MULTIPLE SCATTERING OF PHOTONS USING THE

BOLTZMANN TRANSPORT EQUATION

Jorge E. Fernandez

Laboratory of Montecuccolino-DIENCA

INFN, INFM & Alma Mater Studiorum - University of Bologna

ICXOM-18, Frascati 25-30 September 2005

INTRODUCTION

MULTIPLE SCATTERING

• X-rays penetrate deeply into the matter, and, in a thick medium, give place to a phenomenon known as multiple scattering.

• Multiple scattering models describe the influence of the prevailing interactionsin the x-ray regime (photoelectriceffect, Compton scattering and Rayleigh scattering)

MULTIPLE SCATTERING (cont.)

• Multiple scattering models describe the influence of the prevailing interactionsin the x-ray regime (photoelectriceffect, Compton scattering and Rayleigh scattering)

• The photoelectric effect itself can beconsidered as a ‘scattering process’

Photoelectric effect as ‘scattering’

photoelectric 'scatter ing'=+

characteristic photon

radiativetransition

hν

photoelectron

photoelectric absorption

L3

L2

L1

K

PREVAILING INTERACTIONS IN THE X-RAY REGIME

PRIMARY PHOTON

COHERENT SCATTERING

INCOHERENT SCATTERING

PHOTOELECTRIC EFFECT

RAYLEIGH PHOTON

COMPTONPHOTON

COMPTON ELECTRON

CHAR.X-RAYS

PHOTO ELECTRON

Scattered photons

Scattered electron

DESCRIPTION OF POLARIZATION

WHY POLARIZATION?

Polarization statewave nature of photons

By considering polarization weimprove the model of photon

diffusion

Without polarization photons are considered only as a particles

a good approximation in many cases, but not for phenomena that are influenced

by their wave properties

REPRESENTATION OF POLARIZED RADIATION

Stokes parameters I,Q,U,V (having the dimension of an intensity) can specify:

• Intensity of the beam• Degree of polarization• Orientation of the ellipse of polarization• Ellipticity

Polarization state definition

c) Orientation of the polarization ellipse (angle χ)

d) Ellipticity (expressed by the angle β)

Four parameters: a) The intensity (the square of the electric field) b) The degree of polarization (the fraction of elliptically polarized beam)

Propagation vector

STOKES’ REPRESENTATION OF POLARIZED RADIATION

Definition of STOKES PARAMETERS:

βIVχβIUχβIQ

2sin2sin2cos2cos2cos

===

Degree of polarization:

I)VU(QP

21222 ++=

EXAMPLES OF THE STOKES REPRESENTATION

(1,0,0,1)Circular

(1,0,1,0)Linear (45°)

(1,-1,0,0)Linear ( )

(1,1,0,0)Linear (||)(1,cos2χ,sin2χ,0)Linear (generic)

(1,0,0,0)Unpolarised

Set S (I,Q,U,V)Polarisation state

⊥

INCOMING PHOTON

OUTGOING PHOTON

ATOM )E,ω,rV(

)E,ω,rU(

)E,ω,rQ(

)E,ω,rI(

1n1n1n

1n1n1n

1n1n1n

1n1n1n

+++

+++

+++

+++

nE,nω

nn E,ω

1nE,1nω ++

1nr +)E,ω,rV(

)E,ω,rU(

)E,ω,rQ(

)E,ω,rI(

nnn

nnn

nnn

nnn

COLLISION SCHEME

Modification of the polarization state due to a collision (Stokes representation)

PHYSICAL MODEL

• The photon state is changed by a matrix kerneldepending on the type of the collision a. Thiskernel operates on the polarisation state according to the relationship:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

+

+

+

++

1)(n

1)(n

1)(n

1)(n

(n)

(n)

(n)

(n)

