# Sphere packing in spherical shell

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 at 23:49
• Yes, I meant a spherical shell, thank you – April Apr 15 at 15:52