# 3. Cerenkov Radiation

"*Vision - the art of seeing things invisible*."\....Jonathen Swift

"*And God said, let there be light; and there was light*."\....Genesis
1:1

Cerenkov radiation is emitted when a particle moving in a medium has a
velocity exceeding the velocity of light in that medium. Since the angle
of emission and the intensity of radiation depends on the velocity of
the particle, Cerenkov radiation can be used to determine the velocity
of a charged particle. The emitted Cerenkov light can be converted into
a fast electrical pulse using a PMT (see Section 2). For a particle with
energy \> 1 GeV, only a negligible fraction of its energy is radiated as
photons, making the process "non-destructive". The velocity and momentum
of the incident particle are assumed to be the same in vacuum and in the
medium.

## 3.1 Units

We begin with Maxwell's equations and introduce the key concepts of
index of refraction, skin depth, and plasma frequency. These concepts
find immediate application in "deriving" the results for Cerenkov
radiation, and will also appear in the next Section on "Transition
Radiation". The basic relations are given in Table 3.1 in both CGS and
MKS units. As stated previously, CGS units will normally be used in
calculations. However, it is easy, using the relationships given in
Table 3.1, to convert back and forth from CGS to MKS units. In fact,
numerical calculations will always be quoted in MKS units which are more
widely used in engineering applications. It is important for us to
develop a facility in both systems of units.

The basic difference between the two systems of units, as shown in Table
3.1, is in the relationship between the sources and the potentials.
Clearly, there is an arbitrariness in the definition of charge in terms
of physical quantities such as forces or the work done on particles. In
the Gaussian, CGS, system the static electrical potential V is just the
charge divided by the distance from the charge to the observation point.
By contrast, in MKS units, the electric potential has an additional
factor ![](media/image1.emf). That added factor obviously changes the
relationship between the charge and the potential, and hence the charge
and the physical forces.

