# IIIB. Energy Measurements

The energy of a particle can be destructively detected by initiating a
series of reactions whereby the total particle kinetic energy is
deposited in the detector. This technique is called calorimetry. The
energy is locally deposited so that the particle position is measured to
a degree of accuracy specified by the transverse fluctuations in the
energy deposition. The particle energy is measured to a level of
accuracy specified by the uniformity of the detector medium (the
"constant term") and the level of active sampling of the detector with
respect to total volume (the "stochastic term"). Thus, the calorimetric
technique provides a measurement of $\overrightarrow{p}$, as defined in
Fig. I.1. However, particle identity is lost so that subsequent
redundant measurements are not possible.

# 11. EM Calorimetry

"*A hen's only an egg's way of making another egg*."\....Samuel Butler

"*It grows - it must grow; nothing can prevent it*."\....Mark Twain

In this Section we will begin the discussion of calorimetry by looking
at electromagnetic showers. The reason to begin with the electromagnetic
processes is that they are somewhat better understood and better modeled
than the strong interaction processes which we will study in the
subsequent Section. Calorimetry itself refers to the destructive
detection of the energy of an incident particle. Its root comes from the
Greek word for heat. The idea is to absorb all the energy in a detecting
medium and by that means to record the energy of the incident particle.
We can make redundant measurements of charged particle momenta by first
tracking them in a magnetic field and then absorbing them in a
calorimeter. This redundancy is often very useful in cleaning up the
backgrounds which are always present in the study of a rare process.
Detection in a low mass medium allows for a measurement of the velocity
(Sections 2, 3, 4), momentum (Sections 5 through 8) and secondary decay
vertex (Section 9) of a charged particle. Subsequently the energy can be
absorbed in the calorimeter.

## 11.1 Radiation Length and Critical Energy

The underlying physics of electromagnetic calorimetry has been touched
on previously. The radiative process for electrons is defined by the
Bremsstrahlung cross section and the characteristic length for that is
the radiation length, X~o~. (See Section 10.)

$\begin{array}{r}
(\frac{\text{dE}_{B}}{E})\ \text{\textasciitilde}\ (\rho \text{dx}\frac{)}{X_{o}} \\
X_{o}\text{\textasciitilde}\left\lbrack \text{180}\frac{\text{gm}}{\text{cm}^{2}} \right\rbrack \left\lbrack \frac{A}{Z^{2}} \right\rbrack \\
t = \frac{x}{X_{o}}\text{}
\end{array}$ (11.1)

The radiation length in gm/cm^2^ goes as A/Z^2^. We characterize the
depth in material by expressing it in radiation length units, t. Note
that we implicitly do "particle identification" by observing radiative
processes, since at energies \<100 GeV only electrons and photons are
sufficiently relativistic to emit significant radiation (see Section
13).

The other quantity, which is vitally important for describing the
electron energy loss mechanism, is the critical energy, E~c~.

$\begin{array}{r}
\left( \frac{\text{dE}}{\text{dx}} \right)\text{\textasciitilde} - \frac{E_{C}}{X_{o}} \\
E_{C}\text{\textasciitilde}\frac{\left\lbrack \text{550}\left( \text{MeV} \right) \right\rbrack}{Z} \\
y = \frac{E}{E_{C}}
\end{array}$ (11.2)

The critical energy is that energy above which radiative processes
dominate. We discussed E~C~ in Section 6, showing loss mechanisms in
lead as a function of energy and the 1/Z dependence of the critical
energy. We define the electron energy in units of the critical energy,
y. As an example, in lead the radiation length is 0.56 cm, the critical
energy is about 8 MeV, and the minimum ionization loss is 1.13 MeV
gm/cm2 (Table 1.2). The ionization energy losses are expected to be
effectively universal, (see Section 6).

$\frac{\text{dE}_{I}}{\text{dx}}\text{\textasciitilde}\left\lbrack 3(\frac{\text{gm}}{\text{cm}^{2}}) \right\rbrack\left\lbrack \frac{Z}{A} \right\rbrack$
(11.3)

Minimum ionizing particles should lose about 1.5 MeV gm/cm^2^ in all
elements.

## 11.2 The EM Cascade

Now consider the cascade process which occurs when a high energy
electron hits a block of material. The incoming electron will first
Bremsstrahlung. If the energy of the Bremsstrahlung photon is
sufficiently high it will, in turn, produce an electron-positron pair.
The pair partners will then each Bremsstrahlung. Clearly, the processes
of Bremsstrahlung and pair production imply that a run away "shower"
process will occur leading to a rapid, geometric, increase in the number
of particles with depth.

