# III. Destructive Measurements

Having examined, in Sections 2 through 9, the categories of particle
detection where the energy lost by incident charged tracks is miniscule,
we now turn to detection where the lost energy is a significant fraction
of the kinetic energy that is carried by the incident particle. The
exploration begins in Section 10 with considerations of the radiation of
photons. Radiative processes occur at all energies, but tend to dominate
at the highest energies since radiation is relativistically enhanced.

The applications of destructive readout are looked at in Sections 11 and
12. First, in Section 11, we look at the total absorption of the energy
of particles whose interactions are dominated by electromagnetic
processes (electrons, photons). The "shower" in the detecting medium is
driven by electron Bremmstrahlung and photon pair production. Second, in
Section 12, we look at the total absorption of the energy of strongly
interacting or hadronic particles. Note that, for neutral hadronic
particles, e.g. K~L~ or n, there is no ionization deposit nor is there a
force exerted by $\overrightarrow{E}$ and $\overrightarrow{B}$ fields.
Thus, total energy absorption is the measurement of choice. Therefore,
this detection technique is of major importance for designers of
general-purpose detectors where the goal is to detect all emitted
particles, (see Fig. I.1) both charged and neutral.

# IIIA. Radiation

The emission of radiation by a charged particle is a fundamental
process. An isolated free particle cannot radiate and conserve energy
and momentum (see Section 3). Therefore, since another electromagnetic
vertex, which reduces the rate by a factor of $\alpha$, is needed to
soak up energy (see Section 6, Fig. 6.10), the emission of radiation
dominates all other processes only at high energy, E\>E~c~ (critical
energy).

Radiation is often used to provide a signal in particle detectors. Low
energy photons are detected by observing the electron recoil energies in
$\gamma - e$ elastic scattering (Compton scattering). High energy
electrons which are accelerated emit radiation (Bremmstrahlung). The
probability to emit radiation is enhanced by a factor $\gamma^{6}$ with
respect to non-relativistic considerations. Hence high energy processes
are often dominated by radiation. The processes that are needed to
understand particle detectors are explored in Section 10.

# 10. Radiation and Photon Scattering

"*My candle burns at both ends; it will not last the night; but
oh\-\--it gives a lovely light*."

\.... Edna St. Vincent Millay

Many aspects of radiation and photon scattering are used in particle
detectors. The attempt is made in this Section to "derive" the relevant
formulae. In order to describe the processes, dipole radiation is first
"derived" using the static solutions and dimensional scaling. Thompson
scattering is the first application and it is then extended to higher
energies. The kinematics at high energies (Compton scattering) is
derived. Relativistic four dimensional velocity, acceleration, and
momentum are invoked to extend dipole radiation into the relativistic
regime. The kinematics of photon emission and the virtual photon
frequency spectrum lead to the concept of radiation length and
Bremsstrahlung/pair production. These latter two processes are the
engines that drive EM "showers" and thus are at the heart of EM
calorimetry (see Section 11).

## 10.1 Non-relativistic Radiation

As the first task in this Section we will try to heuristically "derive"
the formula for non-relativistic radiation by making dimensional
substitutions to the well known static solutions for a dipole in order
to go from statics to dynamics. We deal only with the far zone, where an
observation point, O, is at a long distance from the dipole. The scale
for "long" is set by the distance of separation of the charges of the
dipole, the source size. We ignore any higher order multipoles, since we
assume kr\<\<1. As we show in Fig. 10.1, the static field is transverse
which leads to a sin *θ* factor in E where *θ* is the angle between the
axis of the dipole and the vector from the source point to the
observation point.

The static dipole field falls as the cube of the distance between the
source and the observation point. This leads to static solutions where
the energy density falls off as the sixth power of the radius. Clearly,
this can not be a radiative solution.

> ![](media/image1.wmf)
>
> Fig. 10.1: Vector diagram in the far zone for plane waves,
> $(\overrightarrow{E},\overrightarrow{B},\overrightarrow{k})$, caused
> by dipole acceleration. Static situation shown for comparison

We assume that for dipole radiation the electric and magnetic field are
transverse. If the charge distribution is accelerated, with acceleration
a, we make the dimensionless substitution, (ar/c^2^), to the static
dipole field. This substitution leads to an energy density, u, going as
the inverse square of the radius, and a power, [P]{.underline}, which is
independent of radius, as required for a freely propagating radiative
solution. The radiated power is proportional to the charge q squared (or
the fine structure constant) and to the acceleration of the charge
squared.

$\begin{array}{r}
E\text{\textasciitilde}\frac{q\text{sin}\theta}{r^{2}}\left\lbrack \frac{\text{ar}}{c^{2}} \right\rbrack \\
u\text{\textasciitilde}E^{2}\text{\textasciitilde}\left( \text{qa}\text{sin}\frac{\theta}{\text{rc}^{2}} \right)^{2} = \text{energy}\text{density}, \\
\underline{P}\text{\textasciitilde}\text{cr}^{2}E^{2} \\
d\underline{P}\frac{\ }{d}\Omega\  = \ \frac{1}{4\pi}\frac{\left( \text{qa}\ \text{sin}\theta \right)^{2}}{c^{3}} \\
\underline{P} = \frac{2}{3}\frac{\left( \text{qa} \right)^{2}}{c^{3}}
\end{array}$ (10.1)

This expression connects the radiated power and the acceleration a of
the charge q. We can recast the expression for power in terms of the
frequency *ω* of the driving acceleration. We replace the static dipole
b, E \~ qb/r^3^, by the dynamic displacement, d.

$b \rightarrow d,\text{dipole}\text{moment} = D = 2q\lbrack b + d\text{sin}wt\rbrack$
(10.2)

There is a length scale related to the frequency *ω*. We find that the
electromagnetic dipole power goes as the fourth power of the frequency,
*ω*, and the square of the displacement (or alternatively the
acceleration of the dipole). The factor of 1/3 comes from the
integration of the dipole angular distribution,
$\int_{}^{}\text{sin}^{2}\theta d\Omega\text{\textasciitilde}\frac{1}{3}(8\pi)$.
Note that the \< \> means time averaged in Eq. 10.3, and the dot means
time derivative. We use Eq. 10.1, making the replacement
$a \rightarrow {d\omega}^{2}$.

$\begin{array}{r}
\left\langle \underline{P} \right\rangle \rightarrow \left\lbrack q^{2}d^{2}\frac{w^{4}}{c^{3}} \right\rbrack \\
\left\langle \underline{P} \right\rangle = w^{4}\frac{(dD)^{2}}{3}c^{3}, \\
\frac{\left\langle \overset{¨}{D} \right\rangle^{2}}{3}c^{3},\overset{¨}{D}º¶\frac{D}{¶}t^{2}
\end{array}$ (10.3)

Consider an example on the atomic scale. We can estimate the radiated
power for an electron and proton oscillating at a frequency
corresponding to a visible wavelength of *λ* = 3000 $\overset{o}{A}$.
The e-p separation, d, is taken to be an atomic size scale, \~1
$\overset{o}{A}$ , which leads to
$\left\langle \underline{P} \right\rangle = 0\text{.}1\ \text{nW}$. It
only takes \~10 nsec to radiate the \~ 10 eV binding energy E~o~ ,
leaving the atom collapsed to zero size. Indeed, the stability of atoms
was a mystery for classical physics.

