**THE PHYSICS OF**

**PARTICLE DETECTORS**

**Dan Green**

**Fermilab**

#  Introduction

The subject of particle detectors covers those devices by which the
existence and attributes of particles in a detecting medium is made
manifest to us. The full and complete understanding of these devices
requires a good understanding of basic physics. Without that knowledge
we are simply \"mechanics\" without the capacity to advance the state of
the art of detector technology. On the other hand, a rigorous
understanding from first principles is also not an optimal approach. The
\"useful\" understanding of a given device proceeds from an
understanding of what approximations to full rigor are possible. That
understanding can only come from experience and it is the purpose of
this volume to communicate that experience. The aim of this text is to
steer a perilous course between the purely descriptive and the purely
theoretical.

The role of detectors can be visualized by assuming that an interesting
interaction occurs at a point in space and time. From that point several
secondary particles of different masses are emitted with various angles
and momenta as shown in Fig. I.1. It is the job of the detector designer
to measure the time of interaction, *t*, and the vector momenta,
${\overrightarrow{p}}_{i}$, and masses, *M~i~*, of those emitted
particles. The text is organized so as to show the ensemble of tools
available to the designer. Typically, mathematical detail and topics
outside the main scope of the text are relegated to the Appendices.

![](media/image1.wmf){width="2.6798611111111112in"
height="2.529861111111111in"}

Fig. I.1: A schematic representation of the reaction A+B 1+2+3+4+5+6.
The reaction is specified when the vector momentum and mass of each
particle is determined.

A list of the topics covered is given in the Table of Contents. The
first Section is an introduction devoted to a numerical description of
the appropriate size, energy scales, and cross sections for different
processes. The numerical data given in the tables of Section 1 will
constantly be referred to in later Sections.

There then follow eight Sections on \"nondestructive\" measurements, or
those which do not appreciably change the measured particle's position
or momentum. The first subtopic concerns the measurement of time and
velocity. Section 2 starts with the basic physics of the photoelectric
effect and leads into photomultiplier tubes, scintillator and time of
flight measurement. Section 3 describes velocity measurement by way of
the emission of optical photons in the Cerenkov process. Section 4
follows with a discussion of the closely related emission of x-ray
photons in transition radiation which also determines the velocity of a
particle.

The second subtopic within nondestructive readout first lays the physics
groundwork of elastic scattering in Section 5. Section 6 then covers the
application of single scattering to scattering off atomic electrons and
the resulting energy loss. Detection of the energy lost is the physical
basis of many of the techniques used in charged particle detection. The
third subtopic then follows up with the nondestructive measurement of
the position and momentum of charged particles. Section 7 contains a
derivation of the particle trajectory in a magnetic field and the
consequent measurement of momentum. Those measurements have intrinsic
limitations which are explored first in Section 8 in studying diffusion
in gases and wire chambers. Section 9 looks at faster and higher spatial
resolution Si detectors for more accurate position measurements.

The text then switches to destructive measurements, where the particle
to be measured loses a significant fraction of its energy or is fully
absorbed in the detector. First, in Section 10, we lay the physics
foundation by exploring radiation and photon scattering. Then these
concepts are applied in exploring the topic of destructive energy
measurements. Section 11 describes measurements of electron and photon
energy, while Section 12 describes the measurement of the energy of
strongly interacting particles.

Finally, a general-purpose high energy physics detector using all the
previously described techniques is sketched in Section 13. The concept
of multiple redundant measurements is introduced and several examples
are given.

The full set of material in the text is suitable for a 1 year course. If
a 1 semester course is desired, the algebraic details in the Appendices
can be skipped as well as Section 1, the first half of Section 2 and
Sections 5, 6, and 10 assuming that no supplementary physics background
was required.

Note that the subjects covered in the text are strongly limited to
detectors themselves. Exceptions are a brief description of coincidence
circuits in Section 2 and front end noise processing in Section 9. These
brief forays were made since these special topics were tightly connected
to the detectors themselves. However, there is no other discussion of
front end electronics, trigger systems, data acquisition, or computer
programming. In addition, the vital area of detector modeling and Monte
Carlo techniques is only sketched in Appendix K. Probability theory and
statistical analysis appears only briefly in Appendix J. References to
these vital areas are given at the end of the text for readers who want
to go beyond the scope of this volume.

The aim of this text is to describe the full ensemble of particle
detectors from first principles. The goal is to strike a balance between
simply presenting the final result and a full and rigorous derivation
and thus to extract the relevant physics in a clear fashion. Intuition
and order of magnitude numerical estimates are stressed throughout in an
attempt to communicate the insights garnered from experience.

"*Ah, but a mans reach should exceed his grasp or what's a heaven for*."

Robert Browning

"*Curiosity is, in great and generous minds, the first passion and the
last*."

