collisions of charged particles with atoms ECCPA : Calculation of classical and quantum cross sections for elastic Computer Physics Communications

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Computer Physics Communications

www.elsevier.com/locate/cpc

ECCPA : Calculation of classical and quantum cross sections for elastic collisions of charged particles with atoms ^{✩ , ✩✩}

Francesc Salvat

^a,∗

, Josep Llosa

^a

, Antonio M. Lallena

^b

, Julio Almansa

^c

aFacultatdeFísica(FQAandICC),UniversitatdeBarcelona,Diagonal645,08028Barcelona,Catalonia,Spain bDepartamentodeFísicaAtómica,MolecularyNuclear.UniversidaddeGranada,E-18071Granada,Spain

cServiciodeRadiofísicayP.R.,Hosp.Univ.VirgendelasNieves,Avda.delasFuerzasArmadas2,E-18014Granada,Spain

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received21October2021

Receivedinrevisedform18March2022 Accepted1April2022

Availableonline11April2022

Keywords:

Elasticcollisions Classicalscatteringtheory Relativisticcollisions Eikonalapproximation Bornapproximation Partial-waveanalysis WKBphaseshifts

TheFortran program eccpa calculatesdifferentialand integratedcrosssections forelasticcollisions of charged particleswith atoms by using the classical-trajectory methodand several quantum methods andapproximations.Thecollisionsaredescribedwithintheframeworkofthestatic-fieldapproximation, withtheinteractionbetweentheprojectileandthetargetatomrepresented bythe Coulombpotential oftheatomicnucleusscreenedbytheatomicelectrons.Toallowtheuseoffastandrobustcalculation methods,theinteractionisassumedtobethesameinthecenter-of-massframeand inthelaboratory frame.Althoughthisassumptionneglectstheeffectofrelativityontheinteraction,itallowsusingstrict relativistic kinematics.The equationof the relative motion inthe center-of-mass frame is shownto havethesameformasinthenon-relativistictheory,witharelativisticreduced massandaneffective potential.Thewaveequationfortherelativemotion,asobtainedfromthecorrespondenceprinciple,is formallyidenticaltothenon-relativisticSchrödingerequationwiththereducedmassandtheeffective potential,and it reduces to the familiarKlein-Gordonequation whenthe mass ofthe targetatom is muchlargerthanthatofthe projectile.Collisionsofspin1/2projectilesare alsodescribedbysolving theDiracwaveequation.Variousapproximatesolutionmethodsaredescribedandappliedtoageneric potential represented asa sumofYukawa terms, whichallowsa goodpart ofthe calculations tobe performedanalytically.Theprogram eccpa isusefulforassessingthevalidityandtherelativeaccuracyof thevariousapproximations,andasapedagogicaltool.

Programsummary ProgramTitle: eccpa

CPCLibrarylinktoprogramfiles: https://doi.org/10.17632/c3tn9hyfvb.1 Licensingprovisions: CCbyNC3.0

Programminglanguage: Fortran90/95

Nature of problem: The program computes differential cross sections (DCSs) for elastic collisions of charged particles (electrons, positrons, muons, antimuons, protons, antiprotons, and alphas) with neutralatoms.Calculationsareperformedwithinthestatic-fieldapproximationwithscreenedCoulomb potentials expressed as a sum of Yukawa terms with their parameters fitted to approximate the atomicelectrostaticpotentialsresultingfromtheThomas–Fermimodelandfromself-consistentDirac–

Hartree–Fock–Slater calculations. The program eccpa provides DCSs computed with four different approaches:theclassicaltrajectorymethod,theBornapproximation,thepartial-waveexpansionmethod with approximate phase shifts, and the eikonal approximation. The user is allowed to select the atomicnumber ofthe targetatom, the potential model,the kindofprojectile and itskinetic energy.

Calculationresultsarewritteninanumberofoutputfileswithformatssuitedforvisualizationwitha plottingprogram.AJavagraphicaluserinterfaceallowsrunningtheprogramandvisualizingtheresults interactively.

Solution method: A relativistic extension of the classical trajectory method is formulated on the assumption that the interaction in the center-of-mass frame is central, which is a fundamental

✩ ThereviewofthispaperwasarrangedbyProf.StephanFritzsche.

✩✩ ThispaperanditsassociatedcomputerprogramareavailableviatheComputerPhysicsCommunicationshomepageonScienceDirect(http://www.sciencedirect.com/ science/journal/00104655).

*

Correspondingauthor.

E-mailaddress:[email protected](F. Salvat).

https://doi.org/10.1016/j.cpc.2022.108368

0010-4655/©^{2022TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).

requirement of the adopted calculation schemes; the DCS in the laboratory frame is then obtained fromtherelativistic(Lorentz)transformoftheDCScalculatedinthecenter-of-massframe.Thisscheme qualifies as semi-relativistic, because it accounts for relativistic kinematics in a rigorous way, but disregards the differences between the interactions observed from the laboratory and the center-of- massframes.WeconsidertheelementaryquantumformulationbasedontherelativisticSchrödinger(or Klein–Gordon)waveequationobtainedfromthecorrespondenceprinciple.AccurateDCSsforpotential scatteringcanbecomputedbyusingthepartial-waveexpansionmethod,attheexpenseofconsiderable numericalwork.Toavoidthedifficultcalculationofphaseshiftsfromthenumericalsolutionoftheradial waveequation,weadoptasimplifiedstrategythatcombinesthe(first)Bornapproximation,forboththe scatteringamplitudeandthephaseshifts,andtheWentzel–Kramers–Brillouin(WKB)approximationfor the phaseshifts.We alsodescribethe semi-classicaleikonal approximation,whichis knownto yield reliableDCSsforcollisionswithsmallscatteringangles.Thecaseofcollisionsofelectronsandpositrons isconsideredonsimilargrounds,withthescatteringamplitudesobtainedfromtheDiracequation.