## 10.2 Thompson Scattering

Having "derived" the radiated power, we can now consider the problem of
charges accelerated by incident radiation. Thompson scattering is the
non-relativistic scattering of incident photons by electrons of mass m,
charge e. The incident photon energy flux is the energy crossing unit
area in unit time and is given by the Pointing vector,
$\overrightarrow{S}$. Since the plane wave electric and magnetic fields
are perpendicular to each other and to the wave vector,
$\overrightarrow{k}$, the time averaged flux is simply proportional to
the amplitude E~o~ of the electric field squared. (Note that the time
average of (*sin ωt*)*^2^* which is \~
$\left\langle \left| \overrightarrow{E} \right| \right\rangle^{2}$ is
1/2.)

$\begin{array}{r}
\overrightarrow{S} = \frac{c}{4\pi}\left( \overrightarrow{E} \times \overrightarrow{B} \right)\ ,\ (\text{CGS}) \\
\left\langle \left| \overrightarrow{S} \right| \right\rangle = \frac{c}{8\pi}\left| E_{o} \right|^{2}
\end{array}$ (10.4)

The angular distribution of the radiated power follows from Eq. 10.1.

$\begin{array}{r}
\frac{d\underline{P}}{d\Omega} = \frac{e^{2}}{4{pc}^{3}}|a|^{2}\text{sin}^{2}q \\
a = \frac{\text{eE}_{o}}{m}
\end{array}$ (10.5)

Note that *θ* is the angle between the dipole vector or acceleration
vector and the vector to the observation point as in Fig. 10.1. The
Lorentz force, for non-relativistic motion of the target, is just the
incident electric force divided by the target mass. Note that we ignore
the proton charges as sources of power because they are too heavy to
attain significant acceleration.

The cross section, Section 1, is defined to be the probability of
scattering per unit incident flux. We first derive the scattering cross
section in its differential form. Integrating, we then find the total
cross section for the scattering of incident photons by electrons in
subsequent non-relativistic motion. We average over initial photon
polarizations assuming the incoming photons are unpolarized. This is
historically called the Thompson cross section, *σ~T~* (not to be
confused with the total cross section).

$\begin{array}{r}
\frac{{d\sigma}_{T}}{d\Omega} = \left( \frac{e^{2}}{\text{mc}^{2}} \right)^{2}\text{sin}^{2}\theta \equiv \ \frac{\left\langle d\frac{\underline{P}}{d}\Omega \right\rangle}{\left\langle \left| \overrightarrow{S} \right| \right\rangle} \\
\sigma_{T} = \frac{8\pi}{3}(\alpha ƛ)^{2}
\end{array}$ (10.6)

Note that the cross section is proportional to the square of the product
of *α* and the Compton wavelength of the target. The Compton wavelength
of the electron is 0.004 ![](media/image2.emf) (Table 1.1). The Thompson
cross section is 0.665 barns. Note that, as seen in Fig. 1.10, at low
energies the photoelectric effect (see Section 2) dominates while at
higher energies Thompson/ Compton scattering with *σ* \~ *Z* *σ* *~T~*
\~ 10*b* is important.

It is instructive to manipulate the formula for the cross section and to
put it in terms of the geometric cross section, which we discussed in
Section 1. To do that we use the Bohr radius, a~o~, which we know is
roughly the atomic size.

$\begin{array}{r}
s_{T} = \frac{8}{3}\left\lbrack {pa}_{0}^{2} \right\rbrack\alpha^{4} \\
a_{o} = \frac{D}{\alpha}(\text{Section}1) \\
\frac{s_{T}}{{pa}_{0}^{2}}\text{\textasciitilde}\alpha^{4}\text{\textasciitilde}\text{10}^{- 8}
\end{array}$ (10.7)

What we observe is that the Thompson cross section is reduced by the
fourth power of α with respect to the geometric cross section. In
Section 1 we saw that the atomic geometric cross section was roughly
10^8^ barns, so that estimating α as 1/100 we expect about a 1 barn
cross section for *σ~T~*. Numerically, low energy photons incident on a
liquid hydrogen experimental target would have a Thompson mean free path
of \<L\> \~ 0.35 cm.

It is traditional to cast the Thompson cross section in terms of the
geometric cross section of a fictitious "electron size" defined by the
classical self-energy or the classical electron radius (see Table 1.1).
Since at our present level of understanding electrons behave as point
particles down to a length scale of 10^-16^ cm, we will simply omit this
standard classical treatment as somewhat misleading. Interested readers
are referred to the literature given in the references.

## 10.3 Thompson Scattering Off Objects with Structure

At this point we would like to extend the discussion of Thompson
scattering to the case of a target with some internal structure.

We briefly formulate non-relativistic scattering theory and look at the
radial solution\
for a spherically symmetric potential (see Appendix B for slightly more
detail). The radial Schroedinger equation in this situation is given
below. The $Y_{l}^{m}$ are the spherical harmonic solutions that arise
in central force problems in the solution of the angular equations.

$\begin{array}{r}
uº\text{Rr, R} = \text{radial wave function, }y\text{\textasciitilde}\text{RY}_{l}^{m} \\
\frac{d^{2}u}{\text{dr}^{2}} + \left\lbrack k^{2} - U(r) - \frac{l(l + 1)}{r^{2}} \right\rbrack u = 0
\end{array}$ (10.8)

We identify the terms of the differential equation as the radial kinetic
energy, the potential energy, the centrifugal potential and the total
energy, which is proportional to k^2^. Near the origin
$|y|^{2}\text{\textasciitilde}r^{2l}$ so that only $l$ = 0 (S wave)
states are nonzero at r = 0.

$u|_{r \rightarrow o} \rightarrow r^{\ell + 1}$ (10.9)

Let us now consider the scattering of an incoming wave by a square well
of radius a and of depth U~o~. The outgoing scattered solutions are
found by applying boundary conditions requiring the continuity of
interior (r\<a) and exterior (r\>a) solutions and first derivatives.
(See Appendix B.) In the simplified case where ka\<\<1, we have S wave
dominance of the phase shifts (Section 1).

$\begin{array}{r}
\delta_{\ell}\text{\textasciitilde}(\text{ka})^{2\ell + 1} = \text{partial wave phase shift} \\
\delta_{o}\text{\textasciitilde} - \text{ka}\text{>>}\delta_{\ell}
\end{array}$ (10.10)

Using the results of Section 1, we find that the total cross section is
simply proportional to πa^2^.

![](media/image3.emf) (10.11)

Let us now try to generalize the situation. In the limit where the depth
of the well, U~o~, becomes small, the solution is the Mie-Debye-Rayleigh
Law ( Appendix B).

$\begin{array}{r}
k^{2} + U_{o} = K^{2}(\text{interior solution K}) \\
\text{tan}\delta_{o} = - \text{ka}\left\lbrack 1 - \frac{\text{tan}\ (\text{Ka})}{\text{Ka}} \right\rbrack\text{\textasciitilde} - \text{ka}\left\lbrack \frac{\left( \text{Ka} \right)^{2}}{3} \right\rbrack \\
\sigma \rightarrow 4{\pi a}^{2}\left\lbrack \frac{\left( \text{ka} \right)^{2}}{3} \right\rbrack^{2},K\text{\textasciitilde}k \\
\sigma_{\text{RA}}\text{\textasciitilde}\frac{a^{6}}{\lambda^{4}}\text{\textasciitilde}a^{2}\left( \frac{a}{\lambda} \right)^{4},\frac{\sigma_{\text{RA}}}{{\pi a}^{2}}\text{\textasciitilde}\left( \frac{a}{\lambda} \right)^{4}\text{\textasciitilde}\left( \text{ka} \right)^{4}
\end{array}$ (10.12)

The previous result is modified by a factor (ka)^4^. This factor can be
considered to be the \"form factor\" which describes the non-pointlike
behavior of the target scattering center.

