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Compton Shift

Key idea: Apply the Compton wavelength-shift equation, interpret scattering spectra, and connect photon energy–momentum transfer with recoil electrons.

  • Advanced Physics
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Learning objectives

  • Connect blackbody evidence, Planck quantisation, photon momentum, and Compton scattering.

In Compton scattering, a photon transfers energy and momentum to an electron. The scattered photon has lower energy, lower frequency, and greater wavelength. The angle-dependent wavelength shift provides direct evidence for particle-like photon momentum.

1. Model and equation

Assume an electron is initially free and at rest. If the photon is scattered through angle θ relative to its initial direction,

Δλ = λ'-λ = (h/mₑc)(1- cos θ)

The constant

λ_C = h/mₑc = 2.43 × 10⁻¹² m

is the electron Compton wavelength, so Δλ = λ_C(1- cos θ).

Incident photon scattering through angle theta while an initially stationary electron recoils at angle phi
Define the photon angle from its original direction. Energy and both components of momentum are conserved for the photon–electron system.

2. Angle dependence and limits

Compton shift factor versus scattering angle

The wavelength shift divided by the electron Compton wavelength follows one minus cosine theta.

Scroll across the graph to read all labels.

The wavelength shift divided by the electron Compton wavelength follows one minus cosine theta.The wavelength shift divided by the electron Compton wavelength follows one minus cosine theta.
Forward scattering gives no wavelength change; back-scattering gives the maximum shift of two electron Compton wavelengths.
Open full-size graph
View figure data
Values for Compton shift factor versus scattering angle
Scattering angle (°)1 − cos θ
00
300.134
600.5
901
1201.5
1501.866
1802
  • At θ = 0°, Δλ = 0.
  • At θ = 90°, Δλ = λ_C.
  • At θ = 180°, Δλ = 2λ_C = 4.86 pm.

The shift for a free electron depends on angle and electron rest mass, not on incident intensity. The fractional energy loss does depend on the incident wavelength because E = hc/λ.

3. Interpreting observed spectra

A scattering target may produce both shifted and nearly unshifted components. Scattering by weakly bound or effectively free electrons gives the electron Compton shift. Interaction with a tightly bound electron can transfer momentum to the much heavier atom, making h/(Mc) and the resulting shift too small to resolve.

Do not say the unshifted component means “no interaction” in every case; it can reflect coherent or whole-atom scattering with negligible recoil energy.

4. Common mistakes

  • Using the recoil-electron angle φ in the Compton shift equation instead of photon angle θ.
  • Reporting Δλ as the outgoing wavelength. Use λ' = λ + Δλ.
  • Saying a larger wavelength means the scattered photon gained energy. Photon energy is inversely proportional to wavelength.
  • Treating intensity as photon energy. Greater intensity mainly means more photons per unit time.
  • Using a non-relativistic electron energy relation in the derivation without checking recoil energy.

6. Worked Examples

Modelled example 1

Scattering through 90 degrees

Core

Problem

An X-ray photon of wavelength 0.0710 nm scatters through 90° from a free electron. Find the outgoing wavelength.
Study the worked solution

  1. Find the shift

    Method

    Δλ = 2.43 pm.

    Reason

    1- cos 90° = 1.

    Working

    Δλ = (2.43 pm)(1- cos 90°) = 2.43 pm

  2. Add to the incident wavelength

    Method

    λ' = 73.4 pm = 0.0734 nm.

    Reason

    Compton shift is defined as λ'-λ.

    Working

    λ' = 71.0 + 2.43 = 73.4 pm

Guided practice 2

Maximum wavelength shift

About 4 min

Problem

Find the maximum Compton shift for scattering by a free electron.

Try this before viewing the solution

Hints

Hint 1: maximise the angular factor
Find the angle that maximises 1- cos θ.
View solution step by step

  1. Choose back-scattering

    Method

    θ = 180°.

    Reason

    This makes 1- cos θ = 2, its maximum.

    Working

    Δλₘₐₓ = 2λ_C

  2. Evaluate

    Method

    Δλₘₐₓ = 4.86 pm.

    Reason

    The electron Compton wavelength is 2.43 pm.

    Working

    2(2.43 pm) = 4.86 pm

Challenge 3

Energy transferred to the electron

Minimal support

Independent transfer

Using λ = 0.0710 nm and λ' = 0.0734 nm, estimate the electron’s kinetic-energy gain. Use hc = 1.240 keV nm.

Try this before viewing the solution

Hints

Hint 1: use photon energy loss
The electron gains E-E', with photon energy hc/λ.
View solution step by step

  1. Form the energy difference

    Method

    K = hc(1/λ-1/λ').

    Reason

    Energy conservation gives the electron the photon’s lost energy.

    Working

    K = E-E'

  2. Evaluate

    Method

    K ≈ 0.57 keV.

    Reason

    The wavelength increase lowers the photon energy.

    Working

    K = (1.240)(1/0.0710-1/0.0734) = 0.57 keV

7. Evidence statement

Mind stretcher 1: Connect the shift to the evidenceExtension

The wavelength change follows from treating light as particles with E = hf and p = h/λ colliding with electrons while conserving energy and momentum. A wave-only classical model does not produce this angle-dependent transfer relation.

Next: Derivation of the Compton Shift Equation.

Continue with the next resource in this course.

Course and syllabus information
Course
Advanced Physics
Edition
Advanced Physics