The performance of an optical element can be severely degraded by exposure to intense X-ray beams. Keeping them cool is not a big issue, but the main problem is to reduce the unavoidable thermal deformation down to an acceptable minimum. For mirrors and multilayers, finite element analysis (FEA), computer simulations, and the prediction of heat load effects on the performance of optics, are straightforward. Whereas for single crystal monochromators, two complications appear. One is the diffraction angle which can be large and accordingly the penetration depth of the radiation cannot be neglected. The other is that the diffraction process of a deformed crystal is different from that of a perfect crystal, depending on the amount of strain caused by the thermal gradients.

With the steady increase in the X-ray intensity generated by modern storage ring sources we must be capable of predicting the degree to which the optical elements would limit the transfer of the flux gain downstream of the beamline to the sample and of improving the cooling schemes if necessary. For example, the X-ray transmission of a double crystal monochromator could severely suffer from the thermal deformation of the first of the two crystals. Serious discrepancies between the raw FEA results and experimental observations have been reported for cryogenically cooled crystals, fortunately in the sense that the calculations were too pessimistic [1].

As a first step towards elucidating this disagreement we wanted to examine how realistic the calculations could be for a relatively modest thermal load on an ESRF bending magnet source (Optics Beamline BM5) and a silicon crystal plate simply water-cooled from the behind the rear face [2]. The precise power profile of the X-ray beam was carefully measured by calorimetry with and without a 1 mm thick aluminium attenuator. Then rocking curves were taken, recording the intensity reflected by the second (undeformed) crystal in the standard double crystal monochromator as a function of the crystal rotation angle. The incident power was changed by variation of the slit settings that defined the beam size. Any broadening of the rocking curves with increasing power load was then due to the thermal deformation of the first crystal. All the various cases were treated by finite element analysis using the program ANSYS assuming that all the power was absorbed at the surface, i.e. no depth profiling was applied. The output files were directly fed into a finite difference algorithm based on the Takagi-Taupin theory that had been specially developed for a numerical evaluation of diffraction profiles based on discrete values for deformed crystal lattices and then adapted to our present purpose [3]. Here two cases were distinguished: laterally coherent and incoherent beams. Finally, a simple geometrical model was applied where the differentiated profiles of the deformed surface obtained from ANSYS were folded with the perfect-crystal rocking curve.

The results of the calculations and the experiments for the various slit settings are compared in Figures 129 and 130. Figure 129 displays the observations for the (111)-reflection and 20 keV (no Al absorber), and Figure 130 those for a superposition of the (333)- and (444)-reflection corresponding to 20 and 26.4 keV (1 mm Al in the beam), respectively. The abscissa corresponds to the horizontal slit opening that is proportional to the total power: maximum 185 W at 60 mm for Figure 129 and 132 W at 80 mm for Figure 130. The peak-to-valley thermal slope errors obtained from FEA are given as additional information only. They cannot be directly compared with the other calculated results, because they do not include diffraction effects. The results from the simulations are quite close to the experimental values although systematic deviations are observed, in particular for the second case where penetration depth effects are much bigger.

The main conclusions from this study are that the incoherent Takagi-Taupin approach best matches the experiments and that the overall agreement between simulation and reality is reasonably good. We proved that we have developed a simulation tool that works. The same type of calculations has recently been applied to cryogenically cooled crystals where the behaviour of the thermal deformation is more complicated due to non-linear effects. The final goal of obtaining a high level of confidence, similar to that of machine performance predictions has now been reached [4].

[1] D.M. Mills, Presentation at the APS-ESRF-SPring-8 Three-Way Meeting, Argonne, April 1999.
[2] A.K. Freund, J. Hoszowska, J.-S. Migliore, V. Mocella, L. Zhang, C. Ferrero, Proceedings of the International Conference SRI'99, Stanford, Oct. 13-15 (1999), American Institute of Physics, in press.
[3] V. Mocella, PhD Thesis, Université Joseph Fourier, Grenoble (1999).
[4] Results to be published at SRI 2000, Berlin, August 2000.

J. Hoszowska, J-.S. Migliore, V. Mocella, L. Zhang, C. Ferrero, A.K. Freund.