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

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 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

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