2 The Interaction of Radiation with Matter

Transcription

1 Ref. p. 2 39] 2 The Interaction of Radiation with Matter The Interaction of Radiation with Matter Hans Bichsel 2.1 Introduction 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 1 kev for electrons, T 1 MeV for protons). In order to explore and understand the interaction processes they will be described for single particles. Ions are assumed to be nuclei fully stripped of electrons. The particle interactions are with matter. Only two states of matter will be considered: gaseous and condensed. In general the word atom designates molecules, too. Effects related to crystal structure (e.g. channeling) will not be discussed. Electromagnetic interactions mainly are discussed in this chapter. It is useful to think of the interactions in terms of single collisions which occur at random distances x i along a particle track [1]. The collisions result in separate, discrete, random energy losses E i of the particles. Frequently one wants to observe charged particles for a certain distance along their tracks (or paths). Each particle will then experience several collisions in succession. The spacing x i between collisions, the energy losses E i and angular deflections θ i will be random for each collision. For multiple collisions the sum of the energy losses will be designated as Δ = i E i [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 has been done in cloud chambers and should be possible with GEMs [4]. Experiments with electron microscopes are also suitable for this purpose [5, 6]. The comparison of electron energy loss spectra calculated 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: scattering, bremsstrahlung, pair production, Compton effect. inelastic

2 2 2 2 The Interaction of Radiation with Matter [Ref. p elastic collisions where the momentum transfer is large but the energy loss is small, example: Coulomb scattering of charged particles in collisions with nuclei in whole atoms (also called nuclear collisions ). Particle beams are defined to consist of one kind of particles with parallel trajectories extending over a small area and a small spread in speeds 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 condensed matter (local collective excitations); secondary radiation may be produced, such as secondary electrons (called delta-rays), Auger electrons, photons (fluorescence), Cherenkov and transition radiation, bremsstrahlung, etc. elastic (Coulomb or Rutherford) scattering Nuclear reactions and other types of collisions are infrequent [10], they will be discussed mainly in Sects. 3.2 and 3.3, e.g. pair production, meson production, nuclear excitations Observable effects of radiations The energy losses and secondary radiations mentioned in Sect will result in energy deposition in matter. As far as energy transfer is concerned, it is important to distinguish between the effect on the incident particle, described as an energy loss 1 and the effects on the absorber such as production of excitations and secondary radiations (e.g. delta-rays, photons, phonons) which will result 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 example is the observation of light in the retina or in photographic emulsions 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 1 Angular deflections will also occur. Nuclear spallation will not be considered. 2 Thermal effects [11, 12], photographic methods and track etching will not be further mentioned.

3 Ref. p. 2 39] 2 The Interaction of Radiation with Matter 2 3 and also the digital output Q of the electronic analyzing apparatus [13]. Other examples can be found on p. 676 of ref. [14]. Suggestions about symbols for various concepts for TPCs are given on p. 159 of [13]. The original meaning of de/dx was mean energy loss of fast particles per track segment of unit 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 matter was presented as average quantities for particle beams, with a large number of collisions along each particle 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 calculations: sum rules, transforms, various approximations (in particular the use of the Rutherford cross section for inelastic collisions of charged particles with electrons). We are now at a time where calculations with computers permit the solution of many problems with computer-analytic or Monte Carlo methods. These calculations use fewer approximations but they need more detailed information about the absorbers and the collision processes. As an example, for thin absorbers the classical stopping power and Landau functions usually do not give adequate information for practical applications, such as vertex trackers and TPCs[14]. The average quantities have little meaning 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 charged particle collisions. Fermi formulated this in 1924 [22]. 2.3 Description of the most frequent interactions of single fast charged particles The averages of the random distances x i between the collisions along a particle track are called mean free paths λ. They vary widely. For inelastic electronic collisions of protons, e.g. in Si, λ ranges from less than 5 nm to about 250 nm, in gases at NTP λ is a factor of 1000 larger (Tables 2.1 and 2.2). 3 For elastic collisions of electrons 10 <λ(nm) < 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 losses as 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 considered to be half its energy, for a positron it can be its full energy. For heavier particles with rest mass M and speed v there is a maximum energy loss E M to an electron given in the non-relativistic approximation by E M =2mv 2. (2.1) 3 For ions with charge ze the mean free paths are divided by a factor z 2.

