With reference to the discussion in Chapter 14, the extremely low energy levels for the inner-electron states in the atoms of some of the transition metals is demonstrated dramatically in the production of x-rays. Note that the two electrons in the innermost, 1s, shell of an atom of any of the elements (except hydrogen and helium, that is) are located almost entirely inside of the clouds of the rest of the electrons, relatively unscreened from the total charge in the nucleus. An electron in this innermost state is mainly screened only by its fellow 1s electron and thus sees all but a fraction of one of the proton charges of the protons in the nucleus. For the atoms of elements of higher atomic number, this inner electron thus has a negative energy nearly like that of the one-electron ions examined in Chapter 14, roughly in proportion to the number of protons in the nucleus squared (that is, in proportion to the atomic number Z squared). The higher-atomic-number elements thus have very, very highly negative (very low) inner state energies. If one of these inner electrons is knocked into a higher energy state or even out of the atom, another electron may transition into the removed electron's lowest energy state. That requires that this other electron lose energy (to go to the lower, more negative, energy state), and it does so by sending off a very, very high-energy penetrating photon called an x-ray. (X-rays have been used, of course, for many applications, including medical diagnostic imaging and the detection of flaws in the welds of metals and alloys.)

X-rays are typically created by bombarding a metal with electrons that have been accelerated to energies of tens of thousands of electron volts, energies high enough to knock an inner electron completely free of its atom. The metal (held at high positive voltage to attract the bombardment of electrons) and the bombarding electron source (typically a heated tungsten filament) are usually enclosed in an evacuated tube. Once the inner electron is knocked free, the transition described above can take place, in the process radiating an x-ray photon. Of course, the bombardment of a metal involves billions of electrons in billions of atoms, and billions of x-ray photons are emitted.

Two metallic elements commonly used to create x-rays are copper (Z = 29) and tungsten (Z = 74). If one assumes a complete lack of screening (which is not actually the case, but we're approximating here), the energy levels of the inner, n = 1, lowest energy states can be calculated from Schrödinger's formula just as were the energies of the electrons in ions as described in Section (A) of Chapter 14 and displayed in Figure 14.1. For n = 1, in our formula from that chapter,

E = (–13.60 eV) × (Z)²/(n)²,

we get, respectively for copper and tungsten, –11,438 eV and –74,474 eV. The highest measured energies of x-rays emitted from these elements (for transitions from the outermost electrons to these inner, n = 1, electron levels) suggest inner state energy levels of –8,990 eV and –69,550 eV, which are in relatively good agreement with the theoretical values just calculated, considering the complexity of these atoms and that some screening of each inner electron from charge of nucleus is expected from the other inner electron and the partial penetration of the outer electrons. By their very appearance at specific energies these x-rays confirm once again the validity of quantum mechanics in describing the atom. Classical theory would not predict a specific energy at all.