Gaseous Detectors. Bernhard Ketzer University of Bonn

Transcription

1 Gaseous Detectors Bernhard Ketzer University of Bonn XIV ICFA School on Instrumentation in Elementary Particle Physics LA HABANA 27 November - 8 December, 2017

2 Plan of the Lecture 1. Introduction 2. Interactions of charged particles with matter 3. Drift and diffusion of charges in gases 4. Avalanche multiplication of charge 5. Signal formation and processing 6. Ionization and proportional gaseous detectors 7. Position and momentum measurement / track reconstruction B. Ketzer Gas Detectors 2

3 2.3 Fluctuations of Energy Loss Consider single particle: statistical fluctuations of number of collisions energy transfer in each collision range straggling (if stopped in medium) energy straggling (if traversing medium) [H. Bichsel, NIM A 562, 154 (2006)] B. Ketzer Gas Detectors 3

4 Fluctuations of Energy Loss Important quantity in order to understand response of detector: probability density function for energy loss in material of thickness x, determined by collision cross section dσσ/dee nn ee xx Straggling functions Calculation of energy loss distribution: two approaches Convolution method Laplace transform method [Allison, Cobb, Ann. Rev. Nucl. Part. Sc., 253 (1980)] [H. Bichsel, NIM A 562, 154 (2006)] B. Ketzer Gas Detectors 4

5 2.3.1 Convolution Method In each collision, the probability to transfer an energy EE is given by Energy loss Δ for exactly NN cc collisions NN cc -fold convolution of FF(EE) with and B. Ketzer Gas Detectors 5

6 Convolution Method Number of collisions NN cc in layer of thickness xx Linked to CCS through mean free path Mean value Standard deviation Relative width B. Ketzer Gas Detectors 6

7 Convolution Method Pdf for total ionization energy loss in material slice of thickness xx = sum of all FF NNcc ( ), weighted by their Poissonian probability for exactly NN cc collisions Straggling functions Poissonian contribution dominant for very small number of collisions (very thin absorbers) Peak structure vanishing for larger NN cc B. Ketzer Gas Detectors 7

8 Convolution Method Solution for thickness xx: Iterative application of convolution integral (numerical) [Bichsel et al., Phys. Rev. A 11, 1286 (1975)] Monte-Carlo method [Cobb et al., Nucl. Instr. Meth. 133, 315 (1976)] calculate mean number of collisions mm cc from integrated cross section for each trial (particle penetration) choose actual number of collisions from Poisson distribution with mean mm cc total energy loss = sum of energy losses in single collisions, taken from normalized dσ/de distribution FF(EE) B. Ketzer Gas Detectors 8

9 Straggling Functions [H. Bichsel, NIM A 562, 154 (2006)] B. Ketzer Gas Detectors 9

10 Straggling Functions Bethe-Bloch mean energy loss: = 400 ev [H. Bichsel, NIM A 562, 154 (2006)] B. Ketzer Gas Detectors 10

11 2.3.2 Laplace Transform Method [L. Landau, J. Phys. USSR 8, 201 (1944)] Change of energy-loss distribution ff(δ; xx) as a result of the particle passing through a thin elemental layer δx: B. Ketzer 1 st term: probability that the energy loss in x was ( E ), and a collision with energy transfer E occurred in δx, which makes the total energy loss equal to (particle scattered into ) 2 nd term: probability that the energy loss in x was already equal to before entering δx, where a further collision increased the energy loss beyond (particle scattered out of ) Gas Detectors 17

12 Laplace Transform Method Put in form of a transport equation: upper integration limit EE for 1 st term ok, since f x, = 0 for < 0 ( ) Solution: Laplace transform of both sides + solve for ff(ss; xx) + inverse Laplace transform s { ( )} F( xs, ) L f x, = 0 < cc 1 Exact solution, but numerical integration necessary in most cases! B. Ketzer Gas Detectors 18

13 2.3.3 Straggling Functions Remarks to both methods: result determined by dσσ/dee given the same cross section dσσ/dee, both methods are equivalent B. Ketzer Gas Detectors 19

14 Straggling Functions Different approximations, depending on thickness of absorber Characteristic parameter:, Thin absorbers: κ 10 possibility of large energy transfer in single collisions: δ-electrons long tail on high-energy side, strongly asymmetric shape ξξ = scaling parameter (1 st term of Bethe-Bloch eq.) B. Ketzer Gas Detectors m 20

