Exposition on Particle Calorimetry

Michael Kossin

Fall 2010

Contents

I Introduction 2

1 Particles and their classes 2

II Showers 4

2 Electromagnetic showers 52.1 Electron and positron collision events and energy loss mechanisms . . . . . . . . . . . . . . . 5

2.1.1 High energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Low energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Critical energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Photon collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Mean free path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Electromagnetic shower propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Shower maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Energy deposit as a function of depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Moliere radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 Shower profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Hadronic showers 123.1 Electron collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Spallation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Hadronic shower components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Spallation cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Consider the neutron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Invisible energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.7 Hadronic shower profiles and their relation to electromagnetic shower profiles . . . . . . . . . 16

III Calorimetry in general 16

4 Measuring end results 164.1 Recording scintillation events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Homogeneous and sampling calorimeters and energy response 185.1 Energy response of homogeneous calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Energy response of sampling calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1

6 Energy resolution 196.1 Sampling fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.2 Fluctuations due to invisible energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Compensation 207.1 Decreasing electromagnetic shower energy response . . . . . . . . . . . . . . . . . . . . . . . . 207.2 Increasing hadronic shower energy response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

IV Calorimeters at the Large Hadron Collider 21

8 The Large Hadron Collider 21

9 Geometry 23

10 The Compact Muon Solenoid 2410.1 Electromagnetic calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

10.1.1 Main barrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.1.2 Endcaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.1.3 Preshower detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10.2 Hadron calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.3 Zero degree calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.4 Calibration of the CMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11 ATLAS 2811.1 Electromagnetic calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.2 Hadron calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Part I

IntroductionPerhaps the most common use of the term calorimeter is in reference to a device used to measure heattransferred to a medium during various processes, most simply through the measurement of temperaturechange of the medium. In the field of particle physics, a calorimeter is a device which, ideally, measures thetotal energy of a particle. Analogous to the fact that traditional calorimeters can only measure a changein thermal energy if that energy is eventually converted into heat which is transferred to the calorimeter,particle calorimeters can only measure the energy associated with a particle if that energy is transferredfrom the particle to the calorimeter’s sampling medium; since energy is defined as the ability to do work,the energy of a particle can only be measured as that energy is used in a demonstration of the work theparticle is capable of. The purpose of the calorimeter is to cause particles to perform such demonstrationsand measure their final products [12].

1 Particles and their classes

In order to examine how different particles interact, it is important to examine what properties are differentbetween them. We use the term interact to talk about the mechanisms through which particles affect eachother. These fundamental mechanisms are the strong force, the weak force, electromagnetism, and gravity.Mankind’s scientific knowledge has recently grown to the point where we are able to explain how some theseforces affect particles on a subatomic level at low energy levels: particles belonging to a class called bosonscarry forces between other particles. Photons are the carriers of the electromagnetic force, gluons are the

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Figure 1: The different discovered and theorized particles, the forces through which they interact, and relatedtheoretical systems. By Gaetan Landry.

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carriers of the strong force, and the W and Z bosons carry the weak force. A hypothetical particle calledthe graviton may carry the gravitational force. The forces vary in their strengths and the way their effectscarry across a distance. For example, gravity is much weaker than electromagnetism, but the effects of bothgravity and electromagnetism may permeate through an infinite distance, while the effects of the nuclearforce, an artifact of the strong force, are limited to subatomic scales. Not all forces may affect all particles.In fact, there is a theoretical form of matter designated as dark matter which interacts with other matterthrough gravity, and is theorized to interact through the weak force, but is not affected by any other force.More detailed discussion of the mechanisms of force propagation involve studies of topics such as quantumfield theory, and this paper shall proceed no further in this regard. Particles that interact through theelectromagnetic force are said to be charged, and a particle may be either positively charged or negativelycharged. Particles that interact through the strong force are said to be color charged. Particles also carrydifferent values of a property called spin.

Electrons, muons, neutrinos and a few other particles make up the class of particles known as leptons.All leptons carry mass and a spin value of 1/2. Electrons carry negative charge. The muon has the sameelectric charge as the electron but is much more massive. The neutrino carries much less mass than theelectron and no electric charge.

Hadrons are particles made up of quarks. Quarks come in several flavors and may carry positive ornegative charge with a magnitude of 1/3 or 2/3 of the charge of an electron. The different quarks aredifferentiated by their electric charges, their masses, and their weak isospins (a property responsible for howthey interact with the weak force). The different quarks are known as up, down, strange, charm, top, andbottom. Hadrons can be made of two or three quarks; those made of three quarks are called baryons, whilethose made of one quark and one antiquark (a form of antimatter, which shall be discussed shortly) are calledmesons. Hadrons include protons, neutrons, kaons, and pions. The proton carries a positive electric chargewhose magnitude equals that of the electron’s and a spin of 1/2. The neutron carries no electric charge, aspin of 1/2, and a mass slightly greater than that of a proton. Pions are mesons and come in several typesthat carry different electric charges and masses. A pion may carry either a positive or negative charge equalin magnitude to that carried by an electron, or no electric charge, and carries a mass (differently-chargedpions carry slightly different masses) greater than that carried by an electron and less than that carried bya proton. A kaon may also carry a positive or negative charge equal in magnitude to that of an electron, orno electric charge, and carries a mass greater than that of a pion but less than that of a proton. Kaons aremesons, but, unlike pions, they carry a strange quark in addition to an up quark or a down quark.

Antiparticles are those which exhibit the same properties as certain particles except for the fact thatthe electric charge that they hold is opposite in sign. The positron, which is the antiparticle compliment ofthe electron, for example, has the same mass and spin as an electron, but holds a positive charge equal inmagnitude to that of the electron. There exist antiparticle counterparts to particles which carry no electriccharge as well, and, in the case of neutral antihadrons, contain antiquarks which have electric charges thatare opposite to their quark counterparts. Mesons are composed of both quarks and antiquarks. Kaons arecomposed of either an up quark and a strange antiquark, a down quark and a strange antiquark, a strangequark and a down antiquark, or a strange quark and an up quark. A pion can be composed of an up quarkand a down antiquark, which is the antiparticle of the version that is composed of a down quark and an upantiquark. Pions may also be composed of an up quark and an up antiquark or a down quark and a downantiquark.

Part II

ShowersWe shall now discuss the events that occur when high energy particles enter a medium.

As particles move through a medium, they are affected by and affect the medium through collisions. Acollision is an event in which particles or groups of particles come close enough together to affect each other.

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Collisions may result in one or more events depending on the types of particles involved, their energies, andother factors. Showers occur when these events result in subsequent collisions. There are two categoriesof showers that may develop: electromagnetic showers result from the penetration of high energy electrons,positrons, or photons, while hadronic showers result from the penetration of hadrons.

2 Electromagnetic showers

Electromagnetic showers are those that involve electrons, positrons, and photons traveling through a medium.We shall first examine the individual collisions events, then look at how these events are perpetuated throughshowers.

2.1 Electron and positron collision events and energy loss mechanisms

Electrons traveling through a medium lose energy as they collide with atomic electrons through interactionsinvolving the electromagnetic force. The energy lost during each collision can be calculated from the impulseeffect of the electromagnetic force, but this calculation is complicated by the deflection this impulse impartsupon electrons and positrons. Later, we shall examine in greater detail the energy lost by much heavierparticles through calculations simplified by the fact that these particles are hardly deflected during collisionswith electrons.

2.1.1 High energy

Bremsstrahlung Bremsstrahlung, which is German for ”braking radiation”, arises from the fact thataccelerating electric charge sources produce electromagnetic radiation. Since particles much heavier thanelectrons do not undergo significant acceleration when colliding with electrons except when traveling at veryhigh speeds due to relativistic effects, such collisions do not produce significant bremsstrahlung. However,a large portion of the energy lost by high energy electrons and positrons during collisions is due to thisphenomenon. The wavelength of the radiation produced depends on the magnitude of the acceleration ofthe charge source and is continuous: the wavelength can take on an uncountable infinite number of values.Quantum mechanical theory explains that electromagnetic radiation is carried by photons. Photons eachcarry an amount of energy that depends only on the frequency of electromagnetic radiation.

Radiation length relates the energy lost by a high energy electron or positron through bremsstrahlung tothe distance it travels. It is the distance such a particle travels as it loses 1 − 1

e of its energy through thismechanism. The radiation length varies with the material the electron or positron travel through: less densematerials have a longer radiation length than denser materials. When discussing calorimetry, we sometimesuse radiation length as a unit of thickness. When thickness is described in this way, the energy lost by ahigh energy electron or positron due to bremsstrahlung through a given thickness does not depend on thedensity of the penetrated material.

2.1.2 Low energy

At lower energy levels, energy loss is dominated by mechanisms other than bremsstrahlung.

Ionization If an electron imparts enough energy upon an atomic electron to completely overcome thecoulomb force binding that electron to the atom, ionization occurs. The atomic electron leaves the atom,becoming a free electron, and the atom becomes an ion.

