Space lattices

Crystallography

• In 1669 Nicolaus Steno found angles between adjacent prism faces of quartz crystal (interfacial angle), to be 120°.

• In 1780 Carangeot invented the goniometer, a protactor like device used to measure interfacial angles on crystals.

• Law of “constancy of interfacial angels”: angles between equivalent faces of crystals of the same mineral are always the same.The law acknowledges that the size and shape of the crystal may vary.

• In 1784 Rene’Hauy hypothesized the existence of basic building blocks of crystals called integral molecules and argued that large crystals formed when many integral molecules bonded together.

Old view• Crystals are made of small

building blocks• The blocks stack together in a

regular way, creating the whole crystal.

• Each block contains a small number of atoms

• All building blocks have the same atomic composition

• The building block has shape and symmetry of the entire crystal.

We now accept that:• Crystals have basic building

blocks called unit cells• The unit cells are arranged in

a pattern described by points in a lattice.

• The relative proportions of elements in a unit cell are given by the chemical formula of a mineral.

• Crystals belong to one of the seven crystal systems. Unit cells of distinct shape and symmetry characterize each crystal system.

• Total crystal symmetry depends on Unit cell symmetry and lattice symmetry.

Crystal Geometry

1. Crystals

2. Lattice

3. Lattice points, lattice translations

4. Cell--Primitive & non primitive

5. Lattice parameters

6. Crystal=lattice+motif

Matter

Solid Liquid Gas

Crystalline Amorphous

Crystal?

A 3D translationally periodic arrangement of atoms in space is called a crystal.

A two-dimensional periodic pattern by a Dutch artist M.C. Escher

Lattice?

A 3D translationally periodic arrangement of points in space is called a lattice.

A 3D translationally periodic arrangement of atoms

Crystal

A 3D translationally periodic arrangement of points

Lattice

What is the relation between the two?

Crystal = Lattice + Motif

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

Crystal=lattice+basis

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

+

Love Pattern Love Lattice + Heart=

Space Lattice

A discrete array of points in 3-d space such that every point has identical surroundings

Lattice

Finite or infinite?

Primitivecell

Primitivecell

Nonprimitive cell

Cells 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.

Lattice Parameters

Lengths of the three sides of the parallelopiped : a, b and c.

The three angles between the sides: , ,

Conventiona parallel to x-axis

b parallel to y-axis

c parallel to z-axis

Angle between y and z

Angle between z and x

Angle between x and y

The six lattice parameters a, b, c, , ,

The cell of the lattice

lattice

crystal

+ Motif

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

The smallest cell with lattice points at its eight corners has effectively only one lattice point in the volume of the cell.

Such a cell is called PRIMITIVE CELL

Bravais Space Lattices

Conventionally, the finite representation of space lattices is done using unit cells which show maximum possible symmetries with the smallest size.

Symmetries: 1.Translation

2. Rotation

3. Reflection

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 Classes

Arrangement of lattice points in the unit cell

1. 8 Corners (P)

2. 8 Corners and 1 body centre (I)

3. 8 Corners and 6 face centres (F)

4. 8 corners and 2 centres of opposite faces (A/B/C)

Effective number of l.p.

1. Cubic Crystals

• Simple Cubic (P)• Body Centred

Cubic (I) – BCC• Face Centred

Cubic (F) - FCC

2. Tetragonal Crystals

• Simple Tetragonal• Body Centred

Tetragonal

3. Orthorhombic Crystals

• Simple Orthorhombic

• Body Centred Orthorhombic

• Face Centred Orthorhombic

• End Centred Orthorhombic

4. Hexagonal Crystals

• Simple Hexagonal or most commonly HEXAGONAL

5. Rhombohedral Crystals

• Rhombohedral (simple)

6. Monoclinic Crystals

• Simple Monoclinic• End Centred

Monoclinic (A/B)

7. Triclinic Crystals

• Triclinic (simple)

Crystal Structure

Space Lattice + Basis (or Motif)Basis consists of a group of atoms located

at every lattice point in an identical fashionTo define it, we need to specify• Number of atoms and their kind• Internuclear spacings• Orientation in space

Atoms are assumed to be hard spheres