# [4.]{#_Toc437310857}[ ]{#_Toc437310857}[Transition Radiation]{#_Toc437310857}

"*The light of the spirit is invisible, concealed in all
beings.*"\....The Upanishads

In the previous Section we explored the relationship of Cerenkov
emission to ionization loss. The wave vector, k, was shown in Section 3
to be related to the dielectric constant as k \~ (/c). If is real, there
is free photon propagation. If is \> 1, the threshold velocity defined
in Section 3 is \< c. If is \< 1, the threshold velocity is \> c and no
real photon emission is possible in an infinitely long radiator.
However, for finite length radiators the emission angle is not unique,
as discussed in Section 3. Because of the diffraction broadening of the
distribution of emission angles, a particle with velocity \< c can still
emit photons with cos() \< 1. We call this phenomenon the
"sub-threshold" emission of Cerenkov light.

The sub-threshold emission of photons when particles traverse a thin
segment of material is called transition radiation because it is caused
by a transition in the index of refraction between the vacuum and the
material. We will find that the emitted photons are in the X-ray range
of energies and that the number of emitted photons depends on and not .
Therefore, the technique is most useful at very high energies when the
previously discussed methods of velocity measurement become very
difficult.

## [4.1]{#_Toc437310858}[ ]{#_Toc437310858}[Cerenkov Radiation for a Finite Length Radiator]{#_Toc437310858}

Transition radiation occurs when we are below threshold for Cerenkov
radiation, \> 1, but in a finite length, L, medium. Let us consider the
radiation pattern from a thin film due to the passage of a charged
particle. The distance traveled in the film is L, the wavelength in the
medium is , and the angle of emission of the radiation in the medium is
. The geometric definitions of the quantities used are given in Fig.
4.1.

Recall in the previous Section on Cerenkov radiation that for a finite
wave train there is a diffraction pattern due to the fact that the
Cerenkov radiation was only emitted over a finite distance. The
diffraction pattern due to the phase difference was also indicated (Eq.
3.21).

Fig. 4.1: Definition of quantities used in deriving the transition
radiation effect. This effect can be thought of as the pattern due to a
finite length Cerenkov radiator

(4.1)

A very sharp emission angle in the medium, labeled as , is an
approximation to the diffraction pattern of a very long radiator, given
as in Eq. 4.1. For short radiators the problem can be treated as slit
diffraction from the entrance and exit points labeled as A and B in Fig.
4.1. The wavelength in the radiating medium, , is related to the index
of refraction n = , Eq. 3.1, and the photon frequency as discussed in
Section 3.** **

(4.2)

The optical path difference to the observation point, O, shown in Fig.
4.2, leads to a phase difference, , between waves produced from points A
and B this phase difference is related to the geometric path difference.
If is the observation point, then

Fig. 4.2: Interference pattern at observation point O due to amplitudes
A and -A for emission from the front and back of the film with optical
path length difference equivalent to the phase difference between the
amplitudes.

Thus the optical phase difference, which is the argument in the
diffraction pattern, is given in Eq. 4.2 in terms of the radiator
length, L, the emitted wavelength, and of the emitted angle, . Clearly,
the diffraction peak is centered on , the "Cerenkov angle", with an
angular width of . Hence for a very thin radiator, the chance for
sub-threshold radiation is not necessarily negligible.

## [4.2]{#_Toc437310859}[ ]{#_Toc437310859}[Interference Effects]{#_Toc437310859}

The interference of the amplitudes A and -A from points A and B is
controlled by the phase difference , as seen in Fig. 4.2. The intensity,
I, is . Note that if the intensity which is the maximum constructive
interference value, while if the interference is destructive and the
intensity 0. For random phases . Given that the phase at the entrance
and exit transition differs by 180^o^, we need to have a sufficient
optical path length difference to avoid destructive interference. This
point becomes important, as we will see later, in the practical design
of detectors.

The diffraction broadening of the angular distribution, , implies that
even below threshold, , is possible. This observation gives rise to the
name "transition radiation" because what is necessary for real photon
emission is simply a transition in the index of refraction. Putting the
expression in Eq. 4.2 back into Eq. 4.1 we obtain the radiation pattern
with diffraction. We then use the formula for refraction to express the
radiation pattern in terms of the observed angle seen downstream of the
foil as defined in Fig. 4.1.** **

(4.3)

For small angles, , and for high velocities, , we can expand the
expression for and obtain the approximate optical phase difference.

(4.4)

The factor will be motivated as the phase difference between waves in
vacuum in Section 10. The extra term results from the fact that this is
propagation in a medium characterized by a plasma frequency, , which
leads to the additional term beyond what will be found for the vacuum
case. The doubly differential cross section is proportional to and has a
typical dipole pattern, . If phase factors are ignored, it goes roughly
as .

