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Nuclear and Particle Physics

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									      PHYS 490/891 – Winter 2010




Nuclear and Particle Physics




            Lecture Notes




            Wolfgang Rau

          Queen’s University
1                             PHYS 490/891 – Winter 2009                                                L01


1. Introduction
1.1 From Elements to Particles
The question of what the world around us consists of is as old as the
history of the human mind. Many old cultures classified the material
world in a small number of elements. The ancient Greeks e.g.
considered Earth, Air, Fire and Water as the basic elements.
Sometimes additional non-material elements were considered like Idea,
Aether (Ether) or Light.
                                                                       Air

                                                            wet                    hot
     Figure 1: Four Elements in
     ancient Greece                            Water                                        Fire


                                                        cold                      dry

                                                                       Earth
With the advancement of science it was realised that there are quite a
few more distinguishable elements. Nowadays the term elements
usually refers to the roughly 100 chemical elements.
      Group → 1    2     3    4   5    6   7   8   9   10    11   12    13   14   15   16   17     18
      Period ↓
           1

          2

          3

          4

          5

          6

          7


              Lanthanides

                  Actinides


Figure 2: Table of chemical elements (http://en.wikipedia.org/wiki/Chemical_element)

What was earlier considered the four elements could now be viewed as
the states of matter (Earth → solid; Water → liquid; Air → gas; and
perhaps Fire → plasma).
Even though suspected since a long time we learned only at the
beginning of the 20th century for sure that matter is not continuous but
rather composed of smallest quanta, the atoms. Atoms were considered
as small (~ 10    m) solid balls of the respective substance.

    Figure 3: Matter
    composed of atoms
    in different states
                                      solid                  liquid                          gas
2                   PHYS 490/891 – Winter 2009                       L01

The history of the atom as atomos (from Greek: indivisible) did not last
very long: almost at the same time when the existence of atoms was
confirmed the investigation of cathode rays and radioactivity showed
that atoms could emit constituent (sub-atomic) particles. Based on the
work of Thomson, Rutherford, Bohr and others a model of the atom
was developed which involved a tiny (~10       m) positively charged
nucleus and an atomic shell made out of electrons.

       Figure 4: Bohr atom with
       nucleus and orbiting electrons
       (not to scale)



This opened the door to study the atom as such, creating the field of
atomic physics. But at the same time it became clear that radioactivity
involved changes in the atomic nuclei.
So the nuclei themselves are composed objects. The lighter elements
have mostly masses which are roughly integer multiples of the mass of
the hydrogen atom and so might just be composed of such. But this is
not true for all elements. It was realised that there were different types
of atoms for certain elements with the same chemical properties but
different masses (called isotopes). The explanation of isotopism
followed in the 1930s when Chadwick discovered the neutron. With
this, the material components of the matter surrounding us were
identified: all matter is made of atoms; atoms consist of nuclei and
electrons while nuclei are composed of protons (the nuclei of hydrogen
atoms) and neutrons. Protons and neutrons are called nucleons.
Still missing was the answer to the question what it was that held the
nuclei together, overcoming the strong repulsive Coulomb force which
pushes the protons apart.
Also in another aspect this was not the end of the story: in the attempt
to better understand radioactivity – which had led to the conclusion
that nuclei are not fundamental but composed objects – the cosmic
radiation was discovered which in turn held new surprises in the form
of additional particles which were not part of the usual matter. So
beyond atomic physics there was now nuclear physics, trying to
understand the rich phenomenology of atomic nuclei, and the new field
of particle physics.
These three branches of physics are closely connected. The two main
tools used in all three branches to investigate the respective objects are
spectroscopy – investigating the radiation emitted when a system
transitions from an excited state to the ground state (or to another
excited state) – and the scattering of probes (high energetic particles)
off the object of interest, which is governed by the de Broglie relation
between momentum and wavelength: the smaller the object of interest
3                  PHYS 490/891 – Winter 2009                        L01

is the shorter a wavelength and consequently higher momentum of the
probe is required.
Rutherford’s famous experiment which showed the existence of nuclei
was of the latter type while one of the basic inputs for Bohr’s model of
the atom was the Balmer formula which is based on spectroscopy.
Gamma radiation from radioactive nuclei is the exact equivalent to the
emission lines from atoms, telling us about excited states in nuclei,
while Rutherford-like experiments were used to investigate the
structure of nuclei showing that protons and neutrons are themselves
again composed of smaller components.
Scattering experiments beyond Rutherford became possible with the
advancement of the technology of accelerating particles which then
served as probes with sufficient energy to study the small structures.
The same technology is the basis of particle physics where by means of
Einstein’s relation between energy and mass new particles with high
masses can be produced if enough energy is available from the
colliding primary particles. A large variety of new particles has been
found in this type of experiments. Many of these new particles are
compound particles (similar to the nucleons) and again from the decay
of excited states we can learn about their components and interactions.
Today we know of two different types of particles: the quarks, which
are the constituents of the nucleons and other compound particles, but
are never observed as free particles, and the leptons, to which e.g. the
electron and the neutrino belong. There are six of each type (plus their
anti-particles) and they can be grouped into three generations of
families with recurring properties, seen as indication by some that we
are looking at composed instead of elementary objects once more.
The subject of this course is Nuclear and Particle Physics, but we may
from time to time refer back to Atomic Physics and use analogies to
facilitate our understanding of the new phenomena.

