Exercise 2Estimate the magnetic field strength necessary to destroy superconductivity in a sample of lead at 4.2 K.From Table 1, for lead T_{c} = 7.2 K and B_{c}(0) = 0.080 T. From Equation 1,It is interesting to compare the magnetic behaviour of a superconducting element with typical curves for diamagnetic, paramagnetic and ferromagnetic materials. The magnetic behaviour of magnetic materials can be represented by B versus H graphs. Figure 12a shows the behaviour of typical diamagnetic and paramagnetic materials. Note that we have plotted μ_{0}H_{on the horizontal axis rather than H, so that both axes use the same unit (tesla). The straight lines plotted correspond to the relationship B = μμ0H, with μ slightly smaller than unity for the diamagnetic material and slightly greater than unity for the paramagnetic material. The behaviour of a ferromagnet, shown in Figure 12b, is quite different, with B ≫ μ0H, and a highly non-linear and irreversible curve until the magnetisation saturates, after which B increases linearly with H.}Compare these graphs with Figure 13, which shows the B-H curve for a superconducting cylinder of tin, with the field parallel to its axis. The field strength B within the superconductor is zero when μ_{0}H is less than the critical field strength B_{c}; the superconductor behaves like a perfect diamagnetic material and completely excludes the field from its interior. But then B jumps abruptly to a value B_{c}, and at higher fields the tin cylinder obeys the relationship Bμ_{0}H, since the material is weakly diamagnetic in its normal state, with μ = 0.9998. The linear graphs in Figure 12a are similar to those for a superconductor above the critical field strength.2.5 Critical currentThe current density for a steady current flowing along a wire in its normal state is essentially uniform over its cross-section. A consequence of this is that the magnetic field strength B within a wire of radius a, carrying current I, increases linearly with distance from the centre of the wire, and reaches a maximum value of μ_{0}I / 2a at the surface of the wire (see Exercise 3.6). Within a superconductor, however, the magnetic field B is zero.SAQ 2What can you deduce about the current flow in a superconducting wire from the fact that B = 0 within a superconductor?The current density within the bulk of the wire must be zero, since Ampère's law (curl B = μ_{0}J) indicates that a non-zero current density would produce a magnetic field. The current must therefore flow in the surface of the wire.SAQ 3How does the magnetic field just outside the surface of a superconducting wire, radius a, carrying current I, compare with the field just outside the surface of a normal wire with the same radius, carrying the same current?The fields just outside the surface are identical. The currents in both wires are axially symmetric, so the integral version of Ampère's law indicates that the fields just outside the surfaces of the wires are the same.The magnetic field strength B just outside the surface of the wire is μ_{0}I / 2a. It follows that if the current flowing in a superconducting wire is increased, eventually the field strength at the surface of the wire will exceed B_{c} and the sample will revert to its normal state. The maximum current that a wire can carry with zero resistance is known as its critical current, and for a long straight wire the critical current I_{c} is given by I_{c} = 2aB_{c} / μ_{0}. A current greater than I_{c} will cause the wire to revert to its normal state. This critical current is proportional to the radius of the wire.In the previous subsection you saw that the critical field strength is dependent on temperature, decreasing to zero as the temperature is increased to the critical temperature. This means that the superconducting current that a wire can carry will also decrease as the temperature gets closer to the critical temperature. Because of this, in real applications superconductors generally operate at temperatures less than half of the critical temperature, where the critical field strength, and therefore the critical current, is greater than 75 per cent of the maximum value.Now, the current carried by a superconducting wire actually flows in a thin layer at the surface; it cannot be restricted to an infinitesimal layer, because that would lead to an infinite current density. As you will see in Section 3, this means that the magnetic field penetrates into this thin layer, and we derive there relationships between the field and the current density. But in the present context, the point to note is that the transition to the normal state takes place when the magnetic field strength at the surface corresponds to the critical field strength, and this occurs when the current density at the surface reaches a critical current density. This critical current density is much greater than I_{c} / a^{2} because the current flows only in a thin surface layer.The magnetic field