Dual basis

Base duale (Fr). Base duale (It). 双対基底 (Ja).

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 No. 4 of the International Union of Crystallography)
• Chapter 1.1.2 of International Tables for Crystallography, Volume D