Condensed Matter Physics MCQ Quiz - Objective Question with Answer for Condensed Matter Physics - Download Free PDF

 Explanation:

If the potential is periodic with lattice translations 𝑅, i.e., V(𝑉) = V(𝑉 + 𝑅), then the wavefunction has the form:

ψ_k(𝑉) = e^i_k·𝑉 u_k(𝑉)

where u_k(𝑉) is a function with the same periodicity as the lattice:

u_k(𝑉) = u_k(𝑉 + 𝑅)

1. |ψ_k(𝑉)| is constant
   Incorrect – because u_k(𝑉) varies with position, so the modulus of ψ_k(𝑉) is not constant.

2. ψ_k(𝑉) is also an eigenstate of the momentum operator
   Incorrect – ψ_k is not an eigenstate of the momentum operator due to the periodic modulation u_k(𝑉). Only a plane wave e^i_k·𝑉 would be.

3. ψ_k(𝑉) = ψ_k(𝑉 + 𝑅)
   Incorrect – this is not generally true. However, Bloch’s theorem implies:

   ψ_k(𝑉 + 𝑅) = e^i_k·𝑅 ψ_k(𝑉)

4. |ψ_k(𝑉)| = |ψ_k(𝑉 + 𝑅)|
   Correct – using the property above:

   |ψ_k(𝑉 + 𝑅)| = |e^i_k·𝑅 ψ_k(𝑉)| = |ψ_k(𝑉)| 

Solution:

Type-I superconductors are those which show perfect Meissner effect, i.e., no magnetisation above H_c (only one critical field).

Type-II superconductors are those which do not show perfect Meissner effect, but they exhibit a mixed state/vortex state between the lower critical magnetic field and the upper critical magnetic field.

Satisfied by option-3 exactly.

Given:

The dispersion relation for a one-dimensional monatomic lattice chain is given by the equation:

Where: - a is the interatomic spacing, is the wave vector, and - v has the dimension of velocity.

The relation between the phase velocity and the group velocity in the long wavelength limit is given.

Concept:

• The phase velocity is defined as .
• The group velocity is the derivative of with respect to k , i.e., .
• In the long wavelength limit, where , the phase velocity and group velocity become equal for the given dispersion relation.

Calculation:

From the given dispersion relation:

For small k (long wavelength limit), , so the dispersion relation simplifies to:

Therefore, the phase velocity is:

The group velocity is the derivative of \omega with respect to k :

∴ In the long wavelength limit, .

∴ The correct answer is option 1: .

CONCEPT:

The relation between entropy S and themodynamic probability Ω is 

​→ S = k ln Ω 

Frenkel defect: A Frenkel defect is a type of point defect in crystalline solids. The defect forms when an atom or smaller ion leaves its place in the lattice, creating a vacancy and becomes an interstitial by lodging in a nearby location.

Thermodynamic probability of the Frenkel defects is 

Where, N, N' and n are total number of lattice and interstitial sites and number of defects respectively.

change in free energy in creating 'n' Frenkel defects 

EXPLANATION:

The probability of such frenkel defects 

So, the change in entropy 

⇒ ΔS = k ln Ω = 

⇒ ΔS = k ln [ N ln N + N' ln N' - (N - n) - (N' - n)ln(N' - n) - 2n ln n ]

[ Here we have used stirling's approximation i.e ln N! = N ln N - N ]

Now, change in free energy in creating n Frenkel defects 

Since, n N, N'

Hence the correct answer is option 1.

Concept- 

We are using the dispersion relation here which is given by

•  E_k =C√k here k is wave number
• At low temperatures for bosons, E  ∝ ks  
• At low-temperature entropy at constant volume is given by Cv ∝ Td/s  
• s =power of wavenumber in energy dispersion relation and d =dimension

Explanation:

• Ek =C√k
• At low temperatures for bosons, E  ∝ k^s 
• Equate this with given equation s=1/2
• At low-temperature C_v ∝ T^d/s
• Here d is dimension = 2 (Given in the question)
•    
• C_v∝ T⁴

So, the correct answer is option (1).

CONCEPT:

Josephson junction: Two superconductors separated by a very thin strip of an insulator forms a Josephson junction.

A.C. Josephson Effect: when a potential difference V is applied between the two sides of the junction, there will be an oscillation of the tunneling current with angular frequency ω = . This is called the A.C. Josephson Effect.

