Avalanche Statistics
H. Schindler, R. Veenhof
Basics
Since charge amplification is a stochastic process, the size of an electron
avalanche is subject to fluctuations probability P(n x) that an avalancheavalanche is subject to fluctuations probability P(n, x) that an avalanche
initiated by 1 electron comprises n electrons after a distance x.
The shape of the avalanche size distribution can be important for the
detection performance
– contribution to energy resolution
– detection efficiency threshold
In this talk we discuss avalanches in 
uniform fields initiated by single Puniform fields initiated by single
electrons. efficiency
n
Overview
At low fields E/p and moderate gain
electron avalanches exhibitelectron avalanches exhibit
– exponential growth with distance
– exponentially distributed fluctuations
Methylal,
E/p = 70 V cm‐1 Torr‐1
p y
With increasing field the distribution departs 
Methylal,
E/p = 186.5 V cm‐1 Torr‐1
from the exponential shape becoming more 
and more rounded (while the growth remains 
exponential) The relative width decreasesexponential). The relative width decreases,  
the maximum is shifted towards the mean.
At high gain space charge leads to a 
departure from the exponential growth, 
additional deviations from the exponential Methylal,
additional deviations from the exponential
shape are observed.
E/p = 74.8 V cm‐1 Torr‐1
Gain ≈ 107
H. Schlumbohm, Zur Statistik der Elektronenlawinen
im Ebenen Feld. III, Z. Phys. 151 (1958) 563‐576
Low Fields: Yule - Furry Model (I)
Key assumption: probability αΔx of ionising collision within a step Δx
common to all electrons in the avalanchecommon to all electrons in the avalanche
P(n x + Δx) = [1 – nα(x) Δx]P(n x) + (n – 1)α(x)ΔxP(n – 1 x) + O(Δx2)P(n, x + Δx)  [1  nα(x) Δx]P(n, x) + (n 1)α(x)ΔxP(n  1, x) + O(Δx )
no multiplication 1 electron ionises
Letting Δx → 0 yields the differential equation
( ) ( )( ) ( ) ( ) ( ) 1)0(11
d
nPxnnPxxnPnxxnP δαα =−−−=( ) ( )( ) ( ) ( ) ( ) 1,)0,(    ,,,11,
d
nnPxnnPxxnPnxxnP
x
δαα
x d
Substituting →( )∫=
x
ssu
0
dα ( ) ( ) ( ) ( )unnPunPnunP
u
,,11,
d
d
−−−=
( ) 1n
Solution: geometric distribution
W Bl W Ri l L R l di P ti l D t ti ith D ift Ch b S i 2008
( ) ( ) 1
1,
−−−
−=
nuu
eeunP
W. Blum, W. Riegler, L. Rolandi, Particle Detection with Drift Chambers, Springer 2008
W. Legler, Zur Statistik der Elektronenlawinen, Z. Phys. 140 (1955) 221‐240
Low Fields: Yule - Furry Model (II)
Mean and variance are given by ( ) ( )1      ,dexp 2
0
−=⎟
⎠
⎞
⎜
⎝
⎛
∫= nnssn
x
σα
Write P(n, x) in terms of → no explicit dependence on xn
1
1
1
1
),(
−
⎟
⎠
⎞
⎜
⎝
⎛
−=
n
nn
xnP
For P(n x) is
1
( )nPn 20=n
1>>nFor , P(n, x) is
well approximated by an 
exponential distribution
1>>n
(variance equal to mean).
1
1
)(
2
/
≈≈ − σnn
exnP
0.5
1
n
    ,),( 2
≈≈ e
n
xnP
0
0 1 2 3 4 5nn/
Attachment (I)
Many non‐noble gases attach electrons at somewhat lower than ionisation
energies (typically > 5 eV) Some RPC gases massively attach at all energiesenergies (typically > 5 eV). Some RPC gases massively attach at all energies 
(SF6). 
Main effect of attachment: electrons can be lost within the first steps priorp p
to any multiplication. 
To overcome attachment, the initial multiplication must be large.