This "cascade" process will continue until the secondary particles are
no longer energetically capable of multiplying. At that point the
maximum number of shower particles, N~max~, exists. Beyond the depth of
"shower maximum", t~max~, the number of particles dies away due to
ionization range out (Section 6) in the case of the electrons and
Compton scattering (Section 10) and photoelectric absorption (Section 2)
in the case of the photons.

A "cloud chamber" photograph of an electromagnetic cascade developing in
lead plates spaced by a gaseous detection medium is shown in Fig. 11.1.
We can see the shower begin in the upper plates, build up to shower
maximum, and then begin to die away. This is a visual realization of the
words just used to describe the cascade.

![](media/image1.png){width="4.875in" height="5.666666666666667in"}

Fig. 11.1: Cloud chamber photograph of an electromagnetic cascade
developing in spaced Pb plates. (From Ref. 13, with permission.)

A very schematic view of an electromagnetic cascade is shown in Fig.
11.2. The horizontal axis is the depth of the shower in radiation length
units, t, which is, as we said the characteristic scale. In reality
there are fluctuations in the interaction points. We use the mean which
occurs at one X~o~ into the material for the first interaction and, on
average, at depths spaced by one X~o~ for each subsequent interaction.
We also show the number of shower particles, N, as the cascade develops
as a horizontal scale and the fractional energy per particle,
![](media/image2.emf), in the cascade where E refers to the incident
energy.

![](media/image3.png){width="3.9451388888888888in"
height="5.548611111111111in"}

Fig. 11.2: Schematic of an electromagnetic cascade for the first 4
generations indicating depth t, number in the cascade N, and fraction of
the incident energy carried by each particle. Electrons and positrons
are indicated as -, while photons are represented as \-\--.

As the number of particles increases, their energy decreases. For
example, in Section 10 we showed the energy spectrum for electrons in
pair production. Since it is basically uniform from zero to the photon
energy, we again ignore the fluctuations and assign the mean energy of
1/2 the photon energy. A similar approximation is made for
Bremsstrahlung. The fluctuations both in the interaction points and in
the secondary energies are ignored in the interest of clarity.

Let us explore the "decay" kinematics very briefly. Consider a particle
of mass M, energy E, momentum p, which "decays" into massless particles.
The relativistic transformation from the CM (starred) frame to be lab
frame is; $e = g_{M}e \ast (1 + \beta_{M}\text{cos}q \ast )$, where
![](media/image4.emf) is the daughter CM energy, and
![](media/image5.emf). For isotropic decay,
$de = g_{M}\beta_{M}e \ast d(\text{cos}q \ast )$ = constant. The energy
distribution of the daughter is uniform. Clearly,
$\left\langle e \right\rangle = g_{M}e \ast \frac{E}{2},\text{since}\left\langle \text{cos*} \right\rangle = 0,$
and ![](media/image6.emf) is distributed uniformly from
![](media/image7.emf). This discussion should serve to motivate the
model which appears in Fig. 11.2. Other kinematic details are given in
Appendix A.

We observe that the number of particles ![](media/image8.emf) in the
cascade grows geometrically with the depth in the cascade. In what
follows ![](media/image9.emf) and ![](media/image8.emf) are the symbols
used for the particle energy and number of particles as a function of
depth.

![](media/image10.emf) (11.4)

The multiplication processes go on until the mean energy of the
particles in the cascade is equal to the critical energy. As the energy
falls below the critical energy multiplicative processes are no longer
possible and the maximum number of particles is reached. Beyond that
depth the cascade dies off.

$\begin{array}{r}
E_{C}\text{} = \frac{E}{2^{t_{\text{max}}}}\text{} = \text{}\varepsilon(t_{\text{max}})\  = \ \text{yE} \\
t_{\text{max}}\text{}\text{\textasciitilde}\text{ln}(y) \\
N_{\text{max}}\text{\textasciitilde}\frac{E}{E_{C}}\text{} = \text{}N(t_{\text{max}})\text{} = \text{}y
\end{array}$ (11.5)

For example, a 3.2 GeV shower in Pb has shower maximum at a depth of t =
6. At shower

maximum there are about 400 particles in the cascade.

## 11.3 Energy - Linearity and Resolution

The depth at which "shower maximum" occurs, t~max~, increases
logarithmically with the scaled incident energy y. The number of
particles at shower maximum, N~max~, increases linearly with energy,
with a scale factor equal to the critical energy. Therefore, if the
response of a "calorimeter" is proportional to the number of shower
particles, we expect that the calorimeter is a linear device.

