  MAX is a proposed Laue lens gamma-ray telescope taking advantage of Bragg
diffraction in crystals to concentrate incident photons onto a distant
detector. The Laue lens and the detector are carried by two separate satellites
flying in formation. Significant effort is being devoted to studying different
types of crystals that may be suitable for focusing gamma rays in two 100 keV
wide energy bands centered on two lines which constitute the prime
astrophysical interest of the MAX mission: the 511 keV positron annihilation
line, and the broadened 847 keV line from the decay of 56Co copiously produced
in Type Ia supernovae. However, to optimize the performance of MAX, it is also
necessary to optimize the detector used to collect the source photons
concentrated by the lens. We address this need by applying proven Monte Carlo
and event reconstruction packages to predict the performance of MAX for three
different Ge detector concepts: a standard coaxial detector, a stack of
segmented detectors, and a Compton camera consisting of a stack of strip
detectors. Each of these exhibits distinct advantages and disadvantages
regarding fundamental instrumental characteristics such as detection efficiency
or background rejection, which ultimately determine achievable sensitivities.
We conclude that the Compton camera is the most promising detector for MAX in
particular, and for Laue lens gamma-ray telecopes in general.