4 2 4 2 The Interaction of Radiation with Matter [Ref. p The relativistic expression is 2 mc 2 β 2 γ 2 E M = 1+(2γm/M)+(m/M) 2, (2.2) where β = v/c and γ 2 =1/(1 β 2 )andβ 2 =(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 can occur, but are infrequent Narrow beams and straggling of heavy charged particles For beams of particles (kinetic energy T, speed v = βc) traversing an absorber the initial energy spectrum φ(t ) will broaden due to the random number of successive collisions and the large spread in random energy losses E. This process is called straggling. The lateral extent of the beam will broaden due to random multiple elastic collisions, and the number of particles in the beam is reduced by nuclear interactions. Frequency or probability density functions (pdf) are needed to describe 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 absorber is given in Fig Multiple inelastic collisions are seen. For clarity elastic collisions are not 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 x i from the previous collision, then the energy loss E i, see Sects This scheme is similar for all absorbers, all particles and all types of collisions. In successive collisions there is a large spread in energy loss E and angular deflections. Spectra of energy losses are given in Sects For asymmetric pdf (or spectra ) the mean values may not have much meaning, the most probable or median value will be more suitable for a description. Long tracks can be divided into n short segments and average values of the Δ j of the segments can be used. By eliminating a fraction of the largest Δ j a truncated mean [13] can be determined for each track, see Sect 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 therefore can be back-scattered out of absorbers. Examples of tracks of back-scattered electrons are shown in Fig Similar tracks have also been described for low energy protons [23] Relation between track length and energy loss In traveling along a track, Fig. 2.1, the distances between collisions and the energy losses are both random and have no correlation. From Fig. 2.1 it is evident that an exact segment length x can be defined (except for uncertainties of atomic dimensions), but the total energy losses Δ j will have a range of values, as shown in Fig It is not possible to postulate an exact final energy in a

5 Ref. p. 2 39] 2 The Interaction of Radiation with Matter 2 5 Energy loss per collision: E j n j Δj(eV) E t (ev) x=6 λ Δ=ΣΕ Fig Monte Carlo simulation, Sect , of the passage of particles (index j, speed βγ = 3.6) through a segment of P10 gas of thickness x =6λ, whereλ =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 by the symbols showing the location of collisions (total number 56). At each collision point a random energy loss E i is selected from the distribution function Φ(E; βγ), Fig Two symbols are used to represent energy losses: o for E i < 33 ev, for E i > 33 ev. Segment statistics are shown to the right: the number of collisions for each track is given by n j, with a nominal mean value <n>= x/λ = 6. The total energy loss is Δ j = E i, with the nominal mean value Δ = xde/dx= 486 ev, where de/dx is the stopping power, M 1, in Table 2.2. The largest energy loss E t on each track is also given. The mean value of the Δ j is 325 ± 314 ev, much less than Δ. Note that the largest possible energy loss in a single collision is E M = 13 MeV, Eq. (2.2). MC calculation, as described in Fig Correspondingly the straggling functions are different. This is shown in Fig The pdf of energy loss for the particles described in Fig. 2.1 traversing x = 9 cm is given by the solid line in Fig. 2.4, the values of x associated with each energy loss Δ in Fig. 2.3 are given by the dotted line Nuclear interactions Particle deflections by elastic nuclear collisions for heavy particles lead to multiple scattering, [24]. For electrons, see Sect and [25]. Inelastic nuclear collisions are nuclear reactions. For protons with T>20 MeV a rough approximation for the mean free path for nuclear reactions is [9, 27] L 32 A 1/3 g/cm 2, (2.3) about 80 cm for water and 40 cm for Si. A table can be found on p. 110 of [24]. Depending on the type of nuclear interaction there may be only one interaction in which the particle disappears

6 2 6 2 The Interaction of Radiation with Matter [Ref. p z x Fig Monte Carlo simulation of the passage of three electrons with kinetic energy T = 18.6 kev through silicon. Units of x, z are Å. 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 =15nm [25]. Trajectories are projected onto the x, z plane f [Δ] βγ=3.6, Δ<9 kev f [Δ] Δ [kev] Fig 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 of E i 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. Note that the track length for each particle is different, Fig from the beam, and reaction products will remain. For particle beams a description in terms of a beam attenuation will be instructive. To get the energy deposition by the nuclear products a Monte Carlo simulation must be made (see Sect ).

7 Ref. p. 2 39] 2 The Interaction of Radiation with Matter βγ=3.6, Δ=9keV, x=9cm 8000 f [sc] sc Fig The dashed line gives the pdf of energy losses Δ in Ne for a segment length x = 9 cm for the 500,000 particles described in Fig. 2.1, Δ = (sc/10) kev. The peak value is at about 8.5 kev, the FWHM is 4.6 kev. The solid line gives the pdf of the segment length x = sc mm of particles which lost Δ 9 kev, as described in Fig The peak is at 86 mm. The two functions have opposed asymmetry, and they have no correlation. It will be difficult to compare the functions by using stopping power. It may be possible to measure these spectra with a GEM TPC [4, 24]. The quotient of the peaks is 1 kev/cm, much less than M 1 =1.6 kev/cm. 2.4 Photon interactions The principal photon interactions are Rayleigh scattering, photo-electric and Compton effect, pair and meson production and nuclear disintegration. The important energy ranges are 0 to 100 kev for 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 loss spectra 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 Dosimetry [9, 27]. A more detailed theoretical study of Compton profiles can be found in [29]. See also Sect A schematic representation of the photo-absorption of a photon in an atom is given in Fig For photons with energies below the ionization potential excitations are produced. They must be included in the calculations described in Sect photoelectric interaction hf atom e Fig Schematic description of a photo electric interaction of a photon with energy hf = 30 ev (wavelength λ 0.66 Å) with an Ar atom (Z = 18, ionization energy I Z 16 ev). An electron is emitted from the M-shell with an energy E = hf I Z 14 ev in a random direction. 4 In a gas mixture such as Ne, CO 2 and N 2 used for the TPC in ALICE the excited states in Ne extend from 16.7 to 21.6 ev, well above the ionization potentials of the molecules (13.8 and 15.6 ev) and therefore can de-excite by 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.

