Geiger–Marsden Gold Foil Experiment

Concept: Scintillation counts at scattering angle θ follow 1/sin⁴(θ/2), so they fall by about 2×10⁵ between 5° and 150° yet never vanish. Because roughly 1 in 26,000 incident α turns through more than 90° — impossible if the positive charge were spread through the whole atom — that charge must sit in a tiny, massive nucleus.
Sub-question: Nuclear size: the head-on closest approach is d_min = zZ·α_em·ħc/Eₖ = 45.5 fm at 5 MeV. Raising the beam momentum shrinks d_min; when the large-angle counts sag below the dashed point-Coulomb curve the α is grazing the nuclear surface at d_touch = R_Au + R_α ≈ 8.9 fm, so R_Au ≈ 7 fm (diameter ≈ 1.4×10⁻¹⁴ m).
Model notes: Plum pudding is modelled as ≈2.5×10³ random ≈0.02° kicks, so the random walk gives σ = √2500 × 0.02° ≈ 1° at 5 MeV and essentially no counts past a few degrees. The measured (green) points are always generated from the nuclear model — the switch only chooses which prediction curve is drawn over them. Single scattering only: sub-degree multiple scattering is not simulated. Foil: 0.40 µm gold, n·t = 2.4×10²² m⁻².
Serves: Connect the Geiger–Marsden scintillation pattern to the Rutherford scattering formula and contrast it with the plum-pudding expectation.
Misconception: "Positive charge fills the whole atom, so α particles can only be nudged by fractions of a degree — never bounced back."

Principle	Where it appears
Predict first (never gated)	The sim opens paused with an empty screen; the Predict panel sits above every readout. Play is prominent and enabled from load — nothing is locked.
Data are the data	The green points are always generated from the nuclear model. The model switch changes only the overlaid prediction curve, so a wrong model can never "confirm" itself.
Cognitive conflict	The dashed plum-pudding prediction plunges off the bottom of the plot by ~7° while measured θ>90° counts keep arriving.
Invariant made visible	The ×sin⁴(θ/2) toggle collapses the Rutherford data onto one horizontal line at any momentum — the constancy Geiger and Marsden tabulated.
Operational anchors	Live counters: incident α fired, α scattered ≥1°, counts θ>90°, odds per incident α, Eₖ and d_min.
Formal layer on demand	Cross-section, d_min, q, the n·t conversion and the model's stated limits live in the ⚛︎ The Physics revision sheet, opened from the top bar.

Teach me Hide Text

Speed

Geiger–Marsden scattering chamber — counts vs scattering angle

Guided Inquiry

1 · The 1909 apparatus

Here’s the setup Geiger and Marsden built in 1909. An α source fires 5 MeV α particles at a gold foil just 0.40 µm thick, and a ring of ZnS screen flashes wherever one lands. The question they were chasing: how is the positive charge arranged inside an atom?

2 · A spread-out charge

Let’s start with the leading idea of the day. Thomson’s model spreads the atom’s positive charge through its whole volume, so an α passing through feels lots of tiny kicks in random directions. What do you think — how far from the beam can those kicks send it?

3 · Past 90°

Now push that idea as far as it goes. If those tiny random kicks are the only thing deflecting the α, how many do you think will ever turn right around, past 90°?

4 · What gold must hold

Now take a look at the counts past 90° in this run. If random kicks can’t do it, what single feature of the gold atom could turn an α right around?

5 · The angular shape

Let’s test that idea against the data. If the scatterer really is a point charge, what happens to the plot when you multiply every measured count by sin⁴(θ/2)?

6 · Closer approach

One last experiment. Raising the beam momentum from 0.193 to 0.500 GeV/c shrinks how close a head-on α can get. So when that closest distance falls inside the gold nucleus, what do you expect the large-angle counts to do?

