Investigate and analyse the Geiger-Marsden (Rutherford) gold foil experiment and Rutherford's nuclear model of the atom, and Chadwick's discovery of the neutron

Geiger and Marsden (1909, working under Rutherford) directed a beam of alpha particles from a radioactive source at a thin gold foil. A movable detector (a scintillation screen) measured the number of alpha particles scattered through different angles.

Most alpha particles passed through the foil with little or no deflection, as expected for a diffuse positive charge spread over the atom (Thomson's plum-pudding model). However, a small but non-zero fraction were deflected through very large angles, and a few were even back-scattered through more than 90 degrees.

Rutherford famously remarked that this was "as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you." The plum-pudding model could not produce such large deflections, because its smeared-out positive charge gave only small Coulomb forces on the fast alpha particles.

Rutherford (1911) concluded that the positive charge and almost all of the mass of the atom is concentrated in a tiny central nucleus, around $10^{-15}$ m in radius compared with the atom's $10^{-10}$ m. Most alpha particles pass through the (mostly empty) atom; a small fraction encounter the nucleus head-on and recoil. He worked out the angular distribution and showed it matched a $1/r^2$ Coulomb force from a point-like positive charge.

Markers reward the setup, the unexpected back-scattering, the inadequacy of the plum-pudding model, and the nuclear conclusion (small dense positive nucleus, mostly empty atom).

Curie and Joliot (1932) showed that bombarding beryllium with alpha particles produced a neutral radiation that, in turn, ejected protons from paraffin wax. They assumed this radiation was gamma rays.

Chadwick (1932) repeated the experiment but also let the neutral radiation strike a nitrogen target and measured the recoil kinetic energy of the nitrogen nuclei. Applying conservation of energy and momentum, he found that gamma rays would have had to carry implausibly high energies (around 50 MeV) to give the observed recoils, far greater than expected for beryllium-alpha reactions.

A neutral particle of mass approximately equal to the proton, however, could give the observed proton and nitrogen recoils consistently and with reasonable energies. Chadwick concluded the radiation was a stream of neutral particles with mass close to the proton, and named them neutrons.

The reaction is:

$^9_4\text{Be} + ^4_2\text{He} \to ^{12}_6\text{C} + ^1_0\text{n}$.

Markers reward the setup (alpha on beryllium, neutral radiation, recoil targets), the inconsistency of the gamma-ray interpretation, the conservation-of-momentum-and-energy reasoning, and the identification of the neutron with mass close to the proton.

The three observations. (1) The great majority of alpha particles passed straight through the foil with little or no deflection. (2) A small fraction were deflected through moderate angles. (3) A very small fraction (about 1 in 8000) were scattered through more than $90^{\circ}$, a few almost straight back toward the source.

What the large-angle scattering meant. A back-scattered alpha requires a very strong, concentrated Coulomb repulsion, which is impossible if the positive charge is smeared over the whole atom (Thomson's model). Rutherford concluded that the positive charge and almost all the mass are concentrated in a tiny, dense central nucleus, with the rest of the atom mostly empty space.

Marks: one for the "most pass through" result, one for the rare large-angle/back-scattering result, one for the nuclear conclusion (small, dense, positive nucleus).

Reaction. $^{9}_{4}\text{Be} + {}^{4}_{2}\text{He} \to {}^{12}_{6}\text{C} + {}^{1}_{0}\text{n}$ (an alpha particle striking beryllium-9 gives carbon-12 plus a neutron).

Two properties of the neutron. It carries no electric charge (it is neutral), and its mass is very close to that of the proton ($1.0087$ u versus $1.0073$ u).

Marks: one for a correctly balanced equation (mass numbers $9+4 = 12+1$, atomic numbers $4+2 = 6+0$), one for any two correct neutron properties.

At the distance of closest approach the alpha momentarily stops, so all of its kinetic energy has become electrostatic potential energy of the alpha (charge $2e$) and the nucleus (charge $Ze$):

$E_k = \dfrac{k(2e)(Ze)}{r_{\min}}$, so $r_{\min} = \dfrac{2kZe^2}{E_k}$.

Convert the energy: $E_k = 4.0 \times 1.602 \times 10^{-13} = 6.41 \times 10^{-13}$ J.

$r_{\min} = \dfrac{2(8.99 \times 10^9)(79)(1.602 \times 10^{-19})^2}{6.41 \times 10^{-13}} = 5.7 \times 10^{-14}\ \text{m}$ (about $57$ fm).

Marks: one for the energy-conservation statement (KE becomes electrostatic PE), one for $r_{\min} = 2kZe^2/E_k$ with values substituted, one for $r_{\min} = 5.7 \times 10^{-14}$ m with the unit.

The test

Chadwick directed the neutral radiation onto different target nuclei (hydrogen and nitrogen) and measured the recoil speeds of each. He then asked which projectile - a massless gamma-ray photon or a massive neutral particle - could account for the recoils while conserving both momentum and energy.

Ruling out gamma rays

For a photon to give a nitrogen nucleus its observed recoil, conservation of energy and momentum required the photon to carry about $50$ MeV. The beryllium-alpha reaction only releases a few MeV, so such energetic photons were impossible. The gamma-ray hypothesis therefore failed on energy grounds.

