Compton scattering and photon momentum

Planck's E = hν prices energy exchange; the sharper claim, Einstein's of 1916, is that a quantum of a mode with wavevector k also carries directional momentum p = ħk, of magnitude h/λ. Package both as one null four-vector q = (ħω/c, ħk): its self-contraction q⋅q = (ħω/c)² − ħ²|k|² vanishes because ω = c|k|, which is the statement E = pc for a massless particle. In the field language of later units this is enforced by the formalism itself — the mode's creation operator a†ₖ adds exactly ħk to the field's momentum. The classical alternative is definite: a free electron driven by a wave oscillates at the driving frequency and reradiates at that same frequency in every direction, so Thomson scattering predicts no shift at all. The momenta at stake are small but not hopeless: a 71.1 pm X-ray photon carries p = h/λ = 9.32 × 10⁻²⁴ kg m s⁻¹, enough to kick a free electron to a few percent of c.

Conservation is q + p = q′ + p′, four equations. A spectrometer sees only the scattered photon, so eliminate the recoil electron wholesale: write p′ = p + q − q′ and contract each side with itself. The left side is the mass shell, p′⋅p′ = mₑ²c²; on the right, q⋅q = q′⋅q′ = 0 and p⋅p = mₑ²c² cancel it, leaving one scalar equation, p⋅(q − q′) = q⋅q′. Now choose the frame: in the electron's initial rest frame p = (mₑc, 0), so the left side is mₑħ(ω − ω′), while q⋅q′ = (ħ²ωω′/c²)(1 − cos θ) in the metric (+,−,−,−). Divide through and convert frequencies to wavelengths: λ′ − λ = (h/mₑc)(1 − cos θ). The move to remember is the squaring: contracting the conservation law is how collision kinematics deletes whatever the detector does not measure, and it is the same trick that later defines the Mandelstam variables.

A Bragg spectrometer reads wavelengths; a scintillator reads energies. Solve the shift for the scattered energy: E′ = E/[1 + ε(1 − cos θ)] with ε = E/mₑc² and mₑc² = 511 keV, and the electron takes K = E − E′. For the 662 keV line of caesium-137, ε = 1.295. At θ = 180° the denominator is 1 + 2ε = 3.591, so E′ = 184 keV — the backscatter peak, from photons that turned around in the shielding and re-entered the crystal — while K = 478 keV is the Compton edge, the endpoint of the recoil-electron continuum inside the detector. The formula also hides a ceiling: as E → ∞, E′(180°) → mₑc²/2 = 255.5 keV, so no backscattered photon, whatever its source, reaches 256 keV — which is why the backscatter peaks of ¹³⁷Cs, ⁶⁰Co and ²²Na all crowd into the 170–215 keV region however far apart their photopeaks sit.

Nothing in the four-momentum algebra says how many photons scatter through a given angle: conservation laws locate the line, they do not populate it. The rate needs a dynamical theory of the electron–photon coupling — the Klein–Nishina cross-section of 1929, one of the first results of quantum electrodynamics. Its low-energy limit is exactly Thomson's classical formula, dσ/dΩ = (rₑ²/2)(1 + cos²θ) with rₑ = 2.818 fm and total σ_T = 0.665 b, so the classical calculation survives as the ε → 0 limit of the quantum one, as correspondence demands. As ε grows the total cross-section falls — to about 0.26 b at 662 keV — and the angular distribution tilts forward; Compton scattering nevertheless stays the dominant photon interaction from a few hundred keV up to where pair production overtakes it, near 5 MeV in lead and only past 20 MeV in carbon. Keep the division of labour explicit: the shift tests kinematics with ħ in it; the intensity pattern tests the dynamics.

Be precise about the epistemics, because this experiment is often over-claimed. The measured shift is the single length h/mₑc = 2.426 pm: it fixes that ratio, not h itself, and not the existence of photons. A semiclassical calculation — radiation treated as a classical wave, Doppler-shifted once into and once out of the frame of a continuously recoiling electron — reproduces the same Δλ, and the Bohr–Kramers–Slater proposal of 1924 went further, keeping energy and momentum conserved only on statistical average. What killed those readings was not the shift but the correlations. Bothe and Geiger put a recoil-electron counter and a scattered-photon counter in time coincidence and found the pairs arriving together far above chance; Compton and Simon checked in a cloud chamber that each electron's angle matched its photon's through cot φ = (1 + ε) tan(θ/2), event by event. Momentum is conserved in each individual scattering, not merely on average — that is the quantisation this experiment establishes.

Every step so far assumed a free electron at rest, and no target supplies one. Two corrections write themselves into the spectrum. First, quasi-free electrons are not at rest: their initial momenta Doppler-spread the shifted line, and the measured lineshape — the Compton profile J(p_z) — is a one-dimensional projection of the momentum density |φ(p)|², which is why Compton spectroscopy is a working probe of momentum-space wavefunctions in solids. Second, an electron bound more tightly than the energy on offer cannot recoil alone: the whole atom takes the momentum, so mₑ in the shift is replaced by the atomic mass — about 21,900 mₑ for carbon — and that line moves by a negligible 10⁻⁴ pm. That is the unmodified line in Compton's own graphite data. The 17.4 keV Mo Kα photon hands roughly 0.6 keV to an electron at 90°: far above the ~11 eV binding of carbon's valence electrons, comparable to its 284 eV K shell. One target, two populations, two lines.