This present heuristic treatment is not sufficient, although it brings
us quite close to the true result. The final correct expression for the
doubly differential cross section for transition radiation is given
below. ** **

(4.5)

## [4.3]{#_Toc437310860}[ ]{#_Toc437310860}[The Vacuum Phase Shift]{#_Toc437310860}

Comparing to Eq. 4.4, we see that the exact result, Eq. 4.5, can be
thought of as arising from having adjusted Eq. 4.4 by the subtraction
the vacuum phase shift, that we would get for propagation if the thin
film were a vacuum, i.e. no "transition" in the index of refraction. The
factor in the phase difference is proportional to the vacuum factor,
which we will derive in Section 10, . The insertion of the vacuum phase
factor insures that there is no radiation without a transition in the
index of refraction; i.e. if then .

## [4.4]{#_Toc437310861}[ ]{#_Toc437310861}[The Frequency Spectrum]{#_Toc437310861}

The frequency spectrum of transition radiation, in comparison to
Cerenkov emission, is not flat. The emission spectrum has high frequency
components because its very existence depends on the finite width
diffraction pattern, , which is inherent in the Cerenkov process for a
finite length radiator. Typically, Cerenkov light is in the optical
region, while transition radiation is in the x-ray region. This
characteristic high frequency emission is a consequence of needing
enough diffraction broadening to enable sub-threshold Cerenkov emission
to occur at a real angle.

We can obtain an informative expression by integrating Eq. 4.5 over all
angles while ignoring interference by setting . The result for the
photon frequency spectrum is;

(4.6)

We define the frequency scale in terms of the characteristic frequency .
Note that falls off rapidly for . The expression for also diverges at
low frequency, but this has no practical impact, and is an artifact of
the approximations used. A plot of in arbitrary units is shown as a
function of y in Fig. 4.3. The 1/y behavior at small y is evident as is
the steeper fall for y \>\> 1 indicating the high frequency cutoff. For
y \<\< 1, a limiting case is .

Fig.4.3: Frequency spectrum of TR photons, where frequency is defined in
units of

Looking at Eq. 4.6, we see that the integral over a practical range of
frequency , near y \~ 1 is . Thus we argue that in transition radiation
photons per interface are emitted with energies typically y \~ 1/3 or
/3. These approximate considerations lead to some "rules of thumb" for
transition radiation detectors. We can basically read them off from the
phase factor and the coupling given in Eq. 4.4. As will be seen in the
search light effect (Section 10), the typical emission angle is which is
a consequence of special relativity.

(4.7)

The contour of is shown in Fig. 4.4. We have ignored the factor in Eq.
4.5 in making this plot because L is another free variable, and we have
assumed that L is adjusted such that \~ 1. The peaking of the
distribution at and is quite evident. If we accept that y \~ 1, then the
phase difference is .

Fig.4.4: Contour plot of as a function of y = and z . As expected, the
contours peak at y \~ z \~ 1, in agreement with Eq.4.7.

We can heuristically think of the transition radiation as being due to
the changing dipole moment formed by the charge and its "image" charge
on the opposite side of the interface. The image charge is a way of
solving the boundary conditions which is equivalent in effect to the
induced charge density at the interface (see also Section 8) needed to
match the boundary conditions. The charge and image charge pair makes a
dipole which changes with time, and changes direction at the point of
crossing the interface thus causing radiation (see Section 10). The
interface is the point where the pointing of the dipole vector flips
direction indicative of maximum acceleration. A schematic view of the
geometry is shown in Fig. 4.5.

Fig. 4.5: Schematic representation of the dipole image charge - charge
pair for a charged particle approaching a transition in indices of
refraction n~1~ ¹ n~2~.

## [4.5]{#_Toc437310862}[ ]{#_Toc437310862}[Dependence on ]{#_Toc437310862}g and Saturation

Transition radiation is a useful tool in high energy physics because it
has an emitted energy, , which increases with g as opposed to Cerenkov
radiation, where all the radiation patterns collapse together as
approaches 1. That behavior is what makes transition radiation
attractive for particle identification at very high energies where the
Cerenkov technique and time of flight become problematic. However, this
increase in photon energy with g does not continue without limit. As
with all detectors, there are intrinsic limitations.

There is a characteristic length that we define to be the distance, ,
over which phase changes are substantial for fixed physical distance L.
These phase differences must be large since the amplitudes, see Fig.
4.2, are out of phase for the front and back transitions and destructive
interference must be avoided. Therefore, there is a maximum value of
called the saturation value, which depends on foil length, L, and
radiation wavelength, , above which destructive interference begins to
occur. That value of is inversely proportional to . There is therefore a
minimum foil thickness which is needed to avoid saturation. Note that
the phase , which contributes a factor to , is simply .

** **(4.8)

Looking at Eq. 4.8 we might argue that we could go to thicker foils to
increase the optical path difference and thus raise the range over which
transition radiation can be used. However, that increases the self
absorption of the medium which is not a winning strategy. Typically we
must compromise between attenuation in the foil and losses due to
destructive interference. Note that the Z^5^ behavior of the
photoelectric cross section (Section 2) argues for the use of low Z
foils.

