Compton Scattering
Use energy and vector momentum for Compton scattering from an initially stationary free electron.
AP Physics 2 Topic 15.6 · College Board CED printed p. 146, 147 / PDF p. 153, 154 · See this topic in the course outline
A scattered photon changes direction and gives an electron recoil. Momentum is a vector, so energy bookkeeping alone cannot describe the event.
Before you start
- Photon momentum
- Vector components
- Energy conservation
Conserve energy and vector momentum
In the ideal Compton model a photon interacts with a free electron initially at rest. The scattered photon transfers energy and momentum to the recoiling electron. Its outgoing energy and frequency are lower and its wavelength longer for a nonzero scattering angle. This collision description gives evidence for photons as discrete energy and momentum carriers; a continuous-wave model alone does not give the observed wavelength shift.
Choose x along the incoming photon and let the outgoing photon turn by θ toward +y. With in vacuum, momentum conservation gives and . Draw the incoming vector as the sum of the outgoing photon and electron vectors. The electron recoils on the opposite transverse side so total y momentum stays zero.
Total energy including electron rest energy is conserved. Because the electron begins at rest and has the same rest mass after, . The electron's recoil may be relativistic; calculate its momentum components from conservation without assuming K = p²/(2m) unless the small-energy limit is justified. AP includes quantitative and qualitative two-dimensional momentum conservation.
| Vector | x (10⁻²³ kg·m/s) | y (10⁻²³ kg·m/s) |
|---|---|---|
| Incoming photon | 6.630 | 0 |
| Outgoing photon | 0.000 | 5.334 |
| Recoil electron | 6.630 | -5.334 |
| Outgoing sum | 6.630 | 0 |
The incoming photon runs right from the common tail. The outgoing photon runs at +90°. The electron vector begins at the outgoing photon head and ends at the incoming photon head; its physical direction is -38.8° from +x. Translating a vector for a head-to-tail sum does not move its physical scattering origin. The target is a free electron initially at rest.
The angle controls a nonnegative wavelength shift
The electron-at-rest Compton shift is . The electron Compton wavelength h/(mₑc) is 2.43 pm. The shift is zero for forward scattering (0°), 2.43 pm at 90°, and largest, 4.86 pm, for backward scattering (180°).
| θ | Δλ (pm) | λ′ (pm) | Photon energy trend |
|---|---|---|---|
| 0° | 0 | 10.0 | unchanged |
| 60° | 1.215 | 11.215 | lower |
| 90° | 2.43 | 12.43 | lower still |
| 180° | 4.86 | 14.86 | lowest of these |
Because 1 − cos θ cannot be negative, this model cannot produce a shorter outgoing wavelength. The same angle gives the same absolute shift for any incoming λ, but a different fractional energy change. Moving electrons, strongly bound targets, or other scattering processes need other assumptions; specify “free electron initially at rest” when using this formula.
A right-angle photon leaves a diagonal recoil
A vacuum photon of wavelength 10.0 pm travels along +x and scatters at +90° from a free electron initially at rest. Use h = 6.63 × 10⁻³⁴ J·s and h/(mₑc) = 2.43 pm. Find λ′, electron momentum components, and electron kinetic energy.
Δλ = 2.43(1 − cos90°) = 2.43 pm; λ′ = 12.43 pm.
Why this step? The shift is added to the initial wavelength, and the outgoing photon loses energy.
px = h/λ = 6.63 × 10⁻²³ kg·m/s; py = −h/λ′ = −5.33 × 10⁻²³ kg·m/s.
Why this step? The outgoing photon has zero x momentum and positive y momentum; recoil supplies the negative y component.
|pₑ| = √(px² + py²) = 8.51 × 10⁻²³ kg·m/s, directed 38.8° below +x.
Why this step? Add components as vectors rather than subtracting momentum magnitudes.
Kₑ = hc(1/λ − 1/λ′) = 3.89 × 10⁻¹⁵ J ≈ 24.3 keV.
Why this step? Energy transferred to the initially resting electron is the photon energy loss; no nonrelativistic recoil assumption is needed.
Answer and check
λ′ = 12.43 pm; pₑ = (6.63, −5.33) × 10⁻²³ kg·m/s; Kₑ ≈ 24.3 keV.
| Vector | x (10⁻²³ kg·m/s) | y (10⁻²³ kg·m/s) |
|---|---|---|
| Incoming photon | 6.630 | 0 |
| Outgoing photon | 0.000 | 5.334 |
| Recoil electron | 6.630 | -5.334 |
| Outgoing sum | 6.630 | 0 |
The incoming photon runs right from the common tail. The outgoing photon runs at +90°. The electron vector begins at the outgoing photon head and ends at the incoming photon head; its physical direction is -38.8° from +x. Translating a vector for a head-to-tail sum does not move its physical scattering origin. The target is a free electron initially at rest.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
Can this free-resting-electron model produce Δλ = −1 pm or Δλ = 6 pm?
