‘’THE STRUCTURE OF

CRYSTALLINE SOLIDS’’

IE-114 Materials Science and General Chemistry

Lecture-3

Outline

• Crystalline and Noncrystalline Materials

1) Single, Polycrystalline, Non-crystalline solids

2) Polycrystalline Materials

• Crystal Structures

1)Unit cells

2) Metallic crystal structures

4) Crystal Systems (Directions and Planes)

5) Linear and Planar Atomic Densities

Crystal Structures

Material classification can be made based on the regularity or

irregularity of atom or ion arrangement with respect to each other.

1) Crystalline Material

Single crystalline

Polycrystalline

Atoms are situated in a repeating or periodic array

over large atomic distances (Longe range order)

e.g.All metals, some ceramics and polymers

2) Noncrystalline (amorphous) Material

Long range atomic arrangement lacks

in this type of materials.

Single crystal diamond

(schematic view)

Two Dimensional View of Atomic Arrangements

Single crystal Polycrystal Amorphous

Single crystal

of CaF2

Formation of Polycrystals

1) During heavy deformation

2) During solidification from melt

t=t1 t=t2

t=t3 t=t3

Structure of the material

seen under microscope

Completely solid. Material contains

many grains(polycrystalline

material)

Liquid and small crystals Liquid and relatively larger

crystals

Solidification of a pure metal (T=Tm)

t1 t2 t3

(Tm)

Solid crystals

Liquid

Describing Crystal Structure

Unit cells can be imagined as the building block of the crystal structure.

Unit cells in general are paralelepipeds or prisms having three sets of

parallel faces, one is drawn within the aggregate of spheres.

The atoms or ions are thought as solid spheres with their sizes defined. This is called atomic hard sphere model. All atoms are identical in this model.

UNIT CELL: Smallest repeating group

Crystal Structures of Metallic Materials

Atoms tend to be densely packed

Atomic bonding is metallic and nondirectional.

No restrictions as to the number and position of nearest

neighbor atoms

Four simple crystalline structures found in metallic materials:

1) SC (Simple cubic) crystal structure

2) BCC (Body Centered Cubic) crystal structure

3) FCC (Face Centered Cubic) crystal structure

4) HCP (Hexagonal Close-Packed) Crystal Structure

Rare in nature due to poor packing (only Po has this structure)

Close-packed directions are cube edges.

Coordination number = 6

Simple Cubic (SC) Structure

The number of atoms per SC: (1/8)*8 = 1 atom/cell

* Coordination number: number of nearest neighbor or touching atoms.

The relationship between unit cell edge length (a) and atomic radius (R);

a=2R

(Arrangement of atoms in one SC unit cell)

a

Coordination number = 8

--Note: All atoms are identical;

the center atom is shaded

differently only for ease of

viewing.

Body Centered Cubic (BCC) Structure

The number of atoms per BCC: (1/8)*8 + 1= 2 atoms/cell

The relationship between unit cell edge length (a) and atomic radius (R);

a=(4/3)R

BCC unit cell

(Arrangement of atoms in one BCC unit cell)

Face Centered Cubic (FCC) Structure

FCC unit cell

Coordination number = 12

The number of atoms per FCC: (1/8)*8 + 6*(1/2)= 4 atoms/cell

The relationship between unit cell edge length (a) and atomic radius (R);

a=(22)R

--Note: All atoms are identical;

the center atom is shaded

differently only for ease of

viewing.

a (Arrangement of atoms in one FCC unit cell)

Hexagonal Closed Packed (HCP) Structure

Ideally c/a=1.633, but for some metals this ratio deviates from the ideal

value.

HCP Unit Cell

The top and bottom faces of the unit cell have six atoms that form

regular hexagons and a single atom in the center. Another plane provides

three additional atoms is situated between top and bottom planes.

Coordination number = 12

The number of atoms per HCP: (1/6)*12 + 2*(1/2) + 3 = 6 atoms/cell

Cl-

Na+

Structure of Compounds

Compounds often have similar close-packed structures.

Ionic bonding (NaCl)

The number of atoms per SC: [(1/8)*8+(1/2)*6] Cl ions + [(1/4)*12+1] Na ions

(4 Cl- + 4 Na+ ) ions/unit cell

The relationship between unit cell edge length (a) and atomic radius (R);

a=2RCl- + 2RNa+

Polymorphism

Atoms may have more than one type of crystal structure

Example : Iron (Fe)

heating cooling

FCC

BCC

Characteristics of Some Selected Elements

There are 7 different crystal structures;

Cubic, tetragonal, orthorhombic, rhombohedral, monoclinical, triclinic, hexagonal

Note that cubic system is the most symmetric, while triclinic is the least

one.

Crystal Systems

The unit cell geometry: x,y,z coordinate system is established with

its origin at one of the unit cell corners and axes coincide with the

edges of the paralelepiped extending from that corner, the origin.

There are six parameters to define the

geometry of the unit cell:

Three edge lengths : a, b, c

Three interaxial angles : α, β,

Crystal Systems

Also called lattice parameters.

