Mathematical Tools for Physics

12—Tensors 373

Reciprocal Basis
Immediately, when you do the basic scalar product you find complications. If~u=uj~ej, then

~u.~v= (uj~ej).(vi~ei) =ujvi~ej.~ej.

But since the~eiaren’t orthonormal, this is a much more complicated result than the usual scalar product such as

uxvy+uyvy+uzvz.

You can’t assume that~e 1 .~e 2 = 0any more. In order to obtain a result that looks as simple as this familiar form,
introduce an auxiliary basis: thereciprocal basis. (This trick will not really simplify the answer; it will be the
same as ever. It will however be in a neater form and hence easier to manipulate.) The reciprocal basis is defined
by the equation

~ei.~ej=δji =

{

1 ifi=j

0 ifi 6 =j

(31)

The~ej’s are vectors. The index is written as a superscript to distinguish it from the original basis,~ej.

e

1

2

1

e

e

e

2

To elaborate on the meaning of this equation,~e^1 is perpendicular to the plane defined by~e 2 and~e 3 and is
therefore more or less in the direction of~e 1. Its magnitude is adjusted so that the scalar product

~e^1 .~e 1 = 1.

The “direct basis” and “reciprocal basis” are used in solid state physics and especially in describing X-ray
diffraction in crystallography. In that instance, the direct basis is the fundamental lattice of the crystal and the
reciprocal basis would be defined from it. The reciprocal basis is used to describe the wave number vectors of
scattered X-rays.