This is not a completely trivial statement. Human beings do not have
linear sensory tranducers. Our ears and eyes respond logarithmically to
the sound and light levels of our environment respectively. This method
of response allows us to perceive a large dynamic range at the expense
of non-linear sensitivity. For example, the volume control on a radio
appears to be linear, but is simply the dial label; the actual power
level is logarithmic.

Note that the length needed for full shower containment, t~max~, goes as
\~ ln(E) while for tracking the length required for fixed measurement
error rises linearly with energy (dp/p = ap). This fact explains the
preferred use of calorimetry in applications requiring the measurement
of very high energy particles.

We now find a "total path length", L, for all generations in the shower
by converting the sum of all particles in the shower to an integral. We
assume that each particle in the shower goes one X~o~ in depth before
interacting. We find that the total path length in the shower is
proportional to the incident energy.

$\begin{array}{r}
L\text{}\text{\textasciitilde}X_{o}\sum_{i = 1}^{t_{\text{max}}}{N(t)} \\
\frac{L}{X_{o}}\text{}\text{\textasciitilde}\int_{o}^{N_{\text{max}}}{N(t)\text{dt}\text{}\text{\textasciitilde}}\text{}\int_{o}^{t_{\text{max}}}{2^{t}\text{dt}} \\
\text{}\text{\textasciitilde}\ (\frac{E}{E_{C}}\frac{)}{\text{ln}^{2}}\  = \ N_{\text{max}}^{\text{ln2}}
\end{array}$ (11.6)

The number of particles in the shower N(t) is strongly peaked at shower
maximum. Hence it is not too surprising that L is \~ X~0~ N~max~. This
result can be thought of as the N~max~ particles of the last
"generation" going one final X~0~ in depth before they range out by
losing energy E~c~ to ionization. We assume that the detected energy
signal is proportional to L.

Fluctuations in the shower development lead to fluctuations in the
number of particles in the shower. Thus, the path length, L, exhibits
stochastic fluctuations since the cascade is a random process.
Therefore, the fluctuation in the number of particles should go as
$\text{dN}\text{\textasciitilde}\sqrt{N}$ (see Appendix J). We expect
that a measurement of the energy contains a "stochastic term" due to
those fluctuations and consequently that the fractional error in the
calorimeter energy measurements should go as $\frac{1}{\sqrt{E}}$.

$\begin{array}{r}
N_{\text{max}}\  = \ \frac{E}{E_{C}} \\
\frac{\text{dE}}{E}\ \text{\textasciitilde}\ \frac{1}{\sqrt{E}}\ \text{\textasciitilde}\ \frac{\text{dN}}{N}
\end{array}$ (11.7)

For a 3.2 GeV shower in Pb, there is a 5% fluctuation in the 400
particles which exist at shower maximum.

This behavior of the error is in distinction to gaseous tracking
devices, which have fractional error, dp/p \~ p. Therefore, the
fractional momentum error degrades linearly with momentum in a tracking
device of fixed length whereas the fractional energy error for a
calorimeter, in principle, improves with energy. This is another basic
reason why detectors of very high energy particles focus on calorimetry
since it is one of the few detectors whose performance actually improves
with increasing energy.

## 11.4 Profiles and Single Cascades

In Fig. 11.3 is shown the longitudinal energy distribution of six
individual 170 GeV electron showers. This particular device consists
first of a stack of forty lead plates, each 1/8\", 0.57 X~0~, thick
interspersed with scintillator (see Section 2) as the active sampling
element. Each depth segment is read out independently.

![](media/image11.wmf){width="4.961111111111111in"
height="5.895138888888889in"}

Fig. 11.3: Longitudinal energy profiles of 6 individual 170 GeV
electrons incident on a stack of 40 Pb plates of 1/8\" interspersed with
scintillator active elements.

What is striking about these individual electron showers is that the
shower development has few fluctuations. The shape in depth is
essentially the same for each shower reflecting the large number of
particles in the cascade, N~max~ \~ E/E~c~ = 170 GeV/8 MeV \~ 21,250. We
know there is a fluctuation in the first interaction point. Given the
exponential nature of the distribution of free paths, Section 1, we know
that the rms fluctuation in the path length to the first interaction,
t~1~, is X~0~, while the mean first interaction point occurs at a depth
X~0~, i.e. \<t~1~\> = 1, \<~t1~\> = 1. Looking at the events shown in
Fig. 11.3 we see that the main fluctuation, on an event by event basis,
is just the fluctuation in the interaction point which is roughly ± 2
plates. The shower shape at high energy has both energy sharing
fluctuations and interaction point fluctuations of the first and
subsequent generations. However, the shower contains so many particles
that the latter are washed out.