ThenumericalworkissimplifiedbyapproximatingtheinteractionpotentialasasumofYukawaterms, whichallows performing a good part of the calculations analytically. Integrals offunctions givenby analyticalformulasarecalculatedbymeansofanadaptivealgorithmthatcombinesthe20-pointGauss- Legendrequadratureformulawithabisectionscheme;thisalgorithmallowsstrictcontrolofnumerical errorsandgivesresultswitharelativeaccuracybetterthanabout10⁻¹⁰forwell-behavedintegrands.The wholecalculationforagivenenergyoftheprojectiletakesnolongerthanafewsecondsonamodern personalcomputer,quiteirrespectivelyoftheenergyandoftheatomicnumberofthetargetatom.

Additional comments including restrictions and unusual features: The adopted interaction potentials correspondto atoms with pointnuclei. The useof aparameterization insteadof numerical tables of the potential (obtained, e.g.,from atomic structurecalculations) hasa minor effectonthe calculated DCSs.Thiseffectislimitedtolargescatteringangles,wheretheactualDCSdoesdifferfromcalculations withscreenedCoulombpotentialsduetotheeffectofthefinitesizeandstructureoftheatomicnucleus, whichisdisregardedhere.

DCSsobtainedfromthepartial-waveexpansionmethodandwiththeeikonalapproximationprovidea fairlyaccuratedescriptionofcollisionswithsmallandmoderatedeflectionangles.Theycanbeused,e.g., inMonteCarlosimulationsofthetransportoffastchargedparticlesinmatter.Theinformationgenerated bytheprogramallowsassessingtheaccuracyofcalculationswiththevariousapproaches,andpermits identifyingtherangesofvalidityoftheclassicaltrajectorymethodandtheBornapproximation.

©^{2022TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Elastic collisions ofcharged particles withatoms play a central role in theoretical radiationphysics. By definition,elastic collisions areinteractions thatdonot causeexcitation ofthetarget atom,whichusuallyisinits groundstate. Thesecollisions mayproducelarge deflections of the projectile and, consequently, have a direct effecton the structure of trajectoriesof fastcharged particles moving in materialmedia.Inaddition,eachcollisioninvolvesanenergytransferfromtheprojectiletothetargetatomwhichmanifestsastherecoil ofthelatteraftertheinteractionandgivesrisetotheso-callednuclear contributiontothestoppingpower.

Elasticcollisionsareusuallydescribedbymeansofthetheoryofscatteringbycentralpotentials.Thenon-relativistictheoryisstudied inmosttextbooksofbothclassical[1,2] andquantummechanics[3–5].Calculationsnormallyassumethatthepotentialiscentralbecause this is the only case that admits practicable formal or numerical solutions. In the case of collisions with a bare point nucleus, the interaction reduces to the Coulomb potential and the problem can be solved analytically; the corresponding differential cross section (DCS)isgivenbythefamousanalyticalformuladerivedbyRutherford[6],whichwasinstrumentalinthediscoveryoftheatomicnucleus.

The calculationofelasticcollisions withneutralatoms(i.e.,ofthescatteringbyscreened Coulomb potentials)ismoredifficult,because ithastobeperformednumerically.Thenon-relativisticclassicaltrajectorymethod[2] allowsthecalculationoftheDCSthroughasetof quadratures[7].

In aquantum formulation, the scatteringamplitudeisexpressed asa partial-wave series,i.e., asa Legendre serieswithcoefficients determinedbythescatteringphaseshifts,whichrepresenttheeffectofthepotentialontheasymptoticbehaviorofsphericalwaveswith well-definedangularmomentum.Numerical calculationswiththisschemearefeasibleforelectrons andpositrons;theFortran program elsepa[8] calculatesveryaccurate DCSsfortheseparticlesbyusingthe relativistic(Dirac)partial-wave methodwitharbitraryscreened atomicpotentials.Unfortunately,similarcalculationsforheavierprojectilesare notdoablebecausethe shortde Brogliewave lengthsof theseprojectilesmakethenumericalsolutionoftheradialwave equationextremelydifficult;inaddition,thecorrespondingpartial-wave seriesconvergeveryslowly.Consequently,onemusthaverecoursetoapproximationmethods.

Although theessentialsof thetheory of scatteringby centralpotentials andofelastic collisions ofcharged particlesare covered in graduate coursesandtextbooks,comparisons ofresultsfromdifferentapproximationsare hardto findintheliterature. Thesecompar- isons areimportanttoillustratethecapabilitiesandlimitationsofthetheoreticalapproaches. Inthepresentarticlewedescribe several approximatemethodsthatallowthefastcalculationofDCSsforelasticcollisionsofchargedparticleswithneutralatoms.

