Crystal Symmetry Groups PDF

## Chap 2: Groupes ponctuels de symétrie et groupes d'espace ### Introduction - The macroscopic properties of a crystal depend generally on direction, for example : conductivity, electrical susceptibility, piezoelectricity etc. - In certain equivalent directions, one can find the same macroscopic...

## Chap 2: Groupes ponctuels de symétrie et groupes d'espace ### Introduction - The macroscopic properties of a crystal depend generally on direction, for example : conductivity, electrical susceptibility, piezoelectricity etc. - In certain equivalent directions, one can find the same macroscopic properties. ### Example : Conductivity - The conductivity is the same in the two equivalent directions **Δ** and **Δ'** for the same electric field strength **E**. - These directions can be obtained from one another by symmetry operations. ### Symmetry operations - **Symmetry operation:** A geometric operation that transforms an object into itself without any change. - **Point symmetry operations:** Leave at least one point fixed in the crystal. - They include rotations, inversions, roto-inversions (roto-inversion), and roto-reflections (roto-reflection). - The set of point symmetry operations of a crystal forms a **point group** also called **symmetry class**. - There are 32 different point groups. - **Symmetry of position:** The arrangement of atoms inside the crystal lattice. ### Example : Rotation helicoïdal (helicoïdal axis) - The set of symmetry operations combined with translation form a **space group**. - There are 230 space groups (or spatial groups). ### Notation Internationale de Hermann-Maugin (adopted in 1935) - **X:** Direct axis of order X. - **X:** Inverse axis of order X. - **Xm:** Direct axis of order X and a mirror perpendicular to X. - **Xm:** Direct axis of order X and a mirror containing X. - **X2:** Direct axis of order X and a direct axis of order 2 perpendicular to X. - **X3:** Direct axis of order X and a direct axis of order 3 making an angle of 54° with X. - **Xm:** Direct axis of order X, a mirror perpendicular to X, and a mirror containing X. ### Definition of a group - A group is a set of elements (symmetry operations) P, Q, R, S, T, .. that meet the following conditions: - **Closure:** If R ∈ G and S ∈ G then R·S = T, where T ∈ G. - **Associativity:** R·(S·T) = (R·S)·T. - **Neutral element:** G contains a neutral element E, such that ∀R∈G ; R·E = E·R = R. - **Inverse element:** For every element R ∈ G, there exists a corresponding inverse element R-1 ∈ G, such that: R-1·R = R·R-1 = E. - **Order of a group:** The number of elements in a group. ### Classification of point groups in crystal systems | Crystal system | Minimal symmetry | Point groups | Laue groups | |---|---|---|---| | Triclinic | 1 (or 1) | 1, 1 | 1 | | Monoclinic | One axis 2 (or 2 = m) | 2, m, 2/m | 2, m | | Orthorhombic | Three axes 2 (perpendicular) | 222, mm2, mmm | mmm | | Tetragonal | One axis 4 (or 4) | 4, 4, 4/m, 422, 4mm, 42m, mm | 4, 4/m, 4/mmm | | Trigonal (or Rhombohedral) | One axis 3 (or 3) | 3, 3, 32, 3m, ¯3m | 3, 3m | | Hexagonal | One axis 6 (or 6) | 6, 6, 6/m, 6mm, 622, 6m2, 6/mmm | 6, 6/m, 6/mmm | | Cubic | Four axes 3 (or 3) | 23, m3, 432, 43m | m3, m3m | ### Theorems - The combination of symmetry elements leads to the creation of new symmetry elements, such as: - **Theorem 1:** The presence of a two-fold axis and a center of symmetry implies the existence of a mirror plane perpendicular to the axis and passing through the center of symmetry. - It also implies the existence of a center of symmetry located at the intersection of the axis and the mirror plane. - **Theorem 2:** If there exists an n-fold axis and a mirror plane containing the axis, n mirror planes exist perpendicular to the axis and making angles of π/n between them. - **Theorem 3:** If there are n two-fold axes in a plane that intersect at a point, then the normal to that plane is an n-fold axis. - If there exists an n-fold axis and a two-fold axis perpendicular to it, then there exists n-1 additional two-fold axes, each making an angle of π/ n with the others. - **Theorem 4:** The existence of an inverse n-fold axis (n) implies specific existence conditions for various symmetry elements. ### Stereographic projection - This projection is a method for visualising the symmetry relations of a crystal. - The crystal is centered at the point **O** of a sphere. - The North Pole (PN) and South Pole (PS) are defined. - A normal to each crystal face is drawn through the center **O**. - The intersection of the normal with the sphere is labeled **P**. - The stereographic projection of **P** is the point **p** where the line connecting **P** and **PS** intersects the equatorial plane. - The image of a point **R** in the southern hemisphere is projected onto the equatorial plane via the line connecting **R** and **PN**. - The image of point **r** can be represented with a cross on the equatorial plane. ### Figures 2.3, 2.9, 2.10, 2.11: - These figures show examples of rotations, roto-reflections, roto-inversions, helicoïdal axes, and reflection with glide. ### Table of the 230 space groups - This table classifies all the space groups based on their crystal system, Hermann-Maugin notation, and their lattice type. ### Figure 15: - This figure shows examples of symmetry elements, such as 4-fold, 63-fold, 64-fold and 65-fold axes.