Structure of Crystalline Solids PartI

8/11/2019 Structure of Crystalline Solids PartI

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Summary tomic Structure and Chemical Bonding Atom: composed of a nucleus (neutron + proton)

and orbiting electrons around it. Electronic orbitals are quantized and governed by

the laws of quantum mechanics

When two atoms are brought together, depending on favourable energetics, they may form bond and the types of bond is decided primarily by the atomic structure (electronic configuration).

How do the bonding atoms arrange in 3D ?

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Tetrahedron of Materials Science

Processing

Structure

Properties

PerformanceStructureAtomic Structure and BondingStructure of crystalline SolidsImperfections


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Based on the regularity with which atoms or ions are arranged with respect to one another, solids can be classified as:

Crystalline Atoms are packed in periodic array Longrange order exists

Non crystalline (Amorphous) Atoms have no periodic packing Short range order exists

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Crystalline SiO2 Amorphous SiO2Si Oxygen

Examples: Metals Ceramics Polymers (some)

Examples: Glasses Polymers

Si Oxygen

Types of Solids: Based on Atomic

Arrangement


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Ideal crystal: Definition

An ideal crystal is a periodic array of structural units, such as atoms or molecules.

It can be constructed by the infinite repetition of these identical structural units in space.

Crystal structure can be described in terms of a lattice , and an/ a group of atom(s) attached to each lattice point. The /group of atom(s) is the basis .

Crystal

Structure

=

Lattice +

Basis

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6

Lattice


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Lattice

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Lattice and Unit Cell Lattice: An infinite array of discrete points with an arrangement

and orientation that appears exactly the same, from any of the points the array is viewed from.

Unit Cell: The smallest unit which, when repeated in space indefinitely will generate the space lattice.

Primitive cell: A unit cell which must contain precisely one lattice point.

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Crystal Structure

The structure of all crystals can be described in terms of a lattice and an/ a group of atom(s) attached to every lattice point.

The /group of atom(s) is called the basis

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Lattice Basis Crystal structure


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Lattice Parameters of Unit Cell

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A unit cell can be completely described by:The lengths a, b and c of three non coplanar vectors a , b , and c. The angles , , and between the vectors

A three dimensional Bravais lattice consists of all points with position vectors R that can be written as a linear combination of primitive vectors. The expansion coefficients

must be integers.R = n 1a + n 2b + n 3c


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Seven Crystal Systems


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Fourteen Bravais lattice

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Cubic Crystal Structure: Hard Sphere Models

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Each atom is shared by 8 unit cells.

Share of each corner atom to a unit cell =

Number of atoms per unit cell = 8 1

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Simple Cubic Structure: Number of atoms/unit cell

Shared by 8unit cells


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Coordination Number: Number of nearest neighbour or touching atoms

Closed packed directions are the directions along which the hard spheres touch. For simple cubic structure, closed packed direction is along the edge of the unit cell.

Example:

Polonium

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Simple Cubic Structure: Coordination Number


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Atomic Packing Factor (APF) is the fraction of solid sphere volume in a unit cell, assuming hard sphere model

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Simple Cubic Structure: Atomic Packing Factor

.

.

0.52


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Principal Metallic Crystal Structures

Crystal Structure of elemental metals About 90 % of elemental metal crystallizes in one of the

three densely packed structures: (a) body centered cubic (bcc), (b) phase centered cubic (fcc) and (c) Hexagonal close packed (hcp)

These packing allows energy minimization and more tight bonds discussed earlier


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Body Centered Cubic Crystal Structure

BCC:

8

nearest

neighbors

(CN)

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Body Centred Cubic (BCC) Structure

Each corner atom is shared by 8 unit cells. There is one atom at the centre of the unit cell.

Number of atoms per unit cell =

Coordination number = 8

Closed packed direction is along the body diagonals


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Body Centered Cubic Crystal Structure

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,

3

a r

r or a

Relationship between the length of the cube, a and atomic radius r:

No. of atoms in the unit cell:1

8 1 28

N

Atomic packing factor (APF):

vol.of atoms in unit cell100%

vol. of unit cell = 68 %

APF


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Face Centered Cubic Crystal Structure

Metal a (nm) r (nm)

Al 0.4050 0.143Cu 0.3615 0.128

Au 0.4080 0.144Pb 0.4950 0.175

Ni 0.3520 0.125

Pt 0.3930 0.139

FCC is a close packed structure


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Number of atoms per unit cell = 3 + (2 x )+(12 x 1/6) = 6 Coordination number =12 Atomic packing factor = 0.74

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Hexagonal Close Packed (HCP) Structure

What is the ideal c/a ratio in an ideal HCP structure ???


