PHY304 
Particle Physics 
Dr C N Booth 
Particle Physics  Introductory Notes
1. Units of Energy, Momentum and Mass
As in atomic and solid state physics, a useful unit of energy in particle
and nuclear physics is the electron volt.
This is the amount of kinetic
energy gained by an electron when it is accelerated through a potential
difference of one volt.
Normally the energies involved in nuclear
reactions are millions of electron volts (MeV) and in high energy particle
interactions they may be thousands of millions of electron volts or Giga
electron volts (GeV = 10^{9} eV).
A convenient unit for particle
masses makes use of the Einstein massenergy relationship
E = m c^{2}.
This yields a unit mass as energy divided by the square of the velocity
of light, MeV/c^{2} or GeV/c^{2}.
For example
proton mass = 938.3 MeV/c^{2}
electron mass = 0.511 MeV/c^{2}.
This system of units is extended to momentum through the relativistic
relationship for the energy of a particle of rest mass m moving with momentum
p,
E^{2} = p^{2}c^{2}
+ m^{2}c^{4}.
From this, it follows that if we express momentum in the units of energy
divided by the velocity of light (GeV/c), we have a self consistent
system in which the velocity of light is implicitly used, but its value
does not have to be explicitly put in to the calculations.
Using these units, we can write
E^{2} (GeV)^{2} = p^{2}
(GeV/c)^{2} + m^{2} (GeV/c^{2})
^{2}.
Kinetic energy is just expressed in GeV, so
E (GeV) = T (GeV) + m
(GeV/c^{2}).
The velocity of the particle in units of the velocity of light is given
by
and the relativistic gamma factor is given by
.
In the nonrelativistic limit
β → 0
we have p « m.
Thus the expression for kinetic energy reduces as follows
T 
= E − m 


= (p^{2} + m^{2})^{1/2}
− m 


= m (1 + p^{2}/m^{2})^{1/2}
− m 


= p^{2}/2m 
as expected. 
2. Crosssections and Decay Rates
The idea of crosssection arises from the simplest model of a nucleus (or
some other particle) as a completely absorbing sphere of crosssectional
area σ.
Consider a uniform beam of N particles per second per unit area
incident on a thin sheet of material of thickness dt, in which there
are n absorbing nuclei per unit volume.
The effective nuclear area for absorption is then
σndt
per unit area of sheet (assuming that
σndt « 1 so that no nuclei are hidden one behind another).
The rate at which particles are removed from the beam is then just
− dN =
Nσndt (s^{−1})
so the intensity of a beam passing through a thick sheet will decrease
exponentially with distance t into the target
N(t) = N_{0}
exp(−σn t).
This simple model in which the probability of absorption, or some other
interaction, is unity within a certain radius of the centre of a nucleus
and zero elsewhere does not correspond with physical reality, but nevertheless
the crosssection σ
is a very useful way of expressing the overall probability per nucleus
(or other target particle) that a given interaction will occur.
The unit of crosssection used in nuclear and particle interactions
is the barn, b, equal to 10^{−28} m^{2}.
In interactions
between high energy particles, smaller units such as the millibarn (10^{−3}
b) or even picobarn (pb = 10^{−12} b) are often used.
In most cases there are several possible reactions between the incident
and target particles, and the crosssection for each will be different.
These individual crosssections are known as partial crosssections,
and their overall sum is the total crosssection.
After a reaction or scattering has occurred the outgoing particles often
have an anisotropic distribution, with different energies at different
directions.
Then the number of particles scattered per second into
solid angle dΩ at
(θ,
φ)
with respect to the incoming beam is written
N_{0}
σ
n t f(θ,
φ)
dΩ
where ∫f(θ,φ) dΩ = 1
and
f(θ,φ)
is the angular distribution for the process.
The product
σf(θ,
φ)
is usually written as a single function
dσ(θ,
φ)/dΩ
 the differential crosssection.
The partial crosssection
for the process can be obtained by integrating the differential crosssection
over all solid angles.
Very often there is no dependence on
φ
and the integral reduces to
.
Consider the interaction
a + A
→ B
+ b
where a beam of particles of type a strikes a target of type A.
Per unit target particle (A), the transition rate, or interaction
rate, is just
σ Φ,
where Φ
is the flux of incident particles (a).
Thus a useful definition
of the crosssection for the interaction is the transition rate W
per unit incident flux per target particle.
In quantum mechanics,
the value of W is given by the product of the square of the matrix
element M_{fi} between initial (i) and final (f) states
and a density of final states or phase space factor D_{f}.
The matrix element contains all the dynamical features of the interaction
such as its strength, energy dependence and angular distribution.
This formula is often known as Fermi's Golden Rule, and is applicable to
nuclear reactions, decay processes, atomic transitions, etc.
3. Kinematics of Nuclear and Elementary
Particle Reactions
When we discuss a reaction, we normally assume that in the initial state
the incoming particle and target are well separated and so noninteracting,
and that this is also true of the produced particles in the final state.
Thus there are no forces between them and no potential energy term in the
expression for the total energy
E = T + m or
E^{2} = p^{2} + m^{2}.
(Here again, the momenta and masses are expressed in units of energy/c
and energy/c^{2} respectively.)
It is very often useful to consider processes in different frames of reference,
having relative motion between them.
Note that these are just different ways of describing a given process.
If a reaction occurs or is allowed in one frame, then it occurs or is allowed in any other.
Note also that frames of reference are not specifically a feature of special relativity.
Individual kinematic properties of particles (such as velocity, energy or momentum)
will obviously be different in different frames.
Although a calulation can be carried out in any frame, some frames are particularly simple.
For a single particle, the rest frame is often useful.
Here there is no motion, so E = m and p = 0.
For a system of particles, the centreofmass (or C of M) frame may be appropriate.
This is the frame moving with the velocity of the centre of mass of the system.
Although individual particles (in general) have nonzero momentum in this frame,
the vector sum of their momenta must be zero,
Σ_{i} p_{i} = 0).
(This frame is therefore also sometimes known as the "zero momentum frame".)
At the end of a calulation, kinematic properties normally have to be returned in the laboratory
frame, in which the problem was originally posed.
For processes involving elementary particles, it is usually not reasonable
to use nonrelativistic approximations.
As a general example of calculations involving kinematic quantities, consider
the motion of any group of (noninteracting) particles with individual energies
E_{i}
and momenta p_{i}.
This motion can be divided into two components:

