GEOMETRY OF CRYSTALS Space Lattices Crystal Structures Symmetry, Point Groups and Space Groups Acknowledgments: Prof. Rajesh Prasad for a lot of things

Crystal = Lattice + Motif The language of crystallography is one succinctness Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated with each lattice point

An array of points such that every point has identical surroundings Space Lattice An array of points such that every point has identical surroundings In Euclidean space  infinite array We can have 1D, 2D or 3D arrays (lattices) or Translationally periodic arrangement of points in space is called a lattice

A 2D lattice

Crystal = Lattice + Motif Translationally periodic arrangement of points Translationally periodic arrangement of motifs Crystal = Lattice + Motif Lattice  the underlying periodicity of the crystal Basis  atom or group of atoms associated with each lattice points Lattice  how to repeat Motif  what to repeat

Lattice Motif + 

= Crystal                               Courtesy Dr. Rajesh Prasad

Cells A cell is a finite representation of the infinite lattice Instead of drawing the whole structure I can draw a representative part and specify the repetition pattern A cell is a finite representation of the infinite lattice A cell is a parallelogram (2D) or a parallelopiped (3D) with lattice points at their corners. If the lattice points are only at the corners, the cell is primitive. If there are lattice points in the cell other than the corners, the cell is nonprimitive.

Nonprimitive cell Primitive cell Primitive cell Courtesy Dr. Rajesh Prasad

Nonprimitive cell Primitive cell Primitive cell Double Triple Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!!

Centred square lattice = Simple/primitive square lattice Nonprimitive cell Primitive cell b a 4- fold axes Shortest lattice translation vector  ½ [11]

Centred rectangular lattice Maintains the symmetry of the lattice  the usual choice Nonprimitive cell Primitive cell Lower symmetry than the lattice  usually not chosen 2- fold axes

Centred rectangular lattice Simple rectangular Crystal Primitive cell Not a cell MOTIF Shortest lattice translation vector  [10] Courtesy Dr. Rajesh Prasad

Cells- 3D In order to define translations in 3-d space, we need 3 non-coplanar vectors Conventionally, the fundamental translation vector is taken from one lattice point to the next in the chosen direction With the help of these three vectors, it is possible to construct a parallelopiped called a CELL

Different kinds of CELLS Unit cell A unit cell is a spatial arrangement of atoms which is tiled in three-dimensional space to describe the crystal. Primitive unit cell For each crystal structure there is a conventional unit cell, usually chosen to make the resulting lattice as symmetric as possible. However, the conventional unit cell is not always the smallest possible choice. A primitive unit cell of a particular crystal structure is the smallest possible unit cell one can construct such that, when tiled, it completely fills space. Wigner-Seitz cell A Wigner-Seitz cell is a particular kind of primitive cell which has the same symmetry as the lattice.

SYMMETRY SYMMETRY OPERATOR If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation. SYMMETRY OPERATOR Given a general point a symmetry operator leaves a finite set of points in space A symmetry operator closes space onto itself

Symmetry operators Translation Rotation Symmetries Type I Takes object to same form → Proper Type I Rotation Symmetries Roto- reflection Mirror Type II Inversion Takes object to enantiomorphic form → improper Roto- inversion Classification based on the dimension invariant entity of the symmetry operator Operator Dimension Inversion 0D Rotation 1D Mirror 2D

Symmetry operators Symmetries Rotation Mirror Macroscopic Inversion Influence the external shape of the crystal Macroscopic Inversion Symmetries Screw Axes Microscopic Glide Reflection Do not Influence the external shape of the crystal

Ones with built in translation Minimum set of symmetry operators required R  Rotation G  Glide reflection R  Roto-inversion S  Screw axis Ones acting at a point Ones with built in translation

=180 n=2 2-fold rotation axis =120 n=3 3-fold rotation axis If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where =180 n=2 2-fold rotation axis =120 n=3 3-fold rotation axis

=90 n=4 4-fold rotation axis =60 n=6 6-fold rotation axis The rotations compatible with translational symmetry are  (1, 2, 3, 4, 6)

