IGNOU MPH 12 SOLVED ASSIGNMENT

Ques 1.

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. 

Ques 2.

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

Ques 3.

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 . 

Ques 4.

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

Ques 5.

For a linear chain of identical atoms of mass  calculate the maximum value of the angular frequency of the longitudinal wave and the group velocity at , given that the inter-atomic distance is 2.0 A and the spring constant is . 

Ques 6.

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  and a Fermi energy . 

Ques 7.

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. 

Ques 8.

In Sommerfeld free electron theory, show that at a temperature T
(i)  and (ii) 

Ques 9.

For the energy dispersion relation:

calculate the inverse mass tensor. 

Ques 10.

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. 

Ques 11.

Calculate the resistivity of Silicon at  given that its resistivity at  is . 

Ques 12.

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 . 

Ques 13.

The susceptibility of  is  and its density is . Calculate its total polarisability. Assume that the mass number for  is 32. 

Ques 14.

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. 

Ques 15.

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

Ques 16.

For a superconducting specimen, the critical fields are  and  at  and  respectively. Calculate the critical temperature and critical field at . 

Ques 17.

Show that for the superconducting transition in the absence of magnetic field, there is a discontinuity in the specific heat of a superconductor at T_c which can be written as:

Ques 18.

The critical temperature of lead (Pb) of average atomic mass  is . Calculate the critical temperature of a specimen of lead isotope with mass  using the normal isotope effect. 

Ques 19.

Explain the significance of the pseudogap phase in cuprates. 

Ques 20.

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. 

Ques 21.

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

Ques 22.

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 . 

Ques 23.

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

Ques 24.

For a linear chain of identical atoms of mass  calculate the maximum value of the angular frequency of the longitudinal wave and the group velocity at , given that the inter-atomic distance is 2.0 A and the spring constant is . 

Ques 25.

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  and a Fermi energy . 

Ques 26.

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. 

Ques 27.

In Sommerfeld free electron theory, show that at a temperature T
(i)  and (ii) 

Ques 28.

For the energy dispersion relation:

calculate the inverse mass tensor. 

Ques 29.

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. 

Ques 30.

Calculate the resistivity of Silicon at  given that its resistivity at  is . 

Ques 31.

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 . 

Ques 32.

The susceptibility of  is  and its density is . Calculate its total polarisability. Assume that the mass number for  is 32. 

Ques 33.

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. 

Ques 34.

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

Ques 35.

For a superconducting specimen, the critical fields are  and  at  and  respectively. Calculate the critical temperature and critical field at . 

Ques 36.

Show that for the superconducting transition in the absence of magnetic field, there is a discontinuity in the specific heat of a superconductor at T_c which can be written as:

Ques 37.

The critical temperature of lead (Pb) of average atomic mass  is . Calculate the critical temperature of a specimen of lead isotope with mass  using the normal isotope effect. 

Ques 38.

Explain the significance of the pseudogap phase in cuprates. 

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