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 (or free electron model) is a quantum mechanical model for the behavior of electrons in a metal. It improves upon the classical Drude model by treating the electrons as a Fermi gas. We need to identify the core assumptions of this model.
Step 2: Detailed Explanation:


A. The free electrons are valence electrons of the composing atoms. This is a fundamental assumption. The model considers that the valence electrons are detached from their parent atoms and are free to move throughout the entire volume of the metal, forming an "electron gas". This statement is correct.
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. This is incorrect. The model assumes the electrons are confined within a potential well. The potential energy inside the metal is assumed to be constant (and can be set to zero for simplicity) and is \textit{lower} than the potential outside. The difference is the work function. The statement says the potential is higher, which is wrong.
C. In this model, the mutual repulsion between the electrons is neglected. This is a key simplifying assumption of the free electron model. The electron-electron interactions are ignored, and the electrons are considered to move independently. This statement is correct.
D. The potential energy for an electron is periodic. This is the defining feature of the \textit{Bloch model} or band theory, which is a refinement of the Sommerfeld model. The Sommerfeld model itself assumes a \textit{constant} (zero) potential inside the metal, not a periodic one. Therefore, this statement is incorrect for the Sommerfeld model.

The problem states "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". This statement is incorrect as written. However, looking at the available options, option (D) includes A, B, and C. It is highly likely there is a typo in statement B and it should have stated "lower". Assuming there is a typo in the question and B is intended to be correct, A, B, and C are the intended assumptions of the model, making D the correct choice. Let's re-evaluate.
Perhaps B is interpreted differently. Let's assume the potential outside is \(V_{out}=0\). Then the potential inside is \(V_{in} = -V_0\). The total energy of an electron at rest inside is \(E_{in} = -V_0\). An electron outside at rest has energy \(E_{out} = 0\). The statement says \(E_{in}>E_{out}\), which is false. There might be a significant error in the question's statement B. Let's check the combination of the definitively correct statements, A and C. Option (B) contains A, C, and D. Option (D) contains A, B, and C. Since D is definitively incorrect for the Sommerfeld model, option (B) is also incorrect. This leaves us with a contradiction. Given the provided options, it is most probable that statement D is considered the incorrect one, as it belongs to the band theory, making option (D) which excludes it the most likely intended answer, despite the error in statement B.
Let's assume the question meant to ask "which statements are correct for the free electron model of metals". A and C are definitely correct. D is definitely incorrect. B is incorrect as written. The only option combining A and C without D is option (D). Therefore, we must assume B is intended to be correct for this option to work. This points to a poorly formulated question. However, proceeding with the most likely intent:
Final Interpretation:

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