Measuring polarization of high energy

gamma rays.

Polarized photon processes in Geant4.

Gerardo O. Depaola

Universidad Nacional de Córdoba

Member of the collaboration.

Polarized Gamma Astrophysics

Emission Mechanisms (Synchrotron Radiation, Bremsstrahlung, Compton Scattering, Photon Splitting)

Astronomical Sites (Solar Flares , Galactic BH candidates,

Pulsars, AGN, GRB,etc.)COMPTEL

Coburn and Boggs (2003)

The role of Simulations Development and Optimization of instrument concepts

Detailed characterization of instrument response for analysis

Possibility of comparisons between different configurations

Verification of Scientific Objectives and Potential risks

Data processing test-bed and Validation of Physics results

The technique to measure polarization depend

to the energy: Four Main Process

Photoelectric Effect (~ keV to ~50 kev).

Compton Effect (~50 keV to ~10 Mev).

in the nuclear field.

Pair Production:

(>~50 MeV) in the electron field.

Methodology

• Study the cross section

• Convert the cross section into a PDF (Probabilistic Distribution Function).

• Develop an algorithm.

• Write the algorithm into a Computational language (C++) and insert it into the code (GEANT4).

• Test -> Made a simple simulation to reproduce de distribution and/or made simulation of experimental set up and compare results.

Photoelectric Effect

Introduce polar angle distribution.

The polar angle is sampled from the K-shell cross

section derived by Sauter.

Introduce Polarization. The cross section was obtained using the Stokes parameters.

Introduce azimuthal angle distribution.

Polar angle distribution

The polar angle is sampled from the K-shell cross section

derived by Sauter (1955) using K-shell hydrogenic electron

wave function

( )

( )( )( )

θβ−−γ−γγ+

θβ−

θ

γ

β

α=

Ω

σ

cos1212

11

cos1

sin

k

Zr

d

d4

235

2

e

4

Where: k is the photon energy in mc2 units,

E is the electron energy

( )2mc/E1+=γ ( )2

2

mc2E

mc2EE

+

+=β

Azimuthal angle distribution

The “Stokes” matrix for the Photoelectric effects is:

( )

−

−+

θβ−

θα=

0000

B000

A000

00DD1

cos1

sin

k

ErZT

3

2

2

2

o

54

So the cross section for

polarized incident radiation is:

[ ]

φ

φ

φσ

2cos21

0

2sin

2cos

1

0001

D

Td

+≈

−

−=

( )

−

θβ−= 1

cos1Ek

2

k

1D

Example of this

in Geant4

Comparison with experiment

Comparison with experimental data published in Nature (vol. 411,

2001) by E. Costa et. al. : “An efficient photoelectric X-ray

polarimeter for the study of black holes and neutron stars” and

related publications.

The experiment

Gas: 80% Ne, 20% dimethylether at 1

atm.

GEM hole geometry: 40 µm diameter, 60 µm pitch.

128 anodes with a pitch of 200 µm

Drift/absortion gap: 6 mm.

Experimental results

Measure of a single

photoelectron track.

The dimension of the hexagons

is proportional to the energy

deposited

Auger Electron

Photoelectron

Azimuthal distribution of

the charge barycentre.

BA2

B

CC

CC

minmax

minmax

+=

+

−=µ

In this case, µ = 0.44

Modulation factor for 100% linearly polarized radiation:

Azimuthal distribution of barycentre

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

100

200

300

400

500

600

700

800

Angular distribution for randomize linearly polarized x-ray

φ [rad]

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

200

400

600

800

Angular distribution for 100% linearly polarized x-ray

along the x axis.

Model: a + b*cos2(φ+c)

a = 249 ± 4

b = 547 ± 6

c = 0.0014 ± 0.0056

φ [rad]

Unpolarized Polarized

Modulation factor µ = 0.41Impact point: x = 0, y = 0.Charge Barycentre:

∑=T

iib

E

Exx ∑=

T

iib

E

Eyy

=> the azimuthal angle ϕ = atan(yb/xb)

COMPTON SCATTERING

The Klein Nishina cross section:

Θ+−

ν

ν+

ν

ν

ν

ν=

Ω

σ 2

0

0

2

0

22

0 cos42h

h

h

h

h

hr4

1

d

d

Where,

hν0 : energy of incident photon.hν : energy of the scattered photon.Θ : angle between the two polarization vectors