8 2 8 2 The Interaction of Radiation with Matter [Ref. p Gases The quantity used for the description of photon collisions [31] resulting in ionization (continuum excitation states) is the dipole oscillator strength (DOS) f(e; 0) where E = hν = ω is the photon energy and the momentum transfer q = 0. It is related to the photo-ionization cross section σ γ (E) σ γ (E) =4π 2 αa 2 0Rf(E;0) = f(E;0) (nm 2 /ev) (2.4) where a 0 =52.9 pm is the Bohr radius of the H-atom, R=13.59 ev the Rydberg energy. For excitations to excited states with energy E n the symbol f n is used. In gases the excited states contribute of the order of 5% to the total DOS [32]. The absorption coefficient μ(e) for electromagnetic radiation is μ(e) = Nσ γ (E) (2.5) where N is the number of atoms per unit volume [9, 31] Solids Fano [20] described a method to extend the oscillator strength approach to condensed materials by using the complex dielectric constant ɛ(ω) =ɛ 1 (ω)+iɛ 2 (ω) of the absorber, where ω represents the energy loss by the virtual photon [34]. The DOS for solids is related to the dielectric constant as follows [35, 36] f(e;0)=e 2Z πω 2 p ɛ 2 (E) ɛ 2 1 (E)+ɛ2 2 (E) = E 2Z πω 2 p I( 1 ɛ ) (2.6) with the plasma energy Ω p of a free electron gas Ω 2 p =4π 2 e 2 NZ f /m, Ω p =28.8 (ρz f /A) ev (2.7) where ρ(g/cm 2 ) is the density of the solid with atomic number Z and atomic mass A(g). The number Z f is the number of electrons which represent the free electron gas (e.g. Z f =3forAl,4 for Si) Data for DOS Most DOS data are derived from measurements. Examples for several gases are given in Figs. 2.6 and 2.7. Evaluated data for about 30 gases can be found in Berkowitz [32]. They have been used to derive mean excitation energies I [40] (see Appendix A). DOS data have been derived from measurements with electron microscopes [6]. The method is described in [5]. For E>100 ev tables given in [26] are at present the most easily accessible. Calculations of ɛ(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]. 5 A more detailed description can be found in Chapter 5 of [28].

9 Ref. p. 2 39] 2 The Interaction of Radiation with Matter Ar f [E,0] Ne CH E [ev] Fig Dipole oscillator strength f(e;0) for Ar, Ne and CH 4 as function of photon energy E. For Ne, the values f(e;0)/30 are given, for CH 4 f(e;0)/10000 are given. The peak values are approximately 0.32 for Ar, 0.16 for Ne and 0.66 for CH 4, located between E =15 and 20 ev. 0.6 Lee CF 4 f [E,0] HEED Zhang E [ev] Fig Dipole oscillator strength f(e;0) for CF 4 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 electrons of matter resulting in an energy transfer E by the particle in inelastic collisions. The energy is transferred to excited states of atoms, single free electrons (delta rays) or to many electrons as a collective excitation. Photons can also be produced, such as Bremsstrahlung (BMS). Secondary radiations produced are: delta rays, Auger electrons, fluorescence, Cherenkov or transition radiation. Details about their effects are given in Sect In many descriptions of the process the Rutherford or Coulomb collision cross section [9, 44, 45, 46] σ R (E) is used as a first approximation [7, 2, 3, 48].

