Interaction of Radiation of Matter

8/19/2019 Interaction of Radiation of Matter

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Passage of Particles through Matter

Contents

1 Introduction 1

2 The Standard Model 2

3 Charged Particle Interactions 3

3.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1.1 Rate of Energy Loss: dE/dx . . . . . . . . . . . . . . . . . . . . . . . 4

3.1.2 Straggling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.3 The Bragg Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Multiple Coulomb Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Photon Interactions 9

5 Radiative Effects and Electromagnetic Showers 12

6 Hadronic Showers 14

1 Introduction

The interaction of radiation with matter1 is important in a wide range of disciplines, includingparticle and nuclear physics, medical radiology, and possibly even climate change, wherethere is some evidence that cosmic rays play a role in cloud formation. In these notes we will

1The phrase “Interaction of Radiation with Matter” is an old one, dating from the early part of the 20 th

century. Implicit in this phrasing is the notion that radiation and matter are two separate things, whichis not strictly true. In the intervening decades, we have come to understand that much of what we callradiation is really just matter in motion. That understanding is captured in the so-called “Standard Model”of particle physics, which will be used as an organizing principle for these notes.

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Figure 1: The building blocks of the standard model.

cover the main ideas. Additional details can be found in the relevant sections of Refs. [1]and [2].

2 The Standard Model

We provide a short overview of the standard model (SM) of particle physics to put the

various forms of “radiation” in context. The basic features of the SM can be understood byexamining Fig. 1.

The building blocks of the SM are the quarks and leptons, each of which is arranged in aset of three doublets. The doublets in each set are arranged in mass order, with the lightestparticles placed on the left.2 The particles of the higher mass doublets decay to the lowermass doublets and are therefore unstable. The matter around us is thus built from membersof the two lowest doublets—i.e., u and d (“up” and “down”) quarks and the e−. Neutrinosare effectively massless and too weakly bound to other matter to do anything but fly away,and the other quarks and leptons quickly decay to the lowest generation. All quarks and

2At this point, we know that neutrinos have mass, but we don’t know precisely what they are, other than

that they must be very light.

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only the phenomena of ionization and multiple scattering discussed in this section.

3.1 Ionization

All charged particles produce ionization as they pass through matter. In what follows wewill call the ionizing particle the “projectile.” One can understand ionization semi-classicallyas follows. An atom consists of central massive core to which electrons are loosely tethered.The word “loosely” applies, since in most cases the projectile will have a kinetic energy thatis many MeV or more, which is orders of magnitude larger than the typical atomic bindingenergy. As the projectile sweeps past an atom, the atom’s electrons experience an impulsiveforce due to the Coulomb interaction. If the projectile comes close enough to the atom,the force will be large enough to eject an electron—i.e., to ionize the atom. Typically theelectron emerges with kinetic energy of its own, occasionally quite a lot of it. The source of this energy is the projectile, which drops in energy as a consequence. The average energyloss per interaction is tiny, but there are many such interactions, so the projectile ends uploosing a measurable amount energy and can even be brought to rest if the stopping materialis thick enough.

3.1.1 Rate of Energy Loss: dE/dx

The amount of energy lost due to ionization by fast particle of charge z is given by the Betheequation

dE

dx

= −Kz 2 Z

A

1

β 2

1

2 ln

2mec2β 2γ 2W max

I 2 − β 2 − δ (βγ )

2

, (1)

where me is the electron mass, c is the speed of light, β = v/c and γ = E /m are the usualrelativistic variables, Z and A are respectively the atomic number and weight of the targetatom, and K = 4πN Amer

2ec

2 sets the scale for the amount of energy loss. W max is a factorthat takes into account the maximum energy loss, I is the mean excitation energy, and δ (βγ )is a term included to take into account the so-called density effect.

As one can see, a precise account of ionization energy loss is an algebraic extravaganza,so it is useful to distill Eq. 1 down to its essential ingredients. We consider the simple caseof a fast (β > 0.1) charged particle of unit charge traversing water (or any other material of density, ρ = 1, comprising atoms of similar atomic weight). In that limit Eq. 1 becomes

dE dx

− 1β 2 dE dx 0

(1 + ln F (γ )) ∼ − 1β 2

dE dx0

, (2)

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where (dE/dx)0 = 2 MeV/cm. Eq. 2 displays the most important feature of ionizationenergy loss, namely that it scales as 1/β 2. This can be understood by noting that in thelimit of fixed force, the momentum transfer scales like the time of interaction (and hencelike the inverse of the velocity) and the energy transfer scales like the momentum transfersquared ∆E = (∆ p)2/2m. Eq. 2 also includes a term with a logarithmic dependence on the

relativistic γ factor, which can be important at very high energies. This “relativistic rise”correction can be ignored if an approximate answer will suffice.

