Geometry of the reciprocal lattice.

The basis set of the reciprocal lattice vectors are defined by the equation:

ai . bj = 2pd ij { dij = 0 if i # j, dij = 1 if i = j}

Where the a's are the basis vectors of the direct lattice. The b's are then called the fundamental vectors of the reciprocal lattice).

Using the above equation in two dimensions we have:

a1 . b1 = 2p, a2 . b2 = 2p

a1 . b2 = 0, a2 . b1 = 0

From these equations we see that b1 must be perpendicular to a2; and that b2 must be perpendicular to a1. See the 2 top figures for a two-dimensional rectangular lattice in the image below.

If a1 and a2 have an angle q = 90o between them, the conditions are automatically fulfilled and b1 is in the same direction as a1; and b2 is in the same direction as a2. The reciprocal lattice is a set of points in reciprocal space which are connected to a given point by the vectors G = n1b1 + n2b2
where n1 and n2 are integers. It is also rectangular. The magnitudes of the vectors are given by

b1 = 2p /a1, b2 = 2p /a2

Now look at the bottom figures of the same image (above). A general two-dimensional lattice is shown there. If the angle between a1 and a2 is q then the angle between b1 and b2 is 1800 - q.The magnitudes of the reciprocal lattice vectors are:

b1 = (2p /a1) . cos(900 - q) b2 = (2p /a2) . cos (900 - q)

In three dimensions we can define the fundamental reciprocal lattice vectors as:

b1 = 2p (a2 x a3 / a1 . (a2 x a3))

b2 = 2p (a3 x a1 / a1 . (a2 x a3))

b3 = 2p (a1 x a2 / a1 . (a2 x a3))

The term in the numerator is a vector perpendicular to both a2 and a3 and which has magnitude of the area of a parallelogram made by a2 and a3. The term in denominator is a scalar and is equal to the volume of the unit cell formed by a1 , a2 and a3. Overall the equation gives a vector perpendicular to both a2 and a3 with the absolute value of 2p /a1 . Taking the scalar product of a1 with both sides of the equation we get:

a1 . b1 = 2p (a1 . (a2 x a3) / a1 . (a2 x a3 )) = 2p

The vector product a2 x a3 is perpendicular to both a2 and a3 and so:

a2 . b1 = 2p (a2 . (a2 x a3)/ a1 . (a2 x a3 )) = 0

a3 . b1 = 2p (a3 . (a2 x a3)/ a1 . (a2 x a3 )) = 0

Because the scalar product of the two vectors at right angles depends on cos 90o which is zero.

In three dimensions the reciprocal lattice is made up from points connected to some given point defined as the origin by the general vectors

G = n1b1 + n2b2 + n3b3

By analogy with the two-dimensional case, we can see that the fundamental vectors for a three-dimensional simple cubic reciprocal lattice lay in the same direction as those of the direct lattice and have the absolute values given by bi = 2p /ai. And the fundamental vectors for a general three-dimensional reciprocal lattice has a different angle with the vectors of the direct lattice if the direct lattice is not an orthogonal one.

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