Total cross section
Total cross sectionTotal cross section
Attachment Attachment AttachmentAttachment Attachment Attachment
Attachment (II)
Introduce attachment coefficient η [1/cm] 
P(n, x+Δx) = [1 – n(α+η)Δx]P(n, x) + (n–1)αΔxP(n–1, x) + (n+1)ηΔxP(n+1, x) + …
"nothing " happens 1 electron ionises 1 electron attachesnothing  happens 1 electron ionises 1 electron attaches
⎪
⎧ = 0/ nαη
Solution for :
(W. Legler)
( )
1>>= − x
en ηα
( )
( )⎪⎩
⎪
⎨
⎧
>⎟
⎠
⎞
⎜
⎝
⎛
−−
−
=
=
0   ,/1exp
/1
0  ,/
),( 2
n
n
n
n
n
xnP
αη
αη
αη
Considering only avalanches with n > 0, mean and variance are given by
( ) 2
*2*
      ,
/1
n
e
n
x
=
−
=
−
σ
αη
ηα
Distribution remains exponential.
W L l Di St ti tik d El kt l i i l kt ti G b i h h F ld tä k d b i ßW. Legler, Die Statistik der Elektronenlawinen in elektronegativen Gasen, bei hohen Feldstärken und bei großer 
Gasverstärkung, Z. Naturforschg. 16a (1961), 253‐261
Beyond the Yule-Furry Model
To understand/reproduce rounded spectra as observed at higher fields, 
more elaborate models are requiredmore elaborate models are required. 
In particular the assumption that all electrons take part in theIn particular, the assumption that all electrons take part in the
multiplication process with equal probability must be abandoned.
Analytical models incorporating the ionisation threshold and inelastic 
interactions exist (more later)  useful for qualitative understanding
For the quantitative prediction of avalanche spectra, Monte Carlo 
i l ti t it bl hsimulation represents a more suitable approach.
Avalanche Simulation in Garfield
Since mid 2008 Garfield includes routines for microscopic Monte Carlo 
simulation of electron transport using Magboltz cross‐sectionssimulation of electron transport using Magboltz cross‐sections
– MICROSCOPIC_AVALANCHE
– DRIFT MICROSCOPIC ELECTRON 100_ _
Output 50
Ar, E = 30 kV/cm, p = 1atm
– numbers of produced electrons and ions
– electron trajectories
l t di t ib ti 0
μm]
– electron energy distribution
– interaction rates
0
x [
Here we discuss spectra for uniform fields. 
The microscopic tracking procedures can
‐50
p g p
be used with arbitrary geometries though.‐100
0 50 100 150 200
y [μm]
The modification of the field due to space charge is not taken into
account size dependent effects cannot be reproduced.
Experimental Data (I)
Validation of simulation results by comparison with pulse height spectra
measured in parallel‐plate chambersmeasured in parallel‐plate chambers
We use data fromWe use data from
– H. Schlumbohm, Zur Statistik der Elektronenlawinen im Ebenen Feld. III, 
Z. Phys. 151 (1958) 563‐576
– A. H. Cookson and T. J. Lewis, Variations in the Townsend first ionization
coefficient for gases, Brit. J. Appl. Phys.  17 (1966) 1473‐1481
– P Fonte et al Single‐electron pulse‐height spectra in thin‐gap parallel‐plate– P. Fonte et al., Single‐electron pulse‐height spectra in thin‐gap parallel‐plate 
chambers, Nucl. Instr. Meth. A 433 (1999) 513‐517
Avalanches are initiated by single electrons emitted from the cathode due 
to illumination with UV light.
Excursion: Pólya Distribution
1
( )
( )
( )
( ) nn
e
n
n
n
nP /1
1
11 +−
+
⎟
⎠
⎞
⎜
⎝
⎛
1+Γ
+
= θ
θθ
θ
θ
( )nPn
1
θ = 0
θ = 0.5
θ = 1.5
θ = 1.5
Relative variance
falls with increasing shape parameter θ
( )1/1/
22
+= θσ n
0.1
θ = 1
falls with increasing shape parameter θ
By tuning θ good agreement withBy tuning θ good agreement with
measured spectra can be achieved.
nn/
0.01
0 1 2 3 4 5
Physical significance of θ is dubious.
nn/
"Derivation": introduce size dependent multiplication probability
α(n, x) = α(x)[1 + θ/n]
J. Byrne, Statistics of the electron multiplication process in proportional counters, 
Proc. R. Soc. Edinburgh A  66 (1962) 33‐41
A Lansiart and J P Mor cci Amplication ga e se dans n compte r proportionnelA. Lansiart and J.‐P. Morucci, Amplication gazeuse dans un compteur proportionnel
J. Phys. Radium (Supplement) 23/S6 (1962) 102A‐104A
Comparing Spectra At Different Gain (I)
Experimental data are given for gain values ≈ 104 – 106, while reasonable
avalanche sizes for simulation are of order 102 – 103avalanche sizes for simulation are of order 10 – 10 .