Table 3.1

Electromagnetic Relations

+------------------+-----------------------+---------------------------+
| Quantity         | Gaussian CGS          | SI                        |
+------------------+-----------------------+---------------------------+
| Charge:          | 2.997 924 58 x 10^9^  | = 1 C -- 1 A s            |
|                  | esu                   |                           |
| Electron charge  |                       | = 1.602 177 33 x 10^-19^  |
| *e*:             | 4.803 206 8 x 10^-10^ | C                         |
|                  | esu                   |                           |
| Potential:       |                       | = 1 V = 1 J c^-1^         |
|                  | (1/299.792 458)       |                           |
| Magnetic field:  | statvolt (ergs/esu)   | = 1 T = 1 N A^-1^m^-1^    |
|                  |                       |                           |
|                  | 10^4^ gauss = 10^4^   |                           |
|                  | dyne/esu              |                           |
+------------------+-----------------------+---------------------------+
| Lorentz force:   | $$F\                  | $$F\  = \ q(              |
|                  |  = \ q(E\  + \ \frac{ | E\  + \ v\  \times \ B)$$ |
|                  | v}{c}\  \times \ B)$$ |                           |
+------------------+-----------------------+---------------------------+
| Maxwell          | $$\begin{array}{r}    | $$\begin{array}{r}        |
| equations:       | \nabla \cdot          | \na                       |
|                  | D\  = \ 4\text{πρ} \\ | bla \cdot D\  = \ \rho \\ |
|                  | V\  \t                | V\  \times                |
|                  | imes \ H\  - \ \frac{ | \ H\  - \ \frac{\partial  |
|                  | 1}{c}\ \frac{\partial | D}{\partial t}\  = \ J \\ |
|                  |  D}{\partial t}\  = \ | \nabla \cdot B\  = \ 0 \\ |
|                  |  \frac{4\pi}{c}\ J \\ | \nabla \t                 |
|                  | \nab                  | imes E\  + \ \frac{\parti |
|                  | la \cdot B\  = \ 0 \\ | al B}{\partial t}\  = \ 0 |
|                  | \nabla \t             | \end{array}$$             |
|                  | imes E\  + \ \frac{1} |                           |
|                  | {c}\ \frac{\partial B |                           |
|                  | }{\partial t}\  = \ 0 |                           |
|                  | \end{array}$$         |                           |
+------------------+-----------------------+---------------------------+
| Materials:       | $$D\  =               | $$\begin{array}{r}        |
|                  |  \ \varepsilon E,\ H\ | D\  = \ \varepsilon E,\   |
| Permitivity of   |   = \ \frac{B}{\mu}$$ |  H\  = \ \frac{B}{\mu} \\ |
| free space:      |                       | \varepsilon_{0}\  = \ 8   |
|                  | 1                     | \text{.}\text{854}\ \text |
| Permeability of  |                       | {187}\text{.}\text{.}\tex |
| free space:      | 1                     | t{.} \times \text{10}^{-  |
|                  |                       | \text{12}}\ F\ m^{- 1} \\ |
|                  |                       | \mu_{                     |
|                  |                       | 0}\  = \ 4\pi \times \tex |
|                  |                       | t{10}^{- 7}\ N\ A^{- 2}\  |
|                  |                       | \end{array}$$             |
+------------------+-----------------------+---------------------------+
| Fields from      | $$\begin{array}{r}    | $$\begin{array}{r}        |
| potentials:      | E\  = \               | E\  = \  - \nabla V\      |
|                  |  - \nabla V\  - \ \fr |  - \ \frac{1}{c}\ \frac{\ |
|                  | ac{1}{c}\ \frac{\part | partial A}{\partial t} \\ |
|                  | ial A}{\partial t} \\ | B\                        |
|                  | B\  =                 |   = \ \nabla\  \times \ A |
|                  | \ \nabla\  \times \ A | \end{array}$$             |
|                  | \end{array}$$         |                           |
+------------------+-----------------------+---------------------------+
| Static           | $$\begin{array}{r}    | $$\begin{array}{r}        |
| potentials       | V\  = \ \sum_{\       | V\  = \ \frac{            |
|                  | text{ch}\text{arg}\te | 1}{4\text{πε}_{0}}\ \sum_ |
| (coulomb gauge)  | xt{es}}^{}{\ \frac{q_ | {\text{ch}\text{arg}\text |
|                  | {i}}{r_{i}}}\  = \ \i | {es}}^{}{\ \frac{q_{i}}{r |
|                  | nt_{}^{}\frac{\rho\le | _{i}}}\  = \ \frac{1}{4\t |
|                  | ft( r^{'} \right)}{\l | ext{πε}_{0}}\ \int_{}^{}\ |
|                  | eft| r\  - \ r^{'} \r | frac{\rho\left( r^{'} \ri |
|                  | ight|}\ d^{3}x^{'} \\ | ght)}{\left| r\  - \ r^{' |
|                  | A                     | } \right|}\ d^{3}x^{'} \\ |
|                  | \  = \ \frac{1}{c}\ \ | A\  = \ \fra              |
|                  | sum_{\text{currents}} | c{\mu_{0}}{4\pi}\ \sum_{\ |
|                  | ^{}{\ \frac{I_{i}}{r_ | text{currents}}^{}{\ \fra |
|                  | {i}}}\  = \ \frac{1}{ | c{I_{i}}{r_{i}}}\  = \ \f |
|                  | c}\ \int_{}^{}\frac{J | rac{\mu_{0}}{4\pi}\ \int_ |
|                  | \left( r^{'} \right)} | {}^{}\frac{J\left( r^{'}  |
|                  | {\left| r\  - \ r^{'} | \right)}{\left| r\  - \ r |
|                  |  \right|}\ d^{3}x^{'} | ^{'} \right|}\ d^{3}x^{'} |
|                  | \end{array}$$         | \end{array}$$             |
+------------------+-----------------------+---------------------------+
| Relativistic     | $$\begin{array}{r}    | $$\begin{array}{r}        |
| transformations  | E_{\|                 | E                         |
|                  | }^{'}\  = \ E_{\|} \\ | _{\|}^{'}\  = \ E_{\|} \\ |
| (*v* is the      | E_{}^{'}\  = \        | E_{}^{'}\  = \ \gamma \\  |
| velocity of the  |  \gamma\left( E_{}\   |                           |
|                  | + \ \frac{1}{c}\ v\   | \end{array}$$             |
| primed frame as  | \times \ B \right) \\ |                           |
| seen in          | B_{\|                 |                           |
|                  | }^{'}\  = \ B_{\|} \\ |                           |
| the unprimed     | B_{}^{'}\             |                           |
| frame)           | = \ \gamma\left( B_{} |                           |
|                  | \  - \ \frac{1}{c}\ v |                           |
|                  | \  \times \ B \right) |                           |
|                  | \end{array}$$         |                           |
+------------------+-----------------------+---------------------------+
| $$\frac{1}{4\    |                       |                           |
| text{πε}_{0}}\   |                       |                           |
| = \ c^{2}\  \tim |                       |                           |
| es \ \text{10}^{ |                       |                           |
| - 7}\ N\ A^{- 2} |                       |                           |
| \  = \ 8\text{.} |                       |                           |
| \text{987}\ \tex |                       |                           |
| t{55}\text{.}\te |                       |                           |
| xt{.}\text{.} \t |                       |                           |
| imes \ \text{10} |                       |                           |
| ^{9}\ m\ F^{- 1} |                       |                           |
| ;\ \frac{\mu_{0} |                       |                           |
| }{4}\  = \ \text |                       |                           |
| {10}^{- 7}\ N\ A |                       |                           |
| ^{- 2};\ c\  = \ |                       |                           |
|  \frac{1}{\sqrt{ |                       |                           |
| \mu_{0}\varepsil |                       |                           |
| on_{0}}}\  = \ 2 |                       |                           |
| \text{.}\text{99 |                       |                           |
| 7}\ \text{924}$$ |                       |                           |
+------------------+-----------------------+---------------------------+

Similarly, the relationship between the current and the vector potential
$\overrightarrow{A}$ differs. In CGS units the potential
$\overrightarrow{A}$ is simply the current divided by the distance and
the speed of light, whereas in MKS units the permeability of free space
appears as an additional factor, ![](media/image2.emf). Going from
potential $V,\overrightarrow{A}$ to field
$\overrightarrow{E},\overrightarrow{B}$ to Lorentz force
$\overrightarrow{F}$, Table 3.1 shows that there is another factor of c
for CGS units. Since
$\left( 4\text{πε}_{o} \right)\left( \frac{\mu_{o}}{4}\pi \right)$ =
![](media/image3.emf) the conversion becomes clear. Additional
discussion is available in Ref. 3.1. We will freely convert back and
forth in what follows as an exercise in gaining facility with
electromagnetic units.

## 3.2 Index of Refraction

Maxwell\'s equations in the sourceless case, but within a medium defined
by dielectric and dimagnetic constants ![](media/image4.emf) and
![](media/image5.emf), are given below.

