Rutherford Scattering

Source

Lecture: L36 — Central-force scattering & Rutherford's experiment
Simulation: Orbital core with central potential switched to repulsive Coulomb
Concept: A repulsive 1/r potential deflects incoming particles into hyperbolic trajectories. Explore how the deflection angle ψ depends on the impact parameter b and the incoming speed v.
Serves: L36: relate deflection angle to impact parameter b and speed v; investigate what happens as b → 0.
Inquiry: How does ψ depend on b at fixed v? On v at fixed b? What is the closest approach as b → 0?
Units: Natural units throughout: coupling k = 1, mass m = 1. Distances, speeds and angles are dimensionless numbers. b and v are the asymptotic values — measured far from the nucleus, where V = k/r is negligible.

Pedagogy principles applied

ID	Principle	How it shows up here
H2	Structural misconception	Step 4 pairs a wrong "contact required" choice with a canvas overlay comparing the naive straight-line-to-contact path against the true reversing trajectory.
G4	Predict-observe-explain	Each gated step commits a prediction before Fire is enabled; the resulting track and table row are the observation.
A2/C1	Numbers earn their place	Only b, v, ψ, tan(ψ/2), 1/(bv²), r_min appear — each is read by a step.
D2	Labelled symbols	b is drawn as a labelled perpendicular on the aim line; ψ is arced at the exit direction.
E1	Animation honesty	Each shot is integrated once with adaptive RK4, then replayed so the particle decelerates into r_min and accelerates out symmetrically; overall rate scales with the speed control.

Teach me Hide Text

Trajectory (x–y plane)

scattering centre

incoming aim line (b)

current track (ψ)

past tracks

Drag the aim line to set impact parameter b

Guided Inquiry

1 — Orient yourself

Let’s get our bearings first. The red dot is the fixed scattering centre, and the blue dashed line is your aim line.

The aim line’s perpendicular distance from the centre is called the impact parameter b. And the angle between a particle’s outgoing and incoming velocity is its scattering angle ψ.

Give it a try: drag the aim line up or down (or use the b slider), then press Fire. A shot must be completed before you can continue.

2 — Predict: b-dependence

Let’s start with the aim. You’ll fire several particles at the same speed v but with different b. What do you think happens to ψ as b increases?

Now fire a spread of b values and check your answer against the collected table.

3 — Predict: v-dependence

Now for the other knob. Keep b fixed and increase v. What do you expect to happen to ψ this time?

Fire several shots at different v, keeping the same b. Can you spot the quantitative pattern the numbers follow?

4 — Limit & resolve

Now push it to the limit. Aim straight at the centre (b → 0) at a moderate v. What do you think — does the particle ever reach r = 0?

Compare the two paths: the dashed grey one shows the "naive contact" picture, and the yellow one shows what actually happens.

5 — Rare events

One last question, and it’s a big one. A wide beam sprays particles at the target, with every impact parameter equally likely across the area it covers. Among a million of them, which outcome do you think is rare?

Fire a spread of b to see the trend. Just remember it fires seven fixed b values, not a random beam, so it shows you the trend rather than the statistics.

Listen

Controls

▾

Impact parameter b (far from the nucleus) 0.50

0.003.00

Incoming speed v (far from the nucleus) 1.00

0.602.50

Live readouts

▾

ψ (last shot)—

rmin (last)—

Results table

▾

b	v	ψ (°)	r_min	tan(ψ/2)	1/bv²
no shots yet

The last two columns should be proportional. Fire a spread and compare.

Repulsive Coulomb potential

Natural units k = 1, m = 1. Force is F = k/r² directed away from the centre.

Scattering relation (ψ = angle between outgoing and incoming velocity)

Verified per shot to within numerical error. Plot 1/(mbv²) vs tan(ψ/2) — slope k.

Closest approach

Finite for any finite energy — reduces to 2k/(mv²) at b = 0. The particle never reaches r = 0.

Trajectory (repulsive hyperbola, far branch)

All bound-free trajectories in a 1/r potential are conic sections; the repulsive case gives the far branch of a hyperbola.

The Physics

Scattering · what this sim shows, the physics behind it, and every control

At a glance

A charged particle fired past a fixed, repelling centre is deflected by the force alone. This sim lets you choose the aim and the speed of each shot and measures how far it is turned, the kind of measurement that revealed the atomic nucleus in 1911.

