Dresner, Stability of superconductors.2002

Figure 1.6. Surfaces in J-B-T space separating differentkinds of behaviorof a type-II superconductor. Above the larger surface, the superconductorexhibits flux-flow resistivity; below it is superconducting. Above the smaller surface,but below the larger surface, the superconductoris in the mixed state; below the smaller surface it exhibits type-I superconductivity (Meissner effect). Beyondthe base curve in the B-T plane, the superconductor is normal.

1.5. COMPOSITE SUPERCONDUCTORS

Modern commercial wires made of low-temperature superconductors (almost exclusively NbTi or Nb3Sn) consist of many fine filaments of superconductor buried in a matrix of ordinary metal (almost always copper, but for certain special uses CuNi). Such superconductors are called composite superconductors. The number of filaments varies from a few thousand to as much as a hundred thousand, their diameters varying from a few tens of micrometers to a few micrometers. This construction is convenient for several reasons. First, if the superconducting filaments should be driven out of the superconducting region of Fig. 1.6, the current can safely switch to the copper. Since the normal-state resistivity of superconductors is much higher than the residual resistivity of copper, the Joule heat produced in the copper is much lower than if the current continued to flow in the supercon-

ducting filaments. Then there are measures that can prevent the magnet from destroying itself by overheating, which is what would happen if the magnet were wound from the pure superconductors themselves.

Second, the disposition of the superconductor in the form of fine filaments avoids a magnetic instability called a flux jump that is at least one of the causes of the superconductor being driven out of the superconducting state into the normal state. We shall discuss flux jumping in detail in Chapter 3.

Third, the disposition of the superconductor in the form of fine filaments enables them to bend around rather small radii without having their critical current density degraded by the mechanical strain they suffer. This is a very serious consideration in the case of Nb3Sn composites, where tensile strains of only a few tenths of a percent cause substantial reductions in their current-carrying capacity. This is because Nb3Sn is a rather frangible material in which cracks appear that interrupt the path of current flow in the superconductor. Strain degradation is much less of a problem in NbTi composites because NbTi is ductile; these composites can stand tensile strains of several percent without significant degradation of their current-carrying capacity. This is the reason that most superconducting magnets producing fields less than roughly 8 T are wound with NbTi composite conductors. Typical examples of such magnets are those used for focusing beams of charged particles from accelerators and those used in magnetic resonance imaging (MRI) machines. When higher fields are desired, or when it is desired to operate at temperatures higher than that of liquid helium (4.2 K), the critical current density of NbTi becomes so low that it is necessary to use Nb3Sn.

The fourth and last reason for the disposition of the superconductor in the form of fine filaments in a copper matrix is ease of manufacture. For example, NbTi composites are manufactured by stacking NbTi rods in a hexagonal array of holes in a copper billet and then drawing the billet to fine wire. The wires may be restacked and the process repeated several times, the drawings being interspersed with anneals to keep the copper from work-hardening too much. In the case of Nb3Sn composites the most common method is the bronze method, in which the first drawings are of Nb rods in a CuSn (bronze) matrix. Later, the NbCuSn composite wires are surrounded by a Ta diffusion barrier and stacked in a Cu billet. The end result is a composite of Nb wires each encased in an annulus of CuSn that is surrounded by a layer of Ta and all buried in a matrix of copper. The last step is to heat treat the wires: the Sn diffuses into the Nb, reacting with it and forming Nb3Sn; the Ta barrier prevents the mobile Sn from diffusing into the copper and thereby raising its residual resistivity. After the heat treatment such conductors must be handled very carefully because of the great strain sensitivity of the Nb3Sn.

12 CHAPTER 1

1.6. QUENCHING AND STABILITY

Now let us perform a thought experiment in which we wind a solenoid out of such a multifilamentary composite conductor, immerse it in a dewar of liquid helium, and charge it with current. As the current rises, we hear the magnet creaking and groaning. Suddenly, with a great whoosh, all the helium is expelled from the dewar forming a cloud of water vapor in the room. Fearful of the results, we immediately reduce the current in the magnet to zero. What happened?