$\begin{array}{r}
\overrightarrow{\nabla} \cdot \overrightarrow{E}\  = \ 0,\ \overrightarrow{\nabla}\  \times \ \overrightarrow{E}\  + \ \frac{1}{c}\ \frac{\partial\overrightarrow{B}}{\partial t}\  = \ 0,\ D = \varepsilon\overrightarrow{E} \\
\overrightarrow{\nabla} \cdot \overrightarrow{B}\  = \ 0,\text{ \{}\overrightarrow{\nabla}\  \times \ \overrightarrow{B} - \ \frac{\text{με}}{c}\ \frac{\partial\overrightarrow{E}}{\partial t}\  = \ 0,\ \overrightarrow{B} = \mu\overrightarrow{H}
\end{array}$ (3.1)

The content of Maxwell\'s equations in the absence of sources is that
the electromagnetic fields are divergenceless and that the \"rotation\"
or curl of the electric field is due to the time variation of the
magnetic field while the curl of the magnetic field is due to the time
variation of the electric field. In a medium the two curl equations lead
to a second order wave equation with a wave propagation speed *v* that
is not the vacuum velocity *c* but is given by *c* divided by the index
of refraction, *n*. Assuming a plane wave for both
$\overrightarrow{E},\ \overrightarrow{B}$ with exp
$\left\lbrack i\left( \overrightarrow{k} \cdot \overrightarrow{x} - \omega t \right) \right\rbrack$
behavior and substituting into Eq. 3.1.:

$\begin{array}{r}
v = \frac{c}{\sqrt{\text{με}}} \\
k = \sqrt{\text{με}}(\frac{\omega}{c})\text{},\text{}\left\lbrack k^{2} - \text{με}\left( \frac{\omega}{c} \right)^{2} \right\rbrack\text{ \{}\overrightarrow{E} = 0
\end{array}n = \sqrt{\text{με}},k = \frac{\omega}{(}\frac{c}{n}) = \frac{\omega}{v}$
(3.2)

Representative values of n are given in Table 1.2 for different
materials. For gases n - 1 \~10^-4^, while for solids and liquids, n - 1
\~ 0.3. Both the permitivity and permeability of the medium contribute
to the index of refraction, although most often = 1..

## 3.3 Optical Theorum

We can connect the macroscopic propagation of light in the medium,
characterized by *n*, to the microscopic atomic process of scattering
characterized by the total cross section (see Section 1), through the
forward scattering amplitude, A(0), Eq. 1.8. We baldly assert that if
incoherent scattering off atomic electrons with number density n~e~ =
N~o~(Z/A) is the cause of *n,* then;

![](media/image6.emf) (3.3)

The departure of the index of refraction from one is due to scattering.
An imaginary part to A(0) indicates absorption. In Appendix B we argue
that the bound atomic electrons are confined to a region of volume
$\underline{V}\text{\textasciitilde}\text{}\frac{4}{3}\pi a^{3}$.
Hence the index is roughly;

$\begin{array}{r}
n - 1\text{}\text{\textasciitilde}\frac{3}{2}\left\lbrack \frac{1}{\text{ka}} \right\rbrack^{3}\left\lbrack A(0) \right\rbrack \\
\text{Im}(n)\text{\textasciitilde}\frac{3}{2}\left\lbrack \frac{1}{\text{ka}} \right\rbrack^{3}\left\lbrack \frac{\sigma_{T}k^{2}}{4\pi} \right\rbrack
\end{array}$ (3.4)

Thus, for example, for an electromagnetic cross section of size
![](media/image7.emf), see Section 10, the index
$n\ \ i\ s\ \text{\textasciitilde}\ \frac{3}{\text{18}\pi}\ \alpha^{2}\ \left( \frac{\lambda}{a} \right)$.
For optical photons, ![](media/image8.emf), we find
![](media/image9.emf), which is of the right order of magnitude, as seen
in Table 1.2.

## 3.4 Conducting Medium and Skin Depth

Let us consider now the slightly more complicated case where there are
currents in a neutral medium currents that are the sources. For the case
of a medium with conductivity ![](media/image10.emf) (not to be confused
with cross section) with a current density $\overrightarrow{j}$, the
modified Maxwell\'s equation are;

$\begin{array}{r}
\overrightarrow{\nabla}\  \cdot \ (\mu\overrightarrow{H})\  = \ 0 \\
\overrightarrow{\nabla}\  \cdot \ (\varepsilon\overrightarrow{E})\  = \ 0\text{} \\
\overrightarrow{\nabla}\  \times \ \overrightarrow{E}\ \text{} + \ \frac{\mu}{c}\ \frac{\partial\overrightarrow{H}}{\partial t}\  = \ 0 \\
\overrightarrow{\nabla}\  \times \ \overrightarrow{H}\  - \ \frac{\varepsilon}{c}\ \frac{\partial\overrightarrow{E}}{\partial t}\  - \ \frac{4\text{πσ}}{c}\ \overrightarrow{E}\  = \ 0 \\
(\overrightarrow{j}\  = \ \sigma\overrightarrow{E})
\end{array}$ (3.5)

If we again assume a plane wave form for the electric and magnetic
fields, the two curl equations again lead to a wave equation which takes
the form;

$\begin{array}{r}
\overrightarrow{H} = \left( \overrightarrow{k} \times \overrightarrow{E} \right)\frac{c}{\text{μω}} \\
i\left( \overrightarrow{k} \times \overrightarrow{H} \right) + i\varepsilon\frac{\omega}{c}\overrightarrow{E} - \frac{4\text{πσ}}{c}\overrightarrow{E} = 0
\end{array}$ (3.6)

Clearly, Eq. 3.6 implies that $\overrightarrow{H}$,
$\overrightarrow{k}\text{}\text{and}\text{ \{}\overrightarrow{E}$
are again mutually orthogonal. Eliminating the magnetic field
$\overrightarrow{H}$ we again find a second order wave equation. Note
that in Eq. 3.7 if we let the conductivity of the medium go to zero we
recover Eq. 3.2.

