Appendix

Let us examine, as an example, the cubic lattice: since the unit cell constants are $a_0 = b_0 = c_0, \alpha = \beta = \gamma = 90^{\circ}$, the metric tensor G is given by:

$$G = \left[\begin{array} {ccc} g_{11}&0&0\\ 0&g_{11}&0\\ 0&0&g_{11}\end{array}\right].$$

From relation (10) we have:

$$g_{11} \cdot 1 = g_{11} \cdot A^t \cdot 1 \cdot A$$

and consequently:

$$A^t \cdot A = 1.$$

In this particular case the matrices A are such that their inverse A^-1 is equal to their transposed matrix A^t; therefore we can obtain the following relations:

a₁₁a₁₁ + a₂₁a₂₁ + a₃₁a₃₁ = 1

a₁₁a₁₂ + a₂₁a₂₂ + a₃₁a₃₂ = 0

a₁₁a₁₃ + a₂₁a₂₃ + a₃₁a₃₃ = 0

a₁₂a₁₂ + a₂₂a₂₂ + a₃₂a₃₂ = 1

a₁₂a₁₃ + a₂₂a₂₃ + a₃₂a₃₃ = 0

a₁₃a₁₃ + a₂₃a₂₃ + a₃₃a₃₃ = 1

Relations (11), (22), (33) impose the condition that, in each column of the A matrix, one element is equal to $\pm1$, and the other two are equal to zero. Relations (12), (13), (23) impose the same condition for each row, since the element different from zero of each column must lie in a different row from the one occupied by the non-zero element of the other two columns.

In conclusion the symmetry operations compatible with a cubic lattice are represented by the following matrices:

$$\left[\begin{array} {ccc} 1&0&0\\ 0&1&0\\ 0&0&1\end{array}\r... ...ft[\begin{array} {ccc} 0&1&0\\ 1&0&0\\ 0&0&1\end{array}\right],$$

$$\left[\begin{array} {ccc} 0&1&0\\ 0&0&1\\ 1&0&0\end{array}\r... ...eft[\begin{array} {ccc} 0&0&1\\ 0&1&0\\ 1&0&0\end{array}\right]$$

plus those obtained from the above matrices, considering, for each of them, all the possible permutations of one, two and three negative signs. It is not difficult to see that from each of the above six matrices, we can obtain seven others containing negative elements. The symmetry operations compatible with a cubic lattice are, thus, 48 in all. Their respective matrices are shown in Table 3. For each matrix in the table the corresponding symmetry operation and the orientation of the symmetry element, derived as above, are given.

From the table it is seen that the symmetry operation corresponding to a rotation of 60$^{\circ}$, i.e. symmetry element of order 6, is incompatible with the cubic lattice, but is compatible with a different lattice ($a_0 = b_0, c_0, \alpha = \beta = 90^{\circ}, \gamma = 120^{\circ}$). As it is known, all 32 point groups are subgroups of m3m or 6/mmm or both.

Finally, the relation A^tGA = G can be used to derive, if matrix A is known, the metric tensor compatible with the symmetry operation A.

 

Table 3. A matrices for the cubic lattice

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