Crystal symmetry

Symmetry: What it means to you?

Symmetry: How it arrives?

• Derived from Greek ”συμμετρία” / “symmetria“: Aagreement in dimensions, due proportion, arrangement.
• In everyday language, it refers to a sense of harmonious and beautiful proportion and balance.
• In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling.

Symmetric arcades of a portico in the Mosque of Uqba, in Tunisia.

• Symmetry has developed into a huge subject, involving many branches of mathematics, especially geometry and group theory.

Maurits Cornelis Escher (17 June 1898 – 27 March 1972): Dutch graphic artist known for mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, and tessellations.

Relativity, 1953

Crystal Symmetry

• Symmetry defines the order resulting from how atoms are arranged and oriented in a crystal.
• Symmetry operators (there are 13 total) 🡪 actions which result in no change to the order of atoms in the crystal structure.
• Combining different operators gives point groups – which are geometrically unique units.
• Every crystal falls into some point group, which are segregated into 6 major crystal systems.

Crystallography & Symmetry

Robert Hooke (Micrographia, 1664), speculated that crystals have regular geometric shapes because they consist of regular sphere packings.

R. J. Haüy, (Traité de Cristallographie, 1822) explained crystals as stacks of blocks.

With the invention of the goniometer, mineralogists could measure the angles between crystal faces with great precision.

The goniometer that belonged to E. S. Federov, now in the St. Petersburg Mining Institute

Atlas der Krystallformen by Victor Goldschmid (pub. 1913 to 1923) contains 23606 crystal drawings and a short description of each drawing.

Symmetry-equivalent faces have the same labels. Interfacial angles are rendered precisely.

Translational Symmetry

• One can go from any location in the lattice to an identical location by following path composed of integer multiples of the vectors a and b.

A simple Cubic Structure

Simple cubic unit cell

Eight simple cubic unit cells

• Translational Symmetry—a move of one cell in each of 3 axis directions restores the structure

• In addition to the translational symmetry associated with a lattice, most materials have additional point symmetry applied to the basis.

• Point symmetry elements operate to change the orientation of structural motifs.

• A point symmetry operation does not alter at least one point that it operates on.

• Point symmetry elements include

– Rotation axes

– Mirror planes

– Rotation-inversion axes

Point symmetry elements

Rotational Symmetry Operators

• Rotation Axes (1, 2, 3, 4, or 6) – rotation around a rotation axis yields no change in lattice arrangement
• Angle of rotation = 360^o/n where n = Rotation axis notation
• For rotation axes 1, 2, 3, 4, 6, rotation angles are 360, 180, 120, 90, or 60^o

Rotational Symmetry

Al-Pd-Re single quasicrystal

Quasicrystals

• Nobel laureate Dan Shechtman describes the structure of quasicrystals, the discovery of which won him the Nobel Prize in Chemistry in 2011.

https://www.scientificamerican.com/video/the-strange-matter-of-quasicrystals-explained/

A two fold rotation

Three, four and six fold rotations

Rotational Symmetry

For a 3 fold rotation axis

Rotational symmetry of a Cubic unit cell

Rotational symmetry of a Cubic unit cell

3-Fold Rotation

2-Fold Rotation

4-fold Rotation

Mirror symmetry

• Mirror Planes (m) – reflection along a plane.

Examples of mirror symmetry

Leonardo da Vinci's 'Vitruvian Man' (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.

Mirror planes at different orientations

Mirror symmetry of a cube unit cell

Mirror symmetry of a cube unit cell

Inversion symmetry OR Center of symmetry

• Inversion (i) – symmetry with respect to a point, called an inversion center.

Inversion symmetry

Inversion symmetry for cubic unit cell

Implication of center of symmetry

Difference between mirror and inversion

Roto-inversion symmetry

Roto-inversion symmetry

The distinction between proper and improper operations

• Operators producing a ‘right-handed’ replica (proper) and producing a ‘left-handed’ or mirror image replica (improper)
• Left-handed and right-handed objects can not be superimposed by any combination of rotation or translation.

Point Symmetry and Roto-inversions

Symmetry elements of cube

Symmetry elements designation

• Symmetry element is a point of reference about which symmetry operations can take place.
• Symmetry elements can be identities, mirror planes, axes of rotation (both proper and improper), and center of inversion.

Unique symmetry elements

32 Point groups

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