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

What this dot point is asking

NESA wants you to describe the Geiger-Marsden gold foil experiment and Rutherford's analysis that established the nuclear atom (small dense positive nucleus, mostly empty atom), then describe Chadwick's 1932 experiment that identified the neutron as a neutral particle of nearly the proton's mass using conservation of energy and momentum. Together these experiments completed the picture of the atom as a nucleus of protons and neutrons surrounded by electrons.

The answer

Geiger-Marsden gold foil experiment (1909)

Background. Thomson's plum-pudding model (1897) had positive charge smeared diffusely over the atom with embedded electrons. To test it, Hans Geiger and Ernest Marsden (under Rutherford, at Manchester) directed a beam of alpha particles (from radium decay) at a very thin gold foil and measured how many particles scattered into different angles using a movable scintillation detector.

What they expected. Under the plum-pudding model, the smeared positive charge should produce only small Coulomb deflections. Almost all alpha particles should emerge close to the forward direction with a small spread.

What they observed.

• Most alpha particles passed through with almost no deflection (consistent with mostly empty space).
• A small fraction (about 1 in 8000) scattered through angles greater than 90 degrees.
• A handful were back-scattered, returning toward the source.

Rutherford's interpretation. The large-angle scattering events are impossible if the positive charge is smeared over the whole atom. They require the alpha particle to encounter a strong Coulomb repulsion from a very compact positive charge. Rutherford (1911) showed that the angular distribution of scattered alpha particles is exactly what a point-like positive nucleus produces, with a $1/r^2$ Coulomb force.

Rutherford's nuclear model

The picture that emerged:

• Nearly all the mass and all the positive charge of the atom is concentrated in a tiny central nucleus, with radius $\sim 10^{-15}$ m.
• The atom as a whole has radius $\sim 10^{-10}$ m, so the nucleus is $10^{-5}$ of the atomic radius. The atom is mostly empty space.
• Negatively charged electrons orbit the nucleus at relatively large distances.

Two open questions remained:

1. Why don't the orbiting electrons radiate (since accelerating charges in classical electromagnetism should radiate and spiral in)? This was solved by Bohr's 1913 quantised-orbit model, see the related dot point.
2. What balances the Coulomb repulsion between the protons inside the nucleus, and why does the nucleus appear to have more mass than just $Z$ protons? Both pointed to a neutral nuclear constituent.

Chadwick's discovery of the neutron (1932)

Rutherford had postulated as early as 1920 that the nucleus contained, in addition to protons, neutral particles of similar mass that he called "neutrons". The decisive evidence came from a chain of experiments.

The puzzle. Bothe and Becker (1930) observed that bombarding beryllium with alpha particles produced a highly penetrating neutral radiation that, until 1932, was assumed to be high-energy gamma rays. Curie and Joliot (1932) showed that this radiation could eject protons from paraffin wax with surprisingly high kinetic energies.

Chadwick's experiment. James Chadwick (1932) sent the neutral radiation onto various target nuclei (hydrogen, helium, lithium, nitrogen) and measured the recoil kinetic energies of each target. Using conservation of energy and momentum, he tested two hypotheses.

• Hypothesis A: the neutral radiation is gamma rays. To produce the observed nitrogen recoils, the photons would have had to carry about 50 MeV of energy, far more than energetically possible from the beryllium-alpha reaction (which has an available energy of only a few MeV).
• Hypothesis B: the neutral radiation is a stream of massive neutral particles. Treating the collisions as elastic billiard-ball collisions, the kinetic energies of the recoils from different targets were consistent only if the projectile had a mass close to that of the proton.

The neutron hypothesis fitted all the data. Chadwick concluded that beryllium plus alpha gives carbon plus a neutron:

The neutron's mass was later measured as $1.0087$ u, slightly greater than the proton's $1.0073$ u. It has no electric charge.

Mass-and-momentum analysis (sketch)

For a head-on elastic collision of a particle of mass $m$, speed $v_0$, with a stationary target of mass $M$, the target recoils with speed:

Chadwick measured $v$ for hydrogen targets ($M = 1$ u) and for nitrogen targets ($M = 14$ u). The ratio of recoil speeds depends only on $m$ (not $v_0$):

Inserting his measured speeds and solving gave $m \approx 1$ u, confirming the neutron mass close to the proton mass.