Worked answer and reason
No. Since 0 ≤ 1 − cos θ ≤ 2, the allowed shift lies from 0 to 4.86 pm.
A mathematical boundary tests whether the assumed model can explain a result.
Question 2
If the outgoing photon has positive y momentum and the initial photon/electron system has zero y momentum, what is the sign of electron py?
Worked answer and reason
Negative, exactly opposite the outgoing photon’s y component.
Conserve each vector component, not only total energy.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- A Compton photon keeps its momentum because photons always travel at c.
- Speed c stays fixed in vacuum, but energy, wavelength, and momentum magnitude can change in the interaction.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 11 · mixed
- Problem 12 · mixed
- Problem 18 · experimental
Scientific calculator workspace
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For the teacher
- Use the scattering angle between incoming and outgoing photon directions. Require both momentum components; do not use nonrelativistic electron kinetic energy when recoil is relativistic.
Source objectives and scope
- 15.6.A — Describe the interaction between photons and matter using Compton scattering.
AP Physics 2 includes full quantitative and qualitative treatments of conservation of momentum in two dimensions.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- AP Physics 2 Course and Exam Description, official Unit 15 — Official topic/LO/EK scope and boundaries; local pinned pages 143–158, equations and relevant clauses visually checked on 143–144,146,149–150,152–158.
- AP Physics 2 Course and Exam Description Clarifications, effective Fall 2026 — 15.7.B.1 current definition includes decay to a lower energy level of the same nucleus.
- OpenStax University Physics 3, Blackbody Radiation — Verification of Wien peak and integrated spectral intensity versus Stefan–Boltzmann total power.
- OpenStax University Physics 3, Photoelectric Effect — Verification of threshold, maximum kinetic energy, intensity distinctions, and retarding-potential experiment.
- OpenStax University Physics 3, The Compton Effect — Verification of positive electron-at-rest shift and photon energy/momentum accounting.
- OpenStax University Physics 3, Nuclear Binding Energy — Verification of all-nucleon mass defect and atomic-mass convention.
- OpenStax University Physics 3, Radioactive Decay — Verification of exponential survival, half-life, and nuclear transformation balances.
- OpenStax University Physics, Volume 3: 6.1 Blackbody Radiation — Primary textbook verification of the wavelength-based blackbody spectral shape, the distinction between peak height and complete spectrum area, Wien’s displacement law, and Stefan–Boltzmann emitted intensity and power.
Glossary and deeper reading · 7 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Photon
A photon is a quantum of electromagnetic radiation with energy proportional to its frequency. It has zero rest mass and no electric charge but carries energy and momentum. For example, increasing light’s frequency increases the energy of each photon, while increasing the number of same-frequency photons increases the total energy.
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Linear momentum
Linear momentum is a vector describing an object’s motion and equals mass times velocity in the nonrelativistic model. The vector sum of momenta is conserved when the net external impulse on the system is zero. For example, two identical carts moving equally fast in opposite directions have zero total linear momentum but nonzero total kinetic energy.
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Vector component
A vector component describes the part of a vector along a chosen axis. Its signed value changes when the axes change, even though the physical vector remains the same. For example, a diagonal launch velocity can be resolved into horizontal and vertical components that share the same elapsed time.
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Conservation of energy
Conservation of energy means total energy is unchanged when the full isolated system is considered. Energy can transfer between objects or change form, so one selected energy store need not remain constant. For example, friction can reduce a sliding block’s kinetic energy while increasing the thermal energy of the block and floor.
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Frequency
Frequency is the number of complete repeating cycles per unit time. It is measured in hertz, meaning cycles per second, and is the reciprocal of the period. For example, a vibrating string with a higher frequency produces more cycles each second.
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Wavelength
Wavelength is the spatial distance between neighboring points at the same phase of a periodic wave. It connects propagation speed and frequency through speed equal to frequency times wavelength. For example, neighboring crests of a sinusoidal water wave are separated by one wavelength.
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Vector
A vector is a quantity with a magnitude and a direction, represented by an arrow or by components along chosen axes. Vectors add by adding corresponding components rather than simply adding their magnitudes. For example, an eastward displacement followed by a northward displacement produces a resultant pointing northeast. The zero vector has zero magnitude and no unique direction.