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We present a 3-D fused-silica micro-scale shell gyroscope, called the birdbath resonator gyroscope (BRG). The BRGs axisymmetric geometry leads to a good frequency and Q symmetry. The birdbath resonator can be fabricated with a good structural symmetry because its anchor is self-aligned to the rest of the structure. The BRG has n=2 wine-glass modes at 10.5 kHz and has a large frequency separation between the n=2 wine-glass modes and the closest parasitic mode (fparasitic-fn=2/fn=2=0.3), which will potentially lead to a low vibration sensitivity. The equations of motion for 3-D shell gyroscopes are derived and the effective mass and angular gain of the BRG is estimated using finite element method (FEM). The BRG is fabricated using a 3-D micrometer-blow-torching process and assembly on an electrode substrate made with the silicon-on-glass process. The BRG is operated in the force-rebalance mode at vacuum at room temperature and has a scale factor of 27.9 mV/(deg/s), a full-scale range , an angle random walk of 0.106 deg/√h, and a bias stability of 1 deg/h. A large angular gain (0.317) is measured, which is close to the estimated value of 0.25 obtained via FEM. [ABSTRACT FROM AUTHOR]

We present analytic expressions for the zero-order eikonal phase shift and its first-order correction by approximating a distance between two colliding nuclei. This formalism has been applied to elastic scatterings of the 12C + 40Ca and the 12C + 90Zr systems at Elab = 420 MeV, and the 16O + 40Ca and the 16 O + 90Zr ones at Elab = 1503 MeV. The calculated angular distributions, taking into account up to the analytic first-order eikonal phase shift, are found to be in fairly good agreement with the observed data. The reaction cross-sections obtained from the present model produce very excellent agreements with ones of exact first-order eikonal model calculations. We have found that analytic eikonal phase shift including the first-order correction is one theoretical method to the analysis of heavy-ion elastic scattering. [ABSTRACT FROM AUTHOR]