THE 230 SPATIAL GROUPS (3D CRYSTALLOGRAPHIC GROUPS)

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CRYSTAL SYSTEMS (7)

1. TRICLINIC

• Lower symmetry
• 2 groups : P1, P-1

2. MONOCLINIC

• A binary/planar axis
• 13 groups : P2, P2₁, C2, Pm, Pc, Cm, Cc , P2/m, P2₁/m, C2/m, P2/c, P2₁/c, C2/c

3. ORTHORHOMBIC

• Three perpendicular axes
• 59 groups : P222, P2221, P21212, etc.

4. TETRAGONAL

• A quaternary axis
• 68 groups : P4, P4₁, P4₂, P4₃, I4, etc.

5. TRIGONAL (RHOMBOHEDRAL)

• A ternary axis
• 25 groups : R3, R-3, R32, etc.

6. HEXAGONAL

• A senary axis
• 27 groups : P6, P6₁, P6₅, etc.

7. CUBIC

• Maximum symmetry
• 36 groups : P23, F23, I23, P2₁3, etc.

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TYPES OF BRAVAIS NETWORKS (14)

PRIMITIVES (P)

• Points only at vertices

FACE-CENTERED (F)

• Points at vertices + center of faces

BODY-CENTERED (I)

• Points at vertices + cell center

BASE-CENTERED (A, B, C)

• Points at vertices + center of specific faces

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3D SYMMETRY OPERATIONS

TRANSFERS

• 3D Vectors

ROTATIONS

• 1, 2, 3, 4, 6 (own axes)
• 31, 32, 41, 42, 43, 61, 65 (screw)

REFLECTIONS

• m (mirror)
• a, b, c, n, d (sliding planes)

INVESTMENT

• -1 (investment center)
• -3, -4, -6 (turnover-investment)

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NOTABLE EXAMPLES BY SYSTEM

CUBIC SYSTEM (36 groups)

• Fm-3m - Diamond Structure
• Pm-3m - Simple Cubic
• Ia-3d - Garnet structure

HEXAGONAL SYSTEM (27 groups)

• P6₃/mmc - Graphite, magnesium
• P6₃mc - Wurtzite ( ZnS )

TETRAGONAL SYSTEM (68 groups)

• I4₁/ amd - Rutile ( TiO ₂)
• P42/ mmm - Cassiterite

ORTHORHOMBIC SYSTEM (59 groups)

• Pnma - Olivine, perovskite
• Cmcm - β- Tin Structure

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HERMANN-MAUGUIN NOTATION

Format: [Network ][ Main axes][Secondary axes][Tertiary axis]

Example: Pnma

• P → Primitive network
• n → Slip plane (100)
• m → Mirror (010)
• a → Sliding plane (001)

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COMPACT DISTRIBUTION

TRICLINIC: 2 groups ( 0.9%)

MONOCLINIC: 13 groups ( 5.7%)

ORTHORHOMBIC: 59 groups ( 25.7%)

TETRAGONAL: 68 groups ( 29.6%)

TRIGONAL: 25 groups ( 10.9%)

HEXAGONAL: 27 groups ( 11.7%)

CUBIC: 36 groups ( 15.7%)

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SCIENTIFIC APPLICATIONS

CRYSTALLOGRAPHY

• X-ray structural determination
• Neutron/electron diffraction

MATERIALS SCIENCE

• Electrical/mechanical properties
• Phase transformations

MINERALOGY

• Mineral identification
• Structure-property relationship

SOLID CHEMISTRY

• Design of new materials
• Heterogeneous catalysis

BIOLOGY

• Protein crystallography
• Virus structure

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HISTORICAL DATA

• 1891 : Fedorov (Russia) and Schönflies (Germany) independently derived the 230 groups
• 1924 : Hermann-Maugin introduce current notation
• 2011 : "Project 230" is completed - visualization of all groups

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KEY REFERENCES

• International Tables for Crystallography (Vol. A)
• Fedorov-Schoenflies Groups
• Hermann-Maugin (International) notation

Now a flowchart for 2D and another for 3D to determine, by answering questions, which group a figure belongs to.

FLOWCHART: DETERMINATION OF CRYSTALLOGRAPHIC GROUPS

PART 1: PLANE GROUPS (2D) - 17 GROUPS

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PART 2: SPATIAL GROUPS (3D) - 230 GROUPS

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QUICK GUIDE TO KEY QUESTIONS

FOR 2D:

1. What is the rotation order? (6, 4, 3, 2, 1)
2. Does it have reflective mirrors?
3. Are they pure mirrors or sliding mirrors?
4. What is the underlying network like?
5. How are the elements of symmetry related?

