% This is the chapter of my thesis in which I describe the physics of
% particle transport through a medium.  
\def\Moller{M\o{}ller}
\input thesis.sty
\chapter{Background}

EGS, an ``Electron-Gamma Shower'' simulation, uses Monte Carlo methods
to simulate ``the coupled transport of electrons [, positrons,] and
photons in an arbitrary geometry.''  This chapter discusses the ways
in which EGS performs this simulation.

The EGS system simulates a shower of particles, one at a time, as they
pass through various media.  The effects simulated for photons are
pair production, Compton scattering from ``free'' electrons, and
photoelectric absorption.  Those simulated for charged particles
(electrons and positrons) are Coulomb scattering from the nucleus and
from the atomic electrons, electron-positron annihilation, and
bremsstrahlung.\foot{\egs, p.~15} This chapter describes these
physical processes and how they are simulated.

The simulation relies on the {\it cross-sections} of the various
simulated interactions.  The cross-section of an interaction, which
has dimensions of area, is proportional to the probability of the
given interaction occuring during passage through a thin layer of
material at any moment.  The sum of the cross-sections (which are
discussed below) gives the probability of any interaction occurring;
this can also be used to find the mean path length between
interactions.  EGS uses this mean free path length to determine when
to simulate an interaction; it uses the relative cross-sections to
determine which interaction has occured.

A medium in the EGS system is a homogeneous region described by the
cross-sections for various interactions.  These cross-sections are
precalculated as piecewise-linear functions of energy by a program
called PEGS, which bases its calculations on the composition and
density of a substance.  The distribution of the various elements is
assumed to be uniform throughout the medium.  At a given energy, each
medium will have different values of the cross-sections.  Because of
this, EGS uses mean free path lengths (which also change) to determine
how far a particle will travel before its next interaction.

> Photons

A photon passing through a medium is subject to a number of effects.
If the photon has sufficient energy, it may create an electron and
a positron.  The photon may be deflected by Compton scattering from
electrons near its path.  Finally, a photon whose energy exceeds the
ionization energy of an atom in the medium may be absorbed by an
atomic electron, releasing it and ``creating'' (from the perspective
of the simulation) a free electron.

>> Pair Production

A photon, characterized by its energy $\hbar\omega$ and momentum $\vec
p$, can produce an electron and positron under certain conditions.
These conditions are that the energy $\hbar\omega$ be sufficiently
large that it can form the rest mass of the two particles, and that a
nucleus be nearby to permit conservation of both energy and momentum.

Gasciorowicz [1974] characterizes the calculations involved as QED that
are ``beyond the scope of this book.''  The EGS documentation contains
a complicated derivation (pp.~46--48) for the cross-section.  This
derivation makes certain assumptions which are not valid below 50~MeV.
For kinetic energies below this level, the cross-section for pair
production is empirical, taken from Storm and Israel.\foot{\egs,
p.~31}

>> Compton Scattering

In the classical model, Compton scattering is explained by having the
photon accelerate a free electron, which in turn (having been
accelerated) radiates a new photon.  This is Thomson scattering, which
provides a basis for the discussion of Compton scattering.

The determination of the Thomson cross-section $\sigma_T$ follows the
following procedure.\foot{Based on derivations in Bekefi [1977] and
Jackson [1962].} A free electron is to be assumed located at the
origin, with a planar, monochromatic, linearly polarized
electromagnetic wave propagating in the positive $z$ direction.  The
electric field of this wave is thus given by
%
$$\vec E = E_0 \cos(\omega t - kz) \hat x$$
%
The free electron, having mass $m$ and charge $e$, undergoes a
Coulomb acceleration of
%
$$\vec a = (e/m) \vec E = (e/m) E_0 \cos(\omega t - kz) \hat x$$
%
As a result of this acceleration (which is assumed to be
non-relativistic), the electron radiates energy.  The average power
radiated per unit solid angle is given by
%
$$\left\<dP/d\Omega\right\> = {c/8\pi} E_0^2 (e^2/mc^2)^2$$
%
The cross-section $\sigma_T$ is the integral over all solid angles of
the average power radiated per unit solid angle divided by the
time-average incident energy flux.  In this case, the incident energy
flux is given by the Poynting vector, whose time average is
%
$$\<S\> = (c/8\pi) E_0^2$$
%
The Thomson cross-section is therefore given by
%
$$\sigma_T = \int{\left\<dP/d\Omega\right\>/\<S\>} d\Omega
= \int{(c/8\pi) E_0^2 (e^2/mc^2)^2 / (c/8\pi) E_0^2} d\Omega
= {8\pi/3} (e^2/mc^2)^2$$
%
It should be stressed that this is the Thomson cross-section, which
corresponds to the classical, non-relativistic model of
photon/free-electron interaction.  Adjustments are needed to arrive at
the Compton model.

