Nuclear radius from the charge form factor

The electron is the right probe because it has no strong interaction and no structure of its own, so the whole vertex is electromagnetic and one order of perturbation theory is enough. In the first Born approximation the amplitude is the Fourier transform of the potential, so for a static charge distribution the cross-section factorises: dσ/dΩ = (dσ/dΩ)_Mott |F(q)|². The Mott factor is the point-charge answer — Rutherford, with the relativistic spin factor cos²(θ/2) and a recoil denominator — and it holds no nuclear structure whatever. Everything the nucleus contributes lives in |F(q)|². So the first operation on any data set is a division: plot measured cross-section over Mott and you have |F|² on a scale where F(0) = 1 by construction. Nothing has yet been assumed about ρ beyond a static, spherical source.

Angle is what the apparatus turns; q is what the physics depends on. For an ultrarelativistic electron k = E/ħc, and elastic scattering through θ transfers q = 2k sin(θ/2), so with ħc = 197.33 MeV fm a 200 MeV beam at 20° gives q = 0.352 fm⁻¹. Two consequences follow. The reach is largest at backscattering, qₘₐₓ = 2E/ħc: 2.03 fm⁻¹ at 200 MeV, 5.07 fm⁻¹ at 500 MeV. And a truncated transform cannot resolve structure finer than roughly δr ≈ π/qₘₐₓ — 1.55 fm at 200 MeV, 0.62 fm at 500 MeV. That is why radius work is done at several hundred MeV and above: run it lower and you are measuring a blurred nucleus, and no fitting procedure invents detail the kinematics never delivered.

Expand the transform at small q: F(q) = 1 − q²⟨r²⟩/6 + q⁴⟨r⁴⟩/120 − …, a series in q², so ⟨r²⟩ = −6 dF/d(q²) at q = 0 is model-independent — the slope is the radius whatever shape ρ has. That is the number quoted as the rms charge radius: 2.470 fm for ¹²C, 5.501 fm for ²⁰⁸Pb, and it carries no convention at all. The second read is the diffraction pattern. A uniform sphere has F = 3[sin(qR) − qR cos(qR)]/(qR)³, vanishing at qR = 4.4934, so the first minimum gives R = 4.4934/q₁ from angles alone. For ²⁰⁸Pb at 250 MeV, once Coulomb distortion is corrected for, that minimum sits at θ = 30.5°, q₁ = 0.667 fm⁻¹, hence R = 6.74 fm and √(3/5)R = 5.22 fm — 5% short of 5.50. The reads disagree because the second assumed a shape and the first did not.

A detector counts rates, so the measurement is |F(q)|², while for a real spherical density F is real and changes sign at every zero. Those signs are supplied by the model, not by the experiment. Add truncation at qₘₐₓ and the inversion is ill-posed twice over: densities whose transforms agree below qₘₐₓ and differ above it fit the data identically, and so do densities whose transforms differ by a sign flip on one lobe. Practice takes one of two routes. Fit a shape with few parameters — the two-parameter Fermi profile — and quote c and a with their covariance, accepting the shape as an assumption. Or expand ρ in a Fourier–Bessel series inside a fixed cutoff radius, fit the coefficients qₘₐₓ supports, and bound the unmeasured tail, quoting the resulting incompleteness error as a band around ρ(r). Both are honest. Neither is an inversion.

Fit the Fermi profile across the chart and c comes out near 1.1 A¹⁄³ fm with a near 0.55 fm, so the 90-to-10 surface thickness t = 4a ln3 ≈ 2.4 fm hardly changes from carbon to lead: nuclear matter saturates, and the interior density is close to A-independent. But the number you quote for that interior depends on which radius convention you used, and the two in common use are different lengths. For ²⁰⁸Pb the Fermi fit c = 6.60 fm, a = 0.55 fm gives ρ₀ = 3A/(4πc³[1 + (πa/c)²]) = 0.162 nucleons fm⁻³. The sharp-sphere convention R = 1.2 A¹⁄³ fm = 7.11 fm reproduces the same rms radius (√0.6 × 7.11 = 5.51 fm) but spreads 208 nucleons through a bigger volume: 0.138 fm⁻³, 15% lower. Each is right inside its own definition; taking c from one and a density from the other is not.

The Born approximation treats both electron waves as plane waves, which needs Zα ≪ 1. For lead Zα = 82/137 = 0.60, and the failure is quantitative, not cosmetic. An electron at the centre of a uniformly charged sphere sits 3Zαħc/2R ≈ 24.9 MeV deeper than at infinity, so its local wavenumber inside exceeds the asymptotic k — a 10% effect at 250 MeV, 5% at 500 MeV. The diffraction minima therefore appear at smaller angles than plane-wave Born predicts, and such a fit reads the radius about 10% high: 0.7 fm on 7.1 fm, against data good to thousandths of a femtometre. Distortion also partly fills the minima, so they stop being true zeros. The cure is to stop approximating — solve the Dirac equation for the electron in the electrostatic potential of the extended nuclear charge, do a phase-shift analysis, and adjust ρ until the computed cross-section matches.