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## Quarks

Lesson for 16-19

Episodes 538 and 539 of this topic give clear evidence for the size of the nucleus, and for the fact that nucleons are not fundamental particles but contain different parts. This leads onto Gell-Mann and Zweig’s quark model.

## Episode 537: Preparation for the deep scattering and quarks topics

Teaching Guidance for 16-19

Explore the fact that nucleons are not fundamental particles but contain different parts.

## Main aims of this topic

Students will:

- know that Rutherford’s experiment, using alpha particles, cannot probe the nucleus because the alpha particles will interact with the nucleus by the strong nuclear force
- know that electrons, being leptons, do not
feel

the strong nuclear force, and so can probe the nucleus - use electron wavelength and scattering data to calculate the size of the nucleus
- understand that the complex scattering from a nucleus reveals that nucleons are not simple points, but are themselves composed of smaller particles
- describe how hadrons are made from two or three quarks
- deduce the properties of a hadron from the properties of its constituent quarks
- draw Feynman diagrams involving quarks and gluons

## Prior knowledge

Rutherford’s experiment, diffraction, and the quantum nature of electrons (wave-particle duality

).

If you have not covered diffraction already, you will have to modify the suggested approach to take account of this.

### Up next

### Electron scattering

## Episode 538: Electron scattering

Lesson for 16-19

- Activity time 50 minutes
- Level Advanced

This could follow work on Rutherford scattering, where alpha particles are used to identify the nucleus as the region of the atoms containing all the positive charge and most of the mass. Calculation of closest approach of alpha particles gives an estimate of nuclear size, but more direct experimental evidence is given by scattering of electrons.

Lesson Summary

- Student experiment: An optical analogue (10 minutes)
- Discussion: Diffraction graphs (10 minutes)
- Worked example: Calculating nuclear diameter (10 minutes)
- Student questions: Further examples (20 minutes)

## Student experiment: An optical analogue

When waves pass through a collection of spheres, which look

like circles to the oncoming waves, they diffract around them in the same way as through holes of the same size. This means that they produce a diffraction pattern with a minimum at roughly the angle given by the single-slit diffraction equation sin( θ ) = *λ* *d* (it should really be sin( θ ) = 1.22 *λ* *d* for a circle; you could use that form, but the familiar equation is close enough here).

Look at a point source of light (a small circuitry lamp is fine) through a pinhole made in aluminium foil – the smaller the hole the better, so use a fine sewing needle or else pull thin copper wire until it snaps, and push the work-hardened tip through the aluminium foil. A diffraction pattern consisting of a circle with a surrounding ring (more than one if you’re lucky) should be visible. If you look at the point source through a microscope slide dusted with lycopodium powder (cornflour will work, but is not as good, because the particle sizes vary more) you see a similar pattern. You may like to refer to the halo seen around the Moon in the winter, which is due to similar scattering by ice crystals in the upper atmosphere.

The point here is that waves encountering round (spherical) objects will give a diffraction pattern, involving a central maximum, and then a minimum before the secondary maximum of the surrounding ring.

## Discussion: Diffraction graphs

The graph below compares a typical electron diffraction pattern with the Rutherford alpha -scattering pattern and the single slit pattern. Show this graph, and identify the differences.

Episode 538-1: Rutherford alpha-scattering pattern & single slit pattern (Word, 26 KB)

The electron scattering clearly shows features of both Rutherford scattering (due to electrostatic interaction: the fact that the force is attractive rather than repulsive is not important) and diffraction. Concentrating on the minimum, it suggests that the nucleus is behaving as a circular object of diameter *d*.

## Worked examples: Calculating nuclear diameter

The nuclear diameter *d* can be calculated from the electron scattering data. If your students can cope with the maths, you can work through a sample calculation.

