3-D Wave Propagation in Cylindrical Single Crystal Solid-Liquid Bodies
The large difference in the ultrasonic velocity between the solid and the liquid phases of most semiconducting materials results in reflection/refraction of ultrasound at solid-liquid interfaces and an interest in using laser ultrasonics for sensing solid-liquid interfaces during single crystal growth. Using a ray tracing analysis, a set of measured ultrasonic time of flight (TOF) projection data can yield the ray paths connecting the source to the receiver, which can subsequently be used to reconstruct the solid-liquid interface. In previous work  2-D wave propagation in cylindrical single crystal solid-liquid bodies was used to explore the feasibility of using ultrasound to characterize solid-liquid interfaces during vertical Bridgman growth of semiconductor materials. Detailed study of ray paths, wavefronts and TOF for ultrasound propagating in both transverse and diametral planes of liquid-solid single crystal (Ge) bodies was reported. Numerical simulations indicated that the magnitude and direction of the group velocity, the solid:liquid velocity ratio and the curvature of the interface together controlled the ray bending behavior and thus determined the ultrasonic data across the interface. Knowledge of ray paths at the interface enabled reconstruction of the interface using a small set of ultrasonic TOF’s.