2 The Interaction of Radiation with Matter · Ref. p. 2–39] 2 The Interaction of Radiation with Matter 2–1 2 The Interaction of Radiation with Matter Hans Bichsel 2.1 Introduction

Ref. p. 2–39] 2 The Interaction of Radiation with Matter 2–1

2 The Interaction of Radiation with Matter

Hans Bichsel

2.1 Introduction

2.1.1 General concepts

The radiations considered here are

• photons (energies described by hf, hν, or �ω)

• electrons and positrons (rest mass m, charge ±e, speed v and kinetic energy T )

• light ions (mesons, protons, with rest mass M , charge ±e, speed v and kinetic energy T )

• heavier ions (electric charge number 2 ≤ z < 20, rest mass M , speed v and kinetic energy T )

The word particles usually means any of these. Interactions of neutral particles (phonons, neutrons,neutrinos, wimps etc.) will not be discussed.

The emphasis is on high speed charged particles, i.e. β = v/c � 0.05 (kinetic energy T � 1keV for electrons, T � 1 MeV for protons). In order to explore and understand the interactionprocesses they will be described for single particles. Ions are assumed to be nuclei fully strippedof electrons.

The particle interactions are with matter. Only two states of matter will be considered: gaseousand condensed. In general the word atom designates molecules, too. Effects related to crystalstructure (e.g. channeling) will not be discussed.

Electromagnetic interactions mainly are discussed in this chapter. It is useful to think of theinteractions in terms of single collisions which occur at random distances xi along a particle track[1]. The collisions result in separate, discrete, random energy losses Ei of the particles.

Frequently one wants to observe charged particles for a certain distance along their tracks (orpaths). Each particle will then experience several collisions in succession. The spacing xi betweencollisions, the energy losses Ei and angular deflections θi will be random for each collision.

For multiple collisions the sum of the energy losses will be designated as Δ =∑

iEi [2, 3]. Here,E and Δ will be considered as positive numbers, and the residual energy of a particle is T − Δ.

The concept of separate collisions described above can be observed experimentally. This hasbeen done in cloud chambers and should be possible with GEMs [4]. Experiments with electronmicroscopes are also suitable for this purpose [5, 6]. The comparison of electron energy loss spectracalculated for multiple collisions [7] with measured spectra [8] confirms this concept.

Collisions can be divided into three categories:

• interactions where the particles disappear, examples: photo electric effect, nuclear reactions.

• interactions where the particles change energy T and momentum p, examples: inelasticscattering, bremsstrahlung, pair production, Compton effect.

Landolt-BornsteinNew Series I/21B1

2–2 2 The Interaction of Radiation with Matter [Ref. p. 2–39

• “elastic collisions” where the momentum transfer is large but the energy loss is small, exam-ple: Coulomb scattering of charged particles in collisions with nuclei in whole atoms (alsocalled “nuclear collisions”).

Particle beams are defined to consist of one kind of particles with parallel trajectories extendingover a small area and a small spread in speeds.

2.1.2 Types of collisions

For photons the following collisions can be distinguished [9]:

• photo absorption

• Compton scattering

• positron-electron pair production - discussed in Sect. 3.2

• nuclear interactions.

For charged particles two types of collision are most frequent

• inelastic collisions: particles lose energy by excitation and ionization of atoms or condensedmatter (local collective excitations); secondary radiation may be produced, such as secondaryelectrons (called delta-rays), Auger electrons, photons (fluorescence), Cherenkov and transi-tion radiation, bremsstrahlung, etc.

• elastic (Coulomb or Rutherford) scattering

Nuclear reactions and other types of collisions are infrequent [10], they will be discussed mainly inSects. 3.2 and 3.3, e.g. pair production, meson production, nuclear excitations.

2.1.3 Observable effects of radiations

The energy losses and secondary radiations mentioned in Sect. 2.1.2 will result in energy depositionin matter. As far as energy transfer is concerned, it is important to distinguish between the effecton the incident particle, described as an energy loss 1 and the effects on the absorber such asproduction of excitations and secondary radiations (e.g. delta-rays, photons, phonons) which willresult in energy deposition. 2

Photons and neutral particles must produce a charged particle before they can be observed.Low energy photons are an exception because they can produce photo-chemical effects. An exampleis the observation of light in the retina or in photographic emulsions.

2.1.4 Stopping power dE/dx

Attention must be paid to clear definitions of symbols and concepts. In particular the symbol“dE/dx” describes at least six concepts, such as energy loss Δ, ionization J in a track segment

1Angular deflections will also occur. Nuclear spallation will not be considered.2Thermal effects [11, 12], photographic methods and track etching will not be further mentioned.



and also the digital output Q of the electronic analyzing apparatus [13]. Other examples can befound on p. 676 of ref. [14]. Suggestions about symbols for various concepts for TPCs are givenon p. 159 of [13].

The original meaning of “dE/dx” was “mean energy loss of fast particles per track segment ofunit length” and is used correctly only in low energy nuclear physics [11, 15, 16, 17].

2.2 Historical background

In the last century most of our knowledge about interactions of fast charged particles with matterwas presented as average quantities for particle beams, with a large number of collisions along eachparticle track [1, 18, 19, 20, 21].

Most researchers used mathematical-analytic methods to derive averages such as stopping power,ranges, straggling, multiple scattering. Many clever methods were derived to simplify the calcu-lations: sum rules, transforms, various approximations (in particular the use of the Rutherfordcross section for inelastic collisions of charged particles with electrons). We are now at a timewhere calculations with computers permit the solution of many problems with computer-analyticor Monte Carlo methods. These calculations use fewer approximations but they need more detailedinformation about the absorbers and the collision processes. As an example, for thin absorbersthe classical stopping power and Landau functions usually do not give adequate information forpractical applications, such as vertex trackers and TPCs[14]. The average quantities have littlemeaning if segmented tracks of single particles are measured such as in particle identification PID[13].

There is a close relation between photo absorption processes and cross sections for chargedparticle collisions. Fermi formulated this in 1924 [22].

2.3 Description of the most frequent interactions of singlefast charged particles

The averages of the random distances xi between the collisions along a particle track are calledmean free paths λ. They vary widely. For inelastic electronic collisions of protons, e.g. in Si, λranges from less than 5 nm to about 250 nm, in gases at NTP λ is a factor of 1000 larger (Tables 2.1and 2.2). 3 For elastic collisions of electrons 10 < λ(nm) < 2000.

The energy losses E in inelastic collisions have a wide range of values. For most applications,the smallest energy losses are of the order of 10 eV. It is practical to consider small energy lossesas less than 100 eV. Of the order of 80% of all energy losses in single collisions are small [13].The largest energy loss in a single collision for an electron with rest mass m is usually consideredto be half its energy, for a positron it can be its full energy. For heavier particles with rest massM and speed v there is a maximum energy loss EM to an electron given in the non-relativisticapproximation by

EM = 2 m v2. (2.1)

3For ions with charge ze the mean free paths are divided by a factor z2.



The relativistic expression is

EM =2 m c2 β2γ2

1 + (2γm/M) + (m/M)2, (2.2)

where β = v/c and γ2 = 1/(1 − β2) and β2 = (βγ)2/(1 + (βγ)2).The average energy loss 〈E〉 per collision in light elements for 0.05 < βγ < 100 is between 50

and 120 eV.For elastic collisions the energy losses for heavy particles are small. The major effect is an

angular deflection which usually is small. For electrons large deflections and energy losses canoccur, but are infrequent.

2.3.1 Narrow beams and straggling of heavy charged particles

For beams of particles (kinetic energy T , speed v = βc) traversing an absorber the initial energyspectrum φ(T ) will broaden due to the random number of successive collisions and the large spreadin random energy losses E. This process is called straggling. The lateral extent of the beam willbroaden due to random multiple elastic collisions, and the number of particles in the beam isreduced by nuclear interactions. Frequency or probability density functions (pdf) are needed todescribe the properties of the beam along the tracks of the particles.

A schematic representation of the traversal of ten heavy charged particles through a thin ab-sorber is given in Fig. 2.1. Multiple inelastic collisions are seen. For clarity elastic collisions arenot included. The collisions are simulated in a Monte Carlo calculation where for each collision,shown by a symbol, two random numbers are used to give first the distance xi from the previouscollision, then the energy loss Ei, see Sects. 2.5-2.7. This scheme is similar for all absorbers, allparticles and all types of collisions. In successive collisions there is a large spread in energy loss Eand angular deflections. Spectra of energy losses are given in Sects. 2.6-2.8. For asymmetric pdf(or “spectra”) the mean values may not have much meaning, the most probable or median valuewill be more suitable for a description.

Long tracks can be divided into n short segments and average values of the Δj of the segmentscan be used. By eliminating a fraction of the largest Δj a truncated mean [13] can be determinedfor each track, see Sect. 2.10.

2.3.2 Narrow beams of low energy electrons

Because the mass of electrons is small, they can be scattered by as much as π in a single collision.Both elastic and inelastic collisions can cause large deflections of the electrons. Electrons thereforecan be back-scattered out of absorbers. Examples of tracks of back-scattered electrons are shownin Fig. 2.2. Similar tracks have also been described for low energy protons [23].

2.3.3 Relation between track length and energy loss

In traveling along a track, Fig. 2.1, the distances between collisions and the energy losses are bothrandom and have no correlation. From Fig. 2.1 it is evident that an exact segment length x canbe defined (except for uncertainties of atomic dimensions), but the total energy losses Δj will havea range of values, as shown in Fig. 2.4. It is not possible to postulate an exact final energy in a



Energy loss per collision: E

10

9

8

7

6

5

4

3

2

1

x=6 λ Δ=ΣΕ

j nj Δ j(eV) Et(eV)2

8

8

4

4

4

5

5

7

9

56

703

126

82

419

565

95

146

105

930

37

559

68

32

292

502

35

26

34

774

Fig. 2.1. Monte Carlo simulation, Sect. 2.7.1, of the passage of particles (index j, speed βγ = 3.6) througha segment of P10 gas of thickness x = 6λ, where λ = 0.33 mm is the mean free path between collisions.The direction of travel is given by the arrows. Inside the absorber, the tracks are straight lines defined bythe symbols showing the location of collisions (total number 56). At each collision point a random energyloss Ei is selected from the distribution function Φ(E; βγ), Fig. 2.14. Two symbols are used to representenergy losses: o for Ei < 33 eV, × for Ei > 33 eV. Segment statistics are shown to the right: the numberof collisions for each track is given by nj , with a nominal mean value < n >= x/λ = 6. The total energyloss is Δj =

∑Ei, with the nominal mean value 〈Δ〉 = x dE/dx = 486 eV, where dE/dx is the stopping

power, M1, in Table 2.2. The largest energy loss Et on each track is also given. The mean value of theΔj is 325 ± 314 eV, much less than 〈Δ〉. Note that the largest possible energy loss in a single collision isEM = 13 MeV, Eq. (2.2).

MC calculation, as described in Fig. 2.3. Correspondingly the straggling functions are different.This is shown in Fig. 2.4. The pdf of energy loss for the particles described in Fig. 2.1 traversingx = 9 cm is given by the solid line in Fig. 2.4, the values of x associated with each energy loss Δin Fig. 2.3 are given by the dotted line.

2.3.4 Nuclear interactions

Particle deflections by elastic nuclear collisions for heavy particles lead to multiple scattering, [24].For electrons, see Sect. 2.3.2 and [25]. Inelastic nuclear collisions are nuclear reactions. For protonswith T > 20 MeV a rough approximation for the mean free path for nuclear reactions is [9, 27]

L ∼ 32 A1/3g/cm2, (2.3)

about 80 cm for water and 40 cm for Si. A table can be found on p. 110 of [24]. Depending onthe type of nuclear interaction there may be only one interaction in which the particle disappears



x

12500

10000

7500

5000

2500

00 2000 4000 6000

z

Fig. 2.2. Monte Carlo simulation of the passage of three electrons with kinetic energy T = 18.6 keV

through silicon. Units of x, z are A. The electrons enter the detector at x = 0 vertically from the bottom.The initial mean free path between inelastic collisions is λ = 25 nm, between elastic collisions λ = 15 nm[25]. Trajectories are projected onto the x, z plane.

Δ [keV]

f [Δ]

f [Δ]

βγ=3.6, Δ<9 keV

105

104

103

102

101

2 4 6 8 10

105

104

103

102

101

Fig. 2.3. Monte Carlo simulation of the passage of 500,000 particles with speed βγ = 3.6 through Ne(similar to Fig. 2.1). The final energy loss Δ of the particles is 9 keV or less. It is given by the sum ofEi before the next collision produces a sum larger than 9 keV. The spectrum of the energy losses is f(Δ),given by the solid line, the cumulative spectrum is F (Δ), dotted line, with ordinate scale at right. Notethat the track length for each particle is different, Fig. 2.4.

from the beam, and reaction products will remain. For particle beams a description in terms of abeam attenuation will be instructive.