(The virtue of the approach is not that students should be able to reproduce the mathematics, but that they should be able to see that a systematic application of the mathematics related to diffraction and the wave nature of the electron can give the size of the nucleus. Note: The calculation assumes that the rest energy of the electron is negligble.)

energy of electron = 100 MeV

energy of electron = 100 × 10^{6} V × 1.6 × 10^{-19} C

energy of electron = 1.6 × 10^{-11} J

electron momentum, *p*

*p* = *E**c*

*p* = 1.6 × 10^{-11} J3 × 10^{8} m s^{-1}

momentum = 5.3 × 10^{-20} kg m s^{-1}

de Broglie relationship, * λ * =

*h*

*p*

*λ* = 6.6 × 10^{-34} J s^{-1}5.3 × 10^{-20} kg m s^{-1}

wavelength = 1.2 × 10^{-14} m

The first diffraction minimum occurs at about 22 ° , so using the single slit diffraction equation

sin( θ ) = *λ* *d*

rearrange this to get

*d* = *λ* sin( θ )

*d* = 1.2 × 10^{-14} msin(22 ° )

nuclear diameter = 3.3 × 10^{-14} m

## Student questions: Further examples

A structured question will allow your students to follow through this logic for themselves. It should be covered here if you did not do so in Episode 522.

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

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### Up next

### Deep inelastic scattering

## Episode 539: Deep inelastic scattering

Lesson for 16-19

- Activity time 30 minutes
- Level Advanced

At higher electron energies, two things happen:

- The electrons penetrate deeper into the nucleus and scatter off sub-units within protons and neutrons
- The electrons 'lose' energy (they undergo inelastic collisions); this energy is ‘converted’ into mass as a jet of pions is produced

Hence higher energy electrons give us information about the structure of nucleons.

Lesson Summary

- Student experiment: Analogue of electron scattering by quarks (20 minutes)
- Discussion: Deductions from electron scattering (10 minutes)

## Student experiment: Analogue of electron scattering by quarks

Magnets are concealed in a box. These represent charges

which are probed using a freely-suspended magnet.

The important point here is that each of the charges

represented by the magnet poles will affect the trajectory of the target charge

of the moving magnet pole in a more complex way than a single one would.

Episode 539-1: Probing arrangements (Word, 36 KB)

## Discussion: Deductions from electron scattering

Students should appreciate that the necessarily complex analysis of particle paths from deep electron scattering indicates that the neutron and proton are not simple point charges but contain simpler structure within them.

Electrons must be accelerated to very high speeds to penetrate nuclei. The energy involved is so great that some of it can be converted (via *E* = *m* × *c*^{ 2}) into the mass of new particles. This results in the electrons dissipating energy – this is why the scattering is *in*elastic – to produce pions. Jets of pions are typical of the events

seen in particle accelerators colliding protons and anti-protons.

At this stage you could ask the open question: if, as indicated by deep inelastic scattering, neutrons and protons are each made of three particles called quarks, what’s the smallest number of quarks you need?

Obviously you could have any number of quarks, but there must be more than one, or else neutrons and protons would not be different. The simplest model would have two different types.

Assuming there are just two types of quark, then possibilities could be AAA and BBB (chargeless A and +*e*3 for B) or AAB and BBA, which is actually correct, with A being d (-*e*3) and B being u (+2*e*3) The latter (correct) version also explains other particles.

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### Up next

### Quarks and the standard model

## Episode 540: Quarks and the standard model

Lesson for 16-19

- Activity time 90 minutes
- Level Advanced

The quark model, justified by the results of deep inelastic electron scattering, creates relative order out of the chaos of particle classification.

Lesson Summary

- Discussion: Rules for quarks (15 minutes)
- Student activity: Making non-strange hadrons with quark triangles (15 minutes)
- Student questions: Making strange hadrons with quark triangles (15 minutes)
- Discussion: Gluons and the force between quarks (15 minutes)
- Student activity: Constructing Feynman diagrams (20 minutes)
- Discussion: Summing up (10 minutes)

Quarks have three colour

charges, and the rule for stability is that combinations must be colourless. To make the activities accessible for students with colour deficiency, consider labelling the coloured parts of the diagrams.