Compton's adjustment was to permit the transfer of momentum from the
photon to the free electron.  This changes the problem from that of a
free charge accelerated by an incident electric field to that of a
two-particle collision.  Gasciorowicz [1974] quotes the formula of
Klein and Nishina:
%
$$\sigma_C = 2\pi (e^2/mc^2)^2 \left\{{1+x/x^2}
[{2(1+x)/1+2x} - {1/x}\log(1+2x)]
|{1/2x}\log(1+2x) - {1+3x/(1+2x)^2}\right\},$$
%
where $x = {\hbar\omega\char`/mc^2}$.  This is the cross-section used by the
EGS system for Compton scattering.\foot{\egs, p.~52}

>> Photoelectric Effects

A photon that ``collides'' with an electron which is orbiting a
nucleus may knock that electron out of the atom, thereby creating an
ion.  The cross-section for this ``photoelectric'' process cannot be
easily derived.  A general discussion, though, is possible.

\figure
\vskip 7cm
\caption{FIGURE 1.  
\vtop{\baselineskip=13pt%
\hbox{Mass absorption coefficient versus wavelength, platinum.}
\hbox{From page 416 of Gasciorowicz [1974].}}}
\endfigure

The adjoining figure (from Gasciorowicz [1974]% page 416 
) shows the relationship of the cross-section (actually the mass
absorption coefficient, which is proportional to $\sigma_{ph}$) to
wavelength, which is inversely proportional to the energy.  The steep
jumps in the graph correspond to the binding energies of the various
electrons in platinum.  The K-edge indicates the ionization energy for
the $n=1$ electrons, and the L-edges indicate the three pairs of $n=2$
electrons.\foot{\gas{415\it f}}

In the EGS system, the K-edges for the elements through $Z=100$ are
obtained from empirical data of Storm and Israel.\foot{\egs, p.~92}  

> Charged Particles

A charged particle (electron or positron) passing through a medium is
subject to four main interactions.  The particle may undergo Coulomb
scattering with a nucleus, which is an elastic process.  It may
undergo Coulomb scattering by atomic electrons, which is an inelastic
process.  An electron and a positron may annihilate one another,
resulting in an emitted photon.  Finally, bremsstrahlung will occur
when a charged particle emits a photon in the vicinity of a nucleus.

>> Coulomb Scattering, Bremsstrahlung

Atomic nuclei are massive (compared to electrons), and have a stronger
charge.  For these reasons, they can be treated as sources of strong,
static Coulomb fields.  In a particular nucleus/free-particle
interaction, an incident particle is deflected and a photon is
released.  Because of this photon, the deflected particle has less
energy than the incident particle; this loss of energy is known as
{\it bremsstrahlung}, or ``braking radiation.''

Classically, the interaction can be viewed as follows.  The incident
particle is a charge in a static electric field.  This results in a
Coulomb acceleration, which causes the particle to radiate energy.
This energy comes off as a photon.  The acceleration has changed the
direction of the particle's velocity; consequently there will be
conservation of momentum among the three particles, and the nucleus
will recoil slightly.

A quantum description has a virtual photon pass between the nucleus
and the incident particle.  The deflected particle and a photon are
emitted from the incident-particle/virtual-photon collision.  Again,
conservation of momentum has the nucleus recoil slightly.

