Question

In 3-dimensional system, the mean energy of an electron in electron gas at absolute zero is __________________________ of fermi energy, \(E_f(0)\) at absolute zero.

• A. \( \frac{4}{3} \)
• B. \( \frac{5}{3} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{3}{4} \)

Hint:

This is a standard result from the free electron model and is worth memorizing. The factor 3/5 comes directly from the integration of the 3D density of states (\(\propto E^{1/2}\)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question asks for the average energy (\(\langle E \rangle\)) of an electron in a 3D free electron gas at a temperature of absolute zero (T=0 K), expressed as a fraction of the Fermi energy (\(E_F\)). At T=0, all energy states up to the Fermi energy are occupied, and all states above it are empty.
Step 2: Key Formula or Approach:
The average energy \(\langle E \rangle\) is found by integrating the energy of each state, weighted by the density of states \(D(E)\), up to the Fermi energy, and then dividing by the total number of electrons \(N\).
\[ \langle E \rangle = \frac{1}{N} \int_0^{E_F} E \cdot D(E) \, dE \]
For a 3D free electron gas, the density of states is \( D(E) = C E^{1/2} \), where C is a constant.
The total number of electrons is \( N = \int_0^{E_F} D(E) \, dE \).
Step 3: Detailed Explanation:
First, calculate the total number of electrons:
\[ N = \int_0^{E_F} C E^{1/2} \, dE = C \left[ \frac{E^{3/2}}{3/2} \right]_0^{E_F} = \frac{2}{3} C (E_F)^{3/2} \]
Next, calculate the total energy:
\[ E_{\text{total}} = \int_0^{E_F} E \cdot (C E^{1/2}) \, dE = C \int_0^{E_F} E^{3/2} \, dE = C \left[ \frac{E^{5/2}}{5/2} \right]_0^{E_F} = \frac{2}{5} C (E_F)^{5/2} \]
Now, find the average energy per electron:
\[ \langle E \rangle = \frac{E_{\text{total}}}{N} = \frac{\frac{2}{5} C (E_F)^{5/2}}{\frac{2}{3} C (E_F)^{3/2}} \]
\[ \langle E \rangle = \frac{2/5}{2/3} \cdot \frac{(E_F)^{5/2}}{(E_F)^{3/2}} = \frac{3}{5} E_F \]
Step 4: Final Answer:
The mean energy of an electron in a 3D electron gas at absolute zero is \( \frac{3}{5} \) of the Fermi energy.