IGNOU MSCPH MPH 12 Condensed Matter Physics Solved Assignment ENGLISH

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MPH 012 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment

CONDENSED MATTER PHYSICS

Course Code: MPH-012

Assignment Code: MPH-012/TMA/2026

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A
1. a) 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. 

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

c) 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 . 

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

b) 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 . 

c) 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 . 

3. a) 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. 

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

c) For the energy dispersion relation:

calculate the inverse mass tensor. 

d) 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. 

PART B

4. a) Calculate the resistivity of Silicon at given that its resistivity at is . 

b) 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 . 

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

b) 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. 

c) 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

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

b) 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:


c) 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. 

d) Explain the significance of the pseudogap phase in cuprates. 

MPH 12 2025 - English

Course Code: MPH-012

Assignment Code: MPH-012/TMA/2024-25

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) A metallic element has a density of 7.15 g cm^-3, a lattice constant of 2.880 A and an atomic weight of 51.9961. Calculate the number of atoms per unit cell of this element and predict its lattice crystal structure.

b) Show that the volume of the primitive cell in the reciprocal lattice space is inversely proportional to the volume of the primitive cell in the direct lattice.

c) At what angle will a diffracted beam emerge from the (110) planes of a cubic crystal of unit cell length 0.6 nm? Assume diffraction occurs in the first order and that the X-ray wavelength is 0.154 nm ?

2. a) If the potential energy function is expressed as  calculate the inter-molecular distance at which the potential energy is a minimum. Show that in the stable configuration the energy of attraction is eight times the energy of repulsion.

b) Debye temperature for diamond is 2230 K. Calculate the frequency of highest possible lattice vibration in diamond and its molar heat capacity at 20 K.

3. a) Metallic sodium is monovalent and crystallizes in a bcc structure with a lattice constant of 4.25 A. Calculate the number density of conduction electrons and the Fermi energy at 0 K.

b) For a free electron gas in two dimensions, derive the relation between the number density of electrons and the Fermi wave vector.

c) For the energy dispersion relation:

calculate the inverse mass tensor and the group velocity given m₁ = 3m₂.

d) Consider an electron in a bcc lattice with lattice constant b. Show that a wave function of the form:

satisfies the Bloch theorem.

PART B

4 a) Calculate the temperature at which the number of electrons in the conduction band of a semiconductor is four times the number at room temperature. Take the band gap energy to be 1.2 eV with E_c - E_F ≈ E_G.

b) Calculate the total voltage difference (the built-in potential) between the n-type and p-type part for a uniformly doped Silicon p-n junction with N_d = N_a = 10²³ m^-3 at room temperature. The intrinsic carrier density is 1.45 x 10¹⁴ m^-3. Will the built-in voltage increase or decrease with an increase in temperature?

5. a) What is the piezoelectric effect? Describe the piezoelectric effect in barium titanate.

b) Derive the magnetization and susceptibility for the free spin (J=S=1/2). You may use the relation:  

c) Describe the different types of exchange interactions that can give rise to spontaneous magnetic order.

6. a) The critical field for Niobium (Nb) is 10⁵ Am^-1 at 8 K and 2×10⁵ Am^-1at 0 K. Calculate the transition temperature.

b) Calculate the super-electron density for Sn which has a London penetration depth of 34 nm.

c) The transition temperature for an isotope of Mercury (Hg) with a mass of 199u is 4.185K. Calculate the transition temperature for an isotope of Hg with mass number 202u. Take the value of the exponent a to be 0.5.

d) Calculate the wavelength of the photon required to break the Cooper pair in a superconductor with a transition temperature of 1.3 K.

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