All electron of mass m moves in a square lattice of lattice spacing a. Assume that the nearly free electron approximation holds good. If there are two electrons per site, then which of the following is correct?a)system behaves as a metalb)system behaves as an insulationc)both (a) and (b)d)none ot theseCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Answer:

The behavior of a system of electrons in a square lattice can be determined by considering the energy bands formed by the overlapping electron wavefunctions. In the nearly free electron approximation, we assume that the electron wavefunctions are free particle wavefunctions modified by the periodic potential of the lattice.

Explanation:

1. Energy Bands: In a square lattice, the periodic potential leads to the formation of energy bands. The dispersion relation for these energy bands determines the behavior of electrons in the lattice.

2. Two Electrons per Site: In this case, each lattice site is occupied by two electrons. Due to the Pauli exclusion principle, these electrons must have opposite spin states. This means that each energy level in the energy bands can accommodate two electrons with opposite spins.

3. Filling of Energy Bands: As the energy bands are filled with electrons, the behavior of the system depends on the Fermi energy, which is the energy level separating the filled states from the empty states.

4. Metal Behavior: If the Fermi energy lies within a partially filled energy band, the system behaves as a metal. In this case, there are available energy states for electrons to move freely and conduct electricity.

5. Insulator Behavior: If the Fermi energy lies within a completely filled energy band, the system behaves as an insulator. In this case, there are no available energy states for electrons to move freely and conduct electricity.

6. Conclusion: Since there are two electrons per site in the given system, all energy bands will be completely filled. Therefore, the Fermi energy will lie within a completely filled energy band, indicating that the system behaves as an insulator.

Inference: The correct answer should be option 'B' (system behaves as an insulator).

Answer

A

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