Particle Physics

Probing the Nucleus

Rutherford Scattering

Gold is rolled out into a foil a few atoms thick. Radon gas is used to produce a narrow beam of a particles which is fired at the foil. The angle of deflection is plotted against the number of particles. The majority of particles are deflected only slightly or not at all (travel through), a few are deflected fully (back scattering). This suggests a model with a massive, positively charged, small centre and electrons orbiting the nucleus. This became the accepted model for the atom.

Using energy considerations we can calculate a distance of closest aproach. As the particle moves closer to the nucleus, it looses kinetic energy and gains potential energy. Thus we can derive the equation:

where r is the distance of closest approach. This gives a value of about 10-14m, suggesting that nucleons have diameters one order of magnitude less (i.e. 10-15m). Using Coulomb's law we can calculate the electrostatic force between 2 adjacent protons as being equal to 230N. This is a very large force for particles this small, and this suggests that there must be a very strong force holding them together. This is the strong nuclear force.

The Strong Force

This diagram shows the force between 2 nucleons due to the strong force (discovered by Yukawa in the 1930s). Note that a positive force is an repulsion and a negative force an attraction. Thus we can see that the force is repulsive at very short distances (so that the nucleons do not merge into one point), but attractive outside this limit. It acts only over a very small distance (as the a-particles were not captured by the nucleus). It should also be noted that the density of nuclei is constant, and therefore the force must be easily saturated by surrounding nucleons.

Relativistic Equations

In particle physics, due to the small values of most of masses and high values of speeds, we use a different set of units to measure quantities. Energy is measured in MeV (mega electronvolts) or GeV, momentum in MeV/c or GeV/c and energy in MeV/c2 or GeV/c2.

As particles may often be travelling close to the speed of light, it is necessary to take relativistic effects into account. Thus we use the following relativistic equations:

Mass:

(m = mass m0 = rest mass v = speed c = speed of light)

The quantity is called the Lorentz factor, and is often denoted by g.

Momentum:

(p = momentum)

Energy:

The total energy possessed by a body is given by:

It is clear that is the body is at rest then this simplifies to , and if a body has zero rest mass then , and thus v = c.

Non-relativistic Momentum Calculations

If particles are moving at sensible speeds (<90% of the speed of light) then the normal equation for momentum can be used (p = mv). Linear and angular momentum are always conserved in a collision, as well as energy. If motion occurs in more than one dimension, it is necessary to resolve momentum into mutually perpendicular directions.

The law of conservation of momentum demands that, when an atom decays and emits a particle, it recoils. As only one particle is emitted, it must recoil in the opposite direction to the particle's motion. The amount of energy released by any decay process is fixed, and, allowing for different uptake of energy by g rays, the total amount of energy released as kinetic energy is also fixed. As the relative velocity of the two decay products is determined by conservation of momentum, there are a fixed set of possible particle velocities, and thus a fixed set of possible recoil velocities. This can be observed with a particle decays.

However, with b particle decay, this is not the case and a continuous spectrum of velocities is observed. It is also observed that the recoil is not necessarily in the opposite direction to the decay. This is explained by the production of a third particle at the same time, the antineutrino (see later).

Particle Accelerators

Particles are accelerated because they can be used to gather information about the composition of the nucleus. Particles are accelerated using electric fields. This is because the acceleration due to an electric field is in the direction of the field lines, whereas that due to a magnetic field is at 90° to the field lines. Therefore magnetic fields can be used to keep particles moving in a circle, but not to accelerate them.

Drift tube accelerators consist of a set of long tubes along which the particles travel. A constant frequency ac supply is applied to the tubes. Whilst inside the tubes the particles are not accelerated - when between the tubes they are. It is clear that as the particles get faster the tubes must get longer, and therefore to get to high energies the tubes must be very long indeed (up to 2 miles!). Energies of up to 31.5 MeV can be attained for a proton.

Electrons are accelerated using radio waves, and are injected into the linac at very high speeds from a Van der Graff generator.

Proton Synchrotrons are circular accelerators. A magnetic field of varying intensity is used to keep the particles moving in a circle and a varying frequency electric field to accelerate them. The circular nature of the accelerator allows the particles to pass round as many times as necessary, allowing energies of up to 900 GeV to be achieved. It should be noted that particles with opposite charges (e.g. particles and antiparticles) travel in opposite directions and therefore can easily be made to collide.

If particles are accelerated into a stationary target, because momentum must be conserved, most of the energy of the collision needs to go into the kinetic energy of the products. This greatly decreases the energy available for new particle creation (for a 500GeV particle, 96% of the energy is "wasted"). If two beams of particles are collided together, however, there is no initial momentum (if both are moving at the same speed) then all the energy is available for particle creation. However, it is harder to get two beams of moving particles to collide.

