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QUESTION: 1

A monoatomic. cubic material has lattice spacing of a. The sound velocity for longitudinal and transverse phonons is approximately equal. C_{T} = C_{L} = C, is isotropic. The debye frequency of the phonons will be

Solution:

In debye model, the dispersion, related is approximated to be ω = ck and in 3-D case, the density of state is in the form of

where, V= L_{3} and the factor of 3 is due to the tact that there are one longitudinal and two transverse vibration modes for each k state.

If total number of states in solid is N. then

Since, there are three states associated with each atom (2 for transverse and one for longitudinal vibration)

So,

QUESTION: 2

Graphite lias a layered crystal structure in which the coupling between carbon atoms in different lasers is much weaker than the atoms in the same layer. Using Debye theory, the specific heat is proportional to (where T is the temperature)

Solution:

According to debye-model the density of states for a two dimension and lattice is

Hence, total energy.

For

⇒

QUESTION: 3

All X-ray beam of wavelength 1.7 Å is diffracted by a cubic KC1 ciystal of density 1.99 x 10^{3} kg m^{-3}. The interplance spacing for (200) planes and the glancing angle for the second order reflection from these planes will be (given that molecular weight of KC1 is 74.6 amu)

Solution:

For cubic crystal we have

n' = 4 because KC1 has fee structure

a = 6.29 x 10^{-10} m

From Bragg's law, 2d sin θ = nλ

n = 2

QUESTION: 4

Elections of mass m are confined to one dimension. A weak periodic potential described by the fourier series

The energy gap (to first order) at (between the first and second band is)

Solution:

In such problems, the energy bands will be essentially parabolic with gaps at every point of degeneracy. This degeneracy is removed by mixing the corresponding wavefunctions and perturbation potential.

Solve the matrix element, Iii the neighbourhood of k (for lower band) and or the upper band L is the length of sys te m ψ_{1} and ψ_{2} are calculated using extended zone

scheme

W lien we diagnolise the Hamiltonia matrix we get the energy gap,

QUESTION: 5

All electron of mass m moves in a square lattice of lattice spacing a. Assume that the nearly free electron approximation holds good. If there are two electrons per site, then which of the following is correct?

Solution:

With two electrons per site, assuming that the constant energy lines are circles, we obtain

71

This value is larger than π/a. Therefore, we needs to consider the two bands. There will be two fermi surface. a

one in the upper band and other in the lower band. Each of the band is partially filled, with two electrons per site the NFE approximation leads to a metal.

QUESTION: 6

A paramagnetic substance lias 10^{28} atoms/m^{3}. The magnetic moment of each atom is 1.8x10^{-23} Am^{2}. Wliat would be the dipole moment of a bar of this material 0.1 meter long and 1 cnr cross-section placed in a field of 8x 10^{4 }Am^{-1}?

Solution:

QUESTION: 7

A uniform silver wire has a resistivity of 1.54 x 10^{-8} Clm at room temperature and there are 5.8 x 10^{28} conduction electrons per m^{3}. If an electric field of 1 volt cm is applied along the wire, the relaxation time of electron will be

Solution:

Mobility,

Relaxation time τ is given by

QUESTION: 8

Assuming that there are 10^{27} molecules /m^{3} in HCl vapour. The oiientation polarization at room temperature if the vapour is subjected to an electric field of 10^{6} V/m will be (given that dipole moment of HCl molecule is 3.46x10^{-30 }c-m)

Solution:

Orientation polarizability is given by

Hence, the orientational polarization will given by

*Answer can only contain numeric values

QUESTION: 9

Assume that the energy of the two particles in the field of each other is given by where

α and β are cosntants and r is the distance between the centres of paiticles. In stable configuration, the ratio of energy of attraction to the energy of repulsion will be____________(Answer should be an integer)

Solution:

Force,

For stable configuration, F= 0 at r - r_{0}

⇒

E_{1}=Energy of attraction

E_{2} = Energy of repulsion

*Answer can only contain numeric values

QUESTION: 10

Sodium transforms frombcc to hep at about T = 23 K. Assumig that the density remains fixed and c/a ratio is

ideal. The hcp lattice spacing, a [given that the cubic lattice spacing a' = 4.23Å in the cubic phase] is_______Å (Answer should be upto two decimal places)

Solution:

Density of bcc structure is 2 atoms per a^{'3} density of hep structue is 2 atoms per the volume of primitive unit cell V_{0}.

