Space Lattices

Crystal Structures

Symmetry, Point Groups and Space Groups

GEOMETRY OF CRYSTALSGEOMETRY OF CRYSTALS

Acknowledgments: Prof. Rajesh Prasad for a lot of things

Crystal = Lattice + Motif

Motif or basis: an atom or a group of atoms associated with each lattice point

The language of crystallography is one succinctness

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)

Space Lattice

Translationally periodic arrangement of points in space is called a lattice

or

A 2D lattice

a

b

Translationally periodic arrangement of motifs

Crystal

Translationally periodic arrangement of points

Lattice

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

Crystal = Lattice + Motif

+

Lattice

Motif

Crystal

=

Courtesy Dr. Rajesh Prasad

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.

Cells

Instead of drawing the whole structure I can draw a representative partand specify the repetition pattern

Primitivecell

Primitivecell

Nonprimitive cell

Courtesy Dr. Rajesh Prasad

Primitivecell

Primitivecell

Nonprimitive cell

Double

Triple

Symmetry of the Lattice or the crystal is not altered by our choice of unit cell!!

Primitive cell

Nonprimitive cell

4- fold axes

Centred square lattice = Simple/primitive square lattice

Shortest lattice translation vector ½ [11]

a

b

Nonprimitive cell

Primitive cell

Lower symmetry than the lattice usually not chosen

Maintains the symmetry of the lattice the usual choice

2- fold axes

Centred rectangular lattice

Centred rectangular lattice Simple rectangular Crystal

Shortest lattice translation vector [10]

Not a cell

Primitive cell

MOTIF

Courtesy Dr. Rajesh Prasad

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

Cells- 3D

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.

If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.

SYMMETRY

Given a general point a symmetry operator leaves a finite set of points in space

A symmetry operator closes space onto itself

SYMMETRY OPERATOR

Symmetry operators

Symmetries

Type II

Type IRotation

Translation

Inversion

Mirror

Takes object to same form → Proper

Takes object to enantiomorphic form → improper Roto-inversion

Roto-reflection

Classification based on the dimension invariant entity of the symmetry operator

Operator Dimension

Inversion 0D

Rotation 1D

Mirror 2D

Symmetry operators

Symmetries

Microscopic

Macroscopic

Rotation

Mirror

Glide Reflection

Screw Axes

Inversion

Influence the external shape of the crystal

Do not Influence the external shape of the crystal

R Rotation G Glide reflection

R Roto-inversion S Screw axis

Ones with built in translationOnes acting at a point

Minimum set of symmetry operators required

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

0360n

=180

Rotation Axis

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 actingat a point

R R

R + R → rotations compatible with translational symmetry (1, 2, 3, 4, 6)

32 point groups

Along with symmetrieshaving a translation

G + S

230 space groups

Point group symmetry of Lattices →

7 crystal systems

Space group symmetry of Lattices →

14 Bravais lattices

Crystal = Lattice (Where to repeat)

+ Motif (What to repeat)

Previously

=

+

a

a

2

a

Crystal =

Space group (how to repeat)

+ Asymmetric unit (Motif’: what to repeat)

Now

=

+

a

aGlide reflection operator

Usually asymmetric units are regions of space within the unit cell- which contain atoms

Progressive lowering of symmetry in an 1D lattice illustration using the frieze groups

Consider a 1D lattice with lattice parameter ‘a’

a

Unit cell

Three mirror planes The intersection points of the mirror planesgive rise to redundant inversion centres

mmm

Asymmetric Unit

mirror glide reflection

Decoration of the lattice with a motif may reduce the symmetry of the crystal

Decoration with a “sufficiently” symmetric motif does not reduce the symmetry of the lattice

1

2

mmm

mm

Loss of 1 mirror plane

Lattice points

Not a lattice point

g

Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centrethe translational symmetry has been reduced to ‘2a’

2 inversion centres

ii

mg3

4

g

1 mirror plane

m

g

1 glide reflection translational symmetry of ‘2a’

No symmetry except translation

5

6

7

Effect of the decoration a 2D example

4mmRedundant inversion centre

Decoration retaining the symmetry

4mm

Can be a unit cell for a 2D crystal

Two kinds of decoration are shown (i) for an isolated object, (ii) an object which can be an unit cell.

