Measuring Photon Polarization by Pair Production

MEASURING PHOTON POLARIZATION BY PAIR PRODUCTION

Gerardo O. Depaola* and Carlos N. Kozameh*

* Facultad de Matemática Astronomía y Física - Univ. Nac. de Córdoba Ciudad Universitaria 5000 Córdoba, Argentina

ABSTRACT

Recently, a GEANT Monte Carlo code was used to design an outline of the geometry and to simulate the performance of a high energy (10 MeV - 10 GeV) gamma - ray detector (1). It was shown that the incident direction and energy of the incoming photons can be determined from the tracks of the produced electrons - positrons pairs. A natural follow up problem is to study whether this system can be used to detect linearly polarized gamma rays. In principle, this can be done by measuring the azimuthal distribution of the produced pairs since the cross section has a dependence with the vector polarization direction. In this work we first determine the azimuthal angular distribution from the differential cross section for pair production. We then show that the azimuthal distribution of the produced pairs has a very simple angular dependence and can be approximated very accurately by cross section for coplanar events. Finally, we use the simplified cross section to simulate the performance of the detector.

I. INTRODUCTION It is worth mentioning that none of these authors computed the azimuthal probability distribution which is The detection of polarized gamma rays is very necessary to perform Monte Carlo simulations. important in astrophysics since it yields a method to In this work we calculate this spatial azimuthal observe accreting matter near a black hole or neutron star. distribution for high energy gamma - rays (10 MeV - 10 Thus, it is worth asking whether the present detectors that GeV) directly from the differential cross section as in have been designed to measure unpolarized gamma rays Bethe - Heitler (7,8) by integrating over the polar angles (1,2,3) could also be used to detect polarized photons. and the positron (electron) energy. The idea to detect polarization of a gamma - ray The integration is made using a Monte Carlo was brought up in the 50’s, by measuring the ratio of the method and the VEGAS routine of Numerical Recipes (9). number of pairs produced in a plane parallel to To check the method, we calculate the total cross section polarization vector, to the number perpendicular to that and show that it yields the tabulated values. direction. The spatial azimuthal distribution so obtained However, since the differential cross section for this shows that the produced pair tends to be coplanar with the process is a very complicated function of the relevant photon incident momentum k. It thus can be replaced by a angles, several authors used approximations with different, simple function that only depends on the angle between the and sometimes opposite, results. As and example, Berlin produced pair and the polarization vector. et. all (4) calculated a higher probability of pair production We then use this anisotropic azimuthal distribution perpendicular to the polarization vector, whereas Wick (5) on the GEANT Monte Carlo code to simulate the obtained the opposite result. Maximon and Olsen (6), performance of a high energy gamma ray detector (1) using quantum mechanic calculations, found that the previously design for unpolarized rays. It should be azimuthal relation asymmetry depended on the angular remarked that we modified the GEANT Monte Carlo code resolution which one measures these probabilities. Thus, for this purpose since the original code only contained depending on this resolution one might have a higher isotropic azimuthal distribution for pair production. probability of pair production in either direction. A major problem in this determination is multiple scatterings which modify the tracks of the particles and a function of the angles Y and f, where Y is the azimuthal tend to nullify azimuthal information. To minimize these angle measured in a plane orthogonal to k between the effects we test different geometry configurations. polarization vector and the projection of the momentum

