MULTIPLE SCATTERING OF PHOTONS USING THE BOLTZMANN TRANSPORT EQUATION

Transcription

1 MULTIPLE SCATTERING OF PHOTONS USING THE BOLTZMANN TRANSPORT EQUATION Jorge E. Feradez Laboratory of Motecuccolio-DIENCA INFN, INFM & Alma Mater Studiorum - Uiversity of Bologa ICXOM-18, Frascati September 2005

2 INTRODUCTION

3 MULTIPLE SCATTERING X-rays peetrate deeply ito the matter, ad, i a thick medium, give place to a pheomeo kow as multiple scatterig. Multiple scatterig models describe the ifluece of the prevailig iteractios i the x-ray regime (photoelectric effect, Compto scatterig ad Rayleigh scatterig)

4 MULTIPLE SCATTERING (cot.) Multiple scatterig models describe the ifluece of the prevailig iteractios i the x-ray regime (photoelectric effect, Compto scatterig ad Rayleigh scatterig) The photoelectric effect itself ca be cosidered as a scatterig process

5 Photoelectric effect as scatterig hν characteristic photo K L 2 L 1 L 3 photoelectro photoelectric absorptio + radiative trasitio = photoelectric 'scatter ig'

6 PREVAILING INTERACTIONS IN THE X-RAY REGIME PRIMARY PHOTON Scattered photos COHERENT SCATTERING INCOHERENT SCATTERING PHOTOELECTRIC EFFECT RAYLEIGH PHOTON COMPTON PHOTON COMPTON ELECTRON CHAR. X-RAYS PHOTO ELECTRON Scattered electro

7 DESCRIPTION OF POLARIZATION

8 WHY POLARIZATION? Polarizatio state wave ature of photos By cosiderig polarizatio we improve the model of photo diffusio

9 Without polarizatio photos are cosidered oly as a particles a good approximatio i may cases, but ot for pheomea that are iflueced by their wave properties

10 REPRESENTATION OF POLARIZED RADIATION Stokes parameters I,Q,U,V (havig the dimesio of a itesity) ca specify: Itesity of the beam Degree of polarizatio Orietatio of the ellipse of polarizatio Ellipticity

11 Polarizatio state defiitio Four parameters: a) The itesity (the square of the electric field) b) The degree of polarizatio (the fractio of elliptically polarized beam) c) Orietatio of the polarizatio ellipse (agle χ) Propagatio vector d) Ellipticity (expressed by the agle β)

12 STOKES REPRESENTATION OF POLARIZED RADIATION Defiitio of STOKES PARAMETERS: Q = I cos U = I cos 2 β si 2 χ V = I si 2 β Degree of polarizatio: 2 β cos 2 χ P = (Q 2 + U 2 I + V 2 ) 1 2

13 EXAMPLES OF THE STOKES REPRESENTATION Polarisatio state Upolarised Liear (geeric) Liear ( ) Liear ( ) Liear (45 ) Circular Set S (I,Q,U,V) (1,0,0,0) (1,cos2χ,si2χ,0) (1,1,0,0) (1,-1,0,0) (1,0,1,0) (1,0,0,1)

14 INCOMING PHOTON OUTGOING PHOTON ATOM ) E, ω, r V( ) E, ω, r U( ) E, ω, r Q( ) E, ω, r I( ,E ω E, ω 1,E 1 ω r + ) E, ω, r V( ) E, ω, r U( ) E, ω, r Q( ) E, ω, r I( COLLISION SCHEME Modificatio of the polarizatio state due to a collisio (Stokes represetatio)

15 PHYSICAL MODEL The photo state is chaged by a matrix kerel depedig o the type of the collisio a. This kerel operates o the polarisatio state accordig to the relatioship: a H [ r,ω (+ 1), λ (+ 1),ω (), λ () I Q ] U V () () () () = I Q U V (+ 1) (+ 1) (+ 1) (+ 1)

16 PHOTON DIFFUSION IS DESCRIBED BY A VECTOR TRANSPORT EQUATION (THE 1-D EQUATION IS SHOWN HERE) where = ),ω, ( ),ω, ( ),ω, ( ),ω, ( λ λ λ λ z V z U z Q z I f

17 VECTOR TRANSPORT EQUATION where (CONT.) H (S) (S) K = kerel matrix i the meridia plae of referece = scatterig matrix i the scatterig plae of referece

18 η λ ' d PHOTON DIFFUSION IS DESCRIBED BY A VECTOR TRANSPORT EQUATION (THE 1-D EQUATION IS SHOWN HERE) z f( z,ω, λ) = μ( λ)f( z,ω, λ) + 0 4π ' ' ' ' dωu( z) H(ω, λ,ω, λ )f( z,ω, λ ) + δ( z)s(ω, λ) where f = ' I( z,ω, λ) Q( z,ω, λ) U ( z,ω, ) λ V( z,ω, λ)