Eq. 2 can be scaled to other materials by noting that to first order the rate of energy lossdepends only on the number of electrons in the material. For materials comprising light andmedium-light nuclei, if one doubles the density, the energy loss will double with it. Heaviernuclei, however, tend to have more neutrons than protons, so the growth in dE/dx does notquite scale with density.

Figure 2 shows the rate of energy loss by muons in copper as a function of βγ . Muons arechosen because over a very wide range of energies their only significant form of energy loss isionization. Fig. 2 covers nine orders of magnitude in β γ . Eq. 2 covers roughly four of those(0.1 < βγ < 1000). For the low βγ regime, where the binding energy of atomic electronsbecomes important, the Bethe theory breaks down. In the high βγ regime, an additionalenergy-loss mechanism called bremsstrahlung ,4 whereby the muon radiates a photon, kicksin.

3.1.2 Straggling

One frequently encounters the situation where a charged particle passes through a layer of material, creating ionization as it goes. In such cases, the total amount of ionization canbe of interest. Since the ionization process is stochastic (randomly determined), the energyloss will vary from event to event. Fig. 3 shows the expected energy loss distribution for a10 GeV muon passing through a 1.7-mm-thick layer of silicon.

As a consequence of the central limit theorem, the energy loss distribution, which representsthe sum of many collisions, has a roughly Gaussian peak. However, it has a high-energy tail,resulting from single interactions where there is a very large energy loss (every once in awhile, the projectile strikes an atomic electron right on the nose, sending it flying with alarge energy transfer).

4Bremsstrahlung , which is German for “braking radiation,” occurs when a charged particle is accelerated.We will discuss it in Sec. 5.

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Figure 2: Rate of energy loss (stopping power) in copper. Figure is from Ref. [1].

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Figure 3: The peaked curves are the energy loss probability distribution for a 10-GeVmuon passing through a 1.7-mm thick layer of silicon, calculated using two slightly differenttheoretical models. The integral curves labeled M 0(∆)/M 0(∞) and M 1(∆)/M 1(∞) representthe cumulative distribution for the mean number of collisions) and the mean energy loss incrossing the silicon. Figure is from Ref. [1].

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Figure 4: Energy deposition (dose) as a function of depth. Figure is from Ref. [3].

3.1.3 The Bragg Curve

From Eq. 2, it is evident that for β = v/c 1, the rate of energy loss is roughly constant. Itis also evident that for β


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projectile slows down but tends not to change direction. In some interactions, however, theprojectile passes close to the atomic nucleus and interacts with it via Rutherford scattering.In this case, the target is heavy and thus absorbs little energy (recall that ∆E ∆ p2/2M ).In most materials, however, the nucleus has several units of charge, and thus exerts a largeforce on the projectile, causing it to change direction.

As is the case for ionization, a single interaction typically has only a small effect on theprojectile, but there are many interactions, resulting in a non-negligible cumulative effect.Single interactions can deflect the particle to the left or to the right (and/or up or down),but there is no preferred direction, so the multiple interactions result in a random walk indirection space. Once again the central limit theorem comes into play. If a singly-chargedparticle of momentum p moving in the z direction passes through a thin layer of materialhaving thickness L, it will emerge from that material scattered through an angle given bythe following probability density distribution

p(θx,y) = 1√

2πθ0exp−θ

2x,y

2θ20, (3)

where θx,y refers to the scattering angle projected onto either the x or the y plane and θ0 isgiven by

θ0 = 13.6 MeV

βp

L

L0

1 + 0.038ln

L

L0

, (4)

where L0 is an intrinsic property of the scattering medium known as the “radiation length.”

Table 1 lists the radiation length for some representative materials. The radiation lengthdecreases quickly with increasing Z as a result of the Z 2 scaling of the cross section. Gaseousmaterials have far fewer atoms per unit volume, resulting in substantially increased radiationlengths.

4 Photon Interactions

Here we will consider photons with energies above 100 eV, where a particle description (asopposed to a wave description) is generally more convenient. Rayleigh scattering, whereinthe photon scatters off the atom as whole, contributes at low energies, but does not result inabsorption. Interactions of photons with E > 100 eV that result in full or partial absorptionof the photon fall into three general categories:

• Photoelectric absorption, where the photon is completely absorbed by the atom itionizes.