The groaning and creaking we heard was caused by the motion of the currentcarrying conductor in response to the Lorentz force J x B exerted on it by the magnet’s own self-field. That is to say, any turn of the magnet carrying a current density J feels a volumetric force J x B caused by the field B created by the current in the other turns. For a solenoid the net result is a tendency to expand in the radial direction and contract in the axial direction.

While this expansion is going on, individual strands of wire may slip a little from one position to another. When this happens, the Lorentz force does work, and when the slipping segment of wire comes to rest, either because of friction or impact, this work is converted to heat. Now because the specific heats of materials are very small at low temperatures, this heat of slipping, even though small, is often enough to raise the local temperature above the critical temperature. Now a bit of the wire is normal, i.e., resistive. The current traversing it, though it switches to the copper from the superconductor, generates Joule heat at the normal site, which further heats the locale. More conductor goes normal from this heating, and a process of thermal runaway, called a quench, begins.

Now that we have figured out what happened, we ask ourselves what we can do to avoid such quenches. Four things come to mind at once: hold the conductor more tightly to prevent sudden slipping, add more copper to the conductor to reduce the Joule power in the normal state, cool the conductor better, or reduce the operating current in the magnet. Sometimes these solutions conflict with one another. For example, one way of holding the conductor more tightly is to pot it in epoxy, but the epoxy then retards the transfer of heat from the conductor to the helium. Nevertheless, this method is used quite frequently to stabilize small magnets (stored energy < 100 kJ, roughly). Even so, quenches occur with potted magnets, for occasionally the epoxy cracks at points of high stress, and the mechanical energy released is enough to quench the magnet. Typically, if the magnet is recharged after such a quench, it will not quench again or it will quench because of a crack at another point. This is because once a crack develops at the first point, no sudden energy release occurs at that point. Instead, the edges of the crack move gradually under the action of the Lorentz force as the magnet is charged, and the heat generated has time to be conducted away before it can raise the local temperature beyond the critical temperature. With such magnets, after a few

quenches, the magnet can be charged to its full operating current without any further quenches. This process is called training and is typical of potted magnets.

Another strategy to stabilize magnets is to add more copper to the conductor, reserve some void space in the winding for the infiltration of liquid helium, and keep the current density down. This strategy keeps the Joule power in the normal state low and allows good cooling of the conductor. It is possible to provide enough cooling so that after a normalizing event (e.g., conductor motion) the heat produced by the normalizing event and the Joule power can be transferred away from the superconductor and the superconductor can be cooled back down to the superconducting state (called recovery ). Magnets stabilized in this way are called cryostable. They have the enormous advantage of being unconditionally stable, that is, they always recover the superconducting state after any perturbation and so can be operated without interruption. Coupled ineluctably to this advantage is the disadvantage of large size and consequently high cost because of the low current density in the winding. This cost disadvantage may be multiplied when the large size of the magnet forces the apparatus of which it is a part also to be large in scale, as may be the case with fusion machines.

To offset this disadvantage, we can build magnets with less copper in the conductor that operate with higher current densities than cryostable magnets. These magnets are metastable : they recover from small thermal perturbations but quench if exposed to large thermal perturbations. The idea is to stabilize them just against the perturbations that actually occur in the operation of the magnet. Since the spectrum of thermal perturbations is hardly known at all for most magnets, there is some guesswork in the design of metastable magnets, and failure of prototypes to meet specifications is not uncommon.

1.7. THE PHASE DIAGRAM OF HELIUM

So far we have mentioned only the liquid phase of helium as a coolant for the low-temperature superconductors. Historically, it was the first to be used, of course, but it is not the only possibility. Shown in Fig. 1.7 is the phase diagram of helium in the low temperature region. The point C lying roughly at 5.1 K and 0.22 MPa is the critical point and the curve OC is the saturation line separating the liquid phase from the vapor phase. The point P1 at 4.2 K and 0.1 MPa represents helium boiling at atmospheric pressure and is the point that corresponds to liquid helium in an open dewar.