Symmetries acting at a point Point group symmetry of Lattices → 7 crystal systems Symmetries acting at a point R  R 32 point groups Along with symmetries having a translation G + S 230 space groups Space group symmetry of Lattices → 14 Bravais lattices R + R → rotations compatible with translational symmetry (1, 2, 3, 4, 6)

Crystal = Lattice (Where to repeat) + Motif (What to repeat) Previously Crystal = Lattice (Where to repeat) + Motif (What to repeat) a = a +

Space group (how to repeat) + Asymmetric unit (Motif’: what to repeat) Crystal = Space group (how to repeat) + Asymmetric unit (Motif’: what to repeat) Now a = Glide reflection operator a + Usually asymmetric units are regions of space within the unit cell- which contain atoms

a mmm Three mirror planes mirror glide reflection Progressive lowering of symmetry in an 1D lattice  illustration using the frieze groups Consider a 1D lattice with lattice parameter ‘a’ Asymmetric Unit Unit cell a mmm Three mirror planes The intersection points of the mirror planes give rise to redundant inversion centres mirror glide reflection

Decoration of the lattice with a motif  may reduce the symmetry of the crystal 1 mmm Decoration with a “sufficiently” symmetric motif does not reduce the symmetry of the lattice 2 mm Loss of 1 mirror plane

3 mg 4 ii 2 inversion centres Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centre the translational symmetry has been reduced to ‘2a’ 4 ii 2 inversion centres

5 m 1 mirror plane 6 g 1 glide reflection translational symmetry of ‘2a’ 7 No symmetry except translation

Redundant mirrors which need not be drawn Effect of the decoration  a 2D example Two kinds of decoration are shown  (i) for an isolated object, (ii) an object which can be an unit cell. 4mm Redundant mirrors which need not be drawn Redundant inversion centre Can be a unit cell for a 2D crystal 4mm Decoration retaining the symmetry

mm m m

4 No symmetry If this is an unit cell of a crystal → then the crystal would still have translational symmetry

Lattices have the highest symmetry  Crystals based on the lattice can have lower symmetry

Amorphous arrangement No unit cell Unit cell of Triclinic crystal

Object with bilateral symmetry Positioning a object with respect to the symmetry elements mmm Three mirror planes The intersection points of the mirror planes give rise to redundant inversion centres Left handed object Right handed object Object with bilateral symmetry

Positioning a object with respect to the symmetry elements General site  8 identiti-points On mirror plane (m)  4 identiti-points On mirror plane (m)  4 identiti-points Site symmetry 4mm  1 identiti-point Note: this is for a point group and not for a lattice  the black lines are not unit cells

Positioning of a motif w. r Positioning of a motif w.r.t to the symmetry elements of a lattice  Wyckoff positions A 2D lattice with symmetry elements

Number of Identi-points Multi-plicity Wyckoff letter Site symmetry Coordinates 8 g Area 1 (x,y) (-x,-y) (-y,x) (y,-x) (-x,y) (x,-y) (y,x) ((-y,-x) 4 f Lines ..m (x,x) (-x,-x) (x,-x) (-x,x) e .m. (x,½) (-x, ½) (½,x) (½,-x) d (x,0) (-x,0) (0,x) (0,-x) 2 c Points 2mm. (½,0) (0,½) b 4mm (½,½) a (0,0) f g e b d a c Any site of lower symmetry should exclude site(s) of higher symmetry [e.g. (x,x) in site f cannot take values (0,0) or (½, ½)] Number of Identi-points

g e b d a c d e f f Exclude these points Exclude these points

Bravais Space Lattices  some other view points Conventionally, the finite representation of space lattices is done using unit cells which show maximum possible symmetries with the smallest size. Considering Maximum Symmetry, and Minimum Size Bravais concluded that there are only 14 possible Space Lattices (or Unit Cells to represent them). These belong to 7 Crystal systems Or  the technical definition  There are 14 Bravais Lattices which are the space group symmetries of lattices

Bravais Lattice A lattice is a set of points constructed by translating a single point in discrete steps by a set of basis vectors. In three dimensions, there are 14 unique Bravais lattices (distinct from one another in that they have different space groups) in three dimensions. All crystalline materials recognized till now fit in one of these arrangements. or In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations. A Bravais lattice looks exactly the same no matter from which point one views it.

