Physics · Ch 12 — Atoms

Alpha-particle Trajectory

12.2.1

Alpha-particle Trajectory

The Trajectory of an Alpha Particle

When a beam of alpha particles is fired at a thin gold foil, each particle follows a curved path — a trajectory — determined by the repulsive electric force from the nucleus. The key quantity that decides the path is the impact parameter, denoted by $b$ .

The impact parameter is defined as the perpendicular distance between the initial velocity vector of the alpha particle and the centre of the nucleus. Imagine the alpha particle approaching the nucleus along a straight line that would miss the centre by some distance $b$ if there were no force. That distance $b$ is the impact parameter.

In a typical beam, different alpha particles have different impact parameters. Some pass very close to the nucleus (small $b$ ), while others pass far away (large $b$ ). Since all particles in the beam have nearly the same kinetic energy, the only thing that varies from particle to particle is $b$ . This variation causes the beam to scatter in all directions, with different probabilities for different scattering angles.

Note

The scattering angle $\theta$ is the angle through which the alpha particle's direction changes. A head-on collision gives $\theta \approx \pi$ (rebound), while a near-miss gives $\theta \approx 0$ (almost no deflection).

How the Impact Parameter Controls the Trajectory

The relationship between $b$ and the scattering angle $\theta$ is straightforward and physically intuitive:

Small impact parameter — the alpha particle comes very close to the nucleus. The repulsive Coulomb force is strong, so the particle suffers a large deflection. In the extreme case of a head-on collision, $b$ is minimum (effectively zero), and the alpha particle rebounds back along its original path, giving $\theta \approx \pi$ .
Large impact parameter — the alpha particle passes far from the nucleus. The Coulomb force is weak, so the particle goes nearly undeviated, with a small deflection $\theta \approx 0$ . …

Figure 12.4Trajectory of α-particles in the coulomb field of a target nucleus. The impact parameter b and scattering angle θ are also depicted.

Fig. 12.4 — Trajectory of α-particles in the coulomb field of a target nucleus. The impact parameter b and scattering angle θ are also depicted.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Fig. 12.4 is a schematic diagram — not a plot with axes, but a trajectory map showing several incoming alpha particles approaching a single target nucleus from the left. The nucleus is drawn as a small dot at the centre. Each alpha particle follows a different path, and the key variable that determines which path it takes is the impact parameter $b$ .

The impact parameter is defined as the perpendicular distance from the centre of the nucleus to the initial straight-line path of the alpha particle (the line it would follow if no force acted). In the figure, $b$ is marked as a short line segment drawn from the nucleus to the initial velocity line, meeting it at a right angle. The scattering angle $\theta$ — the angle between the incoming and outgoing asymptotic directions — is also labelled on one of the trajectories.

The physical idea is beautifully simple. The alpha particle and the nucleus both carry positive charge, so they repel each other with a Coulomb force that varies as $1/r^2$ . A particle with a small impact parameter passes very close to the nucleus, feels a strong repulsive force, and is deflected through a large angle — in the extreme case of a head-on collision ( $b \approx 0$ ), it rebounds almost straight back ( $\theta \approx \pi$ ). A particle with a large impact parameter stays far from the nucleus, feels only a weak force, and is deflected through a small angle ( $\theta \approx 0$ ). The figure shows several such trajectories side by side, each with a different $b$ , so you can see the continuous variation from nearly undeviated to nearly back-scattered.

Important

The figure makes a crucial point about atomic structure: only a tiny fraction of alpha particles in a beam have a small impact parameter and are scattered through large angles. This tells us that the positive charge and most of the mass of an atom are concentrated in a very small volume — the nucleus. If the positive charge were spread out over the whole atom, large-angle scattering would be far more common.

The textbook uses this geometric picture to derive the distance of closest approach $d$ for a head-on collision. When an alpha particle (charge $+2e$ ) approaches a nucleus of atomic number $Z$ (charge $+Ze$ ) head-on, it slows down as it climbs the Coulomb potential hill, comes momentarily to rest at a distance $d$ , and then reverses direction. Conservation of energy gives:

K = \frac{1}{4\pi\varepsilon_0} \frac{(2e)(Ze)}{d}

where $K$ is the initial kinetic energy of the alpha particle. Solving for $d$ :

d = \frac{1}{4\pi\varepsilon_0} \frac{2Ze^2}{K}

For a 7.7 MeV alpha particle striking a gold nucleus ( $Z=79$ ), this gives $d \approx 3.0 \times 10^{-14}$ m (30 fm). Since the alpha particle reverses without touching the nucleus, the actual nuclear radius must be smaller than this — a powerful upper limit on nuclear size. …

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