The fundamentals of crystal orientation

Abstract

The method described in this paper improves the old methods of crystal orientation, applies new parametric equations for crystallography, and increases the precision and accuracy of measurements. The method applies to inorganic and organic crystals. A breakthrough in crystal orientation happened about 25 years ago when two equations dependent on the Bragg angle and an arbitrary direction in the crystal were developed. Unfortunately, they were analytically insolvable and their unique solution was found numerically. Finding the numerical solution of crystal orientation is challenging from a mathematical point of view. In these conditions the numerical solution was found using the Newton method. The Newton method required a specific programming that limits the full benefit of the method in the laboratory. In recent years, a new numerical technique called GRG (generalized reduced gradient), which can be run on many inexpensive computers, was found to be a good fit for these equations. The solutions that can be found with the GRG method are now completed with additional parametric equations; they are easy to use with computers in many laboratories. In this way, parametrization of nonlinear equations for X-ray crystal orientation determines the positions of a reference surface of the single crystal relative to its crystallographic system and to a goniometer setting with two perpendicular axes of rotation. This approach was successfully validated and checked for different Si wafers with (111) and (004) orientation. The paper shows an innovative approach through the parametric equations in conjunction with exact solutions found with a GRG subroutine. The results of the method demonstrate the potential for new applications in industry and research.