## [4.6]{#_Toc437310863}[ ]{#_Toc437310863}[TRD Foil Number and Thickness]{#_Toc437310863}

As a numerical example, consider the detection of 200 GeV pions. The
factor for pions is 1428 which means that a typical emission angle is
about 0.7 mrad. Using the rule of thumb, the energy of the emitted
photons (Eq. 4.7) is about 8.9 keV which means we\'re in the x-ray
region. Using polyethylene radiators we have of about 21 eV. We can use
other foils, such as Li with = 14 eV, to try to tune . Since , we have
some freedom to change the factor in . However, the Li foils are hard to
work with. If we want to keep from saturating at those frequencies, the
polyethylene foil thickness should be more than 14. Since each foil only
emits \~ a photons we need at least or about 100 foils of material, with
100% detection efficiency, in order to observe just one x-ray per
incident pion. The thickness of polyethylene for the emission of 1
photon is 0.19 cm while at 10 keV the mean free path in carbon is \~ 0.5
gm/cm^2^ (see Section 2) or \~ 0.5 cm indicating the importance of
absorption since the "stack" of foils is \~ \<L\> thick.

A photo of an actual transition radiation detector (TRD) assembly used
in an experiment is shown in Fig. 4.6. This consists of a radiator stack
of many foils of polyethylene followed by a xenon filled proportional
wire chamber (PWC). These chambers are devices which we will discuss in
a later Section. The reason for a detector consisting of a xenon gas
fill is pretty clear from our discussion of the photoelectric effect. We
want the high Z gas in order to have a large absorption cross section
for keV photons which means a large photoelectric cross section, and
(Section 2).

Now, as mentioned before, the TRD is not transparent to its own emitted
photons as was the case for the Cerenkov counter. An approximate
expression for the number of \"useful\" foils, N~EFF~, in a stack of N
foils is given below.

(4.9)

For Cerenkov counters we could simply increase the radiator length to
get more photons. In the present case that simple ploy does not work due
to self absorption.

There is a cut off at low energy due to the very large photoelectric
cross section, , which we mentioned in Section 2. There is also a cut
off at high energies; see Fig. 4.3. Hence the effective number of foils
is a strong function of the energy of the emitted x-ray photons, and
there is a finite frequency range for the useful detection of photons.
We had previously simply assumed this limited detector bandwidth and had
used the estimate around . This discussion provides, after the fact, the
justification for these assumptions.

Fig.4.6: Photograph of a TRD assembly. Each module, like the one being
held, contains \~ 1/ foils followed by a Xe filled chamber. Note the gas
flow indicators showing Xe for the detector (Fermilab E769). (Photo-
Fermilab)

***4.7 TRD Data***

Some data taken with a transition radiation detector exposed to both
pions and electrons of the same energy is shown in Fig. 4.7. The energy
spectrum of both x-rays and ionization is shown. Clearly there is an
exponential falloff with energy in both cases. The electrons are the
relativistic objects with a large g factor and they produce photons in
the 10 to 50 keV range. The emission probability for pions, which have a
much lower g factor due to their increased mass, is much reduced.
However, they do deposit ionization energy (Section 6) with the
occasional large deposition due to ray fluctuations.

Also shown in Fig. 4.7 is the distribution of the statistical likelihood
for electrons and pions showing that there is reasonable separation
between the two. The likelihood (see Appendix J) is made up using
multiple samples of the pattern of detected energy in distinct,
independent, transition radiation detector modules. This technique is in
the spirit of multiple ionization samples to accurately estimate dE/dx
which is discussed in Section 6. One such module in a stack of TRD
modules is shown in Fig. 4.6. The stack itself is visible in the
background. It is clear that TRDs are useful for particle identification
in the very high energy regime, even within the limitations due to
"saturation" phenomena which we have mentioned.

Fig. 4.7: Data taken with a TRD detector showing: a) the mixture of
x-ray and ionization energy for incident pions and electrons. Note the
peak for e at \~30 keV. b) Likelihood function for pions and electrons
using several samplings of the detected energy. (From Ref. 4.10, with
permission.)

## [References]{#_Toc437310864}

Section 4: Transition Radiation

\[1\] W.M. Allison, P.R.S. Wright, Oxford, Dunp, 35/83 (1983).

\[2\] A. Bodek, et al., Z. Physik, C[18]{.underline} 289 (1983).

\[3\] V.L. Ginzburg and I.M. Frank, JETP [16]{.underline} 15 (1946).

\[4\] X. Artru, et al., Phys. Rev. D [12]{.underline} 1289 (1975).

\[5\] C.W. Fabjan and W. Struczinski, Phys. Lett.[ 57B]{.underline} 484
(1976).

\[6\] T. Ludlam, et al., Nucl. Inst. Meth. [180]{.underline} 413 (1981).

\[7\] C.W. Fabjan, et al., Nucl. Inst. Meth. [185]{.underline} 119
(1981).

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