$\begin{array}{r}
\left\lbrack k^{2} - \ (\text{με}\ \frac{\omega^{2}}{c^{2}}\  + \ 4\ \pi i\ \frac{\text{μωσ}}{c^{2}}) \right\rbrack \overrightarrow{E}\  = \ 0 \\
k^{2} = \text{με}\ \left( \frac{\omega}{c} \right)^{2}\left\lbrack 1\  + \ i\ \left( \frac{4\text{πσ}}{\text{ωε}} \right) \right\rbrack
\end{array}$ (3.7)

The complex nature of the square of the wave number implies that the
assumed plane wave solution has an exponentially damped behavior.
Therefore, the electric field inside a conducting medium is not
rigorously zero, as in the static case with infinite conductivity, but
penetrates a finite amount. The characteristic penetration distance,
![](media/image11.emf), which we define to be the "skin depth" is
inversely proportional to the imaginary part of the wave number.

$\begin{array}{r}
\left| \overrightarrow{E} \right|\text{\textasciitilde}e^{\frac{- x}{\delta}}\text{\textasciitilde}e^{- \text{Im}(k)x} \\
\delta  = 1\text{/Im}(k) \\
\frac{\delta}{c}\text{\textasciitilde}\frac{1}{\sqrt{\text{σωμ}}}
\end{array}$ (3.8)

The skin depth approaches 0 as ![](media/image12.emf), since in that
case the charges are infinitely mobile and hence free to move around and
reorient themselves so as to completely exclude the electric field. For
finite conductivity the skin depth is finite and depends on frequency as
![](media/image13.emf). This implies that high frequency waves stay on
the surface of a conductor.

For example, in copper, using the appropriate values for the
conductivity and permeability, we find that the skin depth is 6.6
cm/$\sqrt{f(\text{Hz})}$ when the frequency f is expressed in Hertz. In
copper at 60 Hz the skin depth is roughly 0.85 cm. At 100 MHz the skin
depth is roughly 7.1 microns. Therefore in your home wiring efforts you
can choose to simply crimp the wires since the electric fields will
penetrate through the oxide insulator. On the other hand, it is a good
approximation to say that radio frequency fields stay totally on the
surface of any conductor and require elaborate waveguide "plumbing"
structures.

## 3.5 Plasma Frequency

Now let\'s consider a special case where the conductivity,
![](media/image10.emf), is due to the existence of a number density, ne,
of electrons per unit volume which are free to move. This is a rough
approximation to the behavior of electrons in the medium. Normally, in
this text we use n to denote number density and N denotes number, while
![](media/image14.emf) is used to denote either charge or mass density
or resistivity, (see the Glossary at the end of this volume).

In the presence of an electric field the free charges are accelerated.
The nonrelativistic expression for their acceleration
$\overrightarrow{a}\ \text{}\text{is}\ \text{}e\ \frac{\overrightarrow{E}}{m}$,
(see Table 3.1 for the Lorentz force equation).

$\begin{array}{r}
n_{e} \\
\overrightarrow{a} = \frac{e}{m}\overrightarrow{E}
\end{array}$ (3.9)

For high frequency electric fields the displacement of the electron is
small because the motion is limited by the short period of the harmonic
driving force. Therefore, we obtain an approximate expression for the
velocity by ignoring those small displacements and assuming
$\overrightarrow{x} = 0$ for the spatial part of $\overrightarrow{E}$,
which goes as $e^{i\overrightarrow{k} \cdot \overrightarrow{x}}$, v =
a/. We write the current density $\overrightarrow{j}$ in terms of the
electron number density ![](media/image15.emf) and the velocity
$\overrightarrow{v}$ of the electrons. We can then write down the
conductivity due to the existence of a free electron density. In this
particular case we have a purely imaginary conductivity which decreases
with frequency as ![](media/image16.emf).

$\begin{array}{r}
\overrightarrow{v}\ \text{\textasciitilde}\ \frac{\text{ie}{\overrightarrow{E}}_{o}}{m\omega}e^{- i\omega t},\ \overrightarrow{E}\ \text{\textasciitilde}\ {\overrightarrow{E}}_{o}e^{- i\omega t} \\
\overrightarrow{j}\  = \ n_{e}e\overrightarrow{v}\  = \ \sigma\overrightarrow{E} \\
\sigma\ \text{\textasciitilde}\ i\left\lbrack e^{2}\frac{n_{e}}{m\omega} \right\rbrack
\end{array}$ (3.10)

Looking back to Eq. 3.7 we see that if there is a purely imaginary
conductivity then the relationship between k and ![](media/image17.emf)
for the plane wave solution is real, implying oscillatory propagation as
long as k^2^ \> 0. We define a "plasma frequency".
![](media/image18.emf).

$\begin{array}{r}
k^{2}\text{} = \mu\left( \frac{\omega}{c} \right)^{2}\left\lbrack \varepsilon  - \left( \frac{4{\pi e}^{2}\frac{n_{e}}{m}}{\omega^{2}} \right) \right\rbrack ,\text{}\ (\text{Eq}\text{.}\ \text{}3\text{.}7,\text{}\ \text{Eq}\text{.}\text{}\ 3\text{.}\text{10}) \\
\varepsilon\text{}\text{\textasciitilde}\sqrt{1 - \left( \frac{\omega_{p}}{\omega} \right)^{2}},\text{}\omega  > \omega_{p}\ \text{free}\ \text{}\text{propagation}
\end{array}$ (3.11)

The plasma frequency, ![](media/image19.emf), is dependent only on the
fine structure constant, ![](media/image20.emf), the density of free
electrons and the mass of the electron. For ![](media/image17.emf)
greater than ![](media/image19.emf) the electric permitivity is real and
there is free propagation. For ![](media/image21.emf) the wave is
reflected.

