ADJOINT MONTE CARLO PHOTON TRANSPORT IN CONTINUOUS ENERGY MODE WITH DISCRETE PHOTONS FROM ANNIHILATION

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1 ADJOINT MONT ARLO HOTON TRANSORT IN ONTINUOUS NRGY MOD WITH DISRT HOTONS FROM ANNIHILATION J. ua Hoogenboom Inefaculy Reaco Insiue Delf Univesiy of Technology Mekelweg 5 69 JB Delf The Nehelans j.e.hoogenboom@ii.uelf.nl ABSTRAT The heoy fo alicaion of coninuous enegy ajoin Mone alo o neuon anspo has been exene fo alicaion o phoon anspo. ompon scaeing an he phooelecic effec can be hanle analogous o neuon scaeing an capue. Fo ompon scaeing he so-calle ajoin coss secion which is a basic concep fo coninuous enegy ajoin Mone alo can be calculae analyically fom he Klein-Nishina scaeing funcion an he selecion pocess of enegy an iecion afe an ajoin ompon scaeing is explicily eive. Howeve he phoon pai poucion pocess esuling in wo iscee phoons of 0.5 MeV canno be hanle in he same way. Theefoe a poceue is evise o hanle his pocess by incopoaing an aiional ajoin scaeing sep o aive a he equie iscee enegy of he ajoin paicle fo eneing an ajoin pai poucion eacion. The eivaion of he ajoin anspo equaion fo his case has been given incluing he pobabiliy ensiy funcion fo selecing he enegy afe an ajoin pai poucion pocess. The heoy has been implemene in a compue pogam o emonsae is valiiy. Resuls fo a numeical alicaion ae saisfacoily compae wih a nomal fowa calculaion. Wih he coninuous enegy ajoin Mone alo echnique eeco esponses can be efficienly calculae fo abiaily small enegy inevals an even poin enegy values of a eeco esponse. INTRODUTION Mone alo mehos ae successfully alie in eaco physics suies boh fo neuon an phoon anspo calculaions an Mone alo has become he efeence calculaional meho because of accuae moeling of coss secion an geomey eail. Howeve no evey ype of poblem can be hanle saisfacoy by a egula Mone alo calculaion because of he saisical impecision inheen o he Mone alo echnique. specially hose poblems in which a eeco esponse has o be esimae fo a small eeco eihe geomeically speaking o fo a vey limie enegy sensiiviy ange he saisical pecision of he esul obainable in a easonable compue ime may inhibi pacical alicaion of he Mone alo meho. Fo such cases he soluion of he ajoin anspo equaion by he Mone alo meho so-calle ajoin Mone alo may be a vey effecive soluion. Boh fo neuon an phoon poblems he ajoin Mone alo opion is geneally povie in geneal pupose Mone alo coes like MN bu only fo he muligoup epesenaion of paicle anspo. specially when use as a efeence calculaional meho he muligoup aoximaion may no be

