Casimir energies of spherical plasma shells

We study a model loosely inspired by the giant carbon molecule C-60, namely a negligibly thin spherical shell of radius R, carrying a continuous fluid with mass and charge surface-densities m/a(2) and e/a(2) (plus an inert overall-neutralizing charge distribution), so that a mimics some mean inter-electron spacing, comparable say to the Bohr radius. The Casimir energy B is the total zero-point energy of the exact multipolar normal modes, minus that of empty space, minus the self-energy of the given amount of material at infinite dilution. Subject to a Debye-type cutoff l less than or equal to L on angular momenta l, but needing no frequency cutoff, B is a well-defined function of R, x equivalent to e(2)/mc(2)a, and X equivalent to R/a, expressible in terms of the multipolar phase shifts delta(l)(TE, TM). We consider it only for X much greater than 1 double right arrow L similar to X. Realistically one has x much less than 1, but mu equivalent to 4pixX can be large or small. Then B is always dominated by terms of order (h) over bar roote(2)/ma(3)X(2) stemming from TM modes; but the pattern of corrections as functions of x and X is intricate, and accessible only through the Debye (uniform) expansions of the Bessel functions figuring in the delta(l). Historical interest attaches to Boyer components, far-subdominant parts of B having the form ((h) over barc/R)C-B, where C-B is a pure number. When mu much less than 1 (as in C-60) there are none, because all corrections are at least of order x(1/2) and none are proportional to 1/R. But a Boyer component does exist when mu much greater than 1 (as in macroscopic shells), with C-B = [3/64] - [(9/4096)(pi(2)/8 - 1)] + . . . similar or equal to 0.0464. The two terms come from orders 1 and 2 of the Debye expansion; the contribution from order 0 vanishes because of an apparently fortuituous cancellation between TE and TM. The most precise value proposed so far is C-B = 0.046 1765; but the significance of comparisons is unclear, because previous calculations mistakenly treat the Boyer component as if it included all of B in a hypothetical perfect-reflector limit x --> infinity.