Packing and Coordination Split the Lattices
Metal atoms stack like identical hard spheres, and only a few stacking patterns survive. Two numbers tell them apart: the packing fraction, the share of space the spheres actually fill, and the coordination number, how many neighbors each atom touches. Face-centered cubic and hexagonal close-packed both reach the densest possible 0.74 with 12 neighbors; body-centered cubic settles at 0.68 with 8; and the airy simple cubic fills only 0.52 with 6. From these two numbers the whole crystal follows, right down to a density you can weigh on a scale.
Pick a lattice and the two headline numbers update together. Face-centered cubic (FCC) and hexagonal close-packed (HCP) are the champions, filling 0.74 of space with each atom touching 12 others. Body-centered cubic (BCC) puts one atom in the middle of the cube and reaches 0.68 with 8 neighbors. Simple cubic (SC) stacks atoms only on the corners, so it fills a mere 0.52 with 6 neighbors and is rare in nature. Notice that more contacts means denser packing.
The packing fraction answers a blunt question: once the spheres grow until they touch, what share of the box is solid and what share is empty? Inflate the atoms step by step and the meter climbs, stopping exactly at the structure's value when contact is reached. Even the tightest packing leaves 26 percent of space empty, because round spheres can never fill a volume completely. Simple cubic wastes far more, nearly half.
The coordination number is simply how many nearest neighbors touch a given atom. Click the central atom and every neighbor it presses against lights up, so you can count them. Close-packed FCC and HCP surround each atom with 12 touching spheres, the most any equal spheres can manage. BCC gives 8, and simple cubic gives just 6, one along each axis. A higher coordination number is the local sign of tighter global packing.
Here is the surprise: FCC and HCP share the exact same close-packed layer, and both reach 0.74. What differs is only how the layers stack. If the third layer sits directly above the first, the sequence is ABABAB and you get HCP. If the third layer takes a new offset position before repeating, the sequence is ABCABC and you get FCC. Toggle between them and watch one plane, two stackings, and the same packing fraction.
Now the payoff. Because we know how many atoms n sit in a unit cell and how the radius sets the cube edge a, we can predict density before ever weighing the metal. Each cell holds a mass of n times the atomic mass, packed into a volume a³, so the density is n·M/(NA·a³). Slide the mass and radius and switch the structure: a heavier atom or a tighter lattice pushes the density up. This is how an invisible arrangement of atoms becomes a number you can measure on a balance.