15 Landau Distribution Very thin absorbers: κ 0 (i.e. T max ) single energy transfers sufficiently large to consider e - as free Rutherford particle velocity remains constant Landau distribution [Landau, J. Phys. USSR 8, 201 (1944)] dσ (E )/de / σ B. Ketzer Gas Detectors 21

16 Landau Distribution Very thin absorbers: κ 0 (i.e. T max ) single energy transfers sufficiently large to consider e - as free Rutherford particle velocity remains constant Landau distribution [Landau, J. Phys. USSR 8, 201 (1944)] f ( x, 1 φλ L ) = ( ) ξ λλ = universal parameter, see next page for relation to Δ mm and ξξ Analytical approximation: Moyal distribution [Moyal, Phil. Mag. 46, 263 (1955)]: 1 ( λ 1 λ+ e ) 2 fm ( x, ) = e, 2 π λ = ξ m Note: λλ different from parameter in Landau distr. above B. Ketzer Gas Detectors 22

17 Landau Distribution Universal Landau distribution: Properties of φ(λ): asymmetric: tail up to TT mmmmmm Maximum at λ= FWHM=4.02 λ numerical evaluation B. Ketzer Landau distribution in ROOT: Gas Detectors 23

18 Comparison Landau - Moyal Landau: p 1 =1. p 2 =500. p 3 =100. Moyal: coarse shape correct tail too low! But: tails are important for detector resolution! B. Ketzer Gas Detectors 24

19 Landau Distribution Landau distribution: Uses Rutherford cross section Does not reproduce straggling functions based on more realistic models for CCS for very small NN cc Narrower width also for higher NN cc related to mean free path λλ Rutherford CCS underestimates λλ (overestimates NN cc ) Poisson contribution to straggling function leads to broadening B. Ketzer Gas Detectors 25

20 Symon-Vavilov Distribution Thin absorbers: 0.01 < κ < 10 use correct expression for T max use Mott cross section instead of Rutherford reduces to Landau distribution for very small κ less asymmetric shape for larger κ [S.M. Seltzer, M.J. Berger, Nucl. Sc. Ser. Rep. No. 39 (1964)] B. Ketzer Gas Detectors 26

21 Realistic Straggling Functions Iterative Solution of convolution integral [H. Bichsel, Rev. Mod. Phys. 60, 663 (1988)] PAI Model vs Landau Histogram: data Landau + corrections [Maccabee, Papworth, Phys. Lett. A 30, 241 (1969)] Ar/CH 4 (93/7) Ar/CH 4 (93/7) PAI model [Blunck, Leisegang, Z. Phys, 128, 500 (1950)] [Allison, Cobb, Ann. Rev. Nucl. Part. Sci. 30, 253 (1980)] B. Ketzer Gas Detectors 27

22 2.4 Measurement of Energy Loss Restricted energy loss: de 4π z e n 2 e 1 2mc βγtcut β T cut δ = ln dx T< Tcut ( 4πε ) mc β 2 I 2 T 0 max 2 approaches normal Bethe-Bloch equation for TTcut TTmax Transforms into average total number of e - ion pairs n T along path length x: x de dx = nw T But: actual energy loss fluctuates with a long tail (Landau distribution) mean value of energy loss is a bad estimator use truncated mean of N pulse height measurements along the track: A t N = t 1 Ai Ai+ 1 for i = 1,, N Ai N N, [ 0,1 ] t = t N t t i = 1 Gas Detectors B. Ketzer 28

23 Measurement of Energy Loss Gas Detectors B. Ketzer 29

24 Measurement of Energy Loss B. Ketzer Gas Detectors 30

25 Measurement of Energy Loss Resolution (empirical): N = number of samples Δxx = sample length (cm) p = gas pressure (atm) Gas Detectors B. Ketzer 31

26 Example: ALICE TPC B. Ketzer Gas Detectors 32

27 Example: FOPI GEM-TPC Sum up charge in 5 mm steps Truncated mean: 0-70% Momentum from TPC + CDC PID using de/dx: Resolution ~ 14-17% No density correction First de/dx measurement with GEM-TPC! Gas Detectors [F. Böhmer et al., NIM A 737, 214 (2014)] B. Ketzer 33