The completed atomic picture

After Chadwick:

• The nucleus contains $Z$ protons and $N$ neutrons (collectively, $A = Z + N$ nucleons).
• The nucleus is held together by the strong nuclear force, which acts over very short distances and is independent of electric charge.
• Electrons in number $Z$ surround the nucleus, balancing the charge in a neutral atom.

This sets the stage for the rest of Module 8: nuclear stability (binding energy), radioactive decay (alpha, beta, gamma), and ultimately the quark structure of the nucleons.

Examples in context

Example 1. Rutherford backscatter at the Australian Synchrotron. A $5.5 \text{ MeV}$ alpha particle approaches a gold nucleus ($Z = 79$) head-on. At distance of closest approach $r_{\min}$, kinetic energy converts entirely to electrostatic PE: $E_k = (1/4\pi\varepsilon_0) \cdot 2 Z e^2 / r_{\min}$. Solving: $r_{\min} = (1/4\pi\varepsilon_0) \cdot 2 \times 79 \times (1.6 \times 10^{-19})^2 / (5.5 \times 1.6 \times 10^{-13}) = 9.0 \times 10^{14} \cdot 4.05 \times 10^{-38} / 8.8 \times 10^{-13} = 4.1 \times 10^{-14} \text{ m} = 41 \text{ fm}$, smaller than gold's nuclear radius. This is why Rutherford's 1911 result demanded a tiny dense nucleus: only point-like charges could produce the $\sim 1$ in $20{,}000$ backward-scattered alphas Geiger and Marsden saw.

Example 2. Chadwick's neutron analysis applied to an ANSTO Lucas Heights $\alpha$ + $^9$Be source. An alpha hits $^9$Be: $^4_2 \text{He} + {}^9_4 \text{Be} \to {}^{12}_6 \text{C} + n$. Energy conservation: incident $\alpha$ at $5.30 \text{ MeV}$, products at rest in CM frame approximately. The $n$ recoils carrying $\sim 5.7 \text{ MeV}$ of KE. Chadwick (1932) measured the recoil speed of struck protons in paraffin ($3.3 \times 10^7 \text{ m/s}$) and nitrogen ($4.7 \times 10^6 \text{ m/s}$). From conservation of momentum and KE in elastic collisions, the unknown particle mass solved to $\approx 1.16$ proton masses - neutral and nucleon-like. ANSTO neutron sources today use the same $\alpha$ + Be reaction.

Try this

Q1. State the conclusion drawn by Rutherford from the Geiger-Marsden gold-foil experiment. [2 marks]

• Cue. Atom has a small, dense, positively charged nucleus, with most of the atom empty space.

Q2. Calculate the distance of closest approach of a $4.0 \text{ MeV}$ alpha particle to a gold nucleus ($Z = 79$). [3 marks]

• Cue. $E_k = k \cdot 2 Z e^2 / r$; $r = 8.99 \times 10^9 \times 2 \times 79 \times (1.6 \times 10^{-19})^2 / (4.0 \times 1.6 \times 10^{-13}) = 5.7 \times 10^{-14} \text{ m}$.

Q3. Chadwick's 1932 discovery completed the proton-neutron-electron picture of the atom. (a) Write the nuclear reaction used by Chadwick. (b) Explain how he deduced the neutron's mass using momentum conservation in collisions with protons and nitrogen nuclei. (c) State two properties of the neutron that distinguish it from the proton. [2+3+2 marks]

• Cue. (a) $^4_2 \text{He} + {}^9_4 \text{Be} \to {}^{12}_6 \text{C} + n$. (b) Elastic-collision kinematics: speeds of recoiling H and N nuclei give mass ratio; $m_n \approx m_p$. (c) Neutral charge; slightly larger mass ($1.008 \text{ u}$ vs $1.007 \text{ u}$); unstable when free ($T_{1/2} = 10.2 \text{ min}$).