Lecture 02

Fundamental Properties of Solids

Crystal System Shape of UC Bravais Lattices

P I F C

1 Cubic Cube

2 Tetragonal Square Prism (general height)

3 Orthorhombic Rectangular Prism (general height)

4 Hexagonal 120 Rhombic Prism

5 Trigonal Parallopiped (Equilateral, Equiangular)

6 Monoclinic Parallogramic Prism

7 Triclinic Parallelepiped (general)

14 Bravais Lattices divided into 7 Crystal Systems

P Primitive

I Body Centred

F Face Centred

C A/B/C- Centred

A Symmetry based concept

We will take up these cases one by one(hence do not worry!)

‘Translation’ based conceptSome guidelines apply

3-Dimensional Unit CellsCommon Unit Cells with Cubic

Symmetry

Simple Cubic Body Centered Cubic Face Centered Cubic (SC) (BCC) (FCC)

1 atom/unit cell

(8 x 1/8 = 1)

2 atoms/unit cell

(8 x 1/8 + 1 = 2)

4 atoms/unit cell

(8 x 1/8 + 6 x 1/2 = 4)1 atom/unit cell

(8 x 1/8 = 1)

coordination number 12coordination number 8coordination number 6

Base Centered Cubic

Atom/unit cell:

Coordination number:

Primitive & Conventional Unit CellsUnıt Cell Types

Primitive

A single lattice point per cellThe smallest area in 2 dimensions, orThe smallest volume in 3 dimensions

Simple Simple CCubicubic (sc)(sc)ConventionalConventional Cell Cell == Primitive cell Primitive cell

More than one lattice point per cell Volume (area) = integer multiple of

that for primitive cell

Conventional (Non-primitive)

Body Centered Cubic (bcc)Body Centered Cubic (bcc)ConventionalConventional Cell Cell ≠≠ Primitive cell Primitive cell

1 Cubic Cube P I F C

Lattice point

PI

F

a b c 90

P I F C2 Tetragonal Square Prism (general height)

IP

a b c

90

P I F C3 Orthorhombic Rectangular Prism (general height)

PI

F C

a b c

90

Note the position of ‘a’ and ‘b’

a b c One convention

P I F C4 Hexagonal 120 Rhombic Prism

A single unit cell (marked in blue) along with a 3-unit cells forming a

hexagonal prism

Note: there is only one type of hexagonal lattice (the primitive one)

a b c

90 , 120

P I F C5 Trigonal Parallelepiped (Equilateral, Equiangular)

90

a b c

Symmetry of Trigonal lattices

Rhombohedral

23m

Note the position of the origin and of ‘a’, ‘b’ & ‘c’

P I F C6 Monoclinic Parallogramic Prism

90

a b c a b c

Note the position of ‘a’, ‘b’ & ‘c’

One convention

P I F C7 Triclinic Parallelepiped (general)

a b c

Closed-packed structuresClosed-packed structures There are an infinite number of There are an infinite number of

ways to organize spheres to ways to organize spheres to maximizemaximize the packing fraction. the packing fraction.

There are different ways you can pack spheres together. This

shows two ways, one by putting the spheres in an ABAB…

arrangement, the other with ACAC…. (or any combination of the

two works)

The centres of spheres at A, B, and C positions (from Kittel)

Hexagonal Close PackedCell of an HCP lattice is visualized as

a top and bottom plane of 7 atoms, forming a regular hexagon around a central atom. In between these planes is a half-hexagon of 3 atoms.

Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl

Hexagonal Close PackedThere are two lattice parameters in HCP, a

and c, representing the basal and height parameters respectively. In the ideal case, the c/a ratio is 1.633, however, deviations do occur.

Coordination number for HCP are exactly the same as those for FCC: 12

This is because they are both considered close packed structures.

Hexagonal Close Packed (HCP) Structure:(A Simple Hexagonal Bravais Lattice with a 2 Atom Basis)

The HCP lattice is not a Bravais lattice, because the orientation of the environment of a point varies from layer to layer along the c-axis.

Structure of NaCl

Structure of Cesium Chloride(CsCl)

Carbon structures

Zinic Sulfide Structure

Why are planes in a lattice important?

(A) Determining crystal structureDiffraction methods directly measure the distance between parallel planes of lattice points. This information is used to determine the lattice parameters in a crystal and measure the angles between lattice planes.

(B) Plastic deformationPlastic (permanent) deformation in metals occurs by the slip of atoms past each other in the crystal. This slip tends to occur preferentially along specific lattice planes in the crystal. Which planes slip depends on the crystal structure of the material.

     

(C) Transport Properties In certain materials, the

atomic structure in certain planes causes the transport of electrons and/or heat to be particularly rapid in that plane, and relatively slow away from the plane.

Example: Graphite Conduction of heat is more rapid

in the sp2 covalently bonded lattice planes than in the direction perpendicular to those planes.

Example: YBa2Cu3O7 superconductors

Some lattice planes contain only Cu and O. These planes conduct pairs of electrons (called Cooper pairs) that are responsible for superconductivity. These superconductors are electrically insulating in directions perpendicular to the Cu-O lattice planes.

(GPa)

b

c

a

Unit cell: a volume in space that fills space entirely when translated by all lattice vectors.

The obvious choice:

a parallelepiped defined by a, b, c, three basis vectors with

the best a, b, c are as orthogonal as possible

the cell is as symmetric as possible (14 types)

A unit cell containing one lattice point is called primitive cell.

Unit cellAssuming an ideal infinite crystal we define a unit cell by