Latest version of Bichsel's PID paper - alice tpc test activity at cern

86 Pages·2004·1.54 MB·English

Particle identification with ionization measurements.
Hans Bichsel
Telephone: [phone], FAX: [phone]
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
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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
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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
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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.
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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.
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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
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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
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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
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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.
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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