Page 312 - Electrical Properties of Materials
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294                           Magnetic materials

     11.6. Show that the data for the magnetic susceptibility of  11.10. The energy levels of a free electron gas in the presence
     nickel given below is consistent with the Curie law [eqn  of an applied magnetic field are shown in Fig. 11.34 for abso-
     (11.26)] and evaluate the Curie constant and temperature.  lute zero temperature. The relative numbers of electrons with
     Hence find the effective number of Bohr magnetons per atom.  spins ‘up’ and ‘down’ will adjust so that the energies are equal
                                   –3
       Atomic weight 58.7, density 8850 kg m .       at the Fermi level. Show that the paramagnetic susceptibility
        ◦
       T( C)  500  600   700  800  900               is given by the approximate expression,
       χ m 10 5  38.4  19.5  15  10.6  9.73                         χ m = μ μ 0 Z(E F ),
                                                                         2
                                                                         m
     11.7. An alloy of copper and cobalt consists of spherical  where μ m is the magnetic moment of a free electron. μ 0 the
     precipitates, averaging 10 nm diameter, of pure cobalt in a  free space permeability, and Z(E F ) the density of states at the
     matrix of pure copper. The precipitates form 2 per cent by  Fermi level. Assume that μ m μ 0 H   E F .
     volume of the alloy. Cobalt is ferromagnetic, with saturation
                          –1
     magnetization of 1.4 MA m . Each cobalt precipitate is a
     single domain, and acts as a strong dipole, which responds
     to any external field as a paramagnetic dipole. The effect is      Energy
     called ‘superparamagnetism’. Calculate the susceptibility of                  E , Fermi level
                                                                                    F
     the alloy at 300 K.
     [Hint: The total magnetic moment of each precipitate is     Spin      Spin
     equal to the product of magnetic moment density (saturation  ‘up’    ‘down’
     magnetization) with the volume of the precipitate.]

     11.8. A system of electron spins is placed in a magnetic field
     B = 2 T at a temperature T. The number of spins parallel to the
     magnetic field is twice as large as the number of antiparallel               Density of states
     spins. Determine T.
                                                                              2μ μ H
                                                                               m 0
     11.9. In a magnetic flux density of 0.1 T, at about what
     frequencies would you expect to observe (i) electron spin  Fig. 11.34
     resonance, (ii) proton spin resonance?
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