at the surface of a superconductor may have a contribution from an external source of magnetic field, as well as from the field produced by the current in the wire. This external field will set up screening currents in the surface layer of the material. The transition to the normal state then occurs when the vector sum of the current densities at the surface due to the current in the wire and due to the screening current exceeds the critical current density, or, equivalently, when the magnitude of the vector sum of the magnetic fields that are present at the surface of the wire exceeds the critical field strength.Exercise 3Tin has T_{c} = 3.7 K and B_{c} = 31 mT at T = 0 K. What is the minimum radius required for tin wire if it is to carry a current of 200 A at T = 2.0 K?From Ampère's law, the field strength B at the surface of a wire of radius R carrying a current I is B = μ_{0}I / 2R. From Equation 1, we also have thatso the radius R required is3 Modelling properties of superconductors3.1 A two-fluid modelAs was mentioned earlier, a substantial dose of quantum mechanics would be required to provide a full explanation of the properties of superconductors. This would take us too far away from electromagnetism, and we shall therefore restrict our discussion to aspects that can be discussed using classical concepts of electromagnetism.We shall model the free electrons within a superconductor as two fluids. According to this two-fluid model, one fluid consists of ‘normal’ electrons, number density n_{n}, and these behave in exactly the same way as the free electrons in a normal metal. They are accelerated by an electric field E, but are frequently scattered by impurities and defects in the ion lattice and by thermal vibrations of the lattice. The scattering limits the speed of the electrons, and they attain a mean drift velocity = −eE / m, where is the mean time between scattering events for the electrons and m is the electron mass. The current density J_{n} due to flow of these electrons isInterspersed with the normal electrons are what we shall call the superconducting electrons, or superelectrons, which form a fluid with number density n_{s}. The superconducting electrons are not scattered by impurities, defects or thermal vibrations, so they are freely accelerated by an electric field. If the velocity of a superconducting electron is v_{s}, then its equation of motion isCombining this with the expression for the current density, J_{s} = −n_{s}ev_{s}, we find thatCompare this with Equation 2, which relates current density and electric field in a normal conductor. Scattering of the normal electrons leads to a constant current in a constant electric field, whereas the absence of scattering of the electrons in a superconductor means that the current density would increase steadily in a constant electric field. However, if we consider a constant current flowing in the superconductor, then ∂J_{s}/∂t = 0, so E = 0. Therefore the normal current density must be zero – all of the steady current in a superconductor is carried by the superconducting electrons. Of course, with no electric field within the superconductor, there will be no potential difference across it, and so it has zero resistance.3.2 Magnetic field in a perfect conductorWhen discussing the Meissner effect in Subsection 2.3, we argued qualitatively that a material that just had the property of zero resistance – a perfect conductor rather than a superconductor – would maintain a constant magnetic field in its interior, and would not expel any field that was present when the material became superconducting. We shall now show how that conclusion follows from an application of Maxwell's equations to a perfect conductor. We can then see what additional assumptions are needed to account for the Meissner effect in a superconductor.We assume that the electrons in a perfect conductor (or a proportion of them) are not scattered, and therefore the current density is governed by Equation 3. However, we shall use the subscript ‘pc’ (for perfect conductor) here to indicate that we are not dealing with a superconductor. We are interested in the magnetic field in a perfect conductor, so we shall apply Maxwell's equations to this situation. Faraday's law is valid in all situations,and if we substitute for E using Equation 3, we obtainLooking now at the Ampère-Maxwell law, curl H = J_{f} + ∂D/∂t, we shall assume that our perfect conductor is either weakly diamagnetic or weakly paramagnetic, so thatμ 1 and HB/μ_{0} are very good approximations.We shall also omit the Maxwell term, ∂D/∂t, since this is negligible for the static, or slowly-varying, fields that we shall be considering. With these approximations, the Ampère-Maxwell law simplifies to Ampère's law,where use of the subscript pc for the current density reminds us that the free current J_{f} is carried by the perfectly-conducting electrons. We now use this expression to eliminate J_{pc} from