Up: Event reconstruction in high
The principle of Compton imaging of
-ray photons is illustrated in
Fig. 1. (See von Ballmoos et al. 1989,
for an excellent review of historical Compton telescope configurations.)
An incoming photon of energy E and direction
undergoes a Compton
scatter at an angle
at the position
within a detector,
creating a recoil electron of energy of E1 which is quickly absorbed
and measured by the detector itself. The scattered photon then deposits the
rest of its energy in the instrument in a series of one or more interactions
of energies Ei at the positions
,
until eventually photoabsorbed. Here the total photon energy after each scatter
i, normalized to the electron mass, is defined as
 |
(1) |
where
,
and N is the total number of interactions. The initial
photon direction is related to scatter direction vector
(
after normalization), and the scattered
photon energies Wi by the Compton formula
 |
(2) |
Given the measured scatter direction
and
the angle
implied from the
energy depositions, the equation for
is not unique (if the electron recoil
direction could be measured, it would solve this ambiguity); therefore, the initial
direction of the photon cannot be determined directly, but it can be limited to an
annulus of directions
which satisfy the equation
 |
(3) |
There are two uncertainties in determining the event annulus: the
uncertainty in
due to the finite energy resolution of the detectors, here labelled
,
and the uncertainty in
determined
by the spatial resolution of the
detectors. Both of these uncertainties add to determine the uncertainty (effective width)
of the event annulus
.
From Eq. (2), the derivation
of
is straightforward and yields
![\begin{displaymath}\hspace*{1cm}\left.+\delta W_{i}^{2} \left(\left( \frac{1}{W_...
...}^{2}}\right)^2 -
\frac{1}{W_{i-1}^{4}}\right)\right]^{1/2} ~,
\end{displaymath}](/articles/aas/full/2000/14/h2143/img36.gif) |
(4) |
where,
![\begin{displaymath}\delta W_{i} = \frac{1}{m_{\rm e}c^{2}} \left[\sum_{j=i+1}^{N} \delta E_{j}^{2}\right]^{1/2} ~.
\end{displaymath}](/articles/aas/full/2000/14/h2143/img37.gif) |
(5) |
 |
Figure 2:
Illustration of how uncertainties in the scattered
photon direction
can be translated into
effective uncertainties in the scattering angle  |
In order to simplify the analysis, it is convenient to transform the
uncertainty
into an effective uncertainty in
,
defined as
,
such that
 |
(6) |
The angular resolution
is the effective "wiggle''
of
around its measured
direction due to the uncertainties in the spatial measurements. The spatial
uncertainties are defined as
,
,
.
It is simplest to analyze the situation for each axis separately as shown in
Fig. 2. The uncertainty in the direction of
due to the uncertainty
is given by
 |
(7) |
Likewise for the other axis,
which combine to yield the total uncertainty
given by
 |
(8) |
For detectors with a given energy resolution,
in order to optimize the performance of a Compton telescope one would require that
in the energy range of interest. To first order, this implies that the spatial
resolution in relation to the scale size of the instrument must be comparable to
or less than the energy resolution, i.e.
.
Up: Event reconstruction in high
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