To get the energy deposition by the nuclear products a Monte Carlo simulation must be made(see Sect. 2.7.1).



sc

f [sc]

βγ=3.6, Δ=9keV, x=9cm10000

8000

6000

4000

2000

0 0 50 100 150 200

Fig. 2.4. The dashed line gives the pdf of energy losses Δ in Ne for a segment length x = 9 cm for the500,000 particles described in Fig. 2.1, Δ = (sc/10) keV. The peak value is at about 8.5 keV, the FWHMis 4.6 keV. The solid line gives the pdf of the segment length x = sc mm of particles which lost Δ ≤ 9keV, as described in Fig. 2.3. The peak is at 86 mm. The two functions have opposed asymmetry, andthey have no correlation. It will be difficult to compare the functions by using stopping power. It may bepossible to measure these spectra with a GEM TPC [4, 24]. The quotient of the peaks is 1 keV/cm, muchless than M1 = 1.6 keV/cm.

2.4 Photon interactions

The principal photon interactions are Rayleigh scattering, photo-electric and Compton effect, pairand meson production and nuclear disintegration. The important energy ranges are 0 to 100 keVfor photo absorption, 0.1 to 5 MeV for Compton scattering and above 5 MeV for pair production.These limits increase with Z. Detailed descriptions can be found in [9, 26].

The main need is for photo-absorption data for the use in the calculation of the energy lossspectra for charged particle interactions described in Sects. 2.5 and 2.6.

Compton effect and pair production are described in detail in e.g. Ch 3 of “Radiation Dosime-try” [9, 27]. A more detailed theoretical study of Compton profiles can be found in [29]. See alsoSect. 3.2. A schematic representation of the photo-absorption of a photon in an atom is given inFig. 2.5. For photons with energies below the ionization potential excitations are produced. Theymust be included in the calculations described in Sect. 2.5. 4

photoelectric interaction

hf

e

atom

Fig. 2.5. Schematic description of a photo electricinteraction of a photon with energy hf = 30 eV(wavelength λ ∼ 0.66 A) with an Ar atom (Z =18, ionization energy IZ ∼ 16 eV). An electron isemitted from the M-shell with an energy E = hf −IZ ∼ 14 eV in a random direction.

4In a gas mixture such as Ne, CO2 and N2 used for the TPC in ALICE the excited states in Ne extend from16.7 to 21.6 eV, well above the ionization potentials of the molecules (13.8 and 15.6 eV) and therefore can de-exciteby ionizing the other molecules (Penning effect)[28]. Furthermore in measurements (of, e.g. stopping power and W ,energy needed to produce one ion pair [30]) it is not possible to exclude the effects of the excitations.



2.4.1 Gases

The quantity used for the description of photon collisions [31] resulting in ionization (continuumexcitation states) is the dipole oscillator strength (DOS) f(E; 0) where E = hν = �ω is the photonenergy and the momentum transfer q = 0. It is related to the photo-ionization cross section σγ(E)

σγ(E) = 4π2αa20Rf(E; 0) = 0.01098f(E; 0) (nm2/eV) (2.4)

where a0 =52.9 pm is the Bohr radius of the H-atom, R=13.59 eV the Rydberg energy. For excita-tions to excited states with energy En the symbol fn is used. In gases the excited states contributeof the order of 5% to the total DOS [32]. The absorption coefficient μ(E) for electromagneticradiation is

μ(E) = N σγ(E) (2.5)

where N is the number of atoms per unit volume [9, 31]

2.4.2 Solids

Fano [20] described a method to extend the oscillator strength approach to condensed materials byusing the complex dielectric constant ε(ω) = ε1(ω) + iε2(ω) of the absorber, where �ω representsthe energy loss by the virtual photon [34]. The DOS for solids is related to the dielectric constantas follows [35, 36]

f(E; 0) = E2ZπΩ2

p

ε2(E)ε21(E) + ε22(E)

= E2ZπΩ2

p

�(−1ε

) (2.6)

with the plasma energy Ωp of a free electron gas

Ω2p = 4π�

2e2NZf/m, Ωp = 28.8√

(ρZf/A) eV (2.7)

where ρ(g/cm2) is the density of the solid with atomic number Z and atomic mass A(g). Thenumber Zf is the number of electrons which represent the free electron gas (e.g. Zf = 3 for Al, 4for Si). 5

2.4.3 Data for DOS

Most DOS data are derived from measurements. Examples for several gases are given in Figs. 2.6and 2.7. Evaluated data for about 30 gases can be found in Berkowitz [32]. They have been usedto derive mean excitation energies I [40] (see Appendix A).

DOS data have been derived from measurements with electron microscopes [6]. The method isdescribed in [5].

For E > 100 eV tables given in [26] are at present the most easily accessible.Calculations of ε(ω) have been given for Li and diamond in [41]. The X-ray absorption fine

structure is discussed in [42].Data for solids are given in the Handbook of optical constants of solids [43].

5A more detailed description can be found in Chapter 5 of [28].



E [eV]

f [E,0

]

CH4

Ne

Ar

100

10–2

10–4

10–6

10–8

10–10

10–12

102 104 106

Fig. 2.6. Dipole oscillator strength f(E; 0) for Ar, Ne and CH4 as function of photon energy E. For Ne,the values f(E; 0)/30 are given, for CH4 f(E; 0)/10000 are given. The peak values are approximately 0.32for Ar, 0.16 for Ne and 0.66 for CH4, located between E =15 and 20 eV.

E [eV]

f [E,0

]

CF4

HEED

Lee

Zhang

0.6

0.4

0.2

0.010 20 30 50 70 100 200

Fig. 2.7. Dipole oscillator strength f(E; 0) for CF4 as function of photon energy E. Sources are Zhang[37], Lee [38], HEED [39].

2.5 Interaction of heavy charged particles with matter

The most frequent interactions occur between the electric charge ze of a particle and the electronsof matter resulting in an energy transfer E by the particle in inelastic collisions. The energy istransferred to excited states of atoms, single free electrons (delta rays) or to many electrons as acollective excitation. Photons can also be produced, such as Bremsstrahlung (BMS).

Secondary radiations produced are: delta rays, Auger electrons, fluorescence, Cherenkov ortransition radiation. Details about their effects are given in Sect. 2.9.

In many descriptions of the process the Rutherford or Coulomb collision cross section [9, 44, 45,46] σR(E) is used as a first approximation [7, 2, 3, 48].



For the interaction of a heavy particle with charge ze, spin 0 and speed β = v/c colliding withan electron at rest it can be written as 6

σR(E, β) =k

β2

(1 − β2E/EM )E2

, k =2πe4

mc2· z2 = 2.54955 · 10−19 z2 eVcm2 (2.8)

where m is the rest mass of an electron, and EM ∼ 2mc2β2γ2 the maximum energy loss of theparticle, Eq. (2.2). Note that the mass of the particle does not appear in Eq. (2.8) [9]. This crosssection has been used by Landau [2], Vavilov [3] and Tschalar [48] as an approximation for thederivation of straggling functions. The approximation is quite good for large energy losses, Figs.9, 12, 15. Details are given in Appendix B.

2.5.1 Inelastic scattering, excitation and ionization of atoms orcondensed state matter

For inelastic collisions the electronic structure of the atoms in the absorber (especially bindingenergies of the electrons) is important because energy transfers E change these structures. Thecollisions are also called the inelastic scattering of the particles. Two methods will be discussed inthe following. For a quantitative description the Bethe-Fano method [7, 19, 20, 34, 50, 51] is closestto reality. The Fermi-Virtual-Photon method [7, 19, 20, 34, 50, 51] requires less detailed input.Since the B-F method is accurate to about 1%, ref.[34], it should be used for energy calibration ofdetectors. For the FVP method see Tables 2.3 and 2.4.

For energy losses of electrons with E < EM/2 Eqs. (2.22, 2.24) can be used (Tables 2.3, 2.4).For larger energy losses corrections are derived in [35] or the Moller and Bhaba cross sections areintroduced [52, 51]. The straggling functions given in Figs. 2.23-2.27 will be the same for electronsand heavy particles. The effect of Bremsstrahlung [52, 13] is described in Sect. 2.6.

2.5.1.1 Bethe-Fano (B-F) method

Bethe [19] derived an expression for a cross section doubly differential in energy loss E and mo-mentum transfer K using the first Born approximation for inelastic scattering by electrons of theatomic shells. Fano [20] extended the method for solids. In its nonrelativistic form it can be writtenas the Rutherford cross section modified by the “inelastic form factor” [20, 53]:

σ(E,Q) = σR(E) | F (E,K) |2 E2

Q2, (2.9)

where Q = q2/2m, with q = �K the momentum transferred from the incident particle to theabsorber, and F (E,K) is the transition matrix element for the atomic excitations or ionizations.For large momentum transfers (peak in Fig. 2.8), Q ∼ E.

The relativistic expression is given by Eq. (2.13) in Sect. 2.5.1.2.Usually, F (E,K) is replaced by the generalized oscillator strength (GOS) f(E,K) defined by

f(E,K) =E

Q| F (E,K) |2, (2.10)

and Eq. (2.9) then is written as

σ(E,Q) = σR(E)E

Qf(E,K) . (2.11)

6Additional factors for electrons and positrons and for particles with spin 12

and spin 1 at high speeds (β ∼ 1)are given e.g. in Uehling [49]. An extensive description can be found in Evans [9].



Kao

E=48 Ry

f [E,K

]

0.12

0.10

0.08

0.06

0.04

0.02

0.000 2 4 6 8 10 12 14

Fig. 2.8. Generalized oscillator strength GOS for Si for an energy transfer E = 48 Ry to the 2p-shell electrons [34, 54]. Solid line: calculated with Herman-Skilman potential, dashed line: hydrogenicapproximation[33]. The horizontal and vertical line define the FVP approximation, Sect. 2.5.1.3.

Calculations of GOS have been published by Inokuti [53] and Bote and Salvat [51]. A detailedstudy was made for Al and Si [34, 54]. An example of f(E,K) is shown in Fig. 2.8.

In the limit K → 0, f(E,K) becomes the optical dipole oscillator strength (DOS) f(E, 0)described in Sect. 2.4.1. The cross section differential in E is obtained by integrating Eq. (2.11)over Q,

σ(E; v) = σR(E)∫

Qmin

E f(E,K)dQ

Q(2.12)

with Qmin ∼ E2/2mv2. The dependence on particle speed v enters via Qmin. In our currentunderstanding, this approach to the calculation of σ(E) is closest to reality. 7 Because of thefactor 1/Q in Eq. (2.12), the accuracy of f(E, 0) (Fig. 2.7) enters significantly into the calculationsof cross sections and M0, Eq. (2.25).

2.5.1.2 Relativistic extension of B-F method

The basic equation for the doubly differential cross section is [20, 34]

σ(E,Q) = kRZ

[ | F (E,q) |2Q2(1 +Q/2mc2)2

+| βtG(E,q) |2

Q2(1 +Q/2mc2) − E2/2mc2

]

(1 +Q/mc2) (2.13)

where kR = 2πz2e4/(mc2β2), Q = q2/2m gives the kinetic energy of the secondary electronproduced in the collision, | F (E,q) |2 represents the interaction matrix element for longitudinalexcitations, and | G(E,q) |2 represents that for transverse excitations [51, 34].

Similar to Fano [20], the cross section differential in E is divided into four parts.

• Small Q part: f(E,K) is approximated by f(E, 0), see Fig. 2.8: Q = (Kao)2, and selectQ1 ∼ 1 Ry=13.6 eV

σ1(E, β) = kRZ

Ef(E, 0) ln

Q12mc2β2

E2(2.14)

7A description of the derivation of “Bethe stopping power”, Appendix A.1, can be found in [54].



• High Q part, corresponding to large energy losses E where the binding energy of the atomicelectrons can be neglected. For Fig. 2.8 the delta function approximation is used, resultingin the following expressions [20]

| F (E,q) |2 ∼ 1 +Q/2mc2

1 +Q/mc2δ(E,Q) (2.15)

| βt · G(E,q) |2 ∼ βt1 +Q/2mc2

1 +Q/mc2δ(E,Q) (2.16)

and

β2t =

11 +Q/2mc2

− (1 − β2) (2.17)

are used to get [34]

σh(E) = kRZ

E

(1

1 + s+

s

1 + s− s (1 − β2)

)

(2.18)

with s = E/(2mc2).

• Intermediate Q part: the integral over Q, Eq. (2.12), is calculated numerically [34, 54] withthe Q1 used in Eq. (2.14) and Q2 ∼ 30 keV. The contribution from G(E,q) can be neglectedaccording to Eq. (2.18) since s is small (details in [34]). 8

σ2(E) =∫ Q2

Q1

σ(E,Q) dQ (2.19)

These integrals are calculated once and then stored numerically (Eq. (2.11) in [34, 55]).