Equation 4:We can use a standard vector identity from inside the back cover to rewrite the left-hand side of this equation:The no-monopole law, div B = 0, means that the first term on the right-hand side of this equation is zero, so Equation 6 can be rewritten asThis equation determines how ∂B/∂t varies in a perfect conductor.We shall look for the solution to Equation 7 for the simple geometry shown in Figure 14; a conductor has a boundary corresponding to the plane z = 0, and occupies the region z > 0, with a uniform field outside the conductor given by B_{0} = B_{0}e_{x}.The uniform external field in the x-direction means that the field inside the conductor will also be in the x-direction, and its strength will depend only on z. So Equation 7 reduces to the one-dimensional formwhere we have simplified the equation, for reasons that will soon become clear, by writing The general solution of this equation iswhere a and b are independent of position. The second term on the right-hand side corresponds to a rate of change of field strength that continues to increase exponentially with distance from the boundary; since this is unphysical, we set b = 0. The boundary condition for the field parallel to the boundary is that H_{∥} is continuous, and since we are assuming that μ 1 in both the air and the conductor, this is equivalent to B_{∥} being the same on either side of the boundary at all times. This means that ∂B/∂t is the same on either side of the boundary, soand the field within the perfect conductor satisfies the equationThis indicates that any changes in the external magnetic field are attenuated exponentially with distance below the surface of the perfect conductor. If the distance λ_{pc} is very small, then the field will not change within the bulk of the perfect conductor, and this is the behaviour that we described qualitatively in Subsection 2.3. Note that this does not mean the magnetic field must be expelled: flux expulsion requires B = 0, rather than just ∂B/∂t = 0. So how do we modify the description that we have given of a perfect conductor so that it describes a superconductor and leads to a prediction that B = 0?3.3 The London equationsA simple but useful description of the electrodynamics of superconductivity was put forward by the brothers Fritz and Heinz London in 1935, shortly after the discovery that magnetic fields are expelled from superconductors. Their proposed equations are consistent with the Meissner effect and can be used with Maxwell's equations to predict how the magnetic field and surface current vary with distance from the surface of a superconductor.In order to account for the Meissner effect, the London brothers proposed that in a superconductor, Equation 4 is replaced by the more restrictive relationshipThis equation, and Equation 3 which relates the rate of change of current to the electric field, are now known as the London equations.London equationsIt is important to note that these equations are not an explanation of superconductivity. They were introduced as a restriction on Maxwell's equations so that the behaviour of superconductors deduced from the equations was consistent with experimental observations, and in particular with the Meissner effect. Their status is somewhat similar to Ohm's law, which is a useful description of the behaviour of many normal metals, but which does not provide any explanation for the conduction process at the microscopic level.To demonstrate how the London equations lead to the Meissner effect, we proceed in the same way as for the perfect conductor. First we use Ampère's law, curl B = μ_{0}J_{s}, to substitute for J_{s} in Equation 9, and we obtainwhereBut curl(curl B) = grad(div B) − ∇^{2}B = −∇^{2}B, since div B = 0. SoThis equation is similar to Equation 7, but ∂B/∂t has been replaced by B. The important point to note about this equation is that the only solution that corresponds to a spatially uniform field (for which ∇^{2}B = 0) is the field that is identically zero everywhere. If B were not equal to zero, then ∇^{2}B would not be zero, so B would depend on position. Thus, a uniform magnetic field like that shown in Figure 10b cannot exist in a superconductor.If we consider again the simple one-dimensional geometry shown in Figure 14, then we obtain the solution to Equation 11 by simply replacing the partial time derivatives of the fields in the solution for the perfect conductor (Equation 8) by the fields themselves, that is,Therefore, the London equations lead to the prediction of an exponential decay of the magnetic field within the superconductor, as shown in Figure 15.3.4 Penetration depthThe characteristic length, λ, associated with the decay of the magnetic field at the surface of a superconductor is known as the penetration depth, and it depends on the number density n_{s} of superconducting electrons.We can estimate a value for λ by assuming that all of the free electrons are superconducting. If we set n_{s} = 