Lecture 02 Fundamental Properties of Solids

Crystal SystemShape of UCBravais Lattices PIFC 1CubicCube  2TetragonalSquare Prism (general height)  3OrthorhombicRectangular Prism (general height)  4Hexagonal 120  Rhombic Prism  5TrigonalParallopiped (Equilateral, Equiangular)  6MonoclinicParallogramic Prism  7TriclinicParallelepiped (general)  14 Bravais Lattices divided into 7 Crystal Systems PPrimitive IBody Centred FFace Centred CA/B/C- Centred A Symmetry based concept We will take up these cases one by one (hence do not worry!) ‘Translation’ based concept Some guidelines apply

3-Dimensional Unit Cells Common 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 Cells Unıt Cell Types Primitive A single lattice point per cell The smallest area in 2 dimensions, or The smallest volume in 3 dimensions Simple Cubic (sc) Conventional 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) Conventional Cell ≠ Primitive cell

1CubicCube  PIFC Lattice point P I F

PIFC 2TetragonalSquare Prism (general height)  I P

PIFC 3OrthorhombicRectangular Prism (general height)  P I F C Note the position of ‘a’ and ‘b’ One convention

PIFC 4Hexagonal 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)

PIFC 5TrigonalParallelepiped (Equilateral, Equiangular)  Symmetry of Trigonal lattices Rhombohedral Note the position of the origin and of ‘a’, ‘b’ & ‘c’

PIFC 6MonoclinicParallogramic Prism  Note the position of ‘a’, ‘b’ & ‘c’ One convention

PIFC 7TriclinicParallelepiped (general) 

Closed-packed structures zThere are an infinite number of ways to organize spheres to maximize 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 Packed zCell 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 Packed zThere 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. zCoordination number for HCP are exactly the same as those for FCC: 12 zThis 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 structure Diffraction 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 deformation Plastic (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.

z(C) Transport Properties zIn 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. zExample: Graphite zConduction of heat is more rapid in the sp2 covalently bonded lattice planes than in the direction perpendicular to those planes. zExample: YBa2Cu3O7 superconductors zSome 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 cell Assuming an ideal infinite crystal we define a unit cell by