Question

Temperature (T) dependence of the total specific heat (C_v) for a two dimensional metallic solid at low temperatures is

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is A

To determine the temperature dependence of the total specific heat \( C_v \) for a two-dimensional metallic solid at low temperatures, we need to understand how specific heat behaves in such systems.

At low temperatures, the specific heat of a metal is typically described by the expression:

\[ C_v = \gamma T + \beta T^2 \] 

where:

• \( \gamma T \) is the linear contribution from the electronic states.
• \( \beta T^2 \) is the square contribution from the lattice vibrations (phonons) in two dimensions.

For a two-dimensional metallic solid, at low temperatures, the electronic contribution dominates according to the linear term \( \gamma T \). However, this specific context implies that both electronic and phononic contributions are considered.

Given the options available, the correct graph should represent this behavior where specific heat \( C_v \) initially increases linearly with temperature and shows curvature due to the phononic \( T^2 \) term. The correct graph among the provided options should reflect an initial linear rise followed by a slight curve, indicating the increasing influence of phononic contributions as the temperature increases.

Examining the options, the correct graph for the temperature dependence of specific heat at low temperatures in a two-dimensional metallic solid is:

This graph shows a clear linear rise initially (due to the \( \gamma T \) term) with a subsequent rise indicative of a \( T^2 \) contribution (showing curvature), thus accurately representing the expected behavior of \( C_v \) at low temperatures.