Convenient figures of merit to characterise the "roundness" areConvenient figures of merit to characterise the roundness  are
– shape parameter θ of fitted Pólya function
– relative width of the distribution f = σ2 / 
2
n
To what extent are these parameters size‐dependent?
Hypothesis: for the shape of the distribution is independent of the
mean size (provided that E/p is kept constant!)
1>>n
( )1
H If th fi ld i lt d d i th l h th (
( )nn
n
xnP /
1
),( ϕ=
However: If the field is altered during the avalanche growth (space 
charge), the shape of the distribution does depend on the gap!
Comparing Spectra At Different Gain (II)
2
Garfield: with increasing gap (and thus increasing size)
the Pólya fit parameter approaches an asymptotic value
θ
1
Methane
E/p = 156 V cm‐1Torr‐1
the Pólya fit parameter approaches an asymptotic value 1
Methylal
E/p = 186.5 V cm‐1Torr‐1
Experimental evidence
( )
mean size
0
200 600 1000
P(n)
Methane
E/p = 130 V cm‐1Torr‐1
( )nPn
G. Vidal, J. Lacaze and J. Maurel, Microscopic evolution of the ionizing collision frequency in Townsend 
avalanches, J. Phys. D: Appl. Phys. 7 (1974) 1684‐1698
Is this a universal feature?  use general statistical relations to calculate 
the evolution of the moments of the size distribution.
Excursion: Some Statistics … (I)
Recall: P(n, x) denotes the size distribution after a step x for one primary
electronelectron
Assuming that the individual avalanches evolve independently theAssuming that the individual avalanches evolve independently, the
distribution for k primary electrons is given by k‐fold convolution of
P(n, x)  for large k the size distribution tends to a normal distribution
(central limit theorem)
Th i l l l h i di ib i f 2 i i bThe single electron avalanche size distribution after a step 2x is given by
( ) ( ) ( )xnnPxnPxnP n
n
n
,','2, '*
1'
−∑=
=
where P*n' is the n' ‐fold convolution of P.
n 1
"As one can easily show ": mean and variance after two steps are given by
( ) ( )2
2 xnxn = ( ) ( ) ( )[ ] ( )xxnxnx 22
12 σσ +=( ) ( )2 xnxn = ( ) ( ) ( )[ ] ( )xxnxnx 12 σσ +=
Excursion: Some Statistics … (II)
Mean and variance after k steps are given by
( ) ( )xnkxn
k
= ( ) ( ) ( )∑
−
=
−+
=
1
0
122
k
i
ik
xnxkx σσ
The relative variance evolves as
( ) ( ) ( )⎞⎛ k22
( )
( )
( )
( )
( )
( ) ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
xn
xn
xn
x
kxn
kx
k
11
112
2
2
σσ
The growth of the relative width is bounded (if )( ) 1>xn
Once is large, the relative width barely grows further.n
Experimental Data (II)
Exp. Data Garfield
1
Methylal
E/p = 76 5 V cm‐1Torr‐1
Methylal
E/p = 105 V cm‐1Torr‐1
( )nPn
Schlumbohm (1958):
Observation of
di ff
0.1
E/p = 76.5 V cm 1Torr 1 E/p = 105 V cm 1Torr 1
rounding‐effect at
high E/p in several
gas mixtures. Plots
Gain ≈ 1.4×105Gain ≈ 1.4×105
gas mixtures. Plots 
are available for
methylal, acetone
d l h l
0.01
1
and alcohol.