Excitation When a collision between electrons does not involve energy levels high enough for ionizationto occur, atomic electrons involved in a collision may simply enter an exited state. According to quantummechanical theory, electrons orbiting an atomic nucleus occupy one of a number of states depending on theirenergy. These states have discreet energy requirements: an electron cannot jump from one state to another

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Figure 2: This graph illustrates two definitions of critical energy. The points where the two bremsstrahlunglines cross the ionization line on this graph are the critical energies given by the second definition suggestedby experimental physicist Bruno Rossi. The dotted line shows the approximate bremsstrahlung values givenby the equation which makes the definitions match. From Particle data group.

2 5 10 20 50 100 200

Copper X0 = 12.86 g cm−2

Ec = 19.63 MeV

dE

/dx ×

X0 (

MeV

)

Electron energy (MeV)

10

20

30

50

70

100

200

40

Brems = ionization

Ionization

Rossi: Ionization per X0 = electron energy

Total

Bre

ms

≈ E

Exa

ctbr

emss

trah

lung

unless it has absorbed a certain amount of energy. Conversely, an electron cannot absorb any energy unlessthat energy is enough to cause the electron to rise to the next highest state. The lowest state an electroncan occupy given an atom’s configuration is called its ground state. An electron exited to a state beyond itsground state quickly fall back to its ground state, emitting radiation (photons) in the process. The time ittakes an electron to fall from an exited state to its rest state is called the response time. The frequency of theradiation emitted and the energy of the photons carrying it depend on the difference between the energiesassociated with the atomic electron’s ground state and the exited state.

2.1.3 Critical energy

Critical energy is the energy level at which energy lost by a particle through bremsstrahlung equals thatlost through ionization and excitation during collisions [4]. The reason the relationship between energylost through these two mechanisms depends on the energy of the particle is that energy loss rate per ra-diation length through ionization rises logarithmically while energy loss rate per radiation length due tobremsstrahlung rises linearly. In one definition, the critical energy is the energy level at which the energylost by an electron through bremsstrahlung equals that lost through ionization. Another definition gives thecritical energy as the energy at which the all the energy held by an electron would be lost through ionizationin just one radiation length. These definitions are actually equivalent if we approximate the energy lostthrough bremsstrahlung as the total particle energy divided by the radiation length in the first definition, anapproximation which breaks down at low energy levels. A comparison of these two definitions is illustratedby figure 2.

6

Figure 3: Pair production occurring, with energy being absorbed by a nucleus. We see the electron andpositron diverge from the path of the photon by opposite angles with magnitude θ. By David Horman.

θθ

Nucleus

2.2 Photon collisions

Photons carry energy equal to the frequency of the radiation associated with it multiplied by Planck’sconstant. At low energy, photons deposit energy through the photoelectric effect, but at higher energies aphenomenon known as pair production may occur when a photon enters the vicinity of another particle.

2.2.1 Pair production

Photons, under certain conditions, can be replaced by pairs of electrons and positrons. The fact that thisevent is possible may be counterintuitive: a massless particle disappears, and two particles, both carryingmass and electric charge, suddenly begin to exist. This feeling can be combatted by answering a simplequestion: are any physical laws actually broken when such an event takes place? More specifically, are lawsthat hold that mass, energy, charge, momentum, or angular momentum (which is the sum of the quantummechanical concepts of “spin” and orbital angular momentum) cannot be created or destroyed being violated?Obviously, since an electron has an electric charge that is opposite that of a positron, the amount of electriccharge of the system is conserved during such a transformation: the combined charges of the electron and thepositron equal the charge of the original photon, or zero. Likewise, the spins of the electron and positron,each equalling 1/2, add up to equal the spin of the original photon, or one. It is true that neither massnor energy are conserved if we adhere to the classical understandings of both. However, Einstein’s famousequation E = mc2 tells us that an amount of energy has an equivalence to an amount of mass. A massiveparticle, even one at rest, is said to carry rest energy equal to the particle’s mass multiplied by the square ofthe speed of light. Thus, if a photon becomes an electron and a positron, energy conservation is not violatedsince the energy of the photon can become mass held by the electron and positron, but only if the photonhad an amount of energy that was equal to or greater than the combined rest energies of these two particles.Therefore, if a photon carries less energy than this, pair production cannot occur. If a photon has moreenergy than the combined rest energies of the electron and positron, the electron and positron, upon theircreations, will each have kinetic energies equal to half the difference between the photon’s energy and thecombined rest energies of the particles. Finally, momentum may be conserved during pair production under

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certain conditions as well. Photons carry a positive amount of momentum related linearly to their energies.This momentum can be conserved during production of electron and pairs that are at rest (due to theirgeneses from a photon whose energy equalled the combined rest masses of the positron and electron) onlyif such pair production occurs in the vicinity of a mass-carrying particle or group of particles. Momentumconservation in the event of pair production from a photon possessing more energy than the combinedenergies of the created electron and positron is also impossible in the absence of additional material whichmay absorb momentum, but this concept is a bit less intuitive. We shall examine this concept by lookingat the special case in which a high energy photon produces an electron and a positron that each have thesame velocity relative to the velocity the photon (and the nucleus involved in the collision), and in whichthe photon has more energy than the sum of the rest masses of the electron and positron that are created.Based on relativistic considerations, a photon’s energy Ep is related to its momentum pp by the equation

Ep = ppc, (1)

where c is the speed of light (and the speed at which photons must travel). Also, when considering conse-quences of relativity, the momentum pe of particles with rest mass m traveling at velocity v is given by theequation

pe =mv√

1− (v/c)2

and the energy Ee of such particles is given by

Ee =mc2√

1− (v/c)2.

In this special case, the two massive particles move in paths with opposite angles θ to the original photon’spath. Their momenta in the directions normal to the original photon’s path are therefore canceled if theymove with the same speed, which preserves the direction of the momentum vector if θ is between 90 andzero degrees, and their combined momentum becomes

2 cos θmv√

1− (v/c)2.

The cosine of θ must obviously be less than one due to the above mentioned constraints of θ. For energyto be conserved, we must have the combined energies of the electron and positron equal the energy of thephoton:

Ep =2mc2√

1− (v/c)2,

but if we try to determine the momenta of the photon and the particles created during pair production usingequation 1 we find that

Epc

=2mc√

1− (v/c)2> 2 cos θ

mv√1− (v/c)2

.

We arrive to the result that, in order for energy to be conserved, excess momentum is produced. Pairproduction must involve matter besides the photon and the created pair to carry off excess momentum.This is why pair production occurs in the vicinity of atomic nuclei: it is an event that results from collisionsbetween high-energy photons and atomic nuclei. It is also why, during pair production, the system containingjust the photon and the pair of particles produced loses energy.

When certain particles and antiparticles collide, annihilation occurs: both particles disappear and arereplaced by photons that carry the same amount of energy that the annihilating particles did, including theparticles’ summed rest energies. Therefore energy, like charge, is conserved during annihilation of particlesand antiparticles. Momentum must also be conserved during annihilation events. Similar to the situationabove concerning pair production, and due to the facts that photons carry no mass and may travel at nospeed other than c, momentum conservation places a restriction on the result of annihilation: it must result

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Figure 4: These graphs show the cross sections of several collision events between photons and atoms oftwo different absorbing media with respect to photon energy. We are interested in κnuc, the cross section ofnuclear pair production. Other cross sections shown here are those for collisions involving the photoelectriceffect (σp.e.), Rayleigh scattering (σRayleigh), Compton scattering (σCompton), pair production that transfersmomentum to atomic electrons (κe), and photonuclear interactions (σg.d.r.). From Particle data group.

Photon Energy

1 Mb

1 kb

1 b

10 mb10 eV 1 keV 1 MeV 1 GeV 100 GeV

(b) Lead (Z = 82)- experimental σtot

σp.e.

κe

Cro

ss s

ecti

on (

barn

s/at

om)

Cro

ss s

ecti

on (

barn

s/at

om)

10 mb

1 b

1 kb

1 Mb

(a) Carbon (Z = 6)

σRayleigh

σg.d.r.

σCompton

σCompton

σRayleigh

κnuc

κnuc

κe

σp.e.

- experimental σtot

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in the creation of more than one photon. As an example, if an electron and positron collide at a very lowvelocity, two photons will be produced that travel in opposite directions. The photons will have a radiationfrequency such that their energies equal that of the rest energies of the electron and positron, but the totalmomentum between them will be zero.

There are other processes that effect photons traveling through matter, but pair production is dominantat the high photon energy levels involved in shower production. The cross-section of a pair-production eventrises with the atomic weight A and the atomic number Z of the atomic nucleus involved. As seen in figure4, the cross-section of pair-production events approaches a limit at high photon energy levels. This limit σcan be related to the radiation length X0 defined in subsubsection 2.1.1 by the equation

σ =79

A

X0NA(2)

where NA is Avogadro’s number, being used here as a conversion constant [5].

