= Compton scattering
{c}
{title2=$\lambda'-\lambda=\lambda_C(1-\cos\theta)$}
{wiki}
= Compton effect
{c}
{synonym}
<Compton scattering> redistributes <photon> <energy> and <momentum> in a <collision> with a charged <subatomic particle>, here an <electron>. For a free <electron> initially at rest in the chosen <inertial frame>, an incident <photon> of <energy> $E=hc/\lambda$ leaves with <energy> $E'=hc/\lambda'$ after deflection through $\theta$. The angle $\theta\in[0,\pi]$ is between the incident and outgoing <photon> directions; it is not the <electron> recoil angle. <Four-momentum conservation> gives
$$
E'=\frac{E}{1+\dfrac{E}{m_ec^2}(1-\cos\theta)},
\qquad
\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta),
\qquad
T_e=E-E',
$$
where $m_e$ is the <electron> <rest mass> and $T_e$ its final <kinetic energy>. The <photon> <energy> loss pays for the <electron> recoil: neither <energy> nor <momentum> disappears.
For an original worked example, stipulate $E=70\,\mathrm{keV}$, the rounded <electron> <rest energy> $m_ec^2=511\,\mathrm{keV}$, and the rounded <Compton wavelength> $\lambda_C=2.426\,\mathrm{pm}$; $\mathrm{keV}$ means $10^3$ <electronvolts>. Treat these rounded inputs as the example's constants. They give the following calculated values, with <energies> rounded to three decimal places:
|| <Photon> deflection $\theta$
|| Outgoing <photon> <energy> $E'$ in $\mathrm{keV}$
|| <Electron> <kinetic energy> $T_e$ in $\mathrm{keV}$
|| <Wavelength> increase in $\mathrm{pm}$
| $0^\circ$
| $70.000$
| $0.000$
| $0.000$
| $60^\circ$
| $65.513$
| $4.487$
| $1.213$
| $90^\circ$
| $61.566$
| $8.434$
| $2.426$
| $180^\circ$
| $54.946$
| $15.054$
| $4.852$
Since $0\leq1-\cos\theta\leq2$, every outgoing <photon> has $0<E'\leq E$ and $T_e\geq0$. At zero deflection there is no recoil; backscattering transfers the most <energy>. At fixed $\theta$, the absolute <wavelength> increase is independent of incident <energy>, while the fractional <energy> loss grows with $E/(m_ec^2)$. This makes <Compton scattering> useful for inferring <energy> transfers from directions in <photon> detectors.
\Image[/compton-energy-angle-and-wavelength-shift.png]
{title=Calculated <Compton scattering> <energy> and <wavelength> shift for a $70\,\mathrm{keV}$ <photon>}
{description=Original ideal free-<electron> curves with calculated points at four chosen deflections. Both horizontal axes show <photon> deflection from the incident direction. These are model predictions, not historical measurements or an angular <wave intensity> distribution.}
The free-<electron>-at-rest assumption matters. An <electron> in a <bound state> has <binding energy> and an initial <momentum> distribution. Neglecting these is useful when the transferred <energy> is large compared with the relevant <binding energy> and the initial <momentum> is small compared with the resolved <momentum transfer>. Otherwise the outgoing <photon> <energies> can broaden or depart from the stationary-target formula; an initially moving <electron> can even give <energy> to the <photon>.
A material may also scatter <X-rays> through a <coherent elastic scattering> channel with an effectively unshifted component. When an intact <chemical atom> of <mass> $M$ takes the recoil, the corresponding <Compton wavelength> $h/(Mc)$ is much smaller than the <electron> value. The single free-<electron> curve does not predict that component's weight. More generally, <energy> versus angle is a kinematic constraint; angular <wave intensity> requires a <differential scattering cross-section>, together with target and detector information.
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