Particles and interactions

By the end of this topic you should be able to:

•state the meaning of the term elementary particle;•identify the three classes of elementary, the quarks, the leptons and the exchange particles;•understand the meaning of quantum numbers;•state the meaning of the term antiparticle;•classify particles according to their spin;•understand the Pauli exclusion principle and how it is applied;•understand and apply the Heisenberg uncertainty principle for energy and time;•appreciate the meaning of the term virtual particle,•describe the fundamental interactions;•state the meaning of the term interaction vertex;•understand what is meant by Feynman diagrams;•draw Feynman diagrams in order to represent various physical processes;•apply the Heisenberg uncertainty principle in order to derive the range of an interaction.

In 1896, the British physicist J. J. Thompson discovered the electron, performing experiments with cathode rays.

In 1911, Ernest Rutherford discovered the atomic nucleus. In 1917, (in experiments reported in 1919) Rutherford proved that the hydrogen nucleus is present in other nuclei, a result usually described as the discovery of the proton.

The neutron was not discovered until 1932 when James Chadwick used scattering data to calculate the mass of this neutral particle. Since the time of Rutherford it had been known that the atomic mass number A of nuclei is a bit more than twice the atomic number Z for most atoms and that essentially all the mass of the atom is concentrated in the relatively tiny nucleus. As of about 1930 it was presumed that the fundamental particles were protons and electrons, but that required that somehow a number of electrons were bound in the nucleus to partially cancel the charge of A protons.

Elementary particlesAn elementary particle is called elementary if it is not made out of any smaller component particles.

Particle zoo

QuarksThere are six types (or “flavours”) of quarks. They are denoted by u, d, s, c, b and t, and are called up, down, strange, charmed, bottom and top, respectively. All of these have electric charge. The up quark is the lightest and the top quark is the heaviest. There is solid experimental evidence for the existence of all six flavours of quarks.

A quark can combine with an antiquark to form a meson. Three quarks can combine to form a baryon.

LeptonsThere are six of these: the electron and its neutrino, the muon and its neutrino, and the tau and its neutrino. They are denoted by e-, νe, μ-, νμ, τ-, ντ. The muon is heavier than the electron, and the tau is heavier than the muon. The three neutrinos were once thought to be massless, but there is now conclusive evidence that in fact they have a very small mass.

Exchange particles

This class of elementary particles contains the photon (denoted by γ). The photon is intimately related to the electomagnetic interaction. There are also the particles W+, W- and Z0, called the W and Z bosons. These particles are intimately related to the weak interaction. Then there are eight particles called gluons that are related to the strong interaction. Finally, there is the graviton, which is related to the gravitational force or interaction.

Higgs boson

Standard Model

AntiparticlesIn addition to the elementary particles, we have the antiparticles of all of the above. To every particle there corresponds an antiparticle of the same mass as the particle but of opposite electric charge (and opposite all other quantum numbers).

The existence of antiparticles was predicted theoretically by Paul Dirac in 1928. The first antiparticle to be discovered experimentally was the positron in 1932 by Carl Anderson.

Quantum numbersQuantum ‘numbers’ are numbers (or properties) used to characterize particles. There is one quantum number you know already—the electric charge. Some (but not all) quantum numbers are conserved in interactions. The quantum number for electric charge is always conserved.

Other quantum numbers: flavor, colour, strangeness, baryon number and generation lepton number.

SpinIn classical mechanics, a body of mass m moving along a circle of radiur r with speed v has a property called angular momentum. This is defined to be

L = mvr.

This quantity has units of Js. If the body spins around its own axis, it has additional angular momentum.

Particles appear to have a similar property, measured also in units of Js, and this property was called spin by analogy with a spinning body mechanics. But a particle’s spin is not the same thing as the angular momentum of a spinning body. For elementary particles, spin is a consquence of Einstein’s theory of relativity. The spinning body is just a useful analogy. All known particles have a spin that is a multiple of a basic unit.

Unit of spin = h/2π

h = 6.62 x 10-34 Js

Spin

All the known particles have a spin that is either an integral multiple of the basic unit or a half-integral multiple.

