Groups containing more than one rotation axis

Let us take two symmetry operations: the first one corresponding to a rotation of an angle $\alpha$ around one axis, and the second one to a rotation of an angle $\beta$ around another axis. Let us call $\omega$ the angle between the two axes. Then, the product of the two rotation matrices is also a rotation matrix. The rotation axis of the product matrix is, in general, oriented in a different way than the other two. We can obtain the matrices corresponding to symmetry operations in the following manner: for a given orthonormal vector basis A1A2A3 (Fig. 1), the symmetry operation corresponding to a counter-clockwise rotation of an angle $\alpha$ around the A3 axis is represented by the matrix:


 
Figure 1
\begin{figure} \includegraphics {fig1.ps} \end{figure}

\begin{displaymath} R_3=\left[\begin{array} {ccc} \cos \alpha&-\sin \alpha& 0\\ \sin \alpha&\cos \alpha&0\\ 0&0&1\end{array}\right]\end{displaymath}

If, on the other hand, the rotation takes place around the $A^{\prime}_3$ axis, which lies on the plane determined by A1 and A3 and forms the angle $\omega$ with A3 (Fig. 1), the corresponding symmetry operation is given by:

\begin{displaymath} R_{3^{\prime}} = R_2 \cdot R_3 \cdot R_2^{-1}\end{displaymath}

where:

\begin{displaymath} R_2 = \left[\begin{array} {ccc} \cos \omega&0&\sin \omega\\ 0&1&0\\ -\sin \omega&0&\cos \omega\end{array}\right]\end{displaymath}

represents a counter-clockwise rotation of an angle $\omega$ around A2. We have to bear in mind, in fact, that $R_2 \cdot R_3 \cdot R_2^{-1}$ represents the symmetry operation R3 as it is transformed by the operation R2.

In explicit form we have:

\begin{displaymath} R_{3^{\prime}} = \left[\begin{array} {ccc} \cos \omega&0&\si... ...in \omega\\ 0&1&0\\ \sin \omega&0&\cos \omega\end{array}\right]\end{displaymath}

\begin{displaymath} = \left[\begin{array} {ccc} \cos^2 \omega \cos \alpha + \sin... ...ha&\sin^2 \omega \cos \alpha + \cos^2 \omega\end{array}\right].\end{displaymath}

The counter-clockwise rotation of an angle $\beta$ around the A3 axis is given by the matrix:

\begin{displaymath} R_{3^{\prime\prime}}=\left[\begin{array} {ccc} \cos \beta&-\sin \beta&0\\ \sin \beta&\cos \beta&0\\ 0&0&1\end{array}\right].\end{displaymath}

The combination of two rotations (one of an angle $\beta$ around the A3 axis and the other one of an angle $\alpha$ around the $A_{3^{\prime}}$ axis which forms an angle $\omega$ with A3 and lies on the plane A1A3) is also a rotation, represented by the R matrix, given by:

\begin{displaymath} R = R_{3^{\prime}}R_{3^{\prime\prime}}\end{displaymath}

\begin{displaymath} =\left[\begin{array} {ccc} \cos^2 \omega \cos \alpha + \sin^... ...sin^2 \omega \cos \alpha + \cos^2 \omega\end{array}\right]\cdot\end{displaymath}

\begin{displaymath} \cdot\left[\begin{array} {ccc} \cos \beta& -\sin \beta& 0\\ \sin \beta& \cos \beta& 0\\ 0&0&1\end{array}\right]\end{displaymath}

The trace of the R matrix given by the sum of the elements of the principal diagonal, is:

\begin{displaymath} \left[\begin{array} {ccc} \cos^2 \omega \cos \alpha \cos \be... ...-&-&\sin^2 \omega \cos \alpha + \cos^2 \omega\end{array}\right]\end{displaymath}

i.e.

\begin{displaymath} \mbox{trace} = \cos^2 \omega(\cos \alpha \cos \beta + 1) + \... ... -2 \cos \omega(\sin \alpha \sin \beta)+ \cos \alpha \cos \beta\end{displaymath}

\begin{displaymath} \phantom{MMMMMN} = \cos^2 \omega(\cos \alpha \cos \beta - \c... ...\sin \beta + \cos \alpha \cos \beta + \cos \alpha + \cos \beta.\end{displaymath} (12)

This rotation R must be compatible with the lattice as well. Therefore, the value of the trace, invariant with respect to a base system transformation, must be an integer. The possible values of the trace are: +3, +2, +1, 0, -1. These numbers give the order of the resulting rotation axis.

When we assign to $\alpha$ and $\beta$ in the expression (12) all the possible values, depending upon the order of the rotation axis, we obtain the second degree equations in $\cos \omega$ listed in Table 1, where m is an integer representing the trace of the R matrix.

In Table 1 those solutions for which $\cos \omega$ is greater than 1 are obviously not shown, as well as those that do not give as a result both $\omega$and 180$^{\circ}$ - $\omega$. This last condition is evidently necessary if two axes intersect.

On the basis of the results listed in the table, we can obtain the axis combinations shown in Fig. 2, i.e. the point groups 222, 32, 422, 622, 23, 432.


 
Figure 2
\begin{figure} \includegraphics {fig2.ps} \end{figure}

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