BRAVAIS LATTICES

RAGESH NATH R ST.JOSEPH’S COLLEGE BANGALORE (AUTONOMOUS)

7 CRYSTAL LATTICE We know that a three dimensional space lattice is generated by

repeated translation of three non-coplanar vectors a, b, c. Based on the lattice parameters we can have 7 popular crystal systems.

C TOM THR

Crystal system Unit vector AnglesCubic a= b=c α =β =√=90Tetragonal a = b≠ c α =β =√=90Orthorhombic a ≠ b ≠ c α =β =√=90Monoclinic a ≠ b ≠ c α =β =90 ≠√Triclinic a ≠ b ≠ c α ≠ β ≠√ ≠90 Hexagonal a= b ≠ c α =β=90

√=120Rhombohedral a= b=c α =β =√≠90

BRAVAIS LATTICES In 1850, M. A. Bravais showed that identical

points can be arranged spatially to produce 14 types of regular pattern. These 14 space lattices are known as ‘Bravais lattices’.

Each point in a lattice is called lattice point or lattice site.

Each point in a crsytal lattice represents one constituent particle which may be an atom, a molecule(group of atoms)or an ion.

Lattice points are joined by straight lines to bring out the geometry of the lattice.

UNIT CELL Unit cell is the smallest portion of a crystal lattice

which, when repeated in different directions, generates the entire lattice.

it is characterized by; Its dimensions along the three edges a,b and c.

these edges may or may not be mutually perpendicular.

Angles between the edges α (between b and c) ß (between a and c) and γ (between a and b). Thus a unit cell is characterized by six parameters.

PRIMITIVE AND CENTRED UNIT CELLS

Unit cells can be broadly divided into two categories , primitive and centred unit cells. When constituent particles are present only on the corner positions

of a unit cell. It is called as Primitive unit cell. When a unit cell contains one or more constituent particles present

at the positions other than corners in addition to those at corners, it is called a centred unit cell.

THREE TYPES OF CENTRED UNIT CELLS.1. Body–centred unit cells. Such a unit cell contains one constituent particle(atom, molecule or ion) at its BODY-CENTRE beside the ones that are at the corners.

2. FACE-CENTRED UNIT CELLS Such a unit cell contains one constituent particle present at the CENTRE of each face, besides the ones that are at its corners.

3.End-centred unit cells. In such a unit cell, one constituent particle is present at the centre of TWO OPPOSITE FACES besides the ones present at its corners.

ARRANGEMENT OF LATTICE POINTS IN THE UNIT & NO. OF LATTICE POINTS / CELL.

FOURTEEN BRAVAIS LATTICES

P I F E

90 a b c

Corresponding Examples NaCl, Zinc Blende, Cu

P I F E

a b c

90 White tin, SnO2, TiO2, CaSO4

P I F E

a b c

90 Rhombic sulfur, KNO3, BaSO4

P I F E

a b c

90

4 Monoclinic Parallogramic Prism

Monoclinic sulfur, Na2SO4.10H2O

P I F E

a b c

5 Triclinic Parallelepiped (general)

K2Cr2O7, CuSO4.5H2O, H3BO3

P I F E

a b c

90 , 120

6 Hexagonal 120 Rhombic Prism

Graphite, ZnO, CdS

P I F E

7 Rhombohedral Parallelepiped (Equilateral, Equiangular)

a b c

90

Calcite (CaCO3), Cinnabar (HgS)

4 Monoclinic Parallogramic Prism 5 Triclinic Parallelepiped (general) 6 Hexagonal 120 Rhombic Prism

7 Rhombohedral Parallelepiped (Equilateral, Equiangular)

P I F E

Crystal System Shape of UC Bravais Lattices

NOTE:

The Crystal Systems are defined based on Symmetries (Rotational, Mirror, Inversion etc. forming the Point Groups) and NOT on the geometry of the Unit Cell

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