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Metallic crystals Densely packed Typically for pure metals, only one element is

present

Metallic bonding is non directional Nearest neighbour distances tend to be small to

lower the bond energy

Metals have simplest crystal structure


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Crystal structure and periodic table


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Atomic radii of common metals


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Density computation of metals


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Ceramic crystals Ceramics are composed of at least two elements. Atomic bonding range from purely ionic to purely

covalent.

Degree of ionic character for a two element ceramic depends on the electronegativities ( X A and X B) of the atoms

1002

-exp-1characterionic%2

B A X X


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Ionic crystals Consists of positively charged cations and negatively

charged anions.

Crystal structure is influenced by two factors: Magnitude of electric charge on each component ions Relative sizes of the cations and the anions

Crystal must be electrically neutral

Stable crystals form when anions surrounding a cation are all in contact with the cation.


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Ionic crystals: NaCl structure

Number of Cl ions/unit cell: (8 x 1/8)+(6x1/2) =4

Number of Na+ ions/unit cell: 1 + (12 x ) = 4

Total +ve charge = 4

Total ve charge = 4

Total charge = 4 4 =0Two inter penetrating FCC lattices (one for Na+ ions,

other for Cl ions); with one of them shifted by (,0,0)

4 Cl :(000), (0), (0), (0)4 Na+: (), (00), (00), (00)

rNa+/r Cl = 0.54 6fold co ordination


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Ionic crystals

CsCl structure

rCs+/r Cl = 0.91


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Ionic crystals

CaF2 structure

rCa2+/r F = 0.73


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Exception

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ZnS structure

rZn2+/r S2 = 0.48

When significant covalent character is present

S atoms: (000) (0.5 0.5 0) (0.5 0 0.5) (0 0.5 0.5) Zn atom: (.25 0.25 .75) (0.75

.25 0.25) (.25 .75 .25) (.75 .75 .75) 87 % covalency Similar structure: CdS, InAs,

InSb, ZnSe


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Perovskite Structure

ABO3 Structure

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i i l id


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Interstitial voidsVoids are common in all types of structures i.e also in close packed structure like fcc and hcp

1. Tetrahedral2. Octahedral

O h d l V id


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Octahedral Voids In the octahedral site, there are six

nearest neighbors (ions/atoms) equidistant from the center of the

void

This site is called octahedral since the atoms or ions surrounding the site

forms a eight sided octahedron In Fcc structure, there is one

octahedral site at the center of the

cell and there is one octahedral site on each cube edge of the cell

Therefore, fcc cell has total 4

octahedral sites or one octahedral site/atom This is a copyright material


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This illustration shows the octahedral site in an fcc lattice bottom. We have 12/4 +1 = 4positions per unit cell.

Here we have octahedral sites in the bcc lattice. We have 12/4 + 6/2 = 6 positions per unit cell

Size of the largest sphere that can fit in an octahedral void is 0.414r, where r is the radius of the spheres of the close packed array.

T t h d l V id


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Tetrahedral Voids

In the tetrahedral site, there are four nearest atoms/ionsequidistant from the center of the tetrahedron

A regular tetrahedron is formed when the centers of the fouratoms surrounding the void are joined

In fcc cell tetrahedral sites are located at (0.25, 0.25, 0.25positions)

There are eight tetrahedral sites in fcc cell ie 2 voids/atom

Size of largest sphere that can fit into the tetrahedral void ofa close packed structure = 0.225 r, where r is the radius of asphere of the close packed structure.


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The configuration on top is the tetrahedral position in the fcc

lattice. The black circles denote lattice points, the red circle marks one of the 8 the tetrahedral position.

The picture on the bottom shows the tetrahedral configuration for the bcc lattice. We have (6 4)/2 = 12 positions per unit cell.

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Structure of Crystalline Solids PartI