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Next: Appendix Up: Derivation of the Crystallographic Point Groups Previous: Groups containing more than one rotation

Groups containing type II symmetry operations

To derive the point groups which contain type II symmetry operations as well, it is necessary to remember that the product of two operations of the same type is an operation of type I, while the product of two operations of different type is an operation of type II.

In such point groups the operations of type I, equal in number to those of type II, form a group.

Table 1

Order of the axes1 Trace m Possible values of $\omega$ Order of the resultant axis Orientation2

2-2 $4\cos^2\omega-1-m=0$ +3 0$^{\circ}$, 180$^{\circ}$ 1 -
+2 30$^{\circ}$, 150$^{\circ}$, 210$^{\circ}$, 330$^{\circ}$ 6 010
+1 45$^{\circ}$, 135$^{\circ}$, 225$^{\circ}$, 315$^{\circ}$ 4 010
0 60$^{\circ}$, 120$^{\circ}$, 240$^{\circ}$, 300$^{\circ}$ 3 010
-1 90$^{\circ}$, 270$^{\circ}$ 2 010

3-2 $3\cos^2\omega-1-m=0$ +2 0$^{\circ}$, 180$^{\circ}$ 6 001
+1 35$^{\circ}$16', 144$^{\circ}$44', 215$^{\circ}$16', 324$^{\circ}$44' 4  
0 54$^{\circ}$44$^{\prime}$, 125$^{\circ}$16$^{\prime}$, 234$^{\circ}$44$^{\prime}$ 305$^{\circ}$16$^{\prime}$ 3  
-1 90$^{\circ}$, 270$^{\circ}$ 2 $1/2\,-\sqrt3/2\,0$

4-2 $2\cos^2\omega-1-m=0$ +1 0$^{\circ}$, 180$^{\circ}$ 4 001
0 45$^{\circ}$, 135$^{\circ}$, 225$^{\circ}$, 315$^{\circ}$ 3 $1/\sqrt3\,-1/\sqrt3\,1/\sqrt3$
-1 90$^{\circ}$, 270$^{\circ}$ 2 $1/\sqrt2\,-1/\sqrt2\,0$

6-2 $\cos^2\omega-1-m=0$ 0 0$^{\circ}$, 180$^{\circ}$ 3 001
-1 90$^{\circ}$, 270$^{\circ}$ 2 $\sqrt3/2\,-1/2\,0$

3-3 $9\cos^2\omega-6\cos\omega-3-4m=0$ +3 180$^{\circ}$ 1 -
0 0$^{\circ}$, 109$^{\circ}$28$^{\prime}$, 250$^{\circ}$32$^{\prime}$ 3 001
-1 70$^{\circ}$32$^{\prime}$, 289$^{\circ}$28$^{\prime}$ 2  

4-3 $3\cos^2\omega-2\sqrt3\cos\omega-1-2m=0$ +1 125$^{\circ}$16$^{\prime}$, 234$^{\circ}$44$^{\prime}$ 4  
-1 54$^{\circ}$44$^{\prime}$, 305$^{\circ}$16$^{\prime}$ 2  

6-3 $3\cos^2\omega-6\cos\omega-1-4m=0$ +2 180$^{\circ}$ 6 001
-1 0$^{\circ}$ 2 001

4-4 $\cos^2\omega-2\cos\omega-m=0$ +3 180$^{\circ}$ 1 -
0 90$^{\circ}$, 270$^{\circ}$ 3 $1/\sqrt3\,-1/\sqrt3\,1/\sqrt3$
-1 0$^{\circ}$ 2 001

6-4 $\cos^2\omega-2\sqrt3\cos\omega+1-2m=0$ there are no possible solutions      

6-6 $\cos^2\omega-6\cos\omega+5-4m=0$ +3 180$^{\circ}$ 1 -
0 0$^{\circ}$ 3 001

1The first rotation axis is coincident with A3, the second one with A3'.

2The resulting axis orientation is given by the direction cosines referred to the orthonormal base system A1A2A3 and it is obtained solving the equation (R-1)x=0.

From the 11 groups given above we can obtain 11 other point groups which have as elements the type I operations, plus other operations obtained from these by combining them with the inversion operation, represented by the matrix:

\={1}&0&0\\ 0&\={1}&0\\ 0&0&\={1}\end{array}\right].\end{displaymath}

The centrosymmetric groups so obtained, which have an order double with respect to the order of the groups with which we started, are respectively:

\={1}, 2/m, \={3}, 4/m, 6/m, mmm, \={3}m, 4/mmm, 6/mmm, m3, m3m.\end{displaymath}

It is also possible to obtain groups containing type II symmetry operations but which do not contain the inversion operation. In this case we must first obtain, from the starting groups which contain only type I symmetry operations, the corresponding subgroups, which have order $\frac{1}{2}$ with respect to the starting groups.

From the scheme shown in Table 2 we see that there are 10 subgroups satisfying this condition. So, to obtain the new groups we multiply by the inversion operation all the operations of the starting group which do not belong to the subgroup.

Table 2

\includegraphics {table2.ps}

The sum of the operations obtained in this way, plus the operations belonging to the subgroup, gives all the elements of the new group. The order of the new group is then equal to the order of the starting group.

Let us fully analyse an example: the group 422, of order 8, has the groups 4 and 222 as subgroups of order 4.

In the first case, the subgroup 4 contains the symmetry operations 41, 42, 43, 1; therefore the operations corresponding to a 180$^{\circ}$ rotation around the axis orthogonal to the 4-fold axis are inverted. In this way we obtain mirror planes parallel to the 4-fold axis, and the resulting point group is 4mm.

In the second case, the subgroup 222 contains three 180$^{\circ}$ rotations around three perpendicular axes. The operations inverted in this case are $4^1,
4^3, 2_{110}, 2_{1\={1}0}$. We obtain the operations: $\={4}^1, \={4}^3,
m_{(100)}, m_{(1\={1}0)}$; the resulting point group is $\={4}2m$. Altogether we can derive 10 groups, using the following scheme. (The subgroup utilized is shown in parentheses.)


432 $\relbar\kern-2pt\relbar$ (23) $\longrightarrow$ $\={4}3m$
622 $\relbar\kern-2pt\relbar$ (6) $\longrightarrow$ 6mm
622 $\relbar\kern-2pt\relbar$ (32) $\longrightarrow$ $\={6}m2$
422 $\relbar\kern-2pt\relbar$ (4) $\longrightarrow$ 4mm
422 $\relbar\kern-2pt\relbar$ (222) $\longrightarrow$ $\={4}2m$
6 $\relbar\kern-2pt\relbar$ (3) $\longrightarrow$ $\={6}$
32 $\relbar\kern-2pt\relbar$ (3) $\longrightarrow$ 3m
4 $\relbar\kern-2pt\relbar$ (2) $\longrightarrow$ $\={4}$
222 $\relbar\kern-2pt\relbar$ (2) $\longrightarrow$ mm2
2 $\relbar\kern-2pt\relbar$ (1) $\longrightarrow$ m

Altogether thirty two point groups are possible in three-dimensional space: 11 enantiomorphic; 11 centrosymmetric; and 10 non-enantiomorphic, non-centrosymmetric.

next up previous
Next: Appendix Up: Derivation of the Crystallographic Point Groups Previous: Groups containing more than one rotation

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