X Rays: Atomic Origins and Applications

Each type of atom (or element) has its own characteristic electromagnetic spectrum. X rays lie at the high-frequency end of an atom’s spectrum and are characteristic of the atom as well. In this section, we explore characteristic x rays and some of their important applications.

We have previously discussed x rays as a part of the electromagnetic spectrum in Photon Energies and the Electromagnetic Spectrum. That module illustrated how an x-ray tube (a specialized CRT) produces x rays. Electrons emitted from a hot filament are accelerated with a high voltage, gaining significant kinetic energy and striking the anode.

There are two processes by which x rays are produced in the anode of an x-ray tube. In one process, the deceleration of electrons produces x rays, and these x rays are called bremsstrahlung, or braking radiation. The second process is atomic in nature and produces characteristic x rays, so called because they are characteristic of the anode material. The x-ray spectrum in this figure is typical of what is produced by an x-ray tube, showing a broad curve of bremsstrahlung radiation with characteristic x-ray peaks on it.

A graph of X-ray intensity versus frequency is shown. The curve starts from a point near the origin in the first quadrant and increases. Before the frequency attains its maximum value, two sharp peaks are formed, after which the X-ray intensity decreases sharply to zero at f max. Below the graph appears the equation q V equals h f max.

X-ray spectrum obtained when energetic electrons strike a material, such as in the anode of a CRT. The smooth part of the spectrum is bremsstrahlung radiation, while the peaks are characteristic of the anode material. A different anode material would have characteristic x-ray peaks at different frequencies.

The spectrum in this figure is collected over a period of time in which many electrons strike the anode, with a variety of possible outcomes for each hit. The broad range of x-ray energies in the bremsstrahlung radiation indicates that an incident electron’s energy is not usually converted entirely into photon energy. The highest-energy x ray produced is one for which all of the electron’s energy was converted to photon energy. Thus the accelerating voltage and the maximum x-ray energy are related by conservation of energy. Electric potential energy is converted to kinetic energy and then to photon energy, so that \({E}_{\text{max}}={\text{hf}}_{\text{max}}={q}_{e}V\text{.}\) Units of electron volts are convenient. For example, a 100-kV accelerating voltage produces x-ray photons with a maximum energy of 100 keV.

Some electrons excite atoms in the anode. Part of the energy that they deposit by collision with an atom results in one or more of the atom’s inner electrons being knocked into a higher orbit or the atom being ionized. When the anode’s atoms de-excite, they emit characteristic electromagnetic radiation. The most energetic of these are produced when an inner-shell vacancy is filled—that is, when an \(n=1\) or \(n=2\) shell electron has been excited to a higher level, and another electron falls into the vacant spot. A characteristic x ray (see Photon Energies and the Electromagnetic Spectrum) is electromagnetic (EM) radiation emitted by an atom when an inner-shell vacancy is filled. This figure shows a representative energy-level diagram that illustrates the labeling of characteristic x rays. X rays created when an electron falls into an \(n=1\) shell vacancy are called \({K}_{\alpha }\) when they come from the next higher level; that is, an \(n=2\) to \(n=1\) transition.

The labels \(\mathrm{K, L, M,...}\) come from the older alphabetical labeling of shells starting with \(K\) rather than using the principal quantum numbers 1, 2, 3, …. A more energetic \({K}_{\beta }\) x ray is produced when an electron falls into an \(n=1\) shell vacancy from the \(n=3\) shell; that is, an \(n=3\) to \(n=1\) transition. Similarly, when an electron falls into the \(n=2\) shell from the \(n=3\) shell, an \({L}_{\alpha }\) x ray is created. The energies of these x rays depend on the energies of electron states in the particular atom and, thus, are characteristic of that element: every element has it own set of x-ray energies. This property can be used to identify elements, for example, to find trace (small) amounts of an element in an environmental or biological sample.

Different energy levels are shown in the form of horizontal lines. The line at the bottom shows the energy level for n is equal to one for the K shell. At a distance above this line, another horizontal line shows the energy level for n is equal to two for the shell L. Similarly, other lines are shown for the shells M and N. As we move from bottom to the top, the distance between the lines decreases, and near the end a few lines are shown very close to each other. Each level is labeled according to the characteristic x ray of the shell.

A characteristic x ray is emitted when an electron fills an inner-shell vacancy, as shown for several transitions in this approximate energy level diagram for a multiple-electron atom. Characteristic x rays are labeled according to the shell that had the vacancy and the shell from which the electron came. A \({K}_{\alpha }\) x ray, for example, is produced when an electron coming from the \(n=2\) shell fills the \(n=1\) shell vacancy.

Example: 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 \({E}_{n}=-\cfrac{{Z}^{2}}{{n}^{2}}{E}_{0}\)\((n=1, 2, 3, ...)\). 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

\({E}_{n}=-\cfrac{{Z}^{2}}{{n}^{2}}{E}_{0}\)\((n=1, 2, 3, ...)\) gives the orbital energies for hydrogen-like atoms to be \({E}_{n}=-({Z}^{2}/{n}^{2}){E}_{0}\)_, where \({E}_{0}=13.6 eV\). As noted, the effective \(Z\) is 73. Now the \({K}_{\alpha }\) x-ray energy is given by

\({E}_{{K}_{\alpha }}=\Delta E={E}_{\text{i}}-{E}_{\text{f}}={E}_{2}-{E}_{1},\)

where

\({E}_{1}=-\cfrac{{Z}^{2}}{{1}^{2}}{E}_{0}=-\cfrac{{\text{73}}^{2}}{1}(13.6 eV)=-\text{72.5 keV}\)

and

\({E}_{2}=-\cfrac{{Z}^{2}}{{2}^{2}}{E}_{0}=-\cfrac{{\text{73}}^{2}}{4}(\text{13.6 eV})=-\text{18.1 keV.}\)

Thus,

\({E}_{{K}_{\alpha }}=-\text{18}\text{.1 keV}-(-\text{72.5 keV})=\text{54.4 keV}.\)

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.

X Rays: Atomic Origins and Applications

X Rays: Atomic Origins and Applications

Example: Characteristic X-Ray Energy

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