Mott's formulas for identical charged particles

Units are SI and angles are in the centre-of-mass frame unless marked; nothing here is relativistic, so the course metric diag(1, −1, −1, −1) never enters. Topic 2.07 derived the exact amplitude for two charges Ze, f_C(θ) = −(η/2k) exp[−iη ln sin²(θ/2) + 2iσ₀]/sin²(θ/2), with σ₀ = arg Γ(1 + iη); it is used here, not re-derived. Exchange needs it at π − θ, which swaps sin(θ/2) for cos(θ/2). Multiply f_C(θ) by the complex conjugate of f_C(π − θ): the moduli give (η/2k)²/[sin²(θ/2) cos²(θ/2)], the two minus signs cancel, the constant 2σ₀ cancels, and the logarithms combine into −η ln tan²(θ/2). So Re[f_C(θ) f_C*(π − θ)] = (η/2k)² cos(η ln tan²(θ/2))/[sin²(θ/2) cos²(θ/2)], derived. The first Born amplitude cannot stand in: it is real, so its version of this cosine is 1 at every angle, whereas at η = 3 and θ = 60° the exact one is cos(3 ln(1/3)) = cos(−3.30) = −0.99, the opposite sign.

Topic 2.05 gave spinless bosons |f + f′|² = |f|² + |f′|² + 2 Re(f f′*), and topic 2.06 gave unpolarised spin-half fermions |f|² + |f′|² − Re(f f′*), with f′ = f(π − θ). Insert the Coulomb pieces and both read dσ/dΩ = (η/2k)²[1/sin⁴(θ/2) + 1/cos⁴(θ/2) + c cos(η ln tan²(θ/2))/(sin²(θ/2) cos²(θ/2))], with c = +2 for bosons and −1 for fermions: Mott's 1930 result, derived. Dimensions: ħk = μv makes η/2k = Z²e²/(16πε₀E), with E the centre-of-mass energy, and e²/(4πε₀) = 1.440 MeV fm is an energy times a length, so η/2k is a length and dσ/dΩ comes out in m² sr⁻¹. The first two terms are Rutherford for the particle that turned through θ and for its partner recoiling into the same counter; set c = 0, distinguishable particles, and that classical sum is all that remains. Only the third term carries ħ, inside η. Topic 2.06's spin counting extends c to (−1)^(2s) × 2/(2s + 1), so deuterons, spin 1, get +2/3.

At θ = 90°, tan(θ/2) = 1 and its logarithm vanishes, so the cosine is 1 whatever η is. The reason is that π − θ equals θ there: the direct and exchange amplitudes are one number, logarithmic phase included. Each Rutherford term is 4(η/2k)² and the interference term c × 4(η/2k)², so dσ/dΩ(90°) = 16(η/2k)² for spinless bosons, twice the classical 8(η/2k)², and 4(η/2k)² for unpolarised spin-half fermions, half of it, both derived. Dividing Mott by the classical sum at any angle gives the cleaner ratio 1 + (c/2)[sin²θ/(1 + cos²θ)] cos(η ln tan²(θ/2)), derived, whose envelope is 1 at 90°, 0.6 at 60° and 120° and 1/7 at 30° and 150°: exchange matters most where the two Rutherford terms are equal. For equal masses a centre-of-mass angle θ is θ/2 in the laboratory and dΩ_cm/dΩ_lab = 4 cos θ_lab, both quoted from two-body kinematics, so 90° is the counter at 45°, where the two nuclei leave symmetrically, 45° either side of the beam; at every angle equal masses leave at right angles to each other.

Away from 90° the phase φ(θ) = η ln tan²(θ/2) grows, and dφ/dθ = 2η/sin θ. One full oscillation therefore spans Δθ = π sin θ/η, derived: π/η at 90°, shrinking toward 0° and 180°, where the envelope has already died. For η = 10 the period near 90° is 18.0° in the centre of mass and 9.0° in the laboratory, and the cosine first reaches −1 at exactly 2 arctan(e^(π/20)) = 99.0°, as the linear estimate 90° + 9° says. Counting is quicker still: between 60° and 120° there are η ln 9/(2π) = 0.35η fringes. The opposite limit checks the derivation. As η → 0 the cosine tends to 1 at every angle and Mott's formula becomes the symmetrised first Born result, built from the real amplitude −(η/2k)/sin²(θ/2). Protons at 1 MeV in the laboratory have η = 0.158, a period near 90° of 1140° and a phase of only −0.42 rad at 30°, so their Mott interference is one smooth dip to 1/2 at 90°, with no fringes at all.

Large η is the classical regime: η = kd/2, where d = Z²e²/(4πε₀E) is the head-on closest approach, so η ≫ 1 means an orbit many reduced wavelengths across. Yet the interference term does not shrink as η grows; at 90° it is always c/2 times the classical sum. What changes is its period, π/η, and a real counter accepts a spread of angles. Model the spread as a Gaussian of rms width δθ and take the phase as linear across it, φ(θ + u) ≈ φ(θ) + (2η/sin θ)u. The Gaussian average of cos φ is then cos φ(θ) × exp(−2η²δθ²/sin²θ), so the interference term is multiplied by a visibility V, derived, equal to exp(−2η²δθ²) at 90°. With δθ = 1° in the centre of mass, V = 0.94 at η = 10, 0.58 at η = 30 and 0.002 at η = 100, and V = 1/2 at ηδθ = 0.59. The classical sum returns because no counter resolves angles of order 1/η once η is large, not because the quantum term goes away: here the correspondence limit is an average, not a vanishing.

The measurements that test Mott's formulas are quoted here, not derived. In 1930 Chadwick scattered slow α-particles in helium and found the rate at 45° in the laboratory, 90° in the centre of mass, close to twice the classical value: the spin-0 factor derived above. Measured α–α angular distributions at a few hundred keV follow the spin-0 form, its swings above and below the classical sum included, and depart from it as the energy rises; the departures are read as nuclear phase shifts. Protons are the sharper test, because c = −1 and η = 0.158 at 1 MeV predict one smooth dip. In 1936 Tuve, Heydenburg and Hafstad measured proton-proton scattering below 1 MeV and found departures from Mott's formula far larger than their errors. Breit, Condon and Present read them as an s-wave phase shift: an attractive nuclear force between two protons, close in strength to the neutron-proton singlet force, and by topic 2.06 acting in the singlet alone. The formula also fails where atomic electrons screen the field at small angles, and where relativity adds the Mott factor of Unit 5.