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Quasi-Stationary Temperature Pro?le and Magnetic Flux Jumps in Hard Superconductors

N.A. Taylanov

arXiv:cond-mat/0205081v1 [cond-mat.supr-con] 4 May 2002

Theoretical Physics Department, Scienti?c Research Institute for Applied Physics, National University of Uzbekistan, Tashkent, 700174, Uzbekistan E-mail: taylanov@iaph.tkt.uz Abstract

In the present paper, the temperature distribution in the critical state of hard superconductors is investigated in the quasi-stationary approximation. It is shown that the temperature pro?le can be essentially inhomogeneous in the sample, which a?ects the conditions of initiation of a magnetic ?ux jumps.

Key words: Critical state, ?ux ?ow, ?ux jump, instability. While dealing with instabilities of the critical state in hard superconductors, the character of the temperature distribution T (x, t) and that of the electromagnetic ?eld E(x, t) are of substantial practical interest [1]. This derives from the fact that thermal and magnetic destractions of the critical state caused by Joule self-heating are de?ned by the initial temperature and electromagnetic ?eld distributions. Hence, the form of the temperature pro?le may noticeably in?uence the criteria of critical-state stability with respect to jumps in the magnetic ?ux in a superconductor. Earlier (cf., e.g., [2]), in dealing with this problem, it was usually assumed that the spatial distribution of temperature and ?eld were either homogeneous or slightly inhomogeneous. However, in reality, physical parameters of superconductors may be inhomogeneous along the sample as well as in its cross-sectional plane. Such inhomogeneities can appear due to di?erent physical reasons. First, the vortex structure pinning can be inhomogeneous due to the existence of weak bonds in the superconductor. Second, inhomogenety of the properties may be caused by their dependence on the magnetic ?eld H. Indeed, the ?eld H in?uences many physical quantities, such as the critical current density jc , the di?erential conductivity σd , and the heat conductivity k. In the present paper, the temperature distribution in the critical state of hard superconductors is investigated in the quasi-stationary approximation. It is shown that the temperature pro?le can be essentially inhomogeneous, which a?ects the conditions of initiation of a magnetic ?ux jumps. The evolution of thermal (T ) and electromagnetic (E, H) perturbations in superconductors is described by a nonlinear heat conduction equation [3], ν dT = ?[κ?T ] + j E, dt

1

(1)

a system of Maxwell’s equations, rotE = ? 1 dH , c dt 4π j c (2)

rotH = and a critical-state equation

(3)

j = jc (T, H) + jr (E).

(4)

Here ν = ν(T ) is the speci?c heat, κ = κ(T ) is the thermal conductivity respectively; jc is the critical current density and jr is the active current density. We use the Bean-London critical state model to describe the jc (T, H) dependence, according to which jc (T ) = j0 [1 ? a(T ? T0 )] [4], where the parameter a characterizes thermally activated weakening of Abrikosov vortex pinning on crystal lattice defects, j0 is the equilibrium current density, and T0 is the temperature of the superconductor. The jr (E) dependence in the region of su?ciently strong electric ?elds (E ≥ Ef ; where Ef is the limit of the linear region of the current-voltage characteristic of the sample [2]) can σn Hc2 ηc2 ≈ is the be approximated by a piecewise-linear function jr ≈ σf E, where σf = HΦ0 H πhc is the e?ective conductivity in the ?ux ?ow regime and η is the viscous coe?cient, Φ0 = 2e magnetic ?ux quantum, σn is the conductivity in the normal state, Hc2 is the upper critical magnetic ?eld. In the region of the weak ?elds (E ≤ Ef ), the function jr (E) is nonlinear. This nonlinearity is associated with thermally activated creep of the magnetic ?ux [5]. Let us consider a superconducting sample placed into an external magnetic ?eld H = dH ˙ (0, 0, He) increasing at a constant rate = H=const. According to the Maxwell equation dt (2), a vortex electric ?eld E = (0, Ee , 0) is present. Here He is the magnitude of the external magnetic ?eld and Ee is the magnitude of the back-ground electric ?eld. In accordance with the concept of the critical state, the current density and the electric ?eld must be parallel: E j;. The thermal and electromagnetic boundary conditions for the Eqs. (1)-(4) have the form κ dT dx + w0 [T (0) ? T0 ] = 0 ,

x=0

T (L) = T0 , (5)

dE dx

= 0,

x=0

E(L) = 0 ,

For the plane geometry (Fig.) and the boundary conditions H(0) = He , H(L) = 0, the cHe is the depth of magnetic magnetic ?eld distribution is H(x) = He (L ? x), where L = 4πjc ?ux penetration into the sample and w0 is the coe?cient of heat transfer to the cooler at the equilibrium temperature T0 .

