Metric tensor

Tenseur métrique (Fr). Metrischer Tensor (Ge). Tensor métrico (Sp). Tensore metrico (It). Метрический тензор (Ru). 計量テンソル (Ja).

Definition

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 e_i of a Euclidean space, Eⁿ, the metric tensor is a rank 2 tensor the components of which are:

g_ij = e_i . e_j = e_j.e_i = g_ji.

It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, x = xⁱ e_i and y = y^j e_j is written:

x . y = xⁱ e_i . y^j e_j = g_ij xⁱ y^j.

In a three-dimensional space with basis vectors a, b, c, the coefficients g_ij of the metric tensor are:

g₁₁ = a²; g₁₂ = a . b; g₁₃ = a . c;
g₂₁ = b . a; g₂₂ = b²; g₂₃ = b . c;
g₃₁ = c . a; g₃₂ = c . b; g₃₃ = c².

Because the metric tensor is symmetric, g₁₂ = g₂₁, g₁₃ = 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

( a², b², c², 2b . c, 2a . c, 2a . b )

The inverse matrix of g_ij, g^ij, relates the dual basis, or reciprocal space vectors eⁱ to the direct basis vectors e_i, through the relations:

e^j = g^ij e_j.

Note that g^ikg_kj = δ^k_j, where δ^k_j is the Kronecker symbol, equal to 0 if i ≠ j, and equal to 1 if i = j.

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

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

with:

g¹¹ = b²c² sin² α/ V²; g²² = c²a² sin² β/ V²; g³³ = 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 (a, b, c).

Change of basis

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

e'_j = A_j ⁱ e_i; x'^j = B_i ^j x ⁱ,

where A_j ⁱ and B_i ^j are transformation matrices, transpose of one another. According to their definition, the components g_ij, of the metric tensor transform like products of basis vectors:

g'_kl = A_kⁱA_l^jg_ij.

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 = B_i ^j eⁱ; x'_j = A_j ⁱx_i,

and the components g^ij transform like products of dual basis vectors:

g'^kl = A_i^k A_j^l g^ij.

They are the doubly contravariant components of the metric tensor.

The mixed components, gⁱ_j = δⁱ_j, 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 g_ij and g^ij 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 the g_ij 's and the g^ij 's:

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

V*² = Δ (g^ij) = 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:

g_imt ^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.

See also

• Dual basis
• Reciprocal space
• Metric Tensor and Symmetry Operations in Crystallography (Teaching Pamphlet No. 10 of the International Union of Crystallography)
• Chapter 1.1.3 of International Tables for Crystallography, Volume B
• Chapter 1.1.2 of International Tables for Crystallography, Volume D