Dual basis

Base duale (Fr). Base duale (It)

Definition

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

x_i = x . e_i = x ^j e_j . e_i = x ^j g_ji,

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

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

x ^j = x_i g^ij

where the matrix of the g^ij 's is inverse of that of the g_ij 's (g^ikg_jk = δⁱ_j). The development of vector x with respect to basis vectors e_i can now also be written:

x = x ⁱ e_i = x_i g^ij e_j

The set of n vectors eⁱ = g^ij e_j that span the space Eⁿ forms a basis since vector x can be written:

x = x_i eⁱ

This basis is the dual basis and the n quantities x_i 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:

e_i = g_ij e^j

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

gⁱ_j = eⁱ . e_j = g^ik e_k . e_j = g^ikg_jk = δⁱ_j.

One has therefore, since the matrices g^ik and g_ij are inverse:

gⁱ_j = eⁱ . e_j = δⁱ_j.

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 = A_jⁱ e_i; x'^j = B_i ^j xⁱ,

where A_jⁱ and B_i ^j are transformation matrices, transpose of one another, the dual basis vectors eⁱ and the coordinates x_i transform according to:

e'^j = B_i ^j eⁱ; x'_j = A_jⁱx_i.

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

See also

metric tensor
reciprocal space

The Reciprocal Lattice (Teaching Pamphlet of the International Union of Crystallography)

Section 1.1.2 of International Tables of Crystallography, Volume D