In this episode, students look in detail at Rutherford’s experiment and relate it to a mechanical analogue.

Lesson Summary

• Discussion: Recollecting the significance of Rutherford’s experiment (10 minutes)
• Discussion: Rutherford’s experiment (20 minutes)
• Demonstration: Collisions and momentum (10 minutes)
• Discussion: Rutherford scattering and Coulomb’s law (10 minutes)
• Student experiment or demonstration: The Gravitational model (30 minutes)
• Student questions: Rutherford experiment and atomic structure (optional) (20 minutes)
• Question: Rutherford’s results (optional)
• Discussion: Models in physics (10 minutes)

Discussion: Recollecting the significance of Rutherford’s experiment

As a preparatory task, ask your students to revise what they have previously learned about Rutherford’s α-scattering experiment. What idea of the atom did it suggest? (The nuclear model.) What model of the atom did this replace? (Thomson’s plum pudding model, in which atoms are seen as essentially small balls composed of a mixture of positive and negative electric charge, with no concentration of charge at any particular position.)

Is plum pudding a good name for the model? (Yes, if you see the negative electrons dispersed throughout a spherical lump of continuous positive charge, not so good if the volume of the atom has both positive and negative particles continuously distributed through it – students may well be recalling different pictures from different sources.)

Discussion: Rutherford’s experiment

Now you can present a more advanced exposition of the experiment. Why did Rutherford ask for the experiment to be done? Experiments on the absorption of β particles had also shown that sometimes the β particles were back scattered. Rutherford suggested that Geiger and Marsden should try looking for similar behaviour with α particles. Rutherford thought it was highly unlikely; because α particles are relatively massive compared with electrons, it was predicted that the αs would simply suffer a series of small deflections. They were expected to travel more or less straight through the absorber.However, Rutherford’s main concern was to give Geiger and Marsden something to do that would occupy them and get them some useful hands-on experience, rather than expecting them to get any very exciting results.

Show a diagram of the apparatus. The absorber was a thin gold metal foil. Why use gold? (Thin gold foils, typically 250 atoms thick, were easy to make and readily available.) Why thin? (as are easily absorbed.)

As expected , virtually all the αs went straight through, but about 1:8000 were turned through large angles (reflected or back-scattered). An 8 kBq a source gives one large-angle scattering per second.

The chance of a series of small deflections resulting in a reflection is far too small to account for what was observed.

Rutherford was astonished at the result: It was quite the most incredible event that ever happened to me in my life. It was as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you! (You may find other versions of this quote, because Rutherford described his experience on many different occasions.)

Episode 521-1: Rutherford scattering (Word, 27 KB)

Episode 521-2: Alpha particle scattering experiment (Word, 48 KB)

Demonstration: Collisions and momentum

Use colliding balls to show what happens to a projectile particle hitting a target particle; as the target ball mass gets bigger, the follow through by the projectile gets less. Use a selection of ball bearings, marbles etc on some curtain track, or trolleys loaded with different weights. When the target mass is small relative to the projectile mass, the missile follows through. For equal masses the projectile stops and the target sets off with the speed of the projectile. If the target mass is large, the projectile rebounds. (If your students have already studied momentum, they should be able to predict the outcome of each of these demonstrations.)

The back scattering of αs through large angles implies (i) all the positive charge is concentrated together, and (ii) the mass of the concentrated positive charge must be quite a bit larger than of an α particle.

Discussion: Rutherford scattering and Coulomb’s Law

Rutherford assumed that (i) Coulomb’s Law was obeyed down to very small distances, and that (ii) most of the mass of the nucleus was concentrated into a very small volume – the nuclear atom that resembles a miniature solar system. (Because the analysis works, we can take this as proof that Coulomb’s Law is valid down to distances about the size of a nucleus.)