The differential cross-section of this interaction is given in Jackson
[1962] as
%
$$d\sigma_{\rm nuc} = (2 Z e^2 / pv)^2 {1/(2 \sin(\theta/2))^4}
d\Omega,$$
%
where $e$ is the charge of an electron, $Z$ is the atomic number, $v$
is the velocity, $p$ is the relativistic momentum $\gamma mv$, and
$\theta$ is the scattering angle.  This is not integrated over all
$d\Omega$, however; the actual values used in EGS are empirical.

EGS treats this interaction merely as bremsstrahlung, neglecting the
recoiling nucleus.  If the scattering is significantly large and the
photon cannot be neglected, EGS will also ``create'' a photon.  (The
minimum energy below which the photon is ignored is set by the user.)

>> Coulomb Scattering from Atomic Electrons

A charged particle can undergo an inelastic interaction with an atomic
electron.  Classically, this is a Coulomb interaction; in quantum terms
there is an exchange of a single photon which alters the trajectories
of both particles.  If the incident particle is an electron, the
interaction is \Moller\ scattering.  If the incident particle is a
positron, the interaction is Bhabha scattering.

The derivations of the cross-sections for these two cases are quite
different and quite complicated.  They may be found in the EGS
documentation on pages 56 through~61.

>> Annihilation

A positron passing through a medium can collide with an atomic
electron, resulting in the annihilation of both particles and the
emission of one or two photons.  The nucleus will absorb whatever
momentum the photon(s) fail to carry away.  The EGS documentation
cites Messel and Crawford\foot{\egs, p.~65} as stating that
single-photon annihilation is sufficiently rare that it can be
disregarded in this context.  The documentation also asserts that
annihilation with three or more photons are ``even less likely.''

The two-photon annihilation cross-section is given in the EGS
documentation\foot{EGS, p.~62, citing Heitler.} as
%
$$\sigma_{Annih}(E_0) = (M/N_a\rho) {\pi r_0^2/\gamma+1}
[{\gamma^2+4\gamma+1 / \gamma^2 - 1}
\ln(\gamma+\sqrt{\gamma^2-1}) - {\gamma+3/\sqrt{\gamma^2-1}}],$$
%
where $\gamma={E_0/m}$, $E_0$ is the energy of the incident positron
(in MeV), $n$ is the electron density (in electrons per cm$^3$), $m$
is the electron rest energy (in MeV), and $r_0$ is the classical
electron radius (in cm).  The prefactor $M/N_a\rho$ converts the
macroscopic cross-section to the total cross-section, by adjusting
for the molecular mass, and density.

>> Multiple Scattering

An electron traversing a sample of matter will be buffeted, as it
were, by a number of elastic collisions that leave its energy roughly
unchanged but its direction somewhat altered.  The cross-section for
such interactions is roughly proportional to $Z(Z+\xi_{MS})$, where
$\xi_{MS}$ is what EGS calls the ``multiple-scattering fudge
factor.'' This factor accounts for scattering due to collisions with
atomic electrons, where such collisions are sufficiently minor that
they do not cause a photon to be emitted.  (Collisions in which a
photon has been emitted have been treated earlier under the heading of
\Moller\ scattering.)

> Summary

This section enumerated the interactions that can occur between a
photon or a charged particle and the atoms of a medium through which
the photon or charged particle is passing.  Each interaction was
described physically, and a cross-section for each interaction was
either derived or stated.  The next section describes how these
processes are handled in the EGS simulation.

> Structure of the program

EGS exists as a framework into which the user inserts routines to
create the specific experiment which he wishes to simulate.  The EGS
framework includes routines that handle the simulation of interactions
between the transported particles and the media through which they
travel.  The specific interactions which are simulated were described
above.

The user supplies routines which specify the geometry and the data
collection.  How these routines are combined with the EGS framework is
the topic of the next chapter.  This chapter focuses on the EGS
simulation routines.

