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Avalanche Statistics

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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  1. W. Riegler, H. Schindler, R. Veenhof AvalancheStatistics RD51Collaboration Meeting, 14 October 2008

  2. 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

  3. 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

  4. 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)

  5. 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

  6. 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.“

  7. 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

  8. 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)

  9. Avalanche Growth • The avalanchesizestatisticsisdeterminedbyfluctuations in theearlystages. • After theavalanchesizehasbecomesufficiently large, a stationaryelectronenergydistributionshouldbeattained. Hence, forn 102 – 103theavalancheisexpectedtogrowexponentially. Yule-Furry model Polya

  10. 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

  11. 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

  12. Argon energydistribution 20 kV/cm 30 kV/cm 40 kV/cm 50 kV/cm 60 kV/cm withincreasingfield, theenergydistributionisshiftedtowardshigherenergieswhereionizationis dominant

  13. 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)

  14. Admixtures Ar (80%) + CO2 (20%) Ar (95%) + iC4H10 (5%)

  15. ionizationcross-section (Magboltz) energydistribution (E = 30 kV/cm, p = 1 bar) Whichshapeofσ(ε) yields „better“ avalanchestatistics?

  16. 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

  17. 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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