Sphere packing in spherical shell

Question

What is the maximum number of spheres of radius $R$ that can be packed into a spherical annulus of inner radius $R_i$ and outer radius $R_o$?

Is there an answer for this question? I am not a mathematician, and am overwhelmed by the language in some of the references I've encountered (such as "Sphere Packing Bounds Via Spherical Codes," Cohn and Zhao (2012)). Can anyone please explain this to an engineer?

Specifically for my problem, $R_i=1.25$, $R_o=1.4$, and $R=0.0405$, so the thickness of the annulus is just slightly too thin to have two spheres along a particular radial coordinate. But the annulus is sufficiently large that thousands of spheres should be able to fit inside.

Any help is appreciated!

Ah, my bad, a typo! $R=0.0405$ – April Apr 13 '19 at 23:49 — April, Apr 13 '19 at 23:49
Yes, I meant a spherical shell, thank you – April Apr 15 '19 at 15:52 — April, Apr 15 '19 at 15:52

score 1 · Answer 1 · answered Apr 14 '19 at 00:04

This is the latest result in this domain. Maybe it or its references or its subsequent citations (Google Scholar says it was cited by 16 later papers) could help:

Musin, Oleg R., and Anton V. Nikitenko. "Optimal packings of congruent circles on a square flat torus." Discrete & Computational Geometry 55, no. 1 (2016): 1-20. Earlie arXiv version PDF download.

Sphere packing in spherical shell

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