# Formal definitions for a few lattice packing invariants

I'm a bit hesitant to post this question here because it regards definitions that are likely trivial, and please note that I'm cross-posting this question from Math.Stackexchange (posted nine days ago, one upvote, no comments or answers): http://math.stackexchange.com/questions/406247/understanding-the-definitions-of-lattice-packing-invariants

I hope this is appropriate.

I'm looking for definitions regarding the meaning of a few lattice packing constants, and internet searches haven't helped very much. Specifically, I'd like to know formal definitions for the two- and three-dimensional lattice packing constants: "thickness", "volume", and "center density" (definitions that appear, for example, with a WolframAlpha search for "A5 Lattice" http://www.wolframalpha.com/input/?i=A5+lattice). I would guess here that "volume" refers to the volume of the lattice unit cell.

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Yes, volume is the volume of the fundamental domain (i.e. the lattice unit cell). Center density is the sphere packing density of the lattice divided by the volume of a unit ball. It's given by $r^n/V$ where $r = \min \Lambda /2$ is the radius of the sphere packing, and $V$ is the volume of the lattice. "Thickness" refers to the thickness of the lattice covering, i.e. the ratio $v R^n/V$, where $v$ is the volume of the unit ball, $R$ the covering radius, and $V$ is the lattice volume as above. –  Abhinav Kumar Jun 8 '13 at 13:07
@Abhinav Kumar I'd be happy to accept your comment as an answer. –  WilliamH Jun 8 '13 at 13:51
@Abhinav Kumar To make sure I understand, how is $\Lambda$ defined here? –  WilliamH Jun 8 '13 at 13:56
$\min \Lambda$ is the smallest distance between distinct vectors. –  S. Carnahan Jun 9 '13 at 2:19