2.2.2 Mean free path

A photon carrying radiation of a certain frequency can only exist carrying a certain amount of energy. Aphoton cannot deposit only part of its energy. During any collision event, the photon deposits all of its energyto another entity and vanishes. Thus, a definition similar to to that of ”radiation length”, that is, the distancethrough which a photon loses 1− 1

e of its energy, cannot be meaningful. Instead, we discuss a photon’s meanfree path, which, for photons traveling with a certain energy traveling through a certain material, is definedas the mean distance those photons travel before being involved in collisions and vanishing. From equation2, we find that the mean free path is 9

7 of the radiation length associated with a high-energy electron movingthrough the same material [12].

2.3 Electromagnetic shower propagation

We have now examined the individual processes that perpetuate electromagnetic showers. The fact thatthese chain-reactions occur is intuitively true based on the results of bremsstrahlung, pair production, andannihilation. The initiation of electromagnetic showers only requires a high-energy photon, electron, orpositron to enter a medium. If it is a high-energy electron that enters the medium, the photons producedthrough bremsstrahlung will produce positrons and more electrons through pair production when they collidewith atomic nuclei. Both the newly-created positrons and electrons travel through the medium producingmore photons through bremsstrahlung. These photons may produce even more electrons and positronsthrough pair production, but most photons produced through bremsstrahlung do not carry enough energyto effect pair production, and are destroyed in events such as the photoelectric effect. The positrons continueshower perpetuation when they annihilate, usually with an atomic electron, resulting in the production ofmore high-energy photons. These reactions continue to produce each for some time. It should be obvioushow the penetration of a positron or high-energy photon will result in such a chain reaction as well.

2.3.1 Shower maximum

The particles that perpetuate electromagnetic showers lose energy as they do so. Electrons and positronslose energy through bremsstrahlung whenever they interact with an electric field. Pair production involvesa loss of energy to the atomic nucleus that carries off the excess momentum of the event, and so the totalenergy of the electron and positron that the event produces is slightly less than the energy carried by theinitial photon. The annihilation of electrons and positrons often produces more than two photons, and inthis case the photons each carry less energy than either of the annihilated particles. Eventually, particlesare produced that in no way can increase the number of particles traveling in the shower. We come to seephotons whose energies are less than the combined rest energies of the electron and positron pairs whichcould otherwise be produced, and electrons that move too slowly to produce high-energy photons throughbremsstrahlung. The depth within the medium where particle multiplication in a certain shower ceases is

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Figure 5: This graph shows the total energy deposited by a shower propagating through iron as a fraction ofthe energy of the particle that initiated the shower as a function of the depth in radiation lengths. We alsosee the numbers of particles carrying energy above a certain cutoff value. This data was produced througha computer simulation. From Particle data group.

0.000

0.025

0.050

0.075

0.100

0.125

0

20

40

60

80

100(1

/E

0)d

E/

dt

t = depth in radiat ion lengths

Nu

mbe

r cr

ossi

ng

pla

ne

30 GeV elect ronincident on iron

Energy

Photons× 1/ 6.8

Elect rons

0 5 10 15 20

called the shower maximum. After this point, we see that the number of particles within the shower stopsincreasing. Intuitively, the shower maximum depends on the energy of the particle that initiates the shower;a particle carrying more energy will, in general, produce a shower that propagates further before the energiesof its particles fall below critical levels. The shower maximum, when measured in units such as meters,depends on the material through which the shower propagates. Particles traveling through materials madeof atoms with high atomic numbers will likely experience energy-stripping collisions more often. However,shower maximum is less dependent on the medium when it is given in units of radiation length. The showermaximum occurs where the average energy of the shower’s particles is less than the critical energy.

2.3.2 Energy deposit as a function of depth

As showers develop and involve particles that hold less energy, the rates through which they lose energychange as well as the proportions of the different energy loss mechanisms. Typically, as showers first beginto develop, the amount of particles involved increases at such a rate that the rate at which energy is depositedto the medium increases with depth. Beyond a certain depth, however, the tendency of the shower energyloss rate to increase due to the increase in the particles involved is overtaken by the tendency of the particlesinvolved to carry less energy and deposit less during collision events. As the shower progresses, it loses lessenergy due to bremsstrahlung and pair-production and more energy to other events.

2.3.3 Moliere radius

The Moliere radius ρM is a proportion of the radiation length X0 of a high energy electron moving througha certain medium to that electron’s critical energy εc in the same medium scaled by a factor called thescale energy Es which is related to the electron’s mass and a constant which holds quantum mechanicalinformation concerning the electric force. It is given by the equation

ρM = EsX0

εc.

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The Moliere radius of an absorption medium relates to the maximum width of an electromagnetic showerpropagating through it.

2.3.4 Shower profiles

Since both radiation length and critical energy are a related to the atomic number Z of the atoms thatcompose the medium, the Moliere radius is calculated from an equation which roughly cancels out Z, andso the Moliere possesses a low dependence on Z compared to that possessed by the radiation length. As aresult, the shape of a shower depends heavily on the material through which it propagates: long and narrowshowers develop in low-Z materials while shorter yet wider showers develop in high-Z materials.

If we measure depth in units of radiation length and the distance between a particle and the showercenter in units of Moliere radii, shapes of showers traveling through different media look more similar thanthey do when we measure such values in meters. However, differences still remain in spite of the adjustmentsmade by these units. We see that higher Z materials host electromagnetic showers that have higher showermaximum depths, and that decay more slowly beyond their shower maxima. More radiation lengths of lowerZ materials are therefore needed to contain a given proportion of a shower’s energy. We also find that thelength of a shower’s profile (again, measured in radiation lengths) scales logarithmically with the energy ofthe particle that initiates the shower.

The spread of a shower is caused the paths of electrons being altered, an effect which is to be expecteddue to the electrons’ low masses, and by photons produced by bremsstrahlung, which may travel at highangles relative to the axis of the shower. Transverse properties are those related to the relationship betweenshower components and the distance from the axis of shower development. Transverse shower properties canbe discussed in two ways: lateral shower properties are discussed in relation to a function of a unit volume’sdistance from the shower axis, while radial properties are those discussed in terms of a radial (ring-shaped)slice and a function of that slice’s radius. According to experimental particle physicist Richard Wigmans,however, these two terms are often used interchangeably in papers, presenting a situation of ambiguity whichleads to significant confusion. Radial energy density tends to decrease exponentially as a function of thedistance from the shower’s center. We find that for high Z materials, the radial energy density decreasesmore rapidly as a function of distance from the shower axis than for low Z materials.

3 Hadronic showers

Showers initiated by the penetration of hadrons through a medium develop differently than those initiated byelectrons, positrons, and photons. While we shall first discuss inelastic collisions between penetrating protonsand atomic electrons, we are most interested in phenomena that occur at high energy levels, such as spallation.While the showers discussed previously are perpetuated solely through electromagnetic interactions, hadronicshowers involve phenomena related to the strong nuclear force.

3.1 Electron collisions

If a particle is much heavier than an electron and carries an electric charge, the majority of the energyit loses as it travels through matter is through collisions between the particle and the electrons of themedium’s atoms. While observing these low-energy collisions is not the goal of modern high energy particlecalorimeters, these collisions are one of the results that is directly measured, and shower processes can beextrapolated from these measurements. The amount of energy lost by the traveling particle per unit of pathlength, dE/dx, is called the stopping power. It can be roughly calculated using classical assumptions, butprecise calculation of stopping power requires quantum mechanical and relativistic considerations. We shallproceed through the classical calculation of energy loss, as the process illuminates several aspects of thesetypes of collisions. The following calculations are based on the book by Leo [9].

We shall first calculate the impulse on an atomic electron as a charged particle moving at velocity vcollides with it, then calculate the energy change from that impulse. The minimum distance the electron

12

comes to that particle will be given as b. We start with the definition of impulse:

I =∫

Fdt

We may find the force F due to the electric field E generated by the traveling particle with charge given asa multiple of that of an electron as ze using Coulomb’s law. We only need to consider E⊥, the component ofthe electric force perpendicular to the particle’s path at all points, since any force produced by the particleparallel to its path as it approaches the atomic electron is canceled out by an equal force in the oppositedirection as the particle recedes from the atomic ion:

I = e

∫E⊥dt = e

∫dt

dxdx = e

∫E⊥

dx

v.

The above equations are only valid if the atomic electron is initially at rest and is only moved slightly duringthe collision, and if the penetrating particle is not significantly deviated from its path, a condition whichis satisfied since the penetrating particle we are discussing is very massive compared to the electron. Tocalculate the integral, we use the integral form of Gauss’s Law, which, when expressed using Gaussian units1,is ∫

s

EdA = 4πQenc,

which means that the total electric field experienced by the entirety of a surface (the electric flux throughthat surface) surrounding the source of that field is 4πQ, with Q being the magnitude of the enclosed charge.We imagine that the atomic electron moves (in the reference frame of the penetrating particle) along theoutside of an infinitely long cylinder with radius b and find the electric field experienced by that surface tobe ∫

E⊥2πbdx = 4πze.

So to find the total electric field experienced by the electron, we find the electric flux through a line alongthat cylinder by removing 2πb from both sides of the above equation:∫

E⊥dx =2zeb.

We arrive to the result

I =2ze2

bvSince

I =∫Fdt = m

dv

dtdt =

∫mdv =

∫mdv = m∆v

when mass is not a function of velocity2, and since the change in a electron’s energy is given by

∆U =∫Fdx =

∫mdv

dtdx =

∫mdv

dtvdt =

∫mvdt =

12m(∆v)2,

which again assumes that mass is not a function of velocity, we arrive at a relation between mass andimpulse, and therefore the amount of energy gained by the electron and lost by the traveling particle duringthe collision:

∆U =I

2me=

2z2e4

mev2b2.

This calculation requires modification due to quantum mechanical and relativistic effects which will notbe discussed [9].

1In Gaussian units, distance is expressed in centimeters and mass is expressed in grams. Unlike SI units, Gaussian unitsallow Gauss’s law and many other theories to be expressed without scaling factors, explaining their popularity among theoreticalphysicists.

2An assumption that does not hold and therefore requires that these equations be refined for particles traveling at relativisticspeeds.

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3.2 Spallation

Spallation is the most common result of collisions between high energy hadrons and nuclei of a medium’satoms. Generally, the word “spallation” refers to a process in which an object enters a medium and producesejecta from the medium, referred to as “spall”.

Hadrons and the quarks that compose them interact with each other through the strong force. The strongforce has three aspects, called colors. Full understanding of the concepts of aspects and colors requires astudy of quantum chromodynamics, which is beyond the scope of this paper, but we shall undergo a briefdiscussion using an analogy of the electromagnetic force. The electromagnetic force holds one such aspect: acharge that can either be positive or negative. The strong force, however, has three such degrees of freedom,referred to as red, green, and blue. The strong force is such that its strength increases between particlesas a function of the difference between them. As a result of this property, we never find free quarks. Thestrong force’s interaction with quarks produces a residual effect on the hadrons that they compose, called thenuclear force. However, the nuclear force between two particles which may be affected by it does not increasewith distance, and in fact decreases more rapidly than any other force does as a function of distance. Atvery short distances, however, the nuclear force is stronger than any other force which may act on hadronsby far. As mentioned in section 1, gluons are the mediators of the strong force.

Spallation is facilitated by the strong force. When a hadron collides with an atomic nucleus, it may causethe components of the nucleus to travel through the nucleus itself, colliding with other nuclear hadrons,which may move to produce even more collisions. This phenomenon is known as a fast intranuclear cascade.In a process similar to pair production, certain particles may come into existence if particles move througha nucleus with enough kinetic energy to exceed the rest energies of new particles, and if other requirementsrelated to conservation of mass, momentum, charge, and angular momentum are met. These particles areusually mesons, those which are made of one particle and one antiparticle and that therefore quickly decayinto high energy photons (those that have wavelengths in the gamma radiation spectrum) or other particles.The penetrating hadron itself may transform into new hadrons. Also, the struck nucleus may eject a portionof its neutrons and protons, entering an excited state, and emits hadrons and photons in the process ofreturning to a lower energy state. This process, which follows the fast intranuclear cascade, is known as theevaporation stage.

3.3 Hadronic shower components

Spall particles that do not have short decay times, such as protons, neutrons, and charged pions, mayproceed to undergo spallation events with other nuclei of a medium, and a hadronic shower exists. Alongwith propagation behaviors of these types of showers, we must also consider the proportions of the manydifferent particles that they may involve (remember, when we discussed electromagnetic showers we onlyhad to worry about the propagation of protons, electrons, and positrons). An important consideration, forexample, is the proportion of mesons in a shower at different levels, since mesons quickly decay into photons(or, sometimes, electron and positron pairs) which produce electromagnetic showers. This consideration isimportant since, as we shall discuss later, particle calorimeters almost exclusively measure the effects ofelectromagnetic showers directly. About a third of the mesons ejected from nuclei during a shower’s initialspallation event are neutral pions, and the average number of all mesons produced from spallation increaseslogarithmically with the energy of the collision. To find the portion of shower that is in the form of neutralpions, and therefore the portion of a hadronic shower that propagates as an electromagnetic shower, at ashower generation n, we add the portion of the shower propagating electromagnetically before the generationn to the portion of the shower not propagating electromagnetically multiplied by the fraction fπ0 of spallationparticles that are neutral pions at the energy level of collisions typically occurring during the generation inquestion. For example, if, for simplicity, one third of particles produced through spallation are neutralpions at every generation, then one third of the shower would be an electromagnetic shower after the firstgeneration, 1

3 + 23 ×

13 of the shower would propagate electromagnetically after the second generation, and

so on. The portion of a hadronic shower that has become an electromagnetic shower by a generation n is

14

thereforefem = 1− (1− fπ0)n .

The average fraction of neutral pions produced through spallation fπ0 depends on the atomic number ofatoms in the absorbing medium. The fraction throughout the entire shower also depends on whether theinitial spallation event was started by a pion or a proton: electromagnetic fractions of showers started whena pion impacts a nucleus are about 15% lower than the fractions of showers started when a proton hits anucleus. This is because the type of particles that are leading particles, the particles that leave the nucleusfirst during spallation, depend on the type of particle that started the spallation reaction. A reaction startedby a pion will produce leading particles in the form of pions of types that often interact with absorbermedium atoms before decaying, while a reaction started by a proton will produce leading particles that areprotons, and these effects propagate through the showers. We can determine the percentages of energiescarried by electromagnetic showers from our examinations of the portions of hadron shower particles thatare those which decay electromagnetically.

3.4 Spallation cross sections

Because of the low range of the nuclear force, the cross section of a spallation reaction is extremely smallwhen compared to the cross sections of collisions events discussed previously. The relation for the crosssection of such a collision between a particle with energy E and a nucleus that initially has an atomic massAt that produce a spallation event that results in the atomic nucleus losing particles such that its atomicnumber becomes Zf and its atomic mass becomes Af is

σZf ,Af∼ e−P (AT−Af ) × e−R|Zf−SAf +TA2

f |32

where P , R, S, and T are constants. The total spallation cross section is a sum of the cross sections of allspallation reactions that may occur.

3.5 Consider the neutron

Because they do not carry charge, low-energy neutrons are able to come very close to atomic nuclei andaffect elastic collisions- those in which the sum of the kinetic energies of the neutron and the nucleus remainsconstant and kinetic energy is transferred between the neutron and the nucleus. Depending on the locationof the collision on the nucleus, the energy transferred from the neutron to the nucleus is some portion of themaximum possible transfer amount, which is a function of the nucleus’s atomic number A and is given by

Emax =4A

(A+ 1)2.

Since the energy transferred per collision decreases with atomic number, it is obvious that a medium thathas an abundance of hydrogen will absorb kinetic energy from low-energy neutrons most efficiently.

Free neutrons may spontaneously decay (they have a half-life of about 15 minutes), but they are muchmore likely to be absorbed when traveling through a medium at very low speeds. The neutron is absorbedinto a nucleus of the medium, which puts that nucleus into an excited state. The nucleus then proceedsto emit energy as gamma rays or, in the case of lithium or boron nuclei, an alpha particle, which is amass-carrying boson.

At high energies neutrons may inelastically collide with an atomic nucleus, putting the nucleus into anexcited state without being absorbed.

3.6 Invisible energy

Spallation requires that nuclear particles overcome the nuclear force, and the energy required and expelledin this process is called nuclear binding energy. When spallation occurs, the sum of the energies of the spall

15

particles is less than the that of the particle that produced the spallation event. This difference is the energythat was expended in overcoming the nuclear binding energy. It is not detectable by any means, although itmay be replaced if some of the spall particles join other nuclei in events that produce gamma radiation.

3.7 Hadronic shower profiles and their relation to electromagnetic shower pro-files

The inertias of high-energy hadrons are such that they propagate through an absorbtion medium virtuallyunaffected until they are involved in a nuclear collision. Just as the mean free path tells the average distancea high energy photon will travel in some medium before being destroyed in a collision event, the averagedistance that high energy hadrons travel through a medium is known as the medium’s nuclear interactionlength, and is given by the symbol λint, and the probability that a particle traverses some distance z in thismedium without colliding with a nucleus is

P = ez/λint .

The nuclear interaction length is inversely proportional to the sum of the cross sections of all possiblespallation events and proportional to the weights of atoms with which a penetrating hadron may collide:

σtot ∼A

λint.

Nuclear interaction lengths are generally far greater than the mean free path of photons and the radiationlength of electrons in a given medium. Hadronic showers, therefore, propagate much further into an absorbingmedium than electromagnetic showers in general. Similarly to an electromagnetic shower, the particlesinvolved in a hadronic shower increase in number as the shower progresses through the medium until theirnumber reaches a maximum. The energy carried by the hadronic shower decays less per (SI) unit of distancethan that carried by electromagnetic showers after their respective shower maxima. Hadronic showers arealso larger laterally. We find that as a function of the atomic numbers of the atoms that make up theabsorbtion medium, radiation length drops faster than nuclear interaction length: the cross section of abremsstrahlung event is proportional to Z2 of the medium, while the cross section of hadronic spallationis proportional to Z2/3. High Z materials start bremsstrahlung reactions faster, and are used to separatehadronic showers from electromagnetic showers in media [12].

