Show that for the superconducting transition in the absence of magnetic field, there is a discontinuity in the specific heat of a superconductor at Tc which can be written as:
For a superconducting specimen, the critical fields are and
at
and
respectively. Calculate the critical temperature and critical field at
.
For the hydrogen molecule which has two hydrogen atoms each with one electron occupying the 1s energy level, write the two particle wave functions for the singlet and triplet states. Determine the eigenvalues of the effective Hamiltonian
Discuss the concept of thermodynamic fluctuations in the canonical ensemble. Define energy fluctuations and derive an expression for the mean square fluctuation of energy. Using a classical ideal gas, show that the relative fluctuation in energy varies as and hence, justify why thermodynamic calculations are valid for ordinary macroscopic systems.
For Chromium () vapour at
with a number density of atoms
calculate:
i) The Larmor diamagnetic susceptibility assuming the atomic radius to be 1.2 A.
ii) The Curie paramagnetic susceptibility.
See Answer →Using entropy as a function of temperature and pressure, obtain the first Ehrenfest’s equation.
See Answer →What is meant by Cluster Integrals? Express B2 and B3 in terms of Cluster Integrals.
See Answer →The susceptibility of is
and its density is
. Calculate its total polarisability. Assume that the mass number for
is 32.
In an n-type semiconductor the Fermi level lies below the conduction band at
. Calculate the position of the Fermi level when the temperature is raised to
.
State the Virial theorem and prove Boyle’s law using it.
See Answer →Calculate the resistivity of Silicon at given that its resistivity at
is
.
What is the Bose-Einstein condensation? Write an expression of number of particles in the excited state (Nex) in terms of Bose-Einstein condensation temperature (Tc), total number of bosons (N) in an assembly and temperature T. Show a plot of distribution of bosons as a function of temperature below and above the Tc.
See Answer →Calculate the energy dispersion for s-band in the bcc lattice for the tight binding approximation.
Note: For the central atom located at (0,0,0) in the bcc unit cell, the nearest neighbours are located at , where a is the lattice constant.
Show that the probability that a state with energy above the Fermi level
filled is equal to the probability of a state with energy
below the Fermi level
is empty.
Using the relation of thermodynamic probability
Obtain an expression of Fermi Dirac distribution function. Show a plot of (the occupation index of a state corresponding to energy
) versus
for different temperatures and discuss it. What is the physical interpretation of Fermi energy?
For the energy dispersion relation:
calculate the inverse mass tensor.
State and prove quantum Liouville equation.
See Answer →The time evolution of the quantum state, is given by the Schrödinger wave equation:
where H is the Hamiltonian operator. Using this equation, obtain time evolution of the wave function when the energy eigenvalue is E.
See Answer →In Sommerfeld free electron theory, show that at a temperature T
(i) and (ii)
The time evolution of the quantum state, is given by the Schrödinger wave equation: