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We present analytic and numerical calculations of the localized pinch model for ion collective acceleration by intense relativistic electron beams. These studies demonstrate the model's postulated synchronous acceleration of the ion clump and beam pinch. However, the configuration is found to be phase unstable, terminating the acceleration process before significant ion energy is acheived. Therefore, although the localized pinch model may be an appropriate explanation for some previous experimental observations of collectively accelerated MeV ions, it does not appear to be a suitable mechanism for high energy ion acceleration.