§1.8Scattering Experiments

This is the section that justifies the whole field. Scattering experiments are how every layer of structure — crystals, molecules, atoms, nuclei, nucleons, quarks — was found, and they all work on one principle:

The scattered intensity is the squared Fourier transform of the target’s charge distribution, with momentum transfer as the conjugate variable.

If you have ever looked at a diffraction pattern, you already know this. The rest of the section is making that statement precise and reading the consequences.

Start with optics

Light of wave-vector $k$ falls on a screen whose amplitude transparency is $T(x)$ — how much of the incident amplitude gets through at each point, including any phase shift. The amplitude diffracted at angle θ is the sum over the screen of each element’s contribution with its phase delay $\phi = k_x x$:

That is a Fourier transform, with the transverse wave-vector as the conjugate variable. Nothing here is quantum mechanical yet.

The resolution statement, made sharp: a target of diameter $D$ produces a maximum phase variation $D\phi$ across itself. If

that variation is too small to detect, and the target is indistinguishable from a point. Every “elementary” particle in this book is elementary only in the sense that nobody has yet hit it hard enough to see otherwise.

The same thing, for particles

Scatter an electron off a nucleus’s electrostatic potential. Both the incoming and outgoing electrons are free, so their wave functions are plane waves, and the matrix element becomes

The amplitude is the transform of the potential. Use Poisson’s equation $\nabla^2\varphi = -\rho/\varepsilon_0$ to trade the potential for the charge density — which costs one factor of $1/q^2$ — and you arrive at the master formula:

Rutherford: the case with no structure

Put a point charge at the origin. Its density is a delta function, its transform is a constant, $F(q) = 1$, and (1.91) collapses to

For Rutherford’s own experiment — α particles of a few MeV, so non-relativistic, $E_k = p^2/2m$ and $E \approx m$ — substituting $q = 2p\sin(\theta/2)$ gives the familiar textbook form:

import numpy as np
hbarc = 0.1973269804                                   # GeV fm

E, th = 20.0, np.radians(6)                            # Problem 1.5
q = 2 * E * np.sin(th / 2)                             # ultra-relativistic elastic
print(f"20 GeV e- at 6 deg: q = {q:.4f} GeV -> resolves hbar c/q = {hbarc/q:.4f} fm")

F = lambda x: 3 * (np.sin(x) - x * np.cos(x)) / x**3   # uniform sphere
lo, hi = 3.0, 6.0                                      # bisect for the first zero
for _ in range(200):
    mid = (lo + hi) / 2
    lo, hi = (lo, mid) if F(lo) * F(mid) <= 0 else (mid, hi)
z = (lo + hi) / 2
print(f"uniform-sphere form factor: first zero at qR = {z:.4f}")
for R, name in ((7.0, 'gold  '), (2.7, 'carbon')):
    print(f"  R = {R} fm ({name.strip()}) -> first diffraction zero at q = {z*hbarc/R:.4f} GeV")

a = 0.24                                               # proton dipole parameter
print(f"proton dipole R = {a} fm -> rms radius sqrt(12) R = {np.sqrt(12)*a:.3f} fm; "
      f"|F|^2 at q=1 GeV is {(1/(1+(a/hbarc)**2)**2)**2:.4f}")

for M, name in ((0.938272, 'a proton      '), (197*0.9315, 'a gold nucleus')):
    Ep = 20.0 / (1 + (2*20.0/M) * np.sin(np.radians(30)/2)**2)
    print(f"recoil: 20 GeV e- at 30 deg off {name} -> E' = {Ep:.5f} GeV (E'/E = {Ep/20:.3f})")

print(f"Rutherford sigma(90 deg)/sigma(10 deg) = "
      f"{np.sin(np.radians(5))**4/np.sin(np.radians(45))**4:.3e}")
print(f"Mott/Rutherford at 90 deg = cos^2(45 deg) = {np.cos(np.radians(45))**2:.4f}")

20 GeV e- at 6 deg: q = 2.0934 GeV -> resolves hbar c/q = 0.0943 fm
uniform-sphere form factor: first zero at qR = 4.4934
R = 7.0 fm (gold) -> first diffraction zero at q = 0.1267 GeV
R = 2.7 fm (carbon) -> first diffraction zero at q = 0.3284 GeV
proton dipole R = 0.24 fm -> rms radius sqrt(12) R = 0.831 fm; |F|^2 at q=1 GeV is 0.0265
recoil: 20 GeV e- at 30 deg off a proton       -> E' = 5.18703 GeV (E'/E = 0.259)
recoil: 20 GeV e- at 30 deg off a gold nucleus -> E' = 19.71217 GeV (E'/E = 0.986)
Rutherford sigma(90 deg)/sigma(10 deg) = 2.308e-04
Mott/Rutherford at 90 deg = cos^2(45 deg) = 0.5000

Where this goes next

• §1.9–1.10 introduces the particles this machinery was used to find.
• §5.6–5.7 explains the Mott $\cos^2(\theta/2)$ factor properly, as a consequence of helicity conservation.
• §6.2 Nucleon structure is this section applied at high $q^2$ — and the moment the form factor stops describing a static shape and starts describing point-like constituents.