FOR 3D:

1. What is the crystal system? (measure angles and lengths)
2. What elements of point symmetry exist?
3. Are there helical operations (screw shafts)?
4. Are there slip planes?
5. What type of Bravais network does it feature?

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PRACTICAL RECOMMENDATIONS

For laboratory use:

1. Begin with the crystal system (angles between axes)
2. Identify the Bravais network (P, I, F, A, B, C)
3. Look for main point symmetries
4. Check operations with translation (helical/sliding)
5. Consult international tables to confirm

Practical 2D example:

• Pattern with hexagons → Order 6 → Mirrors? → Yes → p6m

Practical 3D example:

• Cubes with centered faces → Cubic → Lattice F → Complete symmetry? → Yes → Fm-3m

Symmetry in Crystallography

1. General concept of symmetry in crystallography

Crystallography is based on the idea of invariance : a crystal can be superimposed on itself after certain geometric transformations. These transformations are isometries of Euclidean space: rotations, reflections, inversions, and, in ideal crystals, translations .

Symmetry allows:

· Describe a crystal in a minimal and non-redundant way.

· Classify structures.

· Relate physical properties to structure (Curie's law).

2. Symmetries of finite objects (point symmetry)

In 2D there are only:

• Rotations

· Reflections about a straight line

In 3D there are:

• Rotations

· Reflections regarding a plan

• Rotational inversions (rotation + inversion at a point)

A symmetry element is the set of points that remain fixed under the operation (point, line, or plane).

Molecular examples:

· H₂O: 180° rotation axis.

· XeF ₄: axis 4, several axes 2, planes σh , σv , σd .

· SF₆: investment center.

3. Symmetry in crystals

A crystal is conceived as an infinite and periodic object . The periodicity is described by a regime of translations that generates a Bravais lattice .

3.1. Crystal lattice

· Infinite set of congruent points.

· Each point has the same environment.

· It is generated using three basis vectors.

· The maille (unit cell) is the parallelogram/parallelepiped generated by those vectors.

3.2. Motif

Set of atoms within the unit cell. Crystal = Lattice + Motif .

4. Fundamental historical laws

1. Constancy of angles ( Sténon 1669, Romé de l'Isle 1772): The angles between faces of a crystal of the same species are constant.

2. Law of rational indices ( Haüy 1774): The faces of a crystal can be described by integer indices (origin of Miller indices).

3. Bravais ' postulate (1849) : The crystal structure is periodic and can be described by a three-dimensional network.

5. External symmetry of crystals

The crystals exhibit:

· Planes of symmetry

• Rotation axes (only orders 1, 2, 3, 4 and 6)

• Investment centers

The combination of these operations generates the 32 crystallographic classes (point groups).

6. Crystal systems and Bravais lattices

There are 7 crystal systems , defined by the relationships between the cell parameters:

1. Cubic

2. Tetragonal

3. Orthorhombic

4. Trigonal / Rhombhohedral

5. Monoclinical

6. Hexagonal

7. Triclinic

Each system supports certain centering modes : P (primitive), I (body-centered), F (face-centered), A/B/C (base-centered).

Combining + centered systems → 14 Bravais networks .

7. Compatibility between point symmetry and translation

The frequency restricts the permitted operations:

· Only axes 1, 2, 3, 4, 6 are compatible with a network.

· The combination of operations can generate new ones (for example, an axis pair + center → perpendicular plane).

8. Space Groups

A space group combines:

• Point symmetries

· Network translations

• Irreducible operations :

or Helical axes (fractional rotation + translation)

or Glide planes (reflection + parallel translation)

They are described using the Hermann – Mauguin notation (P2₁2₁2₁, C2/c, Fm3m…).

There are 230 space groups .

9. Helical shafts and slip planes

Helical shafts n ₘ

Rotation of order n + translation m/n of the cell vector. Examples: 2₁, 3₁, 4₃, 6₅…

Sliding planes

Reflection + translation of half a cell parallel to the plane.

10. Practical use of space groups

They allow:

· Determine equivalent positions of atoms.

· Classify crystalline structures.

· Interpret diffraction patterns.

• Identify general and special positions (international tables).

11. Conclusion

We have a comprehensive view of symmetry in crystallography , from basic concepts to the mathematical structure of space groups , including:

• Molecular symmetry

• Networks and cells

· Crystal systems

· Bravais Networks

• Specific and spatial operations

• Helical shafts and slip planes

· Classification into 32 specific classes and 230 spatial groups

It is a rigorous synthesis of the geometry that underlies the structure of crystals.