Particles are called bosons if they have an integral spin, and they are called fermions if they have a half-integral spin.

The Pauli exclusion principleIt is impossible for two identical fermions (particles with half-integral spin) to occupy the same quantum state if they have the same quantum numbers.

This is why the inner shell of any atom can contain at most two electrons. Electrons are fermions and so the Pauli exclusion principle applies to them. In the inner shell the one quantum number that can distinguish two electrons is the spin. Since the spin of the electron is ½, there are just two quantum states available: one in which the spin is “up” and another in which it is “down”.

Heisenberg uncertainty principleIn 1928 the German physicist Werner Heisenberg discovered one of the fundamental principles of quantum mechanics.

The uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position x and momentum p, can be known simultaneously. The more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.

The version of the principle that will concern us is that which applies to simultaneous measurements of energy and time. Measurements of the energy of a particle or of an energy level are subject to an uncertainty. This uncertainty is not the result of random or systematic errors.

The measurement of the energy of a particle must be completed within a certain interval of time that we may call Δt. Heisenberg proved that the uncertainty in the measurement of the energy ΔE is related to Δt through

ΔE Δt ≥ h/4π

This says that, the shorter the time interval within which the measurement is made, the greater the uncertainty in the measured value of the energy. To have a very small uncertainty in energy would require a very long time for the measurement of energy.

There is, however, a subtler and more useful interpretation of the energy-time Heisenberg uncertainty principle. We know that total energy is always conserved. But suppose, for a moment, that in a certain process energy conservation is violated. For example, assume that in a certain collision the total energy after the collision is larger than the energy before by an amount ΔE. The Heisenberg uncertainty principle claims that this in fact possible provided the process does not last longer than a time interval Δt given by Δt ≈ h/4πΔE. In other words, energy conservation can be violated provided the time it takes for that to happen is not too long.

Virtual particles

This process actually violates the law of conservation of energy. It cannot take place unless the photon that is emitted is very quickly absorbed by something else so that the energy violation (and the photon itself) becomes undetectable. Precisely because this photon violates energy conservation, it is called a virtual photon.

Interactions and exchange particlesBecause the first electron emitted a photon, it changed direction a bit in order to conserve momentum. Similarly, the second photon also changed direction, since it absorbed a photon. Looked at from a large distance away, the change in direction of the two electrons can be interpreted as the result of a force or interaction between the two electrons.

The electromagnetic interaction is the exchange of a virtual photon between charged particles. The exchanged photon is not observable.

Basic interaction vertices

At a fundamental level, particle physics views an interaction between two elementary particles in terms of interaction vertices. The fundamental interaction vertex of the electromagnetic interaction is:

Not allowed vertices

Feynman diagramsNot just a picture. It represents a very definit mathematical expression called the amplitude of the process. The square of the amplitude gives the probability of the process actually taking place.

For the electromagnetic interaction, the basic vertex is assigned the value √αEM, where αEM ≈ 1/37 and is closely related to the charge of the electron. The amplitude of the diagram is then the product of the √αEM for each vertex that appears.

Since αEM is a small number less than 1, the processes with four interaction vertices are less likely to occur. To a first approximation, it is sufficient to examine the diagram with two vertices only.

Building Feynman diagramsAll you need:

•the basic interaction vertex;•lines with arrows to represent electrons and positrons;•wavy lines to represent photons.

Feynman diagrams for other interactions

Basic interaction vertices for the weak force.

Beta decay

The range of an interactionConsider the diagram in which two particle interact through the exchange of the particle shown by the wavy line. Let the mass of this particle be m.

The fastest the virtual particle can travel is the speed of light c. If R is the range of the interaction, then the virtual particle will reach the second particle in a time no smaller than R/c. The energy that will be exchanged will be of the order of mc2. For the purpose of the estimate, taking uncertainties of order R/c in the time and mc2 in the energy, we then have that by the Heisenberg uncertainty principle:

mc2 x R/c ≈ h/4πand hence the range of the interaction is approximately given by

R ≈ h/4πmc