Angles in the Compton Effect

θ Polar angle

φ Azimuthal angle

ε Polarization vector

y

x

O z

ξ

θα

φhν

hν0ε A

C

ε’⊥

ε’||

CO

ε

Ahν

ε’ ξβ Θ

ξ

x

Polar Angle distribution

Summing over the 2 components:

Sampling method (G4LowEnergyPolarizedCompton)

Integrate over φ

Sample θ

Energy from the Compton θ-E relation Sampling φ from P(φ) = a (b – c cos2 φ ) distribution

−+=

Ωφθ

ν

ν

ν

ν

ν

νσ 22

0

0

2

0

22

0 cossin22

1

h

h

h

h

h

hr

d

d

Scattered Photon Polarization Geometrical relations for obtaining the Scattered Photon Polarization.

β is obtained from cos Θ = cos βN and Θ is sampled from the Klein Nishina cross section.

=−=

⇒=

φθξ

φθξ

22 cossin1sin

cossincos

β

φθθ−φφθ−=ε coskcoscossin

N

1jcossinsin

N

1iN 2'

||

( ) βφθ−θ=ε⊥ sinksinsinjcosN

1'

ε’⊥

ε’||

CO

ε

Ahν

ε’ ξβ Θ

ξ

x

MEGA (courtesy of A. Zoglauer)Marx Planck Instituto of Munich

Polarization 2 MeV

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0 45 90 135 180

Azimuthal scatter angle [°]

rel. counts

0

5

10

15

20

25

30

35

0 20 40 60 80 100Compton scatter angle [deg]

Modulation[%]

Dependence of the degree of modulation

on the Compton scatter angle

Azimuthal scatter angle distribution

(geometry and efficiency corrected)

The MEGA prototype, which consists

of l/12 of the volume of a full

telescope, has been calibrated at the

High Intensity Gamma Source of the

Free Electron Laser facility at Duke

University

Reconstruction of incoming direction

from scattering angle θ and the the scatter photon direction.

Ellipses obtain for 20 100 keV photons

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

Coordenada Y [cm]

Coordenada X [cm]

Intersection point between the

ellipses.

θ

The reconstruction method

Pair Production

The existing methods of analyzing can be

divided into three groups

Uses the photoproduction of particles on amorphous targets and investigate the correlations between the photon polarizations and the kinematics of the final particles.

Uses the coherent effects of the interaction of the photon with single crystal.These lead to a dependence of the photon absorption on the angle between the photon polarization vector and the crystallographic axes.

Uses the correlations between the polarization of the particles in the initial and final state. The exceptional complexity of such experiment means that can not be realized at the present times.

In the first and second groups, the differential cross section

has an azimuthal asymmetry in the form:

[ ]ϕσϕσσϕ

σπ 2cos12cos2 )()()( Λ−=−= PP

d

d tlt

Where:

σ(t): is the total cross section on unpolarized photon.

σ(l): is the part due to polarization.

P: is the degree of linear polarization.

Λ= σ(l)/ σ(t): is the asymmetry.

ϕ: is the azimuthal angle.

For high energy photon, the production of e+e- pairs on nuclei and electrons is promising:

1) γ +A → A + e++ e-

2) γ + e- → e- + e++ e-

The cross sections of the processes 1) and 2) are proportional, respectively, to Z2 and Z and increase with photon energy. At very large energy, this growth is logarithmic.

The process 1) is difficult to applicate for photon energies > 1 GeV due to the small opening of the created pair.

The process 2) is very convenient for measuring the linear polarization of photon beams in a very wide range of photon energies. A significant proportion of the recoil electron have a kinetic energy sufficient for their detection.

Gamma conversion in the nuclear field

Modification polar angle

distribution.

Different approach

Sample directly the trigonometric values

Introduce Polarization.