For example, if visible light, ![](media/image4.emf), is incident on a
typical atom, $a\text{\textasciitilde}1\overset{o}{A}$, the suppression
factor, ka, in the Raleigh cross section is enormous. Raleigh scattering
is something which is evident to us everyday as we look up at the sky
and ask \"if the sun is yellow then why is the sky blue?\". The answer
is that the Raleigh scattering of the sun\'s light in the atmosphere is
very dependent on the wavelength of the incident light. The atmosphere
preferentially scatters short wavelength light (blue) into our eyes.
This phenomenon also tells us why the setting sun reddens. As more
atmosphere is traversed by the sunlight, the shorter wavelengths have an
increased relative chance to be scattered.

## 10.4 Relativistic Photon Scattering

At higher energies, where relativistic effects come into play the
Thompson formula, is no longer valid. One of the contributing Feynman
diagrams for high energy elastic scattering is given in Fig. 10.2. We
can count powers of e in the amplitude and conclude that it goes like
*α*. The cross section will then go as $\alpha^{2}$. There are now two
scales of energy, the center of mass energy, $\sqrt{s}$, and the target
mass, m. Thus a cross section going like *α^2^/s* is possible.

![](media/image5.wmf){width="3.7777777777777777in" height="1.35625in"}

Fig. 10.2: Feynman diagram for photon scattering off electrons.

The high energy cross section, called the Klein-Nishina cross section,
*σ~KN~*, is proportional to the Thompson cross section *σ~T~* but with a
multiplicative factor which at high energies is the dimensionless ratio
of the target electron mass, m, to $\sqrt{s}$ squared. Thus at high
energies, $\sqrt{s} > m_{e}$, the elastic cross section in Fig. 1.10
falls with energy. There are also logarithmic factors which are simply
indicated in parenthesis.

$\sigma_{\text{KN}}\text{\textasciitilde}\frac{3}{8}\sigma_{T}\left( \frac{m}{\sqrt{s}} \right)^{2}\left\lbrack 1 + \text{ln}(\text{  }) \right\rbrack\text{\textasciitilde}\frac{\alpha^{2}}{s}$
(10.13)

As another example, the $e^{+}e^{-} \rightarrow m^{+}m^{-}$cross
section, $s_{\text{ll}}$, also goes as *α^2^/s*.

$s_{\text{ll}} = \frac{4\text{pα}^{2}}{\text{3s}} = \frac{\text{87}\text{nb}}{\left\lbrack s(\text{GeV}^{2}) \right\rbrack},1\text{nb} = \text{10}^{- \text{33}}\text{cm}^{2}$
(10.14)

For example, at a 1 GeV center of mass energy, the cross section is
87nb. An electron incident on a liquid hydrogen target would then have a
2900 km mean free path. This value for \<L\> is much larger than that
for photon Thompson scattering. The fall of the elastic photon-electron
cross section with increasing energy is visible as the behavior of
*σ~coherent~* + *σ~incoherent~* in Section 1.

## 10.5 Compton Scattering

Having looked at non-relativistic elastic scattering, we now consider
the kinematics of the fully relativistic case. To this point in the
discussion we have assumed that the frequency of the incoming photons is
unchanged in the scattering, e.g. Thompson scattering. This is the
principle which allows for clear radio transmission, since reflection in
this case will not cause frequency shifts. However, the constancy of
frequency is no longer valid at high energies. The quantitative scale
for what we mean by high energy is set by the mass of the targets, or by
the \"Compton wavelength\" of the electrons which scatter the incident
radiation.

The kinematic definitions for the scattering of photons of frequency
*ω~o~* off a particle of mass m are given in Fig. 10.3. As hypothesized
by Einstein, we consider the photon to be a particle. We find the quoted
relation by squaring the 2 equations, for energy and momentum
conservation, Eq. 10.15, and subtracting them (see Appendix A).

![](media/image6.wmf){width="3.191666666666667in"
height="1.7034722222222223in"}

Fig. 10.3: Kinematic definitions for photon scattering off a particle of
mass (m).

$\begin{array}{r}
w_{o} + m = e + w \\
{\overrightarrow{k}}_{o} = \overrightarrow{p} + \overrightarrow{k}
\end{array}$ (10.15)

The incident photon, frequency *ω~o~*, strikes the target which is at
rest in the laboratory frame. This causes the target to recoil with
energy *ε* after scattering the photon into final energy *ω*. We know
that energy conservation requires a decrease in the energy of the
photon, which means an increase of the wavelength. The scale for the
increase is set by the Compton wavelength of the electron, $D$.

$\begin{array}{r}
\left( \frac{1}{w} - \frac{1}{w_{o}} \right) = \frac{1}{m}\left( 1 - \text{cos}q \right) \\
\lambda - \lambda_{o} = 2pD(1 - \text{cos}q)
\end{array}$ (10.16)

Since the wavelength of radio waves is meters and the Compton wavelength
of the electron is \~10^-13^m, the frequency shift can be ignored in
radio transmission. However, at incoming photon energies which are
comparable to the rest energy of the electron, 0.511 MeV, (Table 1.1)
the frequency shift becomes a large fraction of the incident energy and
can no longer be ignored. The dynamics also contributes to the
observability of the frequency shift. As seen in Fig. 10.4 one of the
Feynman diagrams for Compton scattering has an exchanged electron. Since
the \"virtual\" electron would like to be as "real" as possible, i.e.
have small momentum transfer, the outgoing electron will take off most
of the energy of the incident photon. Therefore, the dynamics leads to
fast forward electrons, or large energy loss of the photons.

In dealing with photon radioactive sources (see Table 6.1) the
appearance of the full incident photon energy \"Compton peak\" recoil
electron is an everyday experience. The kinematic limit for
back-scattering, ![](media/image7.emf), is that the maximum e energy, or
\"Compton edge\", is T~max~ = ε~max~ -- m = ω~o~\[1-m/(m+2ω~o~)\]  ω~o~
if ω~o~ \>\> m.

![](media/image8.wmf){width="2.5145833333333334in"
height="2.1034722222222224in"}

Fig. 10.4: Feynman diagram for Compton scattering. Fermion exchange
leads to fast, forward electrons.

The exact angular distribution is quoted in Appendix A. Part of the
forward effect is purely kinematic due to the transformation of CM solid
angle, $d\Omega \ast$ , to lab solid angle $d\Omega$.

$p \ast \text{\textasciitilde}\frac{\sqrt{s}}{2}\text{\textasciitilde}\sqrt{\omega_{o}\frac{m}{2}},\ d\Omega = d\Omega \ast (p\text{*/}p)^{2}\text{\textasciitilde}d\Omega \ast \left( \frac{\omega_{o}m}{2\omega^{2}} \right)$
(10.17)

The angular distribution at several values of $w_{o}$ is shown in Fig.
10.5. Note the forward peaking at high energies.

![](media/image9.wmf){width="4.51875in" height="3.395138888888889in"}

Fig. 10.5: Angular distribution shown as (x,y) contour for Compton
scattering at several different incident photon energies,
![](media/image10.emf) = 0.01 MeV, 0.1 MeV and 1.0 MeV.

Some of the actual data taken by Compton showing the appearance of the
primary photon energy and the scattered photon energy is shown in Fig.
10.6. Of course, this was also historically one of the first experiments
which showed the particle nature of light. Referring to Eq. 10.16, we
have $D\lambda = 2pD$ at $q$ = 90^o^. Since
$D_{e} = 0\text{.}\text{0038}\overset{o}{A}$ we expect a shift in
wavelength of \~0.024 ![](media/image11.emf) which, Fig. 10.6, is
exactly what was observed.