10 The Interaction of Radiation with Matter [Ref. p For the interaction of a heavy particle with charge ze, spin 0 and speed β = v/c colliding with an electron at rest it can be written as 6 σ R (E,β)= k (1 β 2 E/E M ) β 2 E 2,k= 2πe4 mc 2 z2 = z 2 evcm 2 (2.8) where m is the rest mass of an electron, and E M 2mc 2 β 2 γ 2 the maximum energy loss of the particle, Eq. (2.2). Note that the mass of the particle does not appear in Eq. (2.8) [9]. This cross section has been used by Landau [2], Vavilov [3] and Tschalär [48] as an approximation for the derivation of straggling functions. The approximation is quite good for large energy losses, Figs. 9, 12, 15. Details are given in Appendix B Inelastic scattering, excitation and ionization of atoms or condensed state matter For inelastic collisions the electronic structure of the atoms in the absorber (especially binding energies of the electrons) is important because energy transfers E change these structures. The collisions are also called the inelastic scattering of the particles. Two methods will be discussed in the following. For a quantitative description the Bethe-Fano method [7, 19, 20, 34, 50, 51] is closest to 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 of detectors. For the FVP method see Tables 2.3 and 2.4. For energy losses of electrons with E<E M /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 are introduced [52, 51]. The straggling functions given in Figs will be the same for electrons and heavy particles. The effect of Bremsstrahlung [52, 13] is described in Sect Bethe-Fano (B-F) method Bethe [19] derived an expression for a cross section doubly differential in energy loss E and momentum transfer K using the first Born approximation for inelastic scattering by electrons of the atomic shells. Fano [20] extended the method for solids. In its nonrelativistic form it can be written as the Rutherford cross section modified by the inelastic form factor [20, 53]: σ(e,q)=σ R (E) F (E,K) 2 E2 Q 2, (2.9) where Q = q 2 /2m, with q = K the momentum transferred from the incident particle to the absorber, 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 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)=σ R (E) E Q f(e,k). (2.11) 6 Additional factors for electrons and positrons and for particles with spin 1 and spin 1 at high speeds (β 1) 2 are given e.g. in Uehling [49]. An extensive description can be found in Evans [9].

11 Ref. p. 2 39] 2 The Interaction of Radiation with Matter E=48 Ry f [E,K] Ka o Fig Generalized oscillator strength GOS for Si for an energy transfer E = 48 Ry to the 2pshell electrons [34, 54]. Solid line: calculated with Herman-Skilman potential, dashed line: hydrogenic approximation[33]. The horizontal and vertical line define the FVP approximation, Sect Calculations of GOS have been published by Inokuti [53] and Bote and Salvat [51]. A detailed study was made for Al and Si [34, 54]. An example of f(e,k) is shown in Fig In the limit K 0, f(e,k) becomes the optical dipole oscillator strength (DOS) f(e,0) described in Sect The cross section differential in E is obtained by integrating Eq. (2.11) over Q, σ(e; v) =σ R (E) Ef(E,K) dq (2.12) Q Q min with Q min E 2 /2mv 2. The dependence on particle speed v enters via Q min. In our current understanding, this approach to the calculation of σ(e) is closest to reality. 7 Because of the factor 1/Q in Eq. (2.12), the accuracy of f(e,0) (Fig. 2.7) enters significantly into the calculations of cross sections and M 0, Eq. (2.25) Relativistic extension of B-F method The basic equation for the doubly differential cross section is [20, 34] [ F (E,q) 2 σ(e,q)=k R Z Q 2 (1 + Q/2mc 2 ) 2 + β t G(E,q) 2 ] Q 2 (1 + Q/2mc 2 ) E 2 /2mc 2 (1 + Q/mc 2 ) (2.13) where k R = 2πz 2 e 4 /(mc 2 β 2 ), Q = q 2 /2m gives the kinetic energy of the secondary electron produced in the collision, F (E,q) 2 represents the interaction matrix element for longitudinal excitations, 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) isapproximated by f(e,0), see Fig. 2.8: Q =(Ka o ) 2, and select Q 1 1 Ry=13.6 ev σ 1 (E,β)=k R Z E f(e,0) ln Q 12mc 2 β 2 E 2 (2.14) 7 A description of the derivation of Bethe stopping power, Appendix A.1, can be found in [54].