F i ith1
Methylal
E/p = 186.5 V cm‐1Torr‐1
Methylal
E/p = 426 V cm‐1Torr‐1
( )nPn
For comparison with 
the simulation, the
avalanche size is 
0.1
45
normalised to the 
mean obtained from 
Pól fitGain ≈ 7×104Gain ≈ 1×105 a Pólya fit.
0 1 2 3 4 5
0.01
0 1 2 3 4 5nn/ nn/
Experimental Data (III)
Cookson and Lewis (1966):
Measurements in methane fitted with Pólya curvesMeasurements in methane, fitted with Pólya curves
Plot E/p Pólya fit parameter θ
[V cm‐1Torr‐1] Cookson/Lewis Garfield
(a) 48.2 0.0 0.13
(b) 51.3 0.1 0.15
( ) 78 9 0 3 0 36(c) 78.9 0.3 0.36
(d) 120.0 0.4 0.76
(e) 156.0 1.0 1.21
(f) 218.0 1.2 1.99
Garfield simulation: Tendency of rounding is
reproduced, shape deviates somewhat from
i lexperimental spectra
avalanche
n0 2×106
avalanche
chains
Experimental Data (IV)
P. Fonte et al. (1999): Measurements with a range of gas mixtures and gaps of  
0.6 mm and 1.2 mm. Data files with parametrisations of the gain spectra 
( bli h d) id d b P l F Th d f 1 2 l k(unpublished) were provided by Paulo Fonte. The data for 1.2 mm gaps looks 
cleaner.
The data show a steep fall of relative RMS with
increasing E field, while Garfield predicts a more g , p
gentle decrease. Example: 90% Ar + 10 % C2H6
(similar disagreement for all other mixtures)
GarfieldGarfield
data 1.2 mm
data 0.6 mm
data 0.6 mmdata 0.6 mm
Compiled from numerous gas mixtures
0.6 mm
1 2 mm
0.6 mm
1 2 mm
Gain seems to be the 
driving factor for the 
Compiled from numerous gas mixtures
1.2 mm 1.2 mm
shape of these 
spectra.
Analytical Models (I)
Legler‘s model: threshold energy for ionisation ionisation probability
depends on electron energy or the distance travelled since the lastdepends on electron energy or the distance travelled since the last 
collision, respectively.
If the electron starts a a
ionisation probability distribution of ionisation
mean free path
with zero kinetic energy,
the minimum step length
b ti t d
α
distance to
α
distance to
can be estimated as
x0 ≈ Ui/E. Exponential growth
requires that a = α / (2exp(‐αx0) – 1).
x0
distance to
previous collision x0
distance to
previous ionisation
q / ( p( 0) )
relative width:
0 6
0.8
1
( )
124
12
00
0
2
2
2
2
−−
−
== −−
−
xx
x
ee
e
n
f αα
α
σ
0
0.2
0.4
0.6
f
W. Legler, Die Statistik der Elektronenlawinen in elektronegativen
Gasen, bei hohen Feldstärken und bei großer Gasverstärkung, Z. Naturforschg. 16a (1961) 253‐261
0 0 0.6αx0
nn/
G. D. Alkhazov, Statistics of Electron Avalanches and Ultimate Resolution of Proportional Counters, Nucl. Instr
Meth. 89 (1970) 155‐165 
Analytical Models (II)
Stepwise evolution: multiplication occurs in 
steps of fixed length x After each step an
ionisation probability
steps of fixed length x0.  After each step an 
electron ionises with probability p or loses 
its energy through another process with probability 1 – p. 
2 3
relative width: 
2
0.8
1
x0 2x0 3x0
Toy MC
(stepwise evolution)
entries
12
1
1 0
2
2
−=
+
−
≈= − x
e
p
p
n
f ασ
0.2
0.4
0.6 f
1000
Alkhazov‘s model 3: distribution of inelastic
lli i b bilit i b t f ti
0 0 0.6αx0
p = 0.05
p = 0.10
p = 0.50
p = 0.60
1000
collision probability given by step‐function; 
probability of ionisation in each inelastic
collision given by p
p  0.10
P = 0.20
p = 0.30
p = 0.40
p
P = 0.70
g y p
( )
( ) ( ) 1112
112)31(
00
00
2
22
−+−+
++−+
≈ −−
−−
xx
xx
epep
epep
f αα
αα
n
100 200 300 400
p
100
G. D. Alkhazov, Statistics of Electron Avalanches and Ultimate Resolution of Proportional Counters, Nucl. Instr
Meth. 89 (1970) 155‐165 
( ) ( )
Analytical Models (III)
Closed‐form expressions for these distributions are not available, but their
shape can be characterised in terms of the moments.
