Proyección estereográfica: elección de los ejes cristalográficos

How to Choose Crystallographic Axes

Identifying Symmetry Elements in Polyhedra

• The process begins with identifying all symmetry elements present in the polyhedron, which is crucial for stereographic projection.

• It is essential to determine if the polyhedron has symmetry planes and their geometric relationships, including angles between them.

• The combinations of symmetry elements are limited and correspond to one of the 32 point groups of symmetry.

Classifying Polyhedra into Crystal Systems

• Once all symmetry elements are identified, the polyhedron can be classified within one of the seven crystal systems.

• For example, in the orthorhombic system, there are only three point groups; group 222 features three binary axes at right angles without any symmetry planes.

Choosing Crystallographic Axes

• The largest binary axis is designated as axis c, while the smallest becomes axis a. This standardization ensures uniformity during practical exercises.

• If a polyhedron has three mutually perpendicular symmetry planes and corresponding binary axes, it belongs to point group 2/m2/m2/m.

Characteristics of Different Crystal Systems

• Cubic system crystals (group 2/m-3) also have three perpendicular symmetry planes but include four ternary inversion axes and tend to exhibit equidimensional habits.

• In contrast, rhombohedral crystals typically appear tabular or prismatic due to their structural characteristics.

Summary of Stereographic Projection Mechanics

• When encountering two perpendicular symmetry planes with a single binary axis where they intersect, classify it under point group 2mm.

• For group 2mm polyhedra, assign crystallographic axis c along the binary axis and choose axes a and b based on directions perpendicular to the symmetry planes.

Steps for Stereographic Projection

• The procedure involves first identifying all present symmetry elements followed by determining the corresponding point group and crystal system.