diff --git a/docs/2_debye_model_solutions.md b/docs/2_debye_model_solutions.md index 6c74132a2aa73106541f6ed9a2b354420f6c9496..6abae06259fdcd81db9bf0a8b5745e2b12f4708d 100644 --- a/docs/2_debye_model_solutions.md +++ b/docs/2_debye_model_solutions.md @@ -106,7 +106,7 @@ ax.legend(); E = \int_{0}^{\omega_D} \hbar\omega n_B(\omega) g(\omega) d\omega \approx \int_{0}^{\omega_D} k_B T g(\omega) d\omega = N_\text{modes} k_B T $$ - where we neglected the zero-point energy. In 3D, we have $N_\text{modes} = 3_p N_\text{atoms}$, so that we recover the Dulong Petit $C_v = dE/dT = 3 k_B$ per atom + where we neglected the zero-point energy. In 2D, we have $N_\text{modes} = 2_p N_\text{atoms}$, so that we recover the 2D law of Dulong–Petit $C_v = dE/dT = 2 k_B$ per atom. 3. In the low temperature limit, the high-energy modes are not excited so we can safely let the upper boundary of the integral go to infinity. For convenience, we write $g(\omega) = \alpha \omega$, with $\alpha = \frac{L^2}{\pi v_s^2}$. We get