X-Rays and the Compton Effect

The photoelectric effect showed that light carries energy in quanta $hf$. It did not show that a quantum also carries momentum, or that it can bounce off a particle and conserve both energy and momentum like an ordinary collision. Compton's 1923 experiment did. The stage was set by X-rays — radiation discovered almost by accident, shown to be electromagnetic waves of very short wavelength, and by then a routine tool for probing crystals.

The nature of X-rays

Roentgen found in 1895 that a cathode-ray tube emits penetrating rays from the point where the electron beam strikes the glass or a metal target. The rays pass through materials opaque to light, darken photographic film, and are not deflected by magnetic fields. Their origin is classical: an electron slammed to a halt in the target is a decelerating charge, and an accelerated charge radiates.

The wavelengths, inferred from slight diffraction broadening, lie near $0.01$–$0.10\text{nm}$ — a thousand times shorter than visible light and comparable to the spacing of atoms in a crystal. A modern X-ray tube accelerates electrons through tens of kilovolts onto a tungsten anode.

The spectrum has two parts. A continuous background is the bremsstrahlung. On top of it sit sharp characteristic lines whose wavelengths are fixed by the target element (labeled $K_{\alpha }$, $K_{\beta }$ historically). The continuous part has a sharp short-wavelength cutoff ${\lambda }_m$ that depends only on the tube voltage, not on the target.

Einstein read the cutoff as an inverse photoelectric effect. The most energetic photon appears when an electron surrenders its entire kinetic energy $eV$ to a single photon. With the few-eV work function negligible against a $20,000\text{eV}$ electron,

This is the Duane-Hunt rule. It gives another independent route to $h/e$, and it is target-independent because it depends only on the electron's energy, not on which atom stops it. The continuous spectrum was thus explained by the quantum hypothesis; the sharp lines are a fingerprint of the atom's inner shells and wait for the nuclear atom.

Bragg diffraction

Because their wavelength matches atomic spacing, X-rays diffract off the regular planes of a crystal. Waves reflected from successive parallel planes travel path lengths differing by $2d\sin \theta$, where $d$ is the plane spacing and $\theta$ the glancing angle. Constructive interference — a bright diffracted beam — requires that difference to be a whole number of wavelengths.

Bragg's law is the workaday tool of X-ray spectroscopy. Compton used exactly such a spectrometer to measure the wavelength shift that carries his name.

The Compton effect

Scattered X-rays had been noticed to come out softer — longer in wavelength, more easily absorbed — than the incident beam. A wave cannot do this: an oscillating electron driven by an incident wave re-radiates at the same frequency. Compton treated the scattering instead as a collision between a photon and a single free electron, with the photon carrying both energy and momentum:

The momentum relation $p=E/c$ is the massless limit of the relativistic energy-momentum relation$E^2=(pc{)}^2+(m{c}^2{)}^2$. The photon strikes a stationary electron, transfers part of its energy and momentum to it, and recoils at reduced frequency.

Deriving the wavelength shift

Let the incident photon have momentum $p_1=h/{\lambda }_1$ along $x$, the scattered photon $p_2=h/{\lambda }_2$ at angle $\theta$, and the recoil electron momentum $p_e$ at angle $\phi$. Conservation of momentum in the two directions:

Isolating the electron terms and squaring to eliminate $\phi$:

Conservation of energy uses the relativistic electron energy $E_e=\sqrt{(p_{e}c{)}^2+(m{c}^2{)}^2}$:

Squaring $E_e^2=(p_{e}c{)}^2+(m{c}^2{)}^2$ and substituting both results, the $(m{c}^2{)}^2$ and quadratic photon terms cancel, leaving

Dividing by $h{f}_1{f}_{2}m{c}^2$ and using $\lambda =c/f$ turns frequencies into wavelengths:

The shift ranges from zero at forward scattering ($\theta =0$) to a maximum of $2{\lambda }_C=0.00486\text{nm}$ at backscattering ($\theta =180^{\circ }$).

Two peaks in the data

Compton scattered the $0.0711\text{nm}$ $K_{\alpha }$ line of molybdenum off graphite and measured the scattered wavelength with a Bragg spectrometer. At each angle he found not one peak but two: a shifted peak at ${\lambda }_2$, and an unshifted peak at the original ${\lambda }_1$.

The unshifted peak comes from photons that scatter off tightly bound inner electrons. Bound so firmly that the whole atom recoils, the effective mass in Compton's formula is the atomic mass — roughly $10^4$ electron masses — so $\Delta \lambda \approx h/(Mc)$ is negligible. The shifted peak comes from the outer, nearly free electrons, where the electron mass applies and the full shift appears. The measured variation of $\Delta \lambda$ with angle matched $(h/mc)(1-\cos \theta )$ exactly.

Worked examples

Shortest X-ray from a picture tube. Electrons accelerated through $25\text{kV}$ striking the tube face produce a Duane-Hunt cutoff

These penetrate matter effectively, which is why such tubes carry shielding.

A Compton measurement. Suppose a $\theta =120^{\circ }$ scattering shifts the wavelength by $1.5\%$. From Compton's equation,

Since $\Delta \lambda =0.015{\lambda }_1$,

At $\theta =75^{\circ }$ the same incident line scatters to

The fractional shift is tiny, which is why Compton scattering is seen with X-rays and gamma rays — where ${\lambda }_1$ is small enough that $\Delta \lambda /{\lambda }_1$ is measurable — and not with visible light.

Evidence for the photon

By 1923 light had three particle-like behaviors on record. The photoelectric effect showed quantized energy exchange; the Compton effect showed quantized momentum exchange in a clean two-body collision obeying relativistic conservation laws; and pair production, the conversion of a photon into an electron-positron pair, shows a photon converting entirely to matter.

None of these erases the wave behavior — diffraction and the Bragg condition are irreducibly wave phenomena, and the same X-rays that scatter as photons also diffract as waves in the very spectrometer that measures the scattering. Light is both, and which face it shows depends on the experiment. Matter turns out to be the same way: the next module gives particles a wavelength and builds the wave-particle duality into a single framework.¹²