In section II we describe the Monte Carlo parameter p+ on that plane, and f is the difference between the and geometry setup, whereas in section III we discuss the azimuthal angles of the pair (see fig. 1.). spatial azimuthal distribution. Results of simulation are shown in section IV. z II. MONTE CARLO k p+ The simulation is done with GEANT (10) Monte q+ q- p- Carlo code since it offers great geometric programation versatility of experimental equipment. x With this code, the geometric design is Y constructed as follows: we first define a mother volume (called BIGM) who plays the role of the outer space. Inside this volume we define our detection system (called CALO) f who is divided in N identical sub-volumes. Each sub- volume (called SUBC), is itself composed by many layers of material. y All of this volumes have a box form and the coordinate system has its axis Z-axis perpendicular to the material layers and the X - Y axis are parallel to the other sides of the box. The incoming directions q and f are Figure 1. Angles Occurring in the Pair Production. measured with respect to the Z and X axis respectively. The basic geometry of the detector system consists of N=15 SUBC’s and the components of each layer are The calculated spatial azimuthal distribution is showed in table 1. practically zero in all ranges of Y and f except for f = p We do simulations varying the number of layers ( see Fig. 2). This indicates that for high energies the and material thickness for many Gamma - ray energy. particles tend to be coplanar with k. From this figure one can also see that at the maximum probability, i. e. for f = p, the pair has a greater TABLE 1. Thickness and Composition of Each Layer probability to be in a plane perpendicular to the Plane. polarization vector. The ratio of the number of pair produced in the 50 mm Lead plane of polarization vector to the number produced 50 mm Silicon perpendicular to this plane ( the asymmetric ratio) is for 30 mm Copper this figure equal to 0.8 which is in coincidence with 50 mm Carbon Maximon and Olsen (6). 30 mm x 2 Copper Repeating this calculation for different values of 50 mm Carbon energies of the incoming ray yields similar results. 30 mm Copper A fit of this Y distribution at f = p shows that we 50 mm Silicon can approximate the azimuthal distribution by the 3 cm Vacuum following formula :

III. SPATIAL AZIMUTHAL DISTRIBUTION ds = A (1 - l cos2 y ) This differential cross section depends on the dy positron energy and the solid angle in the direction of (1) momenta relative to the photon direction k. Thus, to obtain the spatial azimuthal distribution it is necessary to The l parameter represents the degree of integrate in energy and polars angles q+, q- (Fig. 1). asymmetry of the distribution and tends to be constant at In this way we obtain a differential cross section as high energies (Fig. 3) reaching an asymptotic value of 0.2. 16

10 1 ds 8 2 a(Zr0) dfdy 6

3.14 2 2.62 2.09 3.14 2.62 1.57 2.09 y 1.05 1.57 1.05 0.52 f 0.52 0.00

Figure 2. Spatial Azimuthal Distribution of a Pair Created by 100 MeV Photon

plus sign) obtained this formula without providing a hint of how the formula was obtained or why the sign in front 0.22 of the parameter was changed. Their value of l obtained for high energies was 0.3 instead of our calculated parameter. 0.20 It is also worth mentioning that formula 1 is only correct for coplanar events, as one can see from figure 2. To do the Monte Carlo simulation one needs the 0.18 spatial azimuthal distribution. However, it follows from Figure 2 that the cross section is significantly different 0.16 from zero for angles corresponding to coplanar events. l Thus, one can reduce the simulation to only coplanar pairs and use formula 1 to sample the angle Y. 0.14

0.12 IV. SIMULATION RESULTS

0.10 101 102 103 Inserting formula (1) into GEANT Monte Carlo code to replace the isotropic distribution usually used in Energy [MeV] this code yields the simulated performance of the detector. The simulated geometry consists on N detection’s planes explained in section II. Each plane is composed Figure 3. The l Parameter in Function of g - Ray principally by a Pb layer (used to produce the pair Energy electron-positron), and two layers of strip Silicon detector of 100 mm wide (one layer is for the X coordinate and the other for Y coordinate) and 50 mm of thickness used to Using phenomenological considerations Kel’ner detect where the particle crosses the plane. Other (11) (with the correct sign ) and Kel’ner et all (12) (with a materials are used as electrical contacts and to give rigidity to the structure. Note that the parameters used in This effect tends to nullify the original angular this simulation are different from the ones used in direction of the pair. reference (1) since the former simulation parameters are To analyze this problem we do simulation with a not sensible enough to measure photon polarization. fictitious azimuthal distribution. We force the pair to be With the X and Y coordinates of each plane we created in coplanar form and only in one plane ( we used construct the track of the pair. It is worth remarking that: Y = 0 ). 1) Gamma - ray incoming direction is determined by In figure 4 we show the angular distribution, in geometrical means through the bisector method as in other level surface, obtained for 100 MeV gamma - ray with detectors (1). incoming direction perpendicular to the detection planes, 2) The azimuthal distribution is obtained through the in which we compare tree azimuthal distributions: intersection of the plane of the pair with the X-Y plane. isotropic, planar and square cosine. In each distribution When the created charged particles cross the we take the ratio of the number of produced pairs parallel detection planes, they suffer deviations from their original to the polarization vector to the number of pairs directions due to Coulombian multiple scattering. orthogonal to that vector.

