Question

Which of the following statements are correct for the Sommerfeld model:
A. The free electrons are valence electrons of the composing atoms.
B. The potential energy of an electron at rest inside the metal is assumed to be higher than that of an electron outside the metal.
C. In this model, the mutual repulsion between the electrons is neglected.
D. The potential energy for an electron is periodic.
Choose the correct answer from the options given below:

• A. A, B and D only
• B. A, C and D only
• C. B, C and D only
• D. A, B and C only

Hint:

Distinguish between the key models:

\textbf{Drude (Classical):} Free electrons, classical gas.
\textbf{Sommerfeld (Quantum):} Free electrons, quantum Fermi gas, constant potential.
\textbf{Bloch (Band Theory):} Electrons in a periodic potential from the ion lattice.
Knowing which assumption belongs to which model is crucial. "Periodic potential" is the defining feature that separates Bloch theory from the earlier free electron models.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The Sommerfeld model, a quantum mechanical improvement on the Drude model, treats electrons in a metal as a Fermi gas. The key is to identify its fundamental assumptions.
Step 2: Detailed Explanation:


A. Valence electrons become free electrons. The model assumes valence electrons detach from atoms and move freely throughout the metal, forming an "electron gas." This is correct.
B. Potential energy inside the metal is higher than outside. This is incorrect. The model assumes electrons are confined within a potential well, with a constant (often zero) potential energy inside the metal, which is {lower} than the potential outside. The difference is the work function. Therefore, the statement is wrong.
C. Mutual electron repulsion is neglected. This is a crucial simplification. Electron-electron interactions are ignored, and electrons move independently. This is correct.
D. Potential energy is periodic. This defines the {Bloch model} (band theory), a refinement of the Sommerfeld model. The Sommerfeld model assumes a {constant} (zero) potential, not periodic. Thus, this is incorrect for the Sommerfeld model.

The problem's statement B is incorrect as written. If we assume a typo and B should state "lower," then A, B, and C would be intended assumptions, making an option including them the correct choice. Let's re-evaluate assuming a typo in B.
Consider an alternative interpretation. Let \(V_{out}=0\), then \(V_{in} = -V_0\). The total energy inside is \(E_{in} = -V_0\). Outside, \(E_{out} = 0\). The statement claims \(E_{in}>E_{out}\), which is false, highlighting a potential error in statement B. Instead, consider the definitely correct statements A and C. If an option contains A, C, and D, it's wrong because D is incorrect. This creates a contradiction. However, it's more probable that statement D is the incorrect one as it belongs to the band theory. This makes an option excluding D the most likely intended answer, despite the error in B.
Assume the question asks "which statements are correct for the free electron model". A and C are correct, D is incorrect, and B is incorrect as written. Only an option combining A and C without D would work. Therefore, we must assume B is intended to be correct for this option to work. This indicates a poorly formulated question. Proceeding with the most likely intent:
Final Interpretation:

A: Correct.
C: Correct.
D: Incorrect (key feature of band theory, not Sommerfeld).
The only options combining A and C are (B) and (D). But (B) includes D, which is wrong. Thus, (D) is the only possibility, assuming B is intended to be correct despite its flawed wording.
Step 3: Final Answer:
The core assumptions are free valence electrons (A), neglected mutual repulsion (C), and confinement in a constant potential box (opposite of B, and not D). The pair {A, C} is the only certainly correct pair. The combination {A, B, C} is the most plausible intended answer, assuming a typo in B.