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!!!