Lattice Centering and Coordination Number

A primitive lattice has lattice points only at the corners of its unit cell, while centered lattices have additional lattice points either at the center, on faces, or on specific faces. Centered lattices include body-centered, face-centered, and base-centered types. Centered lattices achieve higher packing efficiencies and coordination numbers than primitive lattices.

A face-centered cubic unit cell contains four atoms. This comes from one-eighth of an atom at each of the eight corners, totaling one atom, plus one-half of an atom on each of the six faces, totaling three atoms. Face-centered cubic structures pack atoms most efficiently, occupying 74% of the total volume.

Coordination number is the count of nearest neighbor atoms in direct contact with a given atom. In primitive cubic structures, coordination number is six. Body-centered cubic atoms have a coordination number of eight, while face-centered cubic atoms have a coordination number of twelve, reflecting increasingly efficient packing arrangements.

Body-centered cubic structures have an additional atom positioned at the center of the unit cell. This central atom contacts all corner atoms, creating more efficient space utilization. Body-centered cubic solids occupy approximately 68% of total volume compared to only 52% for simple cubic structures, demonstrating superior packing efficiency.

In a simple cubic lattice, atoms are positioned at the corners of a cubic unit cell, with each atom contacting four neighbors in its own layer, one above, and one below. A simple cubic unit cell contains one atom total. This arrangement represents the least efficient packing, filling only about 52% of the container volume.

The number of atoms in a cubic unit cell depends on lattice centering. Corner atoms contribute one-eighth each since eight unit cells share each corner atom. Face-centered atoms contribute one-half each since two unit cells share each face atom. Center atoms contribute fully. Primitive cubic has one atom, body-centered has two, and face-centered has four atoms.

Packing efficiency increases with lattice centering. Primitive cubic structures pack at 52% efficiency. Body-centered cubic structures achieve 68% efficiency through central atom placement. Face-centered cubic structures achieve maximum efficiency at 74%, with atoms occupying nearly three-quarters of available volume, making them the most densely packed cubic arrangement.