Avalanche Statistics

W. Riegler, H. Schindler, R. Veenhof

RD51 Collaboration Meeting, 14 October 2008 Overview

ƒ The random nature of the electron multiplication process leads to fluctuations in the avalanche size Æ probability distribution P(n, x) that an avalanche contains n electrons after a distance x from its origin. ƒ Together with the fluctuations in the ionization process, avalanche fluctuations set a fundamental limit to detector resolution

Motivation ƒ Exact shape of the avalanche size distribution P(n, x) becomes important for small numbers of primary electrons. ƒ Detection efficiency ∞ η = ∑P()n,x = n nT is affected by P(n, x) η

Outline ƒ Review of avalanche evolution models and the resulting distributions ƒ RlResults from silingleelectron avalhlanchesiliimulations iGfildin Garfield using the recently ilimplemente dmiiicroscopictracking features

Assumptions ƒ homogeneous field E = (E, 0, 0) ƒ avalanche initated by a single electron ƒ space charge and photon feedback negligible Yule-Furry Model

Assumption ƒ ionization probability a (per unit path length) is the same for all avalanche electrons ƒ a = α (Townsend coefficient) ƒ In other words: the ionization mean free path has a mean λ =1/= 1/α and is exponentially distributed ρ −α ()l =αe l

Mean avalanche size

α G = n()x = e x

Distribution ƒ The avalanche size follows a binomial distribution n−1 ( )= 1 ⎛ − 1 ⎞ P n,x ⎜1 ⎟ n()x ⎝ n()x ⎠

ƒ For large avalanche sizes, P(n,x) can be well approximated by an exponential n = 50 1 P()n = exp()− n/n n

− ƒ Efficiency η = e n/n ƒ measurements in methylal by H. Schlumbohm Æ significant deviations from the exponential at large reduced fields

ƒ „rounding-off“ characterized by parameter αx0 (x0 = Ui/E)

E/p = 70 V cm-1 Torr -1 E/p = 186.5 V cm-1 Torr -1

αx0 = 0.038 αx0 =0.19

E/p = 76.5 V cm-1 Torr -1 E/p = 426 V cm-1 Torr -1 αx0 = 0.044 αx0 =0.24

E/p = 105 V cm-1 Torr -1

αx0 = 0.095

H. Schlumbohm, Zur Statistik der Elektronenlawinen im ebenen Feld, Z. Physik 151, 563 (1958) Legler‘s Model

Legler‘ s approach

Electrons are created with energies below the ionization energy eUi and lose most of their kinetic energy after an ionizing collision Æ electron has to gain energy from the field before being able to ionize Æ a depends on the distance ξ since the last ionizing collision a 1.2 a

1.0 Distribution of ionization Legler‘s model gas ≈ Ui x0 mean free path E 0.8

α 0.6 = Yule-Furry a −αx 2e 0 −1 0.4

020.2

l x0 x0 W. Legler, Die Statistik der Elektronenlawinen in elektronegativen Gasen, bei hohen Feldstärken und bei großer Gasverstärkung, Z. Naturforschg. 16a, 253-261 ()(1961)

Mean avalanche size Toy MC ( )∝ αx n x e x0 = 0 μm x0 = 2 μm x0 =1 μm x0 =3 μm Distribution IBM 650

The shape of the distribution is characterized by the parameter αx0 [0, ln2] αx0 1 Æ Yule-Furry With increasing αx0 the distribution becomes more „rounded“, maximum approaches mean Legler‘s Model

n−1 P()n,x = ∫ ρ l dl ()(∑P n − n',x − l P n',x − )(l ) moments of the distribution can be calculated (as shown by Alkhazov) n'=1 Æ allows (very) approximative reconstruction of the distribution (convergence problem) n >> 1 ϕ 1 ν n P()n,x = (), ν = G. D. Alkhazov, Statistics of Electron Avalanches and Ultimate Resolution ϕ ν n()x n of Proportional Counters, Nucl. Instr. Meth. 89, 155-165 (1970) αν νρ ν α ν ν ϕ ∞ ν ' ν 1 ν ⎛ 1 ' ⎞ ν ϕ ()= ∫d ⎜ ln ⎟∫d ' ()()'− '' ν '' no closed-form solution ⎝ ⎠ 0 numerical solution difficult

IBM 650

„Die Rechnungen wurden mit dem Magnettrommelrechner IBM 650 (…) durchgeführt.“ Discrete Steps

ƒ „bumps“ seem to indicate avalanche evolution in steps Distance to first ionization Ar (E = 30 kV//,cm, p = 1 bar) ∝ ƒ an electron is stopped after a typical distance x0 1/E of the order of several μm

x0 ƒ with probability p it ionizes, with probability (1 – p) it loses its energy in a different way after each step αx p = e 0 −1

Mean avalanche size after k steps Distribution

n−1 n = ()1+ p k ()= − + ()− ( ) ( ) k Pk+1 n (1 p)Pk n p ∑Pk n n' Pk n' n'=0

moments can be calculated, but no solution in closed form

p = 1 Æ delta distribution p smallÆ exponential Pόlya Distribution

Pόlya distribution Efficiency 1 1.0

0.1 0.9

0.8 0.01

0.7 m 0.001 m − − ν n Γ()m,mν ϕ(ν )= ν m 1e m , ν = 0.6 η = T Γ() Γ()  4 m n m 10 0.5

z T 1 2 3 4 5 0.2 0.4 0.6 0.8 1.0

Good agreement with experimental avalanche spectra Problem: no (convincing) physical interpretation of the parameter m

Byrne‘s approach: Distribution of ionization mean free path α − ()=α⎛ + m 1⎞ n,x ⎜1 ⎟() space-charge effect ⎝ n x ⎠ 1.0 α Γ 2()m − −α m−1 ρ()l = e m l ()1− e l 0.8 Γ(m)2 J. Byrne, Statistics of Electron Avalanches in the Proportional Counter, 0.6 Nucl. Instr. Meth. 74, 291-296 (1969) 0.4

0.2

l Avalanche Growth

ƒ The avalanche size statistics is determined by fluctuations in the early stages.

ƒ After the avalanche size has become sufficiently large, a stationary electron energy distribution should be attained. Hence, for n 102 – 103 the avalanche is expected to grow exponentially.

Yule-Furry model Polya Simulation

ƒ Microscopic_Avalanche procedure in Garfield available since May 2008 performs tracking of all electrons in the avalanche at molecular level (Monte Carlo simulation derived from Magboltz).

ƒ Information obtained from the simulation – total numbers of electrons and ions in the avalanche – coordinates of ionization events – electron energy distribution – interaction rates

Goal ionization ƒ Investigate impact of – electric field – pressure – gas mixture on the single electron avalanche spectrum

ƒ parallel-plate geometry ƒ electron starts with kinetic energy ε = 1 eV Argon

1.2

1

0.8 What is the effect of the electric field on the avalanche spectrum? 0.6

0.4 RMS/mean gap d adjusted such that ≈ 500 0.2

0 15 20 25 30 35 40 45 50 55 60 E [kV/cm]

E = 30 kV/cm, p = 1 bar E = 55 kV/cm, p = 1 bar

Fit Legler Fit Legler Fit Polya Fit Polya Argon

energy distribution 2.80 2.60 2.40 2.20 20 kV/cm 30 kV/cm 2.00 40 kV/cm 1.80 m 50 kV/cm 1.60 60 kV/cm 1.40 1.20 1.00 15 20 25 30 35 40 45 50 55 60 E [kV/cm]

0.50 0.45 0.40 with increasing field, the energy distribution is shifted 0.35 towards higher energies where ionization is dominant 0.30 00 x 0.25 α 0.20 Fit 0.15 Expected 0.10 0050.05 0.00 15 25 35 45 55 65 E [kV/cm] Attachment

introduce attachment coefficient η (analogously to α)

Mean avalanche size

()α −η n()x = e x effective Townsend coefficient α - η

Distribution for constant α and η η − ⎧ α n 1 = ⎪ η n 0 ⎪ n − /α P()n,x = η − ⎨ −η αα 2 − n 1 ⎪ ⎛ 1 / ⎞ ⎛ n 1 ⎞ > n⎜ ⎟ ⎜ η ⎟ n 0 distribution remains essentially exponential ⎩⎪ ⎝ n − / ⎠ ⎝ n − /α ⎠

W. Legler, Die Statistik der Elektronenlawinen in elektronegativen Gasen, bei hohen Feldstärken und bei großer Gasverstärkung, Z. Naturforschg. 16a, 253-261 (1961) Admixtures

Ar (80%) + CO2 (20%)

2.4

2.2

2

1.8

1.6 m 1.4

1.2

1 15 20 25 30 35 40 45 50 55 60 65 E [kV/cm]

Ar (95%) + iC4H10 (5%) 2.4

2.2

2

1.8

1.6 m 1.4

1.2

1 15 20 25 30 35 40 45 50 55 60 65 E [kV/cm] ionization cross-section (Magboltz)

Ionization energy [eV] Ne 21.56 Ar 15.70 Kr 13.996

energy distribution (E = 30 kV/cm, p = 1 bar)

Which shape of σ(ε) yields „better“ avalanche statistics? Ne Parameters: E = 30 kV/cm, p = 1 bar, d = 0.02 cm m ≈ 3.3 ≈ αx0 030.3

≈ 1070 RMS/ ≈ 0.5 Ar

m ≈ 1.7 ≈ αx0 0.15

Kr ≈ 900 RMS/ ≈ 0.7 m ≈ 1.4 ≈ αx0 0.1

≈ 280 RMS/ ≈ 0.8

Conclusions

ƒ „Simple“ models (e. g. Legler‘s model gas) can provide qualitative insight into the mechanisms of avalanche evolution but are of limited use for the quantitative prediction of avalanche spectra (no analytic solution available or lack of physical interpretation).

ƒ For realistic models, the energy dependence of the ionization/excitation cross-sections and the electron energy distribution have to be taken into account Æ Monte Carlo simulation is a better aproach.

ƒ Avalanche spectra can be simulated in Garfield based on molecular cross-sections. Preliminary results confirm expected tendencies (e.g. better efficiency at higher fields).