## Discussion: Rules for quarks

Students need to be informed of the rules for combining quarks.

Where the electromagnetic interaction is due to the property of electric charge, which can be positive or negative, quark interaction is due to a property which can have three different states. This has been called colour

charge, because the three primary colours red, green and blue add to give white, a colourless combination. These three-quark combinations are the baryons.

Anti-quarks have an anti-colour; you may wish to think of the complementary colours to the three primary colours, so anti-red is cyan, anti-green is magenta and anti-blue is yellow. Three anti-quark combinations are the anti-baryons.

By combining a quark with an anti-quark of the appropriate anti-colour, a two-quark hadron can be produced. These two-quark combinations are the mesons.

## Student activity: Making non-strange hadrons with quark triangles

The quark triangle are constructed so that they can be fitted together in threes, with the 120° vertices together, with the combination red, blue and green giving a baryon. A similar arrangement with anti-red, anti-blue and anti-green gives an anti-baryon. By taking one quark and fitting an anti-quark of the appropriate anti-colour alongside it so that the two long sides coincide, a meson can be constructed.

Episode 540-1: Quark triangles (Word, 121 KB)

Episode 540-2: Quark models (Word, 36 KB)

The document contains two versions of each anti-quark: one version uses the same colours as the related quarks, but with the white and coloured regions of the triangles reversed. The second version has the same pattern of colours on the triangles as the related quarks, but the complementary colours are used instead of red, blue and green: thus anti-red is cyan, anti-blue is yellow and anti-green is magenta. You can choose whichever form you prefer!

Using only u (+2*e*3) and d quarks (-1*e*3) and their anti-quarks, u bar and d bar, in all possible colours, students can quickly use the three-colour rule to construct four possible baryons

(n, p and the unstable Δ ^{-} and Δ ^{++} ) together with their anti-particles. Other non-strange baryons are high energy states of these four, e.g. the Δ ^{+} and Δ ^{0} particles are uud and udd respectively, and can decay to the proton and neutron by emitting the extra energy as a gamma photon

Δ ^{+} → p + γ

Δ ^{0} → n + γ

They should also be able to construct four different mesons, all pions. Two of these are, in fact, the same particle (u + u bar and d + d bar, both π^{0}).

Issue the strange quarks, s and s bar, at this point, with the explanation that they are heavier versions of d and d bar, and that they have strangeness of -1 and +1 respectively. This allows the construction of all the particles in the baryons decuplet and meson octet.

## Student questions: Gluons and the force between quarks

These questions use the u and d quark triangles and their anti-quark triangles.

Episode 540-3: Putting quarks together (Word, 56 KB)

Further questions take a similar approach to explain the baryon decuplet and meson octet. These questions can be used to structure the student activity, or else as homework to consolidate the learning afterwards.

Episode 540-4: Strange quarks (Word, 45 KB)

## Discussion: Gluons and the force between quarks

Just as pions (Yukawa’s mesons) are the particles that bind baryons, the quarks in a baryon are bound by exchanging a particle. The particles here are *gluons* , and we envisage the transfer as exchanging the colour of the two quarks concerned: e.g. a red quark will change into a blue quark by emitting a red-antiblue gluon, which is then absorbed by a blue quark that becomes red.

## Student activity: Constructing Feynman diagrams

If your specification requires it, the Feynman diagrams used for the weak interaction can now be extended to the strong interaction, and the n

and p

of the weak interactions replaced by d

and u

quarks respectively.

Episode 540-5: Feynman diagrams with quarks (Word, 99 KB)

## Discussion: Summing up

This is a conclusion to the entire particle physics sequence. It should be realised that normal matter we see around us consists of two quarks and two leptons, and their anti-particles. Larger mass versions, which are not stable, occur in two generations. The muon and strange quark have already been met, and, together with the muon neutrino and the heavier version of the u quark, the charm quark (c), form the second generation. There is only one more generation, so all matter, whether stable or not, can be described in terms of 6 quarks and 6 leptons.