EXPLANATION:

Current density through thin insulating layer is

∴ The angular frequency of the supercurrent is 

⇒ 

Hence the correct answer is option 1.

Explanation:

• This is the dispersion relation for electrons where  is the speed of particles near the Fermi level in an electron gas. This is actually the case with graphene or a very similar system where we are using a linear dispersion relation.
• The total number of particles (or electrons in this case) with given k up to Fermi wavenumber  in three-dimension can be represented as 
• Since  from the original dispersion relation: Thus, we arrive at  which indicates that  

Concept:

The hopping probability is exponential in the height only of the (free) energy barrier between sites.

Calculation:

i> ∑ piri

= 0.3i - 0.2i + 0.2j - 0.3j

= 0.1i - 0.1j

For N steps, = [i - j]

The correct answer is option (2).

Explanation:

• The two lattices A and B are essentially shifted versions of each other. A can be thought of as the checkerboard pattern where we select all the black squares (those that represent odd sums of indices), whereas B is the lattice that consists of all the white squares (those representing the even sums of indices).
• In a three-dimensional setting, if you translate (shift) lattice A by one unit in any direction (along the x, y or z axis), you will get the lattice points of lattice B.
• Similarly, if you translate lattice B by one unit you will get lattice A. So we can say that lattices A and B are in a relationship called "dual" or "face-centered" where one can be obtained by shifting the other by one unit. This concept is often used in crystal structures where the atoms reside on the vertices of such a lattice point.

CONCEPT:

Hall coefficient: the quotient of the potential difference per unit width of a metal strip in the Hall effect divided by the product of the magnetic intensity and the longitudinal current density

The hall coefficient for a semiconductor having both types of carriers is given as

⇒ RH = 

When the temperature is low p (the no. of holes) is greater than n (no. of electrons)

with the increase in temperature, the no. of holes decreases, and the no. of electrons increases. 

Case I: At low temperature: p >> n, μ_p n

⇒ pμp² > nμn2 

⇒ pμp2 - nμn2 > 0 ⇒ R_H = positive.

Case II: At moderate temperature: p > n

⇒ pμp2 ≈ nμn2 (since, μp n)

⇒ R_H > 0

Case III: At high temperature: p/n ≈ 1

⇒ pμp2 - nμn2 

⇒ RH 

Only, option (4) matches with the above conclusions. 

Hence the correct answer is option 4.

Explanation:

• For a paramagnetic system at thermal equilibrium, the average magnetic moment ⟨M⟩ can be derived using the Boltzmann distribution. The Boltzmann distribution tells us that the probability of a system being in a particular state is proportional to the exponential of the negative energy of that state divided by the product of Boltzmann's constant (k_B) and the temperature (T).
• The magnetic dipole moment μ of an atom or ion in the presence of a magnetic field B has energy -μB. If the magnetic moment per ion (μ±) can take on values ±μB, then the energies of the two states are .
• The probability of being in state with energy E± is given by the Boltzmann factor: 
•  We then normalize by dividing by the sum of the probabilities of all possible states (the partition function Z): 
• This gives us: 
• The average magnetic moment ⟨M⟩ is the sum of each possible magnetic moment multiplied by its Boltzmann-weighted probability:

Simplify to get: 

Now, 

This has following behavior 

as T → 0 ,  → max ,

as T → ∞ ,   → 0,

The curve will be an inverse of hyperbolic tangent.

So, the option (C) curve is the solution.

CONCEPT:

The relation between entropy S and themodynamic probability Ω is 

​→ S = k ln Ω 

Frenkel defect: A Frenkel defect is a type of point defect in crystalline solids. The defect forms when an atom or smaller ion leaves its place in the lattice, creating a vacancy and becomes an interstitial by lodging in a nearby location.

Thermodynamic probability of the Frenkel defects is 

Where, N, N' and n are total number of lattice and interstitial sites and number of defects respectively.

change in free energy in creating 'n' Frenkel defects 

EXPLANATION:

The probability of such frenkel defects 

So, the change in entropy 

⇒ ΔS = k ln Ω = 

⇒ ΔS = k ln [ N ln N + N' ln N' - (N - n) - (N' - n)ln(N' - n) - 2n ln n ]

[ Here we have used stirling's approximation i.e ln N! = N ln N - N ]

Now, change in free energy in creating n Frenkel defects 

Since, n N, N'

Hence the correct answer is option 1.