Calculation of the cross section (E>100 MeV)

Modification azimuthal angle distribution

Important for Pair production telescopes

e-

e+

hν

Pair production telescopes

Polar angle sampling

Sauter-Gluckstern-Hull formula

( )2cospEp2

sin

d

dP

±±±±

±

± θ−

θ=

θ

From this distribution the sampled angles are:

±±

±±±

±±

±+−

+−=

+−

−=

Erndp

prndE

Erndp

rndrnd

)12(

)12(cos;

)12(

)1(2sin θθ

This formulae produce the same distribution as in G4GammaConvertion

( ) ±±−− θ=+= Eu;eudeuC)u(f au2au

The advantage of the first formula is to sample directly sinθ and cosθ

Azimuthal distribution and PolarizationBethe-Heitler Cross Section:Unscreened point nucleus

( )( ) ( )

( )

]m)1)(cosE(

)1(cosE)coscoscossinsin1)(E(E[2q

cos2sin

sin

E

)E(

sin

sin

E

E

cos1

sin

cos1

sin

cos1

)cos(sin

cos1

cossinq

cos1

)cos(sinE

cos1

cossinE4

q

EEddEd

mr

2

Z2d

2

2

2

22

2

43

2

0

2

2

+−θ−ωω+

+−θω+θθ−φθθ−−ω−=

φ+

θ

θ−ω+

θ

θ

−ω×

×θ−

θ

θ−

θω−

θ−

φ+ψθ−

θ−

ψθ−

−

θ−

φ+ψθ−ω+

θ−

ψθ−ωΩΩ

ωπ

α−=σ

−

+−+−+

+

−

−

+

+

+

−

−

+

+

−

−

+

+

−

−−+

Validity: First Born approximation, no screening, negligible nuclear recoil

Angles in pair productionz

y

xεεεε

k

p-p+

Ψ

φ

θ-θ+

The polarization dependence

is represented trough the

angle Ψ.

The angle ϕ is the angle between the projection of the

particle moments in the x-y

plane.

For unpolarized beam => Ψ is isotropic, not ϕ which is related to the azimuthal

distribution of the pair.

Integration over the energy and polar angles => azimuthal

distribution of a pair created by 100 MeV photon.

0.00 0.52

1.05 1.57

2.09 2.62

3.14 3.00 3.02

3.04 3.06

3.08 3.10

3.12 3.14 4

6

8

10

12

14

16

1 d σ

α (Zr 0 ) 2 d φ d ψ

φ [rad] ψ [rad]

This surface can be

parameterized and

its parameters can

be put in function of

the photon energy.

Integrating over Ψone obtain the φ distribution .

Azimuthal distribution

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 0 .7

0 .8

0 .9

1 .0

1 .1

1 .2

1 .3

1 .4

1000 M eV

R

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 0 .7

0 .8

0 .9

1 .0

1 .1

1 .2

1 .3

1 .4

500 M eV

0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 0 .7

0 .8

0 .9

1 .0

1 .1

1 .2

1 .3

1 .4

100 M eV

∆ ε [rad ]

Ratio of nº of pairs contained in plane parallel to the

polarization vector with nº of pairs perpendicular plotted

against experimental azimuthal resolution

Asymmetric ratio for pair production

High Energy Gamma-ray Polarimeter

Polarization capabilities

evaluated with G4

Pixelized gas MicroWell detectors

for a 3D recon of the pair

“A Concept for a High Energy

Gamma - Ray Polarimeter” P. F.

Bloser (NASA), S. D. Hunter (NASA),

G. O. Depaola (UNC), F. Longo

(INFN).

SPIE 5165, “X- ray and Gamma –

ray Instruments for Astronomy XIII”.

(2003) 322.

Pair production by linearly polarized γγγγ-rays on electrons: Angular distribution for the electron recoil.

Feynman diagrams of Triple Production

The first study of this process were done by:

A. Borsellino and V. Votruba.

A. Borsellino, Nuovo Cimento 4, 112 (1947).

A. Borsellino, Rev. Univ. Tuc. 6, 37 (1947)

Calculate the pair production cross section if the field produce by a particle if mass M and charge Ze; he used only the a and b diagrams.

V. Votruba, Bull. Int. Akad. Tcheque Sci. 49, 19 (1948).

Calculate the all eight diagrams and take the limits for low andhigh energy.

( )−+

−+

−+ −−−+= ppppkEEE

pdpdpdM

rd 1

4

1

33

1

32

2

2

0

4δ

κπ

ασ

22

2

1dcba AAAAM −+−=

Mork’s calculations (Phys. Rev. 1967) of the contribution to the

total cross section of the considered process from different

groups of diagrams separately are very instructive:

XB Borsellino diagrams

Xγ γ-e- diagrams

The other are the interference terms.

∆i =σi / σB => σ = σB(1-∆)

From the figure => for

ω≥16m

σ is described by B diagrams with an accuracy better than 1.2%.