2

The condition of applicability of Eqs. (1)-(4) to the description of the dynamics of evolution of thermomagnetic perturbations are discussed at length in [2]. In the quasi-stationary approximation, terms with time derivatives can be neglected in Eqs. (1)-(4). This means that the heat transfer from the sample surface compensates the energy dissipation arising in the viscous ?ow of magnetic ?ux in the medium with an e?ective conductivity σf . In this approximation, the solution to Eq. (2) has the form E= ˙ H (L ? x). c (6)

Upon substituting this expression into Eq. (1) we get an inhomogeneous equation for the temperature distribution T (x, t), d2 Θ ? ρΘ = f (ρ). dρ2 Here we introduced the following dimensionless variables f (ρ) = ?[1 + rωρ] j0 , aT0 Θ= T ? T0 , T0 ρ= L?x , r (7)

and the dimensionless parameters ω =

˙ cκ 1/3 σf H , and, r = , where r characterizes ˙ cj0 aHL2 the spatial scale of the temperature pro?le inhomogeneity in the sample. Solutions to Eq. (7) are Airy functions, which can be expressed through Bessel functions of the order 1/3 [6] Θ(ρ) = C1 ρ1/2 K1/3 2 3/2 2 3/2 + C2 ρ1/2 I1/3 + Θ0 (ρ), ρ ρ 3 3 2 3/2 ρ 3

ρ 0 ρ 0

(8)

Θ0 (ρ) = ρ1/2 K1/3

[1 + rωρ1]ρ1 I1/3

3/2

2 3/2 dρ1 ? ρ 3 1

ρ1/2 I1/3

2 3/2 ρ 3

[1 + rωρ1 ]ρ1 K1/3

3/2

2 3/2 dρ1 , ρ 3 1

where C1 and C2 are integration constants, which are determined by the boundary conditions to be ?w0 LΘ(0) + κ C1 = 0, C2 = w0 L r

1/2

dΘ dρ

ρ=

L r

ρ=

I1/3

d 2 3/2 2 ?3/2 ?2 ρ1/2 I1/3 ρ ρ 3 dρ 3

L r

From the Maxwell equation (2), the temperature inhomogeneity parameter can be expressed in the form 4πνj0 He r α= = 2 ˙ L aHe Htκ

3

1/3

(9)

It is evident that α ? 1 near the threshold for a ?ux jump, when

˙ Htκ νL2 quasi-stationary heating condition << 1; where tκ = is the characteristic time of He κ the heat conduction problem. κ Let us estimate the maximum heating temperature Θm in the isothermal case w = ≥ 1. L The solution to Eq. (7) can be represented in the form Θ(x) = Θm ? ρ0 (x ? xm )2 , 2 (10)

2 aHe ? 1, even under the 4πνj0

near the point at which the temperature is a maximum, x = xm (Fig.). L with the help of the With solution (10) being approximated near the point xm = 2 8 Θm thermal boundary conditions, the coe?cient ρ0 can be easily determined to be L2 and the temperature can be written as Θ(x) = Θm 4 L 1? 2 x? L 2

2

,

(11)

Substituting this solution into Eq.(7), the superconductor maximum heating temperature due to magnetic ?ux jumps can be estimated as j0 + Θm = ˙ ˙ σf H H (L ? xm ) (L ? xm ) c cκT0 . ˙ γ aH ? (L ? xm ) L2 cκ

(12)

For a typical situation when

˙ γ aH << (L ? xm ) the estimation for Θm is L2 cκ (13)

˙ ˙ HL2 σf H (L ? xm ). (L ? xm ) Θm ≈ j0 + c cκT0

Here, the parameter γ ? 1 (for a parabolic temperature pro?le γ ? 8). It is easy to verify ˙ that for typical values of j0 = 106 A/cm2 , H = 104 Gs/sek, and L = 0, 01 cm the heating is su?ciently low: Θm << 1. In the case of poor sample cooling, w = 1 ? 10erg/(cm2sK), the Θm is Θm = ˙ Hj0 L2 ≈ 0, 5; cw0 T0 (14)

i.e., the heating temperature can be as high as δTm = T0 Θm ? 2K. One can see that in the case of poor sample cooling, the heating can be rather noticeable and in?uence the conditions of the thermomagnetic instability of the critical state in the superconductor. Reference

4

[1] Campbell A.M.,and J.E.Evetts. Critical current in superconductors. London, 1972. [2] Mints R.G., Rakhmanov A.L. Instability in superconductors. .: Nauka, 1984. [3] Taylanov N.A. Superconduct. Science and Technology, 14, 326 (2001). [4] Bean C.P. Phys. Rev. Lett. 8, 6, 250(1962). [5] Anderson P.W., and Y.B.Kim, Rev.Mod.Phys. 36, 39 (1964). [6] Kuznetsov D.S. Special functions. .: Moscow. 1965.

5

FIGURE Fig. The distrubation of the temperature pro?le Θ(x).

6

NIZAM A.TAYLANOV, G.B.Berdiyorov Address: Theoretical Physics Department and Institute of Applied Physics, National University of Uzbekistan, Vuzgorodok, 700174, Tashkent, Uzbekistan Telephone:(9-98712),461-573, 460-867. fax: (9-98712) 463-262,(9-98712) 461-540,(9-9871) 144-77-28 e-mail: taylanov@iaph.tkt.uz

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