These observations of real electromagnetic showers lead us to a
one-dimensional parameterization of the longitudinal behavior of the
shower.

$\begin{array}{r}
\frac{\frac{\text{dE}}{E}}{\text{du}}\  = \ \frac{\left\lbrack u^{a}e^{- u} \right\rbrack}{\Gamma}(a + 1) \\
u\  = \ \text{bt},\ b\ \text{\textasciitilde}\ \frac{1}{2} \\
t_{\text{max}}\ \text{\textasciitilde}\ \text{ln}\ y\ \text{\textasciitilde}\ (a\  - \ 1)\frac{\ }{b}
\end{array}$ (11.8)

The depth t is defined with respect to the initial interaction point. In
the shower parameterization the variable u is, up to multiplicative
constants, t. The variable a is proportional to ln(y). The variable b is
weakly energy dependent and weakly dependent on material. The power law
behavior, u^a^, implies a fast geometric rise in the energy deposition
due to the multiplication process in the cascade, whereas the
exponential fall off, e^-u^, is expected at larger depths beyond shower
maximum where the mean particle energy is less than the critical energy.
The gamma factor is there only to normalize the total shower energy to
the incident energy, E.

The shower maximum location, t~max~, moves to greater depths as the
energy increases. For example a 3.2 GeV shower has t~max~ \~ 6, while
t~max~ \~ 9 for a 64 GeV shower. It is also weakly dependant on the
critical energy which goes as 1/Z, Eq. 11.2. Thus, the depth in
radiation length units of the location of shower maximum is somewhat
dependent on the type of material of which the calorimeter is made,
t~max~ \~ ln y = ln (E/E~c~) \~ ln (ZE). As a numerical example for
shower development in lead, with the critical energy of 7.5 MeV, a
photon with = E~c~ has a mean free path of 10 gm/cm2, see Section 1.
Since the radiation length in Pb is 6.4 gm/cm^2^, see Table 1.2, the
mean free path of a photon near the critical energy is about 1.6
radiation lengths. That explains why the exponential fall off given in
Eq. 11.8 is basically in units of radiation lengths up to a numerical
factor of order 1. The range of an electron with ![](media/image12.emf),
is also roughly \~ X~0~.

## 11.5 Sampling Devices

Since the calorimeter is an energy measuring device, its most important
physical characteristic is the energy resolution. Let us consider a
"sampling calorimeter" where the shower is developed in high Z plates
where most of the energy is lost. The shower energy is sampled in thin
plates of low Z material such as scintillator plastic or liquid argon.
The small fraction of the energy which is actively sampled should be
proportional to the total absorbed energy. The basic model for the
sampling calorimeter is that the shower develops in the inert high Z
material and is sampled in the active low Z material.

If we assume that the shower evolves rapidly, i.e. that there are high Z
sampling plates with thickness \> X~0~, then there is no correlation
between consecutive active sampling layers. Thus the total number of
particles traversing the active layers, N~S~, is the total number in the
shower or the total path length, L = E/E~c~, (Eq. 11.6) divided by the
thickness of the inert layers, ![](media/image13.emf), placed between
the active samples. The geometry is shown schematically in Fig. 11.4.

$\begin{array}{r}
N_{S} = \frac{L}{\Delta t} \\
\text{}\ \text{}\text{\textasciitilde}(\frac{E}{E_{C}}\frac{)}{\Delta t}
\end{array}$ (11.9)

The particles traversing the sampling layers are presumed to all
register, either directly by charged particle ionization or indirectly
with photon Compton or photoelectric absorption accompanied by recoil
ionization. The error due to fluctuations in the fraction of the energy
appearing in the active sampling layers, dE, should be the fluctuation
in the total number of sampled particles, N~S~. It is customary to
define what is called the "stochastic coefficient" in the energy
resolution, a~samp~. The stochastic term, due to sampling fluctuations,
depends on the thickness of the inert plates and scales as $\sqrt{Dt}$.
Obviously, if the plates become very thick, the sampled energy is not a
good representation of the full shower energy and the fluctuations are
increased.

![](media/image14.wmf){width="2.0208333333333335in"
height="2.939583333333333in"}

Fig. 11.4: Definitions for sampling calorimetry with active samples, of
thickness $dt$and sampled energy ![](media/image15.emf), and passive
absorber, with $Dt$, ![](media/image16.emf).