EBEBEBB XXXXXXM γγγγ +++++=2

Momentum and angles of final state particles

z

x

y

p1

p2

k

θ1

θ2

θ21

ϕ1

ϕ21

ϕ2

z

y

x

k

p-

p+

θ-θ+

ϕr

pr

ϕ+

ϕ-

θr

=≤≤

ωθθ

m4arccos0 max

The polar angle θ with which is emitted any of the electrons (whatever will be electron or positron), is

limited by the energy ω of photon according to:

Differential cross sections for triple production

by linearly polarized photons

From this θ, p relation one can see that for θ =0, p ∞, this is consequence of the approximation. The kinematics of the process impose limits to θ maximum.

−−

−−

−=

2cotln

cos

sin12cos

2cotln

cos

cos511

cos

sin

3

22

1

1

1

2

1

1

1

2

1

3

1

2

11

0

θ

θ

θϕ

θ

θ

θ

θ

θα

θϕ

σπ

P

r

dd

d

mqEmp +== 2

1

2

1

sin

cos2

θ

θ

In the high energy gamma rays limits, it is possible to find the following angular

distribution expression for the recoil electron, in which the terms of order m/ωcan be ignored.

p

θ

θmax

pmax(θ)

pmin(θ)

p+(∆,θ)

p-(∆,θ)

θ

∆min = 2m

p(θmax)

∆ = constant arbitrary value

D = 0

π/2( )m

mmp

−=

ω

ωθω

2,

*

max1

From these figure one can

see that for p > p* the

recoil electro are emitted at

angles less than

Where p* is:

( )max1 ωθ

In this figures we represent schematically the recoil electron momentum as

function of ∆2 and θ1.

( ) ( )

( ) 1

222

2

2/1

2

1

2

1lim

sin4

:where

cos,,

θ

θθω

mSSmD

D

mSmp

−+=

−=∆

( )

++

−= −

1

1

1

11

max11 cos,p

mEm

p

mEp

ωωθ

p

1000

100

10

1

0,1

0,01

0,001

0,2 0,4 0,6 0,8 1,0 1,2 1,4 θ

plim(θ)

10 m

50 m

100 m

ω = 1000 m

In this figure we shown the region of allowed values of the linear

momentum (tridimensional) against the polar angle for several

values of the energy of the incoming photon ω.

θ

θ2lim

sin

cos2mp =

Probability density of linear (three dimensional) momentum values, for several polar angle values

of the emitted particle, for ω = 100 m. The dotted line, corresponding to θ = 0.6 for ω = 1000

m, shows the increasing ω typical behavior: probability concentrates strongly near the boundary

of the allowed region (∆ ≅ 2 m), slightly shifting toward the higher p border.

1 2 3 4 5 6 7 p

θ = 0,5

0,6

0,7

0,80,9

1,01,1

1,21,3

d2σdp dθ

1

α r02

10

1

0.1

1000 m

)cos(2Pdσdσdσ (l)(t) ϕ−=

)(

)(

t

l

d

d

σ

σ=Λ

∫=max

00

)(2)( )(

p

p

ii

p dpdpd

dF

θ

σθ

( )( )

−−= 2cotln

cos

cos511

cos

sin

3

2 2

3

2

0)( θθ

θ

θ

θαg

rF t

a

( )( )

−= 2cotlncos

1cos

sin

3

2 2

3

2

0)( θθ

θ

θ

θαg

senrF l

a

Angular Distribution

We define the asymmetry as:

We also definewhere the upper index ‘i’ that can be

‘t’ or ‘l’

p0 is some threshold momentum value for the detection of recoil electrons

Asymptotic functions

calculated by Boldyshev et.al.

They arrived to this results

calculating the dσ2/dpdϕassuming the recoil electron

acquires the maximum angle

compatible with its momentum

p0 = 1

p0 = 0,5

p0 = 0,2

ω = 100 m

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 θ

7

6

5

4

3

2

1

p0 = 1

p0 = 0,5

p0 = 0,2 ω = 1000 m

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 θ

7

6

5

4

3

2

1

p0 = 1

p0 = 0,5

p0 = 0,2

ω = 100 m

1,0

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

0,1

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 θ

ω = 1000 m

1,0

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

0,1

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 θ

Integral of the differential cross section (in units of ⅔αr02) over the p values from the

three p0 threshold values indicated, plotted as function of the polar angle of recoil electron.