the absolute motion of the centre of mass of the group,
and

the relative motion of each particle with respect to that centre
of mass.
The total energy and momentum are just
E = Σ_{i} E_{i}
p = Σ_{i} p_{i}
The effective rest mass of the group W is defined by
W^{2} = ( Σ_{i} E_{i})^{2}
−  Σ_{i} p_{i}
^{2}.
The absolute velocity of the centre of mass of the group in the laboratory is
.
W, just like the rest mass of a single particle, is independent
of the relative motion of the observer.
For example, if the observer
is moving with a velocity β
along the z axis (in the laboratory), then employing the Lorentz
transformation, the observed total energy and momentum of the particles
will be
E'  = γ β E − γ β p_{z} 
p_{z}'  = γ β p_{z} − γ β E 
p_{y}'  = p_{y} 
p_{x}'  = p_{x} 
Therefore
(W')^{2}  = (E')^{2} − p'^{2} 
 = γ^{2} E^{2}(1 − β^{2}) − γ^{2} p_{z}^{2}(1 − β^{2}) − p_{y}^{2} − p_{x}^{2} 
 = E^{2} − p^{2} 
 = W^{2} 
i.e. W is invariant  the same in all frames.
The above invariance can be expressed by writing p, E
as the components of a four vector with p_{1} = p_{x},
p_{2} = p_{y},
p_{3} = p_{z} and
p_{4} = iE.
The square of the length of this 4vector is
p^{2} = Σ_{μ} p_{μ}^{2} =
p_{1}^{2} + p_{2}^{2} + p_{3}^{2} + p_{4}^{2} =
p^{2} − E^{2} = −m^{2}
(or −W^{2})
which is relativistically invariant.
The use of 4vectors can
facilitate relativistic calculations of elementary particle interactions,
compressing the conservation of energy and momentum into a single expression.
Note that in transforming to the centre of mass frame (i.e. that frame
in which the centre of mass of the group is at rest, so that
Σ_{i} p_{i}
= 0), we have
W = Σ_{i} E_{i}
= Σ_{i} m_{i}
+ Σ_{i} T_{i}.
The mass of the system is equal to the sum of the masses of the constituents
plus the total "internal" kinetic energy of the group.
Since this
latter must be positive, the mass of the system exceeds the sum of the masses
of the constituents.
Remember that in a decay or interaction, E and p are conserved.
This means that the combination E^{2} − p^{2}
is both invariant and conserved.
It is therefore the same in any frame and at any time  before or after an event.
This is a property we will make frequent use of!
Supplementary Material
For further, nontechnical reading, you might like to consult
The Ideas of Particle Physics,
chapters 1 to 4.
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