Some Properties of Atoms — Question 7

Question 7

A thermal excited fraction includes the degeneracy ratio

A dilute ensemble has levels $E_0=0$ and $E_1=\Delta$, with degeneracies $g_0=1$ and $g_1=3$. It is in thermal equilibrium at a temperature satisfying $k_BT=\Delta/\ln6$. Other levels can be neglected.

1. Find the ratio of excited to ground populations.

2. Find the excited fraction and mean internal energy per atom.

Question 7 - Solution

A thermal excited fraction includes the degeneracy ratio

(a) Method and calculation

$$\frac{N_1}{N_0}=\frac{g_1}{g_0}e^{-\Delta/(k_BT)}=3e^{-\ln6}.$$$$\boxed{N_1/N_0=1/2}.$$

(b) Interpretation and further calculation

$$f_1=\frac{N_1/N_0}{1+N_1/N_0}=\boxed{1/3}.$$$$\boxed{\langle E\rangle=f_1\Delta=\Delta/3}.$$

Physical check

Equal populations per individual state are different from equal populations per energy level. Degeneracy contributes to the level population even when the energy is higher.