2 accepable an one has o esie o coninuous enegy ajoin Mone alo. Alhough he heoy fo coninuous enegy ajoin Mone alo was evelope many yeas ago 3-7 i has no become vey popula fo egula use pobably because his echnique was no incopoae in he mos popula geneal pupose Mone alo coes unil sholy 8. Recenly i has been shown 9 ha incopoaion of coninuous enegy ajoin Mone alo fo neuon anspo in he geneal-pupose coe MN is possible wih o coe moificaion. Inoucing so-calle ajoin coss secions which ae ansfomaion of he nomal coss secions he paicle simulaion pocess fo he ajoin anspo equaion can be mae vey simila o ha of egula neuon hisoy simulaion. The calculaion of he ajoin coss secions an he pobabiliy ables fo selecion of enegy an iecion afe an ajoin collision equies exensive coss secion pocessing o compose he equie coss secion file necessay fo such an ajoin Mone alo calculaion. Up o now coninuous enegy ajoin Mone alo has only been alie o neuon anspo an no o phoon anspo. Alhough he anspo mechanism fo neuons an phoons ae vey much alike an he fomal escipion can be fomulae by ienical anspo equaions in pacice a ifficuly aises because of he pai poucion pocess fo phoons. The phooelecic effec fo phoons is analogous o neuon capue an ompon scaeing is analogous o neuon scaeing alhough he collision mechanics ae iffeen. Howeve hee is no ue analogon of he pai poucion pocess fo phoons. If we esic ouselves o pue phoon anspo an isega elecon anspo he pai poucion pocess is moele as he geneaion of wo new phoons emie in oosie iecions fom he poin whee he ineacion of he oiginal phoon ook place. Boh new phoons have a iscee enegy equal o 0.5 MeV. As fo ajoin anspo he ole of he incog an ougoing enegy in a collision ae inechange he ajoin paicle shoul have an enegy of exacly in oe o allow fo an ajoin pai poucion pocess. In pacice his will neve occu an he pai poucion pocess seems o be exclue fom he ajoin phoon Mone alo simulaion. Theefoe he pai poucion pocess nees sepaae eamen in coninuous enegy ajoin Mone alo simulaion. In his pape he heoeical basis will be eive an a pacical implemenaion an alicaion will be emonsae. Fis he coninuous enegy ajoin Mone alo eamen as alicable fo neuon anspo will be summaize. This is also he basis fo ajoin phoon anspo as fa as he phooelecic effec an ompon scaeing is concene. As he mechanics of ompon scaeing ae iffeen fom neuon scaeing eamen of ompon scaeing in he ajoin simulaion will be pesene. Nex he heoeical an pacical eamen of he pai poucion pocess is eive.. Summay of fowa Mone alo anspo THORY OF ADJOINT MONT ARLO In geneal a eaco physics anspo poblem eihe concening neuons o phoons can be fomulae as calculaing he esponse of a possibly hypoheical eeco wih a ceain esponse funcion. The poblem can be escibe in ems of he neuon flux wih a poin in he phase space bu also in ems of he collision ensiy ψ o he emission ensiy χ. The lae quaniy is he numbe of paicles leaving a collision o he souce. Fo he Mone alo simulaion he inegal Bolzmann equaion fo he emission ensiy is mos suie: χ S K χ wih S he neuon o phoon souce an K he anspo kenel compose of a isplacemen kenel T an a collision kenel 60 :

3 K K T. The collision kenel is a summaion of all paial eacion ypes fo all nuclies in he meium a. I is nomalize o he non-absopion pobabiliy na :. 3 na S T Mone alo simulaion of his equaion is one by sampling he souce S an epeaely sampling he isplacemen kenel T an he collision kenel. This las sep is ofen one by alying a faco na o he paicle weigh an sampling he nomalize fom nom of fo scaeing eacions only. The pocess of sampling q. is visualize in Fig.. n a The eeco esponse R can be escibe in ems of he neuon flux wih a esponse funcion η in ems of he collision ensiy ψ wih a esponse funcion η ψ o in ems of he emission ensiy χ wih a esponse funcion η χ as follows: n o m Fig.. Schemaic iagam fo sampling he emission ensiy χ an he collision ensiy ψ. Ajoin anspo equaions The equaion ajoin o q. is η ψ R ψ η η χ χ. The simples way o esimae R is o aly a collision esimao wih esponse funcion η ψ a evey sample of ψ. 4 wih χ S K χ K K. 6 5 The ajoin equaion 5 ges physical significance if he souce em S is chosen equal o he eeco esponse funcion wih espec o he emission ensiy η χ. Wih his choice fo he souce em S an q.6 fo he ajoin anspo kenel K we can eive an alenaive expession fo he eeco esponse R. Muliplying q. by χ an q.5 by χ inegaing boh expessions ove he phase space an subacion gives R S χ. 7 Hence he oiginal eeco esponse can also be obaine by sampling he soluion of he ajoin anspo equaion 5 an using he oiginal souce S as he eeco esponse funcion of he eeco in he 3

