Transcript sclecture8
Flux line motion In principlea Type II superconductor should be “better” than a Type I for most applications - it remains superconducting to much higher fields ……..but is it? J Flux lines FL FL FL FL FL Flux lines tend to move easily so as to reach equilibrium. But if a current flows in a superconductor above Hc1 there will be a Lorentz force acting between the current (ie charge carriers) and the flux. FL J o So the flux lines move perpendicular to the current and induce an electric field E B v v = velocity of flux lines Now, E is parallel to J, so acts like a resistive voltage, and power is dissipated 1 watt of dissipated power at 4.2K requires 300MW of refrigerator power! The solution is to introduce “pins” by creating defects within the superconductor Lecture 8 Superconductivity and Superfluidity The Bean Model A Type II superconductor without any pinning is said to be reversible - flux enters abruptly at Hc1 and produces a uniform flux density throughout. If there are pinning centres within the Type II superconductor they hold the flux lines back near the surface creating a gradient of flux lines - such superconductors are said to be irreversible surface Lecture 8 Superconductivity and Superfluidity The Bean Model The pins can be thought of as introducing “friction” inhibiting the movement of flux lines into the supercoductor In this respect the superconductor is a little like a sandpile with the flux lines behaving like grains of sand However big we make the sandpile the sloping sides always have the same gradient In an analogous fashion, however large the magnetic field the gradient of flux lines remains constant: This is the basis of the Bean Model Lecture 8 superconductor B2 B2 B* B* B1 B1 D Superconductivity and Superfluidity The Bean Model Remember that curl B=oJ So a field gradient implies that a current is flowing perpendicular to B If B is in the z-direction, and the gradient is in the xdirection, then B in the y-direction o J z y x The Bean Model assumes that the effect of pinning is to: (a) produce a maximum gradient B and therefore to x (b) produce a maximum current density J x B2 B2 B* B* B1 B1 From another viewpoint the Bean model assumes (a) there exists a maximum current density Jc D (b) any emf, however small induces this current to flow The Bean model is therefore also known as the critical state model : only three current states are allowed - zero current for regions that have not felt B and ±Jc (ie the critical current density) depending upon the sense of the emf generated by the last field change Lecture 8 Superconductivity and Superfluidity Critical state model - increasing field +Jc 2B* 2B* B* B* B*/2 B*/2 current density superconductor 2B* and B* 0 B*/2 -Jc D Lecture 8 D Superconductivity and Superfluidity Critical state model - decreasing field First increase B to a value of, say Bo then reduce B to zero again Because the flux density gradient must remain constant, flux is trapped inside the superconducting sample, even at B=0 +Jc a b c Bo e Bo B=0 B=0 D Lecture 8 d current density superconductor a B bo c d e 0 -Jc D Superconductivity and Superfluidity Predictions of the Bean model The magnetisation of a sample can be calculated using the Bean model from diagrams such as the previous ones, with B* as the only free parameter Lecture 8 Superconductivity and Superfluidity Calculating the critical current density 2 Magnetisation . From the Bean model the critical current density is easily calculated from the hysteretic magnetisation loop 2M Jc D in SI units, where Jc is measured in A.m-2 , in A.m-1 and D, the diameter of the sample in m The strength of the pinning force, F, can also be determined: F JB 2M Bapplied D F usually shows a peak, at a field corresponding to that at which the pins “break” Lecture 8 1 0 2M F -1 B -2 -1.0 -0.5 0.0 0.5 1.0 Applied field (T) 0 Superconductivity and Superfluidity Magnetic superconductors Notice that in the 2nd quadrant of the hysteresis loop the magnetisation of the sample is positive, ie strongly (or even very strongly) paramagnetic. 2 Magnetisation . 3 2 b 1 a c So - depending upon the magnetic history of the sample - the superconductor can be attracted to a magnet! ...can become this So this….. 0 -1 4 1 -2 -1.0 -0.5 0.0 0.5 1.0 Applied field (T) 0 netisation (emu) As the magnetic falls away the field decreases from a-b, so the magnetisation increases. The magnet therefore moves closer to the superconductor (b-c) and the -1 field increases, but the magnetisation decreases and the magnet falls away….etc Lecture 8 Superconductivity and Superfluidity Critical current densities Lecture 8 Superconductivity and Superfluidity Critical current densities Lecture 8 Superconductivity and Superfluidity Small Angle Neutron Scattering multi-detector 64x64cm2 R Bragg’s Law: =2dsinq neutron wavelength = 10Å flux lattice spacing, d = 1000Å sinq ~R/L = 1/100 L B scattering angle 2q Superconducting sample incident neutron beam Lecture 9 Flux distribution determined from SANS Superconductivity and Superfluidity Flux lattice melting multi-detector 64x64cm2 R L B scattering angle 2q Superconducting sample incident neutron beam Lecture 9 Superconductivity and Superfluidity