![](media/image12.png){width="1.8805555555555555in"
height="4.347222222222222in"}

> Fig. 10.6: Data taken by Compton showing the appearance of the
> scattered photon at higher wavelength (lower energy) for different
> scattering angles. The shift in ![](media/image13.emf) increases with
> scattering angle, as expected. (From Ref. 10.9 with permission.)

## 10.6 Relativistic Acceleration

The fundamental formula for non-relativistic radiation is given in Eq.
10.1. Since acceleration is not a Lorentz invariant, this is not a
formulation which is easily generalized to the relativistic case. Recall
that in special relativity the normal three dimensional vector equations
of classical physics are generalized to four dimensions. For example,
the position of an event in space and time is labeled by a position
vector, x^μ^, which includes both the spatial position,
$\overrightarrow{x}$, and time, ct, in units of distance.

$x^{\mu} = (\overrightarrow{x},\text{ct})$ (10.18)

The relativistic velocity, U^μ^, is the proper time derivative of the
position.

$\begin{array}{r}
U^{m} = \frac{\text{dx}^{m}}{d}(\frac{s}{c}) \\
g\left( \overrightarrow{v},c \right)
\end{array}$ (10.19)

The proper time, ds, is the relativistically invariant \"distance\"
between two events. The factor of ![](media/image14.emf) (see Appendix
A) that appears is near to 1 at low velocities, ![](media/image15.emf),
and increases without limit as the velocity approaches that of light,
![](media/image16.emf).

$\begin{array}{r}
\text{ds}^{2} = \text{dx}_{\mu}\text{dx}^{\mu} = \text{dx}^{2} - (\text{cdt})^{2} = (\frac{\text{cdt}}{\gamma})^{2} \\
\gamma = \frac{1}{\sqrt{1 - \beta^{2}}},\ \overrightarrow{\beta} = \frac{\overrightarrow{v}}{c}
\end{array}$ (10.20)

The relativistic momentum *p^μ^* is the mass measured with the particle
at rest, the rest mass, m, times the relativistic velocity. The particle
energy is *ε* and the momentum is $\overrightarrow{p}$.

$\begin{array}{r}
p^{\mu} = \text{mU}^{\mu} = (\overrightarrow{p},\frac{\varepsilon}{c}) \\
\gamma m(\overrightarrow{v},c),\overrightarrow{v} = d\frac{\overrightarrow{x}}{\text{dt}} \\
\varepsilon = \gamma\text{mc}^{2} \\
\overrightarrow{p} = \gamma\overrightarrow{\beta}\text{mc} = \overrightarrow{\beta}\frac{\varepsilon}{c}
\end{array}$ (10.21)

The four dimensional generalization of acceleration, *A^μ^*, is the
proper time derivative of the relativistic velocity.

$\begin{array}{r}
A^{\mu} = \frac{\text{dU}^{\mu}}{d}(\frac{s}{c}), \\
\gamma\frac{d}{\text{dt}}\lbrack\gamma(\overrightarrow{v},c)\rbrack \\
\frac{d\gamma}{\text{dt}} = \gamma^{3}(\overrightarrow{\beta} \cdot \overrightarrow{a})\text{, }\ \overrightarrow{a} = d\frac{\overrightarrow{v}}{\text{dt}} \\
A^{\mu} = \gamma^{3}\lbrack\overrightarrow{a} + \gamma^{2}\overrightarrow{\beta}(\overrightarrow{\beta} \cdot \overrightarrow{a}),\ \gamma^{2}(\overrightarrow{\beta} \cdot \overrightarrow{a})\rbrack
\end{array}$ (10.22)

Using Eq. 10.1, we can then use the relativistic acceleration to form a
Lorentz invariant by contracting the acceleration with itself to get the
square of the "length" of the acceleration.

$\begin{array}{r}
\underline{P} = \frac{2}{3}\frac{e^{2}}{c^{3}}A_{\mu}A^{\mu} \\
\frac{2}{3}(\frac{e^{2}}{c^{3}})\gamma^{6}\left\lbrack \left| \overrightarrow{a} \right|^{2} - \left| \overrightarrow{\beta}\text{x }\overrightarrow{a} \right|^{2} \right\rbrack
\end{array}$ (10.23)

Note the dependence of the radiated power on a high power of the
relativistic *γ* factor.. This enhanced radiation is a very important
consequence of relativity. For example, the muon and electron have
identical electromagnetic interactions. However, the muon is \~200 times
heavier. Thus, electrons radiate much more power than muons if they are
prepared to have the same energy, since
$\gamma = \frac{E}{\text{mc}^{2}}$.

## 10.7 Circular and Linear Acceleration

Let's look at the special case of circular motion. In that case the
three dimensional velocity $\overrightarrow{v}$ is perpendicular to the
three dimensional acceleration $\overrightarrow{a}$ which leads us to a
total radiated power which goes as the fourth power of *γ*. The power
radiated is ![](media/image17.emf) larger than the non-relativistic
case, Eq.10.1.

$\left( \underline{P} \right)_{{}_{O}} = \frac{2}{3}\frac{e^{2}}{c^{3}}g^{4}\left| \overrightarrow{a} \right|^{2}$
(10.24)

In looking at the special case of linear acceleration it is convenient
to formulate the power in terms of the proper time rate of change of the
4 dimensional momentum. We can then use the relationship between proper
time, ds, and coordinate time, dt, and the relationships between
momentum and energy given above to formulate the radiated power in terms
of the energy actually supplied by external forces. In making that
substitution we note that in going a linear distance, dx, in time
interval dt the particle instantaneously has velocity *β* since all
these quantities are defined by clocks and rulers set up in the
laboratory reference frame.

$\begin{array}{r}
A_{m} = \left( \frac{\text{gβ}}{m} \right)\frac{d}{\text{dx}}\left\lbrack e\left( \overrightarrow{\beta},1 \right) \right\rbrack\text{, }\text{dx} = \beta\text{cdt} \\
\left( \underline{P} \right)_{\text{LIN}} = \frac{2}{3}\frac{e^{2}}{m^{2}c^{3}}\left( \frac{de}{\text{dx}} \right)^{2}
\end{array}$ (10.25)

Thus in the case of linear acceleration the radiated power is simply
related to the energy supplied by the external forces per unit length.
This fact has immense practical implications in that circular colliders,
such as e^+^e^-^ storage rings, pay an enormous power penalty as *γ*
increases. Therefore, current electron accelerator research is in the
area of linear colliders where we need not pay that added radiative
power bill. This factor first became important in the design of the
Stanford Linear Accelerator Center, SLAC, which is a linear machine and
is the pioneer in the study of linear acceleration and linear colliders.
Note, however, that in terms of 3 dimensional acceleration, Eq. 10.23,
$\left( \underline{P} \right)_{\text{LIN}} = \frac{2}{3}(\frac{e^{2}}{c^{3}})\left| \overrightarrow{a} \right|^{2}g^{6}$,
with acceleration measured by lab clocks and rulers.