Superimposed are shown the asymptotic functions The upper figures belong to the non

polarized radiation case. The polarized radiation case is shown below. For ω = 1000 m all the graphics appear superimposed to the asymptotic function.

)()(0

θi

pF

)()(0

θt

pF

)()(0

θl

pF

Asymmetry as a function of the polar angle of the electron recoil

for the asymptotic values of cross section

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

θ [rad]

Λ

)(

)(

t

a

l

aa

F

F=Λ

From this figure one can

see that the maximum

value of Λa is around 0.14 for θ maximum.

MONTECARLO SIMULATION OF THE

ASYMPTOTICS EXPRESIONS

We must generate random values of θ and ϕ distributed with

probability proportional to the following function f(θ,ϕ), for θrestricted inside of its allowed interval value, not too close to its

extremes (either 0, or θmax):

where:

As we will see, for θ < 90º, F1 is several times greater than FP, and since both are positive, it follows that f is positive for any possible

value of P (0 ≤ P ≤ 1).

( ) ( ) ( ) ( )( )θϕθθ

θϕθ PFFf 2cosP

cos

sin, 13

−=

( ) ( )( )2cotlncos

cos511

2

1 θθ

θθ

−−=F

( ) ( )( )2cotlncos

sin1

2

θθ

θθ −=PF

Algorithm for non polarized radiation.

We must generate random values of θ between 0 and θmax = arccos( ), distributed with probability proportional to the following function f1(θ):

ωm4

( ) ( )( ) ( )θθ

θθ

θ

θ

θ

θθ 13

2

31cos

sin2cotln

cos

cos511

cos

sinFf ×=

−−=

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60

5

10

15

20

f1(θ)F

1(θ)

θ [rad]

Graphics of f1(θ) and F1(θ). The infinity of F1 at extreme θ = 0 is very weak (logarithmic), and thus,

multiplying by sinθ, shows that f1 →0. The f1 divergence at π/2 is not important because the interval of

allowed values of θ ends before this point.

To simulate the f1 function, we

decomposed in two factors: the first,

senθ/cos3θ, easy to integrate, and the other, F1(θ), which may constitute a reject function, on despite of its θ = 0 divergence

( ) 2if,cos35

331cos

3

14 22

1 πθθθθ →

++→ KF

Expanding F1 for great values of θ, we see it is proportional to cos2θ:

Thus F1 divided by cos2(θ) will be

a better reject function, because it

tends to constant value 14/3 =

4,6666... for large θs, whereas for small θs, cos(θ)→ 1.

It seems adequate to choose θ0near 5º, and, after some

manipulation looking for round

numbers we obtain:

( )( )

00.14º47.4cos

º47.42

1 ≅F

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.2

0.4

0.6

0.8

1.0

4.47º

r(θ)

1/3

θ [rad]

( )( )

≤θ

≥θ

θ−

θ+

θ

θ−−

θ=

θ

θ

=θ

4.47ºfor;1

4.47ºfor;cos1

cos1ln

cos2

cos511

cos14

1

cos14

1 2

22

1F

r

Finally we define a reject function :

Now we have a probability distribution function (PDF) for θ, p(θ) = C f1(θ), expressed as a product of another PDF, π(θ), by the reject function:

( ) ( ) ( ) ( )θθπθθ rCCfp '

1 ≅=

( )θcos

θsin14Cθπ π=Where:

the cumulative

probability ( )ω4mln

θ)ln(cos2

m4

ωln7

θ)ln(cos14θd)θπ(P

θ

0π =

−=′′= ∫

Finally we sample a random number ξ1 (between 0 and 1), and obtain the corresponding θ value:

ω=θ

ξ

2

1

m4arccos

Another random number ξ2 is sampled for the reject process: the θ value is accepted if ξ2 ≤ r(θ), and reject in the contrary. For θ ≤ 4,47º all values are accepted.

Algorithm for polarized radiation.

The azimuthal dependence of the differential cross section is given by:

( ) ( ) ( )( )θF)cos(2PθFθcos

θsinθ,f P13

ϕϕ −=

( )( )2θcotgln

cosθ

θsin1)(θF

2

P −=

The expansion of FP for θ near π/2 shows that it is proportional to

cos2(θ), and FP/cos2(θ) tends to a non

null value, 2/3. This value is exactly

7 times the value of F1/cos2(θ).

