The Compton Effect and Photon Interactions

1X-rays that come back longer

In 1922 and 1923 Arthur Compton, at Washington University in St Louis, fired X-rays of a single wavelength at a block of graphite and measured the wavelengths of the X-rays scattered at different angles. Classical wave theory says that the electrons in the graphite should simply be shaken at the frequency of the incoming wave and re-radiate at that same frequency. Instead, Compton found two lines in the scattered radiation: one at the original wavelength and one at a longer wavelength, and the shift grew with the scattering angle. He explained it by treating the X-ray as a particle that bounces off an electron like one billiard ball off another, carrying momentum as well as energy. The Compton effect was the evidence that convinced most physicists that photons are real.

2Deriving the Compton shift

A photon of wavelength $\lambda \backslash lambda$ has momentum $p=\frac{h}{\lambda }p\; =\; \backslash frac\{h\}\{\backslash lambda\}$ and energy $pcpc$. It strikes a free electron at rest (an outer electron of carbon is so loosely bound that it can be treated as free). The photon scatters at angle $\theta \backslash theta$ with a longer wavelength ${\lambda }^{\mathrm{\prime }}\backslash lambda\text{'}$, and the electron recoils at angle $\varphi \backslash phi$.

The recoiling electron can be fast, so its energy must be written relativistically (you will meet this in Unit 10): its total energy $E_{e}E\_e$ and momentum $p_{e}p\_e$ satisfy $E_e^2=p_e^2{c}^2+m^2{c}^{4}E\_e^2\; =\; p\_e^2\; c^2\; +\; m^2\; c^4$, where $mm$ is the electron's rest mass.

Momentum is conserved (a vector equation): ${\vec{p}}_e=\vec{p}-\vec{p}{}^{\mathrm{\prime }}\backslash vec\{p\}\_e\; =\; \backslash vec\{p\}\; -\; \backslash vec\{p\}\backslash ,\text{'}$. Squaring, with the cosine rule,

Energy is conserved: $pc+m{c}^2=p^{\mathrm{\prime }}c+E_{e}pc\; +\; mc^2\; =\; p\text{'}c\; +\; E\_e$, so $E_e=(p-p^{\mathrm{\prime }})c+m{c}^{2}E\_e\; =\; (p\; -\; p\text{'})c\; +\; mc^2$. Squaring,

Put (1) into $E_e^2=p_e^2{c}^2+m^2{c}^{4}E\_e^2\; =\; p\_e^2c^2\; +\; m^2c^4$ and set it equal to (2). The terms $p^2{c}^{2}p^2c^2$, $p^{\mathrm{\prime }2}{c}^{2}p\text{'}^2c^2$ and $m^2{c}^{4}m^2c^4$ cancel, leaving

Divide both sides by $p{p}^{\mathrm{\prime }}mcpp\text{'}mc$: $\frac{1}{p^{\mathrm{\prime }}}-\frac{1}{p}=\frac{1-\cos \theta }{mc}\backslash frac\{1\}\{p\text{'}\}\; -\; \backslash frac\{1\}\{p\}\; =\; \backslash frac\{1\; -\; \backslash cos\backslash theta\}\{mc\}$. Since $\frac{1}{p}=\frac{\lambda }{h}\backslash frac\{1\}\{p\}\; =\; \backslash frac\{\backslash lambda\}\{h\}$,

The shift depends only on the angle, not on the wavelength or on the scattering material. The constant $\frac{h}{mc}=2.43\times {10}^{-12}\backslash frac\{h\}\{mc\}\; =\; 2.43\; \backslash times\; 10^\{-12\}$ m is the Compton wavelength of the electron. The shift is zero at $\theta =0\backslash theta\; =\; 0$, equals $\frac{h}{mc}\backslash frac\{h\}\{mc\}$ at 90° and reaches its maximum, $\frac{2h}{mc}=4.85\backslash frac\{2h\}\{mc\}\; =\; 4.85$ pm, when the photon bounces straight back.

The unshifted line comes from photons scattered by tightly bound inner electrons. Then the whole carbon atom recoils, and its mass is about 22 000 times the electron's, so the shift is far too small to see.

3How photons interact with matter

A photon passing through matter usually travels some distance untouched and then interacts in one event. Four processes matter:

Pair production turns energy into mass. The photon must supply the rest energy of both particles, $2m_e{c}^{2}2m\_ec^2$, and any extra becomes their kinetic energy:

It cannot happen in empty space: a single photon cannot turn into two particles while conserving both energy and momentum. A nearby nucleus takes up the spare momentum while taking almost no energy.

In annihilation of a pair at rest, the total momentum is zero, so two photons must be produced, moving in opposite directions with equal energy $m_e{c}^2=0.511m\_ec^2\; =\; 0.511$ MeV. Hospital PET scanners detect exactly these back-to-back pairs of photons.