Listen

Controls

▾

Beam rate

5.0 MeV α on a 0.40 µm gold foil (n·t = 2.4×10²² m⁻²). At 10 k α/s the rare flashes past 90° arrive about one every few seconds — Geiger and Marsden counted them by eye in a darkened room. Use 100 k α/s to build the pattern quickly.

Show counts-vs-angle plot

Plot style

α momentum 0.193 GeV/c

Natural α sources run 4–9 MeV (RaC′ 7.7 MeV = 0.240 GeV/c).

Readouts

▾

α fired (incident)0

scattered ≥ 1°0

counts θ > 90°0

absorbed by nucleus0

odds of θ > 90°—

kinetic energy Eₖ5.0 MeV

d_min (head-on, θ = 180°)45.5 fm

Odds are per incident α. The 1° line is only where this model starts counting; sub-degree deflections are not simulated.

Screen: upper and lower arcs are opposite azimuthal sides of the same θ.

dσ/dΩ = (zZ·α_em·ħc / 4Eₖ)² · 1/sin⁴(θ/2)

d_min = zZ·α_em·ħc/Eₖ , q = 2p·sin(θ/2) , Eₖ = p²/2m_α

dN / (N₀ dΩ) = n·t · dσ/dΩ

z = 2 (α charge number), Z = 79 (gold), Eₖ = 5.0 MeV, d_min = 45.5 fm

θ is the scattering angle in the LAB frame (lab ≈ CM to 2 % for α on ¹⁹⁷Au); the upper and lower screen arcs are opposite azimuthal sides of the same θ. Solid curve = Rutherford × a phenomenological nuclear-absorption factor: a logistic cut-off of 0.4 fm width removes α whose closest approach reaches d_touch = R_Au + R_α ≈ 8.9 fm (⇒ R_Au ≈ 7 fm). Single scattering only; foil n·t = 2.4×10²² m⁻² (0.40 µm gold).

The Physics

The Geiger–Marsden experiment · what this sim shows, the physics behind it, and every control

At a glance

In 1909 Geiger and Marsden fired α particles at a thin gold foil and counted the flashes they made on a screen, at every angle. This sim repeats that count, lets you change the α energy, and compares it with two pictures of the atom.

What you'll learn

How the number of counts depends on the angle θ through which an α is turned.
What a spread-out positive charge and a concentrated one each predict for large angles.
How the α energy sets the closest distance d_min an α can reach.
How small a structure this kind of experiment can resolve.

The 1909 set-up. An α leaves the foil turned through θ and lands on a screen that flashes where it hits.

What's on screen

α beam (azure): a sample of the α in flight, from the source to the foil.
ZnS screen (green ring, ticks every 30°): each dot is where an α landed. The upper and lower arcs show the same θ.
Counts-vs-angle plot: green bars with ±√N error bars, on a log axis from 10⁻⁸ to 10².
Solid teal curve: the Rutherford prediction, named in the plot legend.
Dashed grey curve: a second prediction, labelled plum pudding.

See itFire the beam at 100 k α/s. Watch for: green dots crowding near the beam direction, and "scattered ≥ 1°" settling at about half of "α fired".

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Beam rate	How many α arrive per second. At 10 k α/s a count past 90° is rare, so 100 k α/s fills the plot faster.	10 k or 100 k α/s opens at 10 k
Show counts-vs-angle plot	Shows or hides the plot. The plot appears once the first α have arrived.	on / off opens on
Plot style	Histogram draws a bar over each 5° bin with ±√N error bars. Points draws one marker per bin.	Histogram / Points opens on Histogram
α momentum	The momentum p of every α, which sets its kinetic energy E_k = p²/2m_α. Changing it clears the run and starts a new experiment.	0.185–0.500 GeV/c opens at 0.193 (5.0 MeV)
Readouts	α fired, scattered ≥ 1°, counts θ > 90°, the odds of θ > 90° (shown as "~1 in" until there are 3 counts), E_k, and d_min, the closest approach for a head-on α.	panel opens collapsed
▶ Play / ⏸ Pause	Starts or freezes the beam. The sim opens paused and empty.	top bar
Speed	Scales the time step, so the beam delivers α faster or slower. The physics is unchanged.	0.25×–4× opens at 1×
↻ Reset	Returns to p = 0.193 GeV/c and 10 k α/s with an empty, paused run.	top bar