Dr. Samuel Johnson, 1750

## General References - A

Mechanics, Electricity and Magnetism and Quantum Mechanics

\[1\] [The Feynman Lectures in Physics]{.underline}, R. Feynman, R.
Leighton, M. Sands, Addison-Wesley Publishing Co., Inc. (1963).

\[2\] [Classical Mechanics]{.underline}, H. Goldstein, Addison-Wesley
Publishing Co., Inc. (1950).

\[3\] [Classical Electricity and Magnetism]{.underline}, W.K.H. Panofsky
and M. Phillips, Addison-Wesley Publishing Co., Inc. (1962).

\[4\] [Quantum Mechanics]{.underline}, E. Merzbacher, John Wiley & Sons,
Inc. (1961).

## General References - B

Textbooks on Particle Detectors

\[1\] [Detectors for Particle Radiation]{.underline}, K. Kleinknecht,
Cambridge University Press (1987).

\[2\] [Experimental Techniques in High Energy Physics]{.underline}, T.
Ferbel, Addison-Wesley Publishing Co., Inc. (1987).

\[3\] [Instrumentation in High Energy Physics]{.underline}, Ed. F.
Sauli, World Scientific (1992).

\[4\] [Instrumentation in Elementary Particle Physics]{.underline}, J.C.
Anjos, D. Hartill, F. Sauli, M. Sheaf, Rio de Janeiro, 1990, World
Scientific Publishing Co. (1992).

\[5\] [Instrumentation in Elementary Particle Physics]{.underline}, C.W.
Fabjan, J.E. Pilcher, Trieste 1987, World Scientific Publishing Co.
(1988).

\[6\] "Particle Detectors" C.W. Fabjan, H.F. Fisher, Repts. Progr. Phys.
[43]{.underline}, 1003 (1980).

# 1. Size, Energy, Cross Section

\"*Beauty depends on size as well as symmetry.*\"\...\...Aristotle

*\"Energy is eternal delight.*\"\...\....W. Blake

Textbooks on detectors often jump directly into a description of the
devices themselves. The relevant descriptive formulae are then simply
given without derivation and readers are instead referred to the
relevant texts. A complementary approach is taken here. A "derivation"
of the relevant physics is always attempted first. Armed with the
derivations, the reader is then introduced to the detector where
approximations which are made are explained along with the reasons why
they are valid. In order to contain the length of the text, all
"derivations" are heuristic and thus either compressed or left to the
Appendices.

Numerical examples are given at regular intervals in order that the
reader be firmly connected to real devices and have a firm grasp of the
appropriate orders of magnitude. We note that \"intuition\" is largely
the result of experience. The judgement that allows a simplifying
approximation to be made usually comes with an appreciation of orders of
magnitude of the quantities involved. In this text, that hard won
\"intuition\" is, it is hoped, passed on to the student.

Detectors function by causing a particle to interact with some detecting
medium. For example, a charged particle might ionize a gas in a device
and the freed charge might be collected as an electrical signal
localized in time, a "pulse", on a detector electrode. To characterize
the detector it is fundamental to understand the probability of
interaction of the particle with the device. The aim of Section 1 is to
provide the basic numerical data needed to later characterize the
interaction probability of the different particles which we wish to
detect.

## 1.1 Units

It is traditional in high energy physics to work in dimensionless units
where $\hslash$ and c are defined to be equal to 1. In those units
momentum (pc), energy (), mass (mc^2^), inverse time
$\left( \frac{\hslash}{t} \right)$ and inverse length
($\frac{\hslash}{x}$) all have the same dimensions, which we take to be
energy. Units for energy are taken to be the electron volt, where one
electron volt, eV, is the energy, ![](media/image2.emf), gained by an
electron of charge e, in dropping through a potential difference,
![](media/image3.emf), of 1 Volt, ![](media/image4.emf) A tabulation of
many of the physical quantities used throughout the text is given in
Table 1.1\[1\].

Table 1.1

Fundamental Physical Constants

![](media/image5.png){width="5.898611111111111in"
height="6.565972222222222in"}

(From Ref. 1.1)

In that table, the speed of light, Planck's constant, the electron
charge, the masses of elementary particles, the fine structure constant,
the classical electron radius, the Compton wavelength of the electron
etc. are gathered together in electron volt and in MKS units. This table
and Table 1.2 contain sufficient numerical information for the needs of
this text. Kinematics, the constraints imposed by energy-momentum
conservation, are worked out in Appendix A.

Table 1.2

![](media/image6.png){width="5.9430555555555555in"
height="7.301388888888889in"}

(From Ref. 1.1)

## 1.2 Planck's Constant

In going from the dimensionless calculations of high energy physics to
dimensional quantities, it is necessary to know the Planck constant and
be able to use it. (See Table 1.1.)

$\begin{array}{r}
hc = 0\text{.}2\text{GeV}\text{fm},1\text{GeV} = \text{10}^{9}\text{eV} \\
\text{2000}\text{eV}\overset{o}{A} \\
1\overset{o}{A} = \text{10}^{- 8}\text{cm},1\text{fm} = \text{10}^{- \text{13}}\text{cm} \\
\text{10}\text{nm}\text{.}
\end{array}$ (1.1)

Planck's constant is given for two different distance scales, the
Angstrom and the Fermi. Those two distance scales are characteristic of
the size of an atom and the size of a nucleus respectively. The
conversion of size to energy leads us to the immediate conclusion that
nuclear energy scales are GeV whereas atomic energy scales are of order
a few electron volts. The ratio of one Angstrom to one Fermi is 100,000
to 1.