Assigning a number density for electrons, which assumes all electrons in
the atom are free, and recalling the relationship between the Bohr
radius, $a_{o}$, and the electron Compton wavelength, $ƛ_{e}$ derived in
Section 1, we have an alternative expression for ![](media/image19.emf)
in terms of the binding energy of the hydrogen atom, E~o~. Note that,
since the factor $N_{o}\rho(\frac{Z}{A})$ is the number of electrons
per mole, and $\pi a_{o^{3}}$ is roughly the volume of an atom, we
expect $\hslash\omega_{P}$ to be equal to the binding energy of the
hydrogen atom times a number of order 1.

$\begin{array}{r}
\hslash\omega_{p} = \ \sqrt{4\text{πα}n_{e}ƛ_{e}}\ \text{}(\text{Eq}\text{.}\ 3\text{.}\text{11}) \\
n_{e}\text{}\  \equiv \ N_{o}\rho(\frac{Z}{A})\text{} \\
a_{o}\text{}\  = \ \frac{ƛ}{\alpha}\text{} \\
\hslash\omega_{p}\text{\textasciitilde}\ \left\lbrack \sqrt{\pi\left( \frac{N_{o}\rho Z}{A} \right)a_{o^{3}}} \right\rbrack E_{o},\ E_{o} = \  - \frac{\text{mc}^{2}}{2}\alpha^{2}
\end{array}$ (3.12)

For a typical density of 1 gm/cm3 and for Z/A = 1/2, $\hslash\omega_{P}$
is \~ 18.8 eV. Specifically, in lithium, which is used in foils of
transition radiation detectors, the plasma energy is 14 eV. We will
return to ![](media/image19.emf) in our discussion of transition
radiation in the next Section.

It is interesting to recall that the Earth's ionosphere is a dilute
plasma with an electron density ![](media/image22.emf). Hence, $w_{p}$
\~ 6 x 10^7^ Hz, and radio waves $\omega < \omega_{p}$ are reflected
from the ionosphere, making short wave radio "bounces" possible. Higher
frequency radio, e.g. FM at ![](media/image17.emf) \~ 100 MHz, is not
reflected as you can infer by noting the line of sight microwave relay
towers that dot our countryside. Higher frequency fields are excluded
from the plasma. Note that numerically,
$\hslash\omega_{p}\  \simeq \ \text{29}\text{eV}$
![](media/image23.emf).

## 3.6 Two "Derivations" of the Cerenkov Angle

We now turn specifically to Cerenkov radiation, having quickly reviewed
Maxwell's equations. A charged particle in uniform unaccelerated motion
in vacuum does not radiate, see Section 10. However, if a particle is
moving with a uniform constant velocity in a medium, its' electric field
will interact with the medium. This interaction can cause the emission
of real photons. Energy and momentum are balanced by the medium. This
concept was discovered theoretically by a Russian physicist, Cerenkov,
in the early part of the 20th century. The effect is now in common use
in detectors used in high energy physics.

A construction for the geometry of the Cerenkov emission angle is given
in Fig. 3.1.

![](media/image24.wmf){width="3.6666666666666665in"
height="3.2666666666666666in"}

Fig. 3.1:Cerenkov cone construction using Huygen\'s principle.

Particles move with constant velocity in a medium whose index of
refraction, n, is greater than 1. Hence, the effective velocity of light
in the medium is less than the vacuum velocity of light. If the particle
velocity exceeds the velocity of light in the medium, a shock wave or
\"Mach cone\" is set up in the medium. The construction shows that the
Hugyens wavelets emitted from each point in time add up constructively
along a line defined by the Cerenkov angle ![](media/image25.emf). Since
the particle energy is constant (non-destructive detection), $c\beta$ is
the constant particle velocity.

![](media/image26.emf) (3.13)

There are common related phenomena that are well known to us. When a
projectile or aircraft exceeds the velocity of sound in air a shock wave
is sent out whose characteristic angles have exactly the same functional
form as the Cerenkov angle. The bow wave of a boat going faster than the
speed of water waves is another familiar example. The Cerenkov angle
just at threshold velocity, when = 1/n, is 0o. As particle velocities
increase the Cerenkov angle opens up into a cone whose maximum angle is
given by ![](media/image27.emf) which occurs as ![](media/image28.emf).

An example is light emission in water, with n = 1.33. The threshold
velocity is \> 0.752, which means T~e~ \> 0.26 MeV and T \> 54 MeV.
Therefore, the emission of Cerenkov light in water casn be used to
distinguish between electrons and muons with kinetic energies in the few
MeV range. This technique is used in neutrino experiments at Kamioka and
elsewhere.

For an alternate "derivation" of the Cerenkov angle consider Fig. 3.2
where we show single photon emission by a charged particle.

![](media/image29.wmf)

Fig. 3.2: Kinematic quantities for photon emission by a charged
particle.

Energy and momentum conservation, (see Appendix A), require that

$\begin{array}{r}
\varepsilon_{o} = \varepsilon + \omega \\
{\overrightarrow{p}}_{o} = \overrightarrow{p} + \overrightarrow{k}
\end{array}$ (3.14)

Squaring both equations, and assuming $\frac{\omega}{c}\  = \ k$, we
find that for free photons;

$w = k_{\text{11}}\beta_{o}$c (3.15)

Therefore, applying this derivation to emission with an index of
refraction, since $\omega = \ \frac{\text{ck}}{n}$, Eq. 3.2,

$\begin{array}{r}
\frac{k}{n}\  = \beta k\text{cos}\theta_{c} \\
n\beta \text{cos}\theta_{c} = 1
\end{array}$ (3.16)

The relationship should hold in general given that the derivation
basically used only energy and momentum conservation. Note that in the
case of vacuum, n = 1, since ![](media/image30.emf) is always less than
1, ![](media/image31.emf) is greater than 1. We thus regain the
statement that free particles in uniform motion do not emit photons.

