Alpha (α) scattering refers to the interaction of alpha particles—helium nuclei consisting of two protons and two neutrons—with atomic nuclei or atoms. The study of α-scattering has played one of the most pivotal roles in the development of modern physics. Historically, Rutherford’s α-scattering experiments in 1909 led to the discovery of the atomic nucleus and gave rise to the planetary model of the atom. These experiments showed that most α-particles pass through thin metal foils with little deflection, while a very small fraction undergo large-angle scattering, revealing the presence of a compact and massive nucleus.

In quantum scattering theory, α-particles interacting with atomic nuclei are treated through the Coulomb potential, since both projectile and target carry positive charge. The long-range nature of the Coulomb field makes α-scattering distinct from short-range nuclear scattering. Classical mechanics provides a useful description in terms of trajectories, impact parameters, and scattering angles, but a complete interpretation requires quantum mechanics, especially at low energies or when nuclear forces become relevant.

The Coulomb interaction between an α-particle of charge $ +2e $ and a nucleus of charge $ +Ze $ is given by:

\[V(r) = \frac{2Ze^2}{4\pi\epsilon_0 r}.\]

This central potential leads to hyperbolic trajectories, where the deflection angle depends on the impact parameter. Rutherford derived the differential cross-section in classical terms and predicted an angular distribution:

\[\frac{d\sigma}{d\Omega} \propto \frac{1}{\sin^4(\theta/2)}.\]

This formula matched experimental data exceptionally well for large impact parameters (i.e., small scattering angles), demonstrating that the Coulomb potential dominates the interaction.

Quantum mechanically, α-scattering is treated using partial waves, similar to other scattering processes. The phase shifts are determined by comparing the asymptotic form of the wavefunction with Coulomb-modified spherical waves. The scattering amplitude is influenced by both the long-range Coulomb field and, if the α-particle approaches close enough, the short-range nuclear potential. This leads to resonances, enhanced cross-sections, and deviations from the simple Rutherford formula at small distances. Such quantum effects have been crucial in understanding nuclear sizes, nuclear charge distributions, and the onset of nuclear forces.

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Supplementary Readings

Coulomb Potential

The α-particle experiences the potential: \(V(r) = \frac{2Ze^2}{4\pi\epsilon_0 r}.\)

Classical Scattering

Using impact parameter $ b $ and scattering angle $ \theta $, \(\cot\left(\frac{\theta}{2}\right) = \frac{2bE}{k},\) where
\(k = \frac{2Ze^2}{4\pi\epsilon_0}.\)

Rutherford Differential Cross-Section

Expressing cross-section in terms of impact parameter: \(\frac{d\sigma}{d\Omega} = \left(\frac{k}{4E}\right)^2 \frac{1}{\sin^4(\theta/2)}.\)

Quantum Mechanical Scattering

The wavefunction in presence of Coulomb potential behaves as: \(\psi(r,\theta) \sim e^{i\mathbf{k\cdot r}} + f_C(\theta)\frac{e^{ikr}}{r}.\)

The Coulomb scattering amplitude: \(f_C(\theta) = -\frac{\eta}{2k\sin^2(\theta/2)} e^{-2i\eta \ln[\sin(\theta/2)]},\) with Sommerfeld parameter: \(\eta = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0\hbar v}.\)

Quantum deviations from Rutherford formula appear when nuclear forces influence scattering, often at large angles.