IGNOU PHE-13 - Physics of Solids

1. Answer in brief: i) Write down the Miller indices of a set of parallel planes which make intercepts in the ratio of 4a1:4a2 on the x and y axes and are parallel to the z-axes. ii) Is it possible to carry out electron diffraction studies in air? Explain. iii) Write down the electronic configuration of the Ge atom. What type of bonding would you expect to find in Ge? iv) List the symmetries observed in a methane molecule. v) The energy of the lowest allowed level for an electron in a 1-D box is 8.0 eV. Can the electron ever have an energy of 200 eV? Justify. vi) Plot the variation of electrical resistivity with temperature for an ideal metal and a superconductor. vii) Show that the group velocity is zero at the zone boundary for a linear diatomic chain. viii) How does the Fermi energy of n- and p- type semiconductors change with dopant concentration? ix) Distinguish between substitutional and interstitial impurities with the help of a diagram. x) In crystal growth why does the nucleation process become more stable as the size of the nucleus increases? 2. a) Calculate the volume of the unit cell for fcc Pb given that its atomic radius is 0.175 nm. b) Determine the boundaries of the first Brillouin zone of a bcc structure with a lattice constant 0.20 nm. c) A crystal of iridium is irradiated by x-rays with a wavelength of 0.721 Å. Calculate the angle of reflection from the (111) plane, given that the lattice constant of iridium is 3.84 Å. d) The direct lattice vectors for a lattice are given by: Obtain the volume of the primitive unit cell and the reciprocal lattice vectors. 3. a) The binding energy of CsCl is 150 kcal mol-1. If the Madelung constant for CsCl is 1.763 and the repulsive exponent n = 10.6, calculate the equilibrium interatomic distance re. (Take ε0 = 8.86 × 10-12 Farad m-1.) b) The Young’s modulus of a linear mono-atomic chain of atoms of mass 10.6 × 10-26kg is 2.0 × 10 11 Nm-2. If the inter-atomic separation is 2.8 Å, calculate the force constant K and the maximum frequency of the atoms. c) The Debye temperatures of NaCl and KCl are 330 K and 220 K respectively. If the heat capacity of KCl at 5 K is 3.8 × 10-2 J mol-1 K-1, calculate the heat capacity of NaCl at 15 K. d) Prove that in a crystal undergoing deformation, the deformed crystal axes are not mutually perpendicular. 4. a) A divalent bcc solid has a lattice constant of 4.5 Å. Calculate its Fermi energy. b) A 2-D square lattice has a side of 1.5 Å. Calculate the momentum and energy of the electron whose wave terminates at the boundary of the first Brillouin zone. c) A semiconductor has the following parameters at temperature T = 300 K: Calculate the energy of the intrinsic Fermi level and the intrinsic carrier concentration. d) Calculate the critical field which would destroy superconductivity at 3 K in Hg which has a critical temperature TC = 4.153 K and Bac(0) = 0.0411 T. 5. a) Determine the magnetic moment of Co2+Fe 3+ 2O4 which has an inverse spinel structure. b) Calculate the mass of boron required to make a silicon crystal with 10 16 cm-3 doping density, if the initial melt load of silicon is 50 kg. The density of silicon in the melt is 2.5 g cm-3 and boron has an atomic weight of 10.8 u. Assume that the equilibrium segregation coefficient k0 is constant throughout the growth process. c) Explain addition and condensation polymerization with an example of each. d) What are ferroelectric materials? Explain with the example of BaTiO3. How are they different from piezoelectric materials?

1. Answer in brief: i) The XeO3 molecule has a trigonal pyramidal structure(like the Ammonia molecule). List its symmetries. ii) List any two missing planes for the fcc lattice. iii) What are the advantages of the neutron diffraction method? iv) Write down the electronic configuration of the Ge atom. What type of bonding would you expect to find in Ge? v) List the independent elastic constants for a cubic crystal and state their significance. vi) If a wave function Ψ(x) is to represent an electron in a crystalline solid, what should be its nature? vii) Write down the energy for the first excited state for electrons in a 3D box. What is the degeneracy of this energy level? viii) What is the effect of dopant concentration on the Fermi energy and carrier concentration of a p-type semiconductor? ix) In which of the following ions do you expect angular momentum quenching and why: Ti 3+, Gd 3+, Ni 2+ x) What is the function of the quartz crystal used in a digital watch? 2. a) A plane intercepts the x-axis at 2a, the y-axis at 3b and the z-axis at 4c. Determine the Miller indices of this plane and the interplanar distance if the lattice parameter is 4.0 Å. b) A metallic element has an atomic weight of 27 u, density 2710kgm-3 and lattice constant 4.05 Å. Predict its crystal structure and calculate the nearest neighbor distance. c) The Bragg angle for reflection from the (110) planes in bcc iron is 22º for an x-ray of wavelength 1.54 Å. Find the lattice parameter for iron (take n = 1). What is the minimum wavelength with which the structure of this unit cell can be probed? d) Show that the reciprocal lattice for a bcc lattice is an fcc lattice. 3. a) The potential energy of a crystal is described by the expression: where ε = 3.12x10-3 eV and ρ0 = 2.5Å. Calculate the minimum potential energy. b) The frequency of the longitudinal optical phonon for NaCl at the centre of the first Brillouin zone is 5rads -1. Calculate the interatomic force constant for this material. (The atomic weight of Na = 23u and Cl = 37u) c) Calculate the Debye specific heat of Molybdenum at 300 K, given that its Debye frequency is 9.74 x 10 13 rads -1. d) The values of the elastic stiffness constants for GaAs are: C11=1.18 x 10 11 Nm -2, C44 = 0.59 x 10 11 Nm -2 and C12 = 0.54 x 10 11 Nm -2 Given that the density of GaAs is 5.32gcm -3 determine the velocity of the transverse and longitudinal elastic waves in the [100] direction. 4. a) Calculate the transition temperature for a superconductor whose energy gap is 1.65x10-3 eV . b) For a Si semiconductor, Nc = 2.8x10 19 cm-3 and NV = 1.04x10 19 cm-3. Determine the position of the Fermi level at room temperature for NA = 10 17 cm-3 and ND = 10 14 cm-3. c) Calculate the Fermi energy and Fermi temperature for potassium metal, whose electron number density is 1.4x10 28 cm-3. d) When 20 mA of current is passed through a specimen under a magnetic field of 0.5 Wbm-2, the measured Hall voltage is 37μV. If the width of the specimen is 0.01 mm, calculate the Hall coefficient. 5. a) The saturation magnetization of fcc Ni is 5.1 x 10 5 A/ m. If the lattice constant for Ni is 3.52 Å, calculate the net magnetic moment per Ni atom in the crystal in units of Bohr magnetons. b) What are fusible alloys? Explain their use in safety sprinklers. c) Explain addition and condensation polymerization with an example of each. d) Explain, with a diagram, the operation of a photovoltaic solar cell.