Case Study 1: Rutherford Scattering — From Classical to Quantum

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Case Study 1: Rutherford Scattering — From Classical to Quantum

The Experiment That Revealed the Nucleus

In 1909, Ernest Rutherford suggested to Hans Geiger and Ernest Marsden, his collaborators at the University of Manchester, that they try firing alpha particles at a thin gold foil and look for large-angle deflections. At the time, the dominant model of the atom was J. J. Thomson's "plum pudding" model, in which the positive charge was smeared uniformly throughout the atom and electrons were embedded like raisins in a pudding. In this model, alpha particles --- helium nuclei, relatively heavy and energetic --- should pass through the atom with at most minor deflections. The positive charge density was too dilute to produce large forces.

The results were shocking. While most alpha particles passed straight through the foil or were deflected by small angles (as expected), a small but significant fraction --- roughly 1 in 8000 --- bounced back at angles greater than $90°$. Rutherford famously recalled: "It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."

Rutherford's Classical Derivation

Rutherford published his analysis in 1911. He modeled the alpha particle as a point charge $Ze$ ($Z = 2$ for helium) approaching a point charge $Z'e$ ($Z' = 79$ for gold), interacting via the Coulomb force. The classical trajectory is a hyperbola, and Rutherford used standard orbit theory to derive the relationship between the impact parameter $b$ (the perpendicular distance between the particle's initial trajectory and the target nucleus) and the scattering angle $\theta$:

$$b = \frac{a}{2}\cot\left(\frac{\theta}{2}\right)$$

where $a = ZZ'e^2/(4\pi\epsilon_0 E)$ is the distance of closest approach for a head-on collision ($\theta = \pi$) and $E$ is the kinetic energy.

The differential cross section follows from $d\sigma = |db/d\theta|\cdot 2\pi b\,d\theta$ and the solid angle element $d\Omega = 2\pi\sin\theta\,d\theta$:

$$\frac{d\sigma}{d\Omega}\bigg|_{\text{classical}} = \left(\frac{a}{4}\right)^2\frac{1}{\sin^4(\theta/2)} = \left(\frac{ZZ'e^2}{4 \cdot 4\pi\epsilon_0 E}\right)^2\frac{1}{\sin^4(\theta/2)}$$

This is the Rutherford formula. Geiger and Marsden verified the $\sin^{-4}(\theta/2)$ angular dependence, the $E^{-2}$ energy dependence, and the $Z'^2$ target dependence with impressive precision, providing conclusive evidence for a compact, massive nucleus at the center of the atom.

The Quantum Calculation

Fifteen years later, quantum mechanics was born. Naturally, one of the first applications of the new theory was to rederive the Rutherford formula from the Schrodinger equation. The result was a surprise: the classical and quantum formulas agree exactly.

Born Approximation Route

The simplest quantum derivation uses the Born approximation. Starting from the Yukawa potential $V(r) = V_0 e^{-\mu r}/r$ (which reduces to the Coulomb potential in the limit $\mu \to 0$), the Born scattering amplitude is:

$$f_{\text{Born}}(\theta) = -\frac{2mV_0}{\hbar^2}\frac{1}{q^2 + \mu^2}$$

Setting $V_0 = ZZ'e^2/(4\pi\epsilon_0)$ and $\mu \to 0$:

$$f_{\text{Born}}(\theta) = -\frac{2m \cdot ZZ'e^2/(4\pi\epsilon_0)}{\hbar^2 \cdot 4k^2\sin^2(\theta/2)} = -\frac{a}{4\sin^2(\theta/2)}$$

where we used $E = \hbar^2 k^2/(2m)$. The differential cross section is:

$$\frac{d\sigma}{d\Omega} = |f|^2 = \left(\frac{a}{4}\right)^2\frac{1}{\sin^4(\theta/2)}$$

This is exactly the classical Rutherford formula.

Exact Quantum Route

The exact quantum solution, first obtained by Gordon and independently by Mott in 1928, uses parabolic coordinates to solve the Coulomb Schrodinger equation exactly. The resulting scattering amplitude is:

$$f_{\text{exact}}(\theta) = -\frac{a}{4\sin^2(\theta/2)}\exp\left[-i\eta\ln\sin^2(\theta/2) + 2i\sigma_0\right]$$

where $\eta = ma/(2\hbar^2 k)$ is the Sommerfeld parameter and $\sigma_0 = \arg\Gamma(1 + i\eta)$. The modulus squared is:

$$|f_{\text{exact}}|^2 = \left(\frac{a}{4}\right)^2\frac{1}{\sin^4(\theta/2)}$$

The extra phase (involving $\ln\sin^2(\theta/2)$) modifies the interference between partial waves but does not affect $|f|^2$. The Rutherford formula is exact --- not an approximation --- for point Coulomb charges.

