Geometry Of Crystals

Prof.devender singh GEOMETRY OF CRYSTALS Vidyanchal academy Roorkee  Space Lattices  Crystal Structures  Symmetry, Point Groups and Space Groups

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

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

 b

•

•

•

•

•

•

 a

Lattice

Crystal

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

=

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 Courtesy Dr. Rajesh Prasad

Cells 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

Double

Primitive cell

Triple

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  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

Takes object to same form → Proper

Type I

Symmetries Type II Takes object to enantiomorphic form → improper

Translation Rotation Mirror

Rotoreflection

Inversion

Rotoinversion

Classification based on the dimension invariant entity of the symmetry operator Operator

Dimension

Inversion

0D

Rotation

1D

Mirror

2D

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

Rotation Axis If an object come into self-coincidence through smallest non-zero rotation angle of θ then it is said to have an n-fold rotation axis where

360 0 n= θ

θ =180°

n=2

2-fold rotation axis

θ =120°

n=3

3-fold rotation axis

θ =90°

n=4

4-fold rotation axis

θ =60°

n=6

6-fold rotation axis

The rotations compatible with translational symmetry are  (1, 2, 3, 4, 6)

Point group symmetry of Lattices →

Symmetries acting at a point

R ⊕ R

32 point groups

7 crystal systems

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)

Previously

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

a

=

a

+ a 2

Now

Crystal = Space group (how to repeat) + Asymmetric unit (Motif’: what to repeat)

a

= a

Glide 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’

Unit cell

Asymmetric Unit

a

mmm Three mirror planes

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

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

Lattice points 3

mg g

Not a lattice point Presence of 1 mirror plane and 1 glide reflection plane, with a redundant inversion centre the translational symmetry has been reduced to ‘2a’

ii

4 2 inversion centres

5

m 1 mirror plane

gg

6 1 glide reflection translational symmetry of ‘2a’

7 No symmetry except translation

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 inversion centre

Can be a unit cell for a 2D crystal

4mm Decoration retaining the symmetry

mm

m

m

4

No symmetry

Lattices have the highest symmetry  Crystals based on the lattice can have lower 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.t to the symmetry elements of a lattice → Wyckoff positions

A 2D lattice with symmetry elements

g

e

f

a

c

1

(x,y)

(-x,-y)

(-y,x)

(y,-x)

(-x,y)

(x,-y)

(y,x)

((-y,-x)

..m

(x,x)

(-x,-x)

(x,-x)

(-x,x)

4

f

4

e

.m.

(x,½)

(-x, ½)

(½,x)

(½,-x)

4

d

.m.

(x,0)

(-x,0)

(0,x)

(0,-x)

2

c

2mm.

(½,0)

(0,½)

1

b

4mm

(½,½)

1

a

4mm

(0,0)

Points

Number of Identi-points

g

Coordinates

Lines

d

8

Site symmetry

Area

b

Multi- Wyckoff plicity letter

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

e

f

b d a

d

c

f

Exclude these points

e

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 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 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. 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/ 8 corners B/ + C 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

2. Tetragonal

P

I

3. Orthorhombic

P

I

4. Hexagonal

P

5. Trigonal

P

6. Monoclinic

P

7. Triclinic

P

F

F

C

C

Courtesy Dr. Rajesh Prasad

14 Bravais lattices divided into seven crystal systems Crystal system

Bravais lattices

1. Cubic

P

I

2. Tetragonal

P

I

3. Orthorhombic

P

I

4. Hexagonal

P

5. Trigonal

P

6. Monoclinic

P

7. Triclinic

P

F

C

F

C

C

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

2. Tetragonal

P

I

3. Orthorhombic

P

I

4. Hexagonal

P

5. Trigonal

P

6. Monoclinic

P

7. Triclinic

P

x

F

C

F F

C

C

Courtesy Dr. Rajesh Prasad

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)

( x − a ) 2 + ( y − b) 2 = r 2 The centre of symmetry of the object does not coincide with the origin

Polar coordinates (ρ , θ )

ρ =r ( x) + ( y ) = r 2

2

2

The type of coordinate system chosen is not according to the symmetry of the object Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)

Mirror

Centre of Inversion

THE 7 CRYSTAL SYSTEMS

TRICLIN

X

X

2=m

1

2 m

X +1

3   m

3

4

3

4 m

6 m

4 2m

6m2

4 2 2 m m m

6 2 2 m m m

6

X 2(2) Xm(m)

N

Xm

X2+1

3

2 2 2 m m m

2 m 3

3m

2 m

4 43m

X3

X 3 ≡ ( X 3 + 1)