WehavewrittenaFortranprogramnamed eccpa thatcomputeselasticcollisionsofelectrons,muons,protons(andtheantiparticlesof thethree) andalphaswithatoms. ThisprogramprovidestheDCS(inthecenter-of-massandthelaboratoryframes)calculatedbyusing thefollowingtheoreticalapproaches:

◦ ^{themethodofclassical}trajectories,

◦ ^{thefirstBorn}approximation,

◦ ^thepartial-waveexpansionmethodwithphaseshiftsobtainedfromtheBornandWKBapproximations,and

◦ ^theeikonalapproximation.

Both the partial-wave expansion method and the eikonal approximation involve a great deal of numerical work for potentials given in numerical form. To alleviate the work without sacrificing the reliability of the calculated DCSs, we consider interaction potentials expressedasasumofYukawa-liketerms,withparametersdeterminedbyfittingrealisticatomicpotentials.Potentialswiththisanalytical formallowperformingagoodpartofthecalculationsanalytically,andreducetheevaluationoftheDCStoasetofintegralsoffunctions ofasinglevariable.

Although eccpa worksforprojectileswitharbitrarykineticenergies,provideditdoesnotencounternumericalconflicts,thecalculated DCSsarereliable(i.e.,consistentwithmeasurements)onlyforalimitedrangeofkineticenergiesoftheprojectile,whichisdeterminedby thesimplificationsoftheunderlyingmodel.Ononehand,thetargetatomisconsideredasarigiddistributionofcharge,andtheprojectile isassumedtotravel“naked” (i.e.,thepossiblecaptureofelectronsby positively-chargedprojectilesisdisregarded).Thesesimplifications are validonlyforprojectileswithkinetic energies largerthanabout2MeV per unifiedmassunit. Ontheother hand,our approximate treatmentofrelativisticeffectsisexpectedtobevalidonlyforprojectileswithkineticenergylessthanabouttheirrestenergy.

Combined with a plotting program to visualize the results, eccpa becomes a useful tool forassessing the validity of the classical trajectorymethodandthequantumapproximations,andalsoforillustratingfundamentalaspectsofthetheory.Asamatteroffact, eccpa providesalltheinformationonelasticcollisionsthatisneededforMonteCarlosimulationofthetransportofchargedparticlesinmatter [9,10], where theevolution of individual particles isdescribed asa sequence ofrandom interaction eventssampled fromthe adopted DCSs.

Thepresentarticleisorganizedasfollows.TheadoptedanalyticalformsoftheatomicpotentialsaredescribedinSection2.InSection3 wederivetherelativisticequationofmotioninthecenter-of-mass(CM)frameandwedescribethecalculationoftheDCSbythemethod of classical trajectories. Section 4 deals with the relativisticwave equation for the relative motion of the colliding particles and with severalmethodsandapproximationsforcomputingtheDCSintheCM frame.ThetransformationoftheDCSfromtheCM frametothe laboratory frame, wherethe target atom isinitially at rest,isconsidered in Section 5.Section 6 deals withcollisions of electrons and positrons, whicharedescribed onthebasis oftheDiracwaveequation withtheaidoftheBornandWKB approximations.The Fortran program eccpa anditsJavagraphicalinterfacearedescribedinSection7.Finally,Section8presentsexampleresults.

2. Interactionpotential

We consider the collisions of charged particles with atoms from the standpoint of the static-fieldapproximation, i.e., as scattering of the projectile by the electrostatic field of the target atom, suitably averaged over directions [11,12]. It should be mentioned that this approximation gives results ingood agreement with measurements only for particles with sufficiently highenergies, higher than about 1 keV in the case of electrons and positrons, ^{2 MeV for protons. The} static-field approximation disregards the effect of the dipole polarizabilityofthetargetatom,thatis,thefact thattheelectricfieldoftheprojectile shiftstheatomicchargesandtheinduced dipolemomentactsbackontheprojectile.Thispolarizabilityeffectisappreciableonlyforslowprojectilesandatsmallscatteringangles (correspondingtolargeimpactparameters),becauseatomicpolarizationiseffectiveonlyunderslowly-varyingelectricfields.Inaddition, resultsfrommeasurementsofelasticcollisions areaffectedbyinelasticinteractions,whichremoveprojectilesfromthe“elasticchannel”.

Finally,inthecaseofprojectileelectrons,elasticcollisionsarealteredbyexchangeeffects,whichresultfromtheindistinguishabilityofthe projectileandtheelectronsinthetargetatom.Intheso-calledoptical-potentialmodels theeffectsofpolarizationandinelasticabsorption, andexchange inthecaseofelectrons, aredescribed approximatelyby meansofeffectivelocalpotentials withan absorptiveimaginary part(see,e.g.,Ref. [13] andreferencestherein).

We assumea neutraltargetatom ofatomicnumber Z ,withapoint nucleusofcharge Z e (e denotestheelementarycharge)atthe originofcoordinates,andasphericallysymmetriccloudof Z electronsinitsgroundstate.Theinteractionpotentialenergybetweenthe projectile(ofcharge Z1e)atr andthetargetatomis

V

(

r

) =

Z1e

ϕ

es

(

r

) ,

(1)

where

ϕ

es(r)istheelectrostaticpotentialoftheatomicchargedistribution,

ϕ

es

(

r

) =

^{Z e} r

−

e

⎛

⎝

¹ r



r 0

ρ (

r^

)

4

π

r^2dr^

+



∞ r

ρ (

r^

)

4

π

r^dr^

⎞

⎠ ,

(2)

and

ρ

(r)istheatomicelectrondensity.Itiscustomarytowrite

ϕ

es

(

r

) =

^{Z e}

r

(

r

),

(3)

where(r),thescreeningfunction,describestheelectrostaticshieldingofthenuclearchargebytheatomicelectrons.

