BRAVAIS LATTICE

Presented By- NRIDUL SINHAB.Sc (Hons.),Physics5th Semester, 2016Roll-1415010231Govt. Degree College, Dharmanagar

CONTENTSIntroductionSpace Lattice, Crystal Lattice and BasisBravais LatticeCrystal Lattice SystemsUnit CellPrimitive and Centred Unit CellsThree Types Of Centred Unit CellsArrangement Of Lattice Points in The Unit &

No. of Lattice Points/CellStructure of Fourteen Bravais LatticesNoteReference

INTRODUCTION Solids are classified broadly into two classes viz.

crystalline and non-crystalline(amorphous).In crystalline solids, the atoms are arranged in periodic manner in all three directions.The crystalline state being the low energy state, most of the solids prefer to be in this state. The crystalline solids are again subdivided into single crystal and microcrystalline(polycrystalline).As the amorphous solids have no directional properties they are known as isotropic substances.Whereas, a crystal having different periodic arrangements in all the three directions, its physical properties vary with direction and, therefore, it is called anisotropic substance.

The branch of science dealing with the study of forms and physical properties of crystalline solids is called crystallography.

Space Lattice,Crystal Lattice and BasisSpace Lattice(or Point Lattice):-It is a mathematical

concept which represents infinite regular and periodic array of points in three dimensional space such that every point has surroundings identical to that of every other point in the array.

Crystal Lattice:-It is defined as a space lattice in which the lattice sites are occupied by atoms or group of atoms known as basis.

Basis:- A basis is defined as a cluster of atoms, ions or molecules identical in composition,arrangement and orientation. The periodic repetition of the basis in three dimensions forms the structure of the crystal.

BRAVAIS LATTICEIn 1884, French Physicist M. Auguste Bravais

proposed the concept of the space lattice and showed that identical points can be arranged spatially to produce 14 types of regular pattern. All possible crystal lattices in three dimensional space can be generated from these 14 lattices. 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 a ion.

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

CRYSTAL LATTICE SYSTEMSThere are 14 distinguishable ways in which

points can be arranged in three dimensional space. Therefore, Bravais lattices are 14 in number and belong to 7 crystal systems as mounted on the following table.

CRYSTAL SYSTEMS Crystal System Lattice

ConstantsInterfacial Angles(Degree)

Cubic 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

UNIT CELL The unit cell of a crystal is defined as the

fundamental elementary pattern of minimum of atoms, molecules or group of molecules which represent fully all the characteristics of the crystal. It is the smallest occupying volume 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 as shown in the following figure.

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 a 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.

PRIMITIVE & CENTRED UNIT CELLS

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

Structure of 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 Parallelogramic Prism

Monoclinic sulfur, Na2SO4.10H2O

P I F E

a b c

5 Triclinic Parallelopiped (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 Parallelopiped (Equilateral, Equiangular)

a b c

90

Calcite (CaCO3), Cinnabar (HgS)

4 Monoclinic Parallelogramic Prism

5 Triclinic Parallelopiped (general)

6 Hexagonal 120 Rhombic Prism

7 Rhombohedral Parallelopiped (Equilateral, Equiangular)

P I F E

Crystal System Shape of UC Bravais Lattices

Note:The Crystal Systems are defined

based on Point Group Symmetries (Rotation,

Reflection, Inversion etc. forming the Point Groups) and

NOT on the geometry of the Unit Cell.

REFERENCESlideshare.ppt Site. Statistical Mechanics & Solid State

Physics By Satyaprakash.Fundamentals of Solid State Physics

By Saxena, Gupta & Saxena.Refresher Course In Physics, Vol.-3

By C.L. Arora.

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