1.2 Scales and Units
All three branches of physics mentioned in the last section have their
own typical length and energy scales. Lengths are measured in units
derived from the meter, while the usual unit for energies is the
electron-volt ( V, the energy a particle with the charge of an electron
gains in a potential of 1 V.
                         1 V 1.602 10            J
The relevant length scales for Atomic Physics is the size of the atom
which is in the order of 10    m     1 Angstrom (or Ångström, ).
The typical energies for atomic processes are from a few V up to
several tens of k V.
The length scale for Nuclear Physics is the size of an atomic nucleus
(order of 10     m ). A very important quantity is the cross section
which is related to the probability for a given interaction to occur. To
4                  PHYS 490/891 – Winter 2009                         L01

first order this can be visualized as the geometrical cross section of the
involved interaction partners. The typical unit in use is the barn (b):
1b       10     m . However the cross section depends very strongly on
the type of interaction and the energy and can range from less than
10      b (fb) to thousands of barns. The energies for nuclear processes
are in the order of M V (10 V; typical binding energy for a nucleon
in a nucleus is 7 8 M V).
The length scale for Particle Physics is at 10     m (1 femto meter, fm
also called fermi) and below. The size of a proton is roughly 1 fm. The
size of elementary particles (like quarks or the electron) is not known.
We only know that it is less than about 10      m without indication
that they have any finite size at all. They might as well be point-like.
To produce new particles the necessary energy is given by Einstein’s
formula
                                           ⁄
Consequently the masses of particles are usually measured in V/ .
Masses of known elementary particles range from below 1 V/ for
neutrinos up to ~170 G V/c for the top quark. Typical energies are
in the G V (10 V) range; the highest energies produced at particle
accelerators to date are about 1 T V (10       V) for protons. For the
description of scattering processes the momentum is needed which is
usually measured in V/c. In many cases the interactions in particle
physics involve highly relativistic particles for which the relationship
between energy and momentum is the same as for photons:                .
The momentum is also relevant to determine the wavelength of a given
particle. According to de Broglie the wavelength is given by
                                       ⁄
with the Planck constant . An easy-to-remember form of this constant
is as reduced Planck constant multiplied by the speed of light:
                     ⁄2      197.3 MeV fm       200 MeV fm.
To resolve structures inside the atomic nucleus (~ 10 fm) one needs
particles with a wavelength in the same range which corresponds to a
momentum of ~20 M V/ or an energy of ~20 M V if electrons are
used, which are highly relativistic in this energy range (note that this is
very close to the typical nuclear binding energies of 7 8 M V
mentioned above). For the typical size of an atom (10       m 10 fm)
we find a corresponding energy of the order of k V which is in the
above discussed range of typical atomic energies.
For theoretical calculations particle physicists often use the so called
natural units where            1. In those units masses and momenta are
measured in V (or M V, G V) instead of V/c or V/c. However it is
easy to get confused in this system so we will mostly account for
factors and (or ) explicitly.
5                  PHYS 490/891 – Winter 2009                        L02


1.3 Relativistic Theories and the four Forces
Schrödinger’s Equation is derived from the non-relativistic relation
between energy and momentum:
                                                                    (1.1)
where is the potential energy (but technically should also include the
rest energy      of the respective particle). The second ingredient is
the observation that for a plain wave we can determine energy and
momentum by derivation with respect to time and space. Using
complex plain waves          exp               we find:

                                   and                              (1.2)
So we can write the above equation (1.1) in terms of operators:

                               with                                 (1.3)
where we call    the Hamilton Operator or Hamiltonian.
We now switch to the correct relativistic energy momentum relation
(we consider here only the case without potential energy)