Fixed: 0.40 µm gold foil (n·t = 2.4×10²² atoms per m²), α charge z = 2, gold Z = 79, single scattering, counts from 1°. The dashed overlay and the special plot views are set by the Guided Inquiry and the See-it buttons, not by a control.

What the sim counts

The plot shows counts per α fired, per steradian, against the angle θ between an α's path before and after the foil. A steradian is a unit of solid angle. Each count is one flash on the screen.

The expected count per α is n·t·dσ/dΩ. Here n·t is the number of gold atoms per area of foil, and the cross-section dσ/dΩ is the target area, per steradian, that turns an α toward θ.
The sim counts single scatterings only, and starts at 1°. At 5.0 MeV about half of all α are turned by at least 1°. That is the "scattered ≥ 1°" readout.
Each bar carries a ±√N error bar. A bin with 100 counts is known to about ±10%. A bin with one count is only a hint.

See itStart from 30,000 α fired, with the beam running at 10 k α/s. Watch for: short error bars on the tall forward bars, long ones on the sparse large-angle bins, and "scattered ≥ 1°" near half of "α fired".

A spread-out charge turns an α by about 1°

In Thomson's plum-pudding atom the positive charge fills the whole atom, so every push on an α is small. The overlay assumes about 2.5×10³ random kicks of about 0.02° each. Random kicks partly cancel: their sum grows as √N, not as N.

The spread of angles after N_kick kicks, and the chance of a turn beyond θ₀.

The typical turn is σ ≈ 1° at 5.0 MeV. It scales as 1/E_k.
The tail falls like a Gaussian. At θ₀ = 90° the exponent is about −4000, a probability near 10⁻¹⁷⁵⁰.
On the plot the dashed curve drops below the 10⁻⁸ floor by about 7°. It is a prediction drawn for comparison. The sim never simulates a plum-pudding atom.

See itFire 3 million α with the dashed curve on. Watch for: green bars continuing out to 180° and about 115 counts past 90°, far beyond where the dashed curve stops (near 7°).

A concentrated charge gives 1/sin⁴(θ/2)

If all the positive charge sits in a tiny core, an α that passes close feels a strong repulsion, and one close pass can turn it a long way. Rutherford's cross-section is the target area for each angle:

The Rutherford cross-section, split into its strength and its angular shape.

Two exact Coulomb orbits at 5.0 MeV. The closer the aim, the larger the turn.

At 5.0 MeV, a miss distance of 22.8 fm turns an α by 90°, and 6.1 fm turns it by 150°.
From 5° to 150° the factor 1/sin⁴(θ/2) falls by 2.4×10⁵. It is steep, and it never reaches zero.
This gives about 1 α in 26,000 turned past 90° at 5.0 MeV, so 3 million α give about 115 counts.
The strength goes as (zZ/E_k)², so doubling p gives 16 times fewer counts at every angle.

See itFire 3 million α at 5.0 MeV. Watch for: the green bars following the solid curve out to 180°, with about 115 counts past 90°.

How close an α gets, and what it resolves

A head-on α (θ = 180°) stops where its kinetic energy has all become Coulomb energy: d_min = zZα_emħc / E_k. It falls as 1/E_k.

p rate

These move the sim's own p and beam-rate controls. Changing p restarts the run.

At 5.0 MeV, d_min = 45.5 fm, about five times the 8.9 fm at which an α touches a gold nucleus (R_Au ≈ 7 fm plus R_α ≈ 1.9 fm).
Multiply every count by sin⁴(θ/2) and a point charge gives one flat line. The sim's flat line says the charge looks like a point from 45 fm outward. It says nothing about structure inside 45 fm.

