The recent WMAP and Planck data have confirmed that exotic dark matter together with the vacuum energy (cosmological constant) dominate in the flat Universe. Many extensions of the standard model provide dark matter candidates, in particular Weakly Interacting Massive Particles (WIMPs). Thus the direct dark matter detection is central to particle physics and cosmology. Most of the research on this issue has hitherto focused on the detection of the recoiling nucleus. 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. We examine here $^{83}$Kr, which offers some kinematical advantages and is currently considered as a possible target. We find appreciable branching ratios for the inelastic scattering mediated by the spin cross sections, with an inelastic event rate of $4.4times 10^{-4}$kg$^{-1}$d$^{-1}$. So, the extra signature of the gamma ray following the de-excitation of these states can, in principle, be exploited experimentally. A brief discussion of the experimental feasibility is given
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