12 The Interaction of Radiation with Matter [Ref. p High Q part, corresponding to large energy losses E where the binding energy of the atomic electrons can be neglected. For Fig. 2.8 the delta function approximation is used, resulting in the following expressions [20] F (E,q) 2 1+Q/2mc2 δ(e,q) (2.15) 1+Q/mc2 β t G(E,q) 2 1+Q/2mc 2 β t δ(e,q) (2.16) 1+Q/mc2 and β 2 t = 1 1+Q/2mc 2 (1 β2 ) (2.17) are used to get [34] σ h (E) = k R Z E with s = E/(2mc 2 ). ( 1 1+s + s ) 1+s s (1 β2 ) (2.18) Intermediate Q part: the integral over Q, Eq. (2.12), is calculated numerically [34, 54] with the Q 1 used in Eq. (2.14) and Q 2 30 kev. The contribution from G(E,q) can be neglected according to Eq. (2.18) since s is small (details in [34]). 8 Q2 σ 2 (E) = σ(e,q) dq (2.19) Q 1 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 in detail by Fano [20]. For longitudinal excitations Eq. (2.14) is replaced by [ ] 1 σ 1 (E,β)=k R Im ln Q 12mc 2 β 2 ɛ(e) E 2 (2.20) This term is equivalent to the third term of Eq. (2.24), except that Q 1 is replaced by E, i.e. in Fig. 2.8 f(e,0) extends to the delta-function instead of only to Q 1 =1. 9 For transverse excitations the contribution is Eq. (47) in [20], using Eqs. ( ) to convert ɛ 2 (ω) into σ γ (E) σ 3 (E; β) = α σ γ (E) β 2 π EZ ln[(1 β2 ɛ 1 ) 2 + β 4 ɛ 2 2 ] 1/2 + α β 2 π 1 N c (β2 ɛ 1 )Θ. (2.21) ɛ 2 where tan Θ = β 2 ɛ 2 /(1 β 2 ɛ 1 ). 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. ( ) σ 4 (E; β) = σ 1 (E; β) + {σ 2 (E) + σ h (E)} + σ 3 (E; β) (2.22) The function calculated with Eq. (2.22) for minimum ionizing particles [20] is shown by the solid line in Fig No Bethe-Fano calculations with improved GOS are available for gases, but see ref. [51]. 8 For particle speeds β<0.1 this approximation will cause errors, especially for M 0. 9 The 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 M 0 in Tables 2.3, 2.4.

13 Ref. p. 2 39] 2 The Interaction of Radiation with Matter σ [E] / ρ [E] E [ev] Fig Inelastic collision cross sections σ(e,v) for single collisions in silicon of minimum ionizing particles (βγ = 4), calculated with different theories. In order to show the structure of the functions clearly, the ordinate is σ(e)/σ R(E). The abscissa is the energy loss E in a single collision. The Rutherford cross section Eq. (2.8) is represented by the horizontal line at 1.0. The solid line was obtained [34] with the Bethe-Fano theory, Eq. (2.22). The cross section calculated with FVP, Eq. (2.24) is shown by the dotted line. The functions all extend to E M 16 MeV, see Eq. (2.1). The moments are M 0 = 4 collisions/μm and M 1 = 386 ev/μm, Table 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) forq<e by 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) and Weizsäcker-Williams approximation. The FVP calculation is based on the use of photo absorption cross section σ γ (E) (where E = ω is the photon energy) and of the dielectric function ɛ(ω) =ɛ 1 (ω)+iɛ 2 (ω) [22, 20]. The differential collision cross section in the non-relativistic approximation is given by [56] [ ] E σ(e,β)=σ R (E,β) Ef(E,0) 2 ln (2mv 2 /E)+ f(e, 0) de (2.23) for E > E M, σ(e) = 0. This model has the advantage that it is only necessary to know the DOS for the absorber, or, equivalently, the imaginary part Im( 1/ɛ) of the inverse of the complex dielectric function ɛ. Data for ɛ can be extracted from a variety of optical measurements [60, 43]. In addition, Im( 1/ɛ) can be obtained from electron energy loss measurements [61]. A detailed description of the relativistic PAI model is given e.g. in [56, 62]. The relativistic cross section is given by σ(e) = α σ γ (E) β 2 π EZ ln[(1 β2 ɛ 1 ) 2 + β 4 ɛ 2 2] 1/2 + α β 2 π 0 1 N c (β2 ɛ 1 ɛ 2 )Θ + α σ γ (E) β 2 β 2 π EZ ln(2mc2 E )+u e(γ) α 1 σ γ (E ) β 2 π E 2 0 Z de (2.24) with β = v/c, σ γ (E) (E/N) ɛ 2 (E) and tanθ = β 2 ɛ 2 /(1 β 2 ɛ 1 ). E

14 The Interaction of Radiation with Matter [Ref. p σ [E] [1 / ev], FVP σ [E] σ R [E] φ [E] φ [E] P10 gas E [ev] Fig Inelastic collision cross section σ(e; βγ) for single collisions in P10 gas of minimum ionizing particles (βγ = 3.6), calculated with FVP theory: solid line. The Rutherford cross section Eq. (2.8) is given by the dash-dotted line. The dotted line represents the cumulative probability density function Φ(E) of Eq. (2.26), see Fig For βγ =3.6 the functions extend to E M 13 MeV, see below Eq The ionization energy for Ar is E I =15.8 ev [32]. Note that σ γ (E) =f(e;0) cm 2 ev, Eq. (2.14) [31]. The function u e (γ) s(1 β 2 ) of Eq. (2.18) 10 is assumed to be equal to 1.0 in [56] because large energy losses E are unimportant in that context. 11 An example of σ(e) for P10 gas is given by the solid line in Fig. 2.10, for Ne in Fig For silicon detectors it is seen in Table 2.1 that Σ t = M 0 calculated with FVP theory differs byabout6to8%fromtheb-ftheory 12 while the difference for M 1 is less than 1%. 13 In measurements of the ionization in TPCs the difference in M 0, Tables 2.3 and 2.4, (see its importance in Eq. (2.27)), may disguise for example the uncertainty of the energy W (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 for energy losses Δ of particles with speed v in track segments of length x the FVP differential single collision cross section spectrum 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 in Fig and for Ne gas in Fig The functions are similar for all speeds, see Figs See Eqs. 2-4 in Uehling [49]. 11 For gases, ɛ 2 and ɛ 1 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 x 1 p 1 will differ from that of a segment of length x 2 p 2 even if x 1 p 1 = x 2 p 2 [62, Fig. 1.20]. 12 The difference is caused by the approximation shown in Fig B-F M 1 differed by less than ±0.5% from experimental measurements [34, 55]. 14 Calculations have also been made for several other gases, but are not given here. Optical data used are described in Sect