4 ajoin game. Fom q.7 i follows ha χ is he impoance of a neuon a fo he eeco esponse. To sample he ajoin funcion χ fom q.5 we have o inepe q.5 as escibing he anspo of some hypoheical paicle wih popeies efine by he kenel K. I uns ou ha he following ansfomaion an χ 8 L K 9 faciliaes he Mone alo simulaion of he ajoin equaion which is ansfome ino S L. 0 Accoing o qs.7 an 8 he eeco esponse R is obaine fom S R..3 Sampling he ajoin anspo equaion To sample q.0 we have o inepe he kenel L in Mone alo ems. To faciliae he sampling of L we efine 6 L T wih T he isplacemen kenel fo he ajoin paicles o simply he ajoin isplacemen kenel an he nomalize ajoin scaeing kenel T T 3 4 wih he nomalizaion faco fo he ajoin scaeing kenel. 5 The faco is given by. The funcion is a kin of macoscopic coss secion an will be calle he ajoin coss secion. Nomally he egula neuon scaeing kenel will be compose of iffeen nuclies an eacion 6 4

5 ypes. Theefoe we inouce he paial micoscopic ajoin coss secion of nuclie A fo eacion ype j as j A j A 7 wih ja he paial iffeenial scaeing coss secion. Now he ajoin micoscopic coss secions can be ae up o fom he oal macoscopic ajoin coss secion N 8 A A wih N A he nuclie ensiy of nuclie A. Sampling he ajoin scaeing kenel can now be one by selecing fis a nuclie popoional wih is macoscopic ajoin coss secion AN A s A nex a eacion ype fo nuclie A popoional o he micoscopic ajoin coss secion ja an finally a new iecion an enegy fom he nomalize pf j j A p j A j A j A To sample he souce em S η χ in q.0 i can be shown ha 6. 9 η χ T η V 0 which means ha we have o sample he eeco esponse funcion wih espec o he paicle flux η an o sample he ajoin isplacemen kenel T. ombining he souce sampling an he ajoin isplacemen kenel sampling he Mone alo simulaion of q.0 can be schemaically visualize as in Fig.. This sampling scheme is analogous o he fowa Mone alo sampling scheme of Fig.. Sampling he ajoin isplacemen kenel T fom q.3 means acking hough he geomey as in he fowa case excep ha acking is in he iecion -W. Alying as a muliplicaion faco fo he paicle weigh is analogous o T Fig.. Schemaic iagam fo sampling he ajoin collision ensiy weighing in lieu of absopion o someimes calle implici capue in he fowa simulaion using he non-absopion pobabiliy. Howeve sampling of he ajoin collision kenel is eally iffeen fom he fowa Mone alo simulaion. Using he concep of micoscopic an macoscopic ajoin coss secions he selecion of a nuclie an a eacion ype is sill analogous bu he collision mechanics in he ajoin simulaion ae iffeen fom he fowa simulaion. Hee also aeas he iffeence beween neuon an phoon anspo. We can euce some popeies of he ajoin paicles fom he kenel. Fom q.4 we see ha he change of enegy an iecion is oosie o ha of neuons. While neuons can only lose enegy in he slowing own enegy ange he ajoin paicle can only gain enegy in a collision. To obain he oiginal eeco esponse accoing o q. we have o aly a collision esimao wih esponse funcion S/ o 5

6 alenaively use a flux ack lengh esimao wih esponse funcion S. 3 ADJOINT OMTON SATTRING Fo he eamen of ompon scaeing he euce ompon wavelengh λ is ofen use insea of he phoon enegy: mc e λ wih m e he elecon mass an c he spee of ligh. Then he iffeenial ompon scaeing coss secion is given by he Klein-Nishina coss secion λ λ π e λ Z λ λ λ λ λ λ λ λ λ λ λ λ wih e he classical elecon aius an Z he aomic numbe of he nucleus. The scaeing angle is uniquely eee by he wavelengh befoe an afe scaeing: µ cos θ λ λ 3 wih µ he cosine of he scaeing angle θ. The ajoin coss secions fo ompon scaeing accoing o q.7 becomes λ λ λ λ λ λ λ λ λ Z e ln. λ λ λ π λ λ λ λ 3 4 This expession is only vali fo λ > o <½. Fo highe enegies he ajoin ompon coss secion woul become infinie as he lowe wavelengh inegaion limi becomes zeo. I is heefoe necessay o inouce a maximum phoon enegy of inees in he sysem m coesponing wih a imum wavelengh λ m. Then he lowe inegaion limi fo λ becomes λ maxλ m λ- an he ajoin coss secion becomes π e Z λ λ λ λ λln λ λ λ 3 3 λ λ. 5 6

