Nuclear size and the charge form factor

An electron has no strong interaction and no measurable size of its own, so it couples to the nucleus through a single exchanged photon whose electron end is fixed by theory rather than fitted: nothing about the probe has to be subtracted before the target can be read. What the beam energy buys is resolution: the probe sees structure of order ħ/q, and since q/ħ = 2E sin(θ/2)/ħc for E ≫ mₑc², with ħc = 197.3 MeV fm, a momentum transfer of 0.634 fm⁻¹ resolves only 1.6 fm. Getting below a fermi means hundreds of MeV — Hofstadter's Stanford measurements of the 1950s ran from about 100 to 550 MeV, and it is no accident that nuclear structure appeared in scattering data only once accelerators reached that range. One further number must be checked before any of this is used: the Born expansion parameter Zα. For ¹²C it is 6/137 = 0.044 and single-photon exchange is an excellent approximation. For ²⁰⁸Pb it is 82/137 = 0.60, and it is not.

The point-charge prediction is Rutherford with two relativistic repairs. Electron spin supplies a factor cos²(θ/2), because helicity conservation forbids exact backscattering of an ultrarelativistic electron, and target recoil supplies 1/[1 + (2E/Mc²) sin²(θ/2)]. Both are smooth and monotonic in θ: the Mott cross-section has no minima, no shoulders, nothing an experimenter could mistake for structure. Nor is the recoil factor doing heavy lifting here — at 250 MeV and 29° it amounts to 1.6 × 10⁻⁴ for ²⁰⁸Pb and 2.8 × 10⁻³ for ¹²C. So divide the measured elastic cross-section by the Mott value point by point in angle. The ratio is |F(q)|²: it starts at 1 in the forward direction and, for ²⁰⁸Pb at 250 MeV, falls by more than two orders of magnitude on the way into a first minimum near 29°. That fall is the entire measurement.

For a spherically symmetric density the angular integral in F(q) collapses the exponential to sin(qr/ħ)/(qr/ħ), and three readings follow. Expand for small argument: F ≈ 1 − q²⟨r²⟩/6ħ², so the initial slope of F against q² gives the mean-square charge radius with no profile assumed — the only genuinely model-independent number here. Next, the positions of the minima fix a radius, once a shape is assumed. Third, data spanning a wide enough range of q can be inverted directly to ρ(r), which is what produced the charge-density curves in every textbook figure. A consistency test comes free with the first Born approximation: F depends on q alone, so 250 MeV data and 500 MeV data must land on one curve when plotted against q rather than against θ. Where they separate, Born is failing and the extracted radius is not trustworthy.

For a uniform sphere F(x) = 3(sin x − x cos x)/x³ with x = qR/ħ, vanishing at x = 4.493, 7.725 and 10.90, the roots of tan x = x — the three-dimensional counterpart of a circular aperture's dark rings, and for the same diffraction reason. One minimum angle therefore fixes R, and the spacing of the later minima tests whether a sphere was a fair description. It is not quite. Real data show the maxima between minima dying away much faster than a sphere predicts, because it is a sharp edge that keeps a diffraction pattern alive. Softening that edge over a diffuseness a multiplies the envelope of F by about (πqa/ħ)/sinh(πqa/ħ), which dies away as exp(−πqa/ħ) at large q; at a = 0.55 fm and q/ħ = 2 fm⁻¹ that is 0.22, so the maxima of the measured |F|² sit some twenty times below a sharp sphere's. Spacing measures the radius; the rate of decay measures the surface. Two independent numbers out of one pattern is exactly why a two-parameter profile is enough.

Fits return the two-parameter Fermi (Woods-Saxon) form ρ(r) = ρ₀/[1 + exp((r − c)/a)] with a ≈ 0.55 fm for essentially everything, so the 90%-to-10% surface thickness t = 4a ln3 = 2.4 fm hardly moves from ¹⁶O to ²³⁸U while c grows as A¹⁄³. Two densities fall out of that, and they are not the same. Normalising, A = (4π/3)ρ₀c³[1 + π²(a/c)²], which for ²⁰⁸Pb with c = 6.62 fm and a = 0.55 fm gives ρ₀ = 0.160 nucleons fm⁻³ — the interior value, with light nuclei within about 10% of it. Separately, the equivalent uniform sphere R = √(5/3)√⟨r²⟩ follows R ≈ 1.2A¹⁄³ fm, a fit good to a few percent above A ≈ 90 that drifts to 1.39A¹⁄³ at carbon; its mean density is A/[(4π/3)(1.2)³A] = 0.138 nucleons fm⁻³, with A cancelling identically. Both constants say one thing — nucleons pack at a fixed density because the force saturates — and they differ by 15% only because the equivalent radius is pushed outward to cover a diffuse tail.

Three limits are built into the method. First, F is the transform of the charge density, so electron scattering locates protons; the neutron distribution needs another probe, and the parity-violating PREX measurement puts the ²⁰⁸Pb neutron rms radius about 0.28 fm outside the proton one. Second, what is measured is ρ_ch, which already has the proton's own 0.84 fm charge radius folded into it; that has to be unfolded before the result is a point-nucleon distribution. Third, one-photon exchange assumes Zα ≪ 1, and at Z = 82 it is 0.60: the incoming and outgoing waves are distorted by the nuclear Coulomb field, and the measured minima are shifted inward by several percent and partly filled in. A defensible heavy-nucleus analysis is an iterated phase-shift calculation against a trial ρ(r), not a single Fourier transform. Quoting a lead radius from a plane-wave fit is a systematic error, not a rounding one.