Second-order electromagnetic eigenfrequencies of a triaxial ellipsoid II

Abstract

Finite-element calculations of electromagnetic eigenvalues are known to converge to the exact solutions in the limit of vanishing element sizes. In an extension of previous work (Mehl 2009 Metrologia 46 554–9) the eigenfrequencies of the TM1n and TE1n (n = 1, 2, ··· 6) modes of triaxial ellipsoids were calculated as a function of mesh size. Higher-accuracy eigenvalues were obtained through a limiting process as the mesh size was reduced; the extrapolation was based on the theoretical convergence rate. The difference between the finite-element eigenfrequencies and the eigenfrequencies predicted by shape perturbation theory is found to be proportional to the cube of the fractional deformation parameter ϵ for all investigated modes. For ellipsoids with axes proportional to 1 : 1.0005 : 1.0010, the cubic term represents a fractional perturbation of the average TM16 eigenvalue k2 by −0.16 × 10 − 6 and the average TE16 eigenvalue by −0.22 × 10 − 6. This work adds support to the correctness of the analytic second-order formula derived in the previous work, and also demonstrates the usefulness of finite-element methods for investigating the quasi-spherical resonators (QSRs) used in measurements of the Boltzmann constant. In principle, the method can be extended to QSRs whose shape differs from triaxial ellipsoids.