An example of the reality of the Cerenkov emission cone is given by data
shown in Fig. 3.3. In this case the photons emitted at the Cerenkov
angle are focused onto a plane and are detected in a gas filled device
(see Section 8). In these particular plots there are several individual
particle passages which are added together. We can see the cone built up
by the individual emitted photons. The "hits" at the center of the
circle are due to ionization energy (see Section 6) deposited by the
incoming particle itself.

![](media/image32.wmf){width="3.1in" height="4.160416666666666in"}

Fig. 3.3: Cerenkov cone for detected photons and ionization products in
a gas filled detector for \~10 individual particle passages. (From Ref.
B.1 and Ref. B.2, with permission.)

A photograph of an actual "ring imaging" device showing the gaseous
detector used to detect the ultraviolet photons by passage through a UV
transmitting window and subsequent photoelectric detection is shown in
Fig. 3.4.

![](media/image33.wmf){width="5.209027777777778in"
height="5.631944444444445in"}

Fig. 3.4: Photo of a ring imaging detector chamber showing the thin UV
transmitting window. (Photo - Fermilab.)

This is a functioning device which has been used in a high energy
physics experiment. The transmission of a CaF~2~ window 3 mm thick is
compared to quartz and LiF in Fig. 3.5 as a function of photon
wavelength and/or photon energy. Clearly these windows pass light down
into the far UV,
$\lambda\text{\textasciitilde}\text{1500}\overset{o}{A}$, which is well
matched to the useful absorption bands of the detecting vapors, TMAE and
TEA.

![](media/image34.png)

Fig. 3.5: Transmission of different window materials as a function of
photon wavelength compared to absorption in different vapors. (From Ref
B.2, with permission.)

The use of such a "Ring Imaging Cerenkov Counter", or RICH, as a
particle identification device is illustrated in Fig. 3.6. Charged
particles are momentum analyzed by measuring their trajectories in a
magnetic field (Section 7). For a given momentum the observed Cerenkov
cone angle, ![](media/image25.emf), is then measured.

![](media/image35.wmf){width="2.7784722222222222in"
height="2.6881944444444446in"}

Fig. 3.6: Plot of observed Cerenkov cone size as a function of momentum.
The bands for , K, and p are evident indicating the utility of a
Cerenkov detector for particle identification. (From Ref. 4.9, with
permission.)

The data showed three distinct bands of correlation between the Cerenkov
angle and the momentum, which correspond to particles of 3 distinct
masses, pions, ![](media/image36.emf) (mass \~ 0.14 GeV), kaons K (mass
\~ 0.5 GeV), and protons p (mass \~ 0.94 GeV). In Fig. 3.6 we see the
angle ![](media/image37.emf) beginning at threshold at 0^o^ and opening
up with increasing momentum to the common maximum Cerenkov angle as
![](media/image28.emf). For example, in helium gas at STP (Table 1.2) we
have ![](media/image38.emf). In a beam with 200 GeV particles, the
Cerenkov angle for pions is 8.7 mrad and for protons it is 7.4 mrad.

Another technique that is used in experiments is to place a Cerenkov
counter in a momentum analyzed beam and change n. Variation of the
counter gas pressure varies the index of refraction (see Eq. 3.3) and
hence the threshold ![](media/image30.emf) value. A plot of the yield of
Cerenkov photons into a fixed angle at fixed beam momentum and variable
n, or ![](media/image30.emf), is shown in Fig. 3.7. The existence of 5
distinct particle species in the negative beam is observed. Clearly, the
Cerenkov counter pressure can be set to "tag" each species and hence to
achieve beam particle identification of even rare components of the
secondary beam. Note that Fig. 3.7 is a semilog plot spanning 5 orders
of magnitude. A resolution of ![](media/image39.emf) can be achieved
before intrinsic limitations such as optical dispersion are reached.

![](media/image40.png){width="2.7416666666666667in" height="2.875in"}

Fig. 3.7: Yield of Cerenkov photons in a negative beam of fixed momentum
as a function of n. Peaks at the value of ![](media/image30.emf)
appropriate to ![](media/image41.emf) are indicated. (From Ref. B.2,
with permission.)

## 3.7 A "Derivation" of the Frequency Spectrum

The emission of Cerenkov radiation is due to the interaction of the
field of a charged particle with a medium. Hence, it is truly a form of
energy loss, dE/dx (see Section 6). A rigorous derivation is not
attempted here. If that is desired, the reader should consult a standard
text, e.g. Ref. 3.1, on electromagnetism or consult Appendix E for an
outline of the derivation. The photons in the case of ionization energy
loss are "virtual", transmitting the forces to the atomic electrons of
the medium. In the Cerenkov effect real photons are emitted.

Energy loss by ionization will be derived in Section 6. For our purposes
now we simply quote the result for the energy loss per unit path length
by ionization, dE~I~/dx up to slowly varying logarithmic factors.

