Particle identification with ionization measurements. Hans Bichsel Telephone: 206-329-2792, FAX: 206-685-4634 e-mail: [email protected] Center for Experimental Nuclear Physics and Astrophysics Box 354290 University of Washington Seattle, WA 98195-4290 December 15, 2004 PC-37 − > [bichsel.PIDNM]PIDN15dd.tex 11:00 15.12.2004 Abstract Charged particle identification can be achieved by measuring the ionization in a medium (gas orcondensed) togetherwiththemeasurement ofthemomentumortheenergy ofparticles. Energy loss of particles and the resulting ionization are related, but not identical. In this study most calculations are made for energy loss and the relation to the resulting ionizationmust be studied experimentally. Someaspectsofthisrelationarediscussed. Adetailedunderstandingoftheenergy loss processes and their stochastic nature is reviewed. Simulationscan be made with analytic and Monte Carlo methods. They can be used to assess the expected performance of a TPC and to applynecessary corrections. For TPCs withthe geometry used in STARand ALICE, an accurate data analysis requires attention to track segmentation. Properties of straggling-functions for Ne, Ar, P10 and Si are similar for equivalent absorber thicknesses and general conclusions given for one absorber willbe valid for the others. The expression “dE/dx” should be abandoned. Specific terms such as ∆, ∆ ,J,Q etc should be used instead. p Contents 1 Introduction 4 2 Models of CCS 7 2.1 Rutherford . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Bethe-Fano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 FVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Calculation and comparison of CCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Integrals of CCS 14 3.1 Cumulative Φ(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Total CCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 3.3 Moments of CCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Algorithms for Landau functions 19 4.1 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Multiple collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Analytic calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 Monte Carlo calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.5 Landau-Vavilov calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Examples of straggling functions for segments 23 5.1 Properties of straggling functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Cumulative straggling functions F(∆;x,βγ) . . . . . . . . . . . . . . . . . . . . . . . . 24 6 Straggling functions for particle tracks 25 7 Scaling of straggling functions 27 7.1 One parameter scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7.2 Two parameter scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8 Dependences of ∆ and w on particle speed 35 p 8.1 Landau and Bethe-Bloch functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.2 Bichsel functions for βγ dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9 Energy deposition and ionization 38 10 Conversion of ionization into pulse-height 40 11 Calibration of TPC 40 12 Comparison of experiments and theory 41 12.1 Comparisons for track segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 12.2 Comparisons for tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 13 Theory of particle identification PID 50 13.1 PID simulations for tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2 13.2 Particle identification for a single track . . . . . . . . . . . . . . . . . . . . . . . . . . 55 13.3 Determination of number of particles of a given mass . . . . . . . . . . . . . . . . . . 56 13.4 Exclusive assignment of particle masses . . . . . . . . . . . . . . . . . . . . . . . . . . 57 13.5 Particle identification - outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 14 Conclusions 58 A Comparison of Bethe-Fano (B-F) theory with FVP 60 B Comparisons of straggling functions for Ar and P10 61 C Optical absorption data 63 D Energy loss and energy deposition functions 64 D.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 D.2 Energy deposition by δ rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 D.3 Monte Carlo calculations for energy deposition . . . . . . . . . . . . . . . . . . . . . . 68 D.