif-fer in direction of current. In the studied case whereτ » 1, instability sets in immediately throughout thevolume of the superconductor (He^Hj), since t}»tH,and the temperature in the specimen can become equal-

Figure 16 shows a graph of y*/r as a function of W. Inparticular, if W«l (we recall that W»f1), Eq. (3.13)implies that

Vs = Wx. (3.14)

As W-*·*>, Eq. (3.13) yields the fundamental approxima-tion of Eq. (3.7).1 0 )

Let us examine now the conditions under which col-lective effects arise in a combined superconductor madeof stabilized elements (b'

FIG. 17. Typical relationship of

magnetic field is varied according to a definite law inexperiments to study the stability of the critical state.An entire set of studies"*·2 7·3 0·3 5·4 9"6 0·8 5·9 1 3 has dealt withthe effect of the rate of growth He of the magnetic fieldon the magnetic-field value Hej at which one observes aflux jump. Figure 17 shows a characteristic relation-ship between He} and He. As we can easily estimate,the relationship Het}«He is known to hold under experi-mental conditions. Thus, as the theory implies, the truestability boundary (the field H}) does not depend on He.This assertion was first formulated in" 5 3 , and Harrisonet al.C3°3 arrived at the same conclusion from their ex-periments. The role of an ac external field is reducedonly to initiating flux jumps. We can easily understandthat this initiation will be effective enough (Η,,^Η,) inthe absence of other "priming" agents if the electricfield Ε that is caused by the variation in He exceeds thevalue EQ in a large enough part of the volume (see Fig.3; E> Eo corresponds to transition to a flux-flow re-gime). This explanation easily allows us to understandalso the climb of the Hej{He) relationship up to a constantvalue that has been found in a number of experiments.u >

Now let us examine the fundamental results of the ex-perimental studies of magnetic instabilities in hard su-perconductors and compare them with the theoreticalresults.

As we have seen, flux jumps arise only when 9jc/dT 10

2 gauss/sec, the flux-jump field ceases to depend on He, and we can naturallyassume here that Hei~Hs (see

C54>5S]). The Η,{Τ) curveconstructed by using (2.14) shows good agreement withexperiment.

It was shown in1·233 that stability does not depend on thevalue of j c in the external-field region where He « Hp (seethe criteria (2.8) and (2.8')). As j c varies by a factorof about three (which corresponded to increasing themagnetic field He from 5 to 30 kilogauss), H, declined byonly 5% (which might, e. g., be explained by the relation-ship of BjjBT to H: 9jc/dT~-jc/T(l-H/HC2). A num-ber of subsequent experiments have observed periodicflux jumps with increasing external field.C20·8».95] Theincrease Δ#β in the external field between successivejumps depended little on He. This evidently confirmsthe conclusion that Hs is independent of j c .

The existing experimental data do not allow us toelucidate the effect of the geometry of the current andfield distribution on stability (see Chap. 2). Existenceof a dependence of Hj on the prehistory involves thefinite nature of the specimen in two dimensions. Hencethe existence of the effect does not depend on the con-crete shape of the conductor. This phenomenon has beentreated qualitatively in a number of cases (see,e.g ./ 7 0 · 7 1 3 ) .

A dependence of the stability on the size of the trans-port current / (He> Hp) has been found in

C57]. Flux jumpswere lacking when /= 0, while they appeared when I> 0,which agrees with the predictions of the theory.

SuttonC723 has studied the stability of a specimen con-sisting of two superconducting plates having differingcritical current densities j e (Fig. 19), where jcfa

FIG. 20. Relationship of the ratioAi = /fJ/fl} to d/dj for the two-layerspecimen. The theoretical curve,is plotted as the solid line; theparameter α = 2.7.

0.5

= ajc(x> 0). The stability boundary in this case is evi-dently a function of the thickness d of the outer layer(we can consider the inner superconductor to be semi-infinite) and of the parameter a. The ratio (ht) of theflux-jump field H* of the two-layer specimen to the flux-jump field Hl} of one of the specimens alone was mea-sured experimentally as a function of the quantity d/d}(Fig. 20, where d, = cH)/4Trjc). The corresponding prob-lem can be easily solved by using (2.13). The solidline in Fig. 20 shows the theoretical curve that we ob-tained. The value a = 2.7 was chosen by least squares.As we see, the theory satisfactorily describes the ex-perimental relationship. We note further that the maxi-mum value in Hzl = 2H

l

l(d~dl, a» 1) for a two-layerspecimen.

The effect of the external thermal conditions on theposition of the stability boundary has been studied

in[24,28,92]_ I n l i n e w U h t h e t h e o r e t i c a l predictions, H,

depends weakly on the nature of the heat removal.

A large number of experiments have measured the timeof development of instability. 120.28-30,53,73-75] T o t h e a c _

curacy with which the experimental and theoretical re-sults can be compared, the agreement between them issatisfactory. We note also that they have observedin 1 3 0 · 7 6 1 an increase in the characteristic time of a fluxjump with increasing conductivity of the normal coating.

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Translated by M. V. King

263 Sov. Phys. Usp., Vol. 20, No. 3, March 1977 R. G. Mints and A. L. Rakhmanov 263