Principles-of-Plasma-Discharges-and-Materials-Proc

CHAPTER3ATOMICCOLLISIONS3.1BASICCONCEPTSWhentwoparticlescollide,variousphenomenamayoccur.Asexamples,oneorbothparticlesmaychangetheirmomentumortheirenergy,neutralparticlescanbecomeionized,andionizedparticlescanbecomeneutral.Weintroducethefunda-mentalsofcollisionsbetweenelectrons,positiveions,andgasatomsinthischapter,concentratingonsimpleclassicalestimatesoftheimportantprocessesinnoblegasdischargessuchasargon.Forelectronscollidingwithatoms,themainprocessesareelasticscatteringinwhichprimarilytheelectronmomentumischanged,andinelas-ticprocessessuchasexcitationandionization.Forionscollidingwithatoms,themainprocessesareelasticscatteringinwhichmomentumandenergyareexchanged,andresonantchargetransfer.Otherimportantprocessesoccurinmoleculargases.Theseincludedissociation,dissociativerecombination,processesinvolvingnegativeions,suchasattachment,detachment,andpositive–negativeionchargetransfer,andprocessesinvolvingexcitationofmolecularvibrationsandrotations.WedeferconsiderationofcollisionsinmoleculargasestoChapter8.ElasticandInelasticCollisionsCollisionsconservemomentumandenergy:thetotalmomentumandenergyofthecollidingparticlesaftercollisionareequaltothatbeforecollision.Electronsandfullystrippedionspossessonlykineticenergy.Atomsandpartiallystrippedionshaveinternalenergylevelstructuresandcanbeexcited,de-excited,orionized,43PrinciplesofPlasmaDischargesandMaterialsProcessing,byM.A.LiebermanandA.J.Lichtenberg.ISBN0-471-72001-1Copyright#2005JohnWiley&Sons,Inc.correspondingtochangesinpotentialenergy.Itisthetotalenergy,whichisthesumofthekineticandpotentialenergy,thatisconservedinacollision.Iftheinternalenergiesofthecollisionpartnersdonotchange,thenthesumofkineticenergiesisconservedandthecollisionissaidtobeelastic.Althoughthetotalkineticenergyisconserved,kineticenergyisgenerallyexchangedbetweenparticles.Ifthesumofkineticenergiesisnotconserved,thenthecollisionisinelas-tic.Mostinelasticcollisionsinvolveexcitationorionization,suchthatthesumofkineticenergiesaftercollisionislessthanthatbeforecollision.However,super-elasticcollisionscanoccurinwhichanexcitedatomcanbede-excitedbyacollision,increasingthesumofkineticenergies.CollisionParametersThefundamentalquantitythatcharacterizesacollisionisitscrosssections(vR),wherevRistherelativevelocitybetweentheparticlesbeforecollision.Todefinethis,weconsiderfirstthesimplestsituationshowninFigure3.1,inwhichafluxG¼nvofparticleshavingmassm,densityn,andfixedvelocityvisincidentonahalf-spacex.0ofstationary,infinitelymassive“target”particleshavingdensityng.Inthiscase,vR¼v.Letdnbethenumberofincidentparticlesperunitvolumeatxthatundergoan“interaction”withthetargetparticleswithinadifferentialdistancedx,removingthemfromtheincidentbeam.Clearly,dnisproportionalton,ng,anddxforinfrequentcollisionswithindx.Hencewecanwritedn¼
snngdx(3:1:1)wheretheconstantofproportionalitysthathasbeenintroducedhasunitsofareaandiscalledthecrosssectionfortheinteraction.Theminussigndenotesremovalfromthebeam.Todefineacrosssection,the“interaction”mustbespecified,forexample,ionizationofthetargetparticle,excitationoftheincidentparticletoagivenenergystate,orscatteringoftheincidentparticlebyanangleexceedingp=2.Multiplying(3.1.1)byv,wefindasimilarequationfortheflux:dG¼
sGngdx(3:1:2)AFIGURE3.1.Afluxofincidentparticlescollideswithapopulationoftargetparticlesinthehalf-spacex.0.44ATOMICCOLLISIONSForasimpleinterpretationofs,lettheincidentandtargetparticlesbehardelasticspheresofradiia1anda2,andletthe“interaction”beacollisionbetweenthespheres.Inadistancedxtherearengdxtargetswithinaunitareaperpendiculartox.Drawacircleofradiusa12¼a1þa2inthex¼constplaneabouteachtarget.Acollisionoccursifthecentersoftheincidentandtargetparticlesfallwithinthisradius.Hencethefractionoftheunitareaforwhichacollisionoccursisngdxpa212.ThefractionofincidentparticlesthatcollidewithindxisthendGG¼dnn¼
ngsdx(3:1:3)wheres¼pa212(3:1:4)isthehardspherecrosssection.Inthisparticularcase,sisindependentofv.Equation(3.1.2)isreadilyintegratedtogivethecollidedfluxG(x)¼G0(1
e
x=l)(3:1:5)withtheuncollidedfluxG0e
x=l.Thequantityl¼1ngs(3:1:6)isthemeanfreepathorthedecayofthebeam,thatis,thedistanceoverwhichtheuncollidedfluxdecreasesto1=eofitsinitialvalueG0atx¼0.Ifthevelocityofthebeamisv,thenthemeantimebetweeninteractionsist¼lv(3:1:7)Itsinverseistheinteractionorcollisionfrequencyn;t
1¼ngsv(3:1:8)andisthenumberofinteractionspersecondthatanincidentparticlehaswiththetargetparticlepopulation.Wecanalsodefinethecollisionfrequencyperunitdensity,whichiscalledtherateconstantK¼sv(3:1:9)3.1BASICCONCEPTS45and,trivially,from(3.1.8)and(3.1.9)n¼Kng(3:1:10)DifferentialScatteringCrossSectionLetusconsideronlythoseinteractionsthatscattertheparticlesbyu¼908ormore.Forhardspheres,takingtheangleofincidenceequaltotheangleofreflection,the908collisionoccursonthex¼458diagonal(seeFig.3.2),thereforehavingacrosssections90¼pa2122,(3:1:11)whichisafactoroftwosmallerthan(3.1.4).Ofcourse,multiplecollisionsatsmallerangles(radiilargerthana12=ffiffiffi2p)alsoeventuallyscatterincidentparticlesthrough908.Thisindeterminacyindicatesthatamoreprecisewayofdetermini