## 1.3 Electromagnetic Units

In the body of the text we will most often use CGS units in symbolic
manipulations and we will freely convert back and forth to MKS units. It
is a fact of life that practicing physicists must acquire a facility
with both systems of units since experimentalists have connections to
both theory and engineering. (See the beginning of Section 3 for a full
explanation.) We will later explicitly give a prescription for
converting between MKS and CGS units.

## 1.4 Coupling Constants

What about the characteristic strength of the forces between particles?
These are defined by dimensionless coupling constants whose size gives
us a measure of the strength of the interaction. For electromagnetism we
have the familiar fine-structure constant, ![](media/image7.emf), which
has the strength of roughly 1/137, Table 1.1. The strong interactions
have a coupling constant ~s~ which is larger because the forces are
stronger. Why is dimensionless? The dimension of U = e^2^/r, indicated
by brackets, is energy so \[e^2^\] = \[energy length\]. The dimension of
$h$ is \[energy time\] and of $hc$ is \[energy length\]. Thus =
e^2^/$hc$ is indeed dimensionless.

$\begin{array}{r}
\alpha = \frac{e^{2}}{hc}\text{\textasciitilde}\frac{1}{\text{137}} \\
\alpha_{s} = \frac{g_{s}^{2}}{hc}\text{\textasciitilde}(\frac{1}{\text{10}} - 1)(\text{Table}1\text{.}1)
\end{array}$ (1.2)

The electromagnetic force is very familiar in everyday life. The
existence of a strong force may be inferred by the observation that
nuclei are bound systems containing one or more protons. Coulomb
repulsion of the protons must then be overcome by another, stronger
force. The strong interaction is responsible for binding the hadronic,
or strongly interacting, particles like the proton together. Nuclei are
then bound together by means of a residual "strong interaction" in
analogy to the Van Der Waals molecular binding of neutral atoms. Our
treatment of the strong interactions will be completely phenomenological
and contained only in the last two Sections. We will not discuss the
weak interactions which are responsible for radioactive decay nor the
gravitational interactions. Detectors using non-destructive methods
usually use only the electromagnetic interaction of a particle with the
detecting medium.

## 1.5 Atomic Energy Scales

Let us now turn towards a description of the energy scales of systems,
beginning with the atomic energy scale. The atom is held together by the
electromagnetic attraction between the nucleus and the \"orbiting\"
electrons. Since the proton is so much heavier than the electron,
m~e~/m~p~ \~ 1/2000 (Table 1.1), it is essentially at rest and there is
only one dynamic mass scale in the problem which is the electron mass,
as illustrated in Fig. 1.1.

![](media/image8.wmf)

Fig. 1.1: Definition of orbital radius and velocity of the electron.

The bound state lowest Bohr energy level is E~o~.

![](media/image9.emf) (1.3)

We know experimentally from spectroscopy that the energy levels of bound
systems are quantized. The hydrogen atom is characterized by a quantum
number n defining an energy E~n~ = -E~o~/n^2^. An example of data for
atomic hydrogen is shown in Fig. 1.2 where the discrete atomic spectrum
of the emitted light is obvious and the 1/n^2^ behavior of the lines is
evident. ![](media/image10.png){width="3.8333333333333335in"
height="4.180555555555555in"}

Fig. 1.2: Discrete emission spectrum for atomic hydrogen. (From Ref.
1.8, with permission) a) visible region, b) full spectrum.

Let us compute the H wavelength to give us confidence. This line is due
to a transition from n = 3 to n = 2 which causes emission of a photon
with energy (Eq. 1.3) of 1.89 eV. Using Planck's constant to convert
energy to frequency, = 2($\hslash c$)/1.89 eV = 6645 $\overset{o}{A}$ as
seen in Fig. 1.2. The ionization continuum where the atom is broken
apart and the electron is freed is also seen in Fig. 1.2.

It is easy to "derive" the Bohr energies by appealing to the
minimization of the total system energy in the ground, or lowest energy,
state. The kinetic, T, plus the potential energy, U, is the total system
energy, E, which is a conserved constant. There is a balance between the
negative attractive potential, whose magnitude increases as the radius
of the system, a, decreases and the positive kinetic energy which also
increases as the radius decreases. Since T goes as 1/a^2^, while U goes
as 1/a, there is a stable minimum. The minimum occurs at the first Bohr
radius, a~o~, which defines the atomic size scale, 0.54 Angstroms, Table
1.1. The Heisenberg uncertainty principle for position and momentum
uncertainties $Dp,Dx$, $DpDx\text{\textasciitilde}h$ supplies the
crucial element of this "derivation". Throughout this text p represents
particle momentum, and we assume that the uncertainty in p,
![](media/image11.emf), is of order p while the uncertainty in position,
![](media/image12.emf), is of order *a*.