Confirming the neutron

Treating the collisions as elastic (billiard-ball) collisions of a projectile of unknown mass $m$ with stationary targets, the ratio of the hydrogen and nitrogen recoil speeds depends only on $m$. The measured ratio was consistent only with $m \approx 1$ u - a neutral particle of almost the proton's mass. Chadwick named it the neutron.

Marks: one for the recoil-measurement method, one for the ~$50$ MeV gamma-ray energy being impossible from the available reaction energy, one for the elastic-collision momentum/energy analysis, one for concluding a neutral particle of mass $\approx m_p$.

(a) As $r$ decreases the potential energy $U$ rises steeply (it follows a $1/r$ Coulomb curve), because the repulsion between the positive alpha and the positive nucleus grows without limit as they approach.

(b) The alpha stops where all its kinetic energy has become potential energy, i.e. where the curve reaches $U = 5.7$ MeV. Reading across from $5.7$ MeV on the graph gives a distance of closest approach of about $40$ fm ($4.0 \times 10^{-14}$ m). It stops there because it cannot climb higher up the potential hill than its total energy allows.

(c) A $4.0$ MeV alpha has less kinetic energy, so its energy line meets the curve at a larger separation - the distance of closest approach is greater (about $57$ fm). A slower, lower-energy alpha cannot penetrate as far into the repulsive field.

Marks: one for describing the steep $1/r$ rise, one for reading $\approx 40$ fm at $U = 5.7$ MeV, one for the KE-becomes-PE explanation, one for correctly reasoning that a lower-energy alpha stops further out.

Band-6 plan. (1) State what the plum-pudding model predicted. (2) Contrast with the observed large-angle scattering and why the smeared charge cannot produce it. (3) State the nuclear model that does. (4) Evaluate: the model succeeded (nucleus) but left real gaps (electron stability, no neutron), so it was necessary but incomplete. End with a judgement.

Model answer. Thomson's model spread the positive charge diffusely across the whole atom, so it predicted that fast alpha particles would feel only weak Coulomb forces and emerge with small deflections. Geiger and Marsden instead found that, while most alphas passed through, a small fraction were scattered through very large angles and a few almost straight back. A diffuse positive charge cannot exert a force large enough to reverse a fast, massive alpha, so the plum-pudding model was untenable.

Rutherford (1911) showed that concentrating the positive charge and nearly all the mass into a tiny central nucleus (radius $\sim 10^{-15}$ m, against the atom's $\sim 10^{-10}$ m) produces exactly the observed $1/r^2$ Coulomb scattering: most alphas miss the nucleus and pass through the mostly empty atom, while the rare near-head-on approach recoils sharply. This explained the data quantitatively.

However, the nuclear model was incomplete. Classically, the orbiting electrons should radiate energy and spiral into the nucleus, so it could not explain the atom's stability (resolved later by Bohr). It also had no account of the extra nuclear mass or of what held like-charged protons together - gaps not closed until Chadwick's neutron and the strong force. So the nuclear model was a necessary and correct advance on the nucleus, but only a partial picture of the whole atom.

Marker's note: the top band contrasts prediction with observation (not just "alphas bounced back"), states the nuclear model quantitatively ($1/r^2$, the radii), AND delivers a genuine evaluation of completeness (names at least the electron-stability gap and the missing neutron). A response that only narrates the experiment, with no judgement of completeness, caps in the middle band.

Band-6 plan. Thesis: the neutron completed the atomic model and unlocked nuclear physics. Argue (1) it completed the proton-neutron-electron nucleus and explained isotopes/mass, (2) the discovery rested on conservation of momentum and energy (method significance), (3) it enabled later physics (fission, the strong force). Weigh and conclude on "significance".

Model answer. Chadwick's discovery was highly significant because it completed the model of the atomic nucleus. Before 1932 the nucleus was known to be small, dense and positive, but its full mass and the existence of isotopes were unexplained. Identifying a neutral particle of almost the proton's mass gave the nucleus its second constituent: $A = Z + N$ nucleons, which at once explained why nuclear mass exceeds the mass of $Z$ protons and why isotopes (same $Z$, different $N$) exist.

The reasoning itself is significant as a model of physics method. Chadwick could not "see" a neutral particle, so he relied on conservation of momentum and energy: he measured the recoil speeds of different target nuclei and showed that a massless gamma ray would need an impossible ~$50$ MeV to produce them, whereas a massive neutral particle of mass $\approx m_p$ fitted every target consistently. The conservation laws turned indirect recoil data into a decisive identification.

The consequences were far-reaching. Because neutrons are uncharged, they are not repelled by nuclei and can be absorbed even at low energies - which made possible artificial transmutation, the discovery of nuclear fission (1938) and nuclear reactors, and sharpened the search for the force (the strong nuclear force) binding nucleons together. Weighing these, the neutron was not a minor addition but the keystone that made both the modern nuclear atom and applied nuclear physics possible; its significance is therefore very high.

Marker's note: the top band makes an explicit judgement of significance (not just "it was important"), ties the discovery to conservation of momentum AND energy as the method, and gives at least one major downstream consequence (fission/reactors or the strong force). Listing facts about the neutron without assessing significance caps below the top band.