Some Properties of Atoms — Question 9

Question 9

A spectroscopic energy network distinguishes direct and cascade photons

An atom has nondegenerate levels at $0$, $2.4$ and $5.9\,\mathrm{eV}$. Assume all downward transitions are allowed and neglect recoil. A preparation puts each of N atoms in the top level. A fraction b decays directly to the ground; the rest follow the two-step cascade.

1. Find all three photon energies.

2. Find the mean number of photons and total radiated energy per initially excited atom.

Question 9 - Solution

A spectroscopic energy network distinguishes direct and cascade photons

(a) Method and calculation

$$E_{20}=5.9-0=\boxed{5.9\,\mathrm{eV}}.$$$$\boxed{E_{21}=3.5\,\mathrm{eV},\quad E_{10}=2.4\,\mathrm{eV}}.$$

(b) Interpretation and further calculation

$$\langle n_\gamma\rangle=b+2(1-b)=\boxed{2-b}.$$$$\langle E_\gamma^{\mathrm{total}}\rangle=b(5.9)+(1-b)(3.5+2.4)=\boxed{5.9\,\mathrm{eV}}.$$

Physical check

Branching changes the photon count and spectrum, but not the total energy released between the same initial and final levels.