7 oss secion Ajoin ompon This ajoin coss secion shows a peculia behavio as can be seen fom Fig. 3. I exhibis a peak wih a isconinuous eivaive a λ λ m o ½ 0.56 MeV. Fo m i goes o zeo as can be expece. In Fig. 3 m 0 MeV was use negy MeV The pobabiliy ensiy funcion fo he enegy afe an ajoin ompon scaeing follows fom q.9 afe inegaion ove iecion. The cumulaive pobabiliy funcion fo becomes Fig. 3. ompon coss secion an ajoin coss secion apa fom a faco π e Z λ λ / λ λ λ λ λ λ λ 3 λln λ λ λ λ λ λ 3 λ λ λ 3 λ ln λ λ λ λ λ λ 3 λ/. λ 6 If ρ is a anom numbe unifomly isibue beween 0 an we selec he enegy by ieaively solving he equaion ρ. The cosine of he scaeing angle follows fom µ-λλ an an azimuhal angle is selece unifomly ove π. 4 ADJOINT AIR RODUTION The iffeenial coss secion fo he pai poucion pocess can be wien as δ. 7 The ela-funcion foces he enegy afe he ineacion o be equal o 0.5 MeV. The faco accouns fo wo phoons pouce in he ineacion. Saighfowa alicaion of he fomula fo he ajoin coss secion of he pai poucion pocess shows ha conains a ela-funcion δ -. Hence i is always zeo accep fo. As in he ajoin simulaion an ajoin phoon will neve have an enegy exacly equal o he ajoin pai poucion pocess will neve be sample. To aive a a wokable scheme he ela-funcion has o be incopoae eepe ino he ajoin hisoy simulaion. In mahemaical ems his implies ha in he Neumann seies aising fom he ajoin anspo equaion he enegy inegaion in he inegal ems conaining he ela-funcion fom ajoin pai poucion ae evaluae analyically an he esul is inepee in Mone alo ems. In pacical ems i means ha we have o consie fis an ajoin ompon scaeing o he enegy afe which he ajoin pai poucion can ake place. In beween he ajoin paicle will move o anohe posiion accoing o he ajoin anspo pocess. The pobabiliy fo he ajoin ompon 7

8 8 scaeing o he enegy is aken ino accoun by muliplying he ajoin paicle weigh by his pobabiliy. Fo a igoous eivaion we sa wih qs.0 an an ecompose he ajoin scaeing kenel in a pa fo ompon scaeing an anohe pa fo pai poucion. Fo convenience we fis wie. V T ζ 8 Then. 4 π δ η ζ 9 In he las em he inegaion ove can simply be caie ou leaing o. 4 π η ζ 30 Now q.8 is use again ogehe wih q.30 o subsiue ino he las inegal leaing o. 4 4 V T V T π η π η ζ 3 Noe ha he las em fom q.30 fo oes no conibue as his enegy is below he heshol fo he pai poucion coss secion. The fis line of q.3 escibes he ajoin phoon anspo wihou pai poucion. The em in he secon line gives he conibuion fom ajoin pai poucion saing fom he ajoin souce η wih he equie enegy. The las lines give he conibuion fom ajoin pai poucion afe an ajoin ompon scaeing o enegy. Fo Mone alo simulaion his em can be inepee as follows. Given a sample of he ajoin collision ensiy a we muliply he paicle weigh by. Nex we selec fom he ajoin ompon scaeing kenel a enegy. As he scaeing angle is uniquely eee by he enegy befoe an afe ompon scaeing we have fom q.3. λ λ λ µ 3 The esuling scaeing kenel is aken ino accoun as anohe paicle weigh muliplicaion faco. Then we selec a new space poin by sampling he ajoin anspo kenel T a. Finally we selec a new iecion isoopically an a new enegy fom he nomalize pobabiliy ensiy