$\begin{array}{r}
T + U = E,DpDx\text{\textasciitilde}h\text{\textasciitilde}\text{pa} \\
\frac{(Dp)^{2}}{2m} + U = E = \frac{h^{2}}{2\text{ma}^{2}} - \frac{e^{2}}{a} \\
¶\frac{E}{¶}a = 0@a = a_{o}
\end{array}$ (1.4)

The other characteristic length scale is the Compton wavelength of the
electron, $D = \frac{h}{\text{mc}}$, which is much smaller than a~o~
(see Table 1.1). The characteristic size in the atomic bound state is
the Compton wavelength divided by ![](media/image7.emf).

$\begin{array}{r}
a_{o} = \frac{D}{\alpha} = 0\text{.}\text{54}\overset{o}{A},E = E_{o} = \frac{- e^{2}}{2}a_{o} \\
D = \frac{h}{\text{mc}} = 0\text{.}\text{004}\overset{o}{A}(\text{Table}1\text{.}1) \\
\beta = \alpha,\beta = \frac{v}{c}
\end{array}$ (1.5)

For example, if the coupling, , went to zero the characteristic size for
the system would become infinite, which means that the system is
unbound. A larger ![](media/image7.emf) would mean tighter binding or a
smaller bound state radius. Plugging the Bohr radius back into the
expression for E we can show that the electron velocity with respect to
that of light, ![](media/image13.emf), is in fact equal to the
fine-structure constant. Therefore, the atom is fairly loosely bound and
its' motion is nonrelativistic, ![](media/image13.emf) \<\< 1.

Perhaps this should not surprising because we know that, for example,
batteries are driven by processes with atomic (or chemical) energy
scales. Hence we expect them to produce of the order of 1 Volt and they
do. As another example, since the atomic energy scales are electron
volts, we expect the light which is emitted in making transitions
between those energy levels to be a few thousand Angstroms which is
indeed the wavelength of visible light, Eq. 1.1. Thus, if we knew only
that human beings see at the thousand Angstrom scale of wavelength, we
could infer that they live in a world with atomic energy levels that
were of the order of electron volts.

## 1.6 Atomic Size

The structure of multielectron atoms is determined by the Fermi
exclusion principle which states that one and only one electron can
occupy a quantum state. Recall that the energy levels in the hydrogen
atom are E~n~ = E~o~/n^2^, n = 1, 2\... Higher level atomic states, with
larger n, have more momentum, and are thus less deeply bound. Since E \~
T \~ p^2^ and p \~ $\frac{¶}{¶}x$, a state with n oscillations of the
wave function will have energy \~1/n^2^ (see Appendix B). As n increases
the electrons are less tightly bound, a~n~ = a~o~n (see Appendix B).
Therefore in atoms lower n values are filled by electrons first.

The angular momentum, *L*, is quantized in units of
$h,L^{2} = l(l + 1)h^{2},l < n$ and the number of projections of angular
momentum along an arbitrary axis is $L_{z} = mh, - l < m < l$ or
$(2l + 1)$ projections. The electron spin, $s_{Z} = \frac{sh}{2}$ can be
![](media/image14.emf) or ![](media/image15.emf). Thus there are
$2(2l + 1)$ electron states for a given $l$. These are called "shells".
Since the centrifugal force pushes us away from the origin, the
effective potential is positive (repulsive) and lower $l$ states lie
lower in energy and hence are filled first (see also Appendix B). The
fact that lower n fill first and that within a given n lower $l$ fill
first is sufficient to understand much of the periodic table of the
elements.

The atomic size, a, and ionization potential I (the energy needed to
free a single electron from the element) of the first, Z = 36, elements
are shown in Fig. 1.3.

![](media/image16.png){width="4.220833333333333in" height="5.2375in"}

Fig. 1.3: Ionization potential and atomic size for atoms with Z 36. The
scales are \~ 10 eV for I and \~ 1 ![](media/image17.emf) for a.

Note that the atomic \"shell structure\" (labeled in the figure) is
clearly mirrored in the behavior of I, which rises until the shell is
filled at the location of a noble gas. That is n=1, $l$=0, 2 states
(1s). For n=2 $l$=0, 2 states (2s) plus $l$=1 or 6 states (2p). For n=3
$l$=0, 2 states (3s), plus $l$=1 or 6 states (3p). At that point n=4
$l$=0, 2 states (4s) intervenes prior to n=3, $l$=2 or 10 states (3d)

The behavior of the size is to decrease with the filling of the shell.
The largest atoms therefore correspond to metals with a single electron
loosely bound outside a closed (noble gas) shell. The variation of a and
I is less than a factor \~4 for Z \< 36. Thus, to lowest approximation,
all atoms have the same size because as Z increases the force on the
electrons becomes stronger but the Pauli exclusion principle forces the
occupation of higher quantum states, n, which are more loosely bound.
The inner closed shells of electrons are tightly bound and largely
chemically inert and only the outer or valence electrons are chemically
active.

The small size of atoms means that we perceive macroscopic bodies as
continuous. For example, for atoms of 10 $\overset{o}{A}$ size, a 1m 1 m
1 m object has 1 billion 1 billion "pixels". Compared to a computer
screen of high resolution this number is enormous. A dust speck 0.1 mm
on a side, just visible to us, contains 100,000 100,000 atomic "pixels".

