Thermal Properties of Solids Questions and Answers

Answer: A

The low-temperature specific heat of a metal is \(\alpha T + \beta T^{3}\).

Two contributions add. The conduction electrons give a term linear in \(T\), because only the electrons within about \(k_{B}T\) of the Fermi level can be excited. The lattice vibrations give the Debye \(T^{3}\) law at temperatures well below the Debye temperature. At the lowest temperatures the linear electronic term dominates, and a plot of \(C/T\) against \(T^{2}\) is a straight line.

\(\beta T^{3}\) alone describes an insulator, with no free electrons. \(\exp(-\alpha/T)\) is the behaviour of a system with an energy gap, such as the Einstein model or a superconductor. There is no \(T^{5}\) term in the specific heat; a \(T^{5}\) law belongs to the electron-phonon resistivity.

Answer: B

The plot is a straight line with a positive intercept, graph (B).

At low temperature the heat capacity of a non-magnetic metal is the sum of an electronic term, linear in \(T\), and the Debye phonon term, proportional to \(T^{3}\):

\[ \begin{aligned} C &= \gamma T + \beta T^{3} \\ \frac{C}{T} &= \gamma + \beta T^{2} \end{aligned} \]

Against \(T^{2}\) this is a straight line of slope \(\beta\) (from the phonons) cutting the axis at \(\gamma\) (from the conduction electrons). This is how the two contributions are separated experimentally.

A line through the origin (D) would describe an insulator with no electronic term. The two curved graphs do not follow from either term.