Polar Point Groups

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Definition

Those point groups for which every operation leaves more than one common point unmoved are known as the polar point groups. The crystallographic 3-dimensional polar point groups are listed in the table below:

Consider the point group 2. All points on the 2-fold axis are left unmoved by the operation of the 180° rotation. A straight line joining points which remain unmoved defines a unique direction in a polar point group. This unique direction is not repeated by the symmetry of the point group. The twofold axis in point group 2 is such a direction. (Note that for point group 1 all directions are unique and that for point group m all directions parallel to the mirror plane are unique.)

The implications of this will be discussed in more detail when dealing with polar space groups. It is sufficient to state at this stage that crystals whose space-group symmetry is associated with one of the polar point groups do not have their origin uniquely fixed in space by the symmetry elements of the space group. For example, in point group 222 all of the 2-fold axes intersect in the same point. By contrast, in point group 2, however, no such point can be defined by the symmetry element so that the origin is not uniquely fixed in space. This origin problem is important in structure analysis and refinement. You will see from the table that this problem does not occur in the cubic crystal system.

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