The cubic growth of the heat capacity of a solid as a function of temperature continues only up to roughly one tenth of the Debye temperature associated with the shortest wavelength excitations in the solid. This may seem surprising, since this implies that at this point the heat capacity becomes sensitive to the absence of modes whose probability of excitation is on the order of e^?10. A derivation of the Debye model parallel to that of the photon gas. Shows that the heat capacity in the Debye model takes the form?

where x is a re scaled energy parameter. At large x,e^x(e^x ? 1)^?2 ? e^?x gives the Boltzmann suppression of states with increasing energy, while x⁴ measures the increase in the number of states available with increasing energy. Compute (approximately) the minimum temperature T/T_D at which the factor x⁴e^?x exceeds one for some x in the range of integration.?

Transcribed Image Text:

Cy/kB = 9(T /Tp)3 | Tp/T dx- (ex – 1)2 0. |