## 10.8 Angular Distribution

The non-relativistic dipole radiation pattern is strongly altered in the
highly relativistic case. However, the details are quite technical and
will not be discussed here. Suffice it to say that much of the basic
physics can be extracted by looking at a photon emitted by a charge with
velocity *β.* In the rest frame of the charge the photon has energy
$e$\* and polar angle $q$\*. In the laboratory frame the factor relating
$e$ and ![](media/image18.emf)\* is, by Lorentz transformation,
$\varepsilon = \gamma(\varepsilon \ast - \beta p\text{*cos}\theta \ast ) = \text{γε} \ast (1 - \beta\text{cos}\theta \ast )$
since photons have $e = p\text{or}v = c$, in all reference frames. This
factor, $(1 - \beta\text{cos}q)$, appears in the angular distributions,
as for example, Eq.10.17. In the limit $\beta$→1 we use the
approximations

$\begin{array}{r}
\frac{1}{\gamma^{2}} = 1 - \beta^{2} \\
\beta \equiv 1 - \text{δβ} \\
\beta^{2} \simeq 1 - 2\text{δβ} \\
\frac{1}{\gamma^{2}} \simeq 2\text{δβ} \\
2(1 - \beta)
\end{array}$ (10.26)

We have previously used these approximations in Section 3 for Cerenkov
radiation.

If this factor alone determined the angular distribution, the
characteristic angle for emission of radiation is 1/γ (see the
discussion on TRD, Section 4). This same factor can be found by looking
at an isotropic emission distribution, in the CM frame, and boosting it
to a rapidly moving frame. This is called the "searchlight effect" in
special relativity.

$\begin{array}{r}
1 - \beta\text{cos}q\text{\textasciitilde}1 - \beta(1 - \frac{q^{2}}{2}) \\
\text{\textasciitilde}\frac{1}{2}\left( \frac{1}{g^{2}} + q^{2} \right) \\
\left\langle q \right\rangle\text{\textasciitilde}\frac{1}{g}
\end{array}$ (10.27)

This factor appeared already in our discussion of transition radiation
detectors in Section 4 as a factor in the vacuum phase angle.

In general the angular distribution is ugly and complex. In the special
case of linear acceleration we find the dipole radiation pattern thrown
forward by the searchlight effect. The angular distribution is shown in
Fig. 10.7 for various values of *β*. The searchlight effect is very
obvious. The general linear motion result approaches, for
non-relativistic motion, *β → o*, the dipole result

![](media/image19.wmf){width="5.801388888888889in"
height="4.356944444444444in"}

Fig. 10.7: Dipole angular distribution as 9x,y) contours for the case of
linear acceleration, showing the tipping forward due to the
\"searchlight\" effect, for ![](media/image20.emf) = 0.01 and
![](media/image21.emf) = 0.3. The growth in the radiated power,
![](media/image22.emf), is evident.

$\begin{array}{r}
d\frac{{\underline{P}}_{\text{LIN}}}{d}\Omega = \frac{{\alpha a}^{2}}{4}{\pi c}^{3}\left\lbrack \text{sin}^{2}\frac{\theta}{(}1 - \beta\text{cos}\theta)^{5} \right\rbrack \\
\ d\frac{{\underline{P}}_{\text{LIN}}}{d}\Omega \rightarrow \frac{{\alpha a}^{2}}{4}{\pi c}^{3}(\text{sin}^{2}\theta)
\end{array}$ (10.28)

## 10.9 Synchrotron Radiation

Let us now turn to the special case of circular motion. We have already
learned that the power radiated in circular motion with respect to the
non-relativistic case is increased by a factor of
![](media/image23.emf). We have also learned that in general there is a
searchlight effect which throws the radiation forward into a cone of
typical angular size 1/*γ.* Now we would like to look at the frequency
spectrum of the radiated photons since we know already from the Compton
effect that the radiated frequency ω need not be the same as the
frequency of the driving acceleration, $w_{o}$.

We simply assert that the relativistic effect is to stretch the emitted
frequency *ω* by a factor \~ ![](media/image24.emf) beyond
![](media/image25.emf). The circular radius is *a* and *β* refers to the
circular velocity

![](media/image26.emf) (10.29)

A rough schematic representation of the frequency distribution of the
intensity of radiation is shown in Fig. 10.8. The quantity dI(*ω*)/d*ω*
is the intensity as a function of frequency defined to be the radiated
energy crossing unit area per unit frequency. The radiative energy loss
per revolution, *∆Ε*, is simply the radiated power,
![](media/image27.emf), times the period of revolution. The power is
proportional to the characteristic frequency, ![](media/image28.emf),
which has a ![](media/image29.emf) factor, times another power of
$\gamma$. Very roughly we can think of synchrotron radiation, or
radiation emitted in relativistic circular motion, as the radiation per
turn of $\text{αγ}$ photons with energies up to ${hw}_{c}$. Hence, in
every turn we lose energy *∆Ε* which goes as $\gamma^{4}$.

Fig. 10.8: Schematic representation of the frequency distribution of the
intensity for circular motion.

$\begin{array}{r}
\text{ΔΕ} = \left( \frac{2\pi a}{c\beta} \right)\underline{P} \\
\text{\textasciitilde}\left( \frac{4\pi}{3}\text{αγ} \right)\left( \gamma^{3}\omega_{o} \right)\text{\textasciitilde}\text{\textasciitilde}\gamma^{4} \\
\text{\textasciitilde}\alpha\text{γω}_{c} \\
\text{\textasciitilde}\ \left\lbrack \text{dI}(\omega\frac{)}{d\omega} \right\rbrack\left\lbrack \Delta\omega\  = \ \omega_{c} \right\rbrack
\end{array}$ (10.30)

An exact calculation of the intensity of radiation is shown in Fig.
10.9. The sharp fall off in emitted frequencies above the characteristic
frequency $\omega_{c}$ survives as a feature of the exact calculation,
as does the basically flat spectrum below $\omega_{c}$.

![](media/image30.png){width="4.46875in" height="3.592361111111111in"}

Fig. 10.9: Frequency distribution for synchrotron radiation where y is
ω/ωc. (From Ref. 1.1.)

Synchrotron emission is very important in certain applications. A
representation of the emission of synchrotron radiation by an electron
spiraling in a magnetic field is shown in Fig. 10.10. The photons are
emitted largely in the plane of the orbit. A contour plot of the
radiation pattern in circular motion is given in Fig. 10.11. The two
plots are for classical and extremely relativistic synchrotron
radiation. The "searchlight" effect, ![](media/image31.emf), is very
evident in the case that ![](media/image32.emf).

![](media/image33.png){width="3.68125in" height="3.683333333333333in"}

Fig. 10.10: Schematic representation of the emission of synchrotron
radiation by an electron spiraling in a magnetic field.

![](media/image34.wmf){width="6.49375in" height="2.928472222222222in"}

Fig. 10.11: The "searchlight" effect in synchrotron radiation. The
x-axis is along $\overrightarrow{a}$. The z-axis is along
$\overrightarrow{\beta}$ and the motion is circular. Shown are
![](media/image35.emf) surfaces with *z(θ,φ)* proportional to *dσ/dΩ.*
a) ![](media/image36.emf). Note the dipole like pattern. b)
![](media/image37.emf). Note the growth in power which goes as
![](media/image38.emf). Note also the strong forward peaking with
![](media/image39.emf).

## 10.10 Synchrotron Applications

There are now several facilities in existence which create synchrotron
light for scientists who use it as a probe of the properties of
materials. Examples in the United States are at Brookhaven National
Laboratory, Argonne National Laboratory and Cornell University. A
dedicated accelerator which magnetically "wiggles" electrons is used to
provide the radiation.

Synchrotron radiation can also be used for "particle identification".
Consider a prepared beam of pions and electrons of the same momentum. If
the beam is bent in a magnet and forward produced x-ray energy photons
are detected, their presence indicates electrons (large
![](media/image40.emf)) and not pions. Assume a 6m beam line magnet with
a field of 20 kG and a 200 GeV incident beam. The radius of curvature
(see Section 7) is, a = 300m. Therefore, the 6m magnet represents only
$\frac{\Delta\varphi}{2}\pi = 0\text{.}\text{003}$ of a revolution. The
electrons in the beam will radiate \~ 24 photons with energy
${hw}_{c}\text{\textasciitilde}\text{53}\text{MeV}$ for a total energy
loss of \~ 1.26 GeV. Such photons are radiated almost straight ahead
(see Fig. 10.10), while the charged beam is bent away by the magnetic
field. Photons of this energy will make e^+^e^-^ pairs which are then
detectable (see Section 1 and Fig. 10.12).

