Abstract: A silicon block (typical size 100 $MUL 100 $MUL (20 - 50) mm$+3$/) cooled by liquid nitrogen has been studied with various incident power densities and spot sizes on the surface. Gaussian power distribution was assumed. Both bottom cooling and side cooling have been considered. The thermal slope error has been minimized by optimizing the cooling conditions (cooling coefficient and bulk temperature of liquid nitrogen) and the thickness of the silicon block. Finite element analysis has been used because the material properties ($alpha@, k) of silicon are highly non linear. The limits of absorbed total power and power density are estimated for both undulator and wiggler beams with various spot sizes and for the requirement in terms of thermal slope error. Correlations between thermal slope error and power, power density have been established. !6
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