Download Particles and interactions

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 particles
An elementary particle is called
elementary if it is not made out of any
smaller component particles.
Particle zoo
Quarks
There 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.
Leptons
There 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
Antiparticles
In 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 numbers
Quantum ‘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.
Spin
In 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 halfintegral spin.
The Pauli exclusion principle
It 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 principle
In 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 particles
Because 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 diagrams
Not 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 diagrams
All 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 interaction
Consider 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/4π
and hence the range of the interaction is approximately given by
R ≈ h/4πmc