Dresner, Stability of superconductors.2002

Figure 3.5. Sketches showing the values of J, B, ∂B/∂t, and E in the slab as it is charged with current.

assumption of an infinitely sharp resistive transition at J = Jc. Suppose, instead, the resistive transition is such that the resistivity r varies as a high power n of the current density. As mentioned earlier, the critical current density Jc is defined as that current density at which the resistivity reaches some fiducial value ρc, typically 10¯13 ohm-m. So our assumption of a power-law resistivity can be written

When J > Jc, r >> ρc and when J < Jc, r << ρc, the transition through r = ρc at J = Jc being very sharp.

Now we can combine Eqs. (3.3.1), (3.3.2), (3.4.1), and Ohm’s law E = ρJ to find the following partial differential equation for the penetration in from the edges of the slab of the z-directed current density J:

(3.4.2)

If we imagine the current to be instantaneously raised to its final, constant value I, the solution of Eq. (3.4.2) we are looking for is subject to the conservation condition

(3.4.3)

Equations (3.4.2) and (3.4.3) have a solution2 in which the current density penetrates a distance a from each edge given by

A typical value for Jc at 4.2 K and zero field for NbTi (ρc= 10-13 ohm-m) is 8 x 109 A/m2 (Larbalestier et al., 1986). For n we choose 50. (Measured values of n for conductors composed of NbTi filaments in a copper matrix are in the range 20–40. Some workers (Warnes and Larbalestier, 1986) have attributed these values to periodic variation in the diameter of the filaments, called sausaging, that arises during processing. Then the measured values of n would be a lower limit to any value intrinsic to the NbTi itself. For the latter we have chosen 50 purely for the purposes of illustration.) According to Eq. (3.4.5), A = 0.5683. Suppose now that the slab is 2 mm thick (d = 1 mm) and is charged to 50% of its critical current, i.e., I= 8.00 x 106 A/m. The extreme critical-state model tells us that the layers of critical current at each edge should be 0.5 mm thick, whereas Eqs. (3.4.4) and Eqs. (3.4.6) give the results in Table 3.1 for the penetration depth a and the ratio J(0,t)/Jc. For these quantities, the results of the critical-state model are in fair agreement. Furthermore, according to Eq. (3.4.6), the current density at five-eighths of the penetration distance a is only 1% less that at the edges of the slab, and at a depth of 96% of a, the decrement in current density is only 5%, so that the current distribution in the layers at each edge is rather uniform, as expected from the critical-state model. Finally, according to the entries in the fourth column, the resistivity in the current layers falls as the current spreads out, dropping far below the fiducial value ρc = 10-13 ohm-m used to define the critical current as time goes on.

This numerical illustration shows that the critical-state model is a good, but hardly perfect, representation of the behavior of type-II superconductors. But in spite of any small drawbacks it might have, it is almost universally used to describe the behavior of superconductors because it enables us to see what is going on at a glance, so to speak.

3.5. FLUX JUMPlNG DEFINED

After the current ramp stops, the current and field distributions according to the critical-state model are as shown in Fig. 3.5. When the current is steady, ∂B/∂t and E are zero.

Steady states are not always stable against small perturbations. A frequently used example of stable and unstable steady states is provided by a weight on the end of a rigid rod pinned at the end opposite the weight (i.e., a rigid pendulum). The steady states are (1) the weight hanging straight down vertically and (2) the weight standing straight up vertically. State (1) is stable against small pushes—the oscillatory motion of the pendulum is damped by friction in the pinned fulcrum so that eventually the weight hangs straight down again and state (1) is restored. But if state (2) is given a slight push, the pendulum falls over and never again returns to state (2), but rather approaches state (1). State (2) is unstable against small perturbations.

Because there are always thermal fluctuations in every system, unstable steady states cannot persist in the laboratory. Now as it happens, the steady state described by Fig. 3.5 is not always stable against small perturbations. Consider, for example, what would happen if the temperature of the sample were to be raised slightly by an amount dT. The critical current density Jc would decrease slightly, and then, since the total current I per unit length remains constant, the sheath of critical current density at the edges of the slab would have to broaden (dashed lines in Fig. 3.6). According to Eq. (3.3.1), the slope of the magnetic field is – oJc. This slope decreases slightly when the temperature rises so that the magnetic field distribution also broadens, as shown by the dashed line in Fig. 3.6b. The changing magnetic

Figure 3.6. Sketches showing the change in the values of J, B, ∂B/∂t, and E in the slab caused by a slight temperature increase. The values before the temperature increase, established by charging the slab with current, are shown as solid lines, the values during the temperature increase are shown as dotted lines.

field creates an electric field (Figs. 3.6c and 3.6d). The electric field vector, being parallel to the current density Jc, does work on the charge carriers that is dissipated in the sample as heat. This heat raises the temperature yet more and suggests the possibility of a thermal runaway. Such a thermal runaway is called a flux jump (Wipf, 1967; Swartz and Bean, 1968) and is undesirable because it may quench the superconductor entirely.