The energy deposited by ionization, in going through a plate of
thickness t at the critical energy E~C~ is just E~C~t. Clearly low E~C~
means more shower particles, N~max~ \~ E/E~c~, which implies a smaller
stochastic term.

$\begin{array}{r}
\left( \frac{\text{dE}}{E} \right)_{\text{samp}} = \frac{1}{\sqrt{N_{S}}}\frac{{ºa}_{\text{samp}}}{\sqrt{E}} \\
a_{\text{samp}}\text{\textasciitilde}\sqrt{E_{C}Dt} \\
\text{\textasciitilde}\sqrt{DE}
\end{array}$ (11.10)

As a simple example, a 1 GeV electron with a critical energy of 7.5 MeV,
has a maximum number of shower particles, N~max~ = 131. With 1/2
radiation length sampling for the thickness of the plates, t = 1/2, we
find the total number traversing the active layers to be 262 which
yields a sampling stochastic term of 6.2% due to fluctuations in the
sampling of the number in the shower. This will later be shown to be a
reasonable scale for the errors observed in typical sampling
calorimeters constructed to measure electromagnetic showers (see Fig.
11.5).

## 11.6 Fully Active Devices

There are other effects that we need to consider in going to a situation
with very fine sampling. The logical extension of fine sampling is the
"fully active" calorimeter where the shower developing medium is itself
active. Examples of this type of calorimeter are crystals, such as BGO,
lead tungstate (PbW0~4~**),** or lead glass.

We have assumed previously that counts in the different active layers
were uncorrelated. A photon of low energy which is in the cascade will
Compton scatter (Section 10) or photoeffect (Section 2) as these are the
two largest cross sections at low energies. Below about 1 MeV in lead
the photoelectric effect dominates. To set the scale, the mean free path
for a 1 MeV photon in lead is about 3 cm, or about 6 X~0~. If we go to
fine sampling, in order to reduce the stochastic term which goes as
\~$\sqrt{Dt}$, the photon can and does cross several sampling layers
invalidating our assumptions.

Another assumption we made was that the shower development took place
entirely in the absorber, while the detection took place in the thin
active layers. Now, clearly, if we go to very fine sampling this
assumption also begins to break down, since significant energy is
deposited in the active medium, ![](media/image15.emf), rather than the
inert absorber, ![](media/image17.emf).

In addition, we have been considering the shower development to be a
one-dimensional problem. In fact, we know that we have multiple
scattering of the shower particles which becomes increasingly important
as their number increases and their energy decreases to \~ E~c~ at
shower maximum. The scattering angle may become very large since E~C~ is
comparable to E~S~ = 21 MeV. The path length then exceeds the one
dimensional path length at normal incidence, ![](media/image18.emf),
since particles traverse the sampling layers at finite angles. A rough
estimate of the increase is

$\begin{array}{r}
Dt \rightarrow \frac{Dt}{\left\langle \text{cos}q_{\text{MS}} \right\rangle} \\
q_{\text{MS}}\text{\textasciitilde}(\frac{E_{S}}{E_{C}})\sqrt{Dt\text{\textasciitilde}1}(\text{Section}5) \\
\text{     }\text{\textasciitilde}(\frac{E_{S}}{E_{C}}p)
\end{array}$ (11.11)

The multiple scattering angle is large, causing a substantial path
length increase and therefore a stochastic coefficient increase.

The changes due to fine sampling and multiple scattering are
incorporated in a theoretically motivated phenomenological formula for
the sampling coefficient.

$\begin{array}{r}
a_{\text{samp}}\text{\textasciitilde}\left\lbrack (1 - W)\left( \frac{DE + dE}{\left\langle \text{cos}q_{\text{MS}} \right\rangle} \right)^{\left( \frac{1 - W}{2} \right)} \right\rbrack \\
W = \frac{dE}{(}dE + DE)
\end{array}$ (11.12)

The ratio of the sampled energy to the total energy lost in a layer is a
parameter, W. Clearly as the parameter ![](media/image19.emf) we are
ignoring the effects of the active sampling layers, and Eq. 11.12
collapses to Eq. 11.10 modified by Eq. 11.11. In the limit
![](media/image20.emf) we have a fully active detector and a~samp~
![](media/image21.emf) 0. This unrealistic limit occurs because in this
particular formulation the only errors that are being considered in the
measurement of the energy are those of a stochastic nature. We will
discuss other sources of error later.