Tofacilitatecalculations,weuseapproximatescreeningfunctionsgivenbythefollowinganalyticalexpression

(

r

) =



3 i=1

Aiexp

(−

air

)

(4)

withparametersdeterminedbyfittingthenumericalscreeningfunctionsobtainedfromdifferentatomicstructurecalculations.Thecorre- spondinginteractionpotentialis

V

(

r

) =

^{Z1Z e2} r



3 i=¹

A_iexp

( −

^air

).

(5)

Analyticalpotentialsofthistype(asumofYukawa-liketerms)havebeenusedbymanyauthorsinstudiesofparticlecollisions[14,15].

Inthecalculationsdescribedbelowweconsiderthefollowingscreeningfunctionmodelsandsetsofparameters.

◦ Thomas–Fermi–Molière(TFM)screeningfunction

The simplesttheoretical framework toobtain approximateatomicelectron densities isthe Thomas–Fermimodelin whichthe electron cloud is considered asa locallyhomogeneous electron gas bound by the screened Coulomb field of the nucleus. This model yields a

“universal”screeningfunction,applicabletoallelements,thatisdeterminedbyadifferentialequation.Becausethisequationneedstobe solvednumerically[16],Molière[17] usedtheform(4) asanapproximationtothenumericalThomas–Fermiscreeningfunctionwiththe setofparameters

A1

=

⁰

.

10

,

A2

=

⁰

.

55

,

A3

=

⁰

.

35

,

a₁

=

⁶

.

0

/

b

,

a₂

=

¹

.

2

/

b

,

a₃

=

⁰

.

3

/

b

,

(6)

where

b

≡ (

3

π )

^2/3 2^7/3

¯

h²

mee²Z^−1/3

=

⁰

.

88534Z^−1/3a0 (7)

isaradialscaleparameterknownastheThomas–Fermiradius,anda0= ¯^h2/(mee²)istheBohrradius,^{h is}¯ ^{thereducedPlanckconstant.}

Theapproximation (6) agreescloselywiththeThomas–Fermiscreeningfunctionintheregionr^{6b wherethelattertakes}appreciable values.

◦ Dirac–Hartree–Fock–Slater(DHFS)screeningfunction

Ingeneral, screeningfunctionsobtainedfromself-consistent atomicelectron densitiesare givennumerically.In thepresentcalculations weusetheanalyticalapproximation(4) withthesetofparametersgivenbySalvatetal.[18],whichwereobtainedbyfittingtheDirac–

Hartree–Fock–Slaterself-consistent electrondensitiesofneutralatoms[19,20].Withtheseparameters,theanalyticalapproximation and the numericalDHFSpotential yield thesame DCSs forscatteringofchargedparticles atsmall angleswhen calculatedwithin the Born approximation[18],whichisexpectedtobeaccurateforprojectileswithsufficientlyhighenergies(seeSection4.2).

◦ ^{TheWentzelpotential}

Inan earlycalculationofelasticscatteringwiththeBorn approximation,Wentzel[21] usedapotentialofthegenericform(5) withthe one-termscreeningfunctionW(r)=^exp(−^ar),

VW

(

r

) =

^{Z1Z e2}

r exp

(−

ar

),

a

=

1

/

R

,

(8)

where R isthe“atomicradius”or“screeninglength”,whichisfrequentlyapproximatedas

R

=

⁰

.

88534Z^−1/3a₀

,

(9)

tocomplywiththeThomas–Fermiscaling.

3. Classicalelasticcollisions

Letusstartwiththeclassicalcalculationofcollisionsofaprojectileparticle(1)(ofmassM1 andcharge Z1e)andatargetatom(2)of atomicnumberZ (andmass M₂),whichinteractthroughthepotential V(r).Tocovertheenergyrangeofinterestinradiationtransport, weuserelativistickinematics.Werecallthatthelinearmomentump,thekineticenergyE,andthetotalenergyW ofaparticleofmass M arerelatedby

(

cp

)

²

+

M²c⁴

= (

E

+

Mc²

)

²

=

W²

.

(10)

Thesequantitiesareconvenientlyexpressedas

p

= β γ

Mcp

ˆ ,

E

= ( γ ₋

1

)

Mc²

,

and W

= γ

Mc²

,

(11)

where

β =

^v c

=



1

+

Mc p

2



₋1/2

=



E

(

E

+

^2Mc2

)

E

+

^{Mc2 (12)}

isthevelocityoftheparticleinunitsofthespeedoflightc,and

γ ₌

1 1

− β

²

=



1

+ 

_p

Mc



2

=

^E

+

^Mc2

Mc² (13)

isitstotalenergyinunitsoftherestenergyMc².

