$word_{count}$: 220

Key relationships between edge length (a) and atomic radius (r): SC: atoms touch along edge, a = 2r. BCC: atoms touch along body diagonal (asqrt(3) = 4r), so a = 4r/sqrt(3) = 2.309r. FCC: atoms touch along face diagonal (asqrt(2) = 4r), so a = 2sqrt(2)r = 2.828r. Packing efficiency = $\frac{Z x 4/3 pi r^3}{a}$^3 x 100. SC: 1 atom, a=2r: PE = pi/6 = 52.36%. BCC: 2 atoms, a=4r/sqrt(3): PE = pisqrt$\frac{3}{8}$ = 68.02%. FCC: 4 atoms, a=2sqrt(2)r: PE = $\frac{pi}{3sqrt(2}$) = 74.05%. HCP also = 74.05%. Void space = 100 - PE. Coordination numbers: SC=6, BCC=8, FCC=12, HCP=12. The density formula d = $\frac{ZM}{N_A x a^3}$ connects structure to measurable properties. a must be in cm: 1 pm = 10^-10 cm, 1 A = 10^-8 cm. Common rearrangements: Z = $dN_{Aa}^3$/M (determine structure), M = $dN_{Aa}^3$/Z (identify element), a = ($\frac{ZM}{dN_A}$)^$\frac{1}{3}$. Distance relationships: body diagonal = asqrt(3), face diagonal = a*sqrt(2), edge = a.