Data on a~samp~ taken with different levels of sampling, W, in
calorimeters are shown in Fig. 11.5 where the vertical axis is the
stochastic coefficient in percent. We see that indeed a 2% stochastic
term can be achieved for fine sampling detectors which are of order 80%
active. If the sampling fraction becomes 10% there\'s a rapid increase
to a \~8% stochastic coefficient. The lines drawn on Fig. 11.5 are just
Eq. 11.12.

![](media/image22.png){width="4.111111111111111in" height="3.69375in"}

W

Fig. 11.5: Stochastic term coefficient as a function of sampling
fraction W for a number of calorimeters. (From Ref. 9, with permission.

A photograph of a typical R&D sampling calorimeter, in this case
installed in a test beam, is shown in Fig. 11.6. This object has lead
plates interspersed with scintillator in its electromagnetic section and
iron plates, rather thicker, interspersed with scintillator in the
following section which is used to intercept strongly interacting
particles or hadrons (see Section 12 and Section 13). The data plotted
in Fig. 11.3 come from the initial 40 lead plates of the most "upstream"
part of this detector. Each layer has independent photomultiplier
(Section 2) readout in order that shower development can be studied. In
contrast, a photograph of a fully active homogeneous calorimeter medium
using lead glass is shown in Fig. 11.7. This is a transparent medium
with high density with respect to ordinary glass. Other homogeneous
media are other glasses and crystals such as BGO.

![](media/image23.wmf){width="6.2in" height="4.983333333333333in"}

Fig. 11.6: Photograph of a typical sampling calorimeter showing
interspersed Pb and scintillator (EM) and Fe and scintillator (HAD)
planes. (Photo - Fermilab.)

![](media/image24.png){width="6.0in" height="4.763194444444444in"}

Fig. 11.7: Photograph of a fully active homogeneous calorimetric medium
(Pb glass). (Photo - Fermilab.)

An energy level diagram for a "generic" crystal detector is shown in
Fig. 11.8. Energy deposit by ionization causes an electron to be excited
from the valence band (VB) to the conduction band (CB). The electron is
quickly trapped by an impurity level ![](media/image25.emf) and a
radiative transfer causes the subsequent emission of a photon of energy,
![](media/image26.emf). Note that this photon energy is \< E~g~ so that
it exits the crystal without absorption. . The crystal is thus
transparent to its own impurity state emission. An example is "thallium
activated" (i.e. ![](media/image27.emf) states) NaI. In NaI the band gap
is E~g~ = 6 eV so that visible light, since it has energy less than the
gap energy, is not absorbed. The thallium impurity state has a decay
time ![](media/image28.emf) \~200 nsec and the emission
![](media/image29.emf) has a spectrum of photons peaked near
![](media/image30.emf) or ![](media/image31.emf) \~ 3.1 eV.

![](media/image32.wmf){width="3.129166666666667in"
height="1.6645833333333333in"}

Fig. 11.8: Energy level diagram for a crystal detector.

## 11.7 Transverse Energy Flow

The shower has, so far, been considered a one-dimensional object, which
is roughly correct. The physics defines the transverse momentum scale
and in this particular case the physics has to do with atomic electrons.
The characteristic transverse momentum in these collisions is the mass
of the electron, m~e~. If we are early in the development of a high
energy shower, typical angles are a few milliradians,
![](media/image33.emf) \~ m~e~/ \~ 0.51 mrad at = 1 GeV. However, as we
get deeper in the shower the cascade particles become softer as they
share the parents energy among themselves. In this case the shower
rapidly becomes more isotropic.

$\begin{array}{r}
\left\langle p_{T} \right\rangle\text{\textasciitilde}m_{e} \\
\left\langle q \right\rangle\text{\textasciitilde}\frac{m_{e}}{e}(t) \\
\left\langle q \right\rangle_{\text{SM}}\text{\textasciitilde}\frac{m_{e}}{E_{C}}
\end{array}$ (Eq. 11.13)

The typical angle of a particle at a depth t is defined by the energy of
the particles in the shower at that depth, ![](media/image34.emf). At
shower maximum the typical energy is the critical energy, so that the
typical angle of a particle at shower maximum is \~ m~e~/E~C~. For
example, in lead, the pair production angle at shower maximum is about
4o.

In fact, multiple scattering dominates the transverse shower development
rather than finite production angles since E~S~ is \>\> m~e~. Hence, we
expect multiple scattering to give the dominant contribution to the
transverse size of a shower (see Eq. 11.11).