Inpracticaltransportstudies,collisionsaredescribedinthelaboratory(L)referenceframe,whereinitially(i.e.,beforethecollision)the targetatomisatrestandtheprojectilemovesinthedirectionofthez axiswithlinearmomentum p_1i.Becausetheinteractioniscentral, trajectoriesofthecollidingparticleslieinaplane,thescatteringplane.Wesetthedirectionofthex axissothatthez- y planecoincides withthescatteringplane.Then, intheLframe,the energy-momentumfour-vectorsoftheprojectile andthetargetbefore thecollision are,respectively,

P1i

= (

W1ic⁻¹

,

0

,

0

,

p1i

)

and P2i

= (

W2ic⁻¹

,

0

,

0

,

0

),

(14) whereWjidenotestheinitialenergyofparticle j,

W_1i

=

^E1i

+

^M1c² and W_2i

=

^M2c²

,

(15)

where E1iisthekineticenergyoftheincidentprojectile.Thefour-momentaofthetwoparticlesafterthecollisionare

P_1f

= (

^W1fc⁻¹

,

0

,

p_1fsin

θ

1

,

p_1fcos

θ

1

)

(16a)

and

P_2f

= (

^W2fc⁻¹

,

0

,

p_2fsin

θ

2

,−

^p2fcos

θ

2

).

(16b)

Thekinematicsofelasticcollisionsissimplerinthecenter-of-mass (CM)frame,inwhichthetotallinearmomentumofthecolliding particlesvanishes.InCMthetwoparticleshaveoppositelinearmomentaofequalmagnitude,andthecollisioncausesonlyarotationof their linearmomentum vectors.Inthenon-relativisticformulation,theinteractioninCM isthesameasintheLframe,andthemotion oftheprojectilerelativetothetargetisthatofasingleparticlehavingthereducedmassofthepairundertheactionoftheinteraction force[2].

WeshallcalculatetheDCSintheCMframebysolvingtherelativistic equationfortherelativemotion,assumingacentralpotentialin theCMframe,andweshallinfertheDCSintheLframebymeansofaLorentzboost.Specifically,weconsiderthattheinteractioninCMis thesameasintheLframe,thatis,wedisregardthetransformationpropertiesoftheelectromagneticfieldunderLorentztransformations (see,e.g.,Ref. [22]).Theassumptionofacentralpotential,whichisessentialtorenderthecalculationsdoable,iscorrectfornon-relativistic projectilesbutitbecomesinconsistentforprojectileswithhighenergies,suchthatthevelocityoftheCM framebecomescomparableto c (partlybecausetheFitzGerald-Lorentzcontractiondestroysthesphericalsymmetryoftheinteraction).Ourapproachqualifiesassemi- relativistic,becauseitimplementsstrictrelativistickinematicsbutdisregardstheeffectsofrelativityontheinteraction.

We consider that the potential V(r) tends to zero at large distances, so that the four-momenta of the particles before and after the interaction are completely defined by their masses andlinear momenta. We useprimes to designatequantities in the CM frame.

The initialfour-momenta ofthetwo particles inCM are P^_ji= (^W_ji^c−1,p^_ji).The CM moveswithrespectto theLframe withconstant velocityvCM=^vCMz.ˆ ThetransformationfromtheCMtotheLframeisaboostinthez direction.Thefour-momentaP= (^{W c−1},p)and P^= (^Wc−1,p^)ofaparticleintheLandCMframesarerelatedby

W

= γ

CM

(

W^

+ β

CMcp^_z

),

p_z

= γ

CM

(

p^_z

+ β

CMW^c⁻¹

),

p_x

=

^p_x

,

p_y

=

^p_y

,

(17)

where

β

CM

=

^vCM

c

, γ

CM

= 

¹

1

− β

_CM²

.

(18)

The velocity vCM of the CM frameis determined by applyingthe transformation (17) to thetotal initial four-momentumin CM, P^_i= P^_1i+^P_2i= (^W_1i^c−1+^W_2i^c−1,0),whichgives

p1i

= γ

CM

β

CM

(

W_1i^

+

W_2i^

)

c⁻¹

,

W1i

+

W2i

= γ

CM

(

W_1i^

+

W_2i^

).

(19) Theseequationsimplythat

β

CM

=

^cp1i W_1i

+

^W2i

.

(20)

Thesquareofthetotalfour-momentumoftheparticles,

s²

≡

^c2

(

P1

+

^P2

)

²

= (

^W1

+

^W2

)

²

−

^c2

(

p1

+

^p2

)

²

,

(21) isaLorentzinvariant.IntheCMframe,s equalsthetotalenergyoftheparticles,s=^W1^+^W₂^.Beforetheinteractionwehave

s²

= (

^W1i

+

^W2i

)

²

−

^c2p2_1i

= (

^M1c²

+

^M2c²

)

²

+

^2M2c²E_1i

.

(22) Theequality(20) impliesthat

γ

CM

=

^W1i

+

^W2i

s

.

(23)

Beforethecollision,intheCMframethecollidingparticleshaveoppositelinearmomenta

p^_1i

= −

^p_2i

≡

^p_i ⁽²⁴⁾

withmagnitude

p^_i

= β

CM

γ

CMM2c

=

^M2c2

s p1i

.

(25)

Theinitialtotalenergiesoftheparticlesaregivenby W_1i^

= 

M²₁c⁴

+

^p_i^{2c2 and W}_1i^

= 

M²₁c⁴

+

^p_i^2c2

.

(26)

After thecollision,when theinteractionenergyhasvanished,the particlesmove awaywithlinearmomentaofthesame magnitudep^_i andwiththesametotalenergies,thatis,

W_1f^

=

W_1i^ and W_2f^

=

W_2i^

.

