  The direct detection of dark matter constituents, in particular the weakly
interacting massive particles (WIMPs), is considered central to particle
physics and cosmology. In this paper we study transitions to the excited
states, possible in some nuclei, which have sufficiently low lying excited
states. Examples considered previously were the first excited states of
$^{127}$I and $^{129}$Xe and $^{83}$Kr. Here we examine $^{125}$Te, which
offers some advantages and is currently being considered as a target.In all
these cases the extra signature of the gamma rays following the de-excitation
of these states has definite advantages over the purely nuclear recoil and, in
principle, such a signature can be exploited experimentally. A brief discussion
of the experimental feasibility is given in the context of the CUORE
experiment.