Particle Detectors

The Geiger-Müller Tube

Particles of radiation enter the tube through the thin mica window and ionise an argon atom. The electron is given great kinetic energy by the very strong force around the anode and causes secondary ionisation. This causes a pulse of up to 108 electrons, which is detected by the counter. It also causes a build-up of positive ions around the anode. The positive ions towards the cathode under the low intensity field. To prevent them causing the release of electrons from the anode (and thus another pulse), a gas such as chlorine is added, which neutralises the argon ions (their energy is taken up dissociating the chlorine molecules). The tube is briefly insensitive after a pulse - during the "dead time" the ions are moving away from the anode, which allows a field to be created again, and during the "recovery time" they are being neutralised by the chlorine molecules. During the second period, pulses are produced but are not sufficiently large to be detected. This limits the count rate to about 1000 Hz. X rays and gamma rays are only very weakly ionising, and are therefore not generally detected by the GM tube.

If the tube is run at too low a voltage, particles are not detected, and if it is run at too high a voltage the quenching process is inefficient. An optimum potential is about 450V.

The Bubble Chamber

A bubble chamber is a large cylinder containing a superheated liquid (a liquid which would normally be a gas at that temperature and pressure). Hydrogen is often used at about -170°C. Charged particles produce ions, which act as nucleating sites for the liquid to boil, forming bubbles. The bubbles show the track of the particle. A magnetic field is applied to the chamber, which produced curved tracks and thus allows momentum, charge, mass, etc. to be calculated. This allows the nature of the observed particles to be detected. The chamber is cleared simply by increasing the pressure, which liquefies the bubbles.

Particle Tracks

When a magnetic field is applied at right-angles to a particle's motion, it is accelerated and performs circular motion. Therefore, , giving the result that mv = Bqr. This means that the momentum of a particle is proportional to the radius of the circular track it produces. This leads to spiral tracks, as particles loose momentum due to collisions.

Slower particles cause more ionisation in a given distance and thus thicker, shorter tracks. Faster particles make thin, less curved tracks. Particle - antiparticle pairs produce a pair of tracks that curve in the opposite directions - in general they have the same energy and so the tracks are very similar.

Fundamental Particles and Forces

Beta Decay

Beta Decay

b- particles are electrons emitted from the nucleus during a decay process. A neutron inside the nucleus becomes a proton and an electron, which is ejected. A g ray is often produced at the same time due to rearrangements within the nucleus as a loss of energy. For example:

is an antineutrino, which is necessary to conserve linear momentum.

The b+ particle is called the positron. It has the same mass as the beta particle but the opposite charge, and is the antiparticle of the electron (and so can be called the anti-electron). Some artificially produced isotopes decay by the production of a b+ particle, e.g.:

(n = neutrino)

Electron capture is another form of beta decay, where a nuclear proton captures an electron from the inner shell of electrons, becoming a proton. A gamma ray and neutrino are produced.

In a beta decay a fixed amount of energy is produced as mass is lost. Some of this energy could be lost as a gamma ray, but a g photon can only have a fixed amount of energy (as it is produced by a promoted electron falling from one energy level to another). This still leaves a quantised set of energies to be lost by the nucleus. By the principle of conservation of momentum this energy should be distributed in a fixed manner (i.e. the beta particle should have fixed energy levels). However, this is not the case. The energy spectrum of beta decay is shown below:

The continuous spectrum suggests that a third particle must be produced to carry off the linear momentum that is not accounted for. This is the antineutrino (neutrino in positron decays), which has no charge and a very small mass (if it has any mass at all). These particles interact very little with matter, and so are difficult to detect. Antineutrinos are detected by the rare event . The neutron and the positron must then be detected.

Antimatter

Each particles of "real" matter has a corresponding antiparticle, with the same mass and spin, but the opposite charge (and lepton number, if a lepton).

When matter and antimatter collide, they annihilate, producing energy. This may be released as photons (2 or more), or as a particle / antiparticle pair (if the energy is high enough). Two or more particles must be produced as mass / energy, charge, momentum and lepton number are conserved. Energy is conserved, as kinetic energy of the particles and the mass / energy of the new particle pair produced. Angular momentum and linear momentum are conserved, necessitating the production of two particles. Lepton number is conserved as an antiparticle has the opposite lepton number to the corresponding matter.