(27)

3.1. RelativemotionintheCMframe

LetusnowconsidertheequationsofmotionofthecollidingparticlesintheCMframe,

˙

p^₁

=

^{F and} p

˙

^₂

= −

^{F (28)}

where

F

= −

^dV

dr^

ˆ

r^

,

(29)

withr^=^r₁−^r₂^{.Dottedquantitiesrepresenttime}derivatives.Therelativevelocity,v^= ˙^r,is v^

=

^v1

−

^v2

=

^c2p1

W₁^

−

^c2p2 W₂^

=

c² W₁^

+

^c2

W₂^

p^

,

(30)

wherep^=^p₁^{isthelinearmomentumofthe}projectile.Hence,theequationfortherelativemotionreads

˙

v^

=

^d

dt



c² W₁^

+

^c2

W₂^

p^



.

(31)

Byvirtueofavisviva theorem,thequantity

S

≡

^W₁^

+

^W₂^

+

^V

(

r^

)

(32)

isaconstantofthemotion.Inaddition,forcentralforces,theangularmomentumL^=^r×^{pisconserved.Because} r^

×

^v

=

c² W₁^

+

^c2

W₂^

L^

,

(33)

the conservation of angular momentum impliesthat the trajectories of the particles lie in the plane of scattering. The values of the constants L^andS ^{aredeterminedbytheinitial}conditions; L^=^bp_i^{,whereb istheimpactparameter(whichtakesthesamevaluesin} L andCM),andS =^{s isgivenbyEq. (22).Withthe aidoftheseconstants ofmotion,thedeflectionangle ofaprojectile withagiven} impactparametercanbeobtainedbyquadrature,instrictanalogywiththeclassicalnon-relativisticformulation[2].

Theequality S

−

^V

(

r^

) = 

M₁²c⁴

+

^p2c2

+ 

M₂²c⁴

+

^{p2c2 (34)}

impliesthat

p^2

(

r^

) =

¹ 4c²

 

S

−

^V

(

r^

) 

2

+



M²₁

−

^M2₂



2

c⁸ [S

−

V

(

r^

)

]²



− (

M²₁

+

^M2₂

)

c²

2

.

(35)

Withsomerearrangements,thisexpressioncanbereducedtothefamiliarnon-relativisticformoftheequationofmotion

p^2

(

r^

) =

^p_i²

−

²

μ

rV_ef

(

r^

)

(36)

where

μ

r

=

^{c−2 W1i W2i}

W_1i^

+

^W_2i^

,

(37)

istherelativisticreducedmassofthecollidingparticlesandV_ef(r^)isaneffectivepotentialgivenby

Vef

(

r^

) =

V

(

r^

) +

Vr1

(

r^

) +

Vr2

(

r^

),

(38) with

V_r1

(

r^

) = −

^V2

(

r^

)

2

μ

rc²

1

−

³

μ

rc² S

(39a)

and

V_r2

(

r^

) = (

M₁²

−

^M2₂

)

²c⁶ 8

μ

rS²



1

−

^V

(

r^

)

S



₋2

−

¹

−

^2V

(

r^

)

S

−

^3V2

(

r^

)

S²



.

(39b)

ThetermsVr1(r^)andVr2(r^)arecorrectionstotheinteractionpotentialthataccountfortheeffectofrelativistickinematics.Thefirst one isproportional to V²(r^);when V(r^)isascreened Coulombpotential, V_r1(r^)divergesasr^{ −2} attheorigin.ConsideringtheTaylor expansionofthefunction(1−^x)⁻²,thesecondcorrectionterm,Vr2(r),isseentobeoforder(V/S)^{3,anditvanisheswhentheprojectile} andthetargetparticleshavethesamemass.

Itis worthobservingthat whenthemass M2 ofthetargettends toinfinity, βCM=^{0 andthe CMframe coincideswiththeLframe.}

Underthesecircumstances,p^_i=^p1i,s ∞^,and

μ

r



W1c⁻²

= γ

1iM1

.

(40)

Thatis,thereducedmassequalstherelativisticmassoftheprojectile,whichincreaseswithp1i. 3.2. ClassicalDCSintheCMframe

Since Eq. (36) has the same form as the corresponding non-relativistic equation, the calculation of the DCS can be performed by followingthesamesteps asinthe classicalnon-relativistictheory(see,e.g., Ref. [2]).The deflectionangle(i.e.,the totalanglesweptby thevectorp^)isafunctionoftheimpactparameterb,oroftheangularmomentum L^=^{pb.Itcanbecalculatedas}

ϑ

^

(

L^

) = π ₋

2



∞ r^₀

L^r^{ −2}



p^_i²

−

²

μ

rV_ef

(

r^

) −

^{L2r −2dr}



,

(41)

wherer^₀isthedistanceofclosestapproach,thatis,thelargestrootoftheequation p^_i²

−

2

μ

rV_ef

(

r^₀

) −

^L2

r^₀²

=

0

.

(42)

ItisworthrecallingthatifVef(r)istheCoulombpotential V_C

(

r

) =

^{Z1Z e2}

r (43)

thedeflectionangleisgivenby[2]

ϑ

_C^

(

L^

) = π ₋

2



∞ r^₀

L^r^{ −1}



r^2p^_i²

−

²

μ

rZ₁Z e²r^

−

^L2dr



=

^{2 arctan}

μ

rZ₁Z e² L^p^_i

.

