Avalanche statistics
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W. Riegler, H. Schindler , R. Veenhof. Avalanche Statistics. RD51 Collaboration Meeting, 14 October 2008 . Overview.

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Avalanche Statistics

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Avalanche statistics

W. Riegler, H. Schindler, R. Veenhof

AvalancheStatistics

RD51Collaboration Meeting, 14 October 2008


Overview

Overview

  • The randomnatureoftheelectronmultiplicationprocessleadstofluctuations in theavalanchesize probabilitydistributionP(n, x) that an avalanchecontainsn electrons after a distancex fromitsorigin.

  • Togetherwiththefluctuations in theionizationprocess, avalanchefluctuationsset a fundamental limittodetectorresolution

    Motivation

  • ExactshapeoftheavalanchesizedistributionP(n, x) becomesimportantforsmallnumbersofprimaryelectrons.

  • DetectionefficiencyisaffectedbyP(n, x)

    Outline

  • Review ofavalancheevolutionmodelsandtheresultingdistributions

  • Resultsfromsingleelectronavalanchesimulations in Garfield usingtherecentlyimplementedmicroscopictrackingfeatures

    Assumptions

  • homogeneousfieldE= (E, 0, 0)

  • avalancheinitatedby a singleelectron

  • spacechargeandphotonfeedbacknegligible

η


Yule furry model

Yule-Furry Model

Assumption

  • ionizationprobabilitya (per unitpathlength) isthe same for all avalancheelectrons

  • a = α (Townsend coefficient)

  • In otherwords: theionizationmeanfreepathhas a meanλ = 1/αandisexponentiallydistributed

    Meanavalanchesize

    Distribution

  • The avalanchesizefollows a binomialdistribution

  • For large avalanchesizes, P(n,x) canbe well approximatedby an exponential

  • Efficiency


Avalanche statistics

  • measurements in methylalby H. Schlumbohm significantdeviationsfromtheexponentialat large reducedfields

  • „rounding-off“ characterizedbyparameterαx0 (x0 = Ui/E)

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

αx0=0.19

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

αx0=0.038

E/p = 76.5 V cm-1 Torr-1

αx0=0.044

E/p = 426 V cm-1 Torr-1

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 Model

Legler‘sapproach

ElectronsarecreatedwithenergiesbelowtheionizationenergyeUiand lose mostoftheirkineticenergy after an ionizingcollision electronhastogainenergyfromthefieldbeforebeingabletoionize

 adepends on thedistanceξsincethe last ionizingcollision

Distribution ofionizationmeanfreepath

Legler‘s model gas

Yule-Furry

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

Meanavalanchesize

Distribution

The shapeofthedistributionischaracterizedbytheparameterαx0  [0, ln2]

αx0 1  Yule-Furry

Withincreasingαx0thedistributionbecomesmore „rounded“,maximumapproachesmean

Toy MC

x0=0 μm

x0=1 μm

x0=2 μm

x0=3 μm

IBM 650


Legler s model1

Legler‘s Model

momentsofthedistributioncanbecalculated (asshownbyAlkhazov)

 allows (very) approximative reconstructionofthedistribution(convergenceproblem)

G. D. Alkhazov, StatisticsofElectronAvalanchesand Ultimate Resolutionof Proportional Counters, Nucl. Instr. Meth. 89, 155-165 (1970)

noclosed-form solution

numericalsolutiondifficult

IBM 650

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


Discrete steps

DiscreteSteps

  • „bumps“ seemtoindicateavalancheevolution in steps

  • an electronisstopped after a typicaldistancex0  1/E ofthe order ofseveralμm

  • withprobabilitypitionizes, withprobability (1 – p) it loses itsenergy in a different way after eachstep

Distancetofirstionization

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

x0

Meanavalanchesize after ksteps

Distribution

momentscanbecalculated, but nosolution in closedform

p = 1  deltadistribution

p smallexponential


P lya distribution

Pόlya Distribution

Pόlyadistribution

Efficiency

Goodagreementwith experimental avalanchespectra

Problem:no (convincing) physicalinterpretationoftheparameterm

Byrne‘sapproach:

Distribution ofionizationmeanfreepath

space-chargeeffect

J. Byrne, StatisticsofElectronAvalanches in the Proportional Counter,Nucl. Instr. Meth. 74, 291-296 (1969)


Avalanche growth

Avalanche Growth

  • The avalanchesizestatisticsisdeterminedbyfluctuations in theearlystages.

  • After theavalanchesizehasbecomesufficiently large, a stationaryelectronenergydistributionshouldbeattained. Hence, forn 102 – 103theavalancheisexpectedtogrowexponentially.

Yule-Furry model

Polya


Simulation

Simulation

  • Microscopic_Avalancheprocedure in Garfield availablesince May 2008 performstrackingof all electrons in theavalancheatmolecularlevel (Monte Carlo simulationderivedfromMagboltz).

  • Information obtainedfromthesimulation

    • total numbersofelectronsandions in theavalanche

    • coordinatesofionizationevents

    • electronenergydistribution

    • interactionrates

      Goal

  • Investigateimpactof

    • electricfield

    • pressure

    • gas mixture

      on thesingleelectronavalanchespectrum

  • parallel-plategeometry

  • electronstartswithkineticenergyε = 1 eV

ionization


Argon

Argon

Whatistheeffectoftheelectricfield on theavalanchespectrum?

gapdadjusted such that<n>  500

E = 30 kV/cm, p = 1 bar

E = 55 kV/cm, p = 1 bar

Fit Legler

Fit Polya

Fit Legler

Fit Polya


Argon1

Argon

energydistribution

20 kV/cm

30 kV/cm

40 kV/cm

50 kV/cm

60 kV/cm

withincreasingfield, theenergydistributionisshiftedtowardshigherenergieswhereionizationis dominant


Attachment

Attachment

introduceattachmentcoefficientη (analogouslytoα)

Meanavalanchesize

Distribution forconstantαandη

effective Townsend coefficientα - η

distributionremainsessentiallyexponential

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


Admixtures

Admixtures

Ar (80%) + CO2 (20%)

Ar (95%) + iC4H10 (5%)


Avalanche statistics

ionizationcross-section (Magboltz)

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

Whichshapeofσ(ε) yields „better“ avalanchestatistics?


Avalanche statistics

Ne

Parameters: E = 30 kV/cm, p = 1 bar, d = 0.02 cm

m 3.3

αx0  0.3

<n>  1070

RMS/<n>  0.5

Ar

m 1.7

αx0  0.15

Kr

<n>  900

RMS/<n>  0.7

m 1.4

αx0  0.1

<n>  280

RMS/<n>  0.8


Avalanche statistics

Conclusions

  • „Simple“ models (e. g. Legler‘s model gas) canprovide qualitative insightintothemechanismsofavalancheevolution but areof limited useforthe quantitative predictionofavalanchespectra (noanalyticsolutionavailableor lack ofphysicalinterpretation).

  • Forrealisticmodels, theenergydependenceoftheionization/excitationcross-sectionsandtheelectronenergydistributionhavetobetakenintoaccount Monte Carlo simulationis a betteraproach.

  • Avalanchespectracanbesimulated in Garfield based on molecularcross-sections. Preliminaryresultsconfirmexpectedtendencies (e.g. betterefficiencyathigherfields).


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