Particular crystallographic direction is shown in a unit cell

The direction is a line between two points or a vector as shown

below:

Crystallographic Directions in

Cubic Crystals

Steps for defining a direction in a crystal system:

1) A vector is positioned such that it passes through the origin of the

coordinate system. Then you can move the vector if you keep the parallelism.

2) The length of the vector projection on each of the three axes is determined in terms of the unit cell dimensions (a, b, c).

3) The three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values.

4) Three indices are enclosed in square brackets as [uvw]

5) Remember to count for positive and negative coordinates based on the origin. When there is a negative index value, then show that by a bar over it, as

This vector has a component in –y direction. [111]

Example1:

Example2:

For cubic crystal structures, several nonparallel directions with different indices are equivalent.

(the spacing of atoms along each direction is the same)

(equivalent directions) <100> Family

FAMILY OF DIRECTIONS, <uvw>

Hexagonal Crystals:

There is a four-axis (Miller-Bravais) coordinate system used for this type

of structures.

Three a1, a2, and a3 axes are placed within a single plane (basal plane) and at 120°

angles to one another.

The z axis is perpendicular to the selected basal plane.

Some directions in HCP crystal structure

Except HCP, crytallographic planes are specified using three MILLER INDICES (hkl). Any two planes parallel to each other are equivalent and have same indices.

The determination of the h,k, and l index numbers are as follows:

1) If the plane passes through the selected origin, then construct a

new parallel plane or change the originto a corner of another unit cell.

2) Plane intersects or parallels each of the axes: the length of each axis is determined by using lattice parameters; a,b, and c.

3) Take the reciprocals of the lattice parameters. Therefore a plane that parallels an axis has a ZERO index. (1/infinity=zero)

4) You may then change these three numbers to the set of smallest integers using a common factor.

5) Report the indices as (hkl).

6) An intercept on the negative side of the origin is indicated by a bar over that index.

Crystallographic Planes in

Cubic Crystals

Example1:

Example2: Example3:

For cubic crystals: Planes and directions having the same indices are

parallel to one another.

A family of planes is formed by all those planes that are crystallographically equivalent, {100}, {111}. (for cubic structure)

{111} =

Family of Planes, {hkl}

Planes in Hexagonal Crystals:

Equivalent planes have the same indices as directions Four-index scheme is

used (hkil) and the index i is calculated by the sum of h and k through

i = - (h+k)

This scheme defines the orientation of

a plane in a hexagonal crystal.

Theoretical Density, ρ

Example:

Copper has an atomic radius of 0.128 nm, FCC crystal structure and an

atomic weight of 63.5 g/mol. Compute its density and compare the

answer with its measured density

Result: theoretical Cu = 8.89 g/cm3

Compare to actual: Cu = 8.94 g/cm3

Theoretical density (ρ) of elements can be found by the use of unit cell and

crystal structure

Theoretical Density of NaCl

NaCl unit cell

Atomic Packing Factor (APF)

Fraction of solid sphere volume in a unit cell

Metals generally have high APF to maximize the shielding provided by electron

cloud.

a

APF for a face-centered cubic(FCC) structure is 0.74

APF for BCC structure = 0.68

APF for HCP structure = 0.74

For FCC structure;

a=(22)R

The number of atoms per SC: (1/8)*8 + 6*(1/2)= 4 atoms/cell

Linear and Planar Density

Linear and planar atomic densities are one and two dimensional

analogs of atomic packing factor (APF).

Linear density: Fraction of line length in a particular

crystallographic direction that passes through atom centers

Planar density: Fraction of total crystallographic plane area that

is occupied by atoms. (The plane must pass through an atom’s center

for particular atom to be included in calculations)

Equivalency of directions and planes is related to the degree of

atomic spacing or atomic packing

Example1:

a= R

Linear density of [100] direction in BCC;

= Line length of atoms within [100] direction / Length of [100] direction

= 2R/4/3R = 0.866

Atomic arrangement on [100] direction;

[100] direction

x y

z

a = 4/3R

Example2:

Atomic arrangement on (110) plane in FCC;

Area of (110) plane;

= 2.a2 = 2(22)R2 = 82R2

Area of atoms inside (110) plane;

= ¼*4*R2 + ½*2*R2 = 2R2

Planar density of (110) plane in FCC = Area of atoms inside (110) plane / area of (110) plane

(110) plane

2.a = 4R

a

Planar density of (110) plane in FCC = 2R2 / 82R2 = 0.555

(110) plane

Anisotropy

Dependency of properties on direction: ‘’ANISOTROPY’’

It is associated with the variance of atomic or ionic spacing

with crystallographic direction.

Substances in which measured properties are independent of

the direction of measurement are ‘’ISOTROPIC’’

Example: For a single crystal material,

different mechanical properties are

observed in [100] and [111] directions

and also in some other directions.

• Incoming X-rays diffract from crystal planes.

• Measurement of: Critical angles, qc,

for X-rays provide atomic spacing, d.

Determination of Crystal Structure

n = 2dSinq

Bragg’s Law

• Atoms may assemble into crystalline or amorphous structures.

• We can predict the density of a material, provided we know the

atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC,

HCP).

• Material properties generally vary with single crystal

orientation (i.e., they are anisotropic),but properties are

generally non-directional (i.e., they are isotropic) in polycrystals

with randomly oriented grains.

Summary