(44)

The numericalcalculationoftheangulardeflectionforthescreenedCoulomb potential (5) faces thesamedifficultiesasinthenon- relativisticcase[23],namely,thedivergenceoftheintegrandatthedistanceofclosestapproachand,inthecaseofrelativelylargeangular momenta,thenearcancellationofthetwotermsontheright-handsideofEq. (41).Itisexpedienttointroducethefunction

(

r^

) =

^rVef

(

r^

)

Z1Z e² (45)

whichplaysarolesimilartothescreeningfunction.Subtractingandadding,respectively,thesecondandthirdquantitiesintheequalities (44) afterreplacingZ with Z(r^₀),andchangingtheintegrationvariabletou= (¹−^r0^/r^)^1/2,wehave

ϑ

^

(

L^

) =

^{2 arctan}



M Z₁Z e²

(

r₀^

)

L^p^_i

 +

⁴



1 0



1



C

(

r₀^

) +

²

−

^u2

− 

¹

C

(

r^₀

) +

²

−

^u2

−

^{C f}

(

u

)



du

,

(46)

where

C

=

²

μ

rZ₁Z e²r^₀

L^2

,

(47)

and

f

(

u

) =

^r₀

(

r^

) − (

^r₀

)

r^

−

^r₀

.

(48)

Thefunction f(u)isapproximatelyconstant foru<0.001,whichcorresponds toradiibetweenr₀^ andr₁^= (¹+¹⁰⁻⁶)r^₀,where f(u) r^₀[^d(r^₀)/dr^₀]^{.Forradiilargerthanr}1, f(u)canbecalculateddirectlyfromtheexpression(48).Thus,theintegrandinEq. (46) isfinite andvariessmoothlyovertheintegrationinterval,andtheintegralcanbecalculatednumerically.

While the deflection angle ϑ^ may take arbitrarily large positive or negative values,the polar scattering angle θ^ is,by definition, limitedtotheinterval[⁰,

π

_].Thetwoanglesarerelatedby

cos

θ

^

=

^cos

ϑ

^

.

(49)

IfthefunctionL^(θ^)issingle-valued,theclassicalDCSinCMis d

σ

^

d



^

=

²

π

b db d



^

=

¹

2 sin

θ

^

 

^db_d

θ

^2

  =

_2p2¹ i sin

θ

^

 

^dL_d

θ

^2

  .

⁽⁵⁰⁾

Whenthefunction L^(θ^)ismulti-valued,theDCSisthesumofcontributionsfromthevariousbranchesofthatfunction, d

σ

^

d



^

=

¹ 2p^_i²sin

θ

^



j

 

^dL_d

θ

^2

 

L^=L^_j

,

(51)

wherethesummationextends overthevalues L^_j oftheangularmomentumthat givedeflectionangles correspondingtothe scattering angleθ^.InthecaseoftheunscreenedCoulombpotential,withthedeflectionfunction(44),theDCSisgivenbytheRutherfordformula

d

σ

_Ruth^

d



^

=

 μ

rZ1Z e² 2p^_i²



2

1

sin⁴

(θ

^

/

2

) .

(52)

Bohr[24] usedqualitativediffraction argumentstoanalyzethevalidityoftheclassicaltrajectorymethod.Assumingthatthefunction L^(θ^)issingly-valued,hisreasoningleadstotheconclusionthattheclassicalmethodisvalid(i.e.,ityieldsthesameDCSasaquantum calculation)when

T_class

(θ ) =

¹

θ

^

¯

h





^d

θ

^ dL^

  

¹

.

(53)

In the caseofa Coulomb potential, thiscondition impliesthat the classicalmethod is validforall angles ifthe absolutevalue of the Sommerfeldparameter

η ≡ μ

rZ1Z e²

¯

hp^_i (54)

ismuchlargerthanunity.InthecaseofscreenedCoulombfields,Bohr’scriterionissatisfiedforscatteringangleslargerthanabout

θ

_clas^

= ¯

^h

p^_iR

,

(55)

whereR istheatomicradius,Eq. (9).Whentheclassicalmethodisvalid,wemayassociateeachimpactparameter(orangularmomentum) toawelldefinedscatteringangle.Generally,smallanglescorrespondtolargeimpactparameters.

The relativistic corrections(39) to thepotential V(r), Eq. (38), are appreciableonly forsmall radii, atwhich the actual interaction potentialdepartsfromthescreenedCoulombpotentialduetothefinitesizeofthenucleus.Thesecorrectiontermshaveaninfluenceon theDCSonlyforrelativelyfastprojectiles,withwavelengthsoftheorderofthenuclearradius,

Rnuc



1

.

2 A^1/3fm

=

2

.

3

×

10⁻⁵A^1/3a0

,

(56)

where A isthemassnumberofthenucleus.Inaddition,theeffectofthecorrectiontermsislimitedtorelativelylargeangles,wherethe DCSisseveralordersofmagnitudesmallerthanatforwarddirections,anditisoutmatched bytheeffectsofthefinitesizeandstructure ofthenucleus[25],whicharedisregardedhere.Forthepurposesofparticletransportcalculations, whicharemostly determinedbythe DCSatsmallandintermediateangles,itisjustifiedtoneglecttherelativisticcorrectionstothepotentialwhentheirconsiderationlargely complicatesthecalculations.