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 D.5 Effects of magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 E Electron ranges in Ar, P10 and Si 71 F Excitation and ionization in P10 gas 73 G Dependence of truncated mean values 74 H Collision cross sections for electrons and heavy ions 74 I Straggling functions for very thin absorbers 74 J Gas Multiplication 78 K Bremsstrahlung 79 3 1 Introduction. Particle Identification PID is based on the fact that the momentum of a particle of mass M is given by pc = Mc2βγ, while ionization depends on particle speed βγ only. We measure momentum and ionization of particles along their tracks to determine their mass M. The ionization in the gas is a stochastic process and we must use probability density functions (pdf) to describe it. These functionsarecalledstraggling-functionsorstraggling-spectra. Inparticlephysicsinsteadof“straggling- functions” the generic expression “Landau functions” is used . Here, Landau function is used only to designate the function described in [35]. An introduction tothe subject isgiven in [24] and in Section 9 of [3]. Much of the past work on PID for the STAR-TPC has been based on empirical information without consideration of many problems that will be described here. Various analytic expressions have been used to correlateexperimental data. In particular,mean values and variances of straggling functions have been used for the data analysis. For segments they should be replaced by “most probable” and “full-width-at-half-maximum”(FWHM).For an exact study of the measurements in a detector we must clearly distinguish measurements for single segments (essentially pad-rows) and for tracks. For segments we discern four stragglingfunctions, that followone from the other in sequence a) the energy loss pdf f(∆) b) the energy deposition pdf g(D) c) the ionization pdf g(J) d) the pulse-height or ADC pdf h(Q). Equivalent pdfs are needed for tracks. These functions will be defined and described. It will be seen that the differences between f(∆) and g(J) are not large, and they have been disregarded in most studies so far. Alargepartofthisstudyisconcerned withdescribingthepropertiesoff(∆)andtheirdependence on particle speed and absorber thickness. It will be shown that the moments of f(∆), e.g. mean value and standard deviation are inappropriate for a description of the functions, especially for thin absorbers. Instead,mostprobableenergyloss∆ andfull-width-at-halfmaximum(FWHM)w should p be studied. I shall give details about many aspects of the problem. The reason for this is that I have been asked about many details over the years and therefore hope to answer questions that have occured to the reader. Mean energy-losses will be calculated where suitable. 4 An extensive review of “Particle Detection with Drift Chambers” has been written by Blum and Rolandi [3]. Other aspects of the subject may be found in “Radiation Detection and Measurement” by Knoll [4]. A general survey of the “electronic” interactions of charged particles with matter can be found in chapter 87 of [1]. Much of the information currently used about the subject is based on empiricaldata. Thepurpose ofthepresentpaperistoreviewaspectsofthephysicsoftheinteractions offastparticleswithmatterforthe applicationtoparticletrackingand identificationand toestablish a comprehensive theoretical foundation. Systematic trends and dependencies will be described and documented. The absorbers considered are Ne, Ar and P10 gas at atmospheric pressure and Si. Measurements, calculations and applications are mainly for the RHIC-STAR TPC. The concepts presented here can be readily applied to other detectors. It is important to always be aware of the fundamental microscopic interaction processes which are described next. It will be shown that the parameters describing the straggling functions do not have simple relationsto particle speed and segment lengths x and track lengths t. In particularconclusions based on the central limit theorem are coarse approximations. Since the “resolution”for experimental data can be aslowas2 or3%,I believe thatcalculationsshould be made withanuncertaintyof1% orless. Therefore few analytic functions can be given for results, and functions are presented in the form of tables and graphs. Scaling procedures will greatly reduce the flood of calculated data. The interactions of fast charged particles with speed β = v/c 1 with matter [1] can be described as the occurrence of random individual collisions along a track in each of which the particles lose a random amount E of energy. The probability of collisions is given by the total collision cross section Σ (βγ),or,equivalently, by the mean freepath λ(βγ)= 1/Σ (βγ)between collisions. The probability t t densityfunction forenergylosses E isdescribed by the differentialcollisionspectrum σ(E;βγ). These functions are discussed in Sect. 2 and 3. The energy-loss interactions along a particle track can be simulated with a Monte Carlo calcula- tion [2, 3], Sect. 4. A simple picture of this process for short track segments is shown in Fig. 1. The totalenergylossinasegmentis∆j = PEi. Other detailsaregiveninthecaption. Allunderstanding of the rest of this paper follows from this model. Foralargenumber of tracksegments,the incidence of ∆ isdescribed bythe energy lossstraggling 1The symbols v,β and βγ willbe used interchangeably to designate particle speed. 