Why Does Classical Equal Quantum?

This agreement is not generic. For most potentials, the classical and quantum cross sections differ, sometimes dramatically (recall the factor-of-4 discrepancy for the hard sphere at low energy). The Coulomb potential is special for several deep reasons:

The $1/r$ potential is the unique power law for which all classical orbits are closed. This is related to the hidden $SO(4)$ symmetry of the Kepler/Coulomb problem (the same symmetry that produces the "accidental" degeneracy of hydrogen energy levels studied in Chapter 5). This extra symmetry prevents quantum corrections from appearing in the cross section.
The Born approximation happens to be exact for the Coulomb potential. Although the Born series does not converge (higher Born terms are individually divergent), the first Born term gives the correct modulus of the amplitude. The higher-order terms contribute only phases.
The de Broglie wavelength does not define a natural length scale for a pure $1/r$ potential. For a Yukawa potential, the range $1/\mu$ provides a length scale, and diffraction effects appear when $\lambda_{\text{dB}} \sim 1/\mu$. The pure Coulomb potential has no intrinsic length, so there is nothing to diffract around.

Where the Classical Picture Breaks Down

Rutherford scattering remains valid as long as:

The nuclear charge is truly point-like on the scale probed. When alpha particles have enough energy to reach the nuclear surface (distance of closest approach $\lesssim R_{\text{nuclear}} \approx 1.2\,A^{1/3}\;\text{fm}$), the cross section deviates from Rutherford. This is precisely how nuclear radii were first measured.

For 7.7 MeV alpha particles on gold ($Z' = 79$):

$$d_{\min} = \frac{2 \times 79 \times 1.44\;\text{MeV}\cdot\text{fm}}{2 \times 7.7\;\text{MeV}} = 29.6\;\text{fm}$$

The gold nuclear radius is approximately $R \approx 1.2 \times 197^{1/3} \approx 7.0\;\text{fm}$. Since $d_{\min} \gg R$, the alpha particle never touches the nucleus, and Rutherford scattering holds. But for 30 MeV alphas, $d_{\min} \approx 7.6\;\text{fm} \approx R$, and deviations begin to appear.

Identical-particle effects are negligible. For alpha-alpha scattering, the projectile and target are identical bosons, and the amplitude must be symmetrized: $f(\theta) + f(\pi - \theta)$. This produces the Mott cross section with interference oscillations absent from Rutherford's formula. These oscillations were first observed by Chadwick and Bieler in 1921 and provided early evidence for quantum effects in nuclear scattering.
The target has no structure. Real nuclei have finite size, excited states, and internal degrees of freedom. At high energies, the alpha can excite the nucleus (inelastic scattering), and the elastic cross section deviates from Rutherford even at angles where the distance of closest approach would seem safe.

Experimental Legacy

Rutherford scattering remains one of the most practically important techniques in physics:

Rutherford Backscattering Spectrometry (RBS) is a standard technique in materials science for determining the composition and depth profile of thin films. A beam of 1--3 MeV He ions is directed at the sample, and the energy and angle of backscattered particles reveal the identity and depth of atoms in the sample. The Rutherford cross section provides absolute normalization without calibration standards.
Nuclear size measurements via the deviation from Rutherford scattering at high energies established the $R = r_0 A^{1/3}$ formula (with $r_0 \approx 1.2\;\text{fm}$) that is foundational in nuclear physics.
The quark model was established by deep inelastic scattering experiments at SLAC in the late 1960s, which showed that high-energy electrons scattered off protons as if the proton contained point-like charged constituents. The cross section at high momentum transfer followed a $1/q^4$ law (reminiscent of Rutherford) rather than the exponential falloff expected from a smooth charge distribution. Friedman, Kendall, and Taylor received the 1990 Nobel Prize for this work --- a direct intellectual descendant of Rutherford's 1911 experiment.

Discussion Questions

If you could design Rutherford's experiment today, what would you change? What additional information could you extract with modern detectors?
The Born approximation gives the correct Rutherford cross section even though it does not converge for the Coulomb potential. How should we interpret an approximation that gives the "right answer" for the "wrong reasons"?
The discovery of quarks via deep inelastic scattering is structurally identical to Rutherford's discovery of the nucleus. In both cases, large-angle scattering reveals point-like constituents inside a composite object. What are the key differences between the two experiments, and what are the analogies?
Rutherford scattering gives an infinite total cross section. Does this mean that every particle in the beam is scattered? How do experimentalists handle this divergence in practice?