Cosine Nx = 2655 Distribution

Ny = 3379

Isotropic Nx = 4006 Distribution

Ny = 4010

Planar Nx = 2742 Distribution

Ny = 4427

Figure 4. Angular Distribution obtained for Cosine, Isotropic and Planar Azimuthal Probabilities. Nx and Ny are the number of pairs in a plane parallel and perpendicular to the polarization vector respectively.

From the results obtained with a planar distribution this run we obtain the same ratio as predicted by formula 1 it is easy to see that the multiple scattering is indeed a for 100 MeV photons. problem that deserve attention. We first analyze the cause On the other hand, reduction in thickness implies of this observed pattern . Since the distribution of the pair loss of detection efficiency. polar angle is inversely proportional to the particle energy, The asyemtric ratio for an incident gamma - ray in their angles are small when compared to the angle that the the z-axis as a function of energy and for three particle scatters due to a hard collision in the intermediate distributions is shown in Table 2. layers. Thus, multiple scattering tend to smooth out the scattered directions as one can see from planar distribution in figure 4. Nevertheless, for the layer thickness used in TABLE 2. Asymmetric Ration in Function of [4] Berlin, T. H. and Madansky L., On the Detection of g- Energy and for Tree Distribution Ray Polarization by Pair Production, Physical Review, vol. 78, p 623-624, 1950.

Distribution Energy [MeV] [5] Wick, G.C., Detection of Gamma - Ray Polarization Model 50 100 500 700 1000 by Pair Production, Physical Review, vol. 81, p 467-468, isotropic 1.0 1.0 1.0 1.0 1.0 1951. planar 0.5 0.6 1.0 1.0 1.0 cosine 0.9 0.8 1.0 1.0 1.0 [6] Maximon, L. C. and Olsen H. A., Measurement of Linear Photon Polarization by Pair Production, Physical Review, vol. 126, p 310-319, 1962. From this table one can see that for low energies the information is disturbed by multiple scattering and for [7] Motz, J. W. , Olsen, H. A. and Koch, H. W., Pair high energies one has the problem that the pair separation Production by Photons, Review of Modern Physics, vol. is too small to be distinguished by the geometry. 41, num. 4, p 581-639, 1969 The change of geometry necessary to distinguish the pair separation involves a big separation of the [8] Heitler, W., The Quantum Theory of Radiation, detection planes or a reduction in the strip’s width. This Oxford University Press, London, 1954 conditions are difficult to realize in an experimental setup so one must measure the ratio for low energies (10 - 200 [9] Press, W. H. , Teukolsky S. A., Vetterling W. T. MeV) looking for an optimal relation between multiple Flannery B. P. Numerical Recipes in Fortran. Cambridge scattering and efficiency. University Press, 1992.

[10] GEANT - Detector Description and Simulation Tool, CERN Program Library Long Writeups Q123. V. CONCLUSIONS [11] Kel’ner, S. R., Dependence of the Azimuthal The first conclusion that one arrives after hours of Asymmetry of Pair-Production Plane on the Photon performance simulation is that the present configuration Polarization, Soviet Journal of Nuclear Physics, vol. 10, and geometry of the high energy gamma ray detectors is num. 3, p 349-351, 1970. not sensible enough to detect photon polarization direction. Thus, the detectors presently operating on space platforms [12] Kel’ner, S. R., Kotov, Y. D. and Logunov, V. M., or on benchmarks will not be able to detect this direction. Methods of measuring polarization of gamma quanta, The new parameters introduced to obtain the Soviet Journal of Nuclear Physics, vol. 21, num. 3, p 313- results tabulated in Table 2 show that at least for these 315, 1975. values of layer’s thickness, separation of the silicon strips, and separation between consecutive planes, it is possible to detect the polarization direction with some degree of accuracy. A word of caution is in order though, such an apparatus will be very expensive with the present technology.

REFERENCES

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[3] Atwood, W.B., Gamma Large Area Silicon Telescope (GLAST). Applying silicon strip detector technology to the detention of gamma rays in space, Nuclear Instruments and Methods, A342, p.302-307, 1994.