Blundell Thermal Physics · Problem 24.2

The Debye model expresses internal energy as an integral over all phonon frequencies, weighted by their thermal occupation. By introducing the dimensionless variable $x = \hbar\beta\omega$ (where $\beta = 1/k_BT$), you transform the frequency integral into a universal form that separates the zero-point energy (the $\frac{9}{8}$ term) from the temperature-dependent contribution. The key is recognizing that this substitution converts the Debye cutoff frequency $\omega_D$ into the dimensionless Debye temperature parameter $x_D$, making the integral dimensionally consistent and revealing how thermal energy scales with temperature through the $1/(e^x-1)$ Bose–Einstein factor.

The Debye model expresses internal energy as an integral over all phonon frequencies, weighted by their thermal occupation. By introducing the dimensionless variable $x = \hbar\beta\omega$ (where $\beta = 1/k_BT$), you transform the frequency integral into a universal form that separates the zero-point energy (the $\frac{9}{8}$ term) from the temperature-dependent contribution. The key is recognizing that this substitution converts the Debye cutoff frequency $\omega_D$ into the dimensionless Debye temperature parameter $x_D$, making the integral dimensionally consistent and revealing how thermal energy scales with temperature through the $1/(e^x-1)$ Bose–Einstein factor.