30.4 X rays: atomic origins and applications  (Page 2/7)

Characteristic x-ray energy

Calculate the approximate energy of a $K_{\alpha }$ x ray from a tungsten anode in an x-ray tube.

Strategy

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.

Solution

$E_n=-\frac{Z^2}{n^2}{E}_0$ $\left(n=\mathrm{1,\; 2,\; 3,\; ...}\right)$ 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,

Discussion

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.

Medical and other diagnostic uses of x-rays

All of us can identify diagnostic uses of x-ray photons. Among these are the universal dental and medical x rays that have become an essential part of medical diagnostics. (See [link] and [link] .) X rays are also used to inspect our luggage at airports, as shown in [link] , and for early detection of cracks in crucial aircraft components. An x ray is not only a noun meaning high-energy photon, it also is an image produced by x rays, and it has been made into a familiar verb—to be x-rayed.