Result

Compton shift

λ^{'} - λ = \frac{h}{m _{e} c} (1 - cos θ)

Statement

The wavelength shift of a photon scattered through angle θ off a free electron.

Symbols and units

$λ, λ^{'}$	incoming, scattered photon wavelength	m
$θ$	scattering angle	degrees
$m_{e}$	electron mass	9.109×10⁻³¹ kg
$h$	Planck constant	6.626×10⁻³⁴ J·s
$c$	speed of light	2.998×10⁸ m/s

When it applies

A photon scattering off a free (or effectively free) electron.

Don't use when

Assumes the photon scatters off a free (or effectively free) electron — an electron tightly bound in an inner atomic shell doesn't behave as free and the shift formula needs modification.

Derivation

Conservation of energy and momentum applied to a photon-electron collision, treating the photon as a particle with momentum p = h/λ, yields the wavelength shift as a function of scattering angle alone.

λ^{'} - λ = \frac{h}{m _{e} c} (1 - cos θ)

Worked examples

The two extremes of scattering angle

θ = 0^{\circ} : Δ λ = 0 θ = 18 0^{\circ} : Δ λ = \frac{2 h}{m _{e} c}

No scattering means no shift; a photon bounced straight back experiences the maximum possible shift, a small but measurable, fixed quantity.

Watch out for

The shift depends only on the scattering angle, not on the incoming wavelength — a fixed shift added to whatever wavelength one starts with. At θ = 0° the shift vanishes; at θ = 180° it is maximum.