Limitations of Liquid Nitrogen Cooling of High Heat Load X-Ray Monochromators

X-ray monochromators, made of single crystals or multilayer coatings, are the most common optical components on many synchrotron beamlines. They intercept the broad-spectrum x-ray (white or pink) beams generated by the radiation source and absorb all but select narrow spectral bands of x-rays, which are diffracted according to Bragg's Law. With some incident beam power in the kW range, minimizing thermally induced deformation detrimental to the performance of the device necessitates the design of optimally cooled monochromators. Monochromator substrate designs have evolved, in parallel with thermal loads of the incident beams, from simple blocks with no cooling, to water cooled (both contact-cooled and internally cooled), and to cryogenically cooled designs where the undesirable thermal distortions are kept in check by operating in a temperature range where the thermomechanical properties of the substrate materials are favorable. Fortuitously, single-crystal silicon at cryogenic temperatures has an exceptionally favorable combination of high thermal conductivity and low thermal expansion coefficient. With further increases in x-ray beam power, partly as a result of the upgrades to the existing synchrotron facilities, the question arises as to the ultimate limits of liquid-nitrogen-cooled silicon monochromators' ability to handle the increased thermal load. In this paper, we describe the difficulties and begin the investigation by using a simple geometric model for a monochromator and obtain analytical solutions for the temperature field. The temperature can be used as a proxy for thermally induced deformation. The significant role of the nonlinear material properties of silicon is examined.