• The last contribution is from low Q excitations in condensed materials. It is described indetail by Fano [20]. For longitudinal excitations Eq. (2.14) is replaced by

σ1(E, β) = kR Im[ −1ε(E)

]

lnQ12mc2β2

E2(2.20)

This term is equivalent to the third term of Eq. (2.24), except that Q1 is replaced by E, i.e.in Fig. 2.8 f(E, 0) extends to the delta-function instead of only to Q1 = 1. 9 For transverseexcitations the contribution is Eq. (47) in [20], using Eqs. (2.4-2.7) to convert ε2(ω) intoσγ(E)

σ3(E;β) =α

β2π

σγ(E)EZ

ln[(1 − β2ε1)2 + β4ε22 ]−1/2 +α

β2π

1N�c

(β2 − ε1|ε|2 )Θ. (2.21)

where tan Θ = β2ε2/(1−β2ε1). This is the same equation as the first two terms in Eq. (2.24).

• The total cross section differential in energy loss E is given by Eqs. (2.14-2.21)

σ4(E;β) = σ1(E;β) + {σ2(E) + σh(E)} + σ3(E;β) (2.22)

The function calculated with Eq. (2.22) for minimum ionizing particles [20] is shown by the solidline in Fig. 2.9. No Bethe-Fano calculations with improved GOS are available for gases, but seeref. [51].

8For particle speeds β < 0.1 this approximation will cause errors, especially for M0.9The effect of the approximation can be seen in Fig. 2.12: for E < 20 eV the FVP σ(E) exceeds the B-F value

considerably, resulting in the larger value of M0 in Tables 2.3, 2.4.



E [eV]

σ [E

] / ρ

[E]

20.0

10.0

4.0

2.0

1.0

0.4

0.2

0.11 3 10 30 100 300 1000 3000 10000 30000 100000

Fig. 2.9. Inelastic collision cross sections σ(E, v) for single collisions in silicon of minimum ionizingparticles (βγ = 4), calculated with different theories. In order to show the structure of the functionsclearly, the ordinate is σ(E)/σR(E). The abscissa is the energy loss E in a single collision. The Rutherfordcross section Eq. (2.8) is represented by the horizontal line at 1.0. The solid line was obtained [34] with theBethe-Fano theory, Eq. (2.22). The cross section calculated with FVP, Eq. (2.24) is shown by the dottedline. The functions all extend to EM ∼ 16 MeV, see Eq. (2.1). The moments are M0 = 4 collisions/μmand M1 = 386 eV/μm, Table 2.1.

2.5.1.3 Fermi-virtual-photon (FVP) cross section

The GOS of Fig. 2.8 has been approximated [22, 35, 56, 57, 58] by replacing f(E,K) for Q < Eby the dipole-oscillator-strength (DOS) f(E, 0) and by placing a delta function (vertical line)at Q = E, as shown in the figure. This approach is here named the Fermi Virtual Photon(FVP) method. It is also known under the names Photo-Absorption-Ionization model (PAI) andWeizsacker-Williams approximation.

The FVP calculation is based on the use of photo absorption cross section σγ(E) (where E = �ωis the photon energy) and of the dielectric function ε(ω) = ε1(ω) + iε2(ω) [22, 20].

The differential collision cross section in the non-relativistic approximation is given by [56]

σ(E, β) = σR(E, β)

[

E f(E, 0) 2 ln (2mv2/E) +∫ E

0

f(E′, 0) dE′]

(2.23)

for E > EM , σ(E) = 0. This model has the advantage that it is only necessary to know theDOS for the absorber, or, equivalently, the imaginary part Im(−1/ε) of the inverse of the complexdielectric function ε. Data for ε can be extracted from a variety of optical measurements [60, 43].In addition, Im(−1/ε) can be obtained from electron energy loss measurements [61]. A detaileddescription of the relativistic PAI model is given e.g. in [56, 62]. The relativistic cross section isgiven by

σ(E) =α

β2π

σγ(E)EZ

ln[(1 − β2ε1)2 + β4ε22]−1/2 +

α

β2π

1N�c

(β2 − ε1|ε|2 ) Θ

+α

β2π

σγ(E)EZ

ln(2mc2β2

E) + ue(γ)

α

β2π

1E2

∫ E

0

σγ(E′)Z

dE′ (2.24)

with β = v/c, σγ(E) ∼ (E/N) · ε2(E) and tanΘ = β2ε2/(1 − β2ε1).



E [eV]

σ [E] [1 / eV], FVP

σR [E]

P10 gas

φ [E] φ [E]

σ [E

]

1.000

0.500

0.100

0.050

0.010

0.005

0.001

1.000

0.500

0.100

0.050

0.010

0.005

0.00110 20 50 100 200 500

Fig. 2.10. Inelastic collision cross section σ(E; βγ) for single collisions in P10 gas of minimum ionizingparticles (βγ = 3.6), calculated with FVP theory: solid line. The Rutherford cross section Eq. (2.8) isgiven by the dash-dotted line. The dotted line represents the cumulative probability density function Φ(E)of Eq. (2.26), see Fig. 2.14. For βγ = 3.6 the functions extend to EM ∼ 13 MeV, see below Eq. 2.8. Theionization energy for Ar is EI = 15.8 eV [32].

Note that σγ(E) = f(E; 0)·1.098·10−16 cm2 eV, Eq. (2.14) [31]. The function ue(γ) ∼ s(1−β2)of Eq. (2.18) 10 is assumed to be equal to 1.0 in [56] because large energy losses E are unimportantin that context. 11 An example of σ(E) for P10 gas is given by the solid line in Fig. 2.10, for Nein Fig. 2.11.

For silicon detectors it is seen in Table 2.1 that Σt = M0 calculated with FVP theory differsby about 6 to 8 % from the B-F theory 12 while the difference for M1 is less than 1%. 13 Inmeasurements of the ionization in TPCs the difference inM0, Tables 2.3 and 2.4, (see its importancein Eq. (2.27)), may disguise for example the uncertainty of the energyW (T ) to produce an electron–ion pair (at least ±2%) and its dependence on particle energy T [30].

For the calculation of straggling functions f(Δ;x, v), Sect. 2.7. for energy losses Δ of particleswith speed v in track segments of length x the FVP differential single collision cross sectionspectrum σ(E; v) is used here, temporarily, as a reference function.

Cross sections calculated with several expressions are given for Si in Fig. 2.9, for P10 gas inFig. 2.10 and for Ne gas in Fig. 2.11. 14 The functions are similar for all speeds, see Figs. 2.13-2.15.

10See Eqs. 2-4 in Uehling [49].11For gases, ε2 and ε1−1 are proportional to the gas pressure p, therefore from Eq. (2.24) we must expect that the

straggling function for a segment of length x1p1 will differ from that of a segment of length x2p2 even if x1p1 = x2p2

[62, Fig. 1.20].12The difference is caused by the approximation shown in Fig. 2.8.13B-F M1 differed by less than ±0.5% from experimental measurements [34, 55].14Calculations have also been made for several other gases, but are not given here. Optical data used are described

in Sect. 2.4.1.



E [eV]

σ [E

] [1/

eV]

Ne βγ=3.6σ [E]

100

10–2

10–4

10–6

10 50 100 500 1000 5000 10000

Fig. 2.11. Inelastic collision cross section σ(E; βγ) for single collisions in Ne gas by ionizing particles withβγ = 3.6, calculated with FVP theory (Eq. (2.24)): solid line. The Rutherford cross section, Eq. (2.8), isgiven by the dashed line, the AliRoot cross section (Appendix B.3) by the dotted line. The coefficient κ(v)for the AliRoot cross section, Eq. (2.49), is chosen to give Σt;A(3.6) =15 collisions/cm [63, Fig. 7.1]. ForEq. (2.8) the function is chosen to cross as shown - the Landau parameterization [2] eliminates the needfor a specific value of k in Eq. (2.8). The K-shell excitation at 850 eV causes a difference in the stragglingfunctions, Fig. 2.28.

E [eV]

σ [E

] / σ

R [E]

Si βγ=0.316

101 102 103 104

125

100

75

50

25

0

Fig. 2.12. Collision cross section for Si, relative to the Rutherford cross section for 14 electrons. Solid line:calculated with B-F approximation, dashed line: with FVP approximation. The horizontal line representsthe Rutherford approximation, Sect. 2.5.1.3. The differences appear to balance to some extent. This indeedis the case for M1, Eq. (2.25) see Table 2.1, but for M0 at E = 20 eV, the difference is 20%, dropping to8.1% at 10 keV.



2.5.2 Integral quantities: moments and central moments

The moments of the collision cross sections are defined by

Mν(v) = N

∫

Eν σ(E; v) dE (2.25)

where N is the number of electrons per unit volume, and ν = 0, 1, 2, 3..... They are obtained as aresult from the computer-analytic calculation for Eq. (2.22) or 2.24. Moments calculated with theanalytic method described in Appendix A are given in Tables 2.3 and 2.4. The moment M0(v),which is also written as Σt(v), is the mean number of collisions per unit track length. It is animportant quantity for the calculation of f(Δ;x, v) (Sect. 2.7) because it is used to calculate mc,the mean number of collisions in a track segment.

From Figs. 2.13 to 2.16 it is evident that the exact shape of σ(E) for small E will greatlyinfluence values of M0. 15 The moment M1(v) is the Bethe-Fano stopping power dT/dx, i.e. themean energy loss per unit track length. Its analytic approximation is Eq. (2.41). 16 (21).

Numerical values of M0(βγ) and M1(βγ) are given in Table 2.1 for Si, calculated with bothmethods given above, and in Table 2.2 for P10 and Ne. From Table 2.1 the differences betweenthe B-F and FVP methods seen in Fig. 2.9 cause quantitative differences in the moments. Thedifference between B-F and FVP is ∼6% for Σt, 0.3-0.9 % for M1 and ∼ 3% for Δp. The classicalformulation of dT/dx and some details are given in Appendix A. The practical use of dT/dx isdescribed in Sects. 2.7.1 to 2.7.3. For Z<20 the average energy loss 〈E〉(β) per collision is between50 and 120 eV.

The higher moments can be used to calculate the shape of straggling functions for large energylosses [64, 48, 46]. But, for thin absorbers, even M1(v) will result in misleading information, seeFig. 2.25. The dependence of M0(βγ) and M1(βγ) on particle speed is shown in Fig. 2.17.

An important function is the cumulative moment

Φ(E, v) =∫ E

0

σ(E′; v) dE′ /∫ ∞

0

σ(E′; v) dE′ (2.26)

needed for Monte Carlo calculations. Examples are shown in Figs. 2.13-2.16.

2.5.2.1 Comparison of moments: Si, Ne, P10

For silicon absorbers it is seen in Table 2.1 that M0 calculated with FVP theory differs by about 6to 8% from the B-F theory. The difference is caused by the GOS approximation shown in Fig. 2.8.For M1 the difference is less than 1%. For Ne and Ar a comparison of the FVP theory of M0 withσtot calculated with the Bethe theory, Appendix A.2, is given in Tables 2.3 and 2.4. A similardifference occurs for M0 for Ne and Ar as for Si.

The stopping power M1 for both gases calculated with FVP differs by less than 1% from ICRU[66].

In measurements of the ionization in TPCs the difference inM0 (see its importance in Eq. (2.33)),may disguise for example the uncertainty of the energy W (T ) to produce an electron–ion pair (atleast ±2%) and its dependence on particle energy T [30].

15Especially important is he choice of a cut-off energy Em for the Rutherford spectrum, Eq. (2.8), since RM0 =k′/Em, Appendix B.1.

16In many publications it is customary to write the particle kinetic energy as E, then the stopping power is dE/dx.Since the expression for σ(E; v) does not contain corrections equivalent to the Barkas and Bloch corrections, thesenames here are not included in the name for the stopping power. On the other hand, Fano [20] formulated theexpression for solids given in Eq. (2.21).



E [eV]

φ [E

;βγ]

βγ0.02262000

1.0

0.8

0.6

0.4

0.2

0.0101 102 103 104

Fig. 2.13. Cumulative energy loss functions Φ(E), Eq. (2.26), for single collisions in Si are shown forseveral values of βγ. The excitation energy for L2 electrons is 100 eV, for K electrons it is 1840 eV [65].A table of the functions is given in [55].

E [eV]

φ [E

] βγ=0.1

1.0

0.8

0.6

0.4

0.2

0.010 20 50 100 200 500 1000

βγ=2.5βγ=7900

Fig. 2.14. Cumulative energy loss functions Φ(E) for single collisions in P10 gas, Eq. (2.26), are shownfor several values of particle speed βγ. For 1.0 ≤ βγ ≤ 7.9 the difference between the functions is no morethan the width of the line. A table of the functions is given in [55].



E [eV]

φ [E

]

βγ=0.316 & 100

1.0000

0.5000

0.1000

0.0500

0.0100

0.0050

0.0010

0.000520 30 50 70 100 200

Fig. 2.15. The cumulative collision cross sections Φ(E; v) of Eq. (2.26) calculated with FVP for Ne fortwo values of βγ. The solid line is for βγ=3.16, the dashed line for βγ=10 000. For E > 20 eV thedifference between the two functions is less than 1%. The difference for 12 < E < 30 eV is the Cherenkoveffect, Sect. 2.9.4. The dotted line for the modified Rutherford cross section (“AliRoot”) is derived fromEq. (2.49).