$\frac{\text{dE}_{I}}{\text{dx}} = \left( \frac{N_{o}\rho Z}{A} \right)\frac{4\text{πα}^{2}}{\beta^{2}c^{2}}\left( ƛ_{e} \right)$
(3.17)

Inserting Eq. 3.12 for the plasma frequency,
$(\hslash\omega_{\rho})^{2} = 4\pi \alpha\left\lbrack N_{o}\rho\left( \frac{Z}{A} \right) \right\rbrack ƛ_{e}$,
Eq. 3.17 can be written in the suggestive form:

$\frac{\text{dE}}{\text{dx}}\  = \ \frac{\alpha}{\beta^{2}}\ \left\lbrack \frac{\hslash\omega_{p}}{\frac{c}{\omega_{p}}} \right\rbrack $
(3.18)

We now assert that the results for ionization can be carried over to
also describe the related process of Cerenkov radiation and ascribe the
energy loss, dE, to the emission of N photons,
$\text{dE} = (\hslash\omega)\text{dN}$ were dN is the number of
emitted photons of frequency ![](media/image17.emf) causing energy loss
dE. Since $\varepsilon^{2} = 1 - (\frac{\omega_{p}}{\omega})^{2}$,
Eq. 3.11:

$\begin{array}{r}
\text{}\frac{\text{dE}}{\text{dx}}\text{}\text{\textasciitilde}\frac{\alpha}{\beta^{2}}\left\lbrack \left( 1 - \varepsilon^{2} \right)\omega^{2} \right\rbrack\ \left( \frac{\hslash}{c} \right) \\
\text{}\frac{\text{dN}}{\text{dx}}\text{}\text{\textasciitilde}\frac{\alpha}{c}\text{}\frac{\left( 1 - \varepsilon^{2} \right)}{\beta^{2}}\omega \\
\text{or} \\
\text{}\frac{d^{2}N}{d\omega\text{dx}}\text{\textasciitilde}\frac{\alpha}{c}\frac{\left( 1 - \varepsilon^{2} \right)}{\beta^{2}}
\end{array}$ (3.19)

This is in no sense a derivation. However, it does manage to show the
deep connection between ionization and Cerenkov radiation. The fact that
![](media/image42.emf) is independent of ![](media/image43.emf) and is
proportional to ![](media/image44.emf) times a function of
![](media/image45.emf) is "demonstrated". In vacuum
![](media/image46.emf) and there is no radiation. The fact that the
function is really ![](media/image47.emf) and not ![](media/image48.emf)
can only appear in a full and rigorous derivation. Assuming such a
derivation, and using ![](media/image49.emf), we then attain the
Cerenkov result.

$\begin{array}{r}
\frac{d^{2}N_{c}}{d\omega\text{dx}}\  = \ \left( \frac{\alpha}{c} \right)\ \text{sin}^{2}\theta_{c}\text{} = \text{}\frac{\alpha}{c}\ \left( 1 - \frac{1}{\beta^{2}\varepsilon} \right) \\
N_{c}\  = \ \frac{\alpha}{c}\ \text{sin}^{2}\theta_{c}\Delta x\Delta\omega \text{.}
\end{array}$ (3.20)

Note that in Eq. 3.20 all energies for the photon have equal
probability. That fact puts a premium on full detection efficiency over
as wide a photon energy range as possible. Indeed, that is the reason
for using UV transmitting \"windows\" which are an important feature of
Cerenkov devices.

A finite length, L, of radiator leads to a more complex emission
distribution. Note that the existence of a diffraction pattern, shown in
Eq. 3.21, is due to the fact that there is a finite coherence length for
the radiation. We cannot have an infinitely precise emission angle
because of the uncertainty relationship inherent in wave phenomenon.
This fact will be of critical importance in the next Section. Note that
diffraction is important, \~ 1, only if L \< . In the case of optical
photons any radiator \> 1 cm is "long". We will revisit this issue in
Section 4, where the radiating foils can be "short".

$\begin{array}{r}
\frac{d^{2}N_{c}}{d\omega d\text{cos}\theta}\  = \ \frac{\alpha}{c}\ \left( \frac{\text{sin}\delta}{\delta} \right)^{2}\text{sin}^{2}\theta\ \left( \frac{L^{2}}{\lambda} \right) \\
\delta \  = \ \left( \frac{1}{n\beta}\  - \ \text{cos}\theta \right)\left( \frac{\pi L}{\lambda} \right)
\end{array}$ (3.21)

In the limit that the radiator becomes very long, the diffraction
pattern, ![](media/image50.emf), becomes extremely sharp and approaches
a "delta function" which can then be integrated over. In this
approximation there are ![](media/image51.emf) photons emitted at the
Cerenkov angle, per unit path length and per unit frequency interval.
Numerically, we can evaluate the factors in Eq. 3.20 to find out that
the yield is 365 photons times ![](media/image52.emf) per eV of
frequency range per centimeter of radiator length.

$\begin{array}{r}
\hslash c\  = \ 2 \times \text{10}^{- 5}\text{eV}\text{cm} \\
\frac{d^{2}N_{c}}{d(\hslash\omega)\text{dx}}\ \text{\textasciitilde}\ \text{365}\text{sin}^{2}\theta_{c}\frac{}{}(\text{eV} \cdot \text{cm})
\end{array}$ (3.22)

## 3.8 Examples and Numerical Values

For example, take a 10 meter long gas radiator filled with N~2~ . The
index (1 - n), is about 3x10^-4^ per atmosphere (see Table 1.2). Visible
photons cover about one eV of frequency range,
$\hslash c\text{\textasciitilde}\text{2000}\ \text{eV}\ \overset{o}{A}$.
We then expect, for fully efficient light collection, 220 photons
collected. A PMT with a 25% quantum efficiency would yield a healthy
current pulse suitable for high speed operation (see Section 2).

![](media/image53.wmf){width="3.5520833333333335in"
height="5.190972222222222in"}

Fig. 3.8: Photo of a gaseous Cerenkov detector showing the mirrors,
light collecting cones and the circular windows of the photomultipliers
(FNAL E687). (Photo - Fermilab.)