![](media/image41.wmf){width="2.472916666666667in"
height="1.8645833333333333in"}

Fig. 10.12: Feynman diagram for photon pair production in the field of a
nucleus with atomic number Z.

As a practical matter the emission of radiation places a limit on the
ease of operation of circular accelerators. For example one GeV
electrons have a loss per turn in a circular orbit of radius 1 meter of
90 keV. The driving frequency, $\omega_{c}$, is of order 10^8^/sec. The
*γ* factor is 2000, which means that the characteristic frequency,
$\omega_{o}$, is of order of 10^17^/sec. Since $h$ is 6.6 x 10^-16^ (eV
⋅ sec), Table 1.1, the energy of the photon is 2.2 keV, i.e. in the
x-ray range. We have roughly 40 photons emitted per turn with energy 2.2
keV for a total energy loss per turn of 90 keV. Thus there is a
substantial radiated power to be supplied by external energy sources.

$\begin{array}{r}
DE(\text{keV}) = \text{90}\lbrack E(\text{GeV})\frac{\rbrack^{4}}{a}(m) \\
{hw}_{c}(\text{keV}) = 2\text{.}2\frac{\left\lbrack E\left( \text{GeV} \right) \right\rbrack^{3}}{a}(m)
\end{array}$ (10.31)

So far we have only talked about electrons, which is perfectly
reasonable since they are the lightest stable charged particle, which
implies that at a given energy they will have the largest *γ* factor.
However, recently, in designing high luminosity multi-TeV proton-proton
colliders, the effects of proton radiation are also beginning to become
important. For example, a 20 TeV proton has a *γ factor* of 20,000. If
the radius of such a machine is 10 kilometers, scaling as in Eq. 10.30,
we find the same loss of roughly 90 keV per turn. Since this new
generation of accelerators utilizes superconducting magnets, and since
that the heat capacity of such magnets is low, the photon heat load due
to proton synchrotron radiation is indeed important and is a vital
consideration in the design. The European High Energy Physics Center,
CERN, is building a 7.5 + 7.5 TeV pp collider (the LHC). Each beam loses
6.9 keV/turn with $\left( {hw}_{c} \right)$ \~ 45 eV. The heat load is
3.7 kW per beam at design beam current which must be removed from the
cryogenic magnets.

## 10.11 Photon Emission Kinematics

Let us now turn to the photon emission probability and the kinematics of
the radiation of a photon by a charged particle. We have already seen
that radiation requires acceleration. Thus, it is not possible for an
isolated free particle to radiate as we proved in Section 3, where we
showed that a particle which does radiate must have
$\omega = \ \text{ck}_{\text{11}}\beta$. Since $\beta < 1$ and
$\omega = \ c\left| \overrightarrow{k} \right|$ for a real photon, this
relation can not be satisfied. It is most nearly satisfied when k \~
k~ll~ or when the photons are emitted in the forward direction.

Since the amplitude, A, in perturbation theory is proportional to the
inverse of the energy difference between the initial and final state,
the largest amplitudes will have the smallest energy difference. The
favored scenario is, therefore, to have soft and co-linear photon
emission,

$A\text{\textasciitilde}\frac{1}{(}\varepsilon - \varepsilon^{'})\text{\textasciitilde}\frac{1}{\omega}$
(10.32)

## 10.12 Photon Frequency Spectrum

In order to quantify the expectation of soft emission, we estimate the
intensity of photons for different frequencies. The energy density, u,
is proportional to the square of the electric field. The impact
parameter is b. The peak electric field of a non-relativistic charged
particle seen at a transverse distance b is e/b^2^. The moving charge
makes a pulse of field in time which therefore implies a frequency
spectrum. The characteristic frequency *ω~c~* is the inverse of the
collision time at low velocities. The fall off of the field with
distance means that there is a finite collision time which is
![](media/image42.emf) for non-relativistic collisions. (See Section 5)

The Fourier components of the field, E(ω), are \~ E∆t \~ 2e/bv. The
field in frequency space is fairly uniform for ω \< ω~c~ = v/b = 1/∆t.
The total energy , U, is found by integrating the energy density u over
all spatial volume.

$\begin{array}{r}
\text{dU}\  = \ \int_{}^{}{\text{du}\  = \ \iint_{}^{}{E(\omega)}}^{2}\ d\omega 2\pi\text{bdb} \\
\  = \ \int_{}^{}{\left\lbrack \frac{\text{dU}}{d\omega} \right\rbrack d\omega}
\end{array}$ (10.33)

Ignoring the logarithmic details, we find a simple behavior for dU/dω by
explicitly substituting for E(ω).

$\frac{\text{dU}}{d\omega}\  = \ 4\frac{\alpha}{\beta^{2}}\left\lbrack \text{ln}() \right\rbrack$
(10.34)

We identify the total field energy of the charged particle as the total
photon number density N~γ~ and the energy of the photon,$\hslash\omega$
.

$\text{dU}\  = \ \hslash\omega\text{dN}_{\gamma}$ (10.35)

Therefore, the number of photons emitted per unit frequency due to the
passage of a charged particle with velocity *ν goes* as 1/*ω which* is a
reflection of the fact that the emitted photons would "like to be" soft,
Eq. 10.32.

$\begin{array}{r}
\frac{\text{dN}_{\gamma}(\omega)}{d\omega}\text{\textasciitilde}\frac{\alpha}{\beta^{2}}\left( \frac{1}{\omega} \right)\lbrack\text{ln}()\rbrack \\
\frac{2\alpha}{\pi}\left( \frac{1}{\beta^{2}} \right)\left( \frac{1}{\omega} \right)\lbrack\text{ln}()\rbrack
\end{array}$ (10.36)

The implication of Eq. 10.36 is that the electromagnetic field of a
moving charge can be represented as the frequency spectrum of its
associated "virtual photons" which is called the Weissacker-Williams
method of virtual quanta We will ignore the modifications to the virtual
photon spectrum due to relativistic motion.

## 10.13 Bremsstrahlung and Pair Production

Now let us consider Bremsstrahlung and pair production. First the
definition; Bremsstrahlung is the emission of photons by charged
particles accelerated in the Coulomb field of a nucleus. The Feynman
diagram for photon pair production in the field of a nucleus of atomic
number Z was given in Fig. 10.12 while the topologically similar diagram
for Bremsstrahlung by charged particles in the field of a nucleus is
given in Fig. 10.13. Note that Fig. 10.13 is just the radiation of a
photon by an electron with an additional interaction with the nuclear
Coulomb field to satisfy energy - momentum conservation and thus allow
the reaction to proceed.

![](media/image43.wmf)

Fig. 10.13: Feynman diagram for Bremsstrahlung in the field of a nucleus
of atomic number Z.

![](media/image44.wmf){width="5.750694444444444in"
height="4.320138888888889in"} The energy distribution of the positron in
photon pair production is shown in Fig. 10.14, for two photon energies.
Clearly, the energy spectrum is roughly uniform from \~ zero to the
maximum allowed energy. We can think of this shape as arising from to
the isotropic "decay" of a virtual "photon" of some nonzero mass (\>2
m~e~) acquired by interaction with the nuclear Coulomb field. Isotropic
"decay" can easily be shown to yield a uniform secondary energy
spectrum. (See Appendix A.)