9 funcion base on. To obain a nomalize pf we inouce he ajoin pai poucion coss secion m. 33 The lowe limi is he heoeical heshol fo pai poucion. The ue limi m is he maximum phoon enegy elevan fo he poblem a han. This ue limi mus be inouce o ge a convegen inegal as fo he ajoin ompon coss secion. Now he nomalize pf fo he enegy selecion is p /. Hence he paicle weigh mus be muliplie by he nomalizaion faco 34 which is an analogon of he weigh faco fo ajoin ompon scaeing. Selecion of fom he nomalize pf p / mus nomally be one by numeical inegaion of o pouce a pobabiliy able fo selecion of. Noe ha he wo phoons aising fom pai poucion ae no sepaaely el wih in he ajoin moe. As long as one is no ineese o eec coelaion effecs of he wo phoons hey can be hanle by he faco in he weigh faco Fom he ange of wavelenghs afe ompon scaeing of q. i follows ha fo o λ λ 3 o /3. Hence he ajoin ompon scaeing funcion is only non-zeo if /3. Inoucing he kenel M T 4π 35 we can ewie q.3 as ζ η M η V M V 36 Simulaion of his Mone alo pocess is visualize in Fig. 4. The las em of q.36 leas o a banch in he loop ove T an. As boh banches has he faco in common he banch can be aken afe alicaion of his weigh faco. In he new banch he faco is aken ino accoun using a weigh faco an selecing he iecion fom q.3 wih a anom azimuhal angle. 9

10 η η M p p q.36 fo ζ an hus he equaion fo has now wo souce ems η an he secon em on he igh han sie of q.36 conaining η. Boh can be sample inepenenly in he aopiae aio. The lae souce em shoul only be consiee if η is non zeo fo. 5 NUMRIAL XAML T To emonsae he coecness of he evelope scheme we suie a phoon anspo case wih ealisic enegy epenence of coss secions. Fo easy hanling of he aa he phooelecic coss secion was aken as a powe epenence / 3 fo he ompon scaeing he Klein-Nishina coss secion was aope an fo he pai poucion coss secion a ahe abiay bu ealisic logaihmic enegy epenence wih a heshol a was assume p0 ln. 37 Mp p Fig. 4. Schemaic iagam fo simulaion of he ajoin collision ensiy x wih ajoin pai poucion This allows an analyic calculaion of he ajoin coss secions fo ompon scaeing an pai poucion. Fom q.7 we have in he case of q.37 p0 m ln m m. 38 Fo he geneal case wih abiay enegy epenence he ajoin coss secions have o be calculae numeically. Table I. ompaison of goup fluxes fom fowa an ajoin coninuous enegy Mone alo calculaions Fowa MN Ajoin negy ange coninuous enegy MeV % % oal

11 Flux MeV hoon enegy MeV goup flux Fig. 5. hoon flux specum wih poin enegy an enegy goup fluxes. Value a 0.5 MeV is off scale. To concenae on he enegy epenence we ook a homogeneous infinie meium. A special coe was wien o simulae he ajoin phoon anspo pocess as ouline in Sec. -4. Alhough he geomeic possibiliies ae limie he enegy epenence of he ajoin anspo pocess is fully implemene. Fo a compaison of he esuls he fowa case was un wih he MN coe using a specially pepae coss secion libay aking ino accoun he above menione enegy epenence of he phoon coss secions. A case is consiee wih a phoon souce beween an MeV an a eeco egiseing he phoon flux fom which he esponse fo a numbe of enegy goups is equese. Table I shows he esuls fom he fowa MN un an he ajoin un each wih 0 4 paicles. Noe ha fo each eeco enegy ange a sepaae ajoin calculaion mus be un. The able shows a goo ageemen beween he fowa an he ajoin esuls which valiaes he coninuous enegy ajoin Mone alo echnique fo phoons. The powe of he ajoin Mone alo meho becomes clea when smalle eeco enegy anges o even poin enegy values ae consiee. The sana eviaion will incease in he fowa calculaion when he enegy ange consiee becomes smalle an smalle bu will aoximaely emain consan in he ajoin calculaion fo a fixe numbe of hisoies. Fig. 4 shows he phoon enegy specum fo he case escibe above wih poin enegy flux values an he goup fluxes given in Table I ecalculae pe uni enegy. The jump in flux value a MeV is ue o he phoon souce ha sas a MeV. The jump a 0.5 MeV is ue o he iscee annihilaion phoons. 6 ONLUSIONS AND DISUSSION In his pape he heoy fo coninuous enegy ajoin Mone alo phoon anspo is pesene. The ajoin coss secion which is a funamenal concep fo coninuous enegy ajoin Mone alo has been eive fo ompon scaeing base on he Klein-Nishina scaeing funcion. Also he pobabiliy ensiy funcion fo he enegy afe an ajoin ompon scaeing is given an he selecion of enegy fom his pf. Of special impoance is he eamen of he iscee enegy of phoons geneae fom he pai poucion pocess which pohibis saighfowa alicaion of he ajoin pocess. The soluion was foun in he inclusion of an aiional em fom he Neumann expansion of he inegal ajoin equaion. This leas o a simulaion scheme wih a ouble banche loop fo he epeae selecion of he ajoin isplacemen kenel an he ajoin scaeing kenel. I also leas o a secon souce em povie he oiginal eeco esponse funcion η is non-zeo a enegy 0.5 MeV. As he iffeenial coss secion fo ompon scaeing is given in an analyic way by he Klein-Nishina fomula he ajoin coss secion can be calculae analyically. This will no longe be ue if a moe geneal coss secion escipion wih sengh funcions is use as efine in he NDF foma. Then a numeical aoach has o be followe. This is he case anyway fo he pai poucion coss secion.