5 Figure1: MonteCarlosimulationofthepassageoftenminimumionizingparticles(indexj,βγ = 3.6) through segmentsof P10gas. The thickness ofthe gas(atNTP) isx = 2mm. The directionof travel is given by the arrows. Inside the gas, the tracks are defined by the symbols showing the location of a collision. The mean free path between collisions is λ = 0.34 mm (see Fig. 7), thus the average number of collisions per track is six. At each collision point a random energy loss E is selected from i the distribution function Φ(E;βγ), Sect. 3.1. Two symbols are used to represent energy losses: o for E < 33 eV, + for E > 33 eV; the mean free path between collisions with E > 33 eV is 2 mm. Segment statisticsare shown to the right: the totalnumber of collisionsfor each track is given by n , j with a mean value < n >= x/λ = 6 and the total energy loss is ∆j = PE, with the nominal mean value < ∆ >= x·dE/dx = 420 eV, where dE/dx is the Bethe-Bloch stopping power. The largest energy loss E on each track is also given. Here, the mean value of the simulated ∆ is 323 eV, much t j less than < ∆ >. Note that the largest possible energy loss in a single collision is E = 13 MeV, max while the probabilityfor E > 40,000eV is one in ten thousand, for E >1 MeV it is 1 in 300,000(see Fig. 5). functionf(∆),Sects. 4and5. Theenergyloss∆resultsinanenergydepositionDinthevolumeunder observation, Sect. 9. The corresponding straggling-function is g(D). This volume here is assumed to include all the space around the track between two planes separated by one segment length x, and perpendicular to the track. For the determination of the track location the lateral extent of the ionization cloud is important, Sect. 6, for PID it is less important. Frequently the difference between D and ∆ is small, Sect. 9 and Appendix D. The experimental observation in a detector is the ionization J caused by D. Here it is assumed that J = D/W, where W is the energy required to produce an electron-ion pair [5], and the corresponding straggling-function is g(J). Further details are given in Appendices D and F. If the assumptions used in the calculations are correct, f(∆) will differ little in shape from g(J). Finally the ionization is amplified in proportional counters and the 6 resulting signals result in a pulse-height Q with the corresponding straggling-function h(Q), Sect. 9. For a given x and speed βγ, energy loss functions (or spectra) f(∆;x,βγ) can be calculated with the convolution method [6]. Such functions will not have the stochastic uncertainties of experimental functions or those calculated with the Monte Carlo algorithm. This is the method used here for calculating energy loss distributions in segments, Sect. 4. In a Time-Projection-Chamber (TPC) the ionization is produced along particle tracks and the measurement of ionization is made for individual track segments x, see Sect. 9. A simulation for full tracks can be made with a Monte Carlo simulation, using f(∆;x,βγ) for each segment: a value ∆ is selected randomly from the distribution function F(∆;x,βγ), Sect. 4.3. The values of ∆ are i i combined into a descriptor C for each track j (e.g. truncated mean C, Sect. 13.1) and a straggling- j function f(C) for tracks is generated. Methods for particle identification (PID) are described in Sect. 13. Calculations relevant to PID have been made and presented over several years by this author [2, 37, 65, 39, 1, 61]. It will be shown that a two parameter scaling procedure implicit in Eq. (14) of Landau [11, 36] is useful in reducing the amount of numerical calculations. We can hope to obtain a more detailedunderstanding forPID fromtheoreticalsimulationsof the TPC than fromwhat we can get from empirical data. An example of experimental data can be seen in Fig. 27.5, p. 010001-212 of [21]. The calculations presented here are obtained with Monte Carlo simulations [2] for individual particle tracks and the subsequent analysis of the distributions for many tracks. The PID analysis is made with truncated mean values and with likelihood values. The “resolution” in PID is defined by “overlap numbers”. They depend strongly on the total length of the track measured, the number of segments in the tracks, the particle speed and the number of tracks for each particle type, Sect. 13. 