Fig. 10.14: Kinetic energy spectrum of the positron scaled to the
maximum allowed energy for incident photon energy of 10 and 100 MeV. The
spectra for nuclear charge Z screened by the atomic electrons and
unscreened are shown. (See references in Section 10 for a definition of
the screening approximations.)

Clearly these diagrams are higher order in the coupling constants than
the photon particle scattering with which we began this Section. The
idea is to now use what we have already learned and factorize the
problem of radiation by a charged particle into distinct pieces.
Consider high speed incident particles scattering off a nucleus. In the
rest frame of those particles, the nucleus is incident at high velocity.
It has Z protons of charge e. We view the electric field of the nucleus
as a distribution of soft virtual photons described by the
Weissacker-Williams formula given in Eq. 10.36. Since the photons are
soft, their wavelengths are larger than the size of the nucleus. Thus,
the fields are coherent over the nucleus. By "coherent" we mean that the
phase of the field does not vary over the size of the nucleus. Therefore
the quantum mechanical amplitude squared has a factor of (Ze)^2^ in
adding up the Z individual amplitudes which are in phase.

The incoming nucleus is equivalent to a distribution of soft photons.
They will Thompson scatter off the projectile. We have already derived
the Thompson cross section, Eq. 10.6, and we know the
Weissacker-Williams distribution, Eq. 10.36. We put them together to
describe the Bremsstrahlung cross section, ![](media/image45.emf), as
the Thompson scattering, ![](media/image46.emf), of the soft virtual
photons, $N_{\gamma}$, describing the coherent field of the nucleus off
the projectile charge.

$\begin{array}{r}
\frac{{d\sigma}_{B}}{d\omega}\text{\textasciitilde}Z^{2}\frac{\text{dN}_{\gamma}}{d\omega}\sigma_{T} \\
\frac{{d\sigma}_{B}}{d\omega}\text{\textasciitilde}\frac{\left( Z^{2}\alpha \right)}{\omega}(\alpha ƛ)^{2}\left\lbrack \text{ln}() \right\rbrack
\end{array}$ (10.36)

Therefore the Bremsstrahlung cross section as a function of frequency
displays the coherent factor ![](media/image47.emf), and goes as 1/*ω.*
There are also logarithmic factors which again we ignore. The cross
section is σ~B~ \~ \[Z^2^α^3^/m^2^\]ln( ). Numerically σ~B~ \~
Z^2^\[0.58 mb\] which is larger than the typical hadronic cross section
(see Section 1).

## 10.14 The Radiation Length

We integrate the Bremsstrahlung distribution over all frequencies and
weight by the energy of the emitted photon to find the total emitted
energy or total radiative energy loss, dE.

$\text{dE}\text{\textasciitilde}\underset{o}{\overset{E}{ò}}(hw)\left( \frac{N_{o}\rho\text{dx}}{A} \right)\left( \frac{{ds}_{B}}{dw} \right)dw$
(10.38)

Cast in terms of a fractional energy loss per path length in gm/cm^2^,
![](media/image48.emf), the radiative mean free path is just X~o~ , the
radiation length.

$\begin{array}{r}
\frac{1}{E}\left( \frac{\text{dE}}{\rho\text{dx}} \right)º\frac{1}{X_{o}},E(x) = E(o)e^{\frac{- \rho x}{X_{o}}} \\

\end{array}$ (10.39)

Using Eq. 10.38,
$\text{dE}\text{\textasciitilde}\left( Z^{2}\alpha \right)(\alpha D)^{2}\left( N_{o}\rho\frac{\text{dx}}{A} \right)E$,
and the radiation length, X~o~, is;

$\begin{array}{r}
X_{o^{- 1}} = \frac{\text{16}}{3}\left( \frac{N_{o}}{A} \right)(Z^{2}\alpha)({\alpha D}_{p})^{2}\lbrack\text{ln}()\rbrack \\
\text{\textasciitilde}(\frac{Z}{A})Z\text{\textasciitilde}Z
\end{array}$ (10.40)

The radiation length, ![](media/image49.emf) is defined to be the mean
free path for emitting Bremsstrahlung radiation. Note that the Compton
wavelength refers to the wavelength of the projectile, $ƛ_{p}$.

We can re-express the Bremsstrahlung cross section in terms of the mean
free path (Section 1).

$\begin{array}{r}
s_{B}º\lbrack\frac{A}{(}N_{o}{\rho X}_{o})\rbrack \\
X_{o} = \left\langle L \right\rangle
\end{array}$ (10.41)

A plot of the radiation length in gm/cm^2^ as a function of the atomic
number (data given in Table 1.2) is shown in Fig. 10.15. Roughly
speaking the functional dependence of ![](media/image50.emf), expressed
in gm/cm^2^, is to decrease as 1/Z. A reasonable representation of the
numerical value is, ![](media/image51.emf) (gm/cm^2^) = \[180(A/Z)\]/Z.
This dependence has an immediate implication. We should use heavier
materials in order to shield against photons. For example, lead is a
standard material used in such shielding applications. Photon sources
are routinely housed in lead "pigs". The radiation length of Pb is 0.56
cm.

![](media/image52.wmf){width="3.689583333333333in"
height="3.2604166666666665in"}

Fig. 10.15: Plot of the radiation length Xo in gm/cm2 as a function of
the atomic number Z. The line, \_\_\_\_, has the functionality 1/Z for
comparison.

The radiation length goes like the square of the mass of the projectile.
Thus electrons will easily radiate whereas muons, which have the same
coupling and are nearly two hundred times heavier, will not tend to
radiate, because it is acceleration which is important in radiation.
Since the forces are the same on the electrons and muons, (because they
have the same charge), the difference is in the acceleration^2^ (since
this is a radiative process) which explains the appearance of the square
of the masses in Eq. 10.40. We return to the particle identification of
muons in Section 13.

For lead, Eq. 10.40 yields a rough estimate of 17 gm/cm^2^ for the
radiation length. Table 1.2 gives us the exact answer of 6.37 gm/cm^2^
or 0.56 cm for lead and incident electrons. Scaling to the muon by the
square of the mass, we estimate that the radiation length for muons on
lead would be 236 meters, which is certainly not a practical way to
shield against muons.

## 10.15 Pair production by Photons

Comparing Figs. 10.12 and 10.13 we can see that photon pair conversion
and Bremsstrahlung are very similar in terms of Feynman diagrams. In
fact, the cross sections are almost equal.

$\sigma_{\text{pair}} = \frac{7}{9}\sigma_{B}$ (10.42)

The roughly energy independent cross section,
$\sigma_{B}\text{\textasciitilde}\left( Z^{2}\alpha \right)\left( \alpha ƛ_{P} \right)^{2}\left\lbrack \text{ln}() \right\rbrack$,
is responsible for the constant high energy photon cross section already
displayed in Section 1.

We can think of the nucleus as supplying a small "virtual mass" to the
photon, $\text{m rSub \{ size 8\{ size 9\{γ\}\} \} } > 2m_{e}$. The
opening angle for the electron-positron pair in this process is
characterized by an angle which is roughly the electron mass divided by
the electron energy. This is typical; masses or other basic physics
quantities provide scales for the transverse momentum, whereas the only
longitudinal energy scale in the problem is the incident energy,
$\theta_{e}\text{\textasciitilde}\frac{m_{e}}{\varepsilon_{\gamma}}\text{\textasciitilde}\frac{p_{T}}{p}$.