Show a diagram to explain how the terms impact parameter p and scattering angle f are defined. Ask: how would you expect the number of αs scattered through angle f to depend upon (i) the impact parameter p, (ii) the charge on the target nucleus Z, and (iii) the energy of the α particles? (As p increases f decreases (force weaker); as Z increases f increases (greater repulsive force); as energy increases, f decreases (less interaction time).)

Episode 521-3: Rutherford’s picture of alpha particle scattering (Word, 298 KB)

Student experiment or demonstration: The Gravitational model

The gravitational model analogue provides a practical way for students to get a feel for the physics of alpha-scattering. Roll a marble past the model to see it deflected. You can change the speed and impact parameter. Students can change these parameters systematically and observe the effects.

The model is designed so that, as the slope of the model gets steeper, the component of gravity parallel to the slope opposing the motion of the ball also gets larger. The actual shape is such that at any position on the slope a distance r from the centre of the hill, the component of a particle’s weight parallel to the slope ~ 1r². (Revision of the relationship between 1r potential and 1r² force can be done here if desired.)

If possible, it’s worth getting several gravitational models so that students can work with them in small groups. (The model can also be used when studying gravity. Turn it upside down to become a potential well so that you can demonstrate orbits, and discuss the difference between bound and unbound particles.)

Episode 521-4: The 1/r hill: Slope and force

Episode 521-5: A model for Rutherford scattering (Word, 37 KB)

A good simulation of alpha particle scattering could be used if desired.

Student questions (optional): Rutherford experiment and atomic structure

Episode 521-6: Rutherford experiment and atomic structure (Word, 28 KB)

Question: Rutherford’s results (optional)

Some actual results are given. Students may plot a graph to test Rutherford’s relation for α -scattering

Episode 521-7: Rutherford scattering data (Word, 41 KB)

Discussion: Models in physics

If time permits, you might have a discussion on the role of models (physical and mathematical) in physics. In what ways is the gravitational model similar to Rutherford scattering? In what ways does it differ? In which ways is the solar system a good model for the nuclear atom? What other models do your students know?

Having established the existence of the nucleus, you can now consider experimental evidence for its size, starting from the Rutherford experiment.

Lesson Summary

• Discussion and worked example: Size of nucleus (15 minutes)
• Discussion: Atomic and nucleus size (10 minutes)
• Student questions: Forces and closest approach (30 minutes)
• Discussion: Atomic number and the charge on a nucleus (5 minutes)
• Discussion: Upper limit of nuclear size (30 minutes)
• Discussion: A puzzle for a future lesson (5 minutes)

Discussion and worked example: Size of the nucleus

You can get an idea of the possible size of the nucleus by thinking about Rutherford’s experiment. Ask: What impact parameter will result in the a particle getting closest to the nucleus? (A head-on collision with p = 0.)

The principle of the conservation of energy is used to calculate the distance of closest approach as a measure for the size of a nucleus. Understanding the calculation that follows depends upon whether the students have covered electric potential and fields. Alternatively it serves as good revision.

Episode 522-1: Alpha particle scattering – distance of closest approach (Word, 29 KB)

Episode 522-2: Distance of closest approach (Word, 107 KB)

When the α is brought momentarily to rest (having climbed as far as it can up the electrostatic hill) work will have been done against the repulsive force from the nucleus. The α​'s kinetic energy is stored in the field around the nucleus. When the speed is zero, all the energy is now stored in the field.

If the α momentarily stops when at a distance d from the (centre of) the nucleus of charge Ze, the energy in the field is:

E_α = 14πε₀ 2e Zed

This equals the initial kinetic energy of the α particle. Rutherford used an α source given to him by Madame Curie. The α energy was ~ 7.7 MeV.

For gold, Z = 79. Solving gives d ~ 3 × 10^-14 m. Compare this with the diameter of gold atoms ~ 3 × 10^-10 m. So a nucleus is at least 10 000 times smaller than an atom. It is important to emphasise that this calculation gives an upper limit on the size of the gold nucleus; we cannot say that the alpha particle touches the nucleus; a more energetic α might get closer still.

