# Metric tensor

A metric tensor is used to measure distances in a space. In crystallography the spaces considered are vector spaces with Euclidean metrics, i.e.ones for which the rules of Euclidean geometry apply. In that case, given a basis ei of a Euclidean spaceEn, the metric tensor is a rank 2 tensor the components of which are:

gij = ei . ej = ej.ei = gji.

It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, x = xi ei and y = yj ej is written:

x . y = xi ei . yj ej = gij xi yj.

In a three-dimensional space with basis vectors abc, the coefficients gij of the metric tensor are:

g11 = a2g12 = a . bg13 = a . c;
g21 = b . ag22 = b2g23 = b . c;
g31 = c . ag32 = c . bg33 = c2;

Because the metric tensor is symmetric, g12 = g21g13 = g31, and g13 = g31. Thus there are only six unique elements, often presented as

g11 g22 g33
g23 g13 g12

or, multiplying the second row by 2, as a so-called G6 ("G" for Gruber) vector

a2b2c2, 2 b . c, 2 a . c, 2 a . b )

The inverse matrix of gijgij, (gikgkj = δkj, Kronecker symbol, = 0 if i ≠ j, = 1 if i = j) relates the dual basis, or reciprocal space vectors ei to the direct basis vectors ei through the relations:

ej = gij ej

In three-dimensional space, the dual basis vectors are identical to the reciprocal space vectors and the components of gij are:

g11 = a*2g12 = a* . b*g13 = a* . c*;
g21 = b* . a*g22 = b*2g23 = b* . c*;
g31 = c* . a*g32 = c* . b*g33 = c*2;

with:

g11 = b2c2 sin2 α/ V2g22 = c2a2 sin2 β/ V2g33 = a2b2 sin2 γ/ V2;

g12 = g21 = (abc2/ V2)(cos α cos β - cos γ); g23 = g32 = (a2bc/ V2)(cos β cos γ - cos α); g31 = g13 = (ab2c/ V2)(cos γ cos α - cos β)

where V is the volume of the unit cell (abc).

### Change of basis

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

e'j = Aj i eix'j = Bi j x i,

where Aj i and Bi j are transformation matrices, transpose of one another. According to their definition, the components gij, of the metric tensor transform like products of basis vectors:

g'kl = AkiAljgij.

They are the doubly covariant components of the metric tensor.

The dual basis vectors and coordinates transform in the change of basis according to:

e'j = Bi j eix'j = Aj ixi,

and the components gij transform like products of dual basis vectors:

g'kl = Aik Ajl gij.

They are the doubly contravariant components of the metric tensor.

The mixed components, gij = δij, are once covariant and once contravariant and are invariant.

### Properties of the metric tensor

• The tensor nature of the metric tensor is demonstrated by the behaviour of its components in a change of basis. The components gij andgij are the components of a unique tensor.
• The squares of the volumes V and V* of the direct space and reciprocal space unit cells are respectively equal to the determinants of thegij 's and the gij 's:

V 2 = Δ (gij) = abc(1 - cos 2 α - cos 2 β - cos2 γ + 2 cos α cos α cos α)

V*2 = Δ (gij) = 1/ V 2.

• One changes the variance of a tensor by taking the contracted tensor product of the tensor by the suitable form of the metric tensor. For instance:

gimt ij..kl.. = t j..klm..

Multiplying by the doubly covariant form of the metric tensor increases the covariance by one, multiplying by the doubly contravariant form increases the contravariance by one.