The angles for photon conversion into pairs can thus be very small.
Often the photon conversion is not even resolved into individual
electrons and positrons (see Section 6). A photo of a photon conversion
in a bubble chamber is shown in Fig. 10.16. We can see, indeed, that the
opening angle is initially rather small and is then increased due to the
opposite sense of the rotation of the momenta of the electron and
positron in the magnetic field.

![](media/image53.png){width="5.291666666666667in"
height="2.4868055555555557in"}

Fig. 10.16: Bubble chamber photograph showing photon conversion into a
particle antiparticle pair. (From Ref. 10.11, with permission.)

In the next section on electromagnetic calorimetry we'll describe an
electromagnetic cascade. Basically an accelerated electron emits
photons. Given sufficiently energetic photons, they will pair produce
electrons and positrons, which will in turn make more Bremsstrahlung
photons. This is clearly a run away process leading to a large number of
electrons and photons of reduced energy. We expect, then, that an
electromagnetic calorimeter which fully absorbs the particles might be
\~ 20 X~o~ "deep" or about 11 cm. Glancing back at Section 1 we can see
that when it becomes energetically possible for photons to make an
electron positron pair, which means photons with energies above twice
the electron rest mass or 1 MeV, the cross section for pair production
rises rapidly. Pair production is more important for high Z elements
than low Z elements as shown by the comparison of carbon and lead shown
in Fig. 1. 10 and by Eq. 10.40.

## 10.16 Pair Production by Charged Particles

A closely related process is pair production by a charged particle. It
may be thought of as Bremsstrahlung of a "virtual" photon which "decays"
into an e^+^e^-^ pair. From a diagrammatic viewpoint, there is another
vertex with respect to Bremsstrahlung where $g \rightarrow e^{+}e^{-}$.
We expect that the pair production cross section is reduced by a factor
$\alpha$. However, integrating over the distribution of the energies of
the pairs, and comparing the two processes at the same value of y, the
ratio of the photon energy to the incident energy, we find (in the high
energy "completely shielded" approximation);

$\begin{array}{r}
\frac{\left( \frac{{d\sigma}_{\text{pair}}}{\text{dy}} \right)}{\frac{{d\sigma}_{B}}{\text{dy}}}\text{\textasciitilde}\frac{\frac{\alpha}{\pi}(\frac{1}{y^{3}})}{(\frac{1}{y})} \\
y \simeq \text{}\text{E rSub \{ size 8\{ size 9\{γ\}\} \} }\text{/}E
\end{array}$ (10.43)

Therefore, the pair production cross section is much larger at low y
then the Bremsstrahlung cross section. This effect is sufficient to
compensate for the power of α that is the cost of the additional
$g \rightarrow e^{+}e^{-}$ vertex. For example the integral of Eq. 10.43
for y \> 5% is ($\frac{\sigma_{\text{pair}}}{\sigma_{B}}$)
~\|\ y\ \>\ 0.05~ = 1.24. Thus in Fig. 6.12 the energy loss in iron for
pair production is comparable to that for Bremsstrahlung.

## 10.17 Strong and EM Interaction Probabilities

The cross section for photons incident on nucleons at high energies
approaches a constant as seen in Section 1. At high energies the main
photon process is pair production.

$\begin{array}{r}
\sigma_{N}\text{\textasciitilde}A^{\frac{2}{3}}ƛ_{p}^{2} \\
\sigma_{B}\text{\textasciitilde}(Z\alpha)^{2}\alpha ƛ_{e}^{2} \\
\frac{\lambda_{1}}{X_{o}}\text{\textasciitilde}(\frac{Z}{A}\frac{)}{\lbrack}5\text{.}1A^{\frac{2}{3}}\rbrack
\end{array}$ (10.41)

It is interesting to note that the nuclear cross section, as we saw in
Section 1, goes as A^2/3^ times the Compton wavelength of the proton
squared. By comparison, the Bremsstrahlung cross section goes as Z^2^
because of coherence. Since this is an electromagnetic process with 3
vertices, (Fig. 10.130, we pay the penalty of a factor *α* ^3^. Even so,
the electromagnetic Bremsstrahlung cross section is comparable to the
nuclear cross section for A\~3 (lithium). A glance at Table 1.2 shows
that for Be and heavier elements the characteristic length for
radiation, ![](media/image54.emf) is less than the characteristic length
for nuclear interaction, ![](media/image55.emf). It is amusing that this
particular electromagnetic process is stronger than the "strong"
interaction because of coherence effects. We will explain how the fact
that $X_{o}\text{>>}\lambda_{I}$ for heavy elements is exploited to
provide calorimetric "particle identification" in Section 13.

## References

Section 10: Radiation and Photon Scattering

\[1\] [Classical Electrodynamics]{.underline}, J.D. Jackson, John Wiley
& Sons, Inc. (1962)

\[2\] [Modern Physics]{.underline}, R.L. Sprooll, John Wiley & Sons,
Inc. (1956).

\[3\] [Relativistic Kinematics]{.underline}, R. Hagedorn, W.A. Benjamin,
Inc. (1964).

\[4\] [Classical Theory of Fields]{.underline}, L.D. Landau and E.M.
Lifshitz, Addison-Wesley (1951).

\[5\] [Conduction of Electricity Through Gases]{.underline}, J.J.
Thompson, G.P. Thomson, Cambridge University Press, 1928.

\[6\] [X-rays in Theory and Experiment]{.underline}, A.H. Compton, S.K.
Allison, van Nostrand Company, Inc. (1935).

\[7\] A.H. Compton, Phys. Rev. [21]{.underline}, 175, (1923).

\[8\] [Quantum Theory of Radiation]{.underline}, W. Heitler, Oxford
University Press, (1954).

\[9\] [Quantum Physics of Atoms, Molecules, Solids, Nuclei and
Particles]{.underline}, Robert Eisberg and Robert Resnick, John Wiley &
Sons, Inc. (1974).

\[10\] [Principles of Modern Physics]{.underline}, Robert B. Leighton,
McGraw-Hill Book Company, Inc. (1959).

\[11\] [Special Relativity]{.underline}, A.P. French, W.W. Norton &
Company, Inc. (1968).

Table of Contents

III\. Destructive Measurements [1](#iii.-destructive-measurements)

IIIA. Radiation [2](#iiia.-radiation)

10\. Radiation and Photon Scattering
[3](#radiation-and-photon-scattering)

10.1 Non-relativistic Radiation [3](#non-relativistic-radiation)

10.2 Thompson Scattering [5](#thompson-scattering)

10.3 Thompson Scattering Off Objects with Structure
[7](#thompson-scattering-off-objects-with-structure)

10.4 Relativistic Photon Scattering [8](#relativistic-photon-scattering)

10.5 Compton Scattering [9](#compton-scattering)

10.6 Relativistic Acceleration [12](#relativistic-acceleration)

10.7 Circular and Linear Acceleration
[14](#circular-and-linear-acceleration)

10.8 Angular Distribution [15](#angular-distribution)

10.9 Synchrotron Radiation [16](#synchrotron-radiation)

10.10 Synchrotron Applications [20](#synchrotron-applications)

10.11 Photon Emission Kinematics [21](#photon-emission-kinematics)

10.12 Photon Frequency Spectrum [22](#photon-frequency-spectrum)

10.13 Bremsstrahlung and Pair Production
[23](#bremsstrahlung-and-pair-production)

10.14 The Radiation Length [25](#the-radiation-length)

10.15 Pair production by Photons [27](#pair-production-by-photons)

10.16 Pair Production by Charged Particles
[28](#pair-production-by-charged-particles)

10.17 Strong and EM Interaction Probabilities
[29](#strong-and-em-interaction-probabilities)

References [30](#references)