10^{29} m^{−3}, a typical free electron density in a metal, then we find thatThe small size of λ indicates that the magnetic field is effectively excluded from the interior of macroscopic specimens of superconductors, in agreement with the experimentally observed Meissner effect.The small scale of the field penetration means that carefully-designed experiments are needed to measure the value of λ. Many experiments have been done with samples that have a large surface to volume ratio to make the penetration effect of the field appreciable. Thin films, thin wires and colloidal particles of superconductors have all been used for this purpose. But it is also possible to use large specimens if the measurement is sensitive to the amount of magnetic flux passing through the superconductor's surface, and not to the ratio of flux excluded by the superconductor to flux through the normal material, which is close to unity.In a classic experiment performed in the 1950s, Schawlow and Devlin measured the self-inductance of a solenoid within which they inserted a long single-crystal cylinder of superconducting tin, 7.4 mm in diameter. They minimised the space between the coil and the tin cylinder, and since no magnetic flux passed through the bulk of the superconductor, the flux was essentially restricted to a thin cylindrical shell of thickness λ at the surface of the cylinder. The inductance of the solenoid was therefore determined mainly by the magnitude of the penetration depth. To measure the inductance, a capacitor was connected in parallel with the solenoid, and the natural angular frequency, _{n} = 1/, of the LC circuit was measured. The precision of the frequency measurement was about one part in 10^{6}, which corresponded to a precision of 0.4 nm in the value of the penetration depth. The result that they obtained for the penetration depth of tin for temperatures much lower than the critical temperature was 52 nm.The number density of superconducting electrons depends on temperature, so the penetration depth is temperature dependent. For T ≪ T_{c}, all of the free electrons are superconducting, but the number density falls steadily with increasing temperature until it reaches zero at the critical temperature. Since λ n_{s}^{-1/2} according to the London model, the penetration depth increases as the temperature approaches the critical temperature, becoming effectively infinite – corresponding to a uniform field in the material – at and above the critical temperature. Figure 16 shows this temperature dependence for tin, which is well represented by the expressionwhere λ(0) is the value of the penetration depth at T = 0 K.3.5 The screening currentThe London equations relate the magnetic field in a superconductor to the superconducting current density, and we derived the dependence of field on position by eliminating the current density. However, if we eliminate the magnetic field instead, we can derive the following equation for the current density:Exercise 4Derive Equation 13 by taking the curl of both sides of Equation 9 and then using Ampère's law to eliminate curl B. Assume that the currents are steady.Taking the curl of both sides of Equation 9 and using Ampère's law, curl B = μ_{0}J_{s}, to eliminate B, we find thatWe now use a standard vector identity to rewrite the curl(curl J_{s}) term:For our steady-state situation, where ∂ρ_{s}/∂t = 0, the equation of continuity, ∂ρ_{s}/∂t + div J_{s} = 0, reduces to div J_{s} = 0, soEquation 13 has exactly the same form as Equation 11. So for the planar symmetry that we discussed earlier – superconducting material occupying the region z > 0 (Figure 14) – the solution for the current density will have the same form as Equation 12, that is,This equation gives no indication of the absolute magnitude or direction of current flow, but we can deduce this by using Ampère's law, curl B = μ_{0}J_{s}. In the planar situation that we are considering, B = B_{x} (z) e_{x}, soThenwhereBut we know that B_{x}(z) = B_{0}e^{−z/λ} (Equation 12), soThus the current that screens the interior of the superconductor from an applied field flows within a thin surface layer, which has a thickness characterised by the penetration depth λ, and the current flows parallel to the surface and in a direction perpendicular to the magnetic field, as shown in Figure 17.Exercise 5The number density of free electrons in tin is 1.5 × 10^{29} m^{−3}. Calculate the penetration depth predicted by the London model, assuming that all of the free electrons are superconducting, and compare the result with the value measured by Schawlow and Devlin.From Equation 10, the penetration depth is given by λ = (m/μ_{0}n_{s}e^{2})^{1/2}, so (in metres)This value, predicted by the London model, is about a quarter of the measured value.The numerical discrepancy between the London model prediction for the penetration depth of tin and the experimentally measured value indicates that this model has limitations. One