2 Principles of Compton imaging

The principle of Compton imaging of $\gamma$-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 $\vec{\hat{p}}$ undergoes a Compton scatter at an angle $\phi _{1}$ at the position $\vec{r}_{1}$ 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 E_i at the positions $\vec{r}_{i}$, until eventually photoabsorbed. Here the total photon energy after each scatter i, normalized to the electron mass, is defined as

$$W_{i}=\frac{1}{m_{\rm e}c^{2}} \sum_{j=i+1}^{N} E_{j} ~,$$ (1)

where $W_{0}=E/m_{\rm e}c^{2}$, and N is the total number of interactions. The initial photon direction is related to scatter direction vector $\vec{r}_{1}'=\vec{r}_{2}-\vec{r}_{1}$( $\vec{\hat{r}}_{1}'$ after normalization), and the scattered photon energies W_i by the Compton formula

$$\vec{\hat{r}}_{1}' \cdot \vec{\hat{p}} = \cos{\phi_{1}} =1 + \frac{1}{W_{0}} - \frac{1}{W_{1}} ~\cdot$$ (2)

Given the measured scatter direction $\vec{\hat{r}}_{1}'$ and the angle $\cos{\phi_{1}}$ implied from the energy depositions, the equation for $\vec{\hat{p}}$ 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 $\vec{\hat{p}}'$ which satisfy the equation

$$\vec{\hat{r}}_{1}' \cdot \vec{\hat{p}}' = \cos{\phi_{1}} ~.$$ (3)

There are two uncertainties in determining the event annulus: the uncertainty in $\phi _{1}$ due to the finite energy resolution of the detectors, here labelled $\delta \phi_{1,E}$, and the uncertainty in $\vec{r}_{1}'$ determined by the spatial resolution of the detectors. Both of these uncertainties add to determine the uncertainty (effective width) of the event annulus $\delta \phi_{1}$. From Eq. (2), the derivation of $\delta \phi_{1,E}$ is straightforward and yields

$$\delta \phi_{i,E} = \frac{1}{\sin{\phi_{i}}} \left[\left( \frac{\delta W_{i-1}^{2}}{W_{i-1}^{4}}\right) \right.$$

$$\hspace*{1cm}\left.+\delta W_{i}^{2} \left(\left( \frac{1}{W_... ...}^{2}}\right)^2 - \frac{1}{W_{i-1}^{4}}\right)\right]^{1/2} ~,$$ (4)

where,

$$\delta W_{i} = \frac{1}{m_{\rm e}c^{2}} \left[\sum_{j=i+1}^{N} \delta E_{j}^{2}\right]^{1/2} ~.$$ (5)

Figure 2: Illustration of how uncertainties in the scattered photon direction $\vec{r}'$ can be translated into effective uncertainties in the scattering angle $\phi$

In order to simplify the analysis, it is convenient to transform the uncertainty $\delta \vec{r}_{1}'$ into an effective uncertainty in $\phi _{1}$, defined as $\delta \phi_{1,r}$, such that

$$\delta \phi_{1} = \sqrt{\delta \phi_{1,E}^{2} + \delta \phi_{1,r}^{2}} ~.$$ (6)

The angular resolution $\delta \phi_{i,r}$ is the effective "wiggle'' of $\vec{\hat{r}}_{i}'$ around its measured direction due to the uncertainties in the spatial measurements. The spatial uncertainties are defined as $\delta x'_{i} = \sqrt{\delta x_{i}^{2} + \delta x_{i+1}^{2}}$, $\delta y'_{i} = \sqrt{\delta y_{i}^{2} + \delta y_{i+1}^{2}}$, $\delta z'_{i} = \sqrt{\delta z_{i}^{2} + \delta z_{i+1}^{2}}$. It is simplest to analyze the situation for each axis separately as shown in Fig. 2. The uncertainty in the direction of $\vec{\hat{r}}_{i}'$ due to the uncertainty $\delta x'_{i}$ is given by

$$\delta \phi_{i,x} \simeq \tan\left(\delta \phi_{i,x}\right) =... ...rt{1-\left(\vec{\hat{r}}_{i}' \cdot \vec{\hat{x}}\right)^2} ~.$$ (7)

Likewise for the other axis,

$$\delta \phi_{i,y} \simeq \left(\frac{\delta y'_{i}}{r_{i}'}\r... ...rt{1-\left(\vec{\hat{r}}_{i}' \cdot \vec{\hat{y}}\right)^2} ~,$$

$$\delta \phi_{i,z} \simeq \left(\frac{\delta z'_{i}}{r_{i}'}\r... ...rt{1-\left(\vec{\hat{r}}_{i}' \cdot \vec{\hat{z}}\right)^2} ~,$$

which combine to yield the total uncertainty $\delta \phi_{i,r}$ given by

$$\delta \phi_{i,r} = \sqrt{\delta \phi_{i,x}^{2} + \delta \phi_{i,y}^{2} + \delta \phi_{i,z}^{2}} ~.$$ (8)

For detectors with a given energy resolution, in order to optimize the performance of a Compton telescope one would require that $\delta \phi_{i,r} \leq \delta \phi_{i,E}$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. $\delta r_{1}'/r_{1}' \leq \delta E_{1}'/E_{1}'$.