Part III

Calorimetry in general

4 Measuring end results

Whether the focus is electromagnetic showers or hadronic showers, a common mechanism that modernparticle calorimeters use to measure energy deposit is scintillation. Scintillation is the light emitted whenan atomic electron moves from an excited state to a less energetic state. Given materials may producescintillation radiation having one of several possible discrete wavelength values depending on the materialcomposing the medium, and photodetectors sensitive to these frequencies may therefore be used to detectscintillation events. Many of the processes that we have discussed in previous sections result, sometimeseventually and indirectly, in scintillation.

More often, though, calorimeters measure the effects of penetrating particles on the charge of their media.A calorimeter designed to measure charge may utilize liquified nobel gasses as its medium, and measures theamount of ionization of the atoms of the element due to charged particle and photon collisions. Electronsfreed from the medium’s atoms are attracted to an anode on one side of a chamber containing the samplingmedium, while the positively charged ions are attracted to a cathode in another location, and current flow

16

Figure 6: A photomultiplier tube. By Colin Eberhardt.

through a conduit linking the anode and cathode due to resultant voltage changes between them is measured.Charge-measuring calorimeters may also utilize semiconductors as the sampling medium. When atoms of thesemiconductor are excited, they become conductors, and electric fields may be used to collect free positive andnegative charges created by the penetration of ionizing particles. Calorimeters that utilize this mechanismare called solid-state calorimeters. The sampling medium of solid state calorimeters may be constructed outof silicon chips.

4.1 Recording scintillation events

All the energy of particles involved in an electromagnetic shower propagating through a homogeneouscalorimeter may absorbed in the excitation of medium in a way that produces scintillation. Most of thephotons produced through bremsstrahlung accomplish this when the collide with electrons of the medium;low energy electrons may produce excitation of an atomic electron through the coulomb force, or may ionizethe atom resulting in a free electron that eventually causes excitation in other atomic electrons. High energyphotons affect pair production, producing particles that cause scintillation through the effects describedabove.

The sampling medium of calorimeters that utilize scintillation for energy measurement consists of a ma-terial that allows light to pass through. There exist boundaries between sections of the sampling mediumthrough which scintillation light may not pass. Light detection systems, tuned to activate upon the obser-vation of radiation with any of the frequencies that may be produced through scintillation of the samplingmedium, are attached to each of the sections separated by opaque boundaries. This setup enables a deter-mination concerning the location of a scintillation event. The detector may record the division in which ascintillation event takes place, but there is no way to determine exactly where the event took place withina single division. Higher granularity results from smaller divisions of the sampling medium and enables thelocation of a scintillation event to be determined with greater precision. A photomultiplier tube (figure 6) isa device which may be attached to a sampling medium division in order to detect light from scintillation. Atthe front of a photomultiplier tube is a plate. When photons interact with this plate, electrons are ejectedfrom the side of the plate facing away from the scintillation medium through the photoelectric effect. Theseelectrons, which are too few in number to be detected directly, are attracted to a positively charged platecalled a dynode. When they collide with this plate, they knock off more electrons, which are then attractedto a second plate, producing a similar result. The dynodes are angled in such a way so as to allow knocked-offand original electrons to continue onto dynodes further back in the tube. The process continues, multiplyingthe number of electrons traveling through the tube, until they produce a measurable current pulse upon

17

an anode at the back end of the tube. Another such device is an avalanche photodiode, which allows a freeelectron ionized through the photoelectric effect to knock other electrons from atoms by accelerating it undera high voltage, producing a current.

Some problems may occur with photodetectors that produce inaccurate energy readings. That is, evenif all the energy of a shower is absorbed by a sampling medium to produce scintillation light, not all of thatenergy will produce recordable results. Photomultiplier tubes may suffer from saturation: electric currentcreated within the tubes upon detection events lowers the potential difference between the dynode plates,meaning that electrons knocked off from the plates during subsequent detection events will not impact platesfurther back in the tube with as much energy as would be expected.

5 Homogeneous and sampling calorimeters and energy response

Homogeneous calorimeters are those in which the material intended to stop incoming particles is also thematerial that produces scintillation light for energy measurement. Sampling calorimeters, on the other hand,are those that utilize one material, the passive medium, to absorb large amounts of energy from penetratingparticles, and another material, the active medium, to measure the energies of particles at different pointsalong their paths so as to ascertain the effect the passive medium has upon them. Sampling calorimetersare designed so that a single particle passes through several boundaries between active and passive media,making use of many possible patterns for staggering the two media classes. Sampling calorimeters are usedto measure the energy content of hadronic showers because the stopping power of scintillation media is toolow to contain hadronic showers in any reasonable volume of the material.

5.1 Energy response of homogeneous calorimeters

Energy response is the portion of the energy carried by a shower that is actually recorded. Phenomenathat affect the energy responses of homogeneous calorimeters are also applicable to sampling calorimeters.In addition to instrumental effects on energy response discussed in subsection 6, the energy response toelectromagnetic showers is decreased by particles that escape the detector. Energy response to hadronicshowers is mitigated by additional problems. For one, the invisible energy, discussed in subsection 3.6, hasan obvious impact. Also, the energy of hadronic showers is mostly detected by scintillation caused by theshowers’ electromagnetic components. This component only makes up a portion of the hadronic shower, andthe size of this portion is dependent on the shower’s energy, as discussed in subsection 3.3.

5.2 Energy response of sampling calorimeters

The sampling fraction, fsamp, of a sampling calorimeter is defined by Wigmans as the fraction of energydeposited by minimum ionizing particles in the sampling medium of the total energy deposited by suchparticles. Minimum ionizing particles are hypothetical particles that have just enough energy to affectionization of the absorbing medium. As mentioned before, scintillation calorimeters, at least in theory, havethe ability to translate all the energy carried by an electromagnetic shower into scintillation light. This isobviously not the case with a sampling calorimeter in which some energy is absorbed in the passive medium.In fact, the portion of the energy contained in an electromagnetic shower that is recorded is less for showersthat carry higher energies in sampling calorimeters in which the sampling medium has a lower Z value thanthat of the active medium, and larger differences in Z produce larger differences in portions of energiesrecorded between higher energy showers and lower energy showers. This is called the transition effect, andis due to the different proportions of mechanisms through which photons and electrons deposit energy in amedium at different energy levels, and the different Z dependencies of these different mechanisms. Thesedependencies are generally examined in terms of the ratio of the energy deposited by an electromagneticshower component to the energy that would be deposited by a minimum ionizing particle: e/mip for electronsor γ/mip for gamma photons.

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6 Energy resolution

6.1 Sampling fluctuations

Sampling fluctuations are inconsistencies in energy measured from showers propagating through samplingcalorimeters, and lower the energy resolution of these devices. Energy resolution is the precision withwhich energy deposited in a calorimeter may be measured. Resolution of sampling calorimeters is loweredby fluctuations in the number of the low energy electrons that would deposit energy into the samplingcalorimeter as scintillation that exist in the sampling and passive portions. Sampling fluctuations are theprimary concern of resolution of sampling calorimeters. There are several equations that offer approximationsof sampling fluctuations’ effects on energy resolution, based on both empirical and theoretical considerations.One such equation was derived by Amaldi and relates the contribution of sampling fluctuations to energyresolution to the average energy lost by a minimum ionizing particle ∆ε, and F (Z)〈cos θ〉 which is a factorthat accounts for empirical observations related to Z dependence:

asamp = 3.2%

√∆ε

F (Z)〈cos θ〉,

and we may find that the energy resolution of a shower with energy E due to sampling fluctuations is

(σ/E)samp =

√∆εE

[1]This equation produces reasonably accurate results concerning energy resolutions of calorimeters with

an iron-based passive medium and a plastic active medium, and also for resolutions of those with an ironpassive medium and a liquid argon active medium, but the results are not consistent with those observedin calorimeters utilizing other materials or those which do not contain very thick scintillator plates. Exper-imentally, we see that energy resolution depends on the relative thickness of the active and passive media.The thickness of the scintillating portions of the calorimeter determines the portion of photons which deposittheir energies in a way that is observed. Apparently, an equation that accounts for the sampling fraction ofa calorimeter is necessary, and empirically, energy resolution is found to scale proportional to

√1/fsamp. We

also find that energy resolution depends on the geometry of the boundaries between passive and samplingmedia. For example, for a given sampling fraction, a calorimeter that is made of alternating plates of passiveand sampling media has less energy resolution than one made of a passive medium interspersed with scin-tillating fibers with thicknesses on the order of a millimeter, and thinner fibers allow for better resolutions.This has to do with the differences in the sum of the areas of the boundaries between sampling and activemedia; higher surface areas allow a higher fraction of low energy electrons to produce scintillation light. Wenow find that, due to sampling fluctuations, the energy resolution of sampling calorimeters scales as

(σ/E)samp = a

√d/fsamp

E,

where d is the thickness of the sampling components.Sampling fluctuations also affect readings from hadronic showers. The energy resolution of hadronic

showers is decreased by the fact that not all of the energy of hadronic showers is detected through scintillation,the fact that charged hadrons affect scintillation in active media, and the fact that individual particles ina hadronic shower generally produce higher energy signals than individual components of electromagneticshowers. Since hadrons penetrate further into media than electromagnetic shower components, hadronicshower components are likely to cross more boundaries between active media and passive media, and thestatistical nature of particle collisions means that sampling fluctuations are increased by a factor of

√N ,

where N is the number of boundaries crossed. We also find that the increase in sampling fluctuations betweenhadronic shower and electromagnetic shower readings is proportional to the thicknesses of the sampling mediameasured in mean interaction lengths (λint). Experimentally, the energy resolution of hadronic showers wasfound to be 0.28 times that of electromagnetic shower based solely on sampling fluctuations.

