Metric tensor

The scalar product of two vectors r₁ and r₂ referred to the same base system consisting of the three non-coplanar vectors $\boldsymbol{\tau}_1$,$\boldsymbol{\tau}_2$, $\boldsymbol{\tau}_3$ is defined as:

$$\textbf{r}_1 \cdot \textbf{r}_2 = (x_1 \boldsymbol{\tau}_1 +... ...bol{\tau}_1 + y_2\boldsymbol{\tau}_2 + z_2\boldsymbol{\tau}_3).$$ (1)

In matrix notation it could be written:

$$\textbf{r}_1 \cdot \textbf{r}_2 = [x_1y_1z_1]\left[\begin{ar... ...ght] \left[\begin{array} {c} x_2\\ y_2\\ z_2\end{array}\right];$$ (2)

it is easy to verify that formulae (1) and (2) are equivalent. Relation (2) can be written more briefly as follows:

$$\textbf{r}_1 \cdot \textbf{r}_2 = r^t_1Gr_2$$ (3)

where r₂ is a column matrix

$$\left[\begin{array} {c}x_2\\ y_2\\ z_2\end{array}\right]$$

and r^t₁ a transposed column matrix [$x_1\,y_1\,z_1$]; 9 is the 3 $\times$ 3 matrix of relation (2) and is called a metric matrix or metric tensor, because its elements $g_{ij} = \boldsymbol{\tau}_i \cdot \boldsymbol{\tau}_j$ are dependent both on the length of the base vectors and on the angles formed by them.

If in (3) we assume r₁ = r₂, we have:

$$\textbf{r}_1 \cdot \textbf{r}_1 = \vert\textbf{r}_1\vert \cdot \vert\textbf{r}_1\vert = r^t_1 Gr_1$$ (4)

and therefore:

$$\vert\textbf{r}_1\vert = \sqrt{r^t_1Gr_1}.$$ (5)

On the other hand, bearing in mind that $\textbf{r}_1 \cdot \textbf{r}_2 = \vert\textbf{r}_1\vert \vert\textbf{r}_2\vert\cos \phi$, where $\phi$ is the angle between $\textbf{r}_1$ and $\textbf{r}_2$, we have:

$$\textbf{r}_1 \cdot \textbf{r}_2 = \vert\textbf{r}_1\vert\vert\textbf{r}_2\vert \cos \phi = r^t_1Gr_2$$ (6)

and finally, using relation (5), we obtain:

$$\cos \phi = \frac{r^t_1Gr_2}{\sqrt{r^t_1Gr_1} \cdot \sqrt{r^t_2Gr_2}}.$$ (7)

Equations (5) and (7) are the rules to obtain the vector lengths and the angles between vectors. The space in which the lengths and the angles between vectors are defined, is called metric space. The metric is given by the G matrix.

Copyright © 1980, 1998 International Union of Crystallography