15 Ref. p. 2 39] 2 The Interaction of Radiation with Matter σ [E] Ne βγ= σ [E] [1/eV] E [ev] Fig Inelastic collision cross section σ(e; βγ) for single collisions in Ne gas by ionizing particles with βγ = 3.6, calculated with FVP theory (Eq. (2.24)): solid line. The Rutherford cross section, Eq. (2.8), is given 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 Σ t;a(3.6) =15 collisions/cm [63, Fig. 7.1]. For Eq. (2.8) the function is chosen to cross as shown - the Landau parameterization [2] eliminates the need for a specific value of k in Eq. (2.8). The K-shell excitation at 850 ev causes a difference in the straggling functions, Fig Si βγ=0.316 σ [E] / σ R [E] E [ev] Fig 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 represents the Rutherford approximation, Sect The differences appear to balance to some extent. This indeed is the case for M 1, Eq. (2.25) see Table 2.1, but for M 0 at E = 20 ev, the difference is 20%, dropping to 8.1% at 10 kev.

16 The Interaction of Radiation with Matter [Ref. p Integral quantities: moments and central moments The moments of the collision cross sections are defined by M ν (v) =N E ν 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 a result from the computer-analytic calculation for Eq. (2.22) or Moments calculated with the analytic method described in Appendix A are given in Tables 2.3 and 2.4. The moment M 0 (v), which is also written as Σ t (v), is the mean number of collisions per unit track length. It is an important quantity for the calculation of f(δ; x, v) (Sect. 2.7) because it is used to calculate m c, the mean number of collisions in a track segment. From Figs to 2.16 it is evident that the exact shape of σ(e) for small E will greatly influence values of M The moment M 1 (v) is the Bethe-Fano stopping power dt/dx, i.e. the mean energy loss per unit track length. Its analytic approximation is Eq. (2.41). 16 (21). Numerical values of M 0 (βγ) andm 1 (βγ) are given in Table 2.1 for Si, calculated with both methods given above, and in Table 2.2 for P10 and Ne. From Table 2.1 the differences between the B-F and FVP methods seen in Fig. 2.9 cause quantitative differences in the moments. The difference between B-F and FVP is 6% for Σ t, % for M 1 and 3% for Δ p. The classical formulation of dt/dx and some details are given in Appendix A. The practical use of dt/dx is described in Sects to For Z<20 the average energy loss E (β) per collision is between 50 and 120 ev. The higher moments can be used to calculate the shape of straggling functions for large energy losses [64, 48, 46]. But, for thin absorbers, even M 1 (v) will result in misleading information, see Fig The dependence of M 0 (βγ) andm 1 (βγ) on particle speed is shown in Fig 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 Comparison of moments: Si, Ne, P10 For silicon absorbers it is seen in Table 2.1 that M 0 calculated with FVP theory differs by about 6 to 8% from the B-F theory. The difference is caused by the GOS approximation shown in Fig For M 1 the difference is less than 1%. For Ne and Ar a comparison of the FVP theory of M 0 with σ tot calculated with the Bethe theory, Appendix A.2, is given in Tables 2.3 and 2.4. A similar difference occurs for M 0 for Ne and Ar as for Si. The stopping power M 1 for both gases calculated with FVP differs by less than 1% from ICRU [66]. In measurements of the ionization in TPCs the difference in M 0 (see its importance in Eq. (2.33)), may disguise for example the uncertainty of the energy W (T ) to produce an electron ion pair (at least ±2%) and its dependence on particle energy T [30]. 15 Especially important is he choice of a cut-off energy E m for the Rutherford spectrum, Eq. (2.8), since R M 0 = k /E m, Appendix B In 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, these names here are not included in the name for the stopping power. On the other hand, Fano [20] formulated the expression for solids given in Eq. (2.21).

