Compton Scatter Polarimetry

The basic physical process used to measure polarization in the 50-300 keV energy range is Compton scattering the cross-section for which is given by,

where,

                                         

Here no is the frequency of the incident photon,  is the frequency of the scattered photon, q is the scattering angle of the scattered photon measured from the direction of the incident photon, and h is the azimuthal angle of the scattered photon measured from the plane containing the electric vector of the incident photon.  For a given value of q, the scattering cross section for polarized radiation reaches a minimum at h = 0° and a maximum at h = 90°.  In the case of an unpolarized beam of incident photons, there will be no net positive electric field vector and therefore no preferred azimuthal scattering angle (h); the distribution of scattered photons with respect to h will be uniform.  However, in the polarized case, the incident photons will exhibit a net positive electric field vector and the distribution in h will be asymmetric.

The ability to measure the polarization of the incident photon beam depends on the asymmetry ratio. This is defined to be the ratio between the maximum in the h distribution to the minimum intensity in the h distribution,

This distribution is plotted as a function of scattering angle (q) for various incident photon energies in Figure 1.  This plot shows that the asymmetry ratio is larger at lower energies and that events with scattering angles between 60° and 120° contain most of the polarization information. 

Figure 1 - The asymmetry ratio (eqn. 3)
for various incident photon energies.

In general, a Compton scatter polarimeter consists of two detectors to determine the energies of both the scattered photon and the scattered electron. One detector, the scattering detector, provides the medium for the Compton interaction to take place. This detector must be designed to maximize the probability of there being a single Compton interaction with a subsequent escape of the scattered photon.  This requires a low-Z material in order to minimize photoelectric interactions. The area of the scattering detector which is exposed to the photon beam is also an important factor in determining the effective area of the polarimeter. The primary purpose of the second detector, the calorimeter, is to aborb the full energy of the scattered photon.

The relative placement of the two detectors defines the scattering geometry. For incident photon energies below 100 keV, the azimuthal modulation of the scattered photons is maximized if the two detectors are placed at a right angle relative to the incident photon beam (q = 90°; c.f., Figure 1). The positioning of the two detectors must also be arranged relatively close to each other so that there is a finite solid angle for scattering to achieve the required detection efficiency. At the same time, a larger separation between the detectors provides more precise scattering geometry information. The accuracy with which the scattering geometry can be measured determines the ability to define the modulation pattern (Figure 2) and therefore has a direct impact on the polarization sensitivity (see below). Here, one must compromise between total efficiency (small detector separation) and the ability to define the modulation pattern (large detector separation).

Figure 2 - The modulation pattern produced by
Compton scattering of polarized radiation. The
minimum of the modulation pattern defines the
plane of polarization of the incident flux.

The ultimate goal of a Compton scatter polarimeter is to measure the azimuthal modulation pattern of the scattered photons.  From equation (1), we see that the azimuthal modulation follows a cos 2h distribution.  More specifically, we can write the integrated azimuthal distribution of the scattered photons as,

where j is the polarization angle of the incident photons. (In practice, a measured distribution must also be corrected for geometrical effects based on the corresponding distribution for an unpolarized beam).  It is customary to define, as a figure-of-merit for the polarimeter, the polarization modulation factor.  For a given energy and incidence angle for an incoming photon beam, this can be expressed as,

where C_p,max and C_p,min refer to the maximum and minimum number of counts registered in the polarimeter, respectively, with respect to h; A_p and B_p refer to the corresponding parameters in equation (4). In this case the ‘p’ subscript denotes that this refers to the measurement of a beam with unknown polarization. In order to determine the polarization of the measured beam, we need first to know how the polarimeter would respond to a similar beam, but with 100% polarization.  This can be done using Monte Carlo simulations. We then define a corresponding modulation factor for an incident beam that is 100% polarized,

Then we can then use this result, in conjunction with the observed modulation factor (µp), to determine the level of polarization in a measured beam,

where P is the measured polarization. The minimum detectable polarization (MDP) can be expressed as,

where n_s is the significance level (number of sigma), S is the total source counting rate, B is the total background counting rate and T is the total observation time. Improved sensitivity to source polarization can be achieved either by increasing the modulation factor (µ₁₀₀) or by increasing the effective area of the polarimeter (thereby increasing the source counting rate, S).