Most materials have a triple point at which solid, liquid, and gaseous phases coexist. Helium, however, does not solidify at pressures less that 4 MPa even at zero temperature. Instead, below about 2.2 K, it enters another liquid phase, called superfluid helium or He-II. The line LL´, called the lambda line, is the phase boundary between ordinary liquid helium (He-I) and the superfluid phase. The point L, lying at about 2.2 K and 5 kPa, is called the lambda point. Superfluid helium has

Figure 1.7.A sketch of the phasediagram of helium at low temperatures. Point C is the critical point. Insteadof a triple point, as is usual in other materials, helium has two liquid phases with vastly different properties. (Redrawn from an original appearing in Dresner (1984, “Superconductor”) with permission of Butterworth-He- inemann, Oxford, England.)

rather different properties from ordinary liquid helium, and indeed from most other materials. For the purposes of this book it suffices to point out that superfluid helium does not obey Fourier’s law of heat conduction in which the heat flux is proportional to the temperature gradient. Instead, in superfluid helium, the heat flux is proportional to the cube root of the temperature gradient (when the heat flux is high enough, as it almost always is in magnet applications). This means that superfluid helium can support a very high heat flux for very small temperature gradients, and this is one of the reasons that it is considered desirable as a coolant for superconducting magnets. A second reason is that at the low temperatures at which it exists, NbTi composite conductors are capable of satisfactory operation up to fields of 12 T, rather than the 8 T typical of NbTi magnets cooled with boiling helium. The French tokamak (a kind of fusion machine) Tore-II Supra is cooled with superfluid helium having a temperature of 1.8 K and a pressure of 0.1 MPa (point P2 in Fig. 1.7).

Magnets have been built that are cooled with supercritical helium, typically helium at a temperature of 4 K and pressures around 1 MPa (point P3). The conductors of such magnets consist of superconducting composite wires braided into cables and placed in a strong jacket, which contains the helium. In some of these conductors, the cable is held tightly by the jacket, which is usually swaged

around it, and the supercritical helium circulates through the interstices of the cable. This kind of conductor, to which a considerable fraction of this book is devoted, is called a cable-in-conduit conductor. Because the strands of the cable are unrestrained except where they cross each other, wire motion is possible in such conductors, though it appears not to cause quenches. In other kinds of internally cooled conductors, the cable is soldered together, and the helium circulates through a tube down its center.

This brief overview of the stability problem shows that it has many aspects: we can have potted, externally cooled, and internally cooled conductors, cooled with either boiling, superfluid, or supercritical helium.

7.8. HIGH-TEMPERATURE SUPERCONDUCTORS

So far we have spoken only of low-temperature superconductors, for which helium is the only coolant. (Actually, this is not perfectly accurate because recently Nb3Sn magnets that were conductively cooled by cryocoolers have been operated at temperatures around 11 K.) The newly discovered high-temperature superconductors, whose critical temperatures are ~ 100 K hold out the possibility of operation in liquid nitrogen (atmospheric boiling temperature, 77 K). Because their behavior is somewhat different from that of the low-temperature superconductors, some description of the differences is now in order.

In the first place, the high-temperature, copper oxide superconductors are granular, in contrast to the low-temperature superconductors NbTi and Nb3Sn. Grain boundaries at which adjacent grains are misaligned with respect to their internal structure are regions of weak superconductivity (so-called “weak links”). These weak links act as bottlenecks to limit the current that can be transported over macroscopic lengths of conductor. In fact, the intragranular critical current density can be three to five orders of magnitude as great as the intergranular critical current density (Larbalestier, 1991). In the design of superconducting magnets, which is a principal interest of this book, the much lower intergrain critical current density is what concerns us. The higher intragrain critical current density is of interest in superconducting electronics, which makes use of thin films of superconductor grown on substrates that orient the grains with their internal structures parallel to one another. But even in the design of magnets, there are cases where the existence of a high intragrain critical current produces noticeable effects (Kwasnitza and St. Clerc, 1993).

