Recoil kinematics and the Compton edge

After the collision there are four numbers to find: the scattered photon's energy E′ and direction θ, and the electron's kinetic energy T and direction φ. Conservation supplies three equations — energy, momentum along the beam, momentum across it — so one unknown survives, and the geometry of the event chooses it. Fix θ and everything else is forced. Energy conservation gives T = E − E′ outright, and the shift derivation already gave E′ = E/[1 + α(1 − cosθ)] with α = E/mₑc², so T = Eα(1 − cosθ)/[1 + α(1 − cosθ)]. For the 662 keV caesium line α = 1.2955; at θ = 90° the bracket is 1 + 1.2955 = 2.2955, so E′ = 662/2.2955 = 288.4 keV and the electron walks off with T = 373.6 keV, well over half the beam. Notice what the formula does at θ = 0: T = 0. There is no transfer without deflection, since a photon that keeps its direction keeps its energy, and that is why the electron continuum starts at zero rather than at some minimum.

The two momentum components carry the direction. Across the beam, pₑ sinφ = (E′/c) sinθ; along it, pₑ cosφ = E/c − (E′/c) cosθ. Divide the first by the second, substitute E′, and the half-angle identities collapse everything to one line: cot φ = (1 + α) tan(θ/2). Read its ends. As θ → 0, tan(θ/2) → 0 and φ → 90°: a barely deflected photon nudges an electron sideways with essentially no energy. As θ → 180°, cot φ → ∞ and φ → 0: the backscatter case throws the electron straight down the beam. In between φ falls monotonically, so the recoil electron is always emitted into the forward hemisphere, 0 ≤ φ < 90°, however hard it is struck — there is no backward recoil to go looking for. The 662 keV beam at θ = 90° gives cot φ = 2.2955 and φ = 23.5°: the photon has turned a right angle while the electron stays close to the beam axis. Raising the beam energy tips every recoil further forward, because the whole angular map is compressed by that single factor (1 + α).

T(θ) rises monotonically with θ, so the electron energies form a continuum from zero up to a ceiling reached at θ = 180°, where 1 − cosθ = 2: Tₘₐₓ = 2αE/(1 + 2α) = 2E²/(mₑc² + 2E). That is the Compton edge — 1117.6 keV for the 1332 keV cobalt line. A detector that stops the electron while the scattered photon escapes records precisely this continuum, so the spectrum shows a broad shelf ending in a step, with the full-energy photopeak stranded above it. The gap between shelf and peak is what the escaping photon removed, E′ₘᵢₙ = mₑc²E/(mₑc² + 2E), 214.4 keV for that line, climbing towards mₑc²/2 = 255.5 keV as the beam hardens but never arriving. The step is abrupt because counts pile up against it: dT/dθ = Eα sinθ/[1 + α(1 − cosθ)]² vanishes at θ = 180°, so every scattering angle beyond about 160° deposits within a few keV of Tₘₐₓ. What rounds the edge off in a real spectrum is detector resolution, not kinematics.

Every step above assumed a free electron at rest, and atomic electrons are neither. The free-electron branch opens only if the collision can pay the binding energy E_B: where T(θ) < E_B the electron cannot be ejected, the recoil is taken up by the whole atom, and mₑ in the shift formula is replaced by the atomic mass M — tens of thousands of times larger even for carbon, so the shift is suppressed by that same factor and the light returns at the incident wavelength. That is Compton's unmodified line. Setting T(θ) = E_B and solving for the angle gives the threshold in closed form, 1 − cosθ = E_B mₑc²/[E(E − E_B)], which names what controls it: the harder the beam and the looser the electron, the smaller the angle beyond which that electron may as well be free. So the unmodified line is strongest near the forward direction, weakens as θ grows, and strengthens with atomic number, since a heavy atom holds a larger fraction of its electrons tightly. Treat the threshold as a sharp angle only inside this free-electron accounting; a real target hands over the coherent fraction gradually. The electrons that do come free are not at rest either, and the component of their orbital momentum along the momentum transfer Doppler-shifts each event, so the modified line arrives as a broadened Compton profile whose width measures the target's electron momentum distribution.

The single parameter α = E/mₑc² decides which description is needed. For α ≪ 1 the largest fractional transfer is 2α/(1 + 2α) ≈ 2α — 7% for a 20 keV photon — so the scattering is elastic to that accuracy and the classical Thomson result holds: dσ/dΩ = (rₑ²/2)(1 + cos²θ), symmetric front to back, independent of energy, integrating to σ_T = (8π/3)rₑ² = 0.665 b per electron. Near α ≈ 1 both the kinematics and the cross-section change, and the angular distribution needs Klein–Nishina in full, dσ/dΩ = (rₑ²/2)(E′/E)²[E′/E + E/E′ − sin²θ], which returns the Thomson expression exactly when E′ = E. At 662 keV it leaves θ = 0 untouched, cuts the 90° value to 33% of its Thomson value and backscatter to 15%, and drops the total to 0.256 b, 38% of σ_T. For α ≫ 1 the cross-section decays roughly as (ln 2α)/α into a narrow forward cone. Two rival processes bracket the Compton window from outside: photoelectric absorption, scaling steeply as about Z⁴/E³, owns the low end, and pair production opens at 1.022 MeV. In water Compton scattering rules from 30 keV to 20 MeV; in lead the window narrows to roughly 0.6–4 MeV.