Solve your IGNOU Doubts

Question:

A proton $(\text{mass} = 1.67 \times 10^{-27} \text{ kg})$ inside a nucleus $(\text{radius} = 10^{-14} \text{ m})$ may have speed upto $2.0 \times 10^8 \text{ ms}^{-1}$ . How many quantum states are available to it?

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Question:

A divalent metal crystallizes in an fcc structure with a lattice constant of 4.5 A. Calculate the number density of conduction electrons and the Fermi velocity.

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Question:

For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.

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Question:

Calculate the temperature at which the lattice contribution to the specific heat and the electronic contribution to the specific heat become equal in a metal which has a Debye temperature of $390 \text{ K}$ and a Fermi energy $11.7 \text{ eV}$ .

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Question:

For a linear chain of identical atoms of mass $10^{-26} \text{ kg}$ calculate the maximum value of the angular frequency of the longitudinal wave and the group velocity at $k=0$ , given that the inter-atomic distance is 2.0 A and the spring constant is $20 \text{ Nm}^{-1}$ .

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Question:

Obtain expression for average energy $U (\equiv \langle E \rangle)$ for the canonical ensemble in terms of partition function Z.

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Question:

Calculate the inter-atomic equilibrium distance r_e for $\text{KCl}$ for which the equilibrium lattice energy is $163.0 \text{ kcal mol}^{-1}$ , $n = 8.6$ and the Madelung constant is 1.75.

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Question:

A metallic crystal has an fcc lattice with a lattice constant $0.4 \text{ nm}$ . 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 $0.154 \text{ nm}$ .

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Question:

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

$$H(q, p) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} + U(q_i)$

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.

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Question:

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

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Question:

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

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Question:

N particles obey Maxwell-Boltzmann distribution. They are distributed among three states with energies $E_1 = 0, E_2 = k_BT,$ and $E_3 = 4k_BT$ . If the equilibrium energy of the system is approximately 3000k_BT, calculate the total number of particles.

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Question:

A monthly demand for commodity is a continuous random distribution with probability distribution function given as:
$$p(x) = \begin{cases} 3N(x^2 - 1) & 1 < x < 2 \\ 0 & \text{elsewhere} \end{cases}$
where N is normalization constant. Obtain the value of N so that the function is normalized and hence, find the mean and the variance.

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Question:

A metallic element has a density of $2.70 \text{ g cm}^{-3}$ , a lattice constant of $4.0495 \text{ Å}$ and an atomic weight $26.9815 \text{ g mol}^{-1}$ . Calculate the number of atoms per unit cell of this element and predict its lattice crystal structure.

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Question:

Show under what condition Poisson distribution tends to Normal distribution.

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Question:

A step-index optical fiber has core refractive index 1.50, cladding refractive index 1.45, and core diameter $3\text{ }\mu\text{m}$ . The wavelength of light propagating in the fiber is $1.56\text{ }\mu\text{m}$ . Calculate the V-number and comment if it supports single-mode or multimode propagation.