Characteristic X-Ray Energy

How do we calculate energies in a multiple-electron atom? In the case of characteristic x rays, the following approximate calculation is reasonable. Characteristic x rays are produced when an inner-shell vacancy is filled. Inner-shell electrons are nearer the nucleus than others in an atom and thus feel little net effect from the others. This is similar to what happens inside a charged conductor, where its excess charge is distributed over the surface so that it produces no electric field inside. It is reasonable to assume the inner-shell electrons have hydrogen-like energies, as given by $E_n=-\frac{Z^2}{n^2}{E}_0$$\left(n=\mathrm{1,\; 2,\; 3,\; ...}\right)$. As noted, a $K_{\alpha }$ x ray is produced by an $n=2$ to $n=1$ transition. Since there are two electrons in a filled $K$ shell, a vacancy would leave one electron, so that the effective charge would be $Z-1$ rather than $Z$. For tungsten, $Z=\text{74}$, so that the effective charge is 73.

$E_n=-\frac{Z^2}{n^2}{E}_0$$\left(n=\mathrm{1,\; 2,\; 3,\; ...}\right)$ gives the orbital energies for hydrogen-like atoms to be $E_n=-(Z^2/n^2)E_0$_, where $E_0=\mathrm{13.6\; eV}$. As noted, the effective $Z$ is 73. Now the $K_{\alpha }$ x-ray energy is given by

where

and

Thus,

This large photon energy is typical of characteristic x rays from heavy elements. It is large compared with other atomic emissions because it is produced when an inner-shell vacancy is filled, and inner-shell electrons are tightly bound. Characteristic x ray energies become progressively larger for heavier elements because their energy increases approximately as $Z^2$. Significant accelerating voltage is needed to create these inner-shell vacancies. In the case of tungsten, at least 72.5 kV is needed, because other shells are filled and you cannot simply bump one electron to a higher filled shell. Tungsten is a common anode material in x-ray tubes; so much of the energy of the impinging electrons is absorbed, raising its temperature, that a high-melting-point material like tungsten is required.