[ PHYS4010 ]Magnetism and Superconductivity

1 (a) Antiferromagnetic materials have zero magnetic moment in zero applied field and small moments in small applied fields. Describe briefly what is meant by the terms antiferromagnetic and diamagnetic. By sketching the inverse susceptibility, show how an antiferromagnet may be distinguished from a diamagnet. [4]

(b) Describe briefly hysteresis in ferromagnets and discuss which magnetic characteristics would lead to the most optimum hysteresis loop shapes for

(i) a permanent magnet material and

(ii) a magnetic sensor material. [4]

(c) Describe the requirements for formation of a Bose-Einstein Condensate (BEC). Explain why liquid helium exhibits a superfluid phase based on a BEC for temperatures up to T = 2.17 K, whereas vapours of alkali metals need to be cooled below 10−6 K. [6]

(d) Wires with cylindrical cross-section, composed from two Type I superconducting materials, are assessed for use in an application. Wire 1 has a diameter of 3.0 mm and, in the superconducting state, is able to conduct a maximum current of 150 A. Wire 2 has a diameter of 1.5 mm and is able to conduct a maximum current of 160 A.

Explaining your reasoning, deduce which of the wires is composed from the material with the highest critical temperature, Tc. [2]

2 (a) A single mole quantity of an inorganic metal salt, X2(SO4)3.8H2O, had its magnetisation measured as the strength of an applied magnetic field, H, was increased from zero to 80, 000 A m−1 at room temperature. The magnetisation of the sample was observed to have increased linearly from zero to 239 A m−1 and was oriented in the same direction as the applied field. Assuming that the magnetisation is due to atom X in the sample, calculate the susceptibility, χ. Taking into account the strength and sign of the susceptibility and the atomic structure of the salt, identify the type of magnetic response exhibited by the sample. [3]

(b) The strength of the magnetisation of the inorganic salt follows the variation described by the expression:

M = NgµBJBJ (x) where x = (JgµBB)/(kBT)

and all terms have their usual meanings. Explain the fundamental basis of the above expression and provide a sketch of the variation of M vs. x. [2]

(c) Given the molar mass of the salt, mM = 746 g mol−1 , and its density, ρ = 3.4 g cm−3 , calculate the effective magnetic moment (in units of Bohr magnetons) of atom X. [3]

(d) Using the formula below for the Lande g-factor, deduce whether the magnetic response arises from 3d electrons (i.e. X a first row transition metal) or 4f electrons (i.e. X a rare earth metal). [4]

(e)Where the terms have their usual meanings. Describe the occurrence of flux vortices and use this expression to show that each vortex is associated with a minimum quantity of magnetic flux, Φ0 = 2.07 × 10−5 Wb. [4]

(f) A slab of Type II superconducting material is imaged and 773 flux vortices are observed across a square area of 2 µm × 2 µm. Calculate the strength of the magnetic field applied to the superconductor. If this is the lowest applied magnetic field strength at which the lattice appears, explain why the lattice would vanish if the applied field strength were reduced. [4]

(g) If the strength of the applied magnetic field is increased to B = 15 T, calculate the spacing of the flux vortices if they are arranged in a hexagonal array. [4]