## 1.7 Atomic Spin Effects

In what follows we will essentially ignore spin effects. The reason we
can do this is that the energy of an electron magnetic moment,
$m = \frac{eh}{2}m_{e}$, in a magnetic field, B, is,
![](media/image18.emf). Numerically ![](media/image19.emf), Table 1.1.
For B = 20 kG, E~B~ \~ 1.2 x 10^-4^ eV . Hence, the spin energies are
normally tiny with respect to E~o~ which allows us to ignore them for
the purposes of describing detectors.

## 1.8 Cross Section and Mean Free Path

Let us now consider the cross section, ![](media/image20.emf), which
describes the probability of collision. It gets its name from the fact
that in some cases it is related to the geometric cross sectional area
of the system which is scattered from. The kinematic definitions of
circular frequency, ![](media/image21.emf), wavelength,
![](media/image22.emf), and wave number, k, for an incident wave packet
scattering off an extended object of radius a are shown in Fig. 1.4. The
linear frequency is f and ![](media/image23.emf). Assuming the velocity
of propagation is c, then ![](media/image24.emf). The wave number k is
related to ![](media/image21.emf) as, ![](media/image25.emf),
![](media/image26.emf).

![](media/image27.wmf){width="3.9118055555555555in"
height="1.5486111111111112in"}

Fig. 1.4: Kinematic quantities for wave packet scattering off an
extended object of size \~a.

The geometric cross section of the scattering centers is classically
simply the cross sectional area of a sphere of characteristic size a.

$\begin{array}{r}
s_{\text{atom}}\text{\textasciitilde}{pa}_{o}^{2}\text{\textasciitilde}3x\text{10}^{8}b,a_{o}\text{\textasciitilde}1\overset{o}{A} \\
s_{\text{nuc}}\text{\textasciitilde}{pa}_{N}^{2}\text{\textasciitilde}\text{31}\text{mb},a_{N}\text{\textasciitilde}1\text{fm} \\
(hc)^{2} = 0\text{.}4\text{GeV}^{2}\text{mb} \\
1\text{mb} = \text{10}^{- \text{27}}\text{cm}^{2},1b = \text{10}^{- \text{24}}\text{cm}^{2}
\end{array}$ (1.6)

The geometric cross section for atoms is expected to be hundreds of
millions of barns, while the cross section for nucleons is expected to
be on the order of tens of millibarns. That ratio of 1010 is simply the
square of the ratios of the characteristic sizes of the systems. For
reference, the dimensionless quantity $(hc)^{2}$ is 0.4 (GeV)2 mb which
means that a system bound with typical energies of GeV has a geometric
cross section of order millibarns. The cross section will be a
fundamental concept in what follows because it quantifies the
probability of a particle to interact with the detecting medium.

It is useful to define a quantity related to ![](media/image20.emf)
which is called the mean free path, \<L\>. The quantity \<L\> is defined
to be the average distance between scatterings. The scatterings are
distributed in distance x of material traversed as
![](media/image28.emf), where ![](media/image20.emf) is the cross
section to scatter off a nucleus, N~o~ is Avagadros number, (see Table
1.1), $\rho$, is the mass density of scattering centers and A is the
atomic weight. Thus $N_{o}\frac{\rho}{A}$ is the number of nuclei per
unit volume of the medium having density $\rho$.

This equation can be easily visualized. Suppose we throw N objects at a
slab of material of thickness dx. We cannot aim at the nuclei, so the
chance to scatter is probabilistic. The number of nuclei per unit area
transverse to the incident x direction is ![](media/image29.emf).
Therefore, if each nucleus has an effective transverse area of , the
probability to scatter is ![](media/image30.emf) which has the solution

$N(x) = N(0)\text{exp}( - N_{o}\frac{\rho xs}{A})$ (1.7)

The mean free path \<L\> is then.

![](media/image31.emf) (1.8)

We define the mean free path in centimeters or in grams per centimeter
squared by using the mass density ![](media/image32.emf). The latter
definition is more compact in that different systems will typically have
mean free paths which are fairly constant in \[gm/cm^2^\] units. See,
for example, the column of entries in Table 1.2 for
![](media/image33.emf) which varies only a factor \~2 over the full
periodic table. As defined, dN/N = -dx/\<L\>, so that \<x\> = \<L\>
where the bracket notation denotes an average over some statistical
distribution. For example, taking ![](media/image34.emf) from Eq. 1.6,
assuming a gas with ![](media/image35.emf) (Table 1.2) and
![](media/image36.emf) we find an atomic mean free path of
![](media/image37.emf) ![](media/image38.emf). In contrast, using
![](media/image39.emf), we find ![](media/image40.emf) in the gas, or
![](media/image41.emf) in a solid of density equal to
![](media/image42.emf). Thus the atomic mean free path in a gas is of
order m, which we refer to in Section 8, while the nuclear mean free
path in a solid is of order meters. Clearly detectors of atoms can be
rather compact.

The atomic cross section for scattering on argon as a function of
projectile energy is shown in Fig. 1.5. It should be noted that,
although there is an energy dependent structure, the estimate given in
Eq. 1.6 is certainly a reasonable representation of the typical size of
the atomic cross section. The two decades shown span the region from 107
to 109 barns.