19 VECTOR TRANSPORT EQUATION where H ( ω, = L( π λ, ω ' (CONT.), λ Ψ ) K ' ) = ( ω, λ, ω = kerel matrix i the meridia plae of referece H K ', λ ' ) L( Ψ = scatterig matrix i the scatterig plae of referece )

20 IMPORTANT PROPERTIES OF THE VECTOR TRANSPORT EQUATION Describes the evolutio of the full polarizatio state (ot oly the itesity of the beam) Isliear (for the Stokes represetatio) Requires the simultaeous solutio of the whole set of trasport equatios Caot be trasformed i a scalar equatio!! (due to the couplig i the scatterig term)

21 THEORETICAL MODELS

22 MODELS Differet degrees of approximatio to describe the diffusio photos: scalar model: photos ever modify a average polarizatio state vector model: trasport of photos startig with arbitrary polarizatio state

23 oe collisio Both models follow a multiple scatterig scheme a Photoelectric effect Rayleigh scatterig Compto scatterig (P) characteristic lies (discrete) (R) Rayleigh peak (discrete) (C) Compto peak (cotiuous) two collisios b a Photoelectric effect Rayleigh scatterig Compto scatterig Photoelectric effect Rayleigh scatterig Compto scatterig (P,P) XRF secodary ehacemet (discrete o XRF lie) (R,P) XRF ehacemet due to scatterig (discrete o XRF lie) (C,P) XRF ehacemet due to scatterig (discrete o XRF lie) (P,R) XRF ehacemet due to scatterig (discrete o XRF lie) (R,R) secod order scatterig (discrete o Rayleigh peak) (C,R) secod order scatterig (cotiuous o Compto peak) (P,C) XRF ehacemet due to scatterig (cotiuous o XRF lie) (R,C) secod order scatterig (cotiuous o Compto peak) (C,C) secod order scatterig (cotiuous o Compto peak)

24 E 0 Scalar trasport equatio Compto peak I (1) I (1) 10-4 Rayleigh peak E E [kev] I (2) I 0 I = Σ i I (i) I (2) E [kev] Compto peak I I SCALAR EQUATION 10-4 Rayleigh peak E E [kev]

25 E 0 E 0 Vector trasport equatio Compto peak I (1) Q (1) U (1) V (1) I (1) 10-4 Rayleigh peak E [kev] + I 0 I = Σ i I (i) I (2) 10-4 I (2) Q (2) U (2) V (2) E [kev] Compto peak I Q U V VECTOR EQUATION I 10-4 Rayleigh peak E E [kev]

26 EXAMPLES

27 LET US SHOW TWO SIMPLE EXAMPLES 1) Scatterig of upolarized radiatio 2) Scatterig of liearly polarized radiatio

28 1) Upolarized Rayleigh scatterig a icidet beam right agle scatterig

29 How scatterig polarizes a beam 90 degrees scatterig Scatterer Upolarized beam

30 SUMMARY FOR UNPOLARIZED RADIATION Upolarized beam (composed by rays with electric vector radomly orieted aroud the propagatio directio) After scatterig the beam is partially (totally) polarized depedig o the type of iteractio ad the scatterig geometry

31 2) Polarized Rayleigh scatterig b Electric vector parallel to the scatterig plae icidet beam right agle scatterig

32 SUMMARY FOR LINEAR POLARIZATION Liearly polarized beam with electric vector parallel to the scatterig plae Almost ull scatterig at 90 degrees

33 COMBINING BOTH PROPERTIES Almost ull scatterig 90 degrees scatterig 90 degrees scatterig Upolarized beam Scatterer = polarizer

34 THE CODES

35 SOLUTION TECHNIQUES The trasport equatio is solved usig a order-of-collisios scheme comparable results for determiistic ad Mote Carlo solutios

36 Determiistic vs. Mote Carlo Solutio Determiistic Mote Carlo (statistical) Scope of the solutio Global Local Accuracy Capability to describe the geometry Number of collisios Developed codes SHAPE MCSHAPE

37 CHARACTERISTICS OF THE CODE MCSHAPE Arbitrary polarizatio state of the source Multi-layer multi-compoet homogeeous targets Moochromatic or polychromatic source Doppler broadeig (for Compto scatterig) Full descriptio of the polarizatio state N-collisios

38 COMPARISON WITH SCALAR VERSION 10-2 Polarizatio depedet (vector model) Average polarizatio Polarizatio depedet (scalar model) Compto peak Itesity (arb. uits) Rayleigh peak E E [kev] The source is upolarized ad moochromatic. The sample is carbo ad ad the scatterig agle is 90.