$\begin{array}{r}
\left\langle p_{T} \right\rangle_{\text{MS}}\text{\textasciitilde}E_{S}\sqrt{t},\sqrt{t}º1 \\
\left\langle q \right\rangle_{\text{SM}}^{\text{MS}}\text{\textasciitilde}\frac{E_{S}}{E_{C}}
\end{array}$ (11.14)

It is traditional to define a transverse size, the Moliere radius, which
can be thought of as the transverse distance that a particle at the
critical energy goes in traversing the last radiation length before it
dies off.

$\begin{array}{r}
r_{M}\text{\textasciitilde}E_{S}\frac{X_{o}}{E_{C}} = \left\langle q \right\rangle_{\text{SM}}^{\text{MS}}X_{o} \\
r_{M}\text{\textasciitilde}\left\lbrack 7(\frac{\text{gm}}{\text{cm}^{2}}) \right\rbrack(\frac{A}{Z})
\end{array}$ (11.15)

The Moliere radius, ![](media/image35.emf), depends on E~S~, the
multiple scattering energy (Section 5), E~C~ the critical energy
(Section 6) and X~0~ the radiation length (Section 10). Referring to
previous Sections for numerical constants, we find that the Moliere
radius is fairly constant expressed in gm/cm2. For most of the periodic
table, where A/Z \~ 2, we have a Moliere radius of about 14 gm/cm2.
Denser materials have physically smaller Moliere radii.

A shower distribution of transverse radius in Moliere units is shown in
Fig. 11.9. In this plot the entire shower has been integrated over in
depth. Comparing Monte Carlo model (Appendix K) results with
measurements on aluminum, lead and copper, we see that the shape in r~M~
units is approximately universal and that 90% of the energy is contained
within about 1 Moliere radius while 95% is contained within about 2
Moliere radii, which, for example, is about 2.3 cm in lead.

![](media/image36.png){width="2.329861111111111in"
height="3.5506944444444444in"}

Fig. 11.9: Transverse electromagnetic cascade energy distribution as a
function of the number of Moliere radii from the incident e direction.
(From Ref. 10, with permission.)

The radius for energy containment clearly depends on where your are in
depth in the shower. In Fig. 11.10 is plotted the transverse
electromagnetic shower energy distribution in a Monte Carlo model as a
function of the depth in the shower. We see that there is a sharp peak
extending out to a depth of \~(0.1 - 0.3) radiation lengths which
broadens as we go deeper in the shower. This is a logarithmic plot so
that even at a depth of 9 radiation lengths there is containment within
x~T~ \~ 0.5 X~0~ because 90% of the energy occurs within that radius.

![](media/image37.png){width="2.7847222222222223in"
height="3.4569444444444444in"}

Fig. 11.10: Transverse electromagnetic cascade energy distribution in
arbitrary units as a function of the number of radiation lengths from
the incident e direction. Curves are shown at different depths in the
shower, displaying the broadening of the shower as the depth increases.
(From Ref. 10, with permission.)

There are other subtle effects due to multiple scattering. As we follow
particles down in energy, we find that \~ 30% of the energy in a shower
is deposited by electrons with energies less than 1 MeV for high Z
absorbers. These particles at the end of the shower are almost isotropic
in angle due to multiple scattering; as many go backwards as forwards.
Therefore, it is important in making computer models of showers (see
Appendix K) to keep track of particles with quite low energies, as they
are important to the total energy deposition mechanism in calorimeters.

In addition to the multiple scattering increase in the path length there
are also delta rays, which we considered in Section 5. They lead to
ionization fluctuations. This effect may dominate over the sampling
fluctuations. It is clearly most important for thin active layers. For
example, if we were sampling with a gaseous detector instead of a
plastic or a liquid, and if the thickness of the sampling layer were
only 10-3 gm/cm2, (\~1 cm of gas), we would have a 40% increase in the
stochastic coefficient, a~samp~, due to the track length fluctuations
caused by the delta rays. In general, small sampling fractions are
dangerous (if inexpensive).

## 11.8 Calibration Methods

Finally there\'s the practical question of the energy calibration of a
calorimeter. How is that performed? In Fig. 11.6 we showed an
unrealistic device, since this is an R&D detector, where each sampling
layer is individually read out. Data taken with that detector, 40 layers
of lead followed by 55 layers of 1" thick iron, for incident muons are
shown in Fig. 11.11. Remember that muons at energies well below the
critical energy simply ionize the material and deposit the minimum
ionizing energy. We need to take into account the relativistic rise that
we discussed previously (Section 6) and the ![](media/image38.emf) ray
fluctuations (Section 5), but these are small effects. Data shown in Fig
11.11 were obtained with 15 GeV incident muons and with 50 GeV incident
muons. The energy deposit per layer is effectively the same at those two
energies as we expect since the ![](media/image39.emf)of the
![](media/image40.emf) has only changed a factor of 3 and we derived a
soft logarithmic dependence on ![](media/image41.emf) as part of the
relativistic rise, (see Section 6).

