The concept of ionic crystal symmetry

1.
The concept of symmetry

The symmetry of a geometric figure means that the figure can be restored to its original shape after a certain geometric element performs a certain operation; the symmetry of a molecule or ion means that the molecule or ion can be restored to its original state after a certain geometric element performs a certain operation , That is, all atoms or ions can overlap with the same kind of atoms or ions
.

For example, after the triangular molecule SO ₃ rotates 180° around the high rotation on the bottom side, the molecule recovers, that is, the SO ₃ molecule has symmetry with respect to the high rotation on the bottom side

Symmetry operation refers to the operation that does not change the distance between any two points in the graph, including rotation, reflection and inversion.
For example, the operation performed by virtue of the height on the bottom edge is rotation
.

The geometric elements on which the symmetry operation depends are called symmetry elements, including the axis of symmetry, the plane of symmetry, and the center of symmetry.

1) Rotation and symmetry axis

Triangular molecule SO ₃ rotates around a straight line that passes through the center (three high intersection points) and is perpendicular to the plane where the triangle is located .
The SO ₃ molecule recovers once every 120° of rotation
.

That is, the SO ₃ molecule is restored once per rotation , or the SO ₃ molecule is restored 3 times per revolution

If a molecule or figure rotates around the axis of symmetry and is restored n times, the axis of symmetry is called the n-fold symmetry axis or the n-fold axis
.

For example, ICI4- is a square structure that rotates around a straight line that passes through the diagonal intersection of the square and is perpendicular to the plane where the square is located.

2) Reflection and symmetry plane

The SF6 molecule with an octahedral structure passes through the 4 F atoms and S atoms in the same plane to form the symmetry plane of the octahedral molecule.
After plane mirror imaging is performed on this plane, the SF ₆ molecular pattern is restored
.

The plane mirror imaging operation is called reflection, and the plane on which the reflection operation relies is a plane of symmetry (Figure 6-2)

Figure 6-2 Schematic diagram of SO ₃ , ICI ₄ ^- and SF ₆ structure

3) Inversion and center of symmetry

For SF ₆ molecules with an octahedral structure , the central atom S is the symmetry center of the molecule
.

That is, all the points (atoms) in the graph are connected to the center and then extended to equal distances in the opposite direction, and the graph is restored