X-ray spectra provide yet another example of the richness and power of the Schrodinger equation and of the model of atomic structure that we derived from it in the preceding ion, In Section 40-8 we discussed x-ray production on the oasis of the photon conzept, With the development of x-ray diffraction techniques (Section 38-7) by von Laue, gg, and others, beginning in 1912, it became possible to measure x-ray wavelengths ite precisely (to within 0,1 % or less), Detailed studies of x-ray spectra showed a continuous spectrum of wavelengths (Fig. – -14), with minimum wavelength (corresponding to maximum frequency and photon
rgy) determined by the accelerating voltage VAC in the x-ray tube, according to the ‘on derived in Section 40-8 for remsstrahlung processes:
Depending on the accelerating voltage and the target element, we may find sharp superimposed on this continuous spectrum, as in Fig, 43-15. These peaks are at erent wavelengths for different elements; they form what is called a characteristic x- . spectrum for each target element. In 1913, the British scientist H. G. J. Moseley .ed these spectra in detail using x-ray diffraction techniques. He found that the most
ase short-wavelength line in the characteristic x-ray spectrum from a particular tar- =~ element, called the Ka line, varied smoothly with that element’s atomic number Z -:g. 43-16). This is in sharp contrast to optical spectra, in which elements with adjat Z values have spectra that often bear no resemblance to each other. Moseley found that the relationship could be expressed in terms of x-ray frequencies a simple formula called Moseley’s law:
But Moseley went far beyond this empirical relationship; he showed how characteristic x-ray spectra could be understood on the basis of energy levels of atoms in the target. His analysis was based on the Bohr model, published in the same year. We will recast it somewhat, using the ideas of atomic structure discussed in Section 43-5. First recall that the outer electrons of an atom are responsible for optical spectra. Their excited states are usually only a few electron volts above their ground state. In transitions from excited states to the ground state, they usually emit photons in or near the visible region. Characteristic x rays, by contrast, are emitted in transitions involving the inner shells of a complex atom. We mentioned these briefly in Section 40-8. In an x-ray tube the electrons may strike the target with enough energy to knock electrons out of the inner shells of the target atoms. These
inner electrons are much closer to the nucleus than are the electrons in the outer shells; they are much more tightly bound, and hundreds or thousands of electron volts may be required to remove them. Suppose one electron is knocked out of the K shell. This process leaves a vacancy, which we’ll call a hole. (One electron remains in the K shell.) The hole can then be filled by an electron falling in from one of the outer shells, such as the L, M, N, … shell. This transition is accompanied by a decrease in the energy of the atom (because less energy would be needed to remove an electron from an L, M, N, … shell), and an x-ray photon is emitted with energy equal to this decrease. Each state has definite energy, so the emitted x rays have definite wavelengths; the emitted spectrum is a line spectrum. We can estimate the energy and frequency of Ka x-ray photons using the concept of screening. A Ka x-ray photon is emitted when
an electron in the L shell (n = 2) drops down to fill a hole in the K shell (n = 1). As the electron drops down, it is attracted b) the Z protons in the nucleus screened by the one remaining electron in the K shell. We therefore approximate the energy by Eq. (43-27), with Zeff = Z – 1, nj = 2, and nr = 1 The energy before the transition.
This relation agrees almost exactly with Moseley’s experimental law, Eq. (43-29). Indeed, considering the approximations we have made, the agreement is better than we have a right to expect. But our calculation does show how Moseley’s law can be understood on the bases of screening and transitions between energy levels.
The hole in the K shell may also be filled by an electron falling from the M or N shell, assuming that these are occupied. If so, the x-ray spectrum of a large group of atoms of a single element shows a series, named the K series, of three lines, called the Ka’ Kp, and K, lines. These three lines result from transitions in which the K-shell hole is ‘filled by an L, M, or N electron, re pectively. Figure 43-17 shows the K series for tungsten (Z = 74), molybdenum (Z = 42), and copper (Z = 29). There are other series of x-ray lines, called the L, M, and N series that are produced after the ejection of electrons from the L, M, and N shells rather than the K shell.
Electrons in these outer shells are farther away from the nucleus and are not held as tightly as are those in the K shell. Their removal requires less energy, and the x-ray photons that are emitted when these vacancies are filled have lower energy than those in the K series.