28 Example: FOPI GEM-TPC Sum up charge in 5 mm steps Truncated mean: 0-70% Momentum from TPC + CDC PID using de/dx: Resolution ~ 14-17% No density correction First de/dx measurement with GEM-TPC! Gas Detectors [F. Böhmer et al., NIM A 737, 214 (2014)] B. Ketzer 34

29 Energy Loss of Charged Particles [Review of Particle Physics, S. Eidelmann et al., Phys. Lett. B 592, 1 (2004)] B. Ketzer Gas Detectors 35

30 Electromagnetic Interaction of Particles with Matter Z 2 electrons, q=-e 0 M, q=z 1 e 0 Interaction with the atomic electrons. The incoming particle loses energy and the atoms are excited or ionized. Interaction with the atomic nucleus. The particle is deflected (scattered) causing multiple scattering of the particle in the material. During this scattering a Bremsstrahlung photon can be emitted. In case the particle s velocity is larger than the velocity of light in the medium, the resulting EM shockwave manifests itself as Cherenkov Radiation. When the particle crosses the boundary between two media, there is a probability of the order of 1% to produce an X ray photon, called Transition radiation. Slide courtesy of W. Riegler B. Ketzer Gas Detectors 36

31 2.5 Multiple Scattering In addition to inelastic collisions with atomic electrons elastic Coulomb scattering from nuclei Rutherford cross section for single scattering (no spin, recoil) 2 ( zze ) dσ 1 = dω 2 Rutherford 2 4θ 4θ ( 4πε 0 ) 4E sin sin 2 2 small angular deflection of incident particle random zigzag path, θ = 0 2 For hadronic projectiles also the strong interaction contributes to multiple scattering contribution to momentum resolution! Gas Detectors B. Ketzer 37

32 Multiple Scattering Theory: Single scattering: very thin absorber such that probability for more than one Coulomb scattering small angular distribution given by Rutherford formula Plural scattering: average number of scatterings N<20 difficult! Multiple scattering: N >20, energy loss negligible statistical treatment Gaussian distribution via Central Limit Theorem non-gaussian tails due to less frequent hard scatterings probability distribution for net angle of deflection as function of thickness of material traversed: Molière distribution (up to θ 30 ) [G. Molière, Z. Naturforsch. 3a, 78 (1948), H.A. Bethe, Phys. Rev. 89, 1256 (1953)] Gas Detectors B. Ketzer 38

33 Multiple Scattering Gaussian approximation: ignore large-angle (>10 ) single scatterings P ( θ ) exp θ d 2 1 plane plane θplane = θ 2 plane 2πθ 2θ θ = θ = θ = θ 2 rms plane plane central limit theorem θ MeV L L z ln βcp X X T = T from fit to Molière distribution X 0 = radiation length of absorber L T = track length in medium (w/o multiple scattering), i.e. straight line or helix Accurate to < 11% for L X 3 T 10 < < [V.L. Highland, NIM 129, 497 (1975)] [G.R. Lynch et al., NIM B 58, 6 (1991)] Gas Detectors B. Ketzer 39

34 Multiple Scattering P ( θ ) θ exp θ d θ 2 1 plane plane plane = 2 plane 2πθ 2θ 0 0 θ MeV L L z ln βcp X X T = T Gas Detectors B. Ketzer 40

35 Literature W. Blum, L. Rolandi, W. Riegler: Particle Detection with Drift Chambers. Springer, H. Spieler: Semiconductor Detector Systems. Oxford C. Leroy, P.-G. Rancoita: Principles of Radiation Interaction in Matter and Detection. World Scientific, Singapore Specialized articles in Journals, e.g. Nuclear Instruments and Methods A, Ann. Rev. Nucl. Part. Sci. C. Grupen, B. Shwartz: Particle Detectors. Cambridge Univ. Press, K. Kleinknecht: Detektoren für Teilchenstrahlung. Teubner, Stuttgart G. F. Knoll: Radiation Detection and Measurement. J. Wiley, New York W. R. Leo: Techniques for Nuclear and Particle Physics Experiments. Springer, Berlin B. Ketzer Gas Detectors 41