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

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

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

(c) In the Debye model, how does the density of states for thesound waves scale with frequency? Using parameters givenpreviously, numerically evaluate the Debye frequency νDcorresponding to the upper cutoff in the density of states.

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