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Table 1: Radiation lengths of selected materials.

Material Z A Density (g/cm3) L0 (cm)

Helium (gas) 2 4 1.66× 10−4

5.6× 105

N2 (gas) 7 14 1.17× 10−3 3.3× 104Water (liquid) — — 1.0 36

Aluminum 13 27 2.7 8.9Iron 26 56 7.9 1.7Lead 82 208 11.4 0.56

• Compton scattering, where the interaction between a photon and an atomic electronresults in a photon of lower energy and a recoil electron.

• e+e− pair production, where the photon’s energy is used to create an e+e− (matter-antimatter) pair.

The first two of these reactions involve the atom or its electrons. Pair production, however,occurs as a result of the extremely high fields that are present near the nucleus, and istherefore most important for high-Z atoms. At energies of E > 5 MeV or so, photonuclear5

interactions, where the photon penetrates the nucleus, breaking it up, or leaving it in anexcited state, also play a role, although they generally represent a small fraction of theoverall interaction cross section.

5For purposes of this discussion, we don’t consider pair production, which takes place in the field of

nucleus to be a nuclear interaction.

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Fig. 5 shows the cross sec-

tion for γ -C and γ -Pb inter-actions as a function of en-ergy. Note that the data areplotted on a log-log scaleand thus represent an ex-tremely wide range in bothenergy and cross section.Note also the qualitativedifference between carbon,a medium light nucleus, and

lead, a heavy nucleus. Forcarbon, the photon crosssection is dominated by thephotoelectric effect for en-ergies up to E γ = 10 keV,where Compton scatteringbegins to play a role. Pairproduction becomes ener-getically possible at E γ =2me = 1022 keV, butCompton continues to dom-

inate until E γ 50 MeV, atwhich point pair productiontakes over. Pair productionis marked as κnuc.

Figure 5: Photon cross sections for carbon and lead as afunction of photon energy. Figure is from Ref. [1].

The situation is qualitatively similar for lead, although the larger size of the lead atomand the large Z of its nucleus means that the photoelectric effect continues to be important

to higher energies, while pair production turns on more quickly near the E γ = 1.022 MeV

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threshold, resulting in a situation where Compton scattering is important only over a com-paratively narrow energy range.

A striking feature of the 1 keV < E γ < 1 MeV energy range for γ -Pb interactions isthe appearance of sudden jumps in the cross section. These absorption “edges,” which aresuperimposed on the general tendency for cross sections to drop with photon energy, occurwhenever the photon energy becomes large enough to ionize electrons in a new electron shell.Carbon exhibits only a single absorption edge, reflecting the simpler shell structure of thecarbon atom.

5 Radiative Effects and Electromagnetic Showers

In the previous section, we saw how pair production could convert a single photon to an e+e−

pair, with each member of the pair having of order half the energy of the original photon.There is nothing to force the e+ and e− to have the same energy, but since the heavy recoilnucleus generally carries off very little energy, E γ E e+ + E e− .

A related process is bremsstrahlung , wherein an e± radiates a photon in the presence of the high electromagnetic field in the vicinity of a nucleus. In this process e± −→ e± + γ with E e± = E e± + E γ . In principle, any charged particle could undergo such a reaction, butat low energies, electrons and positrons are the only particles light enough to do so.6

If a high-energy (E 1 MeV) photon, electron, or positron, impinges on a layer of material, an electromagnetic shower will develop. For purposes of illustration, we assumethe shower starts with a photon. That photon will pair-produce, creating an e+ and ane−. The e+ and e− will undergo bremsstrahlung, creating a final state comprising an e+,

an e−

, and two photons. Each of those particles will undergo either pair production orbremsstrahlung , resulting in a final state with eight particles, and so on.

The mean distance between the production point of a particle and the point at whichit undergoes an interaction will be of order one radiation length. The shower process isstochastic: in some cases, an interaction will occur almost right away, while in others, theparticle travel several radiation lengths until interacting. Moreover, the energy sharingbetween final-state particles will vary significantly from reaction to reaction. The binarydivision picture is only heuristic. Nonetheless, the shower process is such that a single highenergy particle ends up producing many low energy particles. The charged particles in theshower produce ionization and low energy photons undergo the reactions outlined in Section 4

6Close inspection of Fig. 2 reveals the onset of muon bremsstrahlung around pµ = 100 GeV/c.

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above. The result is that the energy of the incident particle ends up in the form of ionizationin the absorbing medium. A class of particle detectors called calorimeters, exploits this tomake precision measurements of the energies of high-energy photons and electrons.

