9781498736367_C000 2017/8/10 15:46 Page i #1 Fundamentals of Charged Particle Transport in Gases and Condensed Matter 9781498736367_C000 2017/8/10 15:46 Page ii #2 Monograph Series in Physical Sciences Recentbooksintheseries: ExchangeBias:FromThinFilmtoNanogranularandBulkSystems SurenderKumarSharma FundamentalsofChargedParticleTransportinGasesand CondensedMatter RobertRobson,RonaldWhite,andMalteHildebrandt 9781498736367_C000 2017/8/10 15:46 Page iii #3 Fundamentals of Charged Particle Transport in Gases and Condensed Matter By Robert E. Robson, Ronald D. White and Malte Hildebrandt 9781498736367_C000 2017/8/10 15:46 Page iv #4 CoverImage:Simulationsofenergydepositionofpositronsinliquidwater,oftenusedinmodelingasa surrogateforhumantissue.Pointsofhigher(lower)energydepositionareindicatedbyblue(red)spheres, whiletrajectoriesofpositronsbetweencollisions,representedbyblacklines,arebiasedtowardsthedirec- tionofanappliedelectricfield.Eventuallythepositronsslowdownsufficientlytoannihilatewiththeelec- tronsofthemedium,producingtwoback-to-backgammarays,asinPET(positronemissiontomography) investigations.(CourtesyofWadeTattersall) CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 ©2018byTaylor&FrancisGroup CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-1-4987-3636-7(Hardback) Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Reasonableefforts havebeenmadetopublishreliabledataandinformation,buttheauthorandpublishercannotassume responsibilityforthevalidityofallmaterialsortheconsequencesoftheiruse.Theauthorsandpublishers haveattemptedtotracethecopyrightholdersofallmaterialreproducedinthispublicationandapologize tocopyrightholdersifpermissiontopublishinthisformhasnotbeenobtained.Ifanycopyrightmaterial hasnotbeenacknowledgedpleasewriteandletusknowsowemayrectifyinanyfuturereprint. ExceptaspermittedunderU.S.CopyrightLaw,nopartofthisbookmaybereprinted,reproduced,trans- mitted,orutilizedinanyformbyanyelectronic,mechanical,orothermeans,nowknownorhereafter invented,includingphotocopying,microfilming,andrecording,orinanyinformationstorageorretrieval system,withoutwrittenpermissionfromthepublishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC),222RosewoodDrive,Danvers,MA01923,978-750-8400.CCCisanot-for-profitorganization thatprovideslicensesandregistrationforavarietyofusers.Fororganizationsthathavebeengranteda photocopylicensebytheCCC,aseparatesystemofpaymenthasbeenarranged. TrademarkNotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andare usedonlyforidentificationandexplanationwithoutintenttoinfringe. LibraryofCongressCataloging-in-PublicationData Names:Robson,R.(Robert),1946-author.|White,Ronald,author.| Hildebrandt,Malte,author. Title:Fundamentalsofchargedparticletransportingasesandcondensed matter/RobertRobson,RonaldWhite,MalteHildebrandt. Description:BocaRaton,FL:CRCPress,Taylor&FrancisGroup,[2017]| Series:Monographseriesinphysicalsciences Identifiers:LCCN2017011666|ISBN9781498736367(hardback;alk.paper)| ISBN149873636X(hardback;alk.paper) Subjects:LCSH:Kinetictheoryofgases.|Transporttheory.|Fluid dynamics.|Condensedmatter. Classification:LCCQC175.13.R632017|DDC533/.7–dc23 LCrecordavailableathttps://lccn.loc.gov/2017011666 VisittheTaylor&FrancisWebsiteat http://www.taylorandfrancis.com andtheCRCPressWebsiteat http://www.crcpress.com 9781498736367_C000 2017/8/10 15:46 Page v #5 ThisbookisdedicatedtoCarola, MarcellaandIsabelle, and tothememoryofBhalaParanjape, EdwardA.Mason,KurtSuchy andPeterNicoletopoulos 9781498736367_C000 2017/8/10 15:46 Page vi #6 9781498736367_C000 2017/8/10 15:46 Page vii #7 Contents MonographSeriesinPhysicalSciences.................................................xvii Preface............................................................................................xix AbouttheAuthors.............................................................................xxi GlossaryofSymbolsandAcronyms.....................................................xxiii 1 Introduction............................................................................... 1 1.1 Boltzmann’sEquation .................................................... 1 1.1.1 Alittlehistory.................................................... 1 1.1.2 Fromthe“golden”eraofgasdischargesto moderntimes..................................................... 1 1.1.3 Transportprocesses:Traditionalandmodern descriptions....................................................... 3 1.1.4 Themeofthisbook............................................. 4 1.2 SolvingBoltzmann’sEquation ........................................ 4 1.2.1 Thepathtosolution............................................ 4 1.2.2 Acomplementaryapproach:Fluidmodelling....... 6 1.3 ExperimentandSimulation ............................................ 6 1.3.1 Anidealizedapparatus....................................... 6 1.4 AboutthisBook ............................................................ 9 AdditionalGeneralReadingMaterials ..................................... 9 I KineticTheoryFoundations 13 2 BasicTheoreticalConcepts:PhaseandConfigurationSpace...........15 2.1 Preliminaries ................................................................. 15 2.1.1 Configurationandvelocityspace......................... 15 2.1.2 Distributionfunctionandaveraging .................... 16 2.1.3 Polarcoordinatesandsymmetries ....................... 18 2.2 PhaseSpaceandKineticEquation ................................... 20 2.2.1 Trajectoriesinphasespace .................................. 20 2.2.2 Kineticequationinphasespace........................... 21 2.2.3 Equilibrium....................................................... 23 vii 9781498736367_C000 2017/8/10 15:46 Page viii #8 viii Contents 2.3 KineticEquationsforaMixture....................................... 