Redundant mirrors

which need not be drawn

mm m m

No symmetry

4

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

Unit cell ofTriclinic crystal

Amorphous arrangementNo unit cell

Positioning a object with respect to the symmetry elements

Three mirror planes The intersection points of the mirror planesgive rise to redundant inversion centres

mmm

Right handed object

Left handed object

Object with bilateral symmetry

Positioning a object with respect to the symmetry elements

Note: this is for a point group and not for a lattice the black lines are not unit cells

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

Positioning of a motif w.r.t to the symmetry elements of a lattice Wyckoff positions

A 2D lattice with symmetry elements

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)

4 e .m. (x,½) (-x, ½) (½,x) (½,-x)

4 d .m. (x,0) (-x,0) (0,x) (0,-x)

2 c

Points

2mm. (½,0) (0,½)

1 b 4mm (½,½)

1 a 4mm (0,0)

a

b

c

d

ef

Number of Identi-points

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)]

g

a

b

c

d

ef

d

Exclude thesepoints

g

Exclude thesepoints

f

Exclude thesepoints

e

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.

Or the technical definition

There are 14 Bravais Lattices which are the space group symmetries of lattices

Considering

1. Maximum Symmetry, and

2. Minimum Size

Bravais concluded that there are only 14 possible Space Lattices (or Unit Cells to represent them).

These belong to 7 Crystal systems

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. of Lattice points / cell

Position of lattice pointsEffective number of Lattice points / cell

1 P 8 Corners = 8 x (1/8) = 1

2 I8 Corners + 1 body centre

= 1 (for corners) + 1 (BC)

3 F8 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

1. Cubic P I F

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

Courtesy Dr. Rajesh Prasad

14 Bravais lattices divided into seven crystal systems

Crystal system Bravais lattices

1. Cubic P I F C

2. Tetragonal P I

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. 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

1. Cubic P I F C

2. Tetragonal P I F

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

Courtesy Dr. Rajesh Prasad

x

The following 4 things are different

Symmetry of theMotif

Crystal

Lattice

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.

Concept of symmetry and choice of axes

(a,b)

222 )()( rbyax

222 )()( ryx

Polar coordinates (, )

r

The centre of symmetry of the object does not coincide with the origin

The type of coordinate system chosen is notaccording to the symmetryof the object

Centre of Inversion

Mirror

Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)

THE 7 CRYSTAL SYSTEMSTHE 7 CRYSTAL SYSTEMS

TRICLINIC MONOCLINIC ORTHORHOMBIC TRIGONAL TETRAGONAL HEXAGONAL

N 2 3 3 5 7 7

X 1 2 3 4 6

X 1 m2 3 4 6

m

3

1X m

2 3

m

4

m

6

)2(2X 222 32 422 622

)(mXm 2mm 3m 4mm 6mm

mX 2mm 3m

2 m3 m24 26m

12 X m

2

m

2

m

2 3

m

2

m

4

m

2

m

2

m

6

m

2

m

2

Increasing Symmetry

In

crea

sing

Sym

met

ry

CUBIC = ISOMETRIC N = 5

X = 2 X = 4 X = 4

3X 23 m34 432

)13(3 XX m

23 m 3

m

43

m

2

m

43

m

2 (m3m)

N is the number of point groups for a crystal system

1. Cubic Crystals

a = b= c = = = 90º

• Simple Cubic (P)

• Body Centred Cubic (I) – BCC

• Face Centred Cubic (F) - FCC

FluoriteOctahedron

PyriteCube

m

23

m

4 432, ,3m 3m,4 23, groupsPoint

[1] http://www.yourgemologist.com/crystalsystems.html[2] L.E. Muir, Interfacial Phenomenon in Metals, Addison-Wesley Publ. co.

[1] [1]GarnetDodecahedron

[1]

Vapor grown NiO crystal

[2]

Tetrakaidecahedron(Truncated Octahedron)

2. Tetragonal Crystalsa = b c = = = 90º

• Simple Tetragonal

• Body Centred Tetragonal

m

2

m

2

m

42m,4 4mm, 422, ,

m

4 ,4 4, groupsPoint

[1] http://www.yourgemologist.com/crystalsystems.html

[1] [1] [1]

Zircon

3. Orthorhombic Crystalsa b c = = = 90º

• Simple Orthorhombic

• Body Centred Orthorhombic

• Face Centred Orthorhombic

• End Centred Orthorhombic

m

2

m

2

m

2 2mm, 222, groupsPoint

[1] http://www.yourgemologist.com/crystalsystems.html

Topaz

[1]

[1]

4. Hexagonal Crystalsa = b c = = 90º = 120º

• Simple Hexagonal

m

2

m

2

m

6 m2,6 6mm, 622, ,

m

6 ,6 6, groupsPoint

[1] http://www.yourgemologist.com/crystalsystems.html

[1] Corundum

5. Rhombohedral Crystalsa = b = c = = 90º

• Rhombohedral (simple)

m

23 3m, 32, ,3 3, groupsPoint

[1] http://www.yourgemologist.com/crystalsystems.html

Tourmaline[1] [1]