In the second place, the copper oxide superconductors are anisotropic with respect to the direction of the magnetic field—that is to say, the critical current density depends on the direction of the magnetic field with respect to the crystal axes. The copper oxide superconductors consist, roughly speaking, of conducting planes of copper and oxygen atoms separated by intercalated, nonconducting planes of atoms. In the conventional crystallographic terminology, the a- and b-axes of the

Figure 1.8. The critical surface of NbTi (based on data of Wilson, 1983; redrawn from an original provided courtesy of Clarendon Press, Oxford, England.)

crystal lie in the conducting planes and the c-axis is perpendicular to them. When the magnetic field is parallel to the a- b plane, the critical current density is several times as large as when the magnetic field is perpendicular to the a-b plane (i.e., parallel to the c-axis). In bulk material, in which the grains are randomly oriented, this anisotropy is not evident. But randomly oriented grains accentuate the weaklink behavior. To suppress the weak-link behavior when Ag/BSCCO conductor is manufactured, the grains are partially aligned by rolling or pressing the conductor into a tape, thereby increasing the critical current density, but at the same time reintroducing anisotropy. The critical current density is then larger when the magnetic field is parallel to the tape surface than when it is perpendicular to it.

In the third place, the dependence of the critical current on magnetic field is somewhat different for the copper oxide superconductors than it is for the low-tem- perature superconductors. A glance at Fig. 1.8, the critical surface for NbTi, shows the nearly hyperbolic behavior of the Jc-B curve required by Eq. (1.4.1). The situation depicted by Fig. 1.9, the critical surface of Ag/BPSCCO (Sato et al., 1993), is quite different. At low temperatures (4.2–20 K, for example), the critical current

Figure 1.9. The critical surface of a Ag/BPSCCO high-temperature superconductor(Sato et al., 1993). (Redrawn from an original appearing in Sato (1993) with permission of Butterworth-Heineman, Oxford, England.)

decreases only gradually with increasing magnetic field up to quite large fields (~ 20–30 T). This means that the constant Bo in Eq. (1.4.1), though only of the order of a few hundredths of a tesla for the low-temperature superconductors, may be tens of teslas at low temperatures for the copper oxide superconductors. (Note that both Figs. 1.8 and 1.9 imply that Bo decreases with increasing temperature!) The insensitivity of Jc of the copper oxide superconductors to B at low temperatures holds out the promise of using them at low temperatures to make high-field magnets. On the other hand, the sharp drop of Jc of the copper oxide conductors with B at high temperatures is at present a stumbling block to the construction of magnets cooled with liquid nitrogen (77 K).

Finally, the temperature ranges over which the high-temperature superconductors are used are both higher and more extensive than those over which the low-temperature superconductors are used. For example, a NbTi conductor cooled

by liquid helium (4.2 K) in a field of 8 T has a critical temperature of 5.6 K. Over the restricted temperature range 4.2–5.6 K, which is the range of importance in determining superconductor stability, it is quite acceptable to assume that the specific heats of the matrix and the superconductor, the matrix resistivity, and the matrix thermal conductivity are all constants independent of temperature. On the other hand, a Ag/BPSSCO conductor cooled to 20 K by a cryocooler in the same 8 T field has a critical temperature of ~60 K, judging from Fig. 1.9. Over the range 20–60 K, the specific heats, the matrix resistivity, and the matrix thermal conductivity change by large amounts, making the assumption of constant properties no longer tenable.

Notes to Chapter 1

1Materials become superconducting because their free electrons interactby locally deforming the atomic lattice that makes up the solid. This interaction is attractive and causes the formation of bound pairs of electrons called Cooper pairs. The Cooper pairs are the current carriers in the superconducting state. In the neighborhood of a defect or impurity, the lattice is changed, the electron–electron interaction is altered, and the Cooper pairs are destroyed.At low temperatures, the superconductingstate has a lower free energy than the normal state (that is why it forms). Therefore, the flux lattice can achieve the lowest free energy by locating the normal cores at the defects rather than by destroying the superconducting state at a point where it otherwise could exist.