An atom is mostly empty (which is why most as went straight through – any electrons would hardly impede the relatively massive high speed α).

Discussion: Atomic and nucleus size

Ask your students to suggest a scale model of the nuclear atom. For example: if a nucleus was 1 mm diameter, an atom would be 10 000 times larger or 10 m in diameter. Choose a suitable position for a 1 mm nucleus (a small ball bearing or ball of Blu-tac). Pace out 5 m (five large steps) to the edge of the atom where the electrons are. NB: textbook diagrams of an atom with a nucleus are not drawn to scale.

Reinforce an accurate picture by getting a student to stand up as a nucleus, estimate their girth (40 cm?) and ask where another student would have stand to be at the edge of the atom. 10⁴  ×  40 cm = 4000 m}, so the radius of this atom is 2 km! Check with a local map to find a named location that students will recognize that is 2 km away.

Further reinforcement: in a solid where atoms are close packed, the distance between adjacent nuclei ~ the size of an atom, i.e. equivalent to two students standing 4 km apart!

So it’s quite amazing that any hit a nucleus at all. Both are a similar size. Cross sectional area presented by a nucleus ~radius², so ~1 × 10^-28 m²}.

Ask: How would you expect the number of reflected as to depend upon the thickness of the metal foil containing the target nuclei? (Imagine the gold atoms in layers, chance of a deflection increases with thickness, but absorption on the way in or back out of the increasingly thick foil will eventually prevent any further increase in the number reflected and detected.

It is of great help if your students can recall the following orders of magnitude:

Radius of atomic nucleus ~ 10 -14^m

Radius of atom ~ 10 -10^m

Student questions: Forces and closest approach

Episode 522-3: Rutherford scattering: Directions of forces (Word, 82 KB)

Episode 522-4: Rutherford scattering: Energy and closest approach (Word, 83 KB)

Discussion: Atomic number and the charge on a nucleus

Rutherford used his data to find the charge of the gold target nucleus. Further experiments to find the charge of Cu, Ag and Pt foils gave:

	Atomic number	A scattering experiment
Cu	29	29.3  ×  e
Ag	47	46.3  ×  e
PT	78	77.4  ×  e

So the electric charge on a nucleus is given by the atomic number  ×  e, i.e. Z e. With one exception (hydrogen, H-1), Z is always less than the atomic mass number. So what accounts for the difference? The atom must be electrically neutral. Rutherford proposed the neutron.

Discussion: Upper limit of nuclear size

Recall that Rutherford’s analysis gives an upper limit on the size on the nucleus (d ~ 1 α particle energy ). The size you measure depends upon the energy of the α particle you use. So we need another approach to find the size of a gold nucleus. Can you think of a better particle to probe the size of a nucleus? (The neutron – being uncharged it will get closer.)

Another technique is the deep inelastic scattering of electrons. Refer back if you have already covered the wave nature of particles (de Broglie waves λ  = hp), or this topic can be inserted here if desired. The electron diffraction apparatus has a basic similarity with a particle scattering. The electrons are fired at a thin film – in this case of graphite.

Rutherford was fortunate that the de Broglie wavelength of the α particles (unknown to him) was quite small, and the coulomb repulsion stops as getting too close – otherwise diffraction effects would have confused the data! (Try the calculation if you have already covered λ  = hp.)

Episode 506-2: Interpreting electron diffraction patterns (Word, 30 KB)

Episode 522-5: Deep inelastic scattering (Word, 56 KB)

Episode 522-6: Electrons measure the size of nuclei (Word, 42 KB)

Discussion: A puzzle for a future lesson

There is a fundamental problem with Rutherford’s model. Ask your class: How can an atom with a central nucleus can be stable – why doesn’t it collapse? According to classical electrodynamics, the electrons should emit radiation as they orbit, and spiral inwards.

(It’s good to leave a class with a puzzle for a future lesson.)