atomic packing factor coordination number
Metal atoms stack like identical hard spheres, and only a few ways of stacking survive. Two numbers split them. The packing fraction, the share of space the spheres actually fill, and the coordination number, how many neighbors one atom touches. The closest stacking reaches 0.74 with 12 neighbors. The stacking with an atom in the middle reaches 0.68 with 8 neighbors, and the one that stacks only the corners reaches 0.52 with only 6 neighbors. The whole crystal follows from these two numbers, and so does a density you can weigh. Two numbers split the lattices.
Pick a lattice and the two signature numbers change together. Face-centered cubic FCC and hexagonal close packing HCP are the closest, filling 0.74 of the space and touching 12 neighbors at each atom. 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 at the corners, fills only 0.52, and has only 6 neighbors, so it is rare in nature. The more places that touch, the tighter the stack. More neighbors mean fewer empty gaps.
The packing fraction answers a plain question. When the spheres grow until they touch, how much of the box is solid and how much is empty? Swell the atoms one step at a time and the meter rises, and it stops exactly at the value of that structure the moment they touch. Even the closest stack leaves 26 percent of the space empty, because round spheres cannot fill a volume completely. The corner-only stack wastes much more than that, nearly half. The round shape is why a gap remains.
The coordination number counts how many neighbors touch a given atom most closely. Press the center atom and every neighbor that touches it lights up, so you can count them. The closest packings surround each atom with 12 spheres that touch it, and spheres of one size cannot have more. The stacking with a filled middle has 8, and the corner-only stacking has only 6, one along each axis. A larger coordination number is the local sign that the whole is tighter. The count of touches is the sign of tightness.
Here is the surprise. The two closest stackings use exactly the same closest layer, and both reach 0.74. The only difference is how the layers are stacked. When the third layer sits directly above the first, one stacking results. When the third layer shifts to a new place before the repeat, the other stacking results. Alternate the two and you see one plane, two ways of stacking, and the same packing fraction at a glance. Only the layers differ. The filling is the same.
Now the payoff. Because you know how many atoms (n) sit in one cell, and how the radius fixes the edge of the cube, you can predict the density before you ever weigh the metal. One cell holds n times the atomic mass inside the volume a³, so the density is n·M/(NA·a³). Push the mass and the radius and change the structure, and a heavier atom or a tighter lattice raises the density. This is how an atomic arrangement you cannot see becomes a number you can weigh. The arrangement becomes the number on the scale.