294 5 Solid State Physics
According to the BCS theory superconductivity is due to a weak binding of two
electrons of equal and opposite momenta and spin to form the so-called a Cooper
pair which behaves as a single particle, a Boson.
An energyEg, called the superconducting energy gap, is required to break the
Cooper pair. AtT=0,
Eg=^3.^53 kTc (5.20)wherekis the Boltzmann constant.5.2 Problems..................................................
5.2.1 CrystalStructure...................................
5.1 Show thatπ/6 of the available volume is occupied by hard spheres in contact
in a simple cubic arrangement.
5.2 Show that
√
3 π/8 of the available volume is occupied by hard spheres in con-
tact in a body-centered cubic arrangement.5.3 Calculate the separations of the sets of planes which produce strong x-ray
diffractions beams at angles 4◦and 8◦in the first order, given that the x-ray
wavelength is 0.1 nm.
5.4 At what angle will a diffracted beam emerge from the (111) planes of a face
centered cubic crystal of unit cell length 0.4 nm? Assume diffraction occurs in
the first order and that the x-ray wavelength is 0.3 nm.
5.5 An x-ray beam of wavelength 0.16 nm is incident on a set of planes of a certain
crystal. The first Bragg reflection is observed for an incidence angle of 36◦.
What is the plane separation? Will there be any higher order reflections?
5.6 In the historical experiment of Davisson and Germer electrons of 54 eV at nor-
mal incidence on a crystal showed a peak at reflection angleθr= 400 .Atwhat
energy neutrons would also show a peak atθr= 400 for the same order.
5.7 Write down the atomic radii r in terms of the lattice constant a, for (a) Simple
cubic structure (b) FCC structure (c) BCC structure (d) Diamond structure.
5.2.2 Crystal Properties ................................
5.8 Show that the Madelung constant for a one-dimensional array of ions of alter-
nating sign with equal distance between successive ions is equal to 2 ln 2.
5.9 Write down the first five terms for the Madelung constant corresponding to the
NaCl crystal.