Question

A homogeneous three-dimensional solid has a heat capacity at constant volume CV that depends on temperature T. Neglecting differences in the transverse and longitudinal waves in the solid, there are 3N vibrational modes, where N is the number of atoms in the solid. Here, the solid has N = 3.01 x 1023 atoms which occupy a total volume V = 18.0 cm3 . There are two transverse shear waves and one longitudinal wave; all waves have the same speed of sound cs. The Debye temperature for this solid is θD = 120.0 K.

(a) In the high-temperature limit T >> θD, write an expression for CV. Evaluate this expression using the parameters at T = 750. K.

(b) In the low-temperature limit, T << θD, write an expression for CV based on the Debye model. Evaluate this expression using the parameters at temperature T = 10.0 K.

(c) In the Debye model, how does the density of states for the sound waves scale with frequency? Using parameters given previously, numerically evaluate the Debye frequency νD corresponding to the upper cutoff in the density of states.

(d) In the Debye model, the Debye frequency νD is related to cs by the density of states and knowing that the total number of modes is 3N and the volume is V. Using your knowledge of the density of states in the Debye model, write an expression to estimate cs. Numerically evaluate this expression. Your answer will be scored based on the scaling of the expression, not based on numerical prefactors.

Answer 1