The Leptons

Lepton

Rest Mass

GeV/c2

Charge

Le

Lm

Lt

e-

0.511

-1

+1

0

0

e+

0.511

+1

-1

0

0

ne

» 0

0

+1

0

0

» 0

0

-1

0

0

m-

105.7

-1

0

+1

0

m+

105.7

+1

0

-1

0

nm

» 0

0

0

+1

0

» 0

0

0

-1

0

t

1784

-1

0

0

+1

t+

1784

+1

0

0

-1

nt

» 0

0

0

0

+1

» 0

0

0

0

-1

Forces acting on Leptons

Leptons are affected by three forces:

  1. The electromagnetic force – if a lepton is charged it is affected in the same way as any other charged particle. It should be noted that in a constant electrostatic field a lepton will move in a parabolic path, whereas in a magnetic field it will move in a circular or spiral path.
  2. The weak force – the weak force is involved in beta decay and in some collisions, decays, etc.
  3. The gravitational force – this is not significant, due to the very small masses of the particles concerned.

Deep Inelastic Scattering

Firing electrons at protons and neutrons produced evidence for substructure in nucleons in much the same way as firing a-particles at atoms provided evidence for substructure within atoms. Low energy electrons are deflected from the nucleons simply by the charge interaction of the particles. Higher energy particles either pass straight through the nucleons or are deflected at very acute angles, indicating point charges within the nucleons. The scattering is deep because it probes deep into the structure of the nucleons and inelastic because the electrons may loose energy in the process.

From this view of the nucleus it appears that momentum is not conserved. However, this is because there are other particles within the nucleons, gluons, which increase the mass. These particles carry the strong force that binds the nucleons together.

The particles that make up the nucleons are called quarks. There are two quarks in protons and neutrons, the up and down quarks. Mesons are made of quark anti-quark pairs, those that only incorporate the up and down quarks and their anti-particles being the p mesons, or pions.

Particle Classification

All particles can be broken down into two classes, fermions and bosons.

Fermions – particles with half-integral spin that make up normal matter.

Bosons – particles with integral spin – these do not obey the Paulii exclusion principle.

Leptons are fermions that are not affected by the strong force. They are all fermions, and therefore obey the Paulii exclusion principle.

Hadrons are all particles affected by the strong force, and could be baryons or mesons:

Baryons are fermions and are made up of three quarks.

Mesons are bosons and are made of two quarks.

From this it is clear that quarks have half-integral spin and are therefore fermions.

The Weak Interaction

The weak interaction enables particles to exchange mass, energy and charge and to interconvert. The weak interaction is felt by leptons and hadrons. The strong interaction binds hadrons together, and may lead to particle interconversion and production of particle – antiparticle pairs. The Yukawa force, which holds the nucleus together, is a remenant of the strong interaction between quarks and gluons. In all interactions, charge and baryon number are conserved. Baryons have baryon number 1, quarks ± a third or two thirds. Mesons and leptons have zero baryon number.

Baryons and Mesons

Baryons and mesons are made of quarks. Baryons are made of three quarks and mesons of two. A meson always consists of a quark-antiquark pair.

 

Quark

Rest Mass

GeV/c2

Baryon Number

Charge

Notes

u

0.33

 

0.33

 

d

0.01

 

0.01

 

c

1.58

1 charm

1.58

-1 charm

s

0.47

-1 strangeness

0.47

1 strangeness

t

180

1 top-ness

180

-1 top-ness

b

4.58

-1 bottom-ness

4.58

1 bottom-ness

As a meson has an overall baryon number of 0, it can disappear without a problem (if charge is conserved).

Symmetry between sets of particles

There is a symmetry between the three generations of leptons and the three generations of quarks:

Quark

Lepton

Antiquark

Antilepton

d

e

/ e+

u

ne

s

m-

m+

c

nm

b

t

t+

t

nt

The symmetry of this set implies that there is some more fundamental underlying structure. It could be that there is a more fundamental set of particles making up the universe than we know of at the moment, or it could be that there is a more basic principle behind the laws of the universe that dictates that the particles are as such. We simply don’t know.

Forces in Particle Physics

Forces arise because of fundamental interactions between particles. When particles come close enough together to interact, they transfer exchange particles. Each interaction represents a different force, and is mediated by different particles. The properties of these exchange particles (or intermediate vector bosons) determines the nature of the force.

 

Interaction

Particle

Symbol

Mass

Range

Decay time

Gravitational

Graviton

g

Zero

Infinite

-

Strong

Gluon

g

Zero

10-15 m

10-23 s

Weak

 

W-, W+, Z0

81, 81, 93

10-18 m

10-10 s

Electromagnetic

Photon

g

Zero

Infinite

10-18 s

The larger mass weak exchange particles have a much shorter range, as they take more energy to create and therefore can only exist for a shorter space of time. Gluons have a short range because they are affected by the strong force and therefore cannot separate from the quarks.

Note that the weak interaction has a longer decay time. This is important, as it means that longer-lived particles tend to decay via the weak interaction.

Feynman diagrams can be used to illustrate forces being mediated by exchange particles. Time is represented by arrows and the position of particles is not relevant. The exchange particles are represented by lines, with different types of lines illustrating different particles. Some examples are shown:

It should be noted that, due to its long range and the fact that it acts on every charged particle, the electromagnetic interaction is the most common of the interactions.

