How To Calculate Interplanar Spacing

When atoms form themselves into lattice structures, as they do in metals, ionic solids and crystals, you can think of them as making geometrical shapes, such as cubes and tetrahedrons. The actual structure a particular lattice assumes depends on the sizes, valencies and other characteristics of the atoms forming it. Interplanar spacing, which is the separation between sets of parallel planes formed by the individual cells in a lattice structure, depends on the radii of the atoms forming the structure as well as on the shape of the structure. There are seven possible crystal systems, and within each system are a number of subsystems, making for a total of 14 different lattice structures. Each structure has its own formula for calculating interplanar spacing.

It makes sense to talk about spacing between planes only if they are parallel to each other. Crystallographers identify a family of parallel planes by their Miller indices. To find them, you choose a plane from the family and note the intercepts of the plane on the x, y and z axes. The Miller intercepts are the reciprocals of the intercepts. When one or more of the intercepts is a fractional number, the convention is to multiply all three indices by a factor that eliminates the fraction. Miller indices are generally denoted by the letters h, k and l. Crystallographers identify a particular plane by enclosing the indices in round brackets (hkl) and show a family of planes by enclosing them in parentheses {hkl}.

The lattice constant of a particular crystal structure is a measure of how closely packed the atoms in the structure are. This is a function of the radius (r) of each of the atoms in the structure as well as the geometric configuration of the lattice. The lattice constant (a) for a simple cubic structure, for example, is a = 2r. A cubic structure that includes an atom in the center of each cube is a body-centered cubic (BCC) structure, and its lattice constant is a = 4R/√3. A cubic structure that includes an atom in the center of each face is a face-centered cubic, and its lattice constant is a = 4r/√2. Lattice constants for more complex shapes are accordingly more complex.

The spacing between planes in a family with the Miller indices h, k and l is denoted by d_hkl. A formula relating this distance to the Miller indices and the lattice constant (a) exists for each crystal system. The equation for a cubic system is:

\(\Big(\frac{1}{d_{hkl}}\Big)^2=\frac{h^2+k^2+l^2}{a^2}\)

For other systems, the relationship is more complicated because you need to define for parameters to isolate a particular plane. For example, the equation for a tetragonal system is:

\(\Big(\frac{1}{d_{hkl}}\Big)^2=\frac{h^2+k^2}{a^2}+\frac{l^2}{c^2}\)

where c is the intercept on the z-axis.