The models reflect (in a simplified way)
energy dependence of electron atom M th l l0 8
1
– energy dependence of electron‐atom
collision cross‐sections
– interplay between ionisation and f
Methylal0.8
other inelastic collisions
The relative width falls with increasing
Exp. Data
Garfield
Legler‘s model
0.4
The relative width falls with increasing
parameter αx0 (x0 ∝ 1/E) and/or increasing
ratio ionisation/excitation. E/p [V/cm]
100 200 300 400
Legler s model
Step model
0
Side note: 
i i ti b bilit
distribution of ionisation
E/p [V/cm]
Which distribution of the
ionisation mean free path
would lead to the
ionisation probability
θ = 0.5
α
θ = 0.5
mean free path
would lead to the
Pólya function?
θ = 1
distance to
previous collision
θ = 1
distance to
previous ionisation
Case Study: Argon
2
energy distribution
p = 1 atm
E = 25 kV/cm
Pólya fit parameter θ
1.5
E = 25 kV/cm
E = 30 kV/cm
E = 40 kV/cm
E = 50 kV/cm
y p
1
E = 60 kV/cm
relative variance f
0.5
0 4
0 5 10 15 20 25 30
electron energy [eV]
20 30 40 50 60
E [kV/cm]
0
distribution of distance
to first ionisation
0.3
0.4
inelexcion
ion
NNN
N
++
p = 1 atm
E = 20 kV/cm
E = 30 kV/cm
E = 40 kV/cm
0.2
E = 40 kV/cm
0.1
ionisation vs. excitation
0 5 10 15 20 25
distance [μm]
20 30 40 50 60
0.0
E [kV/cm]
Case Study: 80% Ar + 20% CO2
2
energy distribution
p = 1 atm
E = 20 kV/cm
1.5
E = 20 kV/cm
E = 30 kV/cm
E = 40 kV/cm
E = 50 kV/cm
relative variance f
1
E = 60 kV/cm
Pólya fit parameter θ
0.5
0 4
electron energy [eV]
20 30 40 50 60
E [kV/cm]
0
0 5 10 15 20 25 30
distribution of distance
to first ionisation
0.3
0.4
inelexcion
ion
NNN
N
++
p = 1 atm
E = 25 kV/cm
E = 30 kV/cm
E = 40 kV/cm
0.2
ionisation vs. excitation
E = 40 kV/cm
0.1
distance [μm]
20 30 40 50 60
0.0
E [kV/cm]
0 5 10 15 20 25
Methylal
CH2(OCH3)2
akaaka
– DMM
– dimethoxy
1000
total
vibration
vibration
vibrationy
methane
– formal
100
excitation
excitation
ionisation
Excitations
6 3 V
10
cm2]
– 6.3 eV
– 8.3 eV
1
σ[10‐16c
Ionisation
– 10 eV
0.1
0.01
0.01 0.1 1 10 100
energy [eV]
Methane
CH4
Dissociation
– 9 eV
100
– 10 eV
– 11 eV 10
– 11.8 eV
Ionisation 1
cm2]
Ionisation
– 12.99 eV
1
σ[10‐16c
0.1
0.01
0.1 1 10 1000.1 1 10 100
energy [eV]
el tot
el mt
vib
vib
vib
vib
vib har
vib har
diss
diss
diss
diss
diss
ion
Argon
Excitation
– 11.55 eV
100
total
elastic
ionisation
excitation (sum)
– 13 eV
– 14 eV 10
excitation (sum)
Ionisation
– 15.7 eV
cm2]
1
15.7 eV
σ[10‐16c
0.1
0 010.01
0.001
0.1 1 10 100
energy [eV]
Comparing Gases (I)
100 % CO2
20 % CO2
10 % CO2
0 % CO2
Comparing Gases (II)
100 % CO2
20 % CO2
10 % CO2
0 % CO2
10 % CH44