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Dual basis

The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows:

Given a basis of n vectors ei spanning the direct space En, and a vector x = x i ei, let us consider the n quantities defined by the scalar products of x with the basis vectors, ei:

xi = x . ei = x j ej . ei = x j gji,

where the gji 's are the doubly covariant components of the metric tensor.

By solving these equations in terms of x j, one gets:

x j = xi gij

where the matrix of the gij 's is inverse of that of the gij 's (gikgjk = δij). The development of vector x with respect to basis vectors ei can now also be written:

x = x i ei = xi gij ej

The set of n vectors ei = gij ej that span the space En forms a basis since vector x can be written:

x = xi ei

This basis is the dual basis and the n quantities xi defined above are the coordinates of x with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:

ei = gij ej

The scalar products of the basis vectors of the dual and direct bases are:

gij = ei . ej = gik ek . ej = gikgjk = δij.

One has therefore, since the matrices gik and gij are inverse:

gij = ei . ej = δij.

These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of reciprocal space are identical.

Change of basis

In a change of basis where the direct basis vectors and coordinates transform like:

e'j = Aji eix'j = Bi j xi,

where Aji and Bi j are transformation matrices, transpose of one another, the dual basis vectors ei and the coordinates xi transform according to:

e'j Bi j eix'j = Ajixi.

The coordinates of a vector in reciprocal space are therefore covariant and the dual basis vectors (or reciprocal vectors) contravariant.

See also

Section 1.1.2 of International Tables of Crystallography, Volume D