Lecture 12 Crystallography Internal Order and 2-D Symmetry Plane Lattices Planar Point Groups Plane Groups

Internal Order and Symmetry Repeated and symmetrical arrangement (ordering) of atoms and ionic complexes in minerals creates a 3-dimensional lattice array Arrays are generated by translation of a unit cell – smallest unit of lattice points that define the basic ordering Spacing of lattice points (atoms) are typically measured in Angstroms (= 10-10 m) About the scale of atomic and ionic radii

Two-Dimensional Plane Lattice Generating an 2D Lattice Array (Plane Lattice) involves translation of a motif in two directions; possible directions not unique Translation in two directions: x and y axes Angle between x and y axes is called g gamma Translation distance: a along x and b along y Replacing motifs with points (or nodes) creates a plane lattice Unit Cell defined by a choice of lengths and directions.

Symmetry Translations (Lattices) 1-D translations = a row A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary 1-D translations = a row

Symmetry Translations (Lattices) 1-D translations = a row A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary 1-D translations = a row a a is the repeat vector

Translations (Lattices) 2-D translations = a net Symmetry Translations (Lattices) 2-D translations = a net

Translations (Lattices) 2-D translations = a net Symmetry Translations (Lattices) 2-D translations = a net A 2-D Unit Cell Unit Cell: the basic repeat unit that, by translation only, generates the entire pattern

Translations (Lattices) 2-D translations = a net Symmetry Translations (Lattices) 2-D translations = a net b a Pick any point Every point that is exactly n repeats from that point is an equipoint to the original

There are 5 Types of Plane Lattices Memorize these names and rules Preferred

Symmetry Elements of Planar Motifs: Planar Point Groups Point groups have labels that are similar to Hermann Mauguin symbols. For example: 2mm shown has the a axis with a two fold rotational axis, and b and c have mirrors A point group is a group of geometric symmetries that keep at least one point fixed. 10 Possible symmetry combinations; called Planar Point Groups Limitations of rotational symmetries: (1,2,3,4, & 6) dark lines added “found mirrors”

Translations The lattice and point group symmetries interrelate, because both are properties of the overall symmetry pattern

Translations The lattice and point group symmetry interrelate, because both are properties of the overall symmetry pattern Choose: Smallest Most orthogonal Most in line with symmetry 2 Nodes per Lattice Vector Most Primitive (non-centered) Good unit cell choice.

Defining a 2-D Unit Cell Rules that help us Choose the: Smallest Most orthogonal Most in line with symmetry Use 2 Nodes per Lattice Vector Pick the Most Primitive (non-centered)

total 17 point groups

Translations There is a new 2-D symmetry operation when we consider translations The Glide Line, g: A combined reflection and translation repeat Step 2: translate Step 1: reflect (a temporary position)

There are 5 unique 2-D plane lattices. There are also 17 2-D Plane Groups that combine translations with compatible symmetry operations. The bottom row are examples of Plane Groups that correspond to each lattice type Note: p refers to a primitive cell, as opposed to c, a 2-end (opposite ends) centered cell. More on this in 3-D

17 Plane Groups 10 H-M Point Groups and 5 Lattices combine to form 17 Plane Groups.

Lecture 13 3-D Crystallography 3-D Internal Order & Symmetry Space (Bravais) Lattices Space Groups So far we examined the five 2-D plane lattices and combined them with the 10 planar point groups to generate the 17 2-D plane (space) groups. Next we study the 14 Bravais 3-D lattices and combine them with the 32 3-D point groups to generate 230 3-D space groups.

3-D Translations and Lattices Different ways to combine 3 axes Translations compatible with 32 3-D point groups (~ crystal classes) 32 Point Groups fall into 6 systems

3-D Translations and Lattices +b g b a Axial convention: “right-hand rule” Different ways to combine 3 axes Translations compatible with 32 3-D point groups (~ crystal classes) 32 Point Groups fall into 6 Crystal Systems

Unit Cell Types in 14 Bravais Lattices P – Primitive; nodes at corners only C – Side-centered; nodes at corners and in center of one set of faces F – Face-centered; nodes at corners and in center of all faces I – Body-centered; nodes at corners and in center of cell

On combining 7 Crystal Classes with 4 possible unit cell types we get 14 Bravais Lattices

P P I = C P C F I a b Triclinic ¹ b ¹ g ¹ c a b c Monoclinic a = g = 90 o ¹ b ¹ I = C c a P Orthorhombic a = b = g = 90 o ¹ C F I There are also several non-primitive lattice choices P = Primitive C = C-face Centered F = all-Face centered I = body-centered One might expect that each choice is possible for every lattice type, but not so. Why not ?

P I P F I (body-centered) a c Tetragonal a = b = g = 90 = a ¹ a 1 c P Tetragonal a = b = g = 90 o = a 2 ¹ I a 1 3 P Isometric a = b = g = 90 o = a 2 F I (body-centered) There are also several non-primitive lattice choices P = Primitive C = C-face Centered F = all-Face centered I = body-centered One might expect that each choice is possible for every lattice type, but not so. Why not ?

Crystal Axes Conventions Triclinic: No symmetry constraints. No reason to choose C (white) when can choose simpler P (blue) Do so by convention, so that all mineralogists do the same

Crystal Axes Conventions +b g b a Axial convention: “right-hand rule”

System Conventions

System Conventions

System Conventions

System Conventions

3-D Space Groups As in the 17 2-D Plane Groups, the 3-D point group symmetries can be combined with translations to create the 230 3-D Space Groups Also as in 2-D there are some new symmetry elements that combine translation with other operations Glides: Reflection + translation Screw Axes: Rotation + translation

A point group is a group of geometric symmetries that keep at least one point fixed. A space group is some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection, rotation and rotoinversion, and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.

230 Space Groups Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal Isometric Notation indicates lattice type (P,I,F,C) and Hermann-Mauguin notation for basic symmetry operations (rotation and mirrors) Screw Axis notation as previously noted Glide Plane notation indicates the direction of glide – a, b, c, n (diagonal) or d (diamond)