E [eV]

βγ=0.1βγ=3.16βγ=3160

τ [E]

101 102 103 104 105

100

10–1

10–2

10–3

10–4

10–5

Fig. 2.16. Probabilities Υ(E) = 1−Φ(E) for single collisions in P10 gas in which the energy loss exceedsa value E for different βγ in P10 gas, see Fig. 2.14.



βγ

M1

100 101 102 103 104

1.8

1.6

1.4

1.2

1.0

L [P10]

Σt [Ne]

Σt [P10]

Fig. 2.17. The dependence on βγ of M0 = Σt(v), Eq. (2.25), in Ne and P10 gas. The Bethe-Fano functionsM1(v) are also given (dotted line for Ne). All functions are normalized to 1.0 at minimum ionization. TheFVP values M1 = dE/dx differ little for the two gases, but M0(Ne) reaches saturation at a higher valuethan M0(P10). The function given by Eq. (2.50) is given by the solid line labeled L[P10]. See Sect. 2.5.2.1about estimated errors.

2.6 Electron collisions and bremsstrahlung

2.6.1 Electronic collisions

For electrons and photons interaction models more sophisticated than described in Sects. 2.4 and 2.5are given in [67]. In general the methods to calculate energy losses Δ, Fig. 2.1, for heavy ions givenin Sects. 2.5 and 2.7 will be reliable for electrons in thin absorbers. Thin means that the numberof collisions in the absorber by the other interactions (BMS, Cherenkov, pair production etc.) areless than ∼ ten. For Si this was considered in [13]. For thick absorbers these interactions becomeimportant and energy loss spectra must be calculated with Monte Carlo simulations.

For 15 < T (MeV) < 1000 the approximation

S = dT/dx = ρ · a · T b MeV/cm, (2.27)

with a and b from Table 2.5, reproduces the “collision” dT/dx of ref. [52] to within 2%.

2.6.2 Bremsstrahlung BMS

The atomic differential cross section for production of BMS of energy E by electrons with kineticenergy T is given by [69] 17

σrad(T,E)dE = 4α r20 Z2 ln

183Z1/3

dE

E≡ χ

dE

E, 4α r20 = 2.32 ·10−27 cm2/nucleus. (2.28)

17A further factor Fe(T, u), u = E/T , is approximated by 1 here [52, 68, 69, 10, 24].



Table 2.1. Integral properties of collision cross sections for Si calculated withBethe-Fano (B-F) and FVP algorithms. Σt = M0[collisions/μm], M1[eV/μm],Eq. (2.25) for heavy particles, Δp in eV for x=8μm. The minimum values forΣt are at βγ ∼ 18, for M1 at βγ ∼ 3.2, for Δp at βγ ∼ 5. The relativistic risefor Σt is 0.1%, for M1 it is 45%, for Δp it is 6%. Computer accessible numericaltables are available in [55].

βγ Σt M1 Δp/x

B-F FVP B-F FVP B-F FVP

0.316 30.325 32.780 2443.72 2465.31 1677.93 1722.920.398 21.150 22.781 1731.66 1745.57 1104.90 1135.680.501 15.066 16.177 1250.93 1260.18 744.60 765.950.631 11.056 11.840 928.70 935.08 520.73 536.510.794 8.433 9.010 716.37 720.98 381.51 394.031.000 6.729 7.175 578.29 581.79 294.54 304.891.259 5.632 5.996 490.84 493.65 240.34 249.251.585 4.932 5.245 437.34 439.72 207.15 215.021.995 4.492 4.771 406.59 408.70 187.39 194.602.512 4.218 4.476 390.95 392.89 176.30 183.063.162 4.051 4.296 385.29 387.12 170.70 177.163.981 3.952 4.189 386.12 387.89 168.59 174.815.012 3.895 4.127 391.08 392.80 168.54 174.636.310 3.865 4.094 398.54 400.24 169.62 175.607.943 3.849 4.076 407.39 409.07 171.19 177.10

10.000 3.842 4.068 416.91 418.58 172.80 178.6612.589 3.839 4.064 426.63 428.29 174.26 180.0615.849 3.839 4.063 436.30 437.96 175.45 181.2419.953 3.839 4.063 445.79 447.44 176.36 182.1425.119 3.840 4.063 455.03 456.68 177.04 182.7931.623 3.840 4.064 463.97 465.63 177.53 183.2839.811 3.841 4.064 472.61 474.27 177.86 183.6150.119 3.842 4.065 480.93 482.58 178.09 183.8363.096 3.842 4.065 488.90 490.55 178.22 183.9579.433 3.842 4.065 496.52 498.17 178.32 184.06

100.000 3.842 4.066 503.77 505.42 178.38 184.10125.893 3.843 4.066 510.66 512.31 178.43 184.15158.489 3.843 4.066 517.20 518.84 178.44 184.17199.526 3.843 4.066 523.40 525.05 178.47 184.18251.189 3.843 4.066 529.29 530.94 178.48 184.18316.228 3.843 4.066 534.91 536.56 178.48 184.21398.107 3.843 4.066 540.28 541.92 178.48 184.22501.187 3.843 4.066 545.43 547.08 178.48 184.22630.958 3.843 4.066 550.40 552.05 178.48 184.22794.329 3.843 4.066 555.21 556.86 178.48 184.22

1000.000 3.843 4.066 559.89 561.54 178.48 184.22



Table 2.2. Integral properties of CCS (Collision Cross Sections), Eq. (2.25),calculated with the FVP algorithm for P10 and the ALICE TPC Ne gas(ρ = 0.91 mg/cm3). Δp and FWHM w given in keV for x = 2 cm, Σt = M0 incollisions/cm, M1 in keV/cm. Computer accessible numerical tables are avail-able in [55].

βγ P10 Ne

Δp w M0 M1 M0 M1

0.316 22.0926 11.4916 211.0726 15.1286 91.8949 9.97030.398 14.6338 8.1277 146.5664 10.7146 64.0487 7.05170.501 9.9505 5.8737 103.9873 7.7376 45.6034 5.08100.631 7.0166 4.3998 76.0672 5.7440 33.4737 3.76960.794 5.1720 3.4753 57.9161 4.4322 25.5713 2.90501.000 4.0128 2.9035 46.2566 3.5811 20.4887 2.34661.259 3.2927 2.5456 38.8999 3.0442 17.2819 1.99361.585 2.8565 2.3229 34.3884 2.7197 15.3192 1.77961.995 2.6043 2.1897 31.7545 2.5369 14.1800 1.65902.512 2.4717 2.1173 30.3570 2.4494 13.5846 1.60023.162 2.4170 2.0853 29.7722 2.4253 13.3474 1.58293.981 2.4133 2.0800 29.7206 2.4432 13.3459 1.59305.012 2.4428 2.0919 30.0180 2.4890 13.4992 1.62116.310 2.4941 2.1147 30.5430 2.5531 13.7539 1.66127.943 2.5593 2.1443 31.2156 2.6291 14.0747 1.7090

10.000 2.6337 2.1779 31.9825 2.7127 14.4387 1.761812.589 2.7137 2.2140 32.8078 2.8011 14.8305 1.817815.849 2.7970 2.2513 33.6658 2.8923 15.2399 1.875719.953 2.8820 2.2892 34.5369 2.9847 15.6598 1.934725.119 2.9674 2.3271 35.4067 3.0772 16.0853 1.994131.623 3.0523 2.3649 36.2903 3.1693 16.5121 2.053539.811 3.1370 2.4024 37.2469 3.2607 16.9365 2.112550.119 3.2162 2.4394 38.0550 3.3485 17.3547 2.170663.096 3.2888 2.4758 38.6576 3.4321 17.7706 2.227679.433 3.3556 2.5116 39.0968 3.5115 18.1605 2.2828

100.000 3.4178 2.5468 39.4162 3.5874 18.5054 2.3353125.893 3.4761 2.5813 39.6515 3.6600 18.7926 2.3845158.489 3.5312 2.6150 39.8283 3.7296 19.0191 2.4301199.526 3.5832 2.6477 39.9648 3.7963 19.1915 2.4722251.189 3.6322 2.6793 40.0725 3.8601 19.3194 2.5108316.228 3.6776 2.7094 40.1590 3.9209 19.4125 2.5464398.107 3.7186 2.7374 40.2288 3.9784 19.4794 2.5792501.187 3.7547 2.7630 40.2850 4.0325 19.5271 2.6098630.957 3.7853 2.7859 40.3296 4.0830 19.5610 2.6383794.328 3.8101 2.8055 40.3634 4.1298 19.5852 2.6651

1000.000 3.8292 2.8219 40.3885 4.1731 19.6024 2.6904



Table 2.3. Comparison of moment M0

for Ne with σtot, ks = 8πa2oRy/mv2.

T [MeV] M0/ks σtot/ks diff%

10 13.59 11.89 14.330 15.65 13.93 12.3

100 17.81 16.06 11.1300 19.69 17.87 10.2500 20.58 18.73 9.9

1000 21.95 20.05 9.53000 24.76 22.81 8.5

10000 28.66 26.70 7.330000 32.57 30.67 6.2

Table 2.4. Comparison of moment M0

for Ar with σtot.

T [MeV] M0/ks σtot/ks diff%

10 34.56 31.02 11.430 39.37 35.82 9.9

100 44.40 40.85 8.7300 48.68 45.13 7.9500 50.70 47.15 7.5

1000 53.81 50.26 7.13000 60.31 56.77 6.2

10000 69.40 65.96 5.230000 78.01 75.30 3.6

For a discussion of the “average properties” of the BMS energy loss in the traversal of fastelectrons through an absorber the moments (Sect. 2.5.2) are used (N = ρNA/A is the number ofnuclei/cm3, Nχ is the inverse of the radiation length)

Mν(T ) ≡ N

∫ T

El

Eνσrad(E;T )dE. (2.29)

We get

M0(T ) = Nχ logT

Elλ(T ) = 1/M0(T ) (2.30)

M1(T ) = Nχ(T − El) (2.31)

M0 is the total collision cross section (CCS) and λ(T ) the mean free path between collisions. Thisis the important quantity for the calculation of BMS spectra. There is little use for the BMS(“radiative”) stopping power M1 and none for M2.

Data for T0 = 1 GeV electrons are given in Table 2.5, assuming El = 100 eV. 18

The contribution to M0 for BMS photons between 0.001 eV and 10 eV for Pb is 18 col/cm andto M1 it is 20 eV/cm. This effect is disregarded.

Table 2.5. Parameters for BMS Eqs. (2.27-2.31), col=collision.

Material Z ρ Nχ M0 λ M1 a b[g/cm3] [1/cm] [col/cm] [cm] [MeV/cm]

H2O 8 1.0 0.022 0.35 2.85 22 1.8 0.0422Si 14 2.329 0.098 1.58 0.63 98 1.6 0.0391Fe 26 7.87 0.55 8.9 0.11 550 1.34 0.0502CsI 55 4.5 0.57 9.1 0.11 570 1.22 0.0495Pb 82 11.36 1.92 31 0.032 1925 1.1 0.0541

In order to demonstrate the statistics of the radiative BMS losses, a MC calculation for 20,000electrons traversing a Si absorber 50 cm thick was made [55]. The small values of M0 lead to largespreads in the pdf, but they are similar in shape. To simplify the simulation, angular deflectionsare neglected and the BMS photons produced are assumed to escape from the absorber. The errorof the calculations is of the order of 10%. Results:

18The electronic energy loss Δ within a distance λ is much larger than El, thus BMS losses below 100 eV willcontribute negligibly to Δ (“infrared divergence”).


Ref. p. 2–39] 2 The Interaction of Radiation with Matter 2–23φ

[Tc]

mTc

Tc

Si, T0=1 GeV e

600

500

400

300

200

100

0 50 100 150 200 250 300

Fig. 2.18. Spectrum φ(Tc) of electronic collision energy losses Tc in Si. The median energy loss mTc is111 MeV. The spectrum φ(Tb) for the sum Tb of the BMS losses is the complement φ(T0 −Tb), with valuesbetween 700 and 1000 MeV.

• The total energy loss of the electrons in the Si is 1000 MeV.

• The pdf for the number of BMS collisions per track is asymmetric, with values between 2and over 90, with a median value of nB = 35 and a FWHM of 35.

• The pdf of the lengths l of tracks of the electrons are asymmetric, with values between 0 andover 50 cm, with a median value lm = 24 cm (approximately equal to nB · λ = 22 cm, butmuch less than the “csda range” R =32.5 cm in [52]), and a FWHM of 24 cm.

• The pdf for the energy losses due to electronic collisions is given in Fig. 2.18. They rangefrom ∼ 6 MeV to 300 MeV, with a median of 110 MeV and FWHM of 120 MeV.

