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(a) *with each node of the reciprocal lattice whose numerical
coordinates have no common divider can be associated a set of direct lattice
planes*

Let *M* be a reciprocal lattice point whose coordinates *h*, *k*, *l* have no common
divider (*M* is the first node on the reciprocal lattice row **OM**), and
*P* a point in direct space. We may write:

(2.5) |

Let us look for the locus of all points *P* of direct space such that the scalar
product should be constant. It is a plane
normal to *O* and passes through the projection *H* of *P* on **OM** (Fig.
2). Using 2.4, we find easily that the equation of this plane in direct space
is given by

(2.6) |

Let us now assume that *P* is a node of the direct lattice:

The locus of *P* is a lattice plane of the direct lattice. Its equation is:

(2.7) |

Since all numbers in the left hand side are integers, we find that *C* is also
an integer. With each value of *C* we may associate a lattice plane and thus
generate a set of direct lattice planes which are all normal to the reciprocal
vector **OM** (Fig. 3). The distance of one of these planes to the origin
is given by:

The lattice planes have, as expected, an equal spacing:

(2.8) |

(2.9) |

*This is the fundamental relation of the reciprocal lattice which shows
that with any node M of the reciprocal lattice whose numerical coordinates have
no common divider we may associate a set of direct lattice planes normal to*
**OM***. Their spacing is inversely proportional to the
parameter along the reciprocal row * **OM**.

In order that the correspondence between direct and reciprocal lattice should be fully established, the converse of the preceding theorem should also be demonstrated. This will be done in paragraph 2.2(c).

It is interesting at this point to give an interpretation to the reciprocal lattice points whose numerical coordinates have a common divider. Let us consider such a point for which:

where
Let *d*_{h1k1l1} be the spacing of the direct lattice planes associated with
*M*. The fundamental law of the reciprocal lattice may be written:

We may also write:

(2.10) |

In other words, with the reciprocal lattice node *M* may be associated a set of
fictitious planes in direct space whose spacing is *n* times smaller than the
real lattice spacing. We shall see that in diffraction by crystal lattices a
reciprocal lattice point may be associated with each Bragg diffraction: if the
coordinates of this point have no common divider, Bragg`s law is satisfied to
the first order (2*d* sin = ); if they have a common divider,
*n*, Bragg`s law is satisfied to the *n*th order (2*d* sin =
*n*), one may also say it is satisfied to the first order for the
fictitious lattice planes of spacing *d*/*n* (2*d*/*n* sin = ) and
this is what is actually always done in practice.

(b) *Miller indices*

Let us consider one particular lattice plane of equation

*hx* + *ky* + *lz* = *C*

We conclude that the lattice plane intercepts, along the three axes, lengths
which are inversely proportional to three integers which have no common divider.
This is the so-called *Law of Rational Indices* or Hauy Law. The three
indices are called the Miller indices.

The planes which are crystallographically the most important ones are the
densest ones, that is those with the largest spacing. Equation (2.9) tells us
that they are associated with the shortest vectors in reciprocal lattice and
that their Miller indices are therefore small. This is the reason why Hauy's
law was also called the law of *simple* rational indices.

(c) *The reciprocal law: to each set of direct lattice planes
corresponds a reciprocal lattice vector*

Let us consider a set of direct lattice planes of equation:

*hx* + *ky* + *lz* = *C*

Since *x*, *y*, *z* may be integers, *h*, *k*, and *l* are also integers. If *C* = 1,
corresponding to the first plane in the family, *h*, *k* and *l* have no common
divider. Let us now consider the reciprocal lattice vector

Its scalar products with the vectors **QR** and **RS** (Fig. 4) are
respectively equal to:

They are both equal to zero, which shows that the reciprocal lattice vector is
normal to the set of direct lattice planes; the scalar product of
**ON**_{hkl} by **OP** where *P* is any direct lattice node in a
plane of the set can be written in the form of equation (2.6). The reciprocal
theorem is thus demonstrated.

**Copyright © 1981, 1998 International Union of
Crystallography**