This set of Engineering Physics Multiple Choice Questions & Answers (MCQs) focuses on “Fermi-Dirac Distribution”.

1. Fermi-Dirac statistics is for the ________

a) Distinguishable particles

b) Symmetrical Particles

c) Particles with half integral spin

d) Particles with integral spin

View Answer

Explanation: The Maxwell-Boltzmann statistics is for the distinguishable particles, which are basically the classical particles like atoms and molecules.

2. The difference between fermions and bosons is that bosons do not obey ______

a) Aufbau principle

b) Pauli’s Exclusion Principle

c) Hund’s Rule of Maximum Multiplicity

d) Heisenberg’s Uncertainty Principle

View Answer

Explanation: The particles that follow Pauli’s Exclusion Principle are defined as Fermions while that don’t are called bosons. Bosons have an integral spin number.

3. The Maxwell-Boltzmann law is given by the expression n_{i} = ________

a) \(\frac{g}{e^{\alpha+\beta E}} \)

b) \(\frac{g}{e^{\alpha+\beta E}-1} \)

c) \(\frac{g}{e^{\alpha+\beta E}+1} \)

d) \(\frac{g}{e^{\alpha+\beta E}+k} \)

View Answer

Explanation: The correct expression for the Maxwell-Boltzmann law is n

_{i}= \(\frac{g}{e^{\alpha+\beta E}+1} \), where α depends on the volume and the temperature of the gas and β is equal to 1/kT.

4. The wave function of fermions is not _________

a) Continuous

b) Single Valued

c) Symmetric

d) Differentiable

View Answer

Explanation: The particles which have antisymmetric wave function having half odd integral spin number and obey Pauli’s principle are called fermions.

5. Fermi-Dirac statistics cannot be applied to ________

a) Electrons

b) Photons

c) Fermions

d) Protons

View Answer

Explanation: Fermi-Dirac Statistics can be applied to particles having half odd integral spin number and obey Pauli’s principle which are electrons, fermions and protons. Photon has an integral spin number.

6. The distribution function is given by ________

a) \(\frac{1}{Ae\frac{E}{nT}+1}\)

b) \(\frac{1}{Ae\frac{E}{nT}-1}\)

c) \(\frac{1}{e\frac{E}{nT}+A}\)

d) \(\frac{1}{e\frac{E}{nT}-A}\)

View Answer

Explanation: The distribution function is given by \(\frac{1}{Ae\frac{E}{nT}+1}\), where A = \(e^{\frac{-E_F}{nT}}\) The energy E

_{F}is called the Fermi energy and is constant for a given system.

7. At T > 0K, the probability of a state with E > E_{F} filled is zero.

a) True

b) False

View Answer

Explanation: At T > 0 K, the probability that a state with E > E

_{F}is filled is

^{1}⁄

_{2}. Hence, fermi energy is the energy at which the probability of occupation is

^{1}⁄

_{2}at any temperature above 0 K.

8. The expression for mean energy is given by ________

a) \(\frac{3}{5}NE_F\)

b) \(\frac{2}{5}NE_F\)

c) \(\frac{3}{5}E_F\)

d) \(\frac{2}{5}E_F\)

View Answer

Explanation: The expression for the mean energy of an electron at absolute zero is given by \(\frac{3}{5}NE_F\). The expression \(\frac{3}{5}E_F\) is called the zero point energy.

9. For all the quantum states with energy greater than Fermi energy to be empty in a Fermi-Dirac system, the temperature should be ______

a) 273 K

b) 373 K

c) 0 K

d) 100 K

View Answer

Explanation: We know that the Fermi-Dirac distribution is given by:

F

_{FD}(E) = \( \frac{1}{e\frac{E-E_F}{nT}+1}\)

For all the quantum states with energy greater than Fermi energy to be empty,

F

_{FD}(E) = 0, for E > E

_{F}and F

_{FD}(E) = 1, for E < E

_{F}

Therefore, for E < E

_{F}\( \frac{1}{e\frac{E-E_F}{nT}+1}=1\)

\(e\frac{E-E_F}{nT}= 0\)

As, E < EF, E- EF < 0. Therefore, to satisfy the given statement,

T = 0 K

Thus, we can define Fermi energy as the energy of the uppermost occupied level at 0 K.

10. What is the relationship between T1 and T2?

a) T1 > T2

b) T1 < T2

c) T1 = T2

d) Insufficient Information

View Answer

Explanation: The given figure shows the variation of Fermi-Dirac distribution with energy E. In this case, T2 > T1, according to the expression: F

_{FD}(E) =\( \frac{1}{e\frac{E-E_F}{nT}+1}\).

11. The density of silver is 10.5 g/cm^{3} and its atomic weight is 108. If each atom contributes one electron for conduction, what is the fermi energy?

a) 2.12 eV

b) 3.31 eV

c) 4.69 eV

d) 5.51 eV

View Answer

Explanation: Fermi Energy, \(E_F = \frac{h^2}{8m} (\frac{3N}{\pi v})^{2∕3}\)

Here, N /V = 10.5 X 6.02 X 10

^{23}/108

= 5.85 X 10

^{28}m

^{-3}

Therefore, \(E_F = \frac{6.626 X 6.626}{8 X 9.1} \times 10^{-19} \times \frac{3 \times 5.585}{\pi}^{2∕3}\)

= 8.816 X 10

^{-19}J

= 5.51 eV.

12. The Fermi energy of a material is 3.45 eV. What is the zero-point energy of the material?

a) 1.02 eV

b) 2.07 eV

c) 3.45 eV

d) 4.16 eV

View Answer

Explanation: As we know, the fermi energy of the material is 3.45 eV.

Now, zero-point energy =\(\frac{3}{5}E_F\)

= 3/5 X 3.45 eV

= 2.07 eV.

13. The average energy of one electron silver is 3.306 eV. What is the fermi-energy of Silver at 0 K?

a) 2.32 eV

b) 3.78 eV

c) 4.12 eV

d) 5.51 eV

View Answer

Explanation: We know, Average Energy, E =\(\frac{3}{5}NE_F\)

Here, E = 3.306 eV, N = 1

Therefore, we get: E

_{F}= 5E/3

= 5.51 eV.

14. In Fermi-Dirac Statistics, one energy state can be occupied by more than one particle.

a) True

b) False

View Answer

Explanation: In Fermi-Dirac statistics, one energy state can be occupied by only one particle. Therefore, when one state is filled, the particle will go on to another state.

15. Which of the following is the curve for Fermi-Dirac statistics?

a) X

b) Y

c) Z

d) None

View Answer

Explanation: As seen the curve the order of the ration of f(K) with E/kT is in the order X > Y > Z. Here, Y is Maxwell-Boltzmann Statistics, X is Bose-Einstein Statistics and Z is Fermi-Dirac Statistics.

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