• The pdf for the energy losses due to BMS collisions is the complement to Fig. 2.18. Theyrange from ∼ 700 MeV to 994 MeV, with a median of 890 MeV and FWHM of 120 MeV.Compare to Fig. 27.23 in ref. [24].

Conclusion: the only “average” quantity needed for dealing with BMS is the total collision crosssection M0. The pdfs described above have no evident relation to M1.

2.7 Energy losses along tracks: multiple collisions andspectra

A quantitative description of the multiple collisions shown in Fig. 2.1 can be given by three quanti-ties: the number of collisions nj for each particle track j, the total energy loss Δj and the maximumenergy loss Et inside the track segment of length xj . The quantity Δj/x does not provide any fur-ther information. If we calculate the collisions for a very large number of particles, with the samespeed and traversing the same segment lengths, we can derive probability density functions P (n)and P (Δ). P (n) is the straggling function for collisions. It is a Poisson distribution [7, 9, 58, 70]

P (n) =mn

c e−mc

n!(2.32)



where mc = x ·M0(v). The straggling function for energy losses is [70, 71]

P (Δ) = f(Δ;x, v) =∞∑

n=0

mnc e

−mc

n!σ(Δ; v)∗n (2.33)

where σ(Δ; v)∗n is the n−fold convolution of σ(E; v), defined by

σ(Δ; v)∗n =∫ Δ

0

σ(E; v) · σ∗(n−1)(Δ −E; v) dE,

σ(Δ; v)∗0 = δ(Δ), σ(Δ; v)∗1 = σ(Δ; v). (2.34)

Some examples of σ(Δ; v)∗n are shown in Figs. 2.19 and 2.20. 19

Clearly straggling functions will depend greatly on the mean number mc of collisions, and to alesser extent on the particle speed βγ. The measurement of straggling functions can be for selectedtracks, such as in a TPC (e.g. Sect. 14 in [13]), or for particle beams [34, 72, 73]. Monte Carloand analytic methods can be used to calculate straggling functions. They are described next.

2.7.1 Monte Carlo method

The interactions occurring during the passage of the particles through matter are simulated oneat a time, collision by collision [65], and include secondary collisions by the δ-rays produced.For the calculations shown in Fig. 2.1, the following procedure was used. A particle j travelsrandom distances xi between successive collisions, calculated by selecting a random number rrand determining the distance to the next collision from the mean free path between collisionsλ = 1/Σt(v)

xi = −λ ln rr. (2.35)

The energy loss Ei is selected with a second random number from the integrated collisionspectrum, Eq. (2.26), shown in Figs. 2.13 to 2.16. This process is repeated until

∑xi exceeds the

segment length x. The total energy loss Δj of the particle is Δj =∑

iEi. To get Ei practically,the inverse function E(Φ;βγ) of Φ(E;βγ) is calculated with e.g. cubic spline interpolation [55].By binning the Δj the straggling function f(Δ) is obtained [55, 74].

The Monte Carlo method can be used for all absorber thicknesses, but with decreasing particlespeed [55] it is necessary to change λ(v) and Φ(E; v) (Figs. 2.13-2.15) at appropriate values of v.It may not be very practical for very thick absorbers, e.g. for the full range of 200 MeV protonsin water (R ∼ 25 cm) the number of collisions is of the order of three million. In order to getreasonable straggling functions, tracks for a million protons may be needed [75]. 20

2.7.2 Analytic methods

In order to use analytic methods we must consider a large number of particles (charge z, massM) with the same speed v traversing the exact same length y of track [78]. 21 The methods aredescribed in a fashion suitable for numerical calculations. The particles traverse an amorphousabsorber consisting of atoms Z,A, as shown in Fig. 2.21. For thin absorbers and monoenergeticcharged particle beams two methods to calculate straggling functions are described next: convolu-tion and transform methods. “Thin” means that the change in v in traversing ξ can be disregarded.

19For the Rutherford cross section these function are shown in [58].20For condensed history MC calculations Landau or Vavilov functions [2, 3, 76] are used frequently [77]. Attention

must be paid to the condition for the applicability of these functions described, in Sect. 2.7.3.21For a single particle track the only possible description is that shown in Fig. 2.1.



σ [E

]*n

E [eV]

n=1 n=2 n=3

0.08

0.06

0.004

0.002

0.0010 20 30 50 70 100

Fig. 2.19. The first three convolution spectra σ(E)∗n defined in Eq. (2.34) for P10-gas are given. Solidline: single collisions, dotted line: two collisions, dashed line: three collisions. To show detail ordinates forσ(E)∗2 and σ(E)∗3 are not to scale.

E [eV]

n=1n=2

n=3

n=4

n=5

0.100

0.050

0.020

0.010

0.005

0.002

0.0010 20 40 60 80 100

σ [E

]*n

Fig. 2.20. Same as Fig. 2.19, but for solid Si, and for n up to 5. The plasmon peak at 17 eV appears ineach spectrum at E = 17n eV, and its FWHM is proportional to

√n. The structure at ∼ 2 eV appears at

2 + 17(n − 1) eV, but diminishes with increasing n. For n = 6 (not shown) the plasmon peak (at 102 eV)merges with the L-shell energy losses at 100 eV, also see Fig. 2.24.

2.7.2.1 Convolutions

With the convolution method [7, 34, 58, 71, 70] the straggling function for short track segments(Fig. 2.1) can be calculated with Eq. (2.33) 22 For a faster calculation the absorber doubling method.can be used. A first step is made with an absorber of thickness x so thin that mc 1 in Eq. (2.33).If the initial energy loss spectrum of particles with speed v is a delta function: f(Δ; 0, v) = δ(Δ),

22For mc = 20, about 40 functions of Eq. (2.34) would be needed.



y+ξy

φ [y,T]

z, M, v

Z, A

shp5–2–1.top

Fig. 2.21. Particles z, M with speed v (kinetic energy T ) traversing segments ξ of an absorber Z, A.Positions on the tracks are defined by the variable y. The energy spectrum at y is φ(y; T ). The directionof travel is along the axis y shown by the arrow. After traversing a thin layer ξ shown in the middle of theabsorber the spectrum is given by φ(y + ξ; T − Δ), where Δ is the energy loss in ξ.

Eq. (2.34) can be written as

f(Δ;x, v) ≈ δ(Δ)(1 −mc) +mc σ(Δ; v) (2.36)

with e−mc ≈ 1 −mc. Then the doubling method can be used (a variant of Eq. (2.34))

f(Δ; 2x) =∫ Δ

0

f(Δ − g;x)f(g;x)dg (2.37)

until the desired thickness x is reached. 23

For thicker absorbers another analytic method is given in Sect. 2.7.3. Because of the structureof the collision spectra, Figs. 2.9-2.11, the straggling functions f(g;x) used in the convolutionsmust be selected according to the classification described in Sect. 2.8.

2.7.2.2 Laplace transforms

The use of Laplace transforms for the solution of the transport equation was introduced by Landau[2] and Vavilov [3]. For details see Appendix B.4. The method was described and compared to theconvolution method in [7]. Here the symbol V (x,Δ, T ) is used to indicate that Vavilov functionsare used. The corresponding symbol in Eq. (2.33) is f(Δ;x, v) where T corresponds to v. 24

The use of fast Fourier transforms has also been suggested. The complexity of the single collisionspectra seen in Figs. 2.9-2.11 has discouraged us from using transforms. 25

23Current programs [55] work well for 〈Δ〉 < 5 MeV or 〈Δ〉 < T/20. One function is calculated in less than 1 s.24For an assessment of the validity of Vavilov functions, the reader may wish to look at Fig. 2.15 in [34]. In

general Vavilov-Fano calculations will agree with convolutions for κ > 0.02.25Analytic methods to determine straggling for various analytic single collision spectra are described in great

detail in the book by Sigmund [46].



φ [y,T]

φ [y+ξ,T]

V [ξ,Δ,T]

δT

Δ

T2T1

T

T

Fig. 2.22. Illustration of the convolution method. The initial energy spectrum φ(y; T ) is shown at thetop. For each narrow band of energies δT (shown by the vertical parallel lines) the straggling spectrumV (ξ, Δ, T + Δ) is added to the spectrum φ(y + ξ; T ) shown at the bottom. T decreases to the right.

2.7.3 Analytic methods for thick absorbers

These are absorbers in which the kinetic energy T of the particles changes considerably inside theabsorber, Fig. 2.21. Straggling functions for thick absorbers can be calculated if the energy loss(straggling) spectrum V (ξ,Δ, T + Δ) for thin absorbers of thickness ξ is known for all T and ξ. 26

For one track segment ξ a convolution integral for each energy interval T, T + δT between T1 andT2 is calculated (s is the range in Δ of V (ξ,Δ, T + Δ), Fig. 2.22)

φ(y + ξ;T ) =∫ s

0

φ(y, T + Δ) × V (ξ,Δ, T + Δ) dΔ, or (2.38)

φ(y + ξ;T − Δ) =∫

φ(y, T ) × V (ξ,Δ, T ) dΔ, (2.39)

where V (ξ,Δ, T ) is calculated with the analytic methods. These equations can be considered astransport equations. The process is illustrated in Fig. 2.22 This method is of the order of 50 timesfaster than corresponding MC calculations.

2.8 Evaluations and properties of straggling functions

Suitable methods must be chosen for different applications. One of the classification that can beused is to sort according to absorber thickness, as is evident from Figs. 2.23-2.26. Four classifica-

26For present purposes (e.g. total range of 200 MeV protons [47] or 3000 MeV C-ions [78] in water) Vavilov-Fanostraggling functions V are used.



Δ [eV]

f [Δ]

1 mm

2 mm

1.0

0.8

0.6

0.4

0.2

0.010 20 50 100 200 500 1000 2000

3 mm 4 mm

5 mm

c

d

Fig. 2.23. Straggling functions for singly charged particles with βγ = 4.48 traversing segments of lengthx = 1 to 5 mm in Ar. The total collision cross-section Σt is 30 collisions/cm. The functions are normalizedto 1.0 at the most probable value. The broad peak at 17 eV is due to single collisions, see Fig. 2.19. For2 collisions it broadens and shifts to about 43 eV, marked c, and for n = 3 it can be seen at d. It maybe noted that the peak at 11.7 eV (if the function is normalized to unit area) is exactly proportional tomce

−mc , as expected from Eq. (2.33). Energy losses to L-shell electrons of Ar, at 250 eV in Fig. 2.10,appear at e, for x = 1 mm they have an amplitude of 0.04. The δ-function at Δ = 0, n = 0, Eq. (2.34), isnot shown. For x = 1 mm, it would be e−mc = 0.05. For x > 2 mm peak c has disappeared, peak d is thedominant contribution and defines the most probable energy loss Δp. The buildup for peak e at 440 to640 eV is the contribution from L-shell collisions. It appears roughly at 250 eV plus Δp. The total crosssection for collisions with E > 250 eV is only 1.7 collisions/cm, thus the amplitude of the peak e is roughlyproportional to x. The Bethe mean energy loss is 250 eV/mm.

tions are described below. Also it is interesting to find for what applications the Rutherford crosssection is sufficient.

2.8.1 Very thin absorbers

Micro pattern detectors, e.g. gas electron multipliers (GEM), permit the observation of tracksegments of the order of 0.1 mm [4, 24]. With mean free paths λ of the same order of magnitude(e.g. for βγ ∼ 0.5, i.e. 50 keV electrons or 100 MeV protons, λ = 0.1 mm in P10 gas, Table 2.2),there will be only one collision on the average per GEM [79]. Therefore the structure of the singlecollision spectrum (Figs. 2.9-2.11 and 2.13-2.16) will cause a large variation of the energy lossspectra for small numbers mc, Eq. (2.32). Examples of such spectra are given in Fig. 2.23 [13].

For Si, a spectrum for particles with βγ = 2.1 traversing x = 1 μm is given in Fig. 2.24. Suchspectra have been described earlier [7, 74] and have been measured [8]. 27 Structures of this typeare also observed in measurements of straggling effects on resonant yields of nuclear reactions,called the “Lewis effect” [80].

Track segments for which mc < 50 must be considered to be very thin, and only the convolutionmethod will produce reliable spectra. Examples: for particles with the smallest values of M0, (i.e.

27The measurements by Perez et al. [8] were made with electrons in Al and agree qualitatively with Fig. 2.24.This paper is a good example of the problems of analytic methods.



Δ [eV]

f [Δ]

2 GeV protons1 μm Si

100

80

60

40

20

00 50 100 150 200 250 300

Fig. 2.24. Straggling in 1 μm of Si (mc = 4), compared to the Landau function (dashed line), see Sect. B.5for explanation. The Bethe-Fano mean energy loss is 〈Δ〉 = 400 eV. Measured straggling functions of thistype are given in [8].