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6.2 Fluctuations due to invisible energy

As mentioned in subsection 3.6, spallation involves the generation of invisible energy, that which is used inthe overcoming of nuclear forces when particles are ejected from nuclei and which cannot be detected bycalorimeters. The amount of energy lost as this invisible energy varies between reactions, depending on thetype of particle striking the atom and the nature of the process in which energy propagates through thenucleus as a result of the collision; it is inherently unpredictable.

7 Compensation

Electromagnetic showers suffer from no effects comparable to the invisible energy loss that guarantees thatthe energy response of a detector to hadronic showers will be less than the minimum ionizing energy re-sponse. In order to compensate for the inherent inefficiencies of measuring the energies of hadronic showers,adjustments should be made so that the energy response to hadronic showers equals the energy responseto electromagnetic showers. This task is obviously impossible in homogeneous calorimeters: the energyresponse to electromagnetic showers cannot be decreased below the energy response to minimum ionizingparticles using any mechanism that does not equivalently decrease the energy response of hadronic showers.Likewise, the energy response to hadronic showers cannot possibly be increased above the energy responseto electromagnetic showers. Sampling calorimeters, however, which do not have perfect energy response toelectromagnetic showers and which present opportunities to adjust relative energy responses to hadronicand electromagnetic showers, offer mechanisms through which the energy responses to electromagnetic andhadronic showers may be equalized.

The ratio e/h gives the ratio of the energy response of a calorimeter to electromagnetic showers to theenergy response to hadronic showers. Without exception, e/h of homogeneous calorimeters is greater than1 due to invisible energy phenomena. In the case where e/h is greater than one, a calorimeter is said to benon-compensating or undercompensating. If e/h is equal to one, a calorimeter is compensating. Finally, ife/h is less than one, the calorimeter is overcompensating. Compensation can either be performed by reducingthe energy response to electromagnetic showers, or by increasing the energy response to hadronic showers.Methods that adjust signal strength of detectors to electromagnetic and hadronic shower components alsoexist, and are called off-line compensation methods. It is important to note that no method of compensationaffects replacement of unobserved energy. As we shall soon find, there are mechanisms which may add extraenergy to the response to hadronic shower components, but there is no way to replace the energy lost.

7.1 Decreasing electromagnetic shower energy response

As mentioned in subsection 5.2, passive media made of atoms with high atomic numbers reduce the energyresponse to electromagnetic showers more profoundly than they reduce the energy response to hadronicshowers. This is because the high Z materials absorb low-energy photons which may produce free electronsthrough the photoelectric effect. The effectiveness of using high Z material to lower the electromagneticshower energy response is increased if the passive material is coated with a certain thickness of metal foil,which prevents free electrons liberated through the photoelectric effect close to the boundaries betweensampling and passive media from escaping and depositing energy into the sampling medium. Changing thethickness of this coating is a way to fine-tune calorimeter compensation.

7.2 Increasing hadronic shower energy response

A sampling medium containing hydrogen atoms can increase the energy response to neutrons freed fromatomic nuclei during spallation events. This is because neutrons lose their kinetic energy through elasticcollisions with atomic nuclei, while it is extremely rare for components of electromagnetic showers to loseenergy through such collisions. If the neutrons collide with a hydrogen nucleus in the sampling medium, thenucleus will begin to move through the medium, perhaps producing signals through coulomb effects.

20

Figure 7: The Large Hadron Collider and the pre-accelerators that feed it. The linear accelerators thatimpart initial accelerations upon protons and lead ions are labeled ”p” and ”pb”. By Arpad Horvath.

Part IV

Calorimeters at the Large Hadron Collider

8 The Large Hadron Collider

This paper shall examine calorimeters observing collisions at the Large Hadron Collider. The Large HadronCollider is a particle accelerator at CERN commissioned in 2008. It is the largest particle accelerator everconstructed, and its capabilities towards energy transfer to particles exceed those of any other such deviceby a factor of seven. As its name suggests, the Large Hadron Collider accelerates and collides protons, andalso lead ions. The particles are accelerated using electric forces, and guided using magnetic forces, througha series of accelerators before being fed into the ring-shaped Large Hadron Collider as shown in figure 7. Thehadrons are produced and travel in bunches- the necessity of this will be explained below. Protons begintheir journey in a duoplasmatron, a device which creates a plasma from a gas then accelerates particlesfrom that plasma [10]. The protons are then accelerated through Linac-2, a linear particle accelerator,leaving with 50 MeV of energy. They are then guided to the Proton Synchrotron Booster a ring acceleratorwhich accelerates them until they possess 1.4 GeV of energy and are then guided to the Proton Synchrotronand further accelerated so that each particle possesses 26 GeV. The protons then enter the Super ProtonSynchrotron, where they receive further acceleration so as to carry 450 GeV of energy before finally beingfed into the Large Hadron Collider, which itself accelerates the particles until they possess energies of up to7 TeV. When lead nuclei are used instead of individual protons, an ECR machine strips electrons from leadatoms in a sample of vapor by free electrons accelerated using electromagnetic radiation. These lead ionsare accelerated through Linac 3, fed to the Low Energy Ion Ring with energies of 4.2 MeV, then continuethrough the Proton Synchrotron, the Super Proton Synchrotron, and on to the Large Hadron Collider in a

21

Figure 8: A magnetic quadrupole. Electrically charged particles moving through the space in the middle ofthe four magnets will experience a force that is in a direction perpendicular to that of the magnetic fieldillustrated by the black lines. The result is that the particles are focused toward a point at the center of thespace surrounded by the field sources. By Wikimedia Commons user Geek3.

manner similar to that of single protons [3].The particle bunches receive their linear acceleration from R-F cavities installed along the particle paths.

R-F cavities produce electric fields that are parallel to the hadron path and oscillate in magnitude at afrequency such that the fields are negative as a positively charged particle bunch approaches the R-F cavityand are positive as the bunch leaves in the opposite direction. The R-F cavities begin producing a negativeelectric field again as a new bunch of particles approaches. To optimally deal with particles with higherspeeds, the R-F cavities are spaced farther apart. This is one reason why hadrons pass through severalaccelerators before they are fed into the LHC: each of the accelerators is optimized for to accelerate particlesthrough a particular energy range. The nature of this acceleration mechanism also leads to the necessityof the particles to be produced and to travel in bunches: the R-F cavities can only impart force onto theparticles in the same direction in which they are already traveling as they move through the cavities if theparticles move through the center of the cavity the moment its electric field changes direction. The majorityof particles in a continuous beam of particles would receive forces in alternating directions. The particles areguided through and between the particle accelerators using dipole magnets. The areas of particle bunchesnormal to their direction of travel are minimized using quadrupole magnets, as illustrated in figure 8.

The particles travel around the Large Hadron Collider in two parallel beams in opposite directions guidedby 1232 dipole magnets and focused by 392 quadrupole magnets. They are kept at a constant speed (after theinitial acceleration) by R-F cavities canceling out the energy lost by the particles through electromagneticradiation associated with their centripetal accelerations through the curve of the ring. The two beamstraveling through the LHC intersect at four points in the vicinities of four detectors designed to collectdata concerning the results of particle collisions. Even though the particle bunches contain extremely highnumbers of particles focused into beams as narrow as one millimeter, the sizes3 of the particles are such that

3The definition of size as it is used here involves a discussion of quantum mechanics and is therefore beyond the scope ofthis paper.

22

the probability that a given particle will be involved in a collision as it passes through a beam intersectionpoint is extremely low. That probability is further reduced by the fact that the particles tend to deflectaway from each other due to the electric charges they carry. Therefore, collision events are distributed alonga time range of several hours. However, due to the large number of hadrons inserted into the LHC, about20 particle collisions occur every time bunches intersect.