4. Quantumtheoryofelasticcollisions

The wave equation forthe motion of the projectile relative to the target atom can be obtainedby following the familiar heuristic procedure basedon thecorrespondenceprinciple [26]. Westart fromtherelativisticequation ofmotionin theCM frame ofreference, Eq. (36) (fortypographicsimplicity,hereweremovetheprimeintherelativepositionvector,i.e.,wesetr=^r₁−^r₂^),

p^2

(

r

) =

^p_i²

−

²

μ

rV_ef

(

r

).

(57)

The time-independent wave equation forfree states(i.e., stateswithpositive energy, E>0) isobtainedfrom theclassical equation by makingthereplacementp^→ −^ih¯∇^andbyconsideringthattheresultingoperatorsactonthewavefunctionψ(r)(see,e.g.,Ref. [26])



− ¯

^h2

2

μ

r

∇

²

+

^Vef

(

r

)



ψ (

r

) =

^pi2 2

μ

r

ψ (

r

) .

(58)

This wave equation hasthesame formasthe non-relativisticSchrödinger equation that describesthescattering ofa particlewiththe relativisticreducedmass

μ

randinitialmomentum p^_ibythepotential Vef(r).Hence,thewavefunctionψ(r)andthescatteringDCScan beevaluatedbyusingthemethodsandapproximationsofnon-relativisticquantumtheory.

ItisinterestingtoconsiderthelimitingformofEq. (58) whenthemassM2 ofthetargetatomtendstoinfinity,sothatβCM^{0 and} theCMframecoincideswiththeLframe.Underthesecircumstances,p^_i=^p1i,s ∞^,

μ

r



^c−2W1i

= γ

_1iM1

,

with

γ

_1i

₌

1

+

p_1i

M1c

2

,

(59)

V_ef

(

r

) =

V

(

r

)



1

−

^V

(

r

)

2

γ

1iM1c²



,

(60)

andthewaveequationbecomes



− ¯

^h2

2

γ

1iM₁

∇

²

+

^V

(

r

)



1

−

^V

(

r

)

2

γ

1iM₁c²



ψ (

r

) =

^p21i

2

γ

1iM₁

ψ (

r

),

(61)

which,asexpected, coincideswiththeKlein-Gordon(orrelativisticSchrödinger)equation forfreestatesofaparticlewithmassM1 and initial momentum p1i in theelectrostatic potential V(r) [3]. Indeed,ourderivation showsthat thesecond termin theeffective Klein- Gordonpotential(60) hasapurelykinematicorigin.

4.1. Partial-waveexpansionmethod

TheDCSforelasticcollisions isdeterminedbytheasymptoticbehaviorofadistortedplanewave,i.e.,asolutionofthewaveequation thatatlarger behavesasaplanewaveplusanoutgoingsphericalwave.Forpotentialssuchthatr Vef(r)vanishesatr= ∞^{,thedistorted} planewavecanbeexpressedintheformofapartial-waveseries

ψ

k

(

r

) = (

²

π )

^{−3/2 1} kr





(

2

 +

¹

)

i^exp

(

i

δ



)

F_k

(

r

)

P_

(

cos

θ

^

),

(62)

wherek=^p_i/¯^{h isthewavevectoroftheincident}projectile,θ^=^cos−1( ˆk· ˆ^r)isthescatteringangle(inCM), P_(cosθ^)aretheLegendre polynomials,andthefunctionsFk(r)aresolutionsoftheradialequation



− ¯

^h2 2

μ

r

d²

dr²

+

^Vef

(

r

) + ¯

^h2 2

μ

r

( +

¹

)

r²



F_k

(

r

) =

^pi2 2

μ

r

F_k

(

r

)

(63)

normalizedsothat F_k

(

r

)

r→∞

∼

sin



kr

−  π

2

+ δ



,

(64)

whereδarethescatteringphase-shifts.Purelyattractive(repulsive)potentialsgivepositive(negative)phaseshifts.Theasymptoticform ofthedistortedplanewave,

ψ

_k

(

r

)

r→∞

∼ (

2

π )

^−3/2exp

(

ik

·

^r

) + (

²

π )

^−3/2exp

(

ikr

)

r f

(θ

^

),

(65)

definesthescatteringamplitude, f(θ^),whichadmitsthefollowingpartial-waveexpansion f

(θ

^

) =

¹

2ik





(

2

 +

1

)

[exp

(

2i

δ



) −

1] P

(

cos

θ

^

).

(66)

TheDCS(inCM)isgivenby d

σ

^

d



^

= 

f

(θ

^

)

²

.

(67)

The case ofscattering by the Coulomb potential, Eq. (43), is of fundamental importance to understand globalproperties ofatomic collisions.Thewaveequation(58) forthispotentialcanbesolvedanalyticallybyusingparaboliccoordinates[12].TheCoulombscattering amplitudeisgivenby

fC

(θ

^

) = − η ^(

¹

⁺

ⁱ

η )

(

1

−

ⁱ

η )

exp

[−

ⁱ

η

ln

(

sin²

(θ

^

/

2

)) ]

2k sin²

(θ

^

/

2

) ,

(68)

where(x)isthecomplexGammafunctionand

η

istheSommerfeldparameter,Eq. (54).TheDCSforscatteringbytheCoulombpotential is