![](media/image42.wmf){width="3.2180555555555554in"
height="2.417361111111111in"}

Fig 11.11: Response of each sampling layer of the calorimeter shown in
Fig. 11.6 to incident muons indicating the potential use of muons as a
means of calibration. a) 15 GeV muons, b) 50 GeV muons.

We could adjust the gain of each photomultiplier tube in the individual
samples to calibrate this detector. If we put a minimum ionizing
particle into the detector, we would then get out the same signal in all
layers. That is one way in which we could use external particles to
calibrate the detector. Note that for financial reasons we normally
construct a homogeneous detector, optically sum the sampling layers
(light pipe, Section 2) and use only a few phototubes or other
transducers.

We could also expose the detector to electrons of different known
energies using a beam of particles prepared by momentum analysis in
magnets (Section 7). Thus we can map out the energy response,
resolution, and linearity of the device using "test beams". The use of
muons would then monitor the initial beam calibration and insure that
the system response is time independent. Recall that muons are available
as the largest component of sea level cosmic rays. (See Table 6.1.)

Electromagnetic calorimeters are major subsystems of all collider
detectors, because photons and electrons are thought to be pointlike
fundamental particles (see Section 13). There-fore, the energy and
position of these particles should be recorded (Fig. I.1). For both
photons and electrons, the EM shower can be localized transversely to a
size which is of order r~M~. This localization with energy improves as
$\frac{1}{\sqrt{E}}$ since it is basically an energy centroid
measurement limited by sampling fluctuations,
$\text{dx}_{T}\text{\textasciitilde}\frac{r_{M}}{\sqrt{E}}$with *E* in
GeV

The total energy is measured with an accuracy which goes as \~
$\frac{1}{\sqrt{E}}$ until limitations in the uniformity of the detector
construction are encountered. At that point a constant fractional error
dE/E b dominates the energy resolution. Hence, for use at arbitrarily
high energies an EM calorimeter should be built to have as uniform a
response as possible leading to as small a constant fractional energy
resolution, b, as possible.

## References

Section 11: EM Calorimetry

\[1\] [High Energy Particles]{.underline}, B. Rossi, Prentice-Hall, Inc.
(1952).

\[2\] Proceedings of the Calorimeter Workshop, M. Atac, Fermilab (1975).

\[3\] Tristan eP Working Group Report, S. Iwata, Nagota University,
DPNU-3-79 (1979).

\[4\] Crystal Calorimeters in Particle Physics, G. Gratta, H. Newman,
R.Y. Zhu, Annual Review of Nuclear and Particle Science,
[44]{.underline} 453 (1994).

\[5\] Liquid-Argon Ionization Chambers as a Total Absorption Detectors,
W.J. Willis, V. Radeka, Nucl. Inst. Meth., [120]{.underline} 221 (1974).

\[6\] [Calorimetry in High Energy Physics]{.underline}, D.F. Anderson,
M. Derrick, H.E. Fisk, A. Para, C.M. Sazama, World Scientific Publishing
Co. (1991).

\[7\] Fluctuations in Calorimetry Measurements, U. Amaldi, Physica
Scripts, [23]{.underline} 409 (1981).

\[8\] D.F. Crawford, H. Messel, Phys. Rev., [128]{.underline} 2352
(1962).

\[9\] "Performance and Limitations of Electromagnetic Calorimeters",
T.S. Virdee in [Calorimetry in High Energy Physics]{.underline}, Capri,
Italy (1991) World Scientific.

\[10\] T. Ferbel, [Experimental Techniques in High Energy
Physics]{.underline}, Addison-Wesley Publishing Co., Inc. (1987).

Table of Contents

IIIB. Energy Measurements [1](#iiib.-energy-measurements)

11\. EM Calorimetry [2](#em-calorimetry)

11.1 Radiation Length and Critical Energy
[2](#radiation-length-and-critical-energy)

11.2 The EM Cascade [3](#the-em-cascade)

11.3 Energy - Linearity and Resolution
[6](#energy---linearity-and-resolution)

11.4 Profiles and Single Cascades [8](#profiles-and-single-cascades)

11.5 Sampling Devices [10](#sampling-devices)

11.6 Fully Active Devices [11](#fully-active-devices)

11.7 Transverse Energy Flow [16](#transverse-energy-flow)

11.8 Calibration Methods [19](#calibration-methods)

References [21](#references)