The term luminosity is important in the study of collision frequency. It is defined as the number ofparticles passing through a unit of area per some unit of time multiplied by the number of particles in thatsame area with which collisions may be experienced per the same unit of time. For the situation describedabove, wherein n bunches of N particles travel in opposite directions around a ring with a revolutionaryfrequency of f and having areas normal to the directions of travel of A, the luminosity L is given by theequation

L = fnN2

A

In the Large Hadron Collider, normal operation involves 2808 bunches per beam initially containing1.1× 1011 protons each traveling around the ring 11245 times per second. At the collision points, the beamsare focused such that their areas normal to their directions of travel are 7 micrometers. According to theabove equation, the luminosity of the Large Hadron Collider is therefore

The collision frequency can be calculated from the luminosity. To do this, we must know the how smallthe area two particles must simultaneously occupy in order for a collision to occur. This area is known ascross section and depends on the energy possessed by the particles. The cross section of a collision is ahypothetical area around a particle through which another particle must pass through in order for a collisionevent to take place, and collisions involving larger cross-sections are more likely to occur given a certainluminosity. When protons travel through the Large Hadron Collider at the maximum speed the device isable to accelerate them to, the cross section of all types of collisions between two protons traveling in oppositedirections is 110 millibarns4, but the cross section of inelastic collisions, the only collisions the instrumentsat the Large Hadron Collider are designed to detect, is only 60 millibarns. The number of inelastic collisionsthat occur per second can be found by multiplying the luminosity by the cross section. The Large HadronCollider produces 6× 108 collisions per second, or one collision every 1.66 nanoseconds on average, or about20 collisions per bunch crossing [2].

When the occurrences of collisions within the collider fall below an acceptable frequency, the magneticdipoles controlling the path of the beam are adjusted so as to cause any hadrons that may remain in thebeam to be jettisoned from the ring and to be safely absorbed by a mass of concrete in the vicinity of theejection point. The LHC may then be prepared for a new batch of particles.

ATLAS (A Toroidal LHC ApparatuS) and CMS (Compact Muon Solenoid) are detectors that collectdata at two of the LHC’s beam intersection points and the remainder of this paper includes discussionsconcerning them. The other beam intersection points exist to produce collisions to be observed by detectorsknown as ALICE (A Large Ion Collider Experiment) and LHCb (Large Hadron Collider beauty), neither ofwhich are discussed in this paper. The unprecedented energy at which the Large Hadron Collider collidesparticles and frequency of those collisions combine with the goals of the experiments involving the detectorsto produce a unique set of requirements on the detectors.

9 Geometry

Pseudorapidity is a measurement of the angle from the beams traveling through a particle accelerator. It isgiven by the symbol η and is defined

η = − ln(

tan(θ

2

)).

4A barn is a unit of area equal to 10−28 square meters, and derives its name from comments made by Manhattan Projectresearchers that the uranium nucleus was as big as the side of a barn [11].

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Figure 9: The components of the Compact Muon Solenoid. From JINST.

Angles between rays that share an endpoint located on the beam and are perpendicular to the beam aregiven by the symbol φ. We shall use r for the distance from the beam, and ρ for the cartesian distance fromthe collision point. A vector r is one that points radially outward from the beam (in a direction perpendicularto the beam).

10 The Compact Muon Solenoid

The Compact Muon Solenoid features several particle detection mechanisms surrounding a point of inter-section between the two beams traveling through the Large Hadron Collider. The individual detectors aremostly cylinders whose axes of extrusion are parallel to the beams, and are concentric so that a single linedrawn at any angle perpendicular to the beam paths passes through most of the detectors, as seen in figure9. The detector closest to the collision point is the inner tracking system. This system’s purpose is to recordthe paths of particles while affecting those particles as little as possible. Surrounding the inner trackingsystem is the electromagnetic calorimeter, a homogeneous calorimeter designed to measure the energy ofelectromagnetic showers only. Outside the electromagnetic calorimeter is the hadron calorimeter, a sam-pling calorimeter that measures the energy of hadronic showers. The component surrounding the hadroncalorimeter is a superconducting magnet which allows the detectors to obtain momentum information fromcharge-carrying particles based on the amount of curvature the magnetic field effects upon them. The coil ofthe superconducting magnet is surrounded by the muon system, components designed to detect properties

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Figure 10: The components of CMS’s electromagnetic calorimeter. From JINST.

of muons that pass through the inner components. The muon system is interspersed with sections of thesuperconducting magnet’s return yoke, the iron sections that connect to the poles of the magnet. Thereare also detectors that do not surround the collision point. Like the other detectors, these are cylinderssurrounding the beam paths, but they are situated so that they are intersected only by particles that leavethe collision point on paths that are close to being parallel to the original beam paths. These detectors areCASTOR (Centauro And Strange Object Research), a quartz calorimeter, and the zero degree calorimeter, aset of two hadron and electromagnetic calorimeters situated between the beam pipes that are able to examineparticles that leave the collision point in paths that are truly parallel to the beams. We shall, of course,focus on the calorimetric components of the CMS. Components of the CMS must be designed to resist theeffects of radiation and magnetic fields.

10.1 Electromagnetic calorimeter

The electromagnetic calorimeter includes a barrel that is a cylinder that surrounds the collision point andranges in pseudorapidity from -1.479 to 1.479, two flat endcaps normal to the beam ranging in pseudorapidityfrom 1.479 to 3.0 and -1.470 to -3.0, and two preshower detectors, with one ranging ranging in pseudorapidityfrom 1.653 to 2.6 and the other ranging from -1.653 to -2.6.

10.1.1 Main barrel

The main barrel of the electromagnetic calorimeter is a homogeneous calorimeter that utilizes lead tungstate(PbWO4) crystals as a stopping and scintillation medium. The crystals were chosen for many reasons.Their short radiation length of 0.89 centimeters and their small Moliere radius of 2.2 centimeters allowlarge portions of electromagnetic showers to be contained in relatively small volumes. The time it takes forscintillation light to decay to very low levels is on the same order of magnitude as the LHC’s bunch crossingperiod, which facilitates resolution between events separated in time. This consideration mitigates pileupnoise, effects from the extremely frequent penetration of particles through parts of the CMS. The crystals

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are transparent to visible light, which allows scintillation light to pass through the crystals to the detectionmechanisms.

When viewed from the side facing the beam, the crystals are roughly square in shape. The crystalsare tightly packed in a cylindrical array, which necessitates that their broadnesses increase as r does. Thedegree of tapering varies slightly with their position in the η coordinate. In order to reduce the probabilityof particles falling through the cracks between crystals, they are twisted three degrees on axes parallel to rso that the cracks between them do not create planes parallel to the beam, and are also angled such thatlines normal to their faces that most nearly face toward the beam are three degrees from r. The tapering ofthe crystals has been shown experimentally to reduce energy resolution because it causes a dependence ofenergy response to showers on the position of the showers in the crystals. This problem, which is related tofocusing of scintillation light, is mitigated by depolishing one face of each crystal. Another contributor tosuch a dependence is the energy lost by light as it travels from the scintillation points to the photodetectorsat the back ends of the crystals: there would be less energy response to showers that develop farther from thephotodetectors [6]. There are 360 crystals arrayed through φ and 170 crystals arrayed through η, meaningthat the barrel houses 61,200 crystals. The positional granularity of this portion of the calorimeter is thevolume of one crystal (detectors cannot discern where an event takes place within a crystal). The crystalsare 23.0 centimeters long, and so they are 23.0 centimeters ÷ 0.89 centimeters, or 25.8 radiation lengthslong.

The number of photons emitted by these crystals during a scintillation event caused by a given energydeposit depends on the temperature of the crystals: less scintillation light is produced at higher temperatures.In order for this phenomenon to not affect energy resolution, the crystals must be kept at a constanttemperature, even as heat is introduced into the crystals by electronics attached to them. This requirementis met through insulating foam separating the crystals from readout electronics, and a thermal screen throughwhich water kept at a constant temperature is pumped separating the crystals from the silicon tracker.

Avalanche photodiodes detect the scintillation light from the lead tungstate crystals. Two such devicesare attached to each crystal. Readings received from the photodiodes are also affected by temperature:higher temperatures result in less amplification. Therefore, the photodiodes must also be kept at a constanttemperature, and one in every ten photodiode has a temperature sensor embedded within it. The photodiodesare designed to avoid the production of false readings as particles that pass through them on their way tothe detectors surrounding the electromagnetic calorimeter. The photodiodes’ amplifications of their receivedsignals is dependent on the voltage used to propel electron avalanches, and so the photodiodes require acustom high voltage power supply.

10.1.2 Endcaps

The endcaps utilize scintillators made of the same material as those in the barrel. The endcaps each house7324 crystals arranged in a flat rectangular grid, the boundary of which is roughly a circle. The crystals inthe endcaps are not tapered. The photodetectors attached to these crystals are vacuum photodiodes, whichare similar to photomultiplier tubes with a single dynode.

10.1.3 Preshower detector

The preshower detectors are located between the electromagnetic calorimeter endcaps and the collision point.Their purpose is to facilitate the identification of neutral pions in the endcaps.

10.2 Hadron calorimeter

The hadron calorimeter is set of calorimeters used to measure the energy of showers initiated by particlesthat penetrated the electromagnetic calorimeter completely.