> Particle Transport

EGS treats a shower one particle at a time.  The current particle is
moved in a straight line until one of the following events takes
place:

\bulletlist
\item The particle interacts with the current medium.
\item The particle reaches a boundary between two media.
\item The particle is terminated by the user's routines.
\endlist

The first event, an interaction with the current medium, is handled by
EGS.  The determination that the particle is about to cross a boundary
is handled by the user routine \verb.HOWFAR..  The decision to end a
particle's simulation (usually because it has entered the detector or
has escaped from the geometry) is also handled by \verb.HOWFAR..  

An interaction with the current medium occurs randomly, based on the
cross-sections of the various interactions.  The probability of an
interaction occurring in a distance $dx$ is given by $dx/\lambda$,
where $\lambda$ is the mean free path length.  That, in turn, depends
on the total cross section $\sigma_t$ by
%
$$\lambda = {M/N_a\rho\sigma_t}$$
%
where $M$ is the molecular mass, $N_a$ is Avogadro's number, and
$\rho$ is the density of the medium.  (These convert $\sigma_t$, the
molecular cross-section, to $\Sigma_t$, the macroscopic total
cross-section.)  If $N_{\lambda}$ is the number of
mean free paths traversed, an exponential probability density function
is obtained:
%
$$Pr\{N_0 < N_\lambda\} = 1 - e^{-N_\lambda}, \qquad \hbox{for}
N_\lambda>0$$
%
EGS arrives at an appropriate distribution of $N_\lambda$'s by
choosing $\zeta$ uniformly between 0 and 1, and setting
%
$$N_\lambda = -\ln\zeta$$
%

Using that distribution, the main loop of the EGS simulation can now
be specified.\foot{This is derived from EGS, p.~15} The steps taken with
photons are the following:
%
\enumeratedlist
\item Select $N_\lambda$.  
\item Let $t_1=\lambda N_\lambda$.  This is the
distance to the next interaction, assuming that the photon will
still be in the same medium.
\item Compute $d$, the distance to the next boundary in the direction
along the photon's path.
\item Take $t_2 = \min(t_1,d)$.  Advance the photon by distance $t_2$.
\item If $t_1$ was less, then exit the loop and terminate.
\item Otherwise, the photon has crossed a boundary.  Subtract
$t_2/\lambda$ from $N_\lambda$ (since the number of mean paths to
travel until the next interaction has not yet been reached).  Permit
data to be collected.  If the new region consists of a different
medium than the old one, recalculate $\lambda$.  In any case, resume
from step 2.
\endlist
%
Step 3 is handled by the user's \verb.HOWFAR. routine.  The data
collection in the final step is done by the user's \verb.AUSGAB.
routine.  When the loop is exited due to an interaction, that
interaction is randomly selected, with the probability of a given
interaction proportional to its cross-section.

Charged particle transport is more difficult, because the
cross-sections depend on the energy of the particle in such a way that
the cross-section grows without bound as the energy approaches zero.
(This is the infrared catastrophe, disguised.)  EGS deals with this by
``discarding'' particles whose energy drops below a user-specified
cutoff.  (These particles are assumed to have come to rest in the
material.)  Interactions that would produce particles or photons with
energies below these cutoffs are lumped together into a ``continuous
loss'' term, and are not treated as discrete events.  To compensate
for this loss of energy, and resulting decrease in total
cross-section, a fictitious event is created whose consequence is to
pass the particle unchanged, but whose cross-section is such that it
increases as the real cross-sections decrease, to maintain a constant
$\sigma_t$ and thus a constant $\lambda$.  The procedure for photons
can now be followed, with provision for ``multiple scattering.''  The
documentation for EGS claims that this procedure is successful in all
but low-energy problems.

When an interaction takes place, the resulting particles (both photons
and electrons/positrons) are placed in lowest-energy-first order on a
stack of particles.  The entire procedure is then repeated.  When the
current particle is discarded, the next one reaches the top of the
stack and is dealt with.  This stack-based loop will terminate when
the stack is exhausted and all the energy of the initial particle has
been accounted for.  Another initial particle is then placed on the
stack for the next simulation.

\bye