What you'll learn

How the scattering angle ψ depends on the aim b and the speed v.
What sets the closest approach r_min of a shot.
What happens when you aim straight at the centre (b = 0).
How a wide beam of particles is shared among small and large deflections.

A shot arrives along the aim line, at distance b from the centre and speed v, and leaves turned through ψ.

What's on screen

Scattering centre (red dot): fixed in place, and it repels the particle.
Aim line (dashed blue): the incoming path; its distance from the centre is b. Drag it to change b.
Current track (yellow): the shot in flight. When it ends, a short arc marks ψ.
Past tracks (grey): every earlier shot, until you clear them.
Live readouts and Results table: ψ and r_min of the last shot, and one row for every shot.

See itFire one shot at b = 0.5, v = 1. Watch for: the yellow track curving away from the red centre, and the ψ readout appearing when the shot ends.

The controls

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

Control	What it changes	Range
Impact parameter b	The perpendicular distance from the centre to the aim line, measured far from the centre. You can also drag the aim line on the stage.	0–3 opens at 0.50
Incoming speed v	The particle's speed far from the centre.	0.6–2.5 opens at 1.00
▸ Fire	Fires one shot with the current b and v.	button
Fire spread of b	Fires seven shots at once at the current v, with b = 0.2, 0.4, 0.7, 1.0, 1.5, 2.0 and 2.5, and adds them to the Results table.	button
Clear tracks	Removes every track and empties the readouts and the table.	button
Live readouts	ψ and r_min of the most recent shot.	panel
Results table	One row per shot: b, v, ψ, r_min, tan(ψ/2) and 1/bv².	panel
▶ Play / ⏸ Pause	Freezes a shot in flight. With none in flight, Play fires one.	top bar
Speed	Replays each shot faster or slower on screen. The angle is unaffected.	0.5×–4×
↻ Reset	Restores b = 0.5 and v = 1, clears the stage, and fires one default shot.	top bar

Fixed: k = m = 1 (natural units), and the centre does not recoil. Each shot is replayed in about 7 s whatever v is, so v cannot be read from how fast the dot moves.

The force and the angle

The centre repels the particle with the Coulomb force. In the sim's natural units (k = 1, m = 1):

The force is felt at every distance. It is weak far away and grows quickly as the particle comes closer.
b and v are measured far away, where the force is negligible and the path is a straight line: b is the perpendicular distance from the centre to that line, and v is the speed along it.
The particle swings past and leaves along a second straight line. The angle between its outgoing and incoming velocity is the scattering angle ψ.
Every path in this force is one branch of a hyperbola.

See itAim far from the centre: b = 2.5, v = 1. Watch for: the track still bending, ending at ψ = 43.6° in the readout.

How b and v set the angle

Energy and angular momentum conservation give the angle in one line (L = mbv is the angular momentum):

b sits in the denominator once, v twice. Doubling b halves tan(ψ/2); doubling v divides it by four.
A smaller tan(ψ/2) means a smaller ψ, so aiming wider or going faster turns the shot less.
Only the product b·v² matters: two shots with the same b·v² leave at the same angle.
In the Results table, tan(ψ/2) and 1/bv² stay proportional, and their ratio is k/m.

b v

These sliders move the sim too.

See itFire the spread of b at v = 1. Watch for: seven tracks fanning out, the closest aim turned back the most (ψ = 157.4° at b = 0.2, 43.6° at b = 2.5).

See itFire b = 0.5 at v = 1, then at v = 2. Watch for: the same aim turned through a much smaller angle at the higher speed, and both ψ values on the card.

The closest approach

Along the path the energy E = ½mv² and the angular momentum L = mbv stay constant. At the closest approach the velocity is perpendicular to the line to the centre, and the two conservation laws give:

e is the eccentricity of the hyperbola. It is at least 1, and equals 1 only when b = 0.
r_min is always larger than b. At closest approach part of the energy is stored as potential energy, so the speed is below v; with L = mbv fixed, the shot must be farther out than b.
Faster shots get closer to the centre. A wider aim keeps the shot farther away.

The default shot (b = 0.5, v = 1): closest approach r_min = 2.12, and ψ = 126.9°.

