How many atoms are found inside a unit cell of an sc, a bcc,...

How many atoms are found inside a unit cell of an sc, a bcc, and an fcc crystal? How far apart in terms of lattice constant $a$ are the nearest neighbor atoms in each case, measured from center to center?

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Key Concepts

Face-Centered Cubic Structure

The face-centered cubic structure includes atoms at the eight corners and at the centers of the six faces of the cube. With each corner atom shared by eight cells and each face atom shared by two cells, the total number of atoms per unit cell is four. The nearest neighbor distance is found along the face diagonal, with adjacent atoms being a distance of a/?2 apart, where a is the lattice constant.

Simple Cubic Structure

The simple cubic structure is characterized by having atoms positioned at each of the eight corners of the cubic unit cell. Because each corner atom is shared by eight adjoining cells, the net contribution is one atom per unit cell. The nearest neighbor distance in this structure is simply the lattice constant a, since adjacent atoms lie directly along the edges of the cube.

Body-Centered Cubic Structure

The body-centered cubic structure features atoms at the eight corners and one atom at the center of the cube. The corner atoms, shared among eight cells, contribute one atom overall, and the center atom is entirely within the cell, making a total of two atoms per unit cell. The nearest neighbor distance is determined by the distance from the central atom to any corner atom, which is half the body diagonal, giving a distance of (?3/2) times the lattice constant a.

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