2 3 m

3

4 2 3 m m

4 2 3 m m

N is the number of point groups for a crystal system

2 1

1. Cubic Crystals a = b= c α = β = γ = 90º • • •

Simple Cubic (P) Body Centred Cubic (I) – BCC Face Centred Cubic (F) - FCC

Point groups ⇒ 23, 4 3m, m 3 , 432,

4 2 3 m m

Pyrite Cube

[1]

[1]

Fluorite Octahedron

Garnet Dodecahedron

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

2. Tetragonal Crystals a=b≠ c α = β = γ = 90º • •

Simple Tetragonal Body Centred Tetragonal

Point groups ⇒ 4, 4 ,

4 4 2 2 , 422, 4mm, 42m, m mmm

Zircon

[1]

[1]

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

3. Orthorhombic Crystals a≠ b≠ c α = β = γ = 90º •

Simple Orthorhombic

•

Body Centred Orthorhombic

•

Face Centred Orthorhombic

•

End Centred Orthorhombic Point groups ⇒ 222, 2mm,

2 2 2 mmm

Topaz

[1]

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

4. Hexagonal Crystals a=b≠ c α = β = 90º •

γ = 120º

Simple Hexagonal

6 6 2 2 Point groups ⇒ 6, 6 , , 622, 6mm, 6 m2, m mmm

[1]

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

5. Rhombohedral Crystals a=b=c α =β =γ

•

≠ 90º

Rhombohedral (simple)

Point groups ⇒ 3, 3 , 32, 3m, 3

2 m

[1]

[1]

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

6. Monoclinic Crystals a≠ b≠ c α = γ = 90º ≠ β • •

Simple Monoclinic End Centred (base centered) Monoclinic (A/C) 2 Point groups ⇒ 2, 2 , m

[1]

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

7. Triclinic Crystals a≠ b≠ c α ≠ γ ≠ β

•

Simple Triclinic

Point groups ⇒ 1, 1

[1]

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

Concept of symmetry and choice of axes

(a,b)

( x − a ) 2 + ( y − b) 2 = r 2 The centre of symmetry of the object does not coincide with the origin

Polar coordinates (ρ , θ )

ρ =r ( x) + ( y ) = r 2

2

2

The type of coordinate system chosen is not according to the symmetry of the object Our choice of coordinate axis does not alter the symmetry of the object (or the lattice)

Mirror

Centre of Inversion

Alternate choice of unit cells for Orthorhombic lattices

2 2 2 mmm

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

Conventional Alternate choice (“orthorhombic”)

z=0& z=1

P

C 2ce the size

1/2

z=½

I

F 2ce the size

F

I 1/2 the size

C

P 1/2 the size

1/2

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

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

2x 2d

2y

(not the operations of the lattice) Artificially introduced 2-folds  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 Hexagonal24 Tetragonal16 Trigonal12 Orthorhombic8

rt e mmys gni s aer c nI

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 α = β = γ = 90º

t

Tetragonal (p = 3, c = 1 , t = 2) a=b≠ c α = β = γ = 90º

r e b mun gni s aer c nI

Orthorhombic1 (p = 4, c = 1 , t = 3) a≠ b≠ c α = β = γ = 90º

Hexagonal (p = 4, c = 2 , t = 2) a=b≠ c α = β = 90º, γ = 120º

Trigonal (p = 2, c = 0 , t = 2) a=b=c α = β = γ ≠ 90º

Orthorhombic2 (p = 4, c = 1 , t = 3) a=b≠ c α = β = 90º, γ ≠ 90º

Monoclinic (p = 5, c = 1 , t = 4) a≠ b≠ c α = γ = 90º, β ≠ 90º

• p = number of independent parameters • c = number of constraints (positive ⇒ “=“) • t = terseness = (p −c) (is a measure of the ‘expenditure’ on the parameters

Triclinic (p = 6, c = 0 , t = 6) a≠ b≠ c α ≠ γ ≠ β ≠ 90º

Orthorhombic1 and Orthorhombic2 refer to the two types of cells

Minimum symmetry requirement for the 7 crystal systems

Crystal system Cubic 23, 43m, m 3 , 432,

4 2 3 m m

6, 6 ,

6 6 2 2 , 622, 6mm, 6 m2, m mmm

4, 4 ,

4 4 2 2 , 422, 4mm, 4 2m, m mmm

3, 3 , 32, 3m, 3

222, 2mm,

2, 2, 1, 1

2 m

Ch

2 m

2 2 2 mmm

Fou

SYMBOL

nD

No.

0

1

1

1/∞

2

∞

{p}

3

5

{p, q}

4

5

6

3

REGULAR SOLIDS IN VARIOUS DIMENSIONS POINT LINE SEGMENT

{p, q, r}

{p, q, r, s}

TRIANGLE {3} TETRAHEDRON {3, 3}

SQUARE {4}

PENTAGON {5}

HEXAGON {6}

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}