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We report a synchrotron x-ray scattering study of the diluted spin-Peierls (SP) material Cu_(1-x)Mg_xGeO_3. We find that for x>0 the temperature T_m at which the spin gap is established is significantly higher than the temperature T_s at which the SP dimerization attains long-range order. The latter is observed only for x<x_c =0.021, while for x>x_c the SP correlation length quickly decreases with increasing x. We argue that impurity-induced competing interactions play a central role in these phenomena.

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