Deterministic and Monte Carlo Codes for Multiple Scattering Photon Transport

Transcription

1 Determnstc and Monte Carlo Codes for Multple Scatterng Photon Transport Jorge E. Fernández 1 1 Laboratory of Montecuccolno DIENCA Alma Mater Studorum Unversty of Bologna Italy Isttuto Nazonale d Fsca Nucleare INFN

2 Summary Introducton Unbased Monte Carlo smulaton of the Compton Profle Determnstc vs Monte Carlo codes for transport calculatons of lne wdth effects Electron contrbutons to photon transport Conclusons

3 Introducton Determnstc and Monte Carlo technques compete to provde the best descrpton of transport problems. However many tmes they demonstrate to be complementary. Ths tal offers three examples from our experence n photon transport whch llustrate the close cooperaton between these two approaches.

4 Summary Introducton Unbased Monte Carlo smulaton of the Compton Profle Determnstc vs Monte Carlo codes for transport calculatons of lne wdth effects Electron contrbutons to photon transport Conclusons

5 Unbased Monte Carlo smulaton of the Compton Profle Ths example shows how a determnstc calculaton has been used to correct a based Monte Carlo algorthm wdely adopted to smulate the Compton profle.

6 Compton profle It s the broadenng of the Compton pea It s produced by the momentum dstrbuton of the electrons n the atom t can be measured qute precsely n synchrotron facltes

7 Intensty [arb. unts] Based Algorthm analytcal profle based algorthm It was dscovered a based behavour n Compton profle MC smulaton at low energes when usng the standard algorthm by Namto 1 used by EGS and MCNP E [ev] The bas was responsble for: the creaton of a false pea n correspondence wth the low energy tal of the profle a wrong Compton profle at low energes 1 Y.Namto S.Ban H.HrayamaNIM A

8 Reason for the bas Wrong samplng of the atomc sub-shells: n shellnumber 1 sub-shells were assumed complete n.e. havng n electrons

9 Unbased samplng Correct samplng of the atomc sub-shells: n 1 Q shellnumber max n J Q sub-shells now are assumed ncomplete Q dq max J Q dq.e. havng n Q max J Q dq electrons

10 Intensty [arb. unts] Results of the unbased algorthm analytcal profle unbased algorthm ths wor The determnstc code was essental to dscover the wrong behavour of the based algorthm E [ev]

11 Summary Introducton Unbased Monte Carlo smulaton of the Compton Profle Determnstc vs Monte Carlo codes for transport calculatons of lne wdth effects Electron contrbutons to photon transport Conclusons

12 Condton to fulfll n order to produce photoelectrc effect Mechansm for producng XRF lnes E 0

13 The wdth of the atomc levels s responsble for the natural wdth of the lnes 1 The wdths of the atomc levels are the recommended values n Campbell and Papp At. Data and Nucl. Data Tables

14 Lorentzan shape of the lne sometmes s used the Half Wdth at Half Maxmum HWHM 1 1 ; E E E E E E E E E E E E E E

15 Emsson of a Lorentzan K-lne K edge E 0 K lne close to the edge K

16 Transport ernel for a Lorentzan lne wavelength regme U ] U [1 ; 4 1 e L P Q K E 0 edge Energy conservaton p g Q e s Emsson probablty Threshold for photoelectrc effect 1 ; Lorentzan dstrbuton

17 MC vs Determnstc descrpton of a Lorentzan lne In MC the energy of the characterstc photon s randomly sampled at every nteracton usng a Lorentzan dstrbuton centered at E 0. One nterestng effect appears when the Lorentzan tal crosses the edge.e. the energy of the emtted photon s hgh enough to produce another vacancy and therefore a self-enhancement effect. Snce the hgh energy tal has always a very low probablty ths case requres refned varance reducton technques n order to get sgnfcant results. The slow asymptotc decrease of the Lorentzan dstrbuton ntroduces a further complcaton to descrbe multple scatterng wth reasonable statstcs. Therefore we propose to use nstead a determnstc method based on the wavelength energy dscretzaton of the Lorentzan dstrbuton.

18 t t Dscretzaton of the Lorentzan dstrbuton wavelength regme 1 ; and use a dscrete δ-expanson for the Lorentzan t t p ; wth coeffcents t t d p.. 1 arctan 1 arctan 1 ; where We defne a new normalzed dstrbuton between the fnte lmts t t arctan ; 1 1 t d t t

19 Dscretzed ernel for the Lorentzan lne wavelength regme t e e t t L P p Q p Q mn ] U [1 4 1 ]U U [1 4 1 wth t max mn energy conservaton cut-off

20 Prmary XRF ntensty of a Lorentzan lne The prmary ntensty of the lne centered at the pea wavelength for an nfnte thcness specmen s computed wthn an nfntely large acquston wndow I 1 0 L I0 Q 4 [1 U ] 0 0 e t p where 0 0 max t 0 0 e photoelectrc collson 0

21 Secondary XRF ntensty of a Lorentzan lne The secondary ntensty of the lne centered at the pea wavelength for an nfnte thcness specmen s computed wthn an nfntely large acquston wndow. I 0 L I0 8 all lnes j t mn Q [1 U ] 0 p 0 0 js mn js max t Int j 0 smn t j 0 max Int j js j e j e j photoelectrc collson t ss mn j p s 0 Q 0 ln 1 js 0 j js 0 js [1 U js e js ] 0 ln 1 js 0 photoelectrc collson js e

22 Intensty a.u. Example of almost symmetrc Lorentzan lnes Sometmes the asymmetry s small E-3 1E-4 prmary Fe K total enh on Fe K 1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 1E Energy ev

