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

An external magnetic field of magnitude H is applied to a type-I superconductor at a temperature below the transition point. Then which one of the following statements is not true for H less than the critical field Hc?

The Bloch theorem states that within a crystal, the wave function $$\psi \left( {\overrightarrow {\bf{r}} } \right)$$ , of an electron has the form