A photograph of a typical large area Cerenkov detector is given in Fig.
3.8. We see the reflecting mirrors which bounce the emitted Cerenkov
light into the light collecting cones on the sides. By looking into the
mirrors we can see the circular PMT windows which are used for detecting
the light.

The relationship between the index of refraction, the Cerenkov angle and
the ![](media/image54.emf) of the particle traversing the detector is
shown below. Note that ![](media/image11.emf) is used in this Section to
denote both the photon optical phase difference in emission, Eq. 3.21,
and the value of 1 - n. In subsequent Sections ![](media/image11.emf)
will refer only to the optical phase difference. Taking the small angle,
high energy, and n \~ 1 limit:

$\begin{array}{r}
n\text{}\  \equiv \ 1 + \delta,\ \text{}1 - \beta\text{\textasciitilde}\frac{1}{2}\gamma^{2}\text{},\ \gamma = \frac{1}{\sqrt{1 - \beta^{2}}} \\
\theta_{c}^{2}\ \text{}\text{\textasciitilde}\ \frac{1}{\gamma_{\text{TH}}^{2}}\text{}\  - \ \text{}\frac{1}{\gamma^{2}} \\
\gamma_{\text{TH}}^{2}\ \text{\textasciitilde}\ \frac{1}{2}\delta
\end{array}$ (3.23)

The threshold gamma factor, ![](media/image55.emf), is proportional to
$\sqrt{\delta}$. The maximum Cerenkov angle, ![](media/image56.emf),
occurs when the particle has extremely large ![](media/image54.emf). The
maximum number of photons, ![](media/image57.emf), thus goes as
1/![](media/image55.emf)^2^. In terms of index of refraction,
![](media/image57.emf) is proportional to ![](media/image11.emf).

$\begin{array}{r}
\theta_{c}^{\text{max}}\text{}\  = \ \frac{1}{\gamma_{\text{TH}}} \\
N_{c}^{\text{max}}\ \text{\textasciitilde}\ \delta \ \text{\textasciitilde}\ \frac{1}{2}\gamma_{\text{TH}}^{2}
\end{array}$ (3.24)

For example, in nitrogen ![](media/image11.emf) is 3x10-4, which means
that ![](media/image55.emf) is 40.8 or that the threshold for pions is
5.6 GeV, for kaons 20.2 GeV and for protons 38.2 GeV. Below those
momenta the incident particles will not radiate photons. The maximum
Cerenkov angle in this gas is only 1.4o so the light is thrown very
forward. That is the reason for the forward mirrors (the hole for
incident beam passage is in the center) seen in Fig. 3.8.

The number of photons is proportional to ![](media/image58.emf).
Increasing the counter length, L, is expensive. The other way to
increase the number of photons is to extend the frequency range
![](media/image59.emf) over which we can capture photons. As an example,
calcium fluoride windows have transmission above 50% for photon energies
of about 9 eV and below (See Fig. 3.5.) or wavelengths above about 1380
![](media/image60.emf). That gives a rather larger frequency range than
the \~ 1 eV visible range we have previously assumed.

$\begin{array}{r}
\lambda \  = \ \frac{\hslash c(2\pi)}{\hslash\omega}\  = \ \frac{\text{12},\text{400}\text{eV}\overset{o}{A}}{E_{\gamma}(\text{eV})} \\
C_{a}F_{2},\text{}E_{\gamma} < \text{}9\text{eV},\text{}\lambda  > \ \text{1380}\overset{o}{A}
\end{array}$ (3.25)

Particle identification for particles with high momenta requires an
attempt to go to lower indices of refraction. Lower indices of
refraction, such as helium, mean fewer photons produced, as we can see
looking at Eq. 3.24. A way to recoup photons is to use a larger
frequency range ![](media/image59.emf). However, at very high momentum
the required resolution, d, cannot be obtained. For example, the
limitation due to dispersion, *n* a function of $\omega$, or optical
aberration limit the achievable resolution in ![](media/image30.emf)
(see Fig. 3.7). Particle identification at very high energies requires a
technique which does not depend on ![](media/image30.emf), since at high
energy ![](media/image61.emf) independent of particle type. Both time of
flight (Section 2) and Cerenkov radiation (Section 3) become
increasingly difficult and the use of "transition radiation" (Section 4)
is called for.

## References

Section 3: Cerenkov Radiation

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

\[2\] P.A. Cherenkov, Phys. Rev. [52]{.underline} 378 (1937).

\[3\] [Nobel Lectures in Physics]{.underline}, P.A. Cherenkov, I.M.
Frank, I.E. Tanm, Elsevier Publishing (1964).

\[4\] [Sub-atomic Physics]{.underline}, H. Fraugnfelder, E.M. Henley,
Prentice-Hall, Inc. (1974).

\[5\] P. Lecompze et. al., Phys. Script [23]{.underline} 377 (1981).

\[6\] J. Litt and R. Meunier, Ann. Rev. Nucl. Sci. [23]{.underline} 1
(1973).

Table of Contents

3\. Cerenkov Radiation [1](#cerenkov-radiation)

3.1 Units [1](#units)

3.2 Index of Refraction [3](#index-of-refraction)

3.3 Optical Theorum [4](#optical-theorum)

3.4 Conducting Medium and Skin Depth
[4](#conducting-medium-and-skin-depth)

3.5 Plasma Frequency [6](#plasma-frequency)

3.6 Two "Derivations" of the Cerenkov Angle
[8](#two-derivations-of-the-cerenkov-angle)

3.7 A "Derivation" of the Frequency Spectrum
[14](#a-derivation-of-the-frequency-spectrum)

3.8 Examples and Numerical Values [16](#examples-and-numerical-values)

References [19](#references)