limitation is that the model is essentially a local model, relating current density and magnetic field at each point. Superconductivity, though, is a non-local phenomenon, involving coherent behaviour of the superconducting electrons that are condensed into a macroscopic quantum state. The characteristic distance over which the behaviour of the superconducting electrons is linked is known as the coherence length, ξ, introduced in Section 1. This distance represents the distance over which the number density of the superconducting electrons changes, and is a measure of the intrinsic non-local nature of the superconducting state. The London local model is a good description if ξ ≪ λ, that is, the coherence length is much shorter than the distance λ over which the fields and current density are changing. Since the penetration depth increases sharply as the temperature approaches the critical temperature (Figure 16), the London model becomes a good approximation in this limit. More importantly, the coherence length of superconductors decreases as the critical temperature increases and as the scattering time for normal electrons decreases. Both of these effects mean that the coherence length is short compared with the penetration depth in alloy and ceramic superconductors, so the London local model is a good approximation in these cases too, and predicted and experimental results for the penetration depth are in good agreement.For pure elements, well below their critical temperatures, the penetration depth is generally much shorter than the coherence length, so a local model is not appropriate. In this limit, the number density of superconducting electrons does not reach the bulk value until a distance of the order of ξ, which is greater than λ, from the surface, and the reduced value of n_{s} accounts for the discrepancy between the predicted and experimental results for the penetration depth of tin discussed earlier.The ratio of the penetration depth to the coherence length is an important parameter for a superconductor, and we shall return to this subject in Subsection 4.2.4 Two types of superconductorPreambleThe two main types of superconducting materials are known as type-I and type-II superconductors, and their properties will be discussed in the remainder of this course. All of the pure elemental superconductors are type-I, with the exception of niobium, vanadium and technetium. The discussion of the effects of magnetic fields and currents on superconductors earlier in this course has been confined to thin cylinders of type-I materials like lead or tin in a parallel magnetic field. In Subsection 4.1 we shall discuss what happens when the magnetic field is perpendicular to cylinders made of these materials.Superconducting alloys and high critical temperature ceramics are all type-II, and these are the materials that are used in most practical applications. In Subsection 4.2, we shall consider the response to a magnetic field of this type of superconductor. Such materials behave quite differently from lead and tin, and this is the reason that they are widely used.4.2 Type-I superconductorsYou saw in Subsection 2.4 that superconductivity in a tin cylinder is destroyed when an applied field with strength B_{0} > B_{c} is applied parallel to the cylinder. However, when the field is applied perpendicular to the cylinder, as shown in Figure 18, the field strength at points A and C is substantially greater than the strength of the applied field at a distance from the cylinder, and this is indicated by the increased concentration of the field lines shown near these points. In fact, it can be shown that the field strength at these points is a factor of two greater than the applied field strength. This means that as the applied field B_{0} is increased, the field at points A and C will reach the critical field strength B_{c} when B_{0} = B_{c}/2.You might think that superconductivity in the cylinder would be completely destroyed at this lower field strength. However, were this to be the case, then the material would be in the normal state with a field strength in its interior of less than B_{c}, which is not possible. Instead, for applied field strengths B_{0} in the range B_{c}/2 < B_{0} < B_{c}, the cylinder splits up into small slices of normal and superconducting material that run parallel to the applied field. This state in which regions of normal and superconducting material coexist in a type-I superconductor is known as the intermediate state, and it is shown schematically in Figure 19. Within the normal regions, B = B_{c}, and in the superconducting regions, B decreases rapidly and is confined to a thin layer, the width of which is determined by the penetration depth, as shown in Figure 20. The number density of superconducting electrons n_{s} increases from zero at the boundary to the bulk value over the coherence length ξ. Note that since n_{s} is not constant in the superconducting