βγ ∼ 15 for Si, βγ ∼ 4 for Ne and P10 gas (see Tables 2.1 and 2.2) very thin thicknesses arexv(Ne) < 40 mm, xv(P10) < 15 mm, xv(Si) < 10 μm. From Figs. 2.23 and 2.24 it is evident thatno simple description (e.g. “Landau type functions”) can be given for straggling functions for verythin absorbers. 28

2.8.2 Thin absorbers

If the mean number mc of collisions is greater than about 50, the straggling functions begin toapproach the Landau-Vavilov shape, but are broader. Again, only the convolution method willproduce reliable spectra. An example is given in Fig. 2.25. It is possible to approximate thesespectra with Vavilov functions by taking into account atomic binding, as shown in Fig. 2.24 andoutlined in [34]. 29 The clearest two-parameter description for the spectra is given by the mostprobable energy loss Δp and the width w of f(Δ). A dependence of Δp on x was derived byLandau [2] and can be seen in Fig. 2.26.

The functions f(Δ) customarily are called “straggling functions” [58] or straggling spectra,except in high energy physics where they are called “Landau functions” in a generic sense. Here,“Landau function” is used only to designate the function described in [2, 81] and shown in Figs. 2.24,2.25 by the dotted line. The upper limit for “thin” absorbers is given by the sensitivity of theconvolutions to the change in speed of the particles in the absorber, e.g. if the mean energy lossexceeds 5 or 10% [48].

28More specific indications about the use of Landau functions can be seen in Fig. 2 of [7], Fig. 15 of [34], Fig. 91.6of [74].

29The Vavilov program is about four times faster than the convolution program, which takes 0.7 sec (1.7 GHzmachine) for one calculation with Eq. (2.37).



Δ [keV]

f [Δ]

[1/ke

v]

<Δ>

Δp

w

0.6

0.4

0.2

0.01 2 3 4 5

Fig. 2.25. The straggling function f(Δ) for particles with βγ = 3.6 traversing 1.2 cm of Ar gas (mc =30 · 1.2 = 36) is given by the solid line. It extends beyond Emax ∼ 2mc2β2γ2 = 13 MeV. The function ismuch broader than the Landau function [81, 82], given by the dotted line. Parameters describing f(Δ) arethe most probable energy loss Δp(x; βγ), i.e. the position of the maximum of the straggling function, at1371 eV, and the full-width-at-half maximum (FWHM) w(x; βγ)=1463 eV. The Bethe mean energy loss is〈Δ〉 = 3044 eV. The mean energy loss for the part of the spectrum shown is 〈Δs〉 = 1724 eV. The medianenergy loss for the part of the spectrum shown is 〈Δm〉 = 1672 eV. The peak of the Landau function is at1530 eV. See Sect. B.7 for an explanation of the difference between the two functions.

Δ/x [eV/cm]

f [Δ/

x]

<Δ/x>

x in Ar

1.0

0.8

0.6

0.4

0.2

0.00 1000 2000 3000 4000

1 cm2 cm4 cm8 cm

Fig. 2.26. Calculated energy loss straggling functions f(Δ/x) [note that Δ/x is called dE/dx in currentliterature of particle physics] for βγ = 3.6 particles, traversing segments x = 1, 2, 4, 8 cm in Ar. Theabscissa is Δ/x. For the full spectrum extending to 13 MeV, the mean energy loss < Δ/x > is the samefor all x. The trend of Δp/x with particle speed is shown in Fig. 2.17. For Si see Fig. 27.7 of [24]. In orderto show the change in shape clearly the functions are scaled to the same peak height.



2.8.3 Thick absorbers

If the particles stop in the absorber the length of the track is called the range. 30 The methoddescribed in Sect. 2.7.3, Eq. (2.30), can be used for any absorber thickness if suitable thin stragglingfunctions V (ξ,Δ;T ) are used. For the practical implementation a large number of V (ξ,Δ;T ) canbe calculated initially for several values of ξ for several ranges of T . An alternative is to use ascaling procedure, Sect. 2.8.2 in [13] to relate different V (ξ,Δ;T ). If the Vavilov-Fano functionscan be used, the mean energy loss values and their distribution functions for a full track should bea good approximation.

Tschalar [48] used the Rutherford cross section and its moments (Appendix B.1) to calculateenergy loss straggling functions for thick absorbers. Good agreement with experimental measure-ments has been found [83].

2.8.4 Comparisons of spectra

For the analysis of ionization data from detectors it is desirable to use straggling functions cal-culated with Eq. (2.33) rather than Gaussians or “parameterized Landau functions”, Fig. 2.27.Functions calculated with the Bethe-Fano method, Sect. 2.5.1.1, agree with measurements to bet-ter than 1% [34]. They should be used to diagnose the operation of the detectors and to calibratethe ionization signals from them absolutely. 31 Because of the errors in the calculation of M0,Tables 2.3 and 2.4, the FVP method is somewhat less suitable for this purpose.

Δ [keV]

π, pc=500 MeV/cx=1.2 cm Arw

f [Δ]

[1/k

eV]

<Δ>

Δp

0.6

0.4

0.2

0.01 2 3 4 5

Fig. 2.27. Energy loss straggling function f(Δ) from FVP theory for particles with βγ = 3.6 traversingsegments of Ar gas with x = 12 mm: solid line. The most probable energy loss is Δp, the median valueis 1.74 keV and the mean energy loss is 〈Δ〉. The dotted line is a “Landau function” (Appendix B.5)parameterized (Examples of the parameters needed are given in Fig. 15 of [34]) to fit at the points definingthe FWHM w. This is a fit similar to the one obtained from ROOT [85].

30The mean track length of a particle beam is frequently called the csda range. The projected range is the meandistance traveled by the particles along the incident direction of the beam.

31Calibrations with radioactive sources can be used as corroboration.



2.9 Energy deposition

The secondary radiations produced in the energy loss process (delta rays, Auger electrons, Brems-strahlung, Cherenkov and transition radiations) will deposit their energy at some distance from thelocation of the collision. Clearly the effects will depend on the volume under observation [86, 87].In some detector geometries typical dimensions are of a 0.1 mm scale [4, 24].

The effective range of the secondary radiation will be needed for the calculation of energydeposition. Some data are given in the following sections. Usually few of the secondary radiationsare produced, in particular note that large energy losses are infrequent, Fig. 2.16.

2.9.1 Ionization

Energy deposition in gases can result in ionization by the formation of electron-ion pairs. In somesolids, electron-hole pairs can be produced. The process is quantified by determining an averageenergy W to produce a pair [30]. 32 The processes are complex [28, 74]. In particular, for mixedgases (P10, ALICE-TPC mix) the possibility of Penning ionization requires the inclusion of excitedstates of the gas atoms in the calculation of collision cross sections. Furthermore it has been foundthat W increases with decreasing particle speed [30, 88]. We have been discouraged by results of”calculations” of W [89]. It is customarily believed that w(β) is constant for β � 0.1. I estimatethe uncertainty of this assumption to be at least ±2% for gases. It may be less for Si [34].

2.9.2 Delta rays

If in an inelastic collision of a particle with an absorber the energy loss E exceeds the ionizationpotentials Jj of the atoms a secondary electron with energy Eδ = E − Jj is emitted. Such anelectron usually is called a delta ray. Evidently delta rays will also collide in the absorber. Ifradiation effects are observed in small volumes (e.g. GEM detectors) some of Eδ may be depositedelsewhere [86].

For energy losses E greater than about 10 keV the collision cross section can be calculatedwith the Rutherford approximation, using Eqs. (2.8) or (2.45). This can be seen in Figs. 2.9 and2.11. Estimates can also be obtained from Φ(E) in Figs. 2.13 to 2.16. As shown in Fig. 2.1 thetracks produced by delta rays will have no regularity. An analytic treatment (such as the restrictedenergy loss, Appendix A.3) may not have much meaning. The practical range of electrons [27, 90]may help in assessing delta ray problems. Values are given in Table 2.6.

The energy deposition by low energy electrons is a complex process [92]. For electrons withT<50 keV the energy deposition extends over a spherical volume tangent to the point of entranceinto the absorber, Fig. 2.2. The diameter of the sphere is of the order of the “practical range,”considerably smaller than the path-length (“continuous-slowing-down-approximation” or CSDArange) calculated with the stopping power [52]. A useful reference is [91].

32In the literature various symbols such as w, ε are also used.



Table 2.6. Practical ranges R (cm for gases, μm for Si) of electronswith kinetic energy T in P10 gas, Si and Ne. The density, ρ =1.56 mg/cm3 for the P10 mixture (90% Ar, 10% CH4), for Ne ρ =0.91 mg/cm3 at 740 torr and 293 K is used. Ranges Rc calculatedwith CSDA for P10 are also given [52]. For E > 50 keV, effectiveranges were calculated with the algorithm given in [91]. For E < 10keV, experimental effective ranges measured for nitrogen [92, 93]were assumed to be the same for argon (with an uncertainty of10%). Between 10 and 50 keV, calculated ranges ([91]) were reducedsmoothly to the experimental value at 10 keV. The uncertainty ofR is about 20%.

T [keV] R(P10) Rc R(Ne) R(Si)

0.1 0.0002 - 0.0003 0.0040.2 0.0004 - 0.0007 0.0080.4 0.0010 - 0.0017 0.0150.7 0.0018 - 0.0030 0.0301.0 0.0031 - 0.0053 0.042.0 0.01 - 0.017 0.104.0 0.03 - 0.05 0.247.0 0.08 - 0.13 0.5510 0.14 0.25 0.24 0.9620 0.42 0.83 0.9 3.040 1.5 2.74 2.6 1070 4.0 7.20 6.7 27

100 7.3 13.1 12 50200 22 40.4 38 160400 63 115 112 450700 136 244 248 960

1000 215 381 396 1520

2.9.3 Auger electrons and photons

Energy losses to inner shells i of atoms will leave vacancies in the shell. If the energy loss E exceedsthe ionization energy Ii of shell i, a delta-ray with energy E−Ii will be produced. The probabilitiesfor these energy losses are small, see Figs. 2.13-2.16. The subsequent electron-rearrangement of theatom can produce the emission of an X-ray or an electron (Auger electron) [94]. For the K-shellin Ne about 2% of the vacancies emit an X-ray, in Si it is 5%, in Ar ∼ 10%. The energy of theemitted particle will be EA = JK −JL, with another Auger electron with EA = JL −JM . Becausethese collisions are infrequent a MC calculation is needed for a simulation of the processes in smallvolumes, also see Sect. 2.9.1.

2.9.4 Cherenkov radiation

If in a material with index of refraction n, the speed v of a particle exceeds the speed of lightcm = c/n, the energy loss of the particle may be converted into photons. This secondary radiationis called Cherenkov radiation CR [95].

An explanation of Cherenkov radiation was given by Fano [20]. Consider the transverse excita-



tions given by Eq. (2.21). If in tanΘ = β2ε2/(1 − β2ε1) the term (1 − β2ε1(E)) is negative, i.e.β2ε1(E)) > 1, a photon with energy ≈ E is emitted at an angle ψ given by cos2ψ = 1/β2ε(E). 33

Data for P10 gas at 1 atm have been extracted from calculations with Eq. (2.22): CR will appearat βγ ∼ 28 with energy E = �ω = 11 eV. At βγ ∼ 100 the range of energies is from 9 to 16 eV, andthe total cross section is about 3% of M0 (Table 2.2), for βγ ∼ 40 it is 1%. The energy loss to CRis 0.1% to 0.4%. This effect is a small contribution to a straggling function, Fig. 2.27. The relationof Cherenkov radiation and density effect is discussed by Crispin and Fowler [96]. Applications toparticle identification are discussed in Sect. 2.3.3.

2.9.5 Transition radiation

Transition radiation is produced at the interface between a dense and a dilute material. It isespecially useful for the detection of particles with high speeds, say γ � 1000.

A description of its theory and practical use is given by Dolgoshein [97]. Its use for particleidentification is presented in Sect. 2.3.3.

2.9.6 Ion beams

Ion beams are mainly used in two applications: radiation therapy and measurements of stoppingpowers and ranges. We consider the direct observation of the energy deposition and its relation tothe energy loss. Consider the particle fluence

Φ(T ;x) = dN/dA (2.40)

where dN is the number of ions crossing an area dA perpendicular to the beam direction. Severalenergy loss processes and their relation to energy deposition have been considered or calculated:

• Multiple scattering. If the beam area is much larger than the lateral spread due to multiplescattering [98] a correction is only needed for the projected track length [99]. For a narrowbeam (“pencil beam”) a more detailed simulation must be made [100].

• Inner shell excitations. The decay to the ground state of the atom can result in the emissionof a photon (“fluorescence radiation”) or an Auger electron (probability for Z < 20 morethan 80%, for Z = 30 about 50%, for Z > 60 more than 90%, [9, 94]).

• Large secondary electron losses.

2.10 Particle ID

If charged particles traverse an absorber in a magnetic field, the momentum of a particle canbe measured from the curvature of its track. Since the energy loss depends only on the particlespeed, the mass of the particle can then be determined. Specific approaches to optimizing themeasurement of the energy loss are given in [13]. Other descriptions for these measurements aregiven in [14].