2 Models of collision cross sections -models It is useful to describe collisioncross sections in relation to the Rutherford cross section, which is the cross section for the collision of two free charged particles. If charged particles collide with electrons bound in atoms, molecules or solids, the cross section can be written as a modified Rutherford cross section. An approximatebut plausiblewayof describingthese interactionsis toconsider the emission ofvirtualphotonsbythefastparticle,whichthenareabsorbed bythematerial. Herethisiscalledthe 7 Fermi-Virtual-Photon method (FVP) [7]. The differential collision cross section then is proportional tothe photoabsorptioncrosssectionof the molecules. Bohr [8] described thisas a“resonance” effect. A more comprehensive approach is given by the Bethe-Fano method [1, 6, 9]. These models are de- scribed here. Binary encounter methods have been used [10, 50], but are not described. Comparison of the models are made at several places. 2.1 Rutherford cross section σ (E) R Much work on stragglingfunctions has been based on the use of the Rutherford cross section [11, 12], see Sect. 4.5. For the interaction of a particle with charge ze and speed β = v/c colliding with an electron at rest it can be written as k (1−β2E/E ) 2πe4 σ (E) = max , k = ·z2 = 2.54955·10−19 z2 eV cm2 (1) R β2 E2 mc2 where m isthe massof an electron,and E ∼ 2mc2β2γ2 the maximumenergy loss2 ofthe particle. max Note that the mass of the particle does not appear in Eq. [1]. Various attempts have been made to take into account that electrons are bound in matter [6, 13, 15, 19] and Sect. 4.5. 2.2 Bethe-Fano cross section Bethe[20]derivedanexpressionforacrosssectiondoublydifferentialinenergylossE andmomentum transfer K using the first Born approximation for inelastic scattering on atoms. Fano [9] extended the method for solids. In its nonrelativistic form it can be written as the Rutherford cross section modified by the “inelastic form factor” [9, 22]: dσ(E,Q)= σ (E)|F(E,K)|2 dQ, (2) R where Q = q2/2m with q = ¯hK the momentum transferred from the incident particle to the absorber and F(E,K) is the transition matrix element for the excitation. Usually, F(E,K) is replaced by the generalized oscillator strength (GOS) f(E,K) defined by E f(E,K)= |F(E,K)|2 . (3) Q An example of f(E,K) is shown in Fig. 2. 2The exact form of E [21] is not important for the present application. max 8 Figure 2: Generalized oscillator strength GOS for Si for an energy transfer (cid:15) = 48 Ry to the 2p- shell electrons [6]. Solid line: calculated with Herman-Skilman potential, dashed line: hydrogenic approximation. The horizontal and vertical line define the FVP approximation, Sect. 2.3. A full set of GOS for H-atoms can be seen in Fig. 10 of [22]. We get dQ dσ(E,Q)= σ (E) E f(E,K) . (4) R Q In the limit K → 0, f(E,K) becomes the optical dipole oscillator strength (DOS) f(E,0). Because of the 1/Qfactorin Eq. 4, the values of the DOS are importantforaccurate cross sections. The cross section differential in E is obtained by integrating Eq. (4) over Q, dQ Z σ(E;v)= σ (E) E f(E,K) (5) r Q Qmin with Q ∼ E2/2mv2. The dependence on particle speed v enters via Q . In our current under- min min standing, this approach to the calculation of σ(E) is closest to reality. The relativistic extension is described in[6]. Adetailedstudyoff(E,K)forallshellsofsolidsiliconandaluminumhasbeen made [6, 23]. Checks have been made that f(E,0) agrees with optical data [6]. Here σ(E,v) calculated with the relativisticversion of Eq. (5) for minimum ionizing particles [9] is shown by the solid line in Fig. 3. No Bethe-Fano calculations are available for gases, but see Sect. 2.4. 9 Figure 3: 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 (E). The abscissa is the energy loss E in a single collision. The R Rutherford cross section Eq. (1) is represented by the horizontal line at 1.0. The solid line was obtained [6] with the Bethe-Fano theory, Eq. (5). The cross section calculated with FVP, Eq. (6) is shown by the dotted line. The dot-dashed line is calculated with a binary encounter approximation [10]. ThefunctionsallextendtoE ∼ 16MeV,seeEq. (1). ThemomentsareM = 4collisions/µm max 0 and M = 386 eV/µm. 1 2.3 Fermi-virtual-photon (FVP) cross section The GOS of Fig. 2 has been approximated [7, 24, 25, 26] by replacing f(E,K) for Q < E by the dipole-oscillator-strength (DOS) f(E,0) and by placing a delta function at Q = E, as shown in the Fig. This approach is here named the Fermi Virtual Photon (FVP) method. It is also known under the names Photo-Absorption-Ionization model (PAI) and Weizs¨acker-Williams approximation. The differential collision cross section in the non-relativistic approximation is given by [24] E σ(E)= σ (E) [E f(E,0) ln (2mv2/E),+Z f(E0,0) dE0] (6) R 0 for E > E , σ(E)= 0. This model has the advantage that it is only necessary to know the DOS for M the absorber, or, equivalently, the imaginary part Im(−1/κ) of the inverse of the complex dielectric function κ. Data for κ can be extracted from a variety of optical measurements [27, 28]. In addition, Im(−1/κ) can be obtained from electron energy lossmeasurements [29]. A detaileddescription of the relativisticPAI model is given e.g. in [3, 24]. The relativisticcross section is given in the form of Eq. 10