We can calculate the secondary temperature rise as follows. According to Eq. (3.3.1), in the left-hand current sheath

where the origin (y = 0) has been placed on the left-hand edge of the slab, and a, the penetration distance, is given by a = I/2Jc. Then,

so that

44 CHAPTER 3

The power dissipation P per unit face area in the left-hand current sheath is then

Now P dt is the heat dQ produced per unit face area by the rise in the temperature dT. Assuming for the moment that this heat is not conducted out of the current layer (adiabatic assumption), it causes a secondary temperature rise dT´ = dQ/Sa, where S = δ Cp is the volumetric heat capacity [J m¯3 K-1] of the superconductor. (Here δ is the density of the superconductor [kg m-3] and Cp is its specific heat [J kg-1 K-1].) Thus

If dT´/dT >1, the critical current sheath continues to penetrate the slab and the current distribution of Fig. 3.5a is unstable to small perturbations. If dT/dT < 1, the penetration slows down and is assumed eventually to stop. The current distribution of Fig. 3.5a is then loosely called stable, although strictly speaking under the adiabatic conditions assumed there is no return to the original state. Stable here simply means that there is no flux jump.

Since a < d, if d is small enough, both sides of Eq. (3.5.7) are < 1 and the slab is stable against flux jumps.

According to Larbalestier et al. (1986), for NbTi, Jc(4.2 K, 0 T) ~ 8 x 109A/m2. Elrod et al. (1982) give b + 3γ /T co2 = 7.5 x 10-3 J/kg-K4, while Wilson (1983) gives δ = 6200 kg/m3, so that S(4.2 K, 0 T) = 3450 J/m3-K. The critical temperature Tco is 9.1 K. If d < 25 µ m, the two sides of Eq. (3.5.7) < 1. Thus thin enough slabs are stable against flux jumps when they are being charged with current. This is one of the main reasons for subdividing the superconductor into fine filaments in commercial conductors.

3.6. VALIDITY OF THE ADIABATIC ASSUMPTION

Eq. (3.5.7) is based on the assumption that the heat generated by the motion of the flux stays in place during the flux jump. We called this the adiabatic assumption. In order to test its validity (cf. Akachi et al., 1981; Lubell and Wipf, 1966; Gandolfo et al., 1968), we need to determine how long a flux jump takes and how long it takes for the heat to diffuse out of the expanding current sheath.

To determine the time scale for a flux jump, let us consider again Eq. (3.4.2). For an ordinary ohmic medium (n = 0), Eq. (3.4.2) becomes

diffusivity are m2/s, and so it connects the distance the current has diffused with

the situation is reversed and the diffusion of heat out of the current sheath is much faster than the progress of the flux jump. In this latter extreme, the material outside the current sheath contributes to the heat capacity that determines the secondary temperature rise dT´.

For a material obeying the Wiedemann–Franz law,

of magnitude smaller than the normal-state resistivity of NbTi. So if after the initial temperature rise dT, the current sheath were to develop its full normal resistance,

would be >> and the adiabatic assumption would be valid.

If the initial temperature rise dT is small enough, however, the current sheath may not develop its full normal resistance and the adiabatic assumption may not be valid. Consider, for example, the conductor of Section 3.4. If the initial perturbing temperature rise dT occurs 1 s after the conductor is charged, the resistivity of the current sheath (at the front face, where it is largest) is 3.47 x 10-15 ohm-m. To raise this resistivity to 6.12 x 10-9 ohm-m, the value at which = the ratio J/Jc must increase by a factor of 1.333 (n = 50). Since J remains constant, Jc must decrease by the same factor. If Jc vanes linearly with T, the temperature rise dT is related to the ratio JCafter/JCbefore as follows (see Fig. 3.7):

Figure 3.7. A sketch justifying Eq. (3.6.3) relating the change in temperature and the critical currents before and after the temperature increase.

If Tco = 9.1 K and T = 4.2 K, dT = 1.22 K. This means that if the initial perturbation

contribute heat capacity (and possibly cooling, if the exterior surface of the conductor is bathed in a cryogen), so that the right-hand side of Eq. (3.5.7) is an upper bound to the ratio dT´/dT. On the other hand, if the initial perturbation is much greater than 1.22 K, >> >> and the adiabatic assumption is valid. In any case, the adiabatic assumption is conservative, which is to say that if it implies that a slab is stable against flux jumping, adding heat conduction to the theory will strengthen this conclusion.