6. Monoclinic Crystalsa b c = = 90º

• Simple Monoclinic• End Centred (base centered) Monoclinic (A/C)

m

2 ,2 2, groupsPoint

[1] http://www.yourgemologist.com/crystalsystems.html

Kunzite

[1]

7. Triclinic Crystalsa b c

• Simple Triclinic

1 1, groupsPoint

[1] http://www.yourgemologist.com/crystalsystems.html

Amazonite[1]

Concept of symmetry and choice of axes

(a,b)

222 )()( rbyax

222 )()( ryx

Polar coordinates (, )

r

The centre of symmetry of the object does not coincide with the origin

The type of coordinate system chosen is notaccording to the symmetryof the object

Centre of Inversion

Mirror

Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)

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

m

2

m

2

m

2

1/2

1/2

Note: All spheres represent lattice points. They are coloured differently but are the same

z = 0 &z = 1

z = ½

Conventional

Alternate choice

(“ortho-rhombic”)

P C2ce the size

IF

2ce the size

FI 1/2 the size

CP 1/2 the size

A consistent alternate set of axis can be chosen for Orthorhombic lattices

Intuitively one might feel that the orthogonal cell has a higher symmetry is there some reason for this?

Artificially introduced 2-folds (not the operations of the lattice)

2x

2y

2d

2d produces this additional pointnot part of the original lattice

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 (---)

This is in addition to our liking for 90!

Cubic48

Tetragonal16

Triclinic2

Monoclinic4

Orthorhombic8

Progressive lowering of symmetry amongst the 7 crystal systems

Hexagonal24

Trigonal12

Incr

easi

ng s

ymm

etry

Superscript to the crystal system is the order of the lattice point group

Arrow marks lead from supergroups to subgroups

Cubic (p = 2, c = 1, t = 1)a = b = c

= = = 90º

Tetragonal (p = 3, c = 1 , t = 2) a = b c

= = = 90º

Triclinic (p = 6, c = 0 , t = 6) a b c

90º

Monoclinic (p = 5, c = 1 , t = 4)a b c

= = 90º, 90º

Orthorhombic1 (p = 4, c = 1 , t = 3) a b c

= = = 90º

Progressive relaxation of the constraints on the lattice parameters amongst the 7 crystal systems

Hexagonal (p = 4, c = 2 , t = 2)a = b c

= = 90º, = 120º

Trigonal (p = 2, c = 0 , t = 2)a = b = c

= = 90º

Orthorhombic2 (p = 4, c = 1 , t = 3) a = b c

= = 90º, 90º

Incr

easi

ng n

umbe

r t

• 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

Orthorhombic1 and Orthorhombic2 refer to the two types of cells

Minimum symmetry requirement for the 7 crystal systems

Crystal system

Characteric symmetry Point groups Comment

Cubic Four 3-fold rotation axes m

23

m

4 432, ,3m 3m,4 23,

3 or 3 in the second place Two 3-fold axes will generate the other two 3-fold axes

Hexagonal One 6-fold rotation axis (or roto-inversion axis) m

2

m

2

m

6 m2,6 6mm, 622, ,

m

6 ,6 6,

6 in the first place

Tetragonal (Only) One 4-fold rotation axis (or roto-inversion axis)

m

2

m

2

m

42m,4 4mm, 422, ,

m

4 ,4 4,

4 in first place but no 3 in second place

Trigonal (Only) One 3-fold rotation axis (or roto-inversion axis)

m

23 3m, 32, ,3 3,

3 or 3 in the first place

Orthorhombic (Only) Three 2-fold rotation axes (or roto-inversion axis)

m

2

m

2

m

2 2mm, 222,

Monoclinic (Only) One 2-fold rotation axis (or roto-inversion axis)

m

2 ,2 2,

Triclinic None 1 1, 1 could be present

nD No. SYMBOL

0 1

1 1/

2 {p}

3 5 {p, q}

4 6 {p, q, r}

5 3 {p, q, r, s}

POINT

LINE SEGMENT

TRIANGLE {3} SQUARE {4} PENTAGON {5} HEXAGON {6}

TETRAHEDRON {3, 3}

OCTAHEDRON {3, 4}

DODECAHEDRON {5, 3}

ICOSAHEDRON {3, 5}

SIMPLEX {3, 3, 3}

16-CELL {3, 3, 4}

120-CELL {5, 3, 3}

600-CELL {3, 3, 5}

HYPERCUBE {4, 3, 3}

DRP

CRN24-CELL {3, 4, 3}

CUBE {4, 3}

REGULAR SIMPLEX {3, 3, 3, 3}

CROSS POLYTOPE {3, 3, 3, 4}

MEASURE POLYTOPE {4, 3, 3, 3}

REGULAR SOLIDS IN VARIOUS DIMENSIONS