Electroweak Unification

At very high energies (in excess of 100GeV) the electromagnetic and weak forces appear to act as one force. This is because the differences between the forces are basically due to the fact that the weak vector bosons have mass, whereas the photon does not. At these high energies, the mass that the bosons have becomes insignificant. The electroweak theory predicted the existence of the Z0 boson and is therefore coming to be regarded as correct.

At even higher energies it may be possible to combine the strong force with the electroweak in a grand unification theory. This would take place at energies over 1015 GeV, which is well beyond the capabilities of the biggest particle accelerators currently available. GUTs also predict that the proton is an unstable particle, with a lifetime of about 1031 years. No evidence for this has yet been found. Therefore GUTs are still very uncertain.

It may also be possible to unify gravity with the other four forces (possibly at energies over 1019 GeV), leading to the idea of one unified field that represented every force. However, this is a long way off in theoretical terms.

Particle Physics and Cosmology

Doppler Shifting of Light

All galaxies produce a spectrum of light with absorption bands due to some of the atoms and molecules present in the stars producing the light. Examples of this are the bands for hydrogen and helium, which are common in stars. For nearby stars, the bands are in the same position as those produced in laboratory experiments – for more distant stars, they are shifted towards the red end of the spectrum. This suggests that the galaxies are moving away from us. The position of the lines in the spectrum allows the recessional velocity of the stars to be calculated and therefore a graph of distance against velocity can be plotted. Hubble was the first person to do this and he found that the velocity was proportional to the distance of the galaxy from Earth. This suggests that the universe is expanding.

The expansion of the universe is predicted by the Big Bang theory, which suggests that the universe was at one time concentrated in one point, and has been expanding ever since. This theory accounts for the expansion of the universe, but it also suggests that the ratio of H to He should be 3 to 1 (it is) and predicts the microwave background radiation.

A Brief History of the Universe

Before 10-43 s – average particle energy > 1019 GeV.

It is believed that all four forces of nature were unified in one force. Very little is known about this period.

After 10-43 s – average particle energy < 1019 GeV.

The strong and electroweak forces are still united and therefore all particles are indistinguishable.

After 10-35 s – average particle energy < 1015 GeV – Quark-Lepton Era

In these very high energies, the universe was a mixture of quarks, leptons and exchange particles. Particles were continually forming and annihilating each other. As the average particle energy is so great, the forces are not strong enough to bind the various particles together, and thus they exist individually.

At 100 GeV, the electroweak force decouples to give the two separate forces. As particles separate, the materialisation of massive particles becomes less and less likely, and so lighter particles form. At about this point, the ratio of matter to antimatter increases above 1.

After 10-5 s – average particle energy < 1GeV – T < 1013 K – Hadron Era

The energy is now low enough for the strong force to bind quarks together into hadrons. As the average energy decreased, it was no longer possible for photons to produce particle pairs, as they did not have enough energy. Annihilations led to an excess of electrons over positrons and baryons over antibaryons. This matter became the universe as we know it.

After 100s – average particle energy < 0.1 MeV – Plasma Era.

The mean energy is now low enough for the Yukawa force to begin operation. Baryons begin to combine into nuclei, especially those of the light elements (isotopes of hydrogen and helium in particular).

After 1013 s – average particle energy < 0.1 eV – T < 3000K – Atomic Era

The average energy is now less than the ionisation energy of the atoms, and thus they start to capture electrons to form atoms. The light photons can now escape from matter and they do so, forming gamma rays with frequencies corresponding to about 3000K. As the universe has expanded by a factor of about 1000 since then the light thus released has stretched by a similar factor and so is now at a 3K level (the background radiation).

Atoms are now surrounded by electrons, and therefore are electrically neutral overall. This means that they can start to group together under gravity, and so form stars, galaxies and so on. The universe continues to expand and cool until the present day.

Open and Closed Universes

All the matter in the universe is attracting all the other matter by universal gravitation, and this is slowing down the rate of expansion of the universe. If there is sufficient mass in the universe, it will stop expanding and start contracting again under the force of its own gravitational attraction, culminating in a "big crunch". This is a closed universe.

If there is insufficient mass, the universe will continue to expand forever. It will cool to roughly absolute zero and undergo the process described as "heat death". This is an open universe.

The point where the density of the universe is just sufficient to prevent collapse is called the critical density. This value is about 6 atoms per cubic metre of space (on average). Visible matter accounts for about 3% of this value – however, there may be much more matter locked up in so-called "dark matter" which is not directly observable.

The three possibilities are shown in the diagram below:

At the moment, an open universe appears to be the most likely situation (although this changes every few months, so you certainly shouldn't take any notice of what I've just written...).