Consider the Atomic Packing Factor (APF) of the following crystal structures:
P. Simple cubic
Q. Body centred cubic
R. Face centred cubic
T. Hexagonal close packed
Which two of the above structures have equal APF?

The valence electrons do not directly determine the following property of a metal

A. electrical conductivity

B. thermal conductivity

C. shear modulus

D. metallic lustre

The solid phase of an element follows van der Waals bonding with inter-atomic potential $$V\left( r \right) = - \frac{P}{{{r^6}}} + \frac{Q}{{{r^{12}}}}$$    where, P and Q are constants. The bond length can be expressed as

A. $${\left( {\frac{{2Q}}{P}} \right)^{ - 6}}$$

B. $${\left( {\frac{Q}{P}} \right)^{ - 6}}$$

C. $${\left( {\frac{P}{{2Q}}} \right)^{ - 6}}$$

D. $${\left( {\frac{P}{Q}} \right)^{ - 6}}$$

A system containing N non-interacting localized particles of spin $$\frac{1}{2}$$ and magnetic moment $$\mu $$ each is kept in constant external magnetic field B and in normal equilibrium at temperature T. The magnetization of the system is

A. $$N\mu \coth \left( {\frac{{\mu B}}{{{k_B}T}}} \right)$$

B. $$N\mu \tanh \left( {\frac{{\mu B}}{{{k_B}T}}} \right)$$

C. $$N\mu \sinh \left( {\frac{{\mu B}}{{{k_B}T}}} \right)$$

D. $$N\mu \cosh \left( {\frac{{\mu B}}{{{k_B}T}}} \right)$$

In a cubic crystal, atoms of mass M₁ lie on one set of planes and atoms of mass M₂ lie on planes interleaved between those of the first set. If C is the forte constant between nearest neighbour planes, the frequency of lattice vibrations for the optical phonon branch with wave vector k = 0 is

A. $$\sqrt {2C\left( {\frac{1}{{{M_1}}} + \frac{1}{{{M_2}}}} \right)} $$

B. $$\sqrt {C\left( {\frac{1}{{2{M_1}}} + \frac{1}{{{M_2}}}} \right)} $$

C. $$\sqrt {C\left( {\frac{1}{{{M_1}}} + \frac{1}{{2{M_2}}}} \right)} $$

D. zero