We are carrying out sensitive X-ray observations with Chandra and XMM of type II quasars selected from the Sloan Digital Sky Survey based on their optical emission line properties. We present observations of four objects at redshifts 0.4 < z < 0.8 and an analysis of the archival data for four additional objects in the same redshift range. Six of the eight were detected in X-rays; five of them have sufficient signal to derive spectral information. All of the detected sources have intrinsic luminosities L(2-10 keV) > 5 x 10^43 erg s^-1. The five with sufficient counts for spectral fitting show evidence for significant absorption (N_H >~ a few x 10^22 cm^-2). At least three of the objects likely have N_H > 10^23 cm^-2; some may be Compton-thick (N_H > 10^{24} cm^-2). In the five objects for which we could fit spectra, the slopes tend to be significantly flatter than is typically observed in AGN; it is possible that this is due either to reprocessing of the nuclear emission or to a line of sight that passes through patchy absorption.
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