Arrangement of lattice points in the unit cell. & No Arrangement of lattice points in the unit cell & No. of Lattice points / cell Position of lattice points Effective number of Lattice points / cell 1 P 8 Corners = 8 x (1/8) = 1 2 I 8 Corners + 1 body centre = 1 (for corners) + 1 (BC) 3 F 8 Corners + 6 face centres = 1 (for corners) + 6 x (1/2) = 4 4 A/ B/ C 8 corners + 2 centres of opposite faces = 1 (for corners) + 2x(1/2) = 2

14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices Cubic P I F Tetragonal P I Orthorhombic P I F C Hexagonal P Trigonal P Monoclinic P C Triclinic P Courtesy Dr. Rajesh Prasad

14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices Cubic P I F C Tetragonal P I Orthorhombic P I F C Hexagonal P Trigonal P Monoclinic P C Triclinic P Courtesy Dr. Rajesh Prasad

Cubic F  Tetragonal I The symmetry of the unit cell is lower than that of the crystal

14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices Cubic P I F C Tetragonal P I F Orthorhombic P I F C Hexagonal P Trigonal P Monoclinic P C Triclinic P x Courtesy Dr. Rajesh Prasad

The following 4 things are different Lattice Motif Symmetry of the Crystal Unit Cell Eumorphic crystal (equilibrium shape and growth shape of the crystal) The shape of the crystal corresponds to the point group symmetry of the crystal

FCT = BCT

Crystal system The crystal system is the point group of the lattice (the set of rotation and reflection symmetries which leave a lattice point fixed), not including the positions of the atoms in the unit cell. There are seven unique crystal systems.

Polar coordinates (, ) Concept of symmetry and choice of axes (a,b) The centre of symmetry of the object does not coincide with the origin Polar coordinates (, ) The type of coordinate system chosen is not according to the symmetry of the object Mirror Our choice of coordinate axis does not alter the symmetry of the object (or the lattice) Centre of Inversion

THE 7 CRYSTAL SYSTEMS

N is the number of point groups for a crystal system

(Truncated Octahedron) Cubic Crystals a = b= c  =  =  = 90º Simple Cubic (P) Body Centred Cubic (I) – BCC Face Centred Cubic (F) - FCC Vapor grown NiO crystal [2] Tetrakaidecahedron (Truncated Octahedron) Pyrite Cube [1] [1] Fluorite Octahedron Garnet Dodecahedron [1] [1] http://www.yourgemologist.com/crystalsystems.html [2] L.E. Muir, Interfacial Phenomenon in Metals, Addison-Wesley Publ. co.

Tetragonal Crystals a = b  c  =  =  = 90º Simple Tetragonal Body Centred Tetragonal Zircon [1] [1] [1] [1] http://www.yourgemologist.com/crystalsystems.html

Orthorhombic Crystals a  b  c  =  =  = 90º Simple Orthorhombic Body Centred Orthorhombic Face Centred Orthorhombic End Centred Orthorhombic [1] Topaz [1] [1] http://www.yourgemologist.com/crystalsystems.html

Hexagonal Crystals a = b  c  =  = 90º  = 120º Simple Hexagonal [1] Corundum [1] http://www.yourgemologist.com/crystalsystems.html

5. Rhombohedral Crystals a = b = c  =  =   90º Rhombohedral (simple) [1] [1] Tourmaline [1] http://www.yourgemologist.com/crystalsystems.html

Monoclinic Crystals a  b  c  =  = 90º   Simple Monoclinic End Centred (base centered) Monoclinic (A/C) [1] Kunzite [1] http://www.yourgemologist.com/crystalsystems.html