3.7. STABILITY AGAINST AN EXTERNAL MAGNETIC FIELD

Another situation in which flux jumping occurs is that of a slab exposed to a rising external magnetic field Bo directed parallel to its faces. Again the field components obey Eqs. (3.3.1) and (3.3.2). As Bo increases, critical shielding currents are induced in the slab that seek to prevent the magnetic flux from entering it (Fig. 3.8). Again the dashed lines show what happens after a small rise dT in temperature. In the left-hand current sheath,

external magnetic field, are shown as solid lines, the values during the temperature increase are shown as dotted lines. The case shown is the case of incomplete penetration.

so that again during the perturbation dT, E is given by Eq. (3.5.3) and P by Eq. (3.5.4). Now dT´/dT is given by Eq. (3.5.7) with a replaced by Bo / o Jc:

Continuing the illustrative example worked at the end of Section 3.5, we find Bo = 0.252 T when dT´/dT = 1. Thus as we raise the external field we expect flux jumps to be possible once it reaches 0.252 T.

The foregoing analysis is based on the slab’s being wide enough that the invading flux does not reach its center, i.e., that d > Bo/ o Jc. This condition is called the case of incomplete penetration. If we substitute Bo= oJc d into Eq. (3.7.2), we obtain Eq. (3.5.7) with a = d. Thus in the example we are pursuing, if d > 25 m, then flux penetration is incomplete up to the flux-jumping threshold.

Figure 3.9. Sketches showing the change in the values of J, B, and E in the slab caused by a slight temperature increase. The values before the temperature increase, established by raising the external magnetic field, are shown as solid lines, the values during the temperature increase are shown as dotted lines. The case shown is the case of full penetration.

Suppose d is smaller than the limit just calculated; then the field components are as shown in Fig. 3.9. This condition is called the case of full penetration. Again in the left-hand sheath, B is given by Eq. (3.7.1), but now for 0 < y < d. The electric field E is now given by Eq. (3.5.3) with a replaced by d, and similarly P is given by Eq. (3.5.4) with a replaced by d. Finally, dT´/dT is given by Eq. (3.5.7) with a replaced by d.

The quantities oJcd and B* = [3 oS(Tc – T)]½ have the dimensions of magnetic induction B. We can therefore summarize the results of this section by means of a graph in which o Jcd is the abscissa and Bo is the ordinate (see Fig. 3.10). The region above the line OP of slope 1 through the origin corresponds to full penetration, the region below it to incomplete penetration. From Eq. (3.7.2) we can see that in the case of incomplete penetration, the stippled region of Fig. 3.10 is stable against flux jumps. From Eq. (3.5.7), with a being replaced by d, we see that in the case of full penetration, the hatched region is stable against flux jumps.

Figure 3.10. A graph summarizing the stability of a slab against flux jumping caused by application of an external magnetic field. The region above the line OP of slope 1 corresponds to full penetration, the region below the line corresponds to incomplete penetration. The shaded regions are stable against flux jumps.

In almost all applications, we are interested in external fields Bo greatly in excess of the field B* , and so we must keep µo Jc d < B* . Thus, as mentioned earlier, the superconductor in modern, commercial conductors is divided into fine filaments whose diameters are well below the limit imposed by the condition µo Jc d < B* = [3µ oS(Tc – T)]1/ 2 .

3.8. TWISTED FILAMENTS

Suppose now we consider a composite slab of multifilamentary superconductor. When this slab is exposed to a rising external magnetic field, critical shielding currents are induced in the filaments near the surface that shield the interior from the invading flux. The situation is quite similar to that just analyzed in Section 3.7 for a slab of pure superconductor, with two small differences. First, the effective critical current is λ Jc,where λ is the volume fraction of superconductor and Jc is the critical current density of the superconductor itself. Second, the resistivity controlling the diffusion of magnetic flux through the composite is that of the matrix (typically copper). Since ρcu at 4.2 K is << ρmean, for copper,and the adiabatic assumption is invalid. If the composite slab is uncooled (the worst case) this simply means that the volumetric heat capacity S is not determined just by that of the material in the critical current sheaths, but must be augmented by the heat capacity of the material in the central parts of the slab as well. But these alterations in Jc and S do not change the order of magnitude of the maximum stable thickness 2d, which is again tens of µ m. Such thin composite conductors are undesirable.