23 Intensty a. u. Secondary Lorentzan contrbutons can be very asymmetrc Sometmes the asymmetry s s large 1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 a Cr K Cr K 1 Fe K 1 Cr K 1 N K 1 Cr K 1 1E-10 1E-11 1E-1 1E-13 1E-14 1E-15 1E-16 1E Energy ev

24 Summary Introducton Unbased Monte Carlo smulaton of the Compton Profle Determnstc vs Monte Carlo codes for transport calculatons of lne wdth effects Electron contrbutons to photon transport Conclusons

25 Electron contrbutons to photon transport The am s to evaluate the contrbuton due to electrons to be ncluded n photon transport codes wthout solvng the complete coupled problem. The code PENELOPE coupled electron-photon Monte Carlo was used to study the effect of secondary electrons nto the photon transport. The ad-hoc code KERNEL was developed to smulate a forced frst collson at the orgn of coordnates. We consdered a pont source of monochromatc photons. The physcs of the nteracton was descrbed usng the PENELOPE subroutne lbrary. All the secondary electrons were followed along ther multple-scatterng untl ther energy become lower of a predefnte threshold value. All photons produced by the electrons at every stage were accumulated. Polarzaton was not consdered.

26 Prevalng photon nteractons n the X-ray regme PRIMARY PHOTON Scattered photons COHERENT SCATTERING INCOHERENT SCATTERING PHOTOELECTRIC EFFECT RAYLEIGH PHOTON COMPTON PHOTON COMPTON ELECTRON CHAR. X-RAYS PHOTO ELECTRON Scattered electron

27 Electron-photon couplng SECONDARY ELECTRON Scattered electrons BREMMSTRAHLUNG INNER SHELL IMPACT IONIZATION PHOTON CONTINUOUS CHAR. X-RAYS PHOTO ELECTRON Scattered photon

28 Descrpton of the KERNEL code Prmary photon N h = N h +1 E=E0 r=x Y Z=0 d=u V W=001 Frst collson YES N h < N max NO Partcles n secondary stac? YES NO END Score of the dstrbutons SI NO Bremsstrahlung or relaxaton? NO Electrons? Smulaton of slowng-down untl E<E abs YES

29 Types of electron contrbuton to photon transport Bremmstrahlung: contrbutes a contnuous dstrbuton Inner shell mpact onzaton: modfes the ntensty of the characterstc lnes

30 Polar angular dstrbuton Inner-shell mpact onzaton Prmary photon source s 100 ev. Blue lnes denote computed values red symbols are error bars. The emsson s sotropc.

31 Spatal dstrbuton 1 Electron range vs photon MFP Parameter as a functon of energy. Bethe range of electrons Mean free-path of photons Value ranges eep always small order of 10-1.

32 Spatal dstrbuton Effectve electron range Effectve range ~ R/4 Au Bethe range

33 Photon emsson Inner shell mpact onzaton Radal dstrbuton Prmary photon source s 100 ev. Blue lnes denote computed values red symbols are error bars. X-axs s r/r All dstrbutons eep below R/3

34 Photon emsson Inner shell mpact onzaton Axal dstrbuton Prmary photon source s 100 ev. Blue lnes denote computed values red symbols are error bars. X-axs s z/r All dstrbutons eep below R/3

35 The correcton as a functon of energy Calculatons were performed for all the lnes of the elements Z=11-9 n the energy range ev. Snce the electrons loose ther energy more effcently n the low energy range the computed contrbuton s hgher for low energy lnes. To compute the correcton for a generc energy the whole nterval was dvded nto 5 energy regons. The best ft of the energy correcton at each energy nterval was computed usng 4 coeffcents.

36 Electron correcton on K-lnes Al Kα ev Low Z 11-0 Na - Ca f E E pe Best ft 3 fe E exp pe ln E 0

37 Electron correcton on L-lnes As Lα ev Medum Z f E E pe Zn - Sn Best ft 3 fe E exp pe ln E 0

38 Electron correcton on M-lnes W Mα ev Hgh Z 6-9 f E E pe Sm - U Best ft 3 fe E exp pe ln E 0

39 Kernel correcton due to nner shell mpact onzaton ] U [1 4 1 e electron electron P Q ] U [1 4 1 e pe electron P Q f Q Q f electron pe To avod data-base dfferences between PENELOPE and other transport codes the electron correcton s computed n unts of the photon contrbuton. pe f Q

40 Corrected ernel comprsng electron contrbutons ] U [ e pe P f Q p g Q e s pe E f ' ln exp 3 0 To compute the correcton for a generc energy the whole nterval s dvded nto 5 regons. The best ft of the energy correcton at each energy nterval requres 4 coeffcents. where

41 Summary Introducton Unbased Monte Carlo smulaton of the Compton Profle Determnstc vs Monte Carlo codes for transport calculatons of lne wdth effects Electron contrbutons to photon transport Conclusons

42 Conclusons Determnstc and Monte Carlo technques demonstrate to be complementary to provde the best descrpton of transport problems. We have shown three dfferent cases n photon transport to llustrate the symboss of MC and Determnstc approaches: 1 Determnstc calculatons were essental to dscover the wrong behavour of a based algorthm used to smulate the Compton profle n largely dffused MC codes. Determnstc calculatons provde a better framewor to descrbe the nfluence of the Lorentzan breath on multple scatterng contrbutons to XRF lnes. 3 Coupled photon-electron MC calculatons were essental to obtan a smple correcton of the photon ernel to nclude the effect of nner shell mpact onzaton from electrons.

43 Jorge Fernandez