regions, the magnetic field strength does not fall exponentially. The proportion of the material in the normal state increases from zero for B_{0} = B_{c}/2 to 100 per cent for B_{0} = B_{c}.The lowest applied field strength at which the intermediate state appears depends on the shape of the specimen and the orientation of the field. Essentially, it is determined by the extent to which the field is deviated by the superconductor, or equivalently, by how much the field strength is enhanced at the edges of the superconductor. For the thin cylinder shown in Figure 21a, the field just outside is essentially the same as the applied field, so there is a direct transition from superconducting to normal state, without the intervening intermediate state. Contrast this with the thin plate oriented perpendicular to the applied field shown in Figure 21b, where the field strength would be greatly enhanced outside the plate's edges if it could not penetrate the plate. Samples like this enter the intermediate state when the applied field strength is a very small fraction of the critical field strength (Figure 21c).Figure 22 shows the pattern of normal and superconducting regions for an aluminium plate in the intermediate state, with the field perpendicular to the surface.4.3 Type-II superconductorsFor decades it was assumed that all superconductors, elements and alloys, behaved in similar ways, and that any differences could be attributed to impurities or defects in the materials. However, in 1957, Abrikosov predicted the existence of a different sort of superconductor, and Figure 23 shows direct evidence for the existence of what are now known as type-II superconductors. A comparison of Figures 23 and 22 indicates that the effect of an applied field on a type-II superconductor is rather different from that for type-I superconductors.For simplicity, we shall consider first a long cylindrical specimen of type-II material, and apply a field parallel to its axis. Below a certain critical field strength, known as the lower critical field strength and denoted by the symbol B_{c1}, the applied magnetic field is excluded from the bulk of the material, penetrating into only a thin layer at the surface, just as for type-I materials. But above B_{c1}, the material does not make a sudden transition to the normal state. Instead, very thin cylindrical regions of normal material appear, passing through the specimen parallel to its axis. We shall refer to such a normal region as a normal core. The normal cores are arranged on a triangular lattice, as shown in Figure 23, and as the applied field is increased, more normal cores appear and they become more and more closely packed together. Eventually, a second critical field strength, the upper critical field strengthB_{c2}, is reached, above which the material reverts to the normal state. The state that exists between the lower and upper critical field strengths, in which a type-II superconductor is threaded by normal cores, is known as the mixed state. As Figure 24 shows, both the upper and lower critical field strengths depend on temperature in a similar way to the critical field strength for a type-I material (Figure 11).The normal cores that exist in type-II superconductors in the mixed state are not sharply delineated. Figure 25 shows how the number density of superelectrons and the magnetic field strength vary along a line passing through the axes of three neighbouring cores. The value of n_{s} is zero at the centres of the cores and rises over a characteristic distance ξ, the coherence length. The magnetic field associated with each normal core is spread over a region with a diameter of 2λ, and each normal core is surrounded by a vortex of circulating current.You can see from Figure 25 that the coherence length ξ, the characteristic distance for changes in n_{s}, is shorter than the penetration depth λ, the characteristic distance for changes in the magnetic field in a superconductor. This is generally true for type-II superconductors, whereas for type-I superconductors, ξ > λ (Figure 20). For a pure type-I superconductor, typical values of the characteristic lengths are ξ ~ 1 μm and λ = 50 nm. Contrast this with the values for a widely-used type-II alloy of niobium and tin, Nb_{3}Sn, for which ξ ~ 3.5 nm and λ = 80 nm.The reason that the relative magnitude of the coherence length and the penetration depth is so important is that when ξ > λ, the surface energy associated with the boundary between superconducting and normal regions is positive, whereas when ξ < λ, this surface energy is negative. Justifying this statement would involve a discussion of the thermodynamics of superconductors, but for our purposes it is sufficient to just look at the consequences. For a positive surface energy, the system will prefer few boundaries and we expect relatively thick normal and superconducting regions, as observed in the intermediate state in type-I materials. Conversely, a