§ 43. Compton Scattering & Relativistic Photon Momentum (Grade 11 · Uni)
Elastic collision of an X-ray photon with a free electron, wavelength shift Δλ = λ_C (1 − cos θ), and recoil momentum. The billiard-like scattering of X-ray quanta proved that a photon carries not only energy hν but also linear momentum h/λ.
Theory Summary & Scientific Mechanism
In 1923, Arthur Holly Compton observed that X-rays scattered by electrons in graphite through angle θ emerge with a longer wavelength λ' > λ₀, where the wavelength shift Δλ = λ' - λ₀ depends solely on the scattering angle θ. Classical Thomson scattering (which predicts λ' = λ₀) could not account for this shift.
Treating the interaction as a relativistic elastic billiard collision between a photon of energy E₀ = (hc)/(λ₀) and momentum p₀ = (h)/(λ₀) and a stationary electron of rest energy mₑ c² ≈ 511 keV, conservation of relativistic 4-momentum (E₀ + mₑ c² = E' + √(pₑ² c² + mₑ² c⁴) and p₀ = p' + pₑ) yields Compton’s scattering formula: Δλ = λ' - λ₀ = λ_C (1 - cos θ).
Here λ_C = (h)/(mₑ c) ≈ 2.426 pm is the Compton wavelength of the electron. Maximum energy transfer to the recoil electron occurs for 180^° backscattering (cos θ = -1), where Δλₘax = 2λ_C ≈ 4.85 pm. Compton’s discovery proved conclusively that photons carry mechanical momentum p = ℏk.
Everyday Analogy
Quantum billiards! When an X-ray photon collides with an electron, it bounces off just like a billiard ball. Because the recoiling electron carries away kinetic energy, the scattered photon loses energy and emerges with a longer wavelength λ' > λ₀!
Interactive Question & Derivation
At what scattering angle θ is the Compton wavelength increase Δλ = λ_C(1 - cos θ) maximal, and what is its value?
- A: For backscattering at θ = 180^° (cos 180^° = -1): the shift is maximal and equals twice the Compton wavelength Δλₘax = 2λ_C ≈ 4.85 pm
- B: For forward scattering at θ = 0^°
- C: The wavelength shift Δλ is independent of the scattering angle θ
Study Hint: From Compton’s formula Δλ = λ_C(1 - cos θ): how does the scattered wavelength λ' compare to λ₀, and at what angle θ is the shift largest?
Proof & Explanation: Since Δλ = λ_C(1 - cos θ) with λ_C = (h)/(mₑ c) = 2.426 pm, the factor 1 - cos θ reaches its maximum value of 2 for head-on backscattering (θ = 180^°), transferring maximum kinetic energy to the recoil electron.
Key Equations & 3D Simulation Parameters
Δλ = λ' - λ₀ = λ_C (1 - cos θ), λ_C = 2.426 pm
- Incident X-ray wavelength λ₀: 2 … 12 pm (default: 5 pm)
- Photon scattering angle θ: 15 … 180 ° (default: 90 °)
- Target collision frequency: 0.8 … 2.4 Hz (default: 1.4 Hz)