Thermal phonon contribution to the reduced internal energy u_th = U_th/N_ik_BT as a function of log (T/T_p) = −log η at β = ħω_ci/k_BT_p = 0, 1, 10, 100, and 10³ (numbers near the lines). The analytical approximation in Eq. (76) (dotted lines) and in Eq. (78) (short-dashed lines) are compared with the numerical results of Baiko (2009) (solid lines for β = 1, 10, and 100).