Calculate the inter-atomic equilibrium distance r_e for  for which the equilibrium lattice energy is ,  and the Madelung constant is 1.75. 

A metallic crystal has an fcc lattice with a lattice constant . Explain whether the following planes are allowed or forbidden for X-ray diffraction:

(100), (111), (210), (220)

Calculate the X-ray diffraction angles for the allowed planes. Assume that diffraction occurs in the first order and the X-ray wavelength is . 

 The Hamiltonian for the non-relativistic, non-interacting, monoatomic ideal gas is given by

where m is the mass of the particle and U(q_i) is the potential energy. Obtain an expression of the phase space volume of the energy shell and hence obtain an expression of number of microstates.

btain the phase space area enclosed by a classical harmonic oscillator for energies ranging from 0 to E.

Show that the reciprocal lattice of a bcc lattice is an fcc lattice. Calculate the magnitude of the shortest non-zero reciprocal lattice vector. 

N particles obey Maxwell-Boltzmann distribution. They are distributed among three states with energies  and . If the equilibrium energy of the system is approximately 3000k_BT, calculate the total number of particles.

 A monthly demand for commodity is a continuous random distribution with probability distribution function given as:

where N is normalization constant. Obtain the value of N so that the function is normalized and hence, find the mean and the variance.

A metallic element has a density of , a lattice constant of  and an atomic weight . Calculate the number of atoms per unit cell of this element and predict its lattice crystal structure. 

Show under what condition Poisson distribution tends to Normal distribution.

 A step-index optical fiber has core refractive index 1.50, cladding refractive index 1.45, and core diameter . The wavelength of light propagating in the fiber is . Calculate the V-number and comment if it supports single-mode or multimode propagation. 

What factors determine the number of modes propagating in an optical fiber?

 Derive the expressions for TE and TM mode fields in an optical waveguide, relating electric and magnetic field components. 

the coherence time of the emitted radiation.
Take the speed of light, . 

the coherence time of the emitted radiation.

the spectral line width in Hertz, and

the coherence length,

A mercury vapour lamp emits light of mean wavelength  with a spectral spread . Calculate

List the components of a laser and discuss each of them briefly.

 On the basis of Fresnel diffraction integral, discuss the conditions which differentiate between Fresnel and Fraunhofer diffraction.

Write the laser rate equation for a two level system and hence prove that the steady state population inversion cannot be achieved in a two level system.