                                                        m c         (1.4)
If we want to write this in terms of operators in the same general form
as in (1.3) we find a Hamiltonian of the form
                                    · ̂                             (1.5)
with the momentum operator ̂ . When Dirac first wrote this equation
down for relativistic electrons he found that and cannot be just
numbers but in the simplest are case 4×4 matrices. Accordingly, the
solution is a vector of wave functions with four components:
                                          x, t
                                          x, t
                            x, t                                    (1.6)
                                          x, t
                                          x, t
Two of the wave functions correspond to the positive energy solution
of equation (1.4) and represent the two spin states ( 1⁄2). The other
two wave functions correspond to the negative energy solution.
Investigating these solutions for an electron it turns out these states
behave exactly like a particle with positive charge, but otherwise the
same properties as the electron. This led to the prediction that for each
particle there should be an antiparticle with the opposite charge. This
was verified for the case of the electron by the discovery of the
positron (the anti-electron) in 1933 and for all other particles as well
since then. In general, antiparticles are symbolized by a bar (e.g. the
proton is symbolised by a while we use for the anti-proton).
6                  PHYS 490/891 – Winter 2009                         L02

This solution is completely symmetric and it is arbitrary which of the
solutions we correlate with particles and which one with anti-particles.
The only reason to call the electron a particle and the positron an anti-
particle is based on the fact that we know electrons longer than
positrons.
In non-relativistic quantum mechanics we usually discuss particles in
certain potentials. However the potentials themselves are not part of
the theory, they are added ad-hoc. A consistent physical theory has to
include a description of these potentials. Theories which deliver such a
description are called field theories since they describe fields in
addition to particles. The best possible description of course should
include relativistic effects and should be consistent with quantum
mechanics, which leads to relativistic quantum field theories.
It turns out that in these theories we find a new phenomenon: not only
the particles are discrete and the states they are in, but also the fields
are quantised. The continuous and quasi-static potential has to be
replaced by quanta which are continuously exchanged between the
interacting particles. Each type of interaction has its own types of
quanta which are exchanged. For the electro-magnetic interaction it
turns out that the exchange particle is the photon.
Beyond the electro-magnetic force there are three other known forces.
Gravity is important at large scales, but is almost always negligible
when dealing with small objects such as elementary particles.
The force that holds nuclei together, overcoming the strong Coulomb
repulsion of the protons, is a consequence of the strong force which
holds to quarks inside the nucleons together, similar to the chemical
forces (like the van-der-Waals force which is a consequence of the
incomplete cancellation of the electric forces inside the atoms or
molecules). The strong force has only a short range and the respective
exchange particles are called gluons. The strong force acts between
particles that carry the “charge” of the strong force, which comes in
three variations (as opposed to the electrical charge which only comes
in two different varieties, referred to as positive and negative) and is
usually referred to as color in analogy to the three colors that produce
white light (red, green, and blue). Be aware that this has nothing to do
with an actual color! For anti-particles we find the respective anti-
colors (anti-red, anti-green, and anti-blue). Quarks, as well as the
gluons themselves carry a strong charge, but leptons do not.
The fourth force is the weak force which is e.g. responsible for the
radioactive beta decay. The exchange particles for this force are the
W , the W and the Z . All three are very heavy and can only exist for
a very short time, which makes this force very short range and weak.
All particles participate in the weak interaction.
While all the particles discussed in the previous section (leptons and
quarks) are fermions, the exchange particles are all bosons.
7                   PHYS 490/891 – Winter 2009                         L03

1.4 Symmetries
Symmetries play an important role in physics since they are always
correlated with conservation laws. The most important conservation
laws (energy, momentum, angular momentum) are correlated to
continuous symmetries: conservation of energy follows from the fact
that a process independent of the point in time at which it occurs; the
corresponding transformation is                    . Conservation of
momentum is a consequence of the fact that it does not matter where in
space a process occurs (transformation:                      and the fact
that the orientation in space does not matter leads to the conservation
of angular momentum (                       ). All these conservation
laws are additive (i.e. the sum of the energies, momenta or angular
momenta in a system is conserved).
In addition to these continuous symmetries there is a set of discrete
symmetries which are relevant for all fundamental interactions. The
most important are Parity, Charge Conjugation, and Time Reversal.
Parity refers to spatial reflection. We introduce the parity operator
which produces the transformation:              . If we apply this
operator to a wave function we find:
                                                                       (1.7)
So if       is an eigenfunction of the eigenvalues can only be 1. If
the potential in Schrödinger’s equation is symmetric:                ,
then the Hamiltonian in (1.3) is symmetric as well and the solution
consequently can be expressed in terms of eigenfunctions of the parity
operator. We therefore find that free particles (this implies   0
which is obviously symmetric) have an intrinsic parity which can be
positive or negative (                or                    ).
For a spherically symmetric problem the solutions are usually given in
polar coordinates , and with
                sin cos ,           sin sin     and           cos .    (1.8)
They can be expressed as a product of a function         of the radius
and a generic set of functions of the angles, the spherical harmonics
      , . The parity operator does not change but affects the angles:
                                   and                    .            (1.9)
From the definition of the spherical harmonics follows:
                ,                   ,                 1        ,      (1.10)
and consequently
                        ,                1        ,            1      (1.11)
From this we can see that if a situation is described by a set of wave
functions             the parity of the combined wave function is the
product of the individual parities. A detailed analysis of the relativistic
8                   PHYS 490/891 – Winter 2009                          L03