17 Ref. p. 2 39] 2 The Interaction of Radiation with Matter φ [E;βγ] βγ E [ev] Fig Cumulative energy loss functions Φ(E), Eq. (2.26), for single collisions in Si are shown for several values of βγ. The excitation energy for L 2 electrons is 100 ev, for K electrons it is 1840 ev [65]. A table of the functions is given in [55] φ [E] βγ=0.1 βγ=2.5 βγ= E [ev] Fig Cumulative energy loss functions Φ(E) for single collisions in P10 gas, Eq. (2.26), are shown for several values of particle speed βγ. For 1.0 βγ 7.9 the difference between the functions is no more than the width of the line. A table of the functions is given in [55].

18 The Interaction of Radiation with Matter [Ref. p φ [E] βγ=0.316 & E [ev] Fig The cumulative collision cross sections Φ(E; v) of Eq. (2.26) calculated with FVP for Ne for two values of βγ. The solid line is for βγ=3.16, the dashed line for βγ= For E > 20 ev the difference between the two functions is less than 1%. The difference for 12 <E<30 ev is the Cherenkov effect, Sect The dotted line for the modified Rutherford cross section ( AliRoot ) is derived from Eq. (2.49) τ [E] βγ=0.1 βγ=3.16 βγ= E [ev] Fig Probabilities Υ(E) = 1 Φ(E) for single collisions in P10 gas in which the energy loss exceeds avaluee for different βγ in P10 gas, see Fig

19 Ref. p. 2 39] 2 The Interaction of Radiation with Matter M 1 L [P10] 1.4 Σ t [Ne] Σ t [P10] βγ Fig The dependence on βγ of M 0 =Σ t(v), Eq. (2.25), in Ne and P10 gas. The Bethe-Fano functions M 1(v) are also given (dotted line for Ne). All functions are normalized to 1.0 at minimum ionization. The FVP values M 1 = de/dx differ little for the two gases, but M 0(Ne) reaches saturation at a higher value than M 0(P10). The function given by Eq. (2.50) is given by the solid line labeled L[P10]. See Sect about estimated errors. 2.6 Electron collisions and bremsstrahlung Electronic collisions For electrons and photons interaction models more sophisticated than described in Sects. 2.4 and 2.5 are given in [67]. In general the methods to calculate energy losses Δ, Fig. 2.1, for heavy ions given in Sects. 2.5 and 2.7 will be reliable for electrons in thin absorbers. Thin means that the number of collisions in the absorber by the other interactions (BMS, Cherenkov, pair production etc.) are less than ten. For Si this was considered in [13]. For thick absorbers these interactions become important 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% Bremsstrahlung BMS The atomic differential cross section for production of BMS of energy E by electrons with kinetic energy T is given by [69] 17 σ rad (T,E)dE =4αr 2 0 Z 2 ln 183 Z 1/3 de E χ de E, 4α r2 0 = cm 2 /nucleus. (2.28) 17 A further factor F e(t,u), u = E/T, is approximated by 1 here [52, 68, 69, 10, 24].

20 The Interaction of Radiation with Matter [Ref. p Table 2.1. Integral properties of collision cross sections for Si calculated with Bethe-Fano (B-F) and FVP algorithms. Σ t = M 0[collisions/μm], M 1[eV/μm], Eq. (2.25) for heavy particles, Δ p in ev for x=8μm. The minimum values for Σ t are at βγ 18, for M 1 at βγ 3.2, for Δ p at βγ 5. The relativistic rise for Σ t is 0.1%, for M 1 it is 45%, for Δ p it is 6%. Computer accessible numerical tables are available in [55]. βγ Σ t M 1 Δ p/x B-F FVP B-F FVP B-F FVP

21 Ref. p. 2 39] 2 The Interaction of Radiation with Matter 2 21 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/cm 3 ). Δ p and FWHM w giveninkevforx =2cm,Σ t = M 0 in collisions/cm, M 1 in kev/cm. Computer accessible numerical tables are available in [55]. βγ P10 Ne Δ p w M 0 M 1 M 0 M