39 EVOLUTION OF THE CODE Developmet of two differet codes: - MCSHAPE0: max. 4 collisios ad aalog calculatio - MCSHAPE1: o limits i umber of collisios First versio: Pascal (1995) Preset versios: 1D ad 3D writte i FORTRAN 90 Platforms: Widows ad LINUX Parallelizatio: MPI (uder LINUX)

40 WEB SITE These codes are goig to be distributed by NEA Data Bak (OECD) ad RSICC (US-DOE)

41 CODES COMPARISON (part 1: Physics) Features Details SHAPE v2.20 D3DSHAPE v1.0 MCSHAPE v2.50 photoelectric effect ~1000 characteristic lies lie width atomic Rayleigh scatterig atomic Compto scatterig Compto profile first collisio oly electro bremsstrahlug foresee i v3 foresee i v3 ope data bases user defied elemets foresee i v3 Physics ifiite thickess targets fiite thickess targets multilayer targets polarizatio represetatio Stokes Stokes source polarizatio state calculated spectrum liear/ upolarised itesity compoet oly upolarised arbitrary full polarizatio state UNIQUE FEATURES! moochromatic source polychromatic source postprocessor exteral detector solid state Si/Ge foresee i v3 reflectio geometry trasmissio geometry

42 CODES COMPARISON (part 2: model ad programmig) Features Details SHAPE v2.20 D3DSHAPE v1.0 MCSHAPE v2.50 Miscellaeous selective computatio of sigle iteractio chais partial partial particle photos photos / electros photos scalar equatio vector equatio Trasport model solutio determiistic determiistic Mote Carlo collisios 1-D spatial geometry 3-D spatial geometry laguage 3 DELPHI 3 FORTRAN usig MCSHAPE3D FORTRAN 90 NEW!! 3D versio of MCSHAPE additioal libraries graphics WINTERACTER Code platform WINDOWS LINUX WINDOWS / LINUX distributio web site alpha testig web site parallelizatio MPICH v1.0 (oly Liux) spectroscopy aalytical chemistry Applicatios radiatio metrology x-ray optics with MCSHAPE3D dosimetry foresee i v2 with MCSHAPE3D radiatio trasport teachig

43 TARGET: 3D - MCSHAPE heterogeeus target -> VOXEL MODEL iterfaced with GAMBIT (FLUENT eviromet) SOURCE: uiform source o a disk uiform source o a rectagle poit source DETECTOR: disk detector rectagular detector plae ifiite detector Collimator i frot of the detector V. Scot, J.E. Feradez, L. Vicze, K. Jasses, submitted to NIM-B (2005)

44 3D MCSHAPE: XRF Tomography θ elemetal siograms Total dimesio: 0.1 x 0.1 x 0.01 cm Compositio: Regio A: C + 0.1%Sr, ρ = 1.0 g/cm 3 Other elemets: Regio B: SiO 2 + 1%Fe, ρ = 2.23 g/cm 3 Regio C: SiO 2 + 1%Ba, ρ = 2.23 g/cm 3 Regio D: SiO 2 + 1%Zr, ρ = 2.23 g/cm 3 X-ray beam x ED-detector Source: eergy: kev type: poit source upolarized Detector: type: disk with 30 mm 2 of total area o collimator V. Scot, J.E. Feradez, L. Vicze, K. Jasses, submitted to NIM-B (2005)

45 3D MCSHAPE: XRF Tomography Full spectrum Sr Ba Zr Fe recostructio Full Sr Ba Zr Fe V. Scot, J.E. Feradez, L. Vicze, K. Jasses, submitted to NIM-B (2005)

46 OPEN PROBLEM #1: COHERENCE Vector trasport equatio behaves liearly oly for a icoheret source Diffusio of coheret radiatio is ot cosidered yet i trasport models used to describe x-ray diffusio

47 OPEN PROBLEM #2: VARIANCE REDUCTION ACTUALLY: Variace reductio o the agular variables is performed usig the average kerel. The Stokes compoets are computed usig weights. MIXED METHOD OPTIMIZED INTENSITY

48 CONCLUSIONS

49 CONCLUSIONS MCSHAPE was developed : - to provide a full descriptio of the polarizatio state evolutio through multiple scatterig collisios - to exted results of the determiistic method to higher orders of collisio

50 CONCLUSIONS The vector MC code MCSHAPE give: - a detailed descriptio of multiple scatterig of the prevailig iteractios i the x-ray regime - full aalysis of the fial state of polarizatio at each collisio umber - for ifiite or fiite, ad sigle or multi-layer multi-compoet targets

51 CONCLUSIONS cot. Good agreemet with experimetal data has bee obtaied for both, upolarized ad polarized sources More detailed tests are beig plaed i the future Experimetal comparisos are welcome!!! As well as scietific cooperatios!!!