(n)(n)1)(n1)(n

VUQI

VUQI

]λ,ω,λ,ω,r[a

H

PHOTON DIFFUSION IS DESCRIBED BY A “VECTOR” TRANSPORT EQUATION

(THE 1-D EQUATION IS SHOWN HERE)

where

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

),ω,(),ω,(),ω,(),ω,(

λλλλ

zVzUzQzI

f

VECTOR TRANSPORT EQUATION (CONT.)

where

= kernel matrix in the meridian plane of reference

= scattering matrix in the scattering plane of reference

)(SH

)(SK

PHOTON DIFFUSION IS DESCRIBED BY A “VECTOR” TRANSPORT EQUATION

(THE 1-D EQUATION IS SHOWN HERE)

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

+

+−=∂∂

∫ ∫∞

),ω,(),ω,(),ω,(),ω,(

f where

),ω(S)(),ω,(f),ω,,ω()(ω

),ω,(f)(),ω,(f

'''

0 4

'

λλλλ

λδλλλλ

λλμλη

π

zVzUzQzI

zzzdd

zzz

''HU

VECTOR TRANSPORT EQUATION (CONT.)

where

)(),,,()(

),,,('

'

Ψ−Ψ−=

=

LKL

H'

'

λλπ

λλ

ωω

ωω

= kernel matrix in the meridian plane of reference

= scattering matrix in the scattering plane of reference

H

K

IMPORTANT PROPERTIES OF THE “VECTOR” TRANSPORT EQUATION

• Describes the evolution of the full polarization state (not only the intensityof the beam)

• Is linear (for the Stokes representation)• Requires the simultaneous solution of

the whole set of transport equations• Cannot be transformed in a scalar

equation !! (due to the coupling in the scattering term)

THEORETICAL MODELS

MODELS

Different degrees of approximation to describe the diffusion photons:

•• scalar modelscalar model: photons never modify an average polarization state

•• vector modelvector model: transport of photons starting with arbitrary polarization state

Both models follow a multiple scattering scheme

a Photoelectric effect Rayleigh scattering Compton scattering(P)

characteristic lines(discrete)

(R)Rayleigh peak

(discrete)

(C)Compton peak(continuous)

bPhotoelectric effect Rayleigh scattering Compton scattering

a

Photoelectric effect(P,P)

XRF secondaryenhancement

(discrete on XRF line)

(P,R)XRF enhancement due to

scattering(discrete on XRF line)

(P,C)XRF enhancement due to

scattering(continuous on XRF line)

Rayleigh scattering(R,P)

XRF enhancement due toscattering

(discrete on XRF line)

(R,R)second order scattering

(discrete on Rayleigh peak)

(R,C)second order scattering

(continuous on Comptonpeak)

Compton scattering(C,P)

XRF enhancement due toscattering

(discrete on XRF line)

(C,R)second order scattering

(continuous on Comptonpeak)

(C,C)second order scattering

(continuous on Comptonpeak)

one collision

two collisions

Scalar transport equation

...+

+

35 40 45 50 55 60 6510-5

10-4

10-3

10-2

Compton peak

Rayleigh peak

E0

I

E [keV]

35 40 45 50 55 60 6510-5

10-4

10-3

10-2

E0

I(2)

E [keV]

35 40 45 50 55 60 6510-5

10-4

10-3

10-2

Rayleigh peak

Compton peak

E0

I(1)

E [keV]

SCALAR EQUATION

I

I(2)

I(1)

I = Σi I(i)I0

Vector transport equation

+

+

...

