Question

The average energy of an electron is related to Fermi energy (at absolute zero) as :

• A. $\bar{E} = \frac{4}{5} E_F$
• B. $\bar{E} = \frac{5}{8} E_F$
• C. $\bar{E} = \frac{3}{5} E_F$
• D. $\bar{E} = \frac{9}{8} E_F$

Hint:

This is a standard derived constant in solid-state physics for 3D systems. For a 1D system, $\bar{E} = \frac{1}{3} E_F$. For a 2D system, $\bar{E} = \frac{1}{2} E_F$. For standard 3D bulk materials, always remember $\bar{E} = \frac{3}{5} E_F$.

Answer 1

The Correct Option is C

Step 1: Concept:
This relates to the free electron gas model of metals at absolute zero (0 K). Even at 0 K, electrons fill available states up to the Fermi energy ($E_F$) due to the Pauli Exclusion Principle. We need the formula for the average kinetic energy of all these electrons.

Step 2: Key Formula or Approach:
The average energy $\bar{E}$ is found by integrating the energy over all occupied states and dividing by the total number of electrons $N$:
\[ \bar{E} = \frac{1}{N} \int_0^{E_F} E \cdot D(E) \, dE \]
For a 3D free electron gas, the density of states $D(E) \propto E^{1/2}$.

Step 3: Step-by-step Explanation:

• Let $D(E) = C E^{1/2}$ where $C$ is a constant.

• The total number of electrons $N$ is the integral of the DOS up to the Fermi level:
\[ N = \int_0^{E_F} C E^{1/2} \, dE = C \left[ \frac{E^{3/2}}{3/2} \right]_0^{E_F} = \frac{2}{3} C E_F^{3/2} \]

• The total energy $E_{total}$ of the system is:
\[ E_{total} = \int_0^{E_F} E \cdot (C E^{1/2}) \, dE = C \int_0^{E_F} E^{3/2} \, dE = C \left[ \frac{E^{5/2}}{5/2} \right]_0^{E_F} = \frac{2}{5} C E_F^{5/2} \]

• Now, find the average energy per electron by dividing the total energy by $N$:
\[ \bar{E} = \frac{E_{total}}{N} = \frac{\frac{2}{5} C E_F^{5/2}}{\frac{2}{3} C E_F^{3/2}} \]
\[ \bar{E} = \left( \frac{2}{5} \times \frac{3}{2} \right) \frac{E_F^{5/2}}{E_F^{3/2}} = \frac{3}{5} E_F \]

Step 4: Final Answer:
The average energy of an electron at 0 K is exactly three-fifths of the Fermi energy. This matches option (C).