Thomson's Plum-Pudding Model

After J. J. Thomson discovered the electron in 1897, the next question was: how are electrons arranged inside the neutral atom? In 1903 Thomson proposed a model in which the atom is a sphere of uniformly distributed positive charge, with tiny negative electrons embedded inside, like raisins in a pudding. The model was the first quantitative attempt to picture atomic structure and could explain a few facts (overall neutrality, emission of electrons under heating), but it failed dramatically when subjected to the alpha-scattering test.

Concept

Thomson's model postulates:

• The atom is a uniform sphere of radius $R \approx 10^{-10}$ m carrying total positive charge $+Ze$.
• $Z$ point electrons of charge $-e$ are embedded inside, free to move.
• In equilibrium the electrons arrange themselves so that the net electric force on each is zero (Thomson predicted ring-like equilibrium positions for several electrons).
• The atom as a whole is electrically neutral.

The positive cloud has volume charge density

$\rho = \frac{Ze}{\tfrac{4}{3}\pi R^3}$

For a single electron at distance $r$ from the centre (with $r < R$), only the positive charge enclosed within $r$ contributes to the force (Gauss's law for a uniform sphere):

$F(r) = \frac{1}{4\pi\varepsilon_0}\frac{(Ze)(e)\, r^3 / R^3}{r^2} = \frac{Ze^2}{4\pi\varepsilon_0 R^3}\, r$

This is a linear restoring force — Thomson's electron behaves like a 3D harmonic oscillator.

Derivation

Electron oscillation frequency in the Thomson sphere. Setting $F = m_e a$ for an electron displaced by $r$ from the centre:

$m_e \ddot r = -\frac{Ze^2}{4\pi\varepsilon_0 R^3}\, r$

Comparing with $\ddot r = -\omega^2 r$:

$\omega = \sqrt{\frac{Ze^2}{4\pi\varepsilon_0 m_e R^3}}$

For hydrogen ($Z=1$, $R \approx 10^{-10}$ m), plugging numbers gives $\omega \sim 10^{16}$ rad/s, corresponding to ultraviolet light. Thomson hoped this would explain spectral lines, but real hydrogen emits a whole series of discrete lines, not a single frequency.

Force on an alpha particle. For Rutherford's later experiment, what scattering does Thomson's model predict? An alpha particle of charge $+2e$ passing through such a diffuse positive blob feels at most a force of order

$F_{\max} \sim \frac{1}{4\pi\varepsilon_0}\frac{(2e)(Ze)}{R^2}$

For gold ($Z=79$, $R \sim 10^{-10}$ m) this is small enough that the deflection of an MeV alpha is at most $\sim 1^\circ$. Thomson's model therefore predicts no large-angle scattering — the very prediction Rutherford falsified.

Worked Example

Estimate the maximum deflection angle of an alpha particle (energy $5$ MeV) passing through a Thomson-style gold atom ($R = 10^{-10}$ m, $Z = 79$).

The impulse delivered during transit time $\Delta t \approx 2R/v$ (where $v = \sqrt{2E/m_\alpha} \approx 1.55\times 10^7$ m/s) is roughly

$\Delta p \sim F_{\max} \Delta t \sim \frac{(2)(79)e^2}{4\pi\varepsilon_0 R^2}\cdot\frac{2R}{v} = \frac{4(79)e^2}{4\pi\varepsilon_0 R v}$

Numerically $\Delta p \sim 2\times 10^{-24}$ kg m/s, while $p = m_\alpha v \sim 10^{-19}$ kg m/s. Hence

$\theta \sim \Delta p / p \sim 2\times 10^{-5}\;\text{rad}\approx 0.001^\circ$

In reality 1 in $\sim 8000$ alphas were deflected by more than $90^\circ$. Thomson's model is wrong by orders of magnitude.

Common Confusions

• Thomson's model has no nucleus. Positive charge is spread over the entire atomic volume, not concentrated at the centre.
• "Plum pudding" does not mean random arrangement. Thomson actually computed stable ring configurations of electrons; the picture of randomly stuck-in plums is a cartoon.
• The model isn't useless — it explained ionisation and approximate atomic size. It was simply incompatible with high-angle scattering and discrete spectra.
• Do not confuse Thomson's atomic model with his $e/m$ measurement, which is independent of the model.

Key Takeaways

• Thomson (1903): atom = uniform positive sphere with embedded electrons.
• Electrons inside a uniform positive sphere feel a linear restoring force and oscillate at a single frequency — predicting only one spectral line.
• The model predicts only tiny ($\sim 0.01^\circ$) deflections of alpha particles.
• Discrete spectra and large-angle alpha scattering both falsified Thomson's model, paving the way for Rutherford's nuclear atom.