![](media/image43.png){width="3.9722222222222223in"
height="3.4583333333333335in"}

Fig. 1.5: Atomic cross section off argon as a function of projectile
energy. (From Ref. B.1, with permission).

## 1.9 Partial Waves and Differential Cross Section

In order to motivate the geometric cross section given in Eq. 1.6 let us
very briefly review the quantum mechanical theory of scattering\[6\].
Some additional detail is given in Appendix B.

The wave function of a scattering state, ![](media/image44.emf),
consists of an incoming plane wave and an outgoing spherical wave with a
scattering amplitude, A, which can be decomposed into individual partial
waves since central forces conserve angular momentum. The wave number is
k which is related to the momentum p as
$\overrightarrow{p}ºh\overrightarrow{k}$. In the spinless case the
conservation of angular momentum means that the quantum number $l$ is
conserved. The elastic differential cross section, or the distribution
function for the scattering angle, ![](media/image45.emf), is related to
the square of the scattering amplitude. The solid angle is the spherical
area element ![](media/image46.emf). Note that ![](media/image47.emf) is
dimensionless and that the dimensions of ![](media/image20.emf) are
inverse momentum squared, or length squared, as expected.

In the case of purely elastic scattering the amplitude is;

$\begin{array}{r}
\psi \  \rightarrow \ e^{\text{ikz}} + \ A(\theta)(\frac{e^{\text{ikr}}}{\text{kr}}) \\
A(\theta)\  = \ \frac{1}{2i}\sum_{\ell}^{}{(2\ell + 1)(e^{i\delta\ell} - 1)P_{l}(\text{cos}\theta)} \\
\frac{d\sigma}{d\Omega}\  = \ \left| A(\theta) \right|^{2}\frac{}{}k^{2}
\end{array}$ (1.9)

The phase shift $d_{l}$ is caused by the scattering interaction;
$d_{l} \rightarrow 0$ means $A(q) \rightarrow 0$. $P_{\ell}$ (cos) is
the Legendre polynomial of index $l$. The total scattering cross
section, ![](media/image20.emf), integrated over all scattering angles
can also be decomposed as a sum over partial waves**.**

$\begin{array}{r}
\sigma  = \frac{4\pi}{k^{2}}\sum_{\ell}^{}{(2\ell + 1)\text{sin}^{2}\delta_{\ell}} \\
\text{} = \frac{4\pi}{k^{2}}\left\lbrack \text{Im}(A(0)) \right\rbrack
\end{array}$ (1.10)

The optical theorem relates the imaginary part of the forward scattering
amplitude, Im(A(0)), to the total cross section, . We can relate the
total cross section to the index of refraction in order to make the
connection to optics. As we will see in Section 3, a wave propagating in
a medium is characterized by an index of refraction n. Since the phase
factor of the associated field is
$i(\overrightarrow{k} \cdot \overrightarrow{x} - wt)$ and
![](media/image48.emf), the attenuation of the wave is
![](media/image49.emf). Since the probability goes as the square of the
wave function, the mean free path is ![](media/image50.emf) =
c/\[2Im(n)\]. (See Section 3.)

Unitarity is another incarnation of the conservation of probability.
Each partial wave, labeled by $l$, has an unitary upper limit which, as
can be seen from Eq. 1.10, occurs when the sine of the phase shift
$d_{l}$ is a maximum. The center of mass (CM) wave number is k, while
the CM energy is ![](media/image51.emf). (See Appendix A.)

$\begin{array}{r}
\sigma_{\ell}\  < \ \frac{4\pi}{k^{2}}\ (2\ell + 1) \\
k\  = \ p^{} \simeq \ \sqrt{s}\frac{}{2},\text{}\ p\  = \ \hslash k,\text{}\ \hslash \  = \ 1 \\
s\  = \ (\text{CM}\ \text{energy})^{2}
\end{array}$ (1.11)

If there are absorptive processes then the phase shift becomes complex,
and we must distinguish between elastic scattering,
![](media/image52.emf), inelastic scattering, ![](media/image53.emf),
and the total cross section, ![](media/image54.emf).\
In the absence of absorption ![](media/image55.emf) A classical example
with non-zero\
inelastic cross section is scattering off a completely absorbing body.
In that case ![](media/image56.emf). The intermediate case of partial
absorption is discussed in the references given at the end of this
Section, e.g. Ref. 6, and in Appendix B.

The ratio of ![](media/image57.emf) is shown in Fig. 1.6 for proton
scattering off nuclei (see Table 1.2). As ![](media/image58.emf)
indicating little absorption, while heavy nuclei look like "black" fully
absorbing objects, since as ![](media/image59.emf) These facts will be
applied in our description of calorimetry in Section 12.

A geometric interpretation of the cross section is possible if the
incoming wave suffers no scattering when it is outside an absorbing
object of radius a but is totally absorbed when it is inside radius *a*.
In that case there is an absorption of all angular momenta,
$\overrightarrow{L}$, up to some maximum which is proportional to the
radius.