7. Triclinic Crystals a  b  c      Simple Triclinic Amazonite [1] Amazonite [1] http://www.yourgemologist.com/crystalsystems.html

Polar coordinates (, ) Concept of symmetry and choice of axes (a,b) The centre of symmetry of the object does not coincide with the origin Polar coordinates (, ) The type of coordinate system chosen is not according to the symmetry of the object Mirror Our choice of coordinate axis does not alter the symmetry of the object (or the lattice) Centre of Inversion

Alternate choice of unit cells for Orthorhombic lattices Alternate choice of unit cell for “C”(C-centred orthorhombic) case. The new (orange) unit cell is a rhombic prism with (a = b  c,  =  = 90o,   90o,   120o) Both the cells have the same symmetry  (2/m 2/m 2/m) In some sense this is the true Ortho-”rhombic” cell

z = 0 & z = 1 Conventional Alternate choice (“ortho-rhombic”) P C 2ce the size I F 2ce the size I 1/2 the size C P 1/2 the size z = ½ Note: All spheres represent lattice points. They are coloured differently but are the same  A consistent alternate set of axis can be chosen for Orthorhombic lattices

2d produces this additional point not part of the original lattice Intuitively one might feel that the orthogonal cell has a higher symmetry  is there some reason for this? 2d produces this additional point not part of the original lattice This is in addition to our liking for 90! The 2x and 2y axes move lattice points out the plane of the sheet in a semi-circle to other points of the lattice (without introducing any new points) The 2d axis introduces new points which are not lattice points of the original lattice The motion of the lattice points under the effect of the artificially introduced 2-folds is shown as dashed lines (---)

Progressive lowering of symmetry amongst the 7 crystal systems Cubic48 Increasing symmetry Hexagonal24 Tetragonal16 Trigonal12 Orthorhombic8 Monoclinic4 Triclinic2 Arrow marks lead from supergroups to subgroups Superscript to the crystal system is the order of the lattice point group

Progressive relaxation of the constraints on the lattice parameters amongst the 7 crystal systems Cubic (p = 2, c = 1, t = 1) a = b = c  =  =  = 90º Increasing number t Tetragonal (p = 3, c = 1 , t = 2) a = b  c  =  =  = 90º Hexagonal (p = 4, c = 2 , t = 2) a = b  c  =  = 90º,  = 120º Trigonal (p = 2, c = 0 , t = 2) a = b = c  =  =   90º Orthorhombic1 (p = 4, c = 1 , t = 3) a  b  c  =  =  = 90º Orthorhombic2 (p = 4, c = 1 , t = 3) a = b  c  =  = 90º,   90º p = number of independent parameters = (p  e) c = number of constraints (positive  “=“) t = terseness = (p  c) (is a measure of the ‘expenditure’ on the parameters Monoclinic (p = 5, c = 1 , t = 4) a  b  c  =  = 90º,   90º Triclinic (p = 6, c = 0 , t = 6) a  b  c       90º Orthorhombic1 and Orthorhombic2 refer to the two types of cells

Minimum symmetry requirement for the 7 crystal systems

REGULAR SOLIDS IN VARIOUS DIMENSIONS   nD No. SYMBOL 1 1/ 2  {p} 3 5 {p, q} 4 6 {p, q, r} {p, q, r, s} POINT LINE SEGMENT TRIANGLE {3} SQUARE {4} PENTAGON {5} HEXAGON {6} TETRAHEDRON {3, 3} OCTAHEDRON {3, 4} DODECAHEDRON {5, 3} CUBE {4, 3} ICOSAHEDRON {3, 5} SIMPLEX {3, 3, 3} 16-CELL {3, 3, 4} 120-CELL {5, 3, 3} DRP 24-CELL {3, 4, 3} HYPERCUBE {4, 3, 3} 600-CELL {3, 3, 5} CRN REGULAR SIMPLEX {3, 3, 3, 3} CROSS POLYTOPE {3, 3, 3, 4} MEASURE POLYTOPE {4, 3, 3, 3}