Volume of unit cell

QUESTION: 11

Atoms are arranged in a one-dimensional chain with lattice spacing a. Each atom is represented by the potential V(x) = . The energy gaps between the bands, assuming that the nearly free electron approximation holds good, is

Solution:

The energy gaps one determined by the fourier components of lattice potential. For the dirac-delta potentially. Tlie matrix elements are

Independent of n.

So, energy gap,

QUESTION: 12

The effective mass tensor (M_{ij}) for electrons in a simple cubic tight binding band at the centre (k = (0, 0,0) of the brillion zone will be

Solution:

The electron energy in tight binding approximation is

Therefore, effective mass tensor,

At (0, 0, 0 )

QUESTION: 13

Consider a two dimensional hexagonal lattice if lattice spacing a = 3 Å and sound velocity c = 10^{3} m/sec. The debye frequency, w_{D} is (provide a numerical value in sec^{-1})

Solution:

The debye wave-number. k_{D} is defined such that the total number of states within a radius of k_{D} equals the total number of states in the Biillion zone.

⇒

⇒

QUESTION: 14

For a diatomic linear chain, the phonon dispersion relation ω (k) has two branches corresponding to + and - sign respectively

There are two atoms in the unit cell with masses and M2 and the force constant of nearest neighbour interaction is F and the effective mass is kept constant. The velocity of sound will be

Solution:

Sound velocity,

Consider acquostic Branch, As q → 0, sin^{2}

⇒

QUESTION: 15

Consider a two dimensional square lattice with one atom of mass m per lattice point interacting with only nearest neighbours with force constant k. Take the phonon dispersion curve to be At high temperature the mean square displacement of an atom from its equilibrium position will be

Solution:

Total internal energy of the lattice

If there are N atoms in lattice

At high temperature, average P.E. equals average K.E., hence half the total energy

QUESTION: 16

For a hexagonal closed packed structure with periodic potential (G being a reciprocal lattice G

vector). If G corrosponding to the first brillion zone face in the C direction C that is normal to the face is in the (001) direction), then the value of V_{G} is

Solution:

If there are n identical atoms in the unit cell with positions. d_{1},d_{2}........dj...,dn

Periodic potential.

where R is the position of lattice point.

Where, is the geomatric structure factor

Unit cell of hcp contains 2 atoms at (0,0,0) and

QUESTION: 17

A type-II superconducting wire of radius R carries current uniformly through its cross-section. If die total current earned by the wire is I. the magnetic energy per unit length of the wire is I. the magnetic energy per unit length of the wire is

Solution:

Magnetic energy,

Magnetic energy per unit length.

QUESTION: 18

The penetrating depth of mercury at 3.5 k is about 750 Å. The penetration depth at OK will be

Solution:

From the formula for penetration depth

*Answer can only contain numeric values

QUESTION: 19

Consider two dimensional electrons subjected to a weak periodic potential coming from a square lattice of spacing a = 5Å. For k vectors for away from the bu llion zone boundaries, the wavefunctions can be well

described by plane waves. Assume we want to write the wavefunction in block form,. Consider a state of energy E and wavevector The lowest energy at this wave number will be ________________eV (upto second decimal places)

Solution:

from schrodiner equation,

⇒

where, k* extends over entire space.

So, we can transform k* into first B.z. by translation

For lowest energy,

= 3.806x 0.5 x 0.5 = 3.806x 0.25

*Answer can only contain numeric values

QUESTION: 20

Assume that the E vs k relationship for electrons in the conduction band of a hypothetical tetravalent n-type semiconductor can be approximated by E = ak^{2} + constant. The cyclotron resonane for electrons in a field B = 0.1 Weber/m^{2} occurs at an angular frequency ω_{e }= 1.8 x 10^{11} rad s^{-1}The value of a is_______x 10^{-38}(in the untis of Jm^{2})

Solution:

Electron in conduction band behaves as if it had an effective mass m* given by

Cyclotron resonance frequency for the electron is

⇒

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