23 2.3.1 Thegeneralkineticequation................................ 23 2.3.2 Diluteparticlesinaneutralmedium.................... 23 2.3.3 Locality,instantaneity,andlinearity..................... 24 2.4 MomentEquations ........................................................ 24 2.4.1 Thegeneralmomentequation............................. 24 2.4.2 Equationofcontinuity........................................ 25 2.5 ConcludingRemarks ..................................................... 25 3 BoltzmannCollisionIntegral,H-Theorem,andFokker–Planck Equation...................................................................................27 3.1 ClassicalCollisionDynamics .......................................... 27 3.1.1 Conservationlaws.............................................. 27 3.1.2 Transformationofcoordinates............................. 28 3.2 DifferentialCrossSection ............................................... 28 3.2.1 Basiccollisionparameters................................... 28 3.2.2 Symmetriesinspaceandtime ............................. 30 3.2.3 Partialcrosssections........................................... 32 3.2.4 Calculationofcrosssections................................ 32 3.3 BoltzmannCollisionIntegral .......................................... 33 3.3.1 Collisionmoment............................................... 33 3.3.2 Fundamentalassumptions.................................. 34 (∂f)(1,2) 3.3.3 Calculating ............................................ 35 ∂t col 3.4 SimpleGas .................................................................... 36 3.4.1 ClassicalBoltzmannkineticequation................... 36 3.4.2 Summationalinvariants...................................... 36 3.4.3 H-theorem,equilibrium,andtheMaxwellian distribution........................................................ 37 3.5 Fokker–PlanckKineticEquation...................................... 39 3.5.1 Smalldeflectioncollisions................................... 39 3.5.2 Coulombscattering............................................ 40 3.6 ConcludingRemarks ..................................................... 41 4 InteractionPotentialsandCrossSections .....................................43 4.1 Introduction .................................................................. 43 4.2 ClassicalScatteringTheory ............................................. 43 4.2.1 Differentialandpartialcrosssections................... 43 4.2.2 Inversepowerlawpotentials............................... 45 4.3 InverseFourth-PowerLawPotential ............................... 47 4.3.1 Polarizationpotential ......................................... 47 4.3.2 Constantcollisionfrequency ............................... 48 4.4 RealisticInteractionPotentials ........................................ 48 4.4.1 TheMason–Schamppotential.............................. 48 4.4.2 Momentumtransfercollisionfrequency............... 49 9781498736367_C000 2017/8/10 15:46 Page ix #9 Contents ix 4.5 CalculationofCrossSectionsforaGeneralInteraction Potential ....................................................................... 50 4.5.1 Transformationofvariables................................. 50 4.5.2 Orbiting,criticalenergy,andcrosssections........... 51 4.5.3 Determinationofϵ ............................................ 52 c 4.6 CrossSectionsforSpecificInteractionPotentials ............... 53 4.6.1 Numericalmethodsandtechniques..................... 53 4.6.2 Powerlawpotentials.......................................... 54 4.6.3 Mason–Schamp(12-6-4)potential........................ 55 4.7 ConcludingRemarks ..................................................... 58 5 KineticEquationsforDiluteParticlesinGases..............................59 5.1 LowDensityChargedParticlesinGases .......................... 59 5.1.1 Freediffusionorswarmlimit.............................. 59 5.1.2 ThelinearBoltzmannkineticequation ................. 59 5.1.3 Momentequations ............................................. 61 5.2 ChargeExchange ........................................................... 61 5.2.1 Collisionmodel.................................................. 61 5.2.2 Polarizationpotentialand Bhatnagar–Gross–Krookequation........................ 62 5.3 CollisionTermforExtremesofMassRatio ....................... 63 5.3.1 Fractionalenergyexchange................................. 63 5.3.2 HeavyionsandRayleighlimit............................. 64 5.3.3 LightchargedparticlesandLorentzgas ............... 66 5.4 InelasticCollisions ......................................................... 71 5.4.1 WangChang–Uhlenbeck–deBoercollision term.................................................................. 71 5.4.2 Semi-classicalandquantumcollisionoperators..... 73 5.4.3 Inelasticcollisiontermforlightparticles............... 75 5.5 Non-Conservative,ReactiveCollisions ............................ 76 5.5.1 Classificationofreactivecollisions....................... 76 5.5.2 Notation............................................................ 77 5.5.3 Particlelosscollisionterm................................... 78 5.5.4 Electronimpactionization................................... 79 5.6 Two-TermKineticEquationsforaLorentzGas ................. 79 5.7 ConcludingRemarks ..................................................... 80 6 ChargedParticlesinCondensedMatter.......................................81 6.1 ChargeCarriersinCrystallineSemiconductors ................. 81 6.2 AmorphousMaterials .................................................... 81 6.2.1 Trappingandtherelaxationfunction.................... 81 6.2.2 Thekineticequationforamorphousmaterials....... 82 6.3 CoherentScatteringinSoft-CondensedMatter ................. 84 6.3.1 Amodelofcoherentscattering............................ 84