I'm learning about lattices and I'd like to confirm the difference between the density $\Delta$ and another density $\delta$. I would greatly appreciate if someone could confirm, correct, or even elaborate on my understanding of the difference written below?
$\Delta$ is the packing density and is used in the lattice (or sphere) packing problem. It calculates how dense we can pack spheres into a space. This is calculated as follows: $$\Delta = \frac{\text{Volume of 1 sphere}}{\text{Volume of the lattice }L}.$$
$\delta$ is the center density of a lattice $L$. If two lattices $L_1$ and $L_2$ have the same density $\delta$, the $L_1$ is isomorphic to $L_2$, i.e. they are the same lattice. This is calculated as follows: $$\delta = \frac{(1/2)\cdot d_{min}(L)}{\text{Volume of the lattice }L},$$ where $d_{min}(L)$ is the minimum distance of the lattice $L$.
So if I were to work on a problem like the sphere packing problem, I would use $\Delta$. But if I were looking for the densest lattice in a specific dimension, then I would use $\delta$.