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Next: 3.2 The volumes of the unit Up: 3. Reciprocal Space and Dual Space Previous: 3. Reciprocal Space and Dual Space

3.1 Definition

Let us now call ei the basic vectors of a vectorial space and xi the coordinates of a given vector x. We may write:

\begin{displaymath}
\textbf{x} = x^i\textbf{e}_i = x^1\textbf{e}_1+x^2\textbf{e}_2+x^3\textbf{e}_3\end{displaymath} (3.1)
using Einstein's summation convention. A change of coordinates may be described by the following relations:

\begin{displaymath}
\textbf{e}^{\prime}_i = A^i_j\textbf{e}_j; \quad
\textbf{e}_...  ... \qquad x^{\prime i} = B^i_jx^i; \quad
x^i = A^i_j x^{\prime j}\end{displaymath}

(3.2)

\begin{displaymath}
A^i_jB^j_k = \delta^i_k \qquad (\delta = 1 \hbox{ if } i = k; \quad
 \delta = 0 \hbox{ if } i \neq k)\end{displaymath}

Quantities with a subscript transform in a change of coordinate like the basic vectors and are called covariant ; those with a superscript transform like the coordinates and are called countervariant . Let us now consider the scalar products:
\begin{displaymath}
x_i = \textbf{x} \cdot \textbf{e}_i = x^j\textbf{e}_i \cdot \textbf{e}_j = x^jg_{ij}\end{displaymath} (3.3)
where we put

\begin{displaymath}
g_{ij} = \textbf{e}_i \cdot \textbf{e}_j \end{displaymath} (3.4)
The nine quantities gij are the components of the so-called metric tensor. We shall show that the three quantities xi are the covariant coordinates of x. The system of equations (3.3) expresses the xi in terms of the xi. We can resolve this system and write

xi = xjgij

(3.5)

where

\begin{displaymath}
g^{ij}g_{jk} = \delta^i_k \end{displaymath} (3.6)
This can always be written since the determinant built on the gij is different from zero by definition of the scalar product.

Let us now introduce the following set of vectors

\begin{displaymath}
\textbf{e}^i = g^{ij}\textbf{e}_j \end{displaymath} (3.7)

This set of vectors constitutes a set of basic vectors. To show this we may simply transform equation (3.1):

\begin{displaymath}
\textbf{x} = x^i\textbf{e}_i = x_jg^{ij}\textbf{e}_j = x_j\textbf{e}^j\end{displaymath} (3.8)

The vectors ej constitute therefore a set of basic vectors and the xj are the coordinates of x with respect to this base. They are called countervariant basic vectors. They are also identical to the basic vectors of the reciprocal space. This can easily be demonstrated by showing that they satisfy the basic relations (2.3) of the reciprocal space vectors. Let us consider the scalar products $\textbf{e}_i \cdot\textbf{e}^j$.Using (3.7), (3.4) and (3.6), we may write:

\begin{displaymath}
\textbf{e}_i \cdot \textbf{e}^j = g^{ik} \textbf{e}_i \cdot \textbf{e}_k =
g^{jk}g_{ik} = \delta^j_i\end{displaymath}

which is indeed identical to 2.3.


next up previous
Next: 3.2 The volumes of the unit Up: 3. Reciprocal Space and Dual Space Previous: 3. Reciprocal Space and Dual Space

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