wave equation discussed above shows that the product of the intrinsic
parities of a particle and its antiparticle is negative. Since the parity of
a single particle cannot be measured, positive parity is arbitrarily
assigned to particles and negative parity to antiparticles.
Charge conjugation (usually symbolised by the charge conjugation
operator ) is the operation which transforms a particle into its
antiparticle. In this operation not only the charge is changed but also a
number of other basic properties (e.g. the magnetic moment or
quantum numbers like the lepton number which we will discuss later).
Like the parity operator the charge conjugation operator has
eigenvalues 1. A charged particle can obviously not be an eigenstate
of this operator but we can construct eigenstates by combining
particles and antiparticles appropriately. The eigenvalue is then again
determined by the angular momentum (in this case including the spin).
The third important discrete symmetry is time reversal. Applied to a
given situation it reverses all processes in time. However it is slightly
different from the other two discussed discrete symmetries: if is the
time reversal operator we have
                                               .                      (1.12)
So this operator does not just reverse the time but it also transforms the
wave function into its complex conjugate.
Most interactions conserve these symmetries. If we have e.g. as initial
state an excited state of an atom and as final state the same atom in the
ground state plus a photon which is emitted in this process then the
parity of initial and final state will be the same. Consequently there are
some transitions in atomic physics which are no allowed because they
would violate the parity conservation law. However there are
interactions which do not conserve parity. As far as we know the
combination of all three of these symmetries (usually referred to as
     ) is always conserved.

1.5 Interactions
In a very condensed form we can write down particle interactions like
chemical reactions. The decay of a neutron can e.g. be written as:
                                                                      (1.13)
All quantities which are conserved in an interaction have to be the
same on both sides of this equation. Like in a mathematical equation
we can shift terms from one side to the other if we make sure that all
conservation laws are obeyed. In a mathematical equation we have to
invert the terms or switch their signs depending on the operation we
perform. Here we have to replace particles by antiparticles. We may
e.g. take away the electron on the right side of (1.13) and compensate
for this by adding an anti-electron (or positron) on the left side:
                                                                      (1.14)
9                    PHYS 490/891 – Winter 2009                      L03

The basic physics which describes both these processes (1.13) and
(1.14) is the same.
Richard Feynman invented a very intuitive way to write down particle
interactions in a graphical form, the so-called Feynman diagrams. In
addition to the initial and final state which we find in the above format
the Feynman diagrams also include the exchange particle. One
possible Feynman diagram for process (1.14) is shown in Fig. 5.

    Figure 5: Feynman diagram
    for process (1.14). Time is
    going from left to right; the
    exchange particle for this
    process is a boson.


Note that the arrows for antiparticles (positron and antineutrino in this
case) point backwards in time. The could be a             going from the
upper vertex to the lower one or a        going the other way. There are
slight variations in the conventions for drawing Feynman diagrams, the
most obvious being that the flow of time might go in a different
direction (e.g. bottom to top) but this is usually easily recognizable.
We will use straight lines for particles and anti-particles (with arrows
forward and backward in time respectively), dashed lines for and ,
wavy lines for photons and spirals for gluons.
The points where lines meet are called vertices and are of special
importance in a Feynman diagram since this is where things happen.
For most purposes we will only deal with Feynman diagrams with
three-line-vertices. There are rules how to convert such diagrams into a
calculation for the probability for the respective process to happen, but
in this course we will use these diagrams mostly just as a convenient
visualisation and for order-of-magnitude estimates. All conservation
laws have to be obeyed at each vertex. However, as we know from
Heisenberg’s Uncertainty Principle, certain quantities may not be well
defined. If e.g. the time between the two interactions in Fig. 5 (the two
vertices) is very short, the energy transferred at each vertex is not very
well defined. In our example the exchange particle is very heavy, so
the incoming positron cannot possibly decay into a neutrino and a
and still conserve energy. However, for a short time we can ‘borrow’
energy to produce the exchange particle. Another way to think of this
is saying the exchange particle is emitted with negative energy. At the
other vertex where the exchange particle is absorbed we have to pay
back the ‘debt’ so that in the total process energy is conserved.
From this perspective we can understand why the weak force with its
heavy exchange particles has only a short range: if the interaction
partner is too far away the exchange particle cannot reach it in time to
pay back the energy debt and has to be re-absorbed by the emitter, so
no interaction happens.

								
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