Figure 6: Schematic viewof electromagnetic shower.

Figure 7: Shower simulated using the program GEANT[4],which accurately models most aspects of particle interac-tions. GEANT is particularly good at simulating such thingsas ionization, multiple scattering, and electromagnetic inter-actions, where the underlying physics is well understood.

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Table 2: Nuclear interaction lengths of selected materials.

Material Z A Density (g/cm3) λint cm

Helium (gas) 2 4 1.66× 10−4

4.3× 105

N2 (gas) 7 14 1.17× 10−3 7.6× 104Water (liquid) — — 1.0 83

Aluminum 13 27 2.7 40Iron 26 56 7.9 17Lead 82 208 11.4 32

6 Hadronic Showers

Particles made from quarks (protons, neutrons, pions, kaons, etc.), interact via the nuclear(strong) force. As its name implies, the strong force is quite strong. It is also short-ranged,so that particles only interact strongly when they come within a fermi (10−15 m) or so of the nucleus. Thus a strongly interacting particle can penetrate fairly deeply into a materialwithout undergoing a nuclear reaction. When an interaction does occur, it can be quitedramatic. For example, a fast pion can be completely absorbed, leaving behind a nucleusthat is so highly excited that it breaks apart, releasing several neutrons in the process.

Just how far a particle goes is sensitive to a range of details including: the species of theprobe particle, its energy, and the nuclear make up and density of the material. Despite thesevariations, it is possible to characterize materials with a nuclear interaction length. Table 2

shows the nuclear interaction lengths for representative materials. Here the variation with Z is not as large as it is for radiation lengths, mainly because the cross section scales roughlylike the geometric size of the nucleus—i.e., A2/3.

Hadronic showers develop in a fashion that is analogous to electromagnetic showers. Giventhe longer interaction lengths (the nuclear interaction length in iron is an order of magnitudehigher than the radiation length in iron), and much greater variation in reaction types,hadronic showers typically are quite ragged in comparison to electromagnetic showers. Indeedfor incident particles less than 10 GeV in energy, the notion of an hadronic shower barelymakes sense. For higher energies, the statistical variations begin to average out and it ispossible to measure the energy of very high energy hadrons using a calorimeter approach

analogous to that used to detect photons and electrons. The technique is especially effectivein detecting jets, which are the remnants of quarks or gluons produced in collisions at high-

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energy colliding beam particle accelerators.

Hadronic showers also occur in nature, and are in fact responsible for the muon flux thatbathes the Earth. Fig. 8 is a schematic representation of such a shower. A galactic cosmic ray(typically a proton, but sometimes a heavier nucleus) impinges upon the upper atmosphereand undergoes a nuclear interaction with an atmospheric molecule. The secondary productsfrom that interaction proceed down through the atmosphere until undergoing interactionsof their own. Many of the particles that are produced are pions (quark-antiquark boundstates of u and d quarks). Neutral pions decay almost immediately to pairs of photons,which initiate electromagnetic showers. Charged pions have mean life times of 27 ns. Sincemany of the pions are highly relativistic, they travel far enough to interact before theydecay. Other charged pions decay via the reaction π± → µ±ν µ. Muons have a mean life of τ = 2.2 µs, corresponding to a proper decay length of cτ = 660 m. They do not undergostrong interactions and thus continue to travel until they decay, “range out” due to ionizationenergy loss, or reach the ground. Interestingly, most muons are produced at altitudes of 10,000 m or above, and would not reach the ground were it not for relativistic time-dilation

effects.

References

[1] Olive, K.A. et al., “Review of Particle Physics,” Chin. Phys. C, 38, 090001(2014). This document, which contains many interesting facts and reviews,can be found online at URL: http://pdg.lbl.gov . The section germaneto this discussion can be found at URL: http://pdg.lbl.gov/2013/reviews/rpp2013-rev-passage-particles-matter.pdf.

[2] Melissinos, A. and Napolitano, J., “Experiments in Modern Physics,” 2nd Ed., AcademicPress (2003).

[3] Particle Therapy Cancer Research Institution (United Kingdom). See URL: http://www.ptcri.ox.ac.uk/research/introduction.shtml

[4] GEANT4 Collaboration, “GEANT4: A simulation toolkit,” Nucl. Instrum. Meth. A 506371 (2003) 250, doi:10.1016/S0168-9002(03)01368-8.

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Figure 8: Schematic depiction of an hadronic shower induced by a galactic cosmic raystriking the upper atmosphere.

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Interaction of Radiation of Matter

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