X Rays and the Ordering of the Elements — Question 2

Question 2

A core vacancy produces line energies from differences of binding energies

In a simplified atom the positive binding energies are $B_K=9.0\,\mathrm{keV}$, $B_L=1.2\,\mathrm{keV}$ and $B_M=.3\,\mathrm{keV}$. A K vacancy is filled from L or M. Neglect relaxation corrections and fine structure.

1. Find the $K_\alpha$ and $K_\beta$ photon energies.

2. Find their wavelengths using $hc=1.240\,\mathrm{keV\,nm}$ and identify the shorter line.

Question 2 - Solution

A core vacancy produces line energies from differences of binding energies

(a) Method and calculation

$$E_n=-B_n,\quad E_{K\alpha}=E_L-E_K=B_K-B_L=\boxed{7.8\,\mathrm{keV}}.$$$$E_{K\beta}=B_K-B_M=\boxed{8.7\,\mathrm{keV}}.$$

(b) Interpretation and further calculation

$$\boxed{\lambda_{K\alpha}=1.240/7.8=.1590\,\mathrm{nm}}.$$$$\boxed{\lambda_{K\beta}=1.240/8.7=.1425\,\mathrm{nm}<\lambda_{K\alpha}}.$$

Physical check

A core-level binding energy is measured to the ionization continuum; an emitted line is a difference between two bound configurations. Real line energies include many-electron rearrangement.