Abstract. We present the results of detailed nuclear shell model calculations of the spin-dependent elastic cross section for neutralinos scattering from 29 Si and 73 Ge. The calculations were performed in large model spaces which adequately describe the configuration mixing in these two nuclei. As tests of the computed nuclear wave functions, we have calculated several nuclear observables and compared them with the measured values and found good agreement. In the limit of zero momentum transfer, we find scattering matrix elements in agreement with previous estimates for 29 Si but significantly different than previous work for 73 Ge. A modest quenching, in accord with shell model studies of other heavy nuclei, has been included to bring agreement between the measured and calculated values of the magnetic moment for 73 Ge. Even with this quenching, the calculated scattering rate is roughly a factor of 2 higher than the best previous estimates; without quenching, the rate is a factor of 4 higher. This implies a higher sensitivity for germanium dark matter detectors. We also investigate the role of finite momentum transfer upon the scattering response for both nuclei and find that this can significantly change the expected rates. We close with a brief discussion of the effects of some of the non-nuclear uncertainties upon the matrix elements. 1 I. IntroductionA host of astronomical evidence points to the existence of large amounts of dark matter in the Universe [1]. Despite the overwhelming amount of evidence for this dark matter's existence, its exact nature remains a mystery. Numerous candidates have been proposed. These include both ordinary baryonic and non-baryonic matter [2]. Among the best motivated, and hence highly favored, of non-baryonic candidates is the lightest supersymmetric particle (LSP). Experimental and theoretical considerations suggest that the LSP is a neutralino,χ, made up of a linear combination of the photino, Z-ino, and 2 higgsinos (or equivalently, the B-ino, neutral W-ino, and 2 higgsinos)The neutralino is an ideal dark matter candidate. The motivation for its existence arises naturally in modern theories of particle physics [3], not as an ad hoc solution to the dark matter problem. For a very large region of supersymmetric parameter space, neutralinos provide densities sufficient to account for the mass-energy density of the Universe [4]. Thẽ χ also possesses the virtue of potential detectability. Its detection may be possible in at least two ways: indirectly, through the products ofχχ annihilation from capture in the Sun or Earth [5], or directly, via elastic neutralino-nucleus,χN , scattering in a detector [6,7]. In either case, theχN elastic scattering cross section is an essential ingredient. In this paper we discuss detailed nuclear structure calculations relevant toχN scattering for two important elements, silicon and germanium.Neutralino-nucleus scattering is governed by physics at several energy scales. The mass and the mixing of theχ , and hence its interactions with quarks, ar...