Assignment 1
1. Show that the Madelung constant for a one-dimensional array of ions of alternating sign
with equal distance between successive ions is equal to 2ln2.
2. A powder pattern is obtained from a face centred cubic lattice with radiation of λ = 1.45
Å. The (220) reflection is observed at Bragg angle θ = 32°. What is the lattice parameter
of unit cell and the radius of the atom?
3. Find an expression for the equilibrium potential energy of an ionic solid.
4. Show the method of calculation of the Madelung constant for NaCl lattice.
5. Relate the value of the repulsive exponent in the interaction to compressibility of NaCl
lattice.
6. Using the Born Haber cycle, derive an expression of lattice energy (U 0) for the formation
of NaCl.
7. Assuming that the potential energy of two particles in the field of each other is given by
U(r) = -(A/r) + (B/r9)
Where A and B are constants, show that for stable configuration, the energy of attraction
is 9 times the energy of repulsion.
8. Applying the concept of 21 symmetry, draw a xy projection of body-centred cubic lattice
which has a diad axis along z and a pair of commas related by the axis is associated with
each lattice point.
9. Show that a twofold rotoinversion is equivalent to a reflection or a reflection through a
plane and is simultaneously a onefold rotoreflection.
10. Draw the stereographic projections for the point groups roto-reflection S 1, S2, S4 and S6.
11. With the help of schematic diagrams only, show that the total rotational axes, reflection
planes and inversion symmetries in a cubic system.
12. Show that for a crystal of cubic symmetry the direction [hkl] is perpendicular to the plane
(hkl).
13. Derive an expression for interplaner spacing in case of three-dimensional lattice.
14. What do you mean by reciprocal lattice? How do you visualise a reciprocal lattice of a
given crystal?
15. Prove that the reciprocal lattice to a simple cubic lattice is a simple cubic lattice itself.
16. Prove that the reciprocal lattice of an FCC lattice is a BCC lattice and vice-versa.
Assignment 1
17. Show that every reciprocal lattice vector is normal to a lattice plane of the direct crystal
lattice.
18. If G is a reciprocal lattice vector, derive the vector form of Bragg’s law.
19. What is meant by atomic scattering factor? Derive an expression for atomic scattering
factor in case of an atom.
20. The spacing between successive (100) planes in NaCl is 2.82 Å. X-ray incident upon the
surface of this crystal, is found to give rise to first order Bragg reflection at a grazing
angle of 8.35°. Calculate the wavelength of the X-ray and the angle at which the second
order Bragg reflection would occur.
21. Show that for metallic sodium (bcc structure) the X-rays diffraction pattern does not
contain lines corresponding to the (hkl) values (100) and (300).
22. What is point defect in crystal? What are the different types of point defects? How are
they caused (explain with suitable sketches)?
23. Show that the number of Frenkel defects in equilibrium at a given temperature is
proportional to (NNi)1/2, where N be number of atoms and Ni be the interstitial atoms.
24. Show that the number of Schottky defects depends on the total number of ionic pairs, the
average energy required to produce defect, and the temperature.
25. If 1 eV is the energy required to move an atom from the crystal’s interior to the surface,
what is the proportion of vacancies present in the crystal at 1000 and at 300K?
26. The electronic configuration of a Cr2+ ion is 3d44S0. Calculate the magnetic susceptibility
for a salt containing 1 Kg mole of Cr2+ ions at 300K.
27. For a magnetic system, what is the relation between adiabatic temperature change and
isothermal entropy change?
28. Using the following Molecular field and Brillouin equation in which λ, N, g, µ B, J and k
are atomic constants,
Bm = λM
and
2J 1
2 J 1 Jg B B 1
1 Jg B B
M NgJ B
Coth
Coth
2J
2 J kT
2 J kT
2J
Determine the Curie temperature and ferromagnetic susceptibility.
0