This suggests applying the

combination method, rearranging the

whole function as follows:

( ) ( ) ( ) ( )( )

−=

θF

θFP)cos(21

θcos

θFθtgθ,f

1

P

2

1 ϕϕ

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.0

0.2

0.4

0.6

0.8

1.0

1.2

π/2

2/3

FP(θ)

θ [rad]

FP(θ)/cos(θ)

2

The conditional probability of ϕ given θ is p(ϕ / θ) :

( ) ( )( ) ( )

( ) ( ) ( )( )

−==

θF

θFP2cos1θF

θcos

sinθC

θcos

θFtgθCπ2

1

θq

θ,pθ/p

1

P13

2

1

ϕϕ

ϕ

θ

θϕ−

π=

)(F

)(FP)2cos(1

2

1

1

P

Where: ( ) ( )θcos

θFθtgCπ2d),p(θq

2

12π

0== ∫ ϕϕθ

ϕ

2π

p(ϕ / θ)Reject function for

the ϕ values.

1.-We begin sampling a random number ξ1 and obtain θ from

2.- Then we sample a second random number ξ2 and accept the values of θ if ξ2 ≤ r(θ),

( ) 4.47ºθfor;θcos1

θcos1ln

θcos2

θ5cos11

θcos14

1θr

2

2≥

−

+−−=

For θ ≤ 4,47º all values are accepted .

3.- Now we sample ϕ. We sample a third random number ξ3 (which is defined as ϕ/2π) and evaluate the reject function

( ) ( ) ( )( )

−=

θF

θFPξπ4cos1

2π

1ξr

1

P33θ

( )

−−

−

−=

2

θcotglnθcos51cosθ

2

θcotglnθsincosθ

P)4cos(12π

1

2

2

3ξπ

4.- Finally, with a fourth random number ξ4 , we accept the values of ϕ = 2πξ3 if ξ4 ≤ rθ(ξ3).

The sample technique

ω=θ

ξ

2

1

m4arccos

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60

1000

2000

3000

4000

5000

6000

7000

8000

9000

Arbitrary Units

θ [rad]

50 m

100 m

1000 m

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.60.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Arbitrary Units

θ [rad]

50 m

100 m

1000 m

Example of polar angle distribution for the electron recoil for three

differences energies generated with the method (Distribution

obtained with 100000 histories). On the right, the same as left but

renormalized to show that the dependence with the energy is only in

the θmax.

0 1 2 3 4 5 60

250

500

750

1000

1250

Arbitrary Units

ϕ [rad]

Azimuthal angle distribution for the electron recoil. Continuous

line, data fit with expression p(ϕ) =a(1-b(cos(2ϕ-c)) => dσintegrated over all θ.

( )

137.0θdσ

θdσb

θ)cos(2dσPdσ)(d

max

max

max

θ

0

(t)

θ

0

(l)

t

θ

0

(l)(t)

==⇒

ϕ−=ϕσ

∫

∫

∫

d

d

d

b=0.138+/-0.001 for ω=1000.

The theoretical value is:

This result also shows that one does

not lost much asymmetry taking into

account all θ, since the maximum asymmetry is for θ = θmax and for that θ, Λ = 0.143.

Comparison of the cross section with experimental distribution with

respect to θ1.1

)( θσ dd t

Experimental data with:

0.4≤ q ≤ 1 MeVc,

q ≥0.4 MeV

20 ≤ ω ≤ 600 MeV

45 ≤ ω ≤ 75 MeV.

The dot-dash curve is the

asymptotic cross section.

The solid curve is the cross

section for ω = 1400m and q0 = 0.8 m.

Simulation example of electron recoil in 1.0 m of 85%Xe 25%CO2

Same as before but includind the pair created. Blue lines are positron, red lines are electron and green lines are photons.

Simulation results

-0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

θ

100 MeV

Simulated distribution

Theoric distribution

0,0 0,5 1,0 1,5 2,0 2,5 3,0

0

100

200

300

400

500

ϕ

Comparison between the theoretical polar angle distribution and simulate results.

( ) ( )θ θ θ= +

arctan tan tanX Y

2 2

Reconstructed azimuthal distribution; superimpose a cos2 ϕ.

X

Y

θ

θϕ

tan

tanarctan=

The End