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Exercises Relativity, quantum and nuclear

Example from Physics, 22.1 The Structure of the Atom

Characteristic X-Ray Energy

Calculate the approximate energy of an X-ray emitted for an n = 2 to n = 1 transition in a tungsten anode in an X-ray tube.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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. That 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

$E_{n} = \frac{Z^{2}}{n^{2}} E_{o} (n = 1, 2, 3, ...)$
22.19

For tungsten, Z = 74, so that the effective charge is 73.
Solution
The amount of energy given off as an X-ray is found using

$Δ E = h f = E_{i} - E_{f},$
22.20

where

$E_{f} = - \frac{Z^{2}}{1^{2}} E_{o} = - \frac{73^{2}}{1} (13.6 e V) = - 72.5 k e V$
22.21

and

$E_{i} = - \frac{Z^{2}}{2^{2}} E_{o} = - \frac{73^{2}}{4} (13.6 eV) = −18.1 keV .$
22.22

Thus,

$Δ E = E_{i} - E_{f} = (−18.1 k e V) - (−72.5 k e V) = 54.4 k e V .$
22.23

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From Physics (1st edition) by OpenStax (Rice University), section 22.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.