Figure references are to the second edition of Modern Physics by Serway, Moses and Moyer (Saunders, 1989).

X-rays were discovered in 1895 by Wilhelm Roentgen. He accelerated electrons (produced in vacuum by thermionic emission from a heated filament) towards a positive metal target (the anode) by applying several thousands of volts, and found that a very penetrating form of radiation is produced at the anode; see Fig. 2.19. Within months, the first medical radiographs were produced, showing bones (which absorb x-rays more strongly) as darker areas on the photographic emulsion. Within a few years it became clear that x-rays are another form of electromagnetic radiation, with wavelengths typically some tens or hundreds of picometers (1 pm = 10^-12 m).

Since atoms in a crystal are spaced regularly by a few hundreds of pm (some tenths of a nanometer, 1 nm = 10^-9 m) a crystal can be used as a kind of 3-dimensional diffraction grating to separate x-rays of different wavelengths. The x-rays penetrate through many layers of atoms but at each layer there is a small probability of the x-rays being scattered elastically (with no change in photon energy or wavelength) by the nuclei of the atoms. In fact, the x-rays can be "reflected" from the atomic planes, as if the latter behaved like a mirror (see Fig. 2.21). However this reflection takes place only for a particular angle of incidence, such that x-ray beams reflected from successive planes of atoms have their electric and magnetic fields in phase .

For a parallel incident beam, the phase is the same at points (A and D in Fig. 2.21) on a wavefront which is perpendicular to the incident direction. The x-rays reflected from plane B must travel an extra distance (AB + BC) further than the x-rays reflected from plane A. If this extra distance is an integral number of wavelengths (nl where n is any integer) the phase at points C and D on the wavefront of the reflected rays will also be the same and the two rays will combine constructively at the detector (e.g. a photographic film). From geometry of the right-angled triangles ABD and BCD, AB = CD = d sin q , where d is the interplanar spacing, so the condition for so-called Bragg reflection of the x-rays is:

n l = 2 d sin q

This equation is known as Bragg's law, after the English physicist W.L. Bragg.

Since the equation is satisfied (for a particular angle of reflection) only for a single value of l , Bragg reflection provides us with of selecting photons of a particular wavelength from an incident beam containing a mixture of wavelengths. In other words, we can produce a monochromatic beam of x-rays, the reflecting crystal acting as a monochromator. If we now slowly rotate the crystal, x-rays of different wavelength will strike a photographic film at different positions (see Fig. 2.22a) and we can record the spectrum of the x-rays. If the film blackening is converted to intensity (using a device called a microdensitometer) the spectrum appears somewhat as shown in Fig.2.22b.

The spectrum recorded from a typical x-ray tube is seen to consist of some sharp peaks (characteristic peaks, or lines as seen on the film) superimposed on a smoothly varying background (the bremststrahlung continuum). The origin of the characteristic x-rays will be discussed later. To understand the origin of the continuum, consider that electrons accelerated within the x-ray tube will travel some distance within the target before being brought to rest. During this time, they will pass close to the positively charged nuclei of several atoms and will be attracted towards each nucleus. This attraction will result in an angular deflection of the path of the electron and a brief period during which the electron undergoes centripetal acceleration. The principles of electromagnetism tell us that any charged particle undergoing acceleration (centripetal or not) must lose energy in the form of electromagnetic radiation, and this is the origin of the bremstrahlung x-rays (braking radiation in German).

If an incoming electron passes an atomic nucleus at a particular distance and decreases its kinetic energy by an amount E , Einstein's quantum principle predicts that an x-ray photon of energy E = hf = hc/lambda will be produced. However, the separation between an electron and a nucleus will be different at each collision and for different electrons, so a continuous distribution of wavelengths is produced. However, we know that the maximum amount of energy Emax which an electron can lose is the entire kinetic energy with which it arrived at the surface of the x-ray anode, equal (in Joules) to e Va where e is the electronic charge and Va the accelerating voltage applied to the x-ray tube.Therefore,

e Va = Emax = h fmax = hc/l min

We therefore predict the bremsstrahlung continuum to have a short-wavelength cutoff at a value lambda min given by this equation. Measurement of the x-ray spectrum confirms this conclusion and therefore provides additional evidence (besides that of the photoelectric effect) of the quantum nature of electromagnetic radiation. Note that the emission of an x-ray photon as an electron passes a nucleus is a momentary interaction taking place on an atomic scale, so a quantum description of x-ray production is appropriate.

If we allow a monochromatic beam of x-rays (wavelength lambda) to fall on a solid specimen such as graphite (see Fig. 2.24a), the x-rays are scattered from both the atomic nuclei and the surrounding electrons in the specimen. Scattering from the nucleus is elastic (as we have discussed) so the spectrum of the scattered x-rays shows a peak at the original wavelength lambda (as in Fig. 2.24b). However, scattering of x-rays from the atomic electrons is inelastic; the x-ray photons lose some of their energy, so their wavelength is increased to a larger value lambda'. This is the Compton effect. Using a rotating-crystal spectrometer (containing a Bragg-reflecting crystal) to measure the spectra of x-rays scattered through several different angles q , Arthur Compton showed in 1922 that the shift in wavelength increases with (theta) as described by the formula:

Moreover, Compton was able to derive this equation by treating the x-rays as particles and applying conservation of energy and momentum to their "collision" with an atomic electron. To achieve the correct wavelength shift, he had to assume that the x-ray photon has a momentum whose magnitude p is given by

p = h/lambda = hf/c

The results of Compton scattering experiments therefore provide further evidence for the particle nature of electromagnetic radiation, as required by a quantum description of interaction of the radiation with (in this case) electrons.

The Compton equation tells us that the wavelength shift (D lambda =lambda '-lambda ) is independent of the original wavelength. Therefore the fractional wavelength shift is inversely proportional to lambda . Taking a scattering angle (theta) of 45 degrees, the equation gives DELTA lambda /lambda = 0.010 for lambda = 71 pm (a typical wavelength from an x-ray tube), so the fractional wavelength change is about 1% (not too hard to detect).

The Compton effect should also take place for visible light but if we substitute lambda = 550 nm (green light) in the equation, we get DELTA lambda /lambda = 1.2 x 10^-6. On the other hand, if we use gamma rays such as those emitted from a radioactive cobalt source, lambda = 1.06 pm and DELTAlambda /lambda = 0.67 , so Compton measurements are often done with gamma rays.

Examination of the spectra of scattered photons reveals that the inelastic Compton peak centred around lambda ' is broader than that of the elastic peak (or the wavelength distribution of the original x-rays). This is because the electrons in the specimen (which scatter the photons) are moving, rather than stationary as it is convenient to assume in deriving the Compton-shift equation. To account for this, extra terms would have to be added to the conservation (of energy and momentum) equations and these terms can be positive or negative since the electrons have a velocities in all directions (they are in orbit around atomic nuclei, according to the simple Bohr model of the atom). The result is a broadening of the Compton peak. In fact, the width and shape of the peak (the "Compton profile") gives us a direct measure of the velocity distribution of electrons in the specimen. We can detect the presence of different shells of electrons around each atomic nucleus and the effect of the periodicity of the atoms (in a crystal) on the motion of the outermost (valence or conduction) electrons. In other words, Compton scattering can be used as an analytical tool for examining the electronic structure of a solid.