Q.Draw a plot of $\alpha$-particle scattering by a thin foil of gold to show the variation of the number of the scattered particles with scattering angle. Describe briefly how the large angle scattering explains the existence of the nucleus inside the atom. Explain with the help of impact parameter picture, how Rutherford scattering serves a powerful way to determine an upper limit on the size of the nucleus.

Concept: Rutherford’s nuclear model — the atom has a tiny, massive, positively charged nucleus; scattering depends on the impact parameter.

Reasoning:

1. Plot: The number $N$ of scattered $\alpha$-particles falls sharply with scattering angle $\theta$. Most particles pass undeflected ($\theta \approx 0$), while very few scatter through large angles ($\theta > 90^\circ$). The curve follows $N \propto 1/\sin^4(\theta/2)$.

2. Large-angle scattering → nucleus: Most $\alpha$-particles pass through empty space and scatter by small angles. A few, however, come very close to a massive positive centre and are repelled back — this is only possible if the atom’s positive charge and most of its mass are concentrated in a tiny nucleus. …

The core idea

Before Rutherford, the atom was imagined as a “plum pudding” — positive charge spread uniformly throughout a sphere, with electrons embedded like raisins. If that were true, $\alpha$-particles (fast, heavy, positively charged) would barely be deflected as they passed through. But Geiger and Marsden observed that about 1 in 8000 $\alpha$-particles bounced back — some through angles greater than $90^\circ$. That was impossible with a diffuse charge. The only explanation: the positive charge and most of the mass must be crammed into a region far smaller than the atom itself — the nucleus.

1. The scattering plot

The graph of number $N$ of scattered $\alpha$-particles versus scattering angle $\theta$ looks like this:

• At small $\theta$ (say $1^\circ$–$10^\circ$), $N$ is huge — most particles are barely deflected.
• As $\theta$ increases, $N$ drops sharply. At $\theta = 90^\circ$, $N$ is already very small.
• Beyond $90^\circ$ (backscattering), $N$ is tiny but non-zero — these are the particles that hit the nucleus nearly head-on.

Mathematically, Rutherford derived:

So the curve falls off steeply, not linearly. A log‑scale plot would show a straight line for this relation — a powerful experimental check.

2. How large‑angle scattering proves the nucleus exists

Imagine firing a bullet at a sheet of tissue paper. If the bullet bounces straight back, there must be something hard behind the paper. Similarly:

• A small deflection ($\theta$ small) means the $\alpha$-particle passed far from any concentrated charge — it felt only a weak Coulomb repulsion from the whole atom.
• A large deflection ($\theta$ near $180^\circ$) means the $\alpha$-particle came very close to a massive, positively charged core — so close that the repulsive force was enormous, reversing its direction.

If the positive charge were spread out, the force at any point inside the atom would be too weak to reverse a fast $\alpha$-particle. Only a point‑like (or very tiny) nucleus can produce the huge electric field needed for backscattering.

3. The impact parameter picture and nuclear size

The impact parameter $b$ is the perpendicular distance between the initial velocity vector of the $\alpha$-particle and the centre of the nucleus. It determines the scattering angle:

• Large $b$ → particle passes far from nucleus → small $\theta$.
• Small $b$ → particle passes close to nucleus → large $\theta$.
• $b = 0$ → head‑on collision → $\theta = 180^\circ$ (direct backscatter).

Rutherford derived the exact relation: