JIF 419 MATERIALS SCIENCE

Webex 1STRUCTURE OF A

CRYSTAL

FUNDAMENTAL CONCEPTS• Crystalline material : atoms self-organize in a periodic 3D

pattern – all metals, many ceramic, certain polymers.

• Noncrystalline (amorphous) material : lacks a systematic atomic arrangement

• Crystal structure

• the manner in which atoms, ions or molecules are spatially

arranged.

• In the simplest crystals, the structural unit is a single

atom. The smallest structural unit may comprise many

atoms or molecules.

• described in terms of a lattice, with a group of atoms

attached identically to every lattice point.

• The group of atoms is called the basis; when repeated

in space it forms the crystal structure.

Crystalline Non-Crystalline

(amorphous)

lattice

UNIT CELLS• basic structural unit or building block that can describe the crystal structure. Repetition of the unit

cell generates the entire crystal.

• is a parallelepiped and can be packed periodically by integer displacements of the unit cellparameters. The unit cell parameters are the three coordinate lengths (axial length) (a, b, c)determined by placing the origin of the coordinate system at a lattice point, and the three angles(interaxial angles) (,,) subtended by the lattice cell axes.

Thus a unit cell with a=b=c and

===90 is a cube. The various

unit cells are generated from the

different values of a,b,c and ,,.

An analysis of how to fill space

periodically with lattice points shows

that only seven different unit cells are

required to describe all the possible

point lattices, and these are termed

the seven crystal systems.

CRYSTAL SYSTEMS • These seven crystal systems are

each defined using primitive unit

cells, in which each primitive cell

only contains a single lattice point

placed at the origin of the unit cell.

• Lattice types:

Primitive (P)

Lattice points lie on the corners

of the unit cell.

Body-Centered (I)

Lattice points lie at the corners

of the unit cell and one lattice

point lies in the middle of the

unit cell

Face Centered (F)

Lattice points lie at the corners

of the unit cell and one lattice

point lies in the middle of

every/each face of the unit cell.

POINTS, DIRECTION AND PLANES

• In crystalline materials, necessary to specify

points, direction and planes within unit cell

and in crystal lattice

• Three numbers (indices) used to designate

points, directions (lines) or planes based on

basic geometric notions

• The three indices are determined by placing

the origin at one of the corners of the unit

cell and the coordinate axes along the unit

cell edges.

• Position of any point in a unit cell is

given by its coordinates or distances

from the x, y and z axes in terms of

the lattice vectors a, b and c.

• Thus the point located at a/2 along x

axis, b/3 along y axis and c/2 along z

axis, as shown in the figure, has the

coordinates 1 1 1

2 3 2

POINT COORDINATES

CRYSTALLOGRAPHIC DIRECTIONS

• Crystallographic direction is defined as a line directed between two points or a

vector passing through the origin.

1. Always establish an origin

2. Determine coordinates of vector tail, pt. 1: x1,

y1, & z1; and vector head, pt. 2: x2, y2, & z2.

3. Subtract the head point coordinates with tail

point coordinates.

4. Normalize coordinate differences in terms of

lattice parameters a, b, and c:

5. Adjust to smallest integer values

6. Enclose in square brackets with no commas

[uvw]

ex:

pt. 1 x1 = 0, y1 = 0, z1 = 0

=> 1, 0, 1/2

=> [ 201 ]

z

x

y

=> 2, 0, 1

pt. 2

headpt. 1:

tail

pt. 2 x2 = a, y2 = 0, z2 = c/2

ab

c

CRYSTALLOGRAPHIC DIRECTIONS

-4, 1, 2

z

x

where the overbar represents a

negative index

[ 412 ]=>

y

Example 2:

pt. 1 x1 = a, y1 = b/2, z1 = 0

pt. 2 x2 = -a, y2 = b, z2 = c

=> -2, 1/2, 1

pt. 2

head

pt. 1:

tail Multiplying by 2 to eliminate the

fraction

CRYSTALLOGRAPHIC PLANES• The orientation of planes or faces in a crystal can be described in terms of their intercepts

on the three axes.

• Crystallographic planes are specified by three Miller indices.

• Miller indices is defined as the reciprocals of the intercepts made by the plane on the three

axes (h,k,l). Any two planes parallel to each other are equivalent and have identical

indices

Equivalence of cartesian coordinates, unit cell

dimension and miller indices

Procedure to determine Miller Indices

1) Identify the locations where the

plane intercepts the x, y, z axes

as the fractions of the unit cell edge

lengths a, b, c.

2) Infinity if the plane is parallel.

3) Take the reciprocal of the

intercepts.

4) Clear any fraction but do not reduce

to lowest terms.

6) Use parentheses to indicate

planes (hkl) again with a bar over

for the negative indices.

7) Families are indicated by {hkl}

Intercept: (1, 1, ∞)

Miller indices: (110)Intercept: (1, 1, 1)

Miller indices: (111)

Intercept: (1/2, 1, 0)

Miller indices: (210)

A B C

METALLIC CRYSTAL STRUCTURES• The atomic bonding in metals is non-directional

• no restriction on numbers or positions of nearest-

neighbour atoms

• large number of nearest neighbours and dense atomic

packing

• Important characteristic of crystal structure

• Coordination number – number of nearest neighbour

to a particular atom in the crystal (touching atoms)

• Atomic Packing Factor (APC) – the sum of the

sphere volumes of all atoms within a unit cell divided

by the volume of unit cell

• The most common types of unit cells are the

• Faced Centered Cubic (FCC)

• Body Centered Cubic (BCC)

• Hexagonal Close-Packed (HCP)

FACE-CENTERED CUBIC STRUCTURE (FCC)

• FCC has a unit cell of cubic geometry

• Atoms located at each corner and the

centers of all the cubic faces

• The coordination number is 12

• The atomic packing factor (APF) is 0.74

• FCC can be represented by a stack of

close-packed planes (planes with highest

density of atoms)

Two representations of the FCC unit cell

Hard sphere unit cell

representation

Reduced-sphere unit

cell representation

close-packed

structure

FCC coordination number

Hard sphere unit cell representation shows that:

• Number of atoms per unit cell, n = 4. (For an atom

that is shared with m adjacent unit cells, we only

count a fraction of the atom, 1/m)

6 face atoms shared by two cells: 6 x 1/2 = 3

8 corner atoms shared by eight cells: 8 x 1/8 = 1

1

3

2

4

F

C

Hard sphere unit cell representation

Using Pythagoras’ theorem we have

a2 + a2 = (4R)2

Thus by simplifying we have

For length,

a = 2R 2 = 2.83R

For volume,

V = a3 = (2R 2)3 = 8 × 2 × R3 2 = 16 R3 2

FCC unit cell length and volume

Atomic Packing Factor for FCC

APF = vol of atoms in unit cell/total unit cell

vol

Note that there are 4 atoms per FCC unit cell

Volume of atoms = 4 × =

Total unit cell volume = 16 R3 2

34

3R 316

3R

APF = = 0.74 3

3

16

3

16 2

R

R

BODY CENTERED CUBIC STRUCTURE (BCC)• The bcc has a cubic unit cell with atoms located

at all eight corners and one atom at the cube center

• The single atom at the center is wholly contained within the unit cell

• The coordination number is 8

• The atomic packing factor is 0.68

• Corner and center atoms are equivalent

Hard sphere unit cell

representation; the unit

cell contains 2 atoms

Reduced-sphere unit cell

representation

BCC COORDINATION NUMBER

Hard sphere unit cell representation shows that:

• Number of atoms per unit cell, n = 2

• Center atom (1) shared by no other cells: 1 x 1 = 1

• 8 corner atoms shared by eight cells: 8 x 1/8 = 1

Atomic Packing Factor for BCC

length = 4R =

a = 4 R

3

Close-packed directions:

3 a

a

a2

a3

APF = 2 (4/3 π ) R3

((4/√3) R)3

= 2 (4/3 π ) R3

(64/3√3)R3

= 0.6802

• The hard spheres touch one

another along cube diagonal

• the cube edge length,

a= 4R/√3 = 2.31 R

The number of nearest neighbours for each structure is 6, 8, and 12 respectively

Crystal Structure Coordination

Number

Atomic Packing

Factor

Simple Cubic (SC) 6 0.52

Body Centered

Cubic (BCC)

8 0.68

Face Centered Cubic

(FCC)

12 0.74

COMPARISON OF CRYSTAL STRUCTURE