$\begin{array}{r}
\ell_{\text{max}}\ \text{\textasciitilde}\ \text{ka}\ \text{}(\overrightarrow{L}\  \equiv \ \overrightarrow{r}\  \times \ \overrightarrow{p},\ \overrightarrow{L}\text{\textasciitilde}\hslash\ell,\ \overrightarrow{p} = \hslash\overrightarrow{k}) \\
\sigma_{T}\text{}\text{\textasciitilde}\sum_{0}^{\ell\text{max}}\sigma_{\ell}\  = \ \frac{4\pi}{k^{2}}\ \sum_{0}^{(\text{ka})}{(2\ell + 1)} \\
\text{}\ \text{}\text{\textasciitilde}\ 4{\pi a}^{2}
\end{array}$ (1.12)

![](media/image60.wmf){width="3.3604166666666666in" height="3.0125in"}

Fig. 1.6: Ratio of elastic to inelastic cross section for proton
scattering off nuclei of different atomic weight A. A "transparent"
nucleus has ![](media/image61.emf) while a "black" nucleus has
![](media/image62.emf).

Indeed, the cross section is proportional to the geometric cross section
of the object. The fact that the cross section exceeds the physical area
is due to the quantum mechanical wave nature of the scattering.
Basically in the presence of absorption there must be elastic "shadow"
scattering. At high energies the absorptive diffraction pattern is
confined to an angle ![](media/image63.emf) where
$q\text{\textasciitilde}\frac{{Dp}_{T}}{p}\text{\textasciitilde}\frac{\lambda}{a}\text{\textasciitilde}\frac{h}{\text{ap}},$
since the uncertainty principle implies that there is in irreducible
momentum impulse ![](media/image64.emf) given to the wave in scattering
off an object of size a, ${Dp}_{T}\text{\textasciitilde}\frac{h}{a}$.

## 1.10 Nuclear Scales of Energy and Size

We now turn to the cross section and size scale characteristic of the
atomic nucleus. Basically we think of a nucleus as containing Z protons
plus (A-Z) neutrons or A "nucleons" in total. If we consider them to be
spherical objects with a size of order the Compton wavelength of a
proton, $ƛ_{p}$, packed together in a nucleus of volume [V]{.underline},
it is clear that the size of the nucleus, a~N~, scales as the atomic
weight, A, to the 1/3 power.

$\begin{array}{r}
\underline{V} = \frac{4p}{3}a_{N}^{3} = A\frac{4p}{3}D_{P}^{3} \\
a_{N}\text{\textasciitilde}D_{P}A^{\frac{1}{3}}
\end{array}$ (1.13)

Notice that the Compton wavelength of a proton, $ƛ_{P}$, is about 0.2
Fermi. Referring to Eq. 1.4, $a = \frac{D}{\alpha}$, and knowing that
the strong coupling constant *~s~* is large, we expect a proton size
\~$D_{P}$. Using the derived geometrical interpretation of the total
cross section, we expect scaling of the nuclear cross section,
![](media/image65.emf), as the 2/3 power of the atomic weight, and the
scaling of the mean free path, \<L\> as the 1/3 power of the atomic
weight. In Table 1.2 \<L\> is given, in gm/cm^2^ units, as the column
labeled ~I~.

$\begin{array}{r}
s_{N}\text{\textasciitilde}{pa}_{N}^{2} \\
\text{\textasciitilde}A^{\frac{2}{3}} \\
\left\langle L \right\rangle\text{\textasciitilde}A^{\frac{1}{3}}(\text{Eq}\text{.}1\text{.}8) \\
\lambda\text{\textasciitilde}(\text{35}\frac{\text{gm}}{\text{cm}^{2}})A^{\frac{1}{3}}
\end{array}$ (1.14)

For example, a detection device which absorbed all the energy in an
incident proton might take \~10~I~ since the secondary debris may itself
interact. Thus, a steel device (Hadron Calorimeter - Section 12) could
be expected to be \~ 1.6m \"deep\".

## 1.11 Nuclear Cross Section

In Fig. 1.7 is shown the proton-protron and antiproton-proton scattering
cross section as a function of energy. The geometric cross section is
about 30 millibarns which is comfortably within an order of magnitude of
the observed total cross section over a substantial range of energy.

The nuclear cross section, atomic number Z, and atomic weight *A* are
given in Table 1.2 along with several other properties of materials. For
different materials the total cross section, ![](media/image54.emf), the
inelastic cross section, ![](media/image53.emf), the inelastic mean free
path, ![](media/image66.emf) and other quantities which will be
discussed later, (the ionization, the ionization per unit length and the
radiation length) are all given in Table 1.2. This table serves as the
source of data for many of the numerical calculations given in later
Sections of the text. For example, using Table 1.2 we find that a 1m
long liquid hydrogen target has a 14% (![](media/image67.emf)/A = 0.14)
probability that an incident proton will interact in traversing it.

![](media/image68.png){width="4.429166666666666in"
height="5.4944444444444445in"}

Fig. 1.7: Proton/antiproton cross section on proton as a function of
projectile energy. (From Ref. 1.1.)

