Rotary-Inversion Symmetry: Mirror Planes

Rotary-Inversion Symmetry

II. Mirror Planes

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Mirror Symmetry

A more common everyday example of a symmetry operation that changes the handedness of an object is a mirror. The action of a mirror is illustrated in the figure below where the black thick vertical line indicates the position of a mirror perpendicular to the plane of the screen.

In an everyday mirror, the real object on one side is equidistant to the virtual image on the other, the only difference between object and image being the change of handedness. Mirror symmetry in crystallography has the same properties, but relates real objects. In the above figure, the mirror is shown perpendicular to the conventional Cartesian X-axis direction. If one chooses the origin of space to lie on the mirror plane, then the coordinates of the molecule on the left-hand side are related to those on right-hand side by the symmetry operator -x,y,z.

What is the symmetry operator corresponding to a mirror plane perpendicular to the Cartesian Y axis?

Crystallographers use the notation "m" for mirror planes. The operation of mirror symmetry is equivalent to a rotation of 180° about an axis perpendicular to the mirror plane followed by inversion at the point where axis and plane intersect. The notation for this operation is "-2", i.e. a mirror plane is simply the special case of a twofold rotary-inversion axis.

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Author(s): Jeremy Karl Cockcroft