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Cryogenically cooled, single‐crystal silicon, x‐ray monochromators offer much better thermal performance than room‐temperature silicon monochromators. The improved performance can be quantified by a figure‐of‐merit equal to the ratio of the thermal conductivity to the coefficient of thermal expansion. This ratio increases by about a factor of 50 as the temperature is decreased from 300 to 100 K. An extensive thermal and structural finite element analysis is presented for an inclined, liquid‐nitrogen‐cooled, Si monochromator crystal diffracting 4.2 keV photons from the  planes using undulator A at the Advanced Photon Source. The angular size of the beam accepted on the crystal was chosen to be 50 μrad vertically and 120 μrad horizontally. The deflection parameter, K, was 2.17 for all cases. The peak power density at normal incidence to the beam was calculated to be 139 W/mm2, and the total power was 750 W at a distance of 30 m from the source for a positron current of 100 mA. The crystal was oriented in the inclined geometry with an inclination angle of 85° for all cases. The performance of the crystal was investigated for beam currents of 100, 200, and 300 mA. The calculated peak slopes of the diffraction plane over the extent of the beam footprint were -1.17, -2.35, and 0.33 μrad, and the peak temperatures were 88.2, 102.6, and 121.4 K, respectively. The variation in the Bragg angle due to change in d spacing across the beam footprint was less than 1 μrad for all cases. These results indicate that a properly designed, cryogenically cooled, inclined silicon monochromator can deliver the full brilliance of undulator A at even the highest machine currents. © 1995 American Institute of Physics.