A vectorial analysis for the numerical simulation of the surface profiles of etched singly-rotated quartz plates

Abstract

An extended method of graphical simulation for moving surface profiles is proposed in which the characteristic trajectory of a moving profile element is determined using the geometrical properties of the slowness dissolution vector L and of the vector T tangent to the polar diagram of the dissolution slowness. This numerical treatment presents two main advantages. Firstly, it is now possible to track completely the changes in the shape of the etch profiles with the orientation θ 0 of the reference surface for θ 0 varying from 0° to 360°. Secondly, the typical etch rate-orientation relationships which give rise to etch profiles with convex-concave or concave-convex shape are identified precisely by a systematic investigation of the theoretical results. The comparison of the theoretical etch profiles with the Z' etch profiles produced by repeated etchings on some singly-rotated quartz plates shows a complete agreement. A precise drawing of the etch rate-orientation relationships related to singly-rotated quartz crystals etched in a concentrated bifluoride solution is proposed, taking into account the results of the numerical simulation. The corrected etch rate vs. orientation plot is found to satisfy all the features required for the formation of etch profiles which exhibit the typical shapes observed experimentally. From the consistency between the experimental results and the theoretical results deduced from a kinematic theory of the dissolution, we can conclude that the shape of the etch profiles is essentially determined by the orientation of the quartz plate in agreement with a great number of previous experimental works.