Anisotropy and crystal orientation of silicon--application to the modeling of a bent mirror

Matrix formula and MATLAB algorithm are proposed to calculate the stiffness coefficient matrix C, the Young's modulus, shear modulus and Poisson ratio for the silicon crystal in any orientation. Results for Si(110) and Si(311) are given as an example. The anisotropic material properties of the silicon have been used in the mirror width profile optimization for the nano-imaging end-station ID22NI at the ESRF. As the Si(110) is used as the substrate of this multilayer coated KB mirror, the silicon crystal axis [0 0 1] is proposed to orient to the mirror axis. This is the case to have low stress in the mirror and low bending forces from actuators.