For a three-dimensional crystal having N primitive unit cells with a basis of p atoms, the number of optical branches is

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