VU

Q

35 40 45 50 55 60 6510-5

10-4

10-3

10-2

Compton peak

Rayleigh peak

E0

I

E [keV]

35 40 45 50 55 60 6510-5

10-4

10-3

10-2

E0

I(2)

E [keV]

35 40 45 50 55 60 6510-5

10-4

10-3

10-2

Rayleigh peak

Compton peak

E0

I(1)

E [keV]

VECTOR EQUATION

I(2) Q(2)

U(2)

V(2)

V(1)U(1)Q(1)

I(1)

I = Σi I(i)I0

I

EXAMPLES

LET US SHOW TWO SIMPLE EXAMPLES

1) Scattering of unpolarized radiation2) Scattering of linearly polarized radiation

incidentbeam

right anglescattering

a

1) Unpolarized Rayleigh scattering

How scattering polarizes a beam

Scatterer

Unpolarized beam

90 degrees scattering

SUMMARY FOR UNPOLARIZED RADIATION

Unpolarized beam(composed by rayswith electric vectorrandomly oriented

around the propagation

direction)

After scattering the beam is

partially (totally) polarized

depending on the type of

interaction and the scattering

geometry

2) Polarized Rayleigh scattering

incidentbeam right angle

scattering

bElectric vector parallel to the

scattering plane

SUMMARY FOR LINEAR POLARIZATION

Linearly polarizedbeam with

electric vectorparallel to the

scattering plane

Almost nullscattering at 90 degrees

Scatterer = polarizer

•

•Unpolarized beam•

90 degreesscattering•

Almost nullscattering

90 degreesscattering

COMBINING BOTH PROPERTIES

THE CODES

SOLUTION TECHNIQUES

The transport equation is solved usingan order-of-collisions scheme

comparable results for deterministicand Monte Carlo solutions

MCSHAPESHAPEDeveloped codes

Number of collisions

Capability to describe the geometry

Accuracy

LocalGlobalScope of the solution

Monte Carlo (statistical)DeterministicSolution

Deterministic vs. Monte Carlo

CHARACTERISTICS OF THE CODE MCSHAPE

• Arbitrary polarization state of the source• Multi-layer multi-component

homogeneous targets• Monochromatic or polychromatic source• Doppler broadening (for Compton

scattering)• Full description of the polarization state• N-collisions

COMPARISON WITH SCALAR VERSION

The source is unpolarized and monochromatic.The sample is carbon and and the scattering angle is 90°.

35 40 45 50 55 60 6510-5

10-4

10-3

10-2

Compton peak

Rayleigh peak

E0

Polarization dependent (vector model) Average polarization Polarization dependent (scalar model)

Inte

nsity

(arb

. uni

ts)

E [keV]

EVOLUTION OF THE CODE

• Development of two different codes:- MCSHAPE0: max. 4 collisions and analog calculation - MCSHAPE1: no limits in number of collisions

• First version: Pascal (1995)• Present versions: 1D and 3D written in

FORTRAN 90• Platforms: Windows and LINUX• Parallelization: MPI (under LINUX)

WEB SITE http://shape.ing.unibo.it

These codes are going to be distributed by NEA Data Bank (OECD) and RSICC (US-DOE)

CODES COMPARISON (part 1: Physics)

⌧ transmission geometry

⌧ ⌧ ⌧reflection geometry

foreseen in v3 solid state Si/Geexternal detector

⌧ postprocessorpolychromatic source

⌧ ⌧ ⌧monochromatic source

full polarizationstate

intensitycomponent onlycalculated spectrum

arbitraryunpolarised linear/ unpolarised

source polarizationstate

StokesStokespolarizationrepresentation

⌧ multilayer targets

⌧ ⌧ finite thickness targets

⌧ ⌧ ⌧infinite thickness targets

foreseen in v3 user defined elements

⌧ ⌧ ⌧open data bases

foreseen in v3 ⌧ foreseen in v3 electron bremsstrahlung

⌧ ⌧ first collision onlyCompton profile

⌧ ⌧ ⌧atomic Compton scattering

⌧ ⌧ ⌧atomic Rayleigh scattering

⌧ ⌧line width

⌧ ⌧ ⌧~1000 characteristic lines

⌧ ⌧ ⌧photoelectric effect

Physics

MCSHAPE v2.50D3DSHAPE v1.0SHAPE v2.20DetailsFeatures

UNIQUE FEATURES!