The main part of the hadron calorimeter is the barrel, which is located in the cylindrical volume betweenthe electromagnetic calorimeter and the superconducting magnet. This barrel is a sampling calorimeter,which uses layers brass plates oriented so that their normal vectors are perpendicular to the particle beam

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as passive media, and plastic scintillators as active media. There are 16 layers of scintillators separated bybrass plates. The flat scintillation and brass layers create concentric polygons when viewed from the endsof the barrel cylinder, and layers cover larger areas as r increases. The outermost layer is thicker to preventleakage of late-developing showers. The innermost layer is a scintillator, and its purpose is to detect showersthat developed in the material between the electromagnetic calorimeter and the hadron calorimeter. In total,there are roughly 70,000 separate scintillation tiles in the barrel of the hadron calorimeter, each connectedto a separate photodetector.

Outside the superconducting magnet, at low pseudorapidity values, is the outer hadronic calorimeter,in place to capture shower components not absorbed by any of the calorimeters closer to the beam and toensure that the calorimeters are hermitic. The outer hadronic calorimeter uses plastic scintillator platesas its sampling medium, and the iron of the superconducting magnet as its passive medium. It is a sam-pling calorimeter with two sampling layers: the first sampling layer is located outside the solenoid of thesuperconducting magnet, and the second layer is located outside the magnet’s first return yoke. The outercalorimeter extends through a low pseudorapidity region because particles leaving the collision point at lowpseudorapidities are those which traverse through less material as they travel through a cylindrical volumeextending between two constant r values. Mathematically, a particle leaving a collision point on a path thatmakes an angle of θ with the particle beam passing through a cylinder with an inner radius of r1 and anouter radius of r2 is (r2 − r1) sin θ.

The endcaps of the hadron calorimeter are, again, sampling calorimeters that utilize plastic plates as thescintillation medium and brass as the passive medium.

Light from the plastic scintillator plates sandwiched between the brass passive media in the hadroncalorimeter is carried to photodetectors though optical fibers run through grooves in the scintillator plates.These fibers are made of light carrying media coated with reflective cladding. The fibers carry the light tophotodiodes for detection.

10.3 Zero degree calorimeter

To ensure that the CMS is a hermetic detector, there exist electromagnetic and hadron calorimeters thatspan pseudorapidity between 8.3 and infinity and between -8.3 and negative infinity: they are able to detectshowers that develop parallel to the beam path. This is possible because the zero degree calorimeters arelocated between the two beam pipes. The importance of the hermetic nature of the CMS in all directionslies in one of the missions of the system. In order to detect particles which are practically invisible to anyof the detectors, their existences may be inferred by detecting unequal energy and momentum distributionsof detectable particles. As a crude example, if we observe that more energy was carried by detectableparticles into a certain half of a sphere with a center at the beam intersection point, then we would knowthat invisible particles carried energy into that sphere’s opposite hemisphere. This type of analysis is onlypossible if detectable particles do not escape from the CMS without being detected [8].

10.4 Calibration of the CMS

The Compact Muon Solenoid was calibrated using a relatively low-energy test beam of particles pointedat production plates which produced particles directed into the detectors designed to mimic trajectoriesof particles moving away from the beam intersection point. The test beams were produced in the SPS,and entered the detectors in two modes: a high energy mode, ranging from 15 to 350 GeV/c, and a verylow energy mode, carrying particles with energy less than 9 GeV/c. The particles were first passed throughseveral counters to identify their types with certainty. Time of flight counters measured the speeds of particlesby simply measuring the time it took for the particles to traverse a certain distance.

The barrel sections of the electromagnetic and hadron calorimeters were calibrated using beams of 50GeV/c electrons, and the outer hadron calorimeter was calibrated using a 150 GeV/c electron beam.

The relative responses to different particles throughout the collection of CMS’s detectors was measured.It was found that the response to protons is 47 percent of the response to electrons, and that the response tocharged pions is 62 percent of the response to electrons. It was found that the response to positively charged

27

pions was larger than the response to negatively charged pions; this difference is due to the decay productsof the particles. Furthermore, the difference in the responses to these two particles increases as their energiesincrease. While the electromagnetic calorimeter was designed to measure showers started by electrons,positrons, and photons, it was expected that a portion of protons and hadrons would initiate showers in thisdetector. It was found that 59 percent of pions that penetrated the electromagnetic calorimeter barrel and65 percent of the protons that penetrated started a shower reaction. However, these results were producedusing the low energy test beam, and it was found that the portion of energy deposited by hadrons in theelectromagnetic calorimeter barrel decreased as the energy of the particles was increased.

The differences in the electromagnetic/hadronic shower energy responses between the electromagneticcalorimeter barrel, the hadron calorimeter barrel, and the outer hadron calorimeter were measured so thatcorrections may be applied to determine the total energy of particle systems passing through all of thesedetectors. Corrections were made for non-linear energy responses: those that changed as a function of energy,once these functions were determined through calibration.

An important step was to parameterize the ratio of energy response to pions to the energy response toelectrons in the hadron calorimeter barrel as a function of the average energy of the particles in the barrel[13].

11 ATLAS

Unlike the calorimeters of the Compact Muon Solenoid, the calorimeters that are part of the ATLAS sys-tem are mainly designed to measure ionization. This is done by detecting the effect that hadronic andelectromagnetic showers have on the electric charge of samples of liquid argon.

11.1 Electromagnetic calorimeters

The innermost calorimeter of ATLAS is its electromagnetic calorimeter. It features a liquid argon samplingmedium housed by accordion-shaped lead plates. The waves of the accordion shape are oriented so thatparticles traveling in in the r direction pass through several waves, and so that the crest of a wave makes acircle around the particle beam in the barrel portion of the electromagnetic calorimeter. In the electromag-netic calorimeter’s endcaps, the waves of the accordion geometry run so that the crests of waves extend inlines perpendicular to the particle beam. In the endcaps, the gaps between the accordion-shaped plates varywith η so that the energy response of this part of the calorimeter is not dependent on η. Each accordionlayer consists of a lead stopping medium, and a liquid argon sampling medium. Between the lead and theliquid argon is a stainless steel coating glued to the lead. Housed in the middle of the liquid argon samplingchambers are plates that hold the readout electronics. The outer layer of these plates is copper which isconnected to a high-voltage power supply, and which attracts electrons freed through ionization by particlespassing though the liquid argon. This layout is seen in figure 12.

11.2 Hadron calorimeters

The ATLAS hadron calorimetry system features both scintillator-based hadronic shower detectors and liquidargon based ionization detectors.

A scintillator tile-based sampling hadron calorimeter surrounds the electromagnetic accordion calorimeterbarrel. Steel is used as the stopping medium. Light from the scintillator tiles is carried to photomultipliertubes by optical fibers running along the edges of the tiles. There are eleven layers of scintillating tiles, andthe area of the tiles increases as r does. This barrel features a total of 460,000 separate scintillating tiles.

The endcaps of the hadron calorimeter system are liquid argon ionization detectors, but unlike those of theelectromagnetic calorimeter, they feature flat absorber plates. There are also forward calorimeters attachedto ATLAS that utilize liquid argon ionization detectors, and detect particles traveling at pseudorapiditiesranging to infinity. Like the CMS, ATLAS is intended to be a hermetic detector for detectable particles. In

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Figure 11: The accordion sampling geometry of ATLAS’s calorimeters. From JINST.

η

η = 0

Strip cells in Layer 1

Square cells inLayer 2

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Figure 12: A closeup of a liquid argon chamber. From ATLAS Technical Design Report CERN/LHCC 96-40.

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the forward calorimeter, the liquid argon medium surrounds electrodes in the shape of rods that run throughthe medium parallel to the beam direction [7].

References

[1] Ugo Amaldi. Fluctuations in calorimetry measurements. Physica Scripta, 23(4a), 1981.

[2] Unknown authorship. LHC Collisions. http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/collisions.htm.

[3] Michel Chanel. LEIR: THE LOW ENERGY ION RING AT CERN.

[4] K. Nakamura et al. Particle data group. Review of Particle Physics, 2010.

[5] K. Nakamura et al. Passage of particles through matter. Review of Particle Physics, 2010.

[6] D.J. Graham and C. Seez. Simulation of Longitudinal Light Collection Uniformity in PbWO4 Crystals.

[7] Institute of Physics Publishing and SISSA. The ATLAS Experiment at the CERN Large Hadron Collider,August 2008.

[8] Institute of Physics Publishing and SISSA. The CMS experiment at the CERN LHC, August 2008.

[9] William Leo. Techniques for Nuclear and Particle Physics Experiments. Springer-Verlag, New YorkBerlin Heidelberg, 1994.

[10] Richard Scrivens. Cern hadron linacs. http://linac2.home.cern.ch/linac2/default.htm, January2008.

[11] Doreen Wackeroth. Cross section. High Energy Physics made painless, 2002.

[12] Richard Wigmans. Calorimetry: Energy Measurement in Particle Physics. Clarendon Press, Oxford,2000.

[13] Efe Yazgan. The CMS barrel calorimeter response to particle beams from 2 to 350 gev/c. In Journalof Physics: Conference Series 160, 2009.

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