12 Alhough an analyic coss secion escipion fo pai poucion was aope hee his was only one fo emonsaion puposes. In pacice a numeical inegaion has o be pefome o calculae he ajoin pai poucion coss secion. Fo he selecion of he enegy afe an ajoin pai poucion eacion a common echnique is o calculae in avance ables of equipobable values fo he selecion vaiable. This will simplify he pocess of implemenaion in exising geneal-pupose Mone alo anspo coes. Noneheless he ouble banching in he simulaion pocess of Fig. 4 as well as he secon souce em will always equie consieable exensions of such coes o inclue he coninuous enegy ajoin Mone alo phoon simulaion. Ajoin Mone alo calculaions ae especially useful an efficien in case of a small eeco an a elaively lage phoon souce. Small nee no only o efe o geomeically small bu may also efe o a small enegy ange. In fac hee is no poblem in calculaing eeco esponses a poin enegy values in ajoin Mone alo while his povies majo poblems in a fowa calculaion. This avanage of an ajoin calculaion was clealy emonsae wih a numeical example. RFRNS. J.F. Biesmeise MN- A Geneal Mone alo N-aicle Tanspo oe Vesion 4B epo LA- 65-M Los Alamos Naional Laboaoy 997. J J.. Wagne.L. Remon II S.. almag J.S. Henicks MN Muligoup Ajoin apabiliies epo LA-704 Los Alamos Naional Laboaoy B. iksson e al. Mone alo Inegaion of he Ajoin Neuon Tanspo quaion Nucl. Sc. ng J.. Hoogenboom FOUS A non-muligoup ajoin Mone alo coe wih impove vaiance eucion oc. NAR meeing of a Mone alo suy goup epo ANL-75-/NAR-L-8 p. 44 Agonne Naional Laboaoy J.. Hoogenboom Ajoin Mone alo Mehos in Neuon Tanspo alculaions DSc. hesis Delf Univesiy of Technology Delf The Nehelans J.. Hoogenboom A acical Ajoin Mone alo Technique fo Fixe-Souce an igenfuncion Neuon Tanspo oblems Nucl. Sc. ng R.J. Bissenen oninuous enegy ajoin Mone alo og. Nucl. negy AA Technology MBND A Mone alo ogam fo Geneal Raiaion Tanspo Soluions Use Guie fo Vesion 9D The ANSWRS Sofwae ackage J.. Hoogenboom oninuous negy Ajoin Mone alo in MN wih Mino oe xensions In: oc. M& 99 confeence Mai 6-30 Sepembe I. Lux L. Koblinge Mone alo aicle Tanspo Mehos: Neuon an hoon alculaion R ess 99.. A.B. hilon e al. inciples of Raiaion Shieling enice Hill F. Rose.L. Dunfo Daa Fomas an oceues fo he valuae Nuclea Daa File NDF-6 epo BNL-NS 44945/NDF-0 Rev. 0/9 Bookhaven Naional Laboaoy 99.