33In ref.[56] Θ is given as Θ = arg(1 − ε1β2 + iε2β2).



2.11 Discussion and recommendations

The evaluation of the interactions of radiations with matter is done at present with computerprograms [55], many based on Monte Carlo calculations, such as GEANT 4 [101], PENELOPEand many others. In principle though it is probably more appropriate to calculate intermediateresults or functions with computer analytic methods, then use tabulated functions for furtheranalysis of a given process. If a MC simulation is used and if computer analytic calculations arepossible, it would be a good idea to compare results of the two methods. Example: simulate thetruncated mean of the energy loss of particles of a given speed for a segmented track (e.g. ina TPC). First step: calculate the probability density functions for the energy losses analytically(Figs. 2.13-2.16), then use a MC simulation to obtain the spectra for the truncated mean [13].

It must be kept in mind that evaluated quantities, such as stopping power, range, classicalstraggling parameters can only give qualitative information for a single particle. The actual processis of the nature shown in Fig. 2.1. For thin segments the energy loss function is best describedby the most probable energy loss and the width of the straggling function, Figs. 2.23-2.27. For asingle particle there will be one single point on the abscissa of e.g. Fig. 2.27, located between 1and 2 keV with probability of 50%. For very thin segments there is even less information available:Fig. 2.23.

2.12 Appendix

A Stopping power and track length

A.1 Stopping power M1

Bethe [19] derived the method described in Sects. 2.5.1.1 and 2.5.1.2. for atoms and calculated thefirst moment M1(v), Eq. (2.25), by using sum rules and defining the logarithmic mean excitationenergy I. The nonrelativistic result is [54]

dE

dx≡M1(v) =

kN

β2Z ln

EM · 2mc2β2

I2(2.41)

with I given by

Z ln I =∫

f(E, 0) lnE dE (2.42)

It must be kept in mind that dE/dx is an average of the energy losses per unit segment lengthfor single particles, Δ/x, as shown in Fig. 2.1. From Fig. 2.23-2.26 it is evident that dE/dx has alimited use for thin absorbers.

The use of the sum rules introduces an error which can be corrected with the shell corrections[54]. Other effects are treated with the Barkas and Bloch corrections etc. [102, 103, 104, 105]. Thedensity effect, Cherenkov radiation etc. are implicit in Eq. (2.21), Sect. 2.9.4 [20].

In practice there will be little need to calculate M1(v) for the uses described in this book.Tables for protons and alpha particles for many absorbers are available [66, 21], also for muons[10].

A concept often used in particle physics is “minimum ionizing particles”. This usually refersto the minimum value of M1(βγ). For Si this minimum is at βγ ∼ 3.5, but for Σt = M0 it is atβγ ∼ 16, and for Δp/x it is at βγ ∼ 4.5 (Table 2.1), for gases near βγ ∼ 3.5, Table 2.2.



Stopping power for electrons and positrons need a different analytic expression than heavy ions.Equations and tables are given in ICRU-37 [52], also see Sect. 2.6.

A.2 Total collision cross section M0

Analytic calculations for M0 with the Bethe-Fano equations given in Sects. 2.5.1.1 and 2.5.1.2have been made [28, 53, 106, 107, 108]. Because of the complexity of the optical data, no simpleexpression similar to Eq. (2.41) is available. The analytic calculations differ considerably from theFVP calculations, Sect. 2.5.2.1. Calculations of M0 (also called inverse mean free path) have beenmade by several groups [21] (ORNL, Barcelona) with computer-analytic methods.

A.3 Restricted energy loss

The concept of restricted energy loss was introduced in the consideration of biological effects ofradiations. Conceptually it is an energy deposition rather than an energy loss and therefore mustbe defined for a limited volume around a particle track. The customary definition, Eq. (27.7) in[24]

M1(v) ≈ k

β2z2Z

12

lnEcut · 2mc2β2γ2

I2(2.43)

is valid for a cylindrical volume around the track with a radius given by the practical range ofelectrons with energy Ecut, see Table 2.6, Sect. 2.9.2. Because of the small number of energeticdelta rays, Fig. 2.16, a MC calculation should be used for practical applications.

A.4 Ranges

Mean track lengths t (also called path lengths) for particles with initial kinetic energy Tu and finalmean energy Tl can be calculated with the “CSDA approximation”

t =∫ Tu

Tl

dT ′

dT ′/dx(2.44)

where dT/dx represents M1(v) of Eq. (2.41). Customarily if Tl = 0, t is called the particle range.Corrections for multiple scattering must be considered [99, 109], and nuclear interactions willreduce the particle fluence, see Sect. 2.3.4. Range data for muons can be found in [10], for pions,protons and alpha particles in [54, 66].

If Tl is finite, there will be an energy spectrum [110, 78] which does not correlate well with thetrack-straggling, see Fig. 2.4, Sect. 2.3.3.

B Rutherford type cross sections

B.1 Rutherford cross section and moments

The Rutherford cross section is given by Eq. (2.8). In order to calculate the moments, Eq. (2.25),integration limits must be defined. The upper limit must be EM given in Eq. (2.2), a lower limitEm, defined e.g. in Appendix B.4, is used.

M0(v) =k′

β2

[

(1Em

− 1EM

) − β2

EMlnEM

Em

]

∼ k′

β2

1Em

(2.45)



M1(v) =k′

β2

[

lnEM

Em− β2

EM(EM − Em)

]

∼ k′

β2

[

lnEM

Em− β2

]

=k′

β2

[

lnE2

M

I2− β2

]

(2.46)

From Figs. 2.9-2.11 it is clear that no plausible values of Em can be defined from knowledge aboutatomic structure.

M2(v) =k′

β2

[

(EM − Em) − β2

2EM(E2

M − E2m)

]

∼ k′ · 2mc2 1 − β2/21 − β2

(2.47)

where k′ = kNAZ/A = 0.1535 Z/A MeV cm2/g for an absorber Z,A.The Landau [2] method to deal with Em is given in Sect. B.4.

B.2 Rutherford cross section with shell structure

Inokuti [53] suggested an approximation by modifying the Rutherford cross section for each electronshell s with νs electrons as follows:

σs(E) = σR(E) · νs · (1 +4U3E

) (2.48)

to be used for E � Js where Js is the ionization energy of shell s and with a parameter U similar tothe average kinetic energy of the electrons in the shell. An application of this approach is describedin Sect. III.B of [34]. This structure appears in Fig. 2.9 at least for the K-shell. It is likely thatthe use of this approximation would improve the FVP method shown in Fig. 2.8 and Eq. (2.24),see ref. [35].

B.3 Modified Rutherford cross section

In the “ALICE Technical Design Report of the Time Projection Chamber” [63], a modified Ruther-ford cross section was used, “AliRoot MC”

σA(E; v) =κ(v)β2E2.2

(2.49)

with the restriction EI < E < Eu, where EI is the ionization energy of the gas and Eu is chosenarbitrarily to be 10 keV. κ(v) is a parameter chosen to give the value Σt(βγ = 3.6) = 15 colli-sions/cm as suggested in [63, Fig. 7.1]. This cross section is shown by the dotted line in Fig. 2.11and κ(v)/E2.2 appears to be a reasonable approximation to the FVP σ(E; v). Note that theRutherford cross section differs by its slope. The exact choice of Eu is not critical as the readermay find by calculating Σt;A with Eq. (2.45). Further details are described in [55, 112].

To get a better approximation for Σt(v) for the scheme proposed in [63] the “Bethe-Blochcurve” was used to define the function Σt;A(v). This function was obtained from measurements ofparticle tracks in P10 (mainly by Lehraus et al., [113]). It is approximated by

f(βγ) =P1

βP4{P2 − βP4 − ln[P3 +

1(βγ)P5

]} (2.50)

and is shown as L[P10] in Fig. 2.17. Clearly this function also differs substantially from Σt;F (v).In addition it differs conceptually from Σt(v). The reason for the difference is that Eq. (2.50) wasdetermined from experimental measurements of truncated mean values < C > for tracks. It alsodiffers from the Bethe-Fano function M1(βγ) = dE/dx(βγ) shown in Fig. 2.17.

Monte Carlo calculations of straggling functions with the model of ref. [63] are compared tothe analytic calculation with FVP σ(E) in Fig. 2.28. The differences seen are explained by thedifferences in the σ(E), Fig. 2.11. Predominant is the excess of σ(E) over σA(E) for 30 < E(eV ) <300 resulting in the shift to greater Δ of the FVP function.



βγ=3.6, x=7.5 mm

80000

60000

40000

20000

00.0 0.5 1.0 1.5 2.0

βγ=3.6, x=7.5 mm

Δ [keV]

f [Δ]

Fig. 2.28. Straggling functions for particles with βγ = 3.6 traversing Ne segments with x = 7.5 mm. Solidline: analytic FVP calculation, Σt = 13.3 collisions/cm. × Monte Carlo calculation with ALICE TPCalgorithm, with Σt = 15/cm. Even though the Σt is somewhat larger, the MC calculation has a smallerΔp. A Monte Carlo calculation with FVP σ(E) gives the same function as the analytic calculation [55].The K-shell energy losses above Δ = 1.3 keV do not appear in the MC calculation.

We see that the AliRoot [63] Monte Carlo calculation does not produce accurate stragglingfunctions. Its attractiveness is that the functions used are analytic.

A modified approach which is just as simple as the current method but will produce moreaccurate straggling functions consists of

• replacing in Fig. 2.17 the function L[P10] by Σt[Ne]

• replacing in Fig. 2.15 the function given by the dotted line (from Eq. (2.49) by the FVPfunctions. A single tabulated function might be sufficient [55, 114].

B.4 Landau-Vavilov-Fano Laplace transform method

The straggling functions calculated by Landau and Vavilov [2, 3] were based on two approximations:the use of the Rutherford spectrum, Eq. (2.8), for the collision cross section and the requirementthat the stopping power, Eq. (2.46), (written as α, Eq. 4 in [2]) give the Bethe stopping power,Eq. (2.41). In order to achieve this, Em was chosen as 34

Em =I2

EM(2.51)

The value for M0 then is M ′0 = k′

β2 · I2

EM. It is larger than the Bethe-Fano value.

Example: Ar, βγ = 3.6 β2 = 0.93, I = 190 eV, ρ = 0.0016 g/cm3, k′/β2 = 120 eV/cm. ThenEm = 0.0027 eV, M ′

0 = 44000 col/cm and M ′1 = 2.5 keV/cm. Clearly M ′

0 is much greater thangiven in Table 2.2, while M1 is quite close. This is plausible from a look at Fig. 2.10. Since theconvolution method, Eq. (2.32), is equivalent to the Laplace transform method [7], the relativevariance of the Poisson distribution, m−1/2

c , as well as that of the convolutions, Fig. 2.19, will be34For the Vavilov [3] result this will result in the correct mean value for the straggling functions if the number of

collisions is large enough, Fig. 2.15 in [34].


References for 2 2–39

much smaller for the Rutherford cross section, and the Landau functions are narrower, as seen inFig. 2.25.

Functions calculated with modifications taking atomic binding into account [20, 115, 116, 117]are too wide, Fig. 2.15 in [34]. The calculation of the correction term D in [117] 35 is incorrect. Thecorrect term is given by Fano (p.42[20]), [34]. If account is taken of the limit on E in Eq. (2.34)(E must be smaller than Δ in the calculation of δ2 [34]) the Vavilov-Fano functions are fairapproximations for thin absorbers, Figs. 11-13 in ref. [34].

As far as calculation time is concerned they are similar to the convolution calculations. Becausethe mean energy loss ξ is used as a parameter this method has the advantage that it will be quiteaccurate for thicker segments. Because of the change in particle speed v, the upper limit of validityof all thin layer methods is reached if the mean energy loss exceeds 5 or 10%, see Fig. 2 of [48].

References for 2

[1] N. Bohr, Phil. Mag. xxv(1913) p.10, and Phil. Mag. xxx(1915) p.581.[2] L. Landau, J. Phys. VIII (1944) 201.[3] P. V. Vavilov, Soviet Physics JETP 5 (1957) 749.[4] F. Sauli, Nucl. Inst. Meth. A 580 (2007) 971, and Ann. Rev. Nucl. Part. Sci. 49 (1999) 341.[5] C. Brion - http://www.chem.ubc.ca/personnel/faculty/brion/.[6] J. Cooper - ftp://ftp.chem.ubc.ca/pub/cooper/nitrogen/.[7] H. Bichsel and R. P. Saxon, Phys. Rev. A 11 (1975) 1286.[8] J. Ph. Perez, J. Sevely and B. Jouffrey, Phys. Rev. A 16 (1977) 1061.[9] R. D. Evans, The Atomic Nucleus, McGraw Hill (1967)

[10] D. E. Groom, N. V. Mokhov and S. I. Striganoff, Atomic Data and Nucl. Data Tables 78(2001) 183.