negative surface energy favours formation of as much boundary between normal and superconducting regions as possible, and this is what happens in the mixed state in type-II materials with many narrow normal cores. The lower limit for the diameter of the cores is 2ξ, as shown in Figure 25, because ξ is the distance over which n_{s} can vary.This energy argument does not explicitly indicate how much magnetic flux passes through each of the normal cores. However, quantum mechanical arguments show that the magnetic flux linking any superconducting loop must be quantised, and that the quantum of magnetic flux ish / 2e = 2.07 × 10^{−15}T m^{2}where h = 6.63 × 10^{−34}J s is Planck's constant. In fact, each of the normal cores shown in Figure 23 contains just one quantum of flux, since this is more favorable energetically than having two or more quanta of flux in a core.The quantisation of flux in a superconductor is of particular importance in SQUIDs, the superconducting quantum interference devices that are at the heart of the magnetometers used for measuring the magnetic fields produced by currents in the brain. A SQUID contains a small loop of superconductor with a weakly superconducting link in it, and the quantisation of flux in the loop causes its electrical properties to depend on the flux applied to it, with a periodicity equal to the flux quantum. The very small magnitude of the flux quantum is responsible for the sensitivity of the device to very small magnetic fields.A final point is worth noting about the quantum of flux: the factor of 2 in the denominator of the expression h / 2e is a consequence of the coupling of pairs of electrons in a superconductor and their condensation into a superconducting ground state. There is charge −2e associated with each of these electron pairs.Critical currents in type-II superconductorsThe high values of the upper critical field strength B_{c2} of many type-II superconducting alloys make them very attractive for winding coils for generating high magnetic fields. For example, alloys of niobium and titanium (NbTi_{2}) and of niobium and tin (Nb_{3}Sn) have values of B_{c2} of about 10 T and 20 T, respectively, compared with 0.08 T for lead, a type-I superconductor. However, for type-II materials to be usable for this purpose, they must also have high critical currents at high field strengths, and this requires some help from metallurgists to overcome a significant problem.This problem is related to the interaction between the current flowing through a type-II superconductor in the mixed state and the ‘tubes’ of magnetic flux that thread through the normal cores. The electrons will experience a Lorentz force, perpendicular to both the current density and the magnetic field. We can regard this as a mutual interaction between the electrons and the flux in the normal cores, as a result of which each normal core experiences a force that is in the opposite direction to the Lorentz force on the electrons. The directions of the magnetic field, current and forces are shown in Figure 26.This Lorentz force can cause the cores and their associated magnetic flux to move, and the flux motion will induce an emf that drives a current through the normal cores, somewhat like an eddy current. Energy is therefore dissipated in the normal cores, and this energy must come from the power supply. The energy dissipation means that the flow of electrons is impeded, and therefore there is a resistance to the flow of the current.Flux motion is therefore undesirable in type-II superconductors, and the aim of the metallurgists who develop processes for manufacturing wire for magnets is to make flux motion as difficult as possible. This is done by introducing defects into the crystalline structure, particularly by preparing the material in such a way that it comprises many small crystalline grains with different orientations and small precipitates of different composition. These defects effectively pin the normal cores in position – they provide a potential barrier to motion of the cores, so that the force on the cores must exceed a certain value before the cores can move. The more of these flux pinning centres that are present, and the greater the potential barrier they provide, the greater will be the current required to set them in motion, i.e. the greater the critical current. So, unlike a normal conductor, for which improving the purity and reducing imperfections in the crystal structure lead to better conductivity, with type-II superconductors the inclusion of impurities and defects in the crystal structure can improve the critical current and make the material more suitable for use in electromagnets.Undoubtedly the largest use of superconducting material for a single project is in the Large Hadron Collider at CERN, due for commissioning in 2007. The 27 km-long accelerator tunnel contains 1232 superconducting magnets that are responsible for steering the particle beams around