22 The Interaction of Radiation with Matter [Ref. p Table 2.3. Comparison of moment M 0 for Ne with σ tot, k s =8πa 2 ory/mv 2. T [MeV] M 0 /k s σ tot /k s diff% Table 2.4. Comparison of moment M 0 forarwithσ tot. T [MeV] M 0 /k s σ tot /k s diff% For a discussion of the average properties of the BMS energy loss in the traversal of fast electrons through an absorber the moments (Sect ) are used (N = ρn A /A is the number of nuclei/cm 3, Nχ is the inverse of the radiation length) We get M ν (T ) N T E l E ν σ rad (E; T )de. (2.29) M 0 (T )=Nχlog T E l λ(t )=1/M 0 (T ) (2.30) M 1 (T )=Nχ(T E l ) (2.31) M 0 is the total collision cross section (CCS) and λ(t ) the mean free path between collisions. This is the important quantity for the calculation of BMS spectra. There is little use for the BMS ( radiative ) stopping power M 1 and none for M 2. Data for T 0 = 1 GeV electrons are given in Table 2.5, assuming E l = 100 ev. 18 The contribution to M 0 for BMS photons between ev and 10 ev for Pb is 18 col/cm and to M 1 it is 20 ev/cm. This effect is disregarded. Table 2.5. Parameters for BMS Eqs. ( ), col=collision. Material Z ρ Nχ M 0 λ M 1 a b [g/cm 3 ] [1/cm] [col/cm] [cm] [MeV/cm] H 2 O Si Fe CsI Pb In order to demonstrate the statistics of the radiative BMS losses, a MC calculation for 20,000 electrons traversing a Si absorber 50 cm thick was made [55]. The small values of M 0 lead to large spreads in the pdf, but they are similar in shape. To simplify the simulation, angular deflections are neglected and the BMS photons produced are assumed to escape from the absorber. The error of the calculations is of the order of 10%. Results: 18 The electronic energy loss Δ within a distance λ is much larger than E l, thus BMS losses below 100 ev will contribute negligibly to Δ ( infrared divergence ).

23 Ref. p. 2 39] 2 The Interaction of Radiation with Matter Si, T 0 =1 GeV e 400 φ [T c ] m T c T c Fig Spectrum φ(t c) of electronic collision energy losses T c in Si. The median energy loss mt c is 111 MeV. The spectrum φ(t b ) for the sum T b of the BMS losses is the complement φ(t 0 T b ), with values between 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 2 and over 90, with a median value of n B =35andaFWHMof35. The pdf of the lengths l of tracks of the electrons are asymmetric, with values between 0 and over 50 cm, with a median value l m = 24 cm (approximately equal to n B λ = 22 cm, but much 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 They range from 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 They range from 700 MeV to 994 MeV, with a median of 890 MeV and FWHM of 120 MeV. Compare to Fig in ref. [24]. Conclusion: the only average quantity needed for dealing with BMS is the total collision cross section M 0. The pdfs described above have no evident relation to M Energy losses along tracks: multiple collisions and spectra A quantitative description of the multiple collisions shown in Fig. 2.1 can be given by three quantities: the number of collisions n j for each particle track j, the total energy loss Δ j and the maximum energy loss E t inside the track segment of length x j. The quantity Δ j /x does not provide any further information. If we calculate the collisions for a very large number of particles, with the same speed 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)

24 The Interaction of Radiation with Matter [Ref. p where m c = x M 0 (v). The straggling function for energy losses is [70, 71] m n c e mc P (Δ) = f(δ; x, v) = σ(δ; v) n (2.33) n! n=0 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 and Clearly straggling functions will depend greatly on the mean number m c of collisions, and to a lesser extent on the particle speed βγ. Themeasurement of straggling functions can be for selected tracks, such as in a TPC (e.g. Sect. 14 in [13]), or for particle beams [34, 72, 73]. Monte Carlo and analytic methods can be used to calculate straggling functions. They are described next Monte Carlo method The interactions occurring during the passage of the particles through matter are simulated one at 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 travels random distances x i between successive collisions, calculated by selecting a random number r r and determining the distance to the next collision from the mean free path between collisions λ =1/Σ t (v) x i = λ ln r r. (2.35) The energy loss E i is selected with a second random number from the integrated collision spectrum, Eq. (2.26), shown in Figs to This process is repeated until x i exceeds the segment length x. The total energy loss Δ j of the particle is Δ j = i E i. To get E i practically, the inverse function E(Φ; βγ) ofφ(e; βγ) is calculated with e.g. cubic spline interpolation [55]. By binning the Δ j the straggling function f(δ) is obtained [55, 74]. The Monte Carlo method can be used for all absorber thicknesses, but with decreasing particle speed [55] it is necessary to change λ(v) andφ(e; v) (Figs ) at appropriate values of v. It may not be very practical for very thick absorbers, e.g. for the full range of 200 MeV protons in water (R 25 cm) the number of collisions is of the order of three million. In order to get reasonable straggling functions, tracks for a million protons may be needed [75] Analytic methods In order to use analytic methods we must consider a large number of particles (charge z, mass M) with the same speed v traversing the exact same length y of track [78]. 21 The methods are described in a fashion suitable for numerical calculations. The particles traverse an amorphous absorber consisting of atoms Z, A, as shown in Fig For thin absorbers and monoenergetic charged particle beams two methods to calculate straggling functions are described next: convolution and transform methods. Thin means that the change in v in traversing ξ can be disregarded. 19 For the Rutherford cross section these function are shown in [58]. 20 For 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 For a single particle track the only possible description is that shown in Fig. 2.1.