CRISTAL STRUCTURS

MATERIAL ENGINEER

Short-Range Order versus

Long-Range Order

Short-range order - The regular and predictable arrangement of the atoms over a

short distance - usually one or two atom

spacings.

Long-range order (LRO) - A regular repetitive arrangement of atoms in a solid which extends

over a very large distance.

Levels of atomic arrangements

Levels of atomic arrangements

in materials:

(a) Inert monoatomic gases

have no regular ordering of

atoms:

(b,c) Some materials, including

water vapor, nitrogen gas,

amorphous silicon and silicate

glass have short-range order.

(d) Metals, alloys, many

ceramics and some polymers

have regular ordering of

atoms/ions that extends through

the material.

Lattice, Unit Cells, Basis, and Crystal

Structures

Lattice - A collection of points that divide space into smaller equally sized segments.

Basis - A group of atoms associated with a lattice point.

Unit cell - A subdivision of the lattice that still retains the overall characteristics of the entire lattice.

Atomic radius - The apparent radius of an atom, typically calculated from the dimensions of the unit

cell, using close-packed directions (depends upon

coordination number).

Packing factor - The fraction of space in a unit cell occupied by atoms.

Basis

NET BASE

UNIT CELD

ATOMIC

RADIOS

PACKING FACTOR

The fourteen types of

Bravais lattices

grouped in seven

crystal systems. The

actual unit cell for a

hexagonal

(c) 2003 Brooks/Cole Publishing / Thomson Learning

Characteristic of the crystal system

Definition of the

lattice

parameters and

their use in

cubic,

orthorhombic,

and hexagonal

crystal systems.

(c) 2003 Brooks/Cole Publishing / Thomson Learning

Commoun crystal systems

For the 14 crystal system,

FACE CENTERED CUBIC (FCC)

BODY CENTERED CUBIC (BCC)

HEXAGONAL CLOSE PACKED (HCP)

They are the commun system on the nature, about 98%

of the metals exibiht it

FACE CENTERED CUBIC- FCC

CC o FCC

(Face cubic center)

Aluminium

Cooper

Gold

Nickel

Iron

FACE CENTERED CUBIC- FCC

How many atoms are in the unit celd?

What is the coordination number?

What is the packing factor?

What is the ratio lattice parameter and the atomic ratio?

Un octavo de tomo por

celda unidad

Each atom is in contact with other 12

Coordination number: Atoms or ions which are in contact with each

other

FACE CENTERED CUBIC- FCC

Atoms number in FCC

atoms = (1/2)x6 + (1/8)x8 = 4 atoms per unit celd

1/8

1/2

4

240

ra

Relationship between Atomic Radius and Lattice

Parameters

By Pitagora

h2= c2 + c2

But h = 4r y c = ao

Then (4r)2 = 2ao2

ao2 = (4r)2 / 2

FACE CENTERED CUBIC- FCC

Atomic Packing factor: Is the fraction of volume in a crystal

structure that is occupied by atoms

APF = Volume of everyone atoms on the unit celd

Volum of unit celd

Atoms are supposed as rigid sphere

FACE CENTERED CUBIC- FCC

BODY CENTERED CUBIC - BCC

All atoms are the

same element and

same size

Chrom

Molibden

Tantalium

Wolframium

Iron

BODY CENTERED CUBIC - BCC

How many atoms are in the unit celd?

What is the coordination number?

What is the packing factor?

What is the ratio lattice paremeter and the atomic ratio?

BODY CENTERED CUBIC - BCC

No atomos = (1/8)x8 + 1

2 atoms in BCC

1/8

1

Coordination number = 8

by Pitgoras

h2= c12 + c22

But h = 4r c1 = ao2

c2 = ao

(4r)2 = ao 2 + (ao 2 )2

(4r)2 = 3ao 2

BODY CENTERED CUBIC - BCC

3

40

ra

Relationship between Atomic Radius and Lattice

Parameters

Atomic packing factor

BODY CENTERED CUBIC - BCC

2

Hexagonal close packedHCP (Hexagonal

close packed)

Magnesium

Berilium

Cobalt

Titanium

Cinc

The unit cell can be

a complete hexagon

or one prism for the

six

HCP (a complete hexagon)

The unit cell can be a complete hexagon or one prism for

the six

Atoms by cell

To the hexagon

2*1/2= 1 face

3 center

6*1/3*1/2*2= 2 corner

Total 6 atoms by cell

HCP (a complete Hexagon)

Packing factor = 0.74 the same value in everyone FCC

Number coordination = 12

On HCP there are four different axes, a1, a2 a3

y c and the calculus of the atomic ratio is more

complicated to calculate

a1

a2

a3

c

Change of the cristal

structure

Change of the

volume

Change of the mechanical

properties

Allotropy - The characteristic of an element being able to exist in more than one crystal structure, depending

on temperature and pressure.

Polymorphism - Compounds exhibiting more than one type of crystal structure.

Allotropic or Polymorphic

Transformations

Fusion point

Deltha iron FCC

Gamma iron BCC

Alpha iron FCC

Room temperature

Determining the Density of BCC Iron

SOLUTION

Atoms/cell = 2, a0 = 0.2866 nm = 2.866 10-8 cm

Atomic mass = 55.847 g/mol

Volume of unit cell = = (2.866 10-8 cm)3 = 23.54 10-24 cm3/cell

Avogadros number NA = 6.02 1023 atoms/mol

3

0a

3

2324/882.7

)1002.6)(1054.23(

)847.55)(2(

number) sadro'cell)(Avogunit of (volume

iron) of mass )(atomicatoms/cell of(number Density

cmg

Determine the density of BCC iron, which has a lattice

parameter of 0.2866 nm.

Determining the Density of BCC IronDetermine the density of aluminium, which has a lattice parameter of

0.2866 nm.

2