Three lattice points define a crystallographic plane. Suppose this plane intersects the three crystallographic axes X, Y and Z at the three lattice points (p, 0, 0), (0, q, 0), (0, 0, r) with integer p, q and r. If the largest common integer factor of p, q and r is 1 and if m = pqr is the least common multiple of p, q and r, then the equation of the plane is:
x′/pa + y′/qb + z′/rc = 1 (1)
Introducing the fractional coordinates x, y and z (x = x′/a, y = y′/b and z = z′/c) the equation of the plane becomes:
x/p + y/q + z/r = 1 (2)
Multiplying both sides by pqr we obtain:
qrx + pry + pqz = pqr (3)
which can be rewritten as:
hx + ky + lz = m (4)
where h = qr, k = pr, l = pq and m = pqr. By varying m over all integer numbers from −∞ to +∞ we can construct a family of planes parallel to the plane expressed in equation 1.
The rational properties of all points being the same, there will be a plane of the family passing through each lattice point. For the same reason each lattice plane is identical to any other within the family through a lattice translation.
Equation (4) defines, as m is varied, a family of identical and equally spaced crystallographic planes. The three indices h, k, l, called Miller indices, define the family. To indicate that a family of lattice planes is defined by a sequence of three integers, these are included within braces: (hkl). Equations (4) is useful to deduce a simple interpretation of the three indices h, k and l (i.e. they indicate that the planes of the family divide a in h parts, b in k parts, and c in l parts).