their circular paths. Each of these magnets is 15 m long, has a mass of 35 tonnes and produces a magnetic field strength of 8.5 T. The coils in each magnet are made from about 6 km of niobium-titanium cable, with a mass of about a tonne, and will be maintained at a temperature of 1.9 K using liquid helium. To construct an accelerator with a similar specification using non-superconducting magnets would have required a 120 km tunnel and phenomenal amounts of power to operate.ConclusionSection 1 Superconductivity was discovered in 1911, and in the century since then there have been many developments in knowledge of the properties of superconductors and the materials that become superconducting, in the theoretical understanding of superconductivity, and in the applications of superconductors.Section 2 A superconductor has zero resistance to flow of electric current, and can sustain a current indefinitely. The magnetic flux remains constant in a completely superconducting circuit, since changes in the flux from the field applied to the circuit are balanced by changes to (persistent) currents induced in the circuit. For each superconductor there is a critical temperature T_{c} below which the material is superconducting.Superconductors also exhibit perfect diamagnetism, with B = 0 in the bulk of the material. The exclusion of magnetic field is known as the Meissner effect. An external magnetic field penetrates for a short distance into the surface of a superconducting material, and a current flows in the surface layer to screen the interior of the material from the applied field. Superconductivity is destroyed when the magnetic field strength exceeds a critical value for the material. The critical field strength falls to zero as the temperature is raised to the critical temperature. A superconducting specimen will have a critical current I_{c} above which the material reverts to the normal state. This critical current corresponds to the field strength exceeding the critical field strength in some region of the specimen.Section 3 The two-fluid model of a superconductor regards some of the conduction electrons as behaving like normal electrons and some like superconducting electrons. For T ≪ T_{c}, all of the conduction electrons are superconducting electrons, but the proportion of superconducting electrons drops to zero at the critical temperature.For a perfect conductor (which has R = 0), Maxwell's equations predict that the magnetic field cannot change, except in a thin surface layer. This does not predict the Meissner effect. The London equations are relationships between current density and magnetic field that are consistent with the Meissner effect:When combined with Maxwell's equations, they lead to the prediction that the magnetic field strength and the surface current decrease exponentially below the surface of a superconductor, over a characteristic distance called the penetration depth λ, which is typically tens of nanometres. The London equations are local relationships and therefore are strictly valid only when λ ≫ ξ, where the coherence length ξ is the characteristic distance over which n_{s} varies.Section 4 There are two types of superconductors, type-I and type-II. For a type-I material in the form of a thin specimen parallel to the field, there is an abrupt transition to the normal state at the critical field strength B_{c}. When the field is inclined to the surface of a type-I material, the material exists in the intermediate state over a range of field strengths below B_{c}. In this state there are thin layers of normal and superconducting material, with the proportion of normal material rising to unity at field strength B_{c}. In type-I materials, the coherence length ξ is greater than the penetration depth λ, and the surface energy of the boundary between superconducting and normal material is positive, which favours a course subdivision into regions of normal and superconducting material.A type-II superconductor has two critical field strengths, B_{c1} and B_{c2}, between which the material is in the mixed state. In this state the superconductor is threaded by thin cores of normal material, through which the magnetic field passes. The coherence length ξ is shorter than the penetration depth λ, and the surface energy of the boundary between superconducting and normal material is negative, which favours a fine subdivision into regions of normal and superconducting material. To take advantage of the high values of B_{c2} to produce high magnetic fields with superconducting magnets, it is essential to pin the normal cores to inhibit their motion.Keep on learningStudy another free courseThere are more than 800 courses on OpenLearn for you to choose from on a range of subjects. Find out more about all our free courses.Take your studies furtherFind out more about studying with The Open University by visiting our online prospectus. 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