See itFire b = 1.5 at v = 1. Watch for: the r_min readout at 2.80, larger than the aim distance b = 1.5.

A head-on shot never reaches the centre

At b = 0 there is no angular momentum, so the shot runs straight at the centre and climbs the potential hill V = k/r. It stops where all its kinetic energy has become potential energy:

½mv² = k / r_min ⇒ r_min = 2k / (mv²)

V grows without limit as r → 0, so any shot with finite energy stops first. At v = 1, r_min = 2.00; even at the sim's top speed (v = 2.5) it is 0.32.
The shot then retraces its path: ψ = 180° with no contact.
A head-on shot is also the closest approach at its speed: for b > 0, r_min = (k/mv²)(1 + e) is larger.

Key idea. Coming straight back is not a collision. The repulsion turns the particle round before it gets there.

See itAim dead centre: b = 0, v = 1. Watch for: the track going in and coming back out along the same line, with ψ = 180.0° and r_min = 2.00 in the readouts.

Large deflections are rare

A wide, uniform beam sends equal numbers of particles through equal areas. A shot turns by more than ψ only if it is aimed inside the circle b < k / (mv² tan(ψ/2)). That circle's area is the cross-section σ = πb².

turned by more than	aim inside b <	target area πb²	compared with 90°
90°	1.00	3.14	1
150°	0.27	0.23	14 times smaller
170°	0.09	0.024	130 times smaller

At v = 1, k = m = 1. Doubling v shrinks every one of these areas by a factor of 16.

Large angles need a small bullseye, so they are rare. Small angles come from the large area outside.
The sim's spread fires seven chosen values of b, not a random beam. It shows the trend, not the counts.

See itFire b = 1.00, then b = 0.27, at v = 1. Watch for: ψ = 90.0° for the first shot and 149.8° for the second, with the area ratio on the card.

Where it matters

The nucleus (1911). Geiger and Marsden found that of order 1 in 10⁴ α-particles fired at a thin metal foil came back by more than 90°. Rutherford showed this needs a tiny, dense, positive core: head-on, a 7.7 MeV α-particle on gold stops no closer than about 30 fm, thousands of times smaller than the atom.
Rutherford backscattering spectrometry. Ions of a few MeV fired at a thin film, and counted by energy as they bounce back, reveal which elements are present and at what depth.
Spacecraft flybys. Gravity is the attractive version of the same 1/r² force. The path bends toward the planet, and tan(ψ/2) keeps the same form, with k = GMm.

Head-on closest approach for a 7.7 MeV α-particle on gold, in real units (charges 2e and 79e). It is an upper bound on the nuclear size.

Go deeper: the Rutherford cross-section

Counting particles per unit solid angle, with E = ½mv², gives dσ/dΩ = (k/4E)² / sin⁴(ψ/2). It rises steeply toward small angles, matching section 07. This holds only while the particle stays outside the nucleus. At high enough energy it reaches the nuclear force and the Coulomb law fails. The same formula follows from quantum mechanics.

Summary

In one line. A repelling 1/r centre turns a shot by ψ with tan(ψ/2) = k/(mbv²); the shot always stops short of the centre, and large angles come from a small target area.

Key points

tan(ψ/2) = k/(mbv²): a smaller b or a smaller v gives a larger ψ.
The closest approach r_min = (k/mv²)(1 + e) exceeds b and is at least 2k/mv²; at b = 0 it equals 2k/mv².
The cross-section for ψ above a given angle is πb², so large deflections are rare.

Common mix-ups

"A particle that comes straight back must have hit something." The repulsion turned it round at r_min = 2k/mv², before it reached the centre.
"Faster particles are deflected more, because they carry more energy." They spend less time near the centre and are turned less: tan(ψ/2) falls as 1/v².
"Every deflection angle is equally likely." Likelihood follows target area πb², and large angles need a small b.

Things to try on your own

Fire a spread at v = 1 and check in the Results table that tan(ψ/2) tracks 1/bv².
Fire head-on shots at v = 0.6 and v = 2.5, and compare each r_min with 2/v².
Find another (b, v) with about the same ψ as the default shot: keep b·v² near 0.5, e.g. v = 1.41, b = 0.25.