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Results and Conclusion:
We manufactured gyroscope rotors in both fused silica and single crystal silicon. The rotors deviated from the perfect sphere by less then 1 micro inch (25 nm) peak to valley. The shape of the rotors was measured and represented by an expansion in spherical harmonics (see Fig. 9) To study the effects of the shape on the motion of the gyroscope one also uses spherical harmonics. Based on the requirements of the experiment, it is possible to establish specifications for the spherical harmonics coefficients and the coefficients obtained from the rotor measurements had to satisfy these specifications. The maximum peak to valley height for the L=2 component was specified to be less than 1 micro inch (25 nm), and for higher values of L, either L even (4£L£16 ) or odd (3£L£15), less than 0.7 micro inch (~18 nm).
It was found that for silicon spheres the crystalline structure contributes mostly to the L=4 harmonic and therefore this was the most difficult specification to satisfy. The best results obtained for the sum of the deviations due to the even harmonics L>2 were 0.3 to 0.4 micro inch (8 to 10 nm). Somewhat better results were obtained for the sum of the deviations due to the odd harmonics, which were typically 0.3 micro-inch (8 nm). This includes approximately 0.1 micro inch (2.5 nm) of residual uncorrected spindle error. For L=2 the peak to valley height was typically less then 0.5 micro inch (12nm). For overall sphericity, the best results were 0.6 to 0.7 micro-inch (15 to 18 nm).
Twelve fused silica spheres and eight silicon spheres were fabricated, satisfying all the requirements for the experiment. After extensive testing of all completed gyroscope spheres, two silicon and two fused silica gyroscope spheres were selected for the scientific mission.
There are other applications where spheres of similar precision are needed. Examples include silicon spheres for a precise determination of Avogadro's number and one kilogram silicon spheres to replace the present weight standard (see ref.21) We expect that the experience gained in this work will also prove useful in other applications.