8.5 Atomic spectra and x-rays  (Page 4/12)

X-ray production provides an important test of quantum mechanics. According to the Bohr model, the energy of a $K_{\alpha }$ X-ray depends on the nuclear charge or atomic number, Z . If Z is large, Coulomb forces in the atom are large, energy differences ( $\text{Δ}E$ ) are large, and, therefore, the energy of radiated photons is large. To illustrate, consider a single electron in a multi-electron atom. Neglecting interactions between the electrons, the allowed energy levels are

where n = 1, 2, …and Z is the atomic number of the nucleus. However, an electron in the L ( $n=2$ ) shell “sees” a charge $Z-1$ , because one electron in the K shell shields the nuclear charge. (Recall that there is only one electron in the K shell because the other electron was “knocked out.”) Therefore, the approximate energies of the electron in the L and K shells are

The energy carried away by a photon in a transition from the L shell to the K shell is therefore

where Z is the atomic number. In general, the X-ray photon energy for a transition from an outer shell to the K shell is

or

where f is the frequency of a $K_{\text{α}}$ X-ray. This equation is Moseley’s law    . For large values of Z , we have approximately

This prediction can be checked by measuring f for a variety of metal targets. This model is supported if a plot of Z versus $\sqrt{f}$ data (called a Moseley plot    ) is linear. Comparison of model predictions and experimental results, for both the K and L series, is shown in [link] . The data support the model that X-rays are produced when an outer shell electron drops in energy to fill a vacancy in an inner shell.