[11] H. H. Andersen, C. C. Hanke, H. Sorensen and P. Vajda, Phys. Rev. 153 (1967) 338.[12] A. Giuliani and S. Sanguinetti, Materials Science and Engineering R11 (1993) 1.[13] H. Bichsel, Nucl. Inst. Meth. A 562 (2006) 154-197.[14] M. Harrison, T. Ludlam and S.Ozaki, eds., Nucl. Inst. Meth. A 499 (2006) 1-880.[15] C.Tschalar and H. Bichsel, Phys. Rev. 175 (1967) 476, and C.Tschalar, Ph.D. dissertation,

University of Southern California, 1967.[16] H. H. Andersen, J. F. Bak, H. Knudsen and B. R. Nielsen, Phys. Rev. A 16 (1967) 338.[17] N. Shiomi-Tsuda, N. Sakamoto, H. Ogawa and U. Kitoba, Nucl. Inst. Meth. B 159 (1999)

123.[18] J. J. Thomson, Phil. Mag. xxiii(1912) 449.[19] H. Bethe, Ann. Phys. 5 (1930) 325.[20] U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1.[21] http://physics.nist.gov/PhysRefData/Star/Text/contents.html.[22] E. Fermi, Z. Phys. 29 (1924) 3157, and Nuovo Cimento 2 (1925) 143.[23] Ohya, J. Phys. Soc. Japan 60 (1991) 4089.[24] Review of Particle Physics, Phys. Lett. B667 (2008) 1, and http://pdg.lbl.gov.[25] Z. Chaoui, Appl. Phys. Let. 88 (2006) 024105 and Z. Chaoui and H. Bichsel, Surface and

Interface Analysis 86 (2006) 664.

35D corresponds to the term K2r in [115] and δ2 [34].


2–40 References for 2

[26] N. J. Carron, An Introduction to the Passage of Energetic Particles through Matter. Taylorand Francis ( A division of Harcourt) 2002.

[27] F. H. Attix and Wm. C. Roesch, Eds, Radiation Dosimetry, 2nd ed., Academic Press, NewYork, London (1968).

[28] Atomic and molecular data for radiotherapy and radiation research Final report of aco-ordinated research programme, IAEA-TECDOC-799, May 1995, IAEA Vienna

[29] B. J. Bloch and L. B. Mendelsohn, Phys. Rev. A9 (1974) 129.[30] ICRU report 31, Average energy required to produce an ion pair, Inter. Commission on

Radiation Units, 7910 Woodmont Ave., Washington D.C. 20014 (1979).[31] U. Fano and J. W. Cooper, Rev. Mod. Phys. 40 (1968) 441.[32] J. Berkowitz, Atomic and molecular photo absorption, Academic Press ( A division of

Harcourt) 2002.[33] M. C. Walske, Phys. Rev. 101 (1956) 940 and Phys. Rev. 88 (1952) 1283.[34] H. Bichsel, Rev. Mod. Phys. 60 (1988) 663.[35] J. M. Fernandez-Varea, F. Salvat, M. Dingfelder and D. Liljequist, Nucl. Instr. Meth. B

229 (2005) 187-217.[36] D. Y. Smith, M. Inokuti, W. Karstens and E. Shiles, Nucl. Instr. Meth. B 250 (2006) 1.[37] W. Zhang, G. Cooper, T. Ibuki and C.e. Brion, Chem. Phys. 137 [1989] 391.[38] L.C. Lee, E. Phillips and D.L. Judge, J. Chem. Phys. 67 [1977] 1237.[39] I. Smirnov: projects-CF4-HEED cf4photoabs.dat, http://ismirnov.web.cern.ch/ismirnov/heed,

e-mail: [email protected].[40] S. Kamakura et al., J. of Applied Physics 100 (2006) 064905.[41] G. F. Bertsch. J.-I. Iwata, A. Rubio and K. Yabana, Phys.Rev. B 62 (2000) 7998.[42] J. J. Rehr and R. C. Albers, Rev. Mod. Phys. 72 (2000) 621.[43] E. D. Palik, Handbook of optical constants of solids, three volumes, Academic Press, New

York, 1985, 1991 and 1998.[44] H. Bichsel, Passage of Charged Particles Through Matter, ch. 8d in Amer. Inst. Phys.

Handbook, 3rd ed., D. E. Gray, ed. (McGraw Hill, 1972).[45] C. Leroy and P.-G. Rancoita, Principles of Radiation Interaction in Matter and Detection,

World Scientific (2004)[46] P. Sigmund, Particle Penetration and Radiation Effects, Springer (2006).[47] I. Gudowska, private communication (2009).[48] Ch. Tschalar, Nucl. Inst. Meth. 61 (1968) 141.[49] E. A. Uehling, Penetration of of heavy charged particles in matter, Ann. Rev. Nucl. Sci. 4

(1954) 315.[50] E. Fermi, Phys. Rev. 57 (1940) 485.[51] D. Bote and F. Salvat, Phys. Rev. A 77 (2008) 042701.[52] ICRU report 37, Stopping Power of electrons and Positrons, International Commission

on Radiation Units and Measurements, 7910, Woodmont Ave., Bethesda, MD 20814.Washington D.C. 20014 (1979).

[53] M. Inokuti, Rev. Mod. Phys, 43, 297 (1971), and 50, 23 (1978).[54] H. Bichsel, Phys. Rev. A 65 (2002) 052709, Phys. Rev. A 46 (1992) 5761 and Phys. Rev.

A 23 (1982) 253.[55] H. Bichsel, http://faculty.washington.edu/hbichsel.[56] W.W.M. Allison and J. H. Cobb, Ann. Rev. Nucl. Part. Sci. 30 (1980) 253.[57] D. Liljequist, J. Phys. D 16 (1983) 1567.[58] E. J. Williams, Proc. Roy. Soc. A 125 (1929) 420.


References for 2 2–41

[59] H. Bichsel, Scanning Microscopy Supplement 4 (1990) 147. The values of M0 are in unitsof collisions/10nm (not collisions/μm).

[60] J. W. Gallagher, C. E. Brion, J. A. R. Samson and P. W. Langhoff, J. Phys. Chem. Ref.Data 17 (1988) 9.

[61] H. Raether, “Excitation of plasmons and interband transitions by electrons”, SpringerVerlag, NY. (1980).

[62] W. Blum, L. Rolandi, Particle Detection with Drift Chambers, second printing, Springer-Verlag, (1994), and W. Blum, L. Rolandi, M. Regler, third printing, Springer-Verlag, (2007).

[63] ALICE Technical Design Report of the Time Projection Chamber, CER/LHCC 2000-001,ALICE TDR 7 (7 January 2000).

[64] H. Bichsel, Phys. Rev. B 1 (1970) 2854.[65] H. Bichsel, Nucl. Inst. Meth. B 52 (1990) 136.[66] ICRU report 49, Stopping Powers and Ranges for Protons and Alpha Particles, Interna-

tional Commission on Radiation Units and Measurements, 7910 Woodmont Ave., Bethesda,MD 20814 (1993).

[67] F. Salvat and J.M. Fernandez-Varea, Metrologia 46 (2009) S112.[68] W. Heitler, The Quantum Theory of Radiation, second edition, Oxford University Press

(1949).[69] A.N. Kalinovskii, N.V. Mokhov and Yu.P. Nikiyin, Passage of High-Energy Particles

through Mstter, American Institute of Physics, New York (1989).[70] A. M. Kellerer, G. S. F. Bericht B-1, Strahlenbiologisches Institut der Universitat Munchen

(1968).[71] J. R. Herring and E. Merzbacher, J. Elisha Mitchell Sci. Soc. 73 (1957) 267.[72] J. J. Kolata, T. M. Amos and H. Bichsel, Phys. Rev. 176 (1968) 484.[73] J. F. Bak, et al, Nucl. Phys. B 288 (1987) 681[74] H. Bichsel, Charged-Particle-Matter Interactions, Ch. 91 in Springer Handbook of Atomic,

Molecular, and Optical Physics, G. W. F. Drake (Ed.), 2nd ed., Springer (2006).[75] I. Gudowska et al., Phys. Med. Biol. 49 (2004) 1933, and I. Gudowska and N. Sobolevsky,

Radiation Measurements 41 (2006) 1091.[76] M. J. Berger and S. M. Seltzer. Studies in penetration of charged particles in matter, Nat.

Acad. Sci.-Nat. Res. Council Publication 1133 (1964).[77] M. J. Berger, Monte Carlo Calculations of the Penetration and Diffusion of Fast Charged

Particles, Methods in computational physics 1 (1963) 135.[78] H. Bichsel, T. Hiraoka and K. Omata, Rad. Res. 153 (2000) 208.[79] H. Bichsel, Advances in Quantum Chemistry 46 (2004) 329.[80] H. W. Lewis, Phys. Rev. 125 (1962) 937.[81] W. Boersch-Supan, J. Res. Nat. Bur. Stand. 65B (1961) 245.[82] B. Rossi, High-Energy Particles, Prentice Hall, Inc. Englewood Cliffs, N.J. (1952).[83] C. Tschalar and H. D. Maccabee, Phys. Rev. B 1 (1970) 2863.[84] Y. Fisyak, private communiaction (2008).[85] R. Brun, et al., ROOT - An Object Oriented Data Analysis Framework, Nucl. Inst. Meth.

A 389 (1997) 81. See also http://root.cern.ch.[86] H. Bichsel, Proceedings of the Ninth Symposium on Microdosimetry (May, 1985), Radiation

Protection Dosimetry 13 (1985) 91.[87] M.Pfutzner et al., Nucl Inst Meth B 86 (1994) 213.[88] D. Combecher, Rad. Res. 84 (1980) 189[89] H. Bichsel and M. Inokuti, Radiat. Res. 67 (1976) 613.


2–42 References for 2

[90] L. V. Spencer, The theory of electron penetration , Phys. Rev. 98 (1955) 1597, and Energydeposition by fast electrons, Nat. Bur. Std. (U.S), Monograph 1 (195*).

[91] E. J. Kobetich and R. Katz, Phys. Rev. 170 (1968) 391.[92] E. Waibel and B. Grosswendt, Stopping power and energy range relation for low energy

electrons in nitrogen and methane, 7th Symposium on Microdosimetry, 8-12 September1980, Oxford/UK, EUR 7147 (Commission of European Communities).

[93] B. G. R. Smith and J. Booz, Experimental results on W-values and transmission of lowenergy electrons in gases, 6th Symposium on Microdosimetry, Brussels, Belgium, May22-26, 1978, EUR 6064 (Commission of European Communities).

[94] W. Bambynek, et al. Rev. Mod. Phys 44 (1972) 716.[95] V. Grichine, Nucl. Inst. Meth. A 453 (2000) 597.[96] A. Crispin and G. N. Fowler, Rev. Mod. Phys. Rev. 42 (1970) 290.[97] B. Dolgoshein, NIM A 326 (1993) 434.[98] H Bichsel and T. Hiraoka, Nucl. Instr. Meth. B 66 (1992) 345.[99] H. Bichsel and E. A. Uehling, Phys. Rev. 119 (1960) 1670.

[100] H. Bichsel, K. M. Hanson and M. E. Schillaci, Phys. Med. Biol. 27 (1982) 959.[101] S. Agostinelli et al., GEANT 4 - a simulation toolkit, Nucl. Inst. Meth. A 506 (2003) 250.[102] H. Bichsel, Phys. Rev. A 41 (1990) 3642.[103] A. H. Soerensen, Nucl. Inst. Meth. B 264 (2007) 240.[104] S. P. Ahlen, Rev. Mod. Phys. 52 (1980) 121.[105] J. Lindhard and A. H. Soerensen, Phys. Rev. A 53 (1996) 2443.[106] M. Inokuti, R.P Saxon and J.L. Dehmer, Int. J. Radiat. Phys. Chem., 7 (1975) 109.[107] M. Inokuti and D.Y. Smith, Phys. Rev. B 25 (1982) 61.[108] R. P. Saxon, Phys. Rev. A 8 (1973) 839.[109] H. Bichsel, Phys. Rev. 120 (1960) 1012.[110] H. Bichsel and T. Hiraoka, Int. J. Quantum Chemistry.: Quantum Chem. Symp. 23 (1989)

565.[111] M. Gryzinski, Phys. Rev. 138 A (1965) 336.[112] H. Bichsel, Nucl. Inst. Meth. A 565 (2006) 1.[113] I. Lehraus, R. Matthewson and W. Tejessy, IEEE Trans. Nucl. Science NS-30 (1983) 50.[114] Peter Christiansen, private communication (2008).[115] O. Blunck and S. Leisegang, Z. Physik 128 (1950) 500.[116] O. Blunck and K. Westphal, Z. Physik 130 (1951) 641.[117] P. Shulek et al, Sov. J. Nucl. Physics 4 (1967) 400


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2 The Interaction of Radiation with Matter · Ref. p. 2–39] 2 The Interaction of Radiation with Matter 2–1 2 The Interaction of Radiation with Matter Hans Bichsel 2.1 Introduction