CODES COMPARISON (part 2: model and programming)

MCSHAPE v2.50D3DSHAPE v1.0SHAPE v2.20DetailsFeatures

⌧ ⌧ ⌧radiation transport teaching

with MCSHAPE3D foreseen in v2 dosimetry

with MCSHAPE3Dx-ray optics

⌧ ⌧ ⌧radiation metrology

⌧ ⌧ ⌧analytical chemistry

⌧ ⌧ ⌧spectroscopy

Applications

MPICH v1.0 (onlyLinux)parallelization

web site alpha testingweb site distribution

WINDOWS / LINUX LINUX WINDOWS platform

WINTERACTER graphicsadditional libraries

FORTRAN 90 FORTRAN 77 DELPHI language

Code

using MCSHAPE3D⌧ 3-D spatial geometry

⌧ ⌧1-D spatial geometry

100 3 3 collisions

Monte Carlo deterministicdeterministicsolution

⌧ ⌧vector equation

⌧ ⌧scalar equation

photonsphotons / electronsphotonsparticle

Transport model

partialpartial⌧selective computation of single interaction chains Miscellaneous

NEW!!3D version

of MCSHAPE

3D - MCSHAPE• TARGET:

– heterogeneus target -> VOXEL MODELVOXEL MODEL– interfaced with GAMBIT (FLUENT environment)

• SOURCE:– uniform source on a disk– uniform source on a rectangle– point source

• DETECTOR:– disk detector– rectangular detector– plane infinite detector– Collimator in front of the detector

V. Scot, J.E. Fernandez, L. Vincze, K. Janssens, submitted to NIM-B (2005)

3D – MCSHAPE: XRF Tomography• Total dimension: 0.1 x 0.1 x 0.01 cm• Composition:

Region A: C + 0.1%Sr, ρ = 1.0 g/cm3

Other elements: Region B: SiO2 + 1%Fe, ρ = 2.23 g/cm3

Region C: SiO2 + 1%Ba, ρ = 2.23 g/cm3

Region D: SiO2 + 1%Zr, ρ = 2.23 g/cm3

•• SourceSource::energy: 59.54 keVenergy: 59.54 keVtype: point sourcetype: point sourceunpolarizedunpolarized

•• DetectorDetector: : type: disk with 30 mmtype: disk with 30 mm22

of total areaof total areano collimatorno collimator

x

θ

ED-detector

X-ray beam

elemental sinograms

V. Scot, J.E. Fernandez, L. Vincze, K. Janssens, submitted to NIM-B (2005)

3D – MCSHAPE: XRF Tomography

reconstruction

120 ×

3°

Full Sr Ba Zr Fe

Full spectrum Sr Ba Zr Fe

V. Scot, J.E. Fernandez, L. Vincze, K. Janssens, submitted to NIM-B (2005)

OPEN PROBLEM #1: COHERENCE

• Vector transport equation behaveslinearly only for an incoherent source

• Diffusion of coherent radiation is notconsidered yet in transport modelsused to describe x-ray diffusion

OPEN PROBLEM #2: VARIANCE REDUCTION

ACTUALLY:• Variance reduction on the angular

variables is performed using the average kernel.

• The Stokes components are computed using weights.

MIXED METHODOPTIMIZED INTENSITY

CONCLUSIONS

CONCLUSIONS

• MCSHAPE was developed :

- to provide a full description of the polarization state evolution through multiple scattering collisions

- to extend results of the deterministic method to higher orders of collision

CONCLUSIONS

The vector MC code MCSHAPE give:

- a detailed description of multiple scattering of the prevailing interactions in the x-ray regime

- full analysis of the final state of polarization at each collision number

- for infinite or finite, and single or multi-layer multi-component targets

CONCLUSIONS cont.

• Good agreement with experimental data has been obtained for both, unpolarized and polarized sources

• More detailed tests are being planned in the future

• Experimental comparisons are welcome!!!

• As well as scientific cooperations!!!