In order to display the expected functional dependence of the inelastic
cross section, that quantity is plotted as a function of the atomic
weight in Fig. 1.8. It\'s clear that the expected power law dependence
of the cross section going as A^2/3^ describes the data well.

![](media/image69.wmf){width="4.098611111111111in"
height="4.222916666666666in"}

Fig. 1.8: Proton cross section on nuclei, as a function of atomic
weight, A. The straight line has A^2/3^ functionality.

## 1.12 Photon Cross Section

In the case of photons scattering off protons, a compilation of the
energy dependence of the total cross section is shown in Fig. 1.9. Note
that the scale is about 0.2 millibarns which, compared to the nucleon
scattering case, is reduced by a factor of about 100. As shown in Eq.
1.2, this is roughly the ratio of the strong to the electromagnetic
coupling constant. The factor of can be thought of as the probability
that a photon will virtually disassociate into a strongly interacting
particle.

![](media/image70.png){width="4.9430555555555555in"
height="3.3673611111111112in"}

Fig. 1.9: Photon cross section on nucleons as a function of photon
energy. (From Ref. 1.1.)

The energy dependence of the cross section after some low energy
structure at ![](media/image51.emf) \~ 2-3 GeV is reasonably constant
over several decades in energy. In analogy to the transition from
nucleon-nucleon to nucleon-nucleus cross sections, in Fig. 1.10 we show
the cross section of photons on nuclei.

![](media/image71.png){width="4.892361111111111in" height="2.49375in"}

Fig. 1.10: Photon cross section on carbon and lead as a function of
photon energy. (From Ref 1.1)

The high energy behavior is fairly straight forward. The energy
dependence above about 1 GeV is very slight. However, the low energy
behavior of the photon-nucleus cross section is rather complicated. At
energy scales of a few keV the photoelectric effect, which will be
discussed in Section 2, is the dominant process. In the intermediate
range, at energy scales of order 1 MeV, the Compton cross section,
discussed in Section 10, becomes the dominant process. At high energies
radiative processes such as pair production, which are constant in
energy, dominate. We will discuss all of these processes in subsequent
Sections. The cross section for photons on nuclei above a few MeV is of
order \~(10-100)b which is intermediate between atomic (Fig. 1.5) and
nuclear (Fig. 1.8) cross sections.

Some interaction rates are remarkably small. For example, we are all
bathed in a sea of photons of temperature 2.3 ^o^K with a blackbody
frequency spectrum which are a relic of the "big bang" creation of the
universe. The number density of these photons is \~400 photons/cm^3^
which implies flux of \~ 1.2 10^13^ photons/(cm^2^ sec). However, since
these photons are below the energy threshold to excite any bound atoms
(photon energy \~ kT = 1/40 eV @ T = 300^o^ or kT \~ 1/4000 eV, k =
Boltzmann constant, see Table 1.1), our atoms are transparent to the
photons. Therefore, we do not experience those photons. They only appear
in sensitive detectors. For example they can be seen as the "snow" on
our personal photon detectors -- our T.V. sets. For example, if you tune
to a blank channel, some of the speckles you see are relic photons from
the birth of the Universe.

## References

Section 1: Size, Energy, Cross Section

\[1\] Review of Particle Properties, Phys. Rev D. Particles and Fields
50, August 1, 1994.

\[2\] [Twentieth Century Physics]{.underline}, J. Norwood, Jr.,
Prentice-Hall, Inc. (1976).

\[3\] [Elementary Modern Physics]{.underline}, A.P. Arya, Addison-Wesley
(1974).

\[4\] [Principles of Modern Physics]{.underline}, R.B. Leighton,
McGraw-Hill Book Company, Inc. (1959).

\[5\] [Atomic Physics]{.underline}, M. Born, Hafnor Publishing Company
(1935).

\[6\] [Quantum Mechanics]{.underline}, L.I. Schiff, McGraw-Hill Book
Company, (1955).

\[7\] [An Introduction to Quantum Physics]{.underline}, A.P. French and
Edwin F. Taylor, W.W. Norton & Company, Inc. (1978).

Table of contents

I. Introduction [2](#introduction)

General References - A [5](#general-references---a)

General References - B [5](#general-references---b)

1\. Size, Energy, Cross Section [6](#size-energy-cross-section)

1.1 Units [6](#units)

1.2 Planck's Constant [9](#plancks-constant)

1.3 Electromagnetic Units [9](#electromagnetic-units)

1.4 Coupling Constants [9](#coupling-constants)

1.5 Atomic Energy Scales [10](#atomic-energy-scales)

1.6 Atomic Size [13](#atomic-size)

1.7 Atomic Spin Effects [15](#atomic-spin-effects)

1.8 Cross Section and Mean Free Path
[15](#cross-section-and-mean-free-path)

1.9 Partial Waves and Differential Cross Section
[18](#partial-waves-and-differential-cross-section)

1.10 Nuclear Scales of Energy and Size
[20](#nuclear-scales-of-energy-and-size)

1.11 Nuclear Cross Section [21](#nuclear-cross-section)

1.12 Photon Cross Section [23](#photon-cross-section)

References [26](#references)