See itShow the counts multiplied by sin⁴(θ/2). Watch for: the bars and the curve lying on one flat horizontal line.

Reaching the nucleus changes the tail

A head-on α reaches the 8.9 fm touching distance when d_min = 8.9 fm, at E_k ≈ 25.6 MeV (p ≈ 0.437 GeV/c). Rutherford's formula assumes a point charge, so beyond that point it can no longer be exact.

For a turn of θ, b(θ) is the miss distance and d(θ) the closest approach.

Every angle has its own closest approach d(θ), and the smallest is d_min. A large-angle α gets closer than a small-angle one.
The sim keeps an α with a probability that falls from 1 to 0 as d(θ) drops through 8.9 fm, over about 0.4 fm. That is a simple cut-off, not a nuclear-force calculation.
At 0.500 GeV/c (33.5 MeV, d_min = 6.8 fm), α turned beyond about 90° are mostly lost, so the tail sags below the dashed point-Coulomb curve. From p ≈ 0.376 GeV/c (d_min < 12 fm) the sim swaps the plum-pudding curve for this point-Coulomb reference.

See itFire 3 million α at 0.500 GeV/c. Watch for: the solid curve bending below the dashed one, and no counts past 90°.

Where it matters

1909–1911. Geiger and Marsden found that of order 1 in 10⁴ α were turned back by a thin metal foil. Rutherford explained this in 1911 with a nucleus. The sim gives 1 in 26,000 for this foil at 5.0 MeV.
Nuclear size. A 7.7 MeV α, from a natural source, comes within 29.5 fm of a gold nucleus head-on. Counts that follow Rutherford's law at that energy put the nucleus inside this distance. Gold's nucleus is about 7 fm, in an atom about 10⁴ times wider.
Materials analysis. Rutherford backscattering sends MeV helium ions at a solid and reads composition and depth from the energies and counts of the ions that return.
Inside the proton. In the late 1960s electron scattering at SLAC used the same strategy: compare the angular pattern with a point-charge prediction. The departures showed point-like constituents in the proton.

Go deeper: small angles and screening

The Coulomb force has no range, so dσ/dΩ grows without limit as θ → 0 and the total cross-section is infinite. In a real atom the electrons screen the nucleus beyond about the atom's size, which cuts the divergence off. The sim does not model the electrons and starts counting at 1°.

Summary

In one line. A concentrated charge gives counts that fall as 1/sin⁴(θ/2) out to 180°, a spread-out one gives none past a few degrees, and the α energy sets how close an α can come.

Key points

Counts per α per steradian are n·t·dσ/dΩ, with dσ/dΩ = (zZα_emħc/4E_k)² / sin⁴(θ/2).
The closest approach is d_min = zZα_emħc/E_k, 45.5 fm at 5.0 MeV.
A flat sin⁴(θ/2)-weighted plot means a point charge, from d_min outward.
Once d_min is below 8.9 fm, the large-angle tail sags.

Common mix-ups

"An α that turns back has hit the nucleus." At 5.0 MeV it turns around 45.5 fm out, about 6.5 gold radii away. The push is Coulomb repulsion at a distance.
"A faster α scatters more." The cross-section goes as 1/E_k², so a faster α is turned less often.
"The dashed curve is a second data set." It is a prediction. All the green counts come from the sim's Rutherford model.

Things to try on your own

Set p to 0.240 GeV/c (7.7 MeV) and fire 3 million α. Expect about 48 counts past 90° (1 in 62,000), against 1 in 26,000 at 5.0 MeV.
Set p to 0.386 GeV/c, twice the opening value, and fire 3 million α. Expect about 7 counts past 90° instead of 115.
Raise p from 0.40 GeV/c until the sim labels "α now grazes the nuclear surface", at p ≈ 0.437 GeV/c.