CRYSTALLINE STATE

INTRODUCTIONElectro

nic structur

e

Bonding

State of aggrega

tion

Octet stability Primary:1. Ionic2. Covalent3. Metallic4. Van der Waals

Secondary:1. Dipole-dipole2. London dispersion3. Hydrogen

GasLiquidSolid

STATE OF MATTER

GAS LIQUID SOLID

• The particles move rapidly

• Large space between particles

• The particles move past one another

• The particles close together

• Retains its volume

• The particles are arranged in tight and regular pattern

• The particles move very little

• Retains its shape and volume

CLASSIFICATION OF SOLID BY ATOMIC ARRANGEMENT

Ordered

• regular• long-range• crystalline• “crystal”• transparent

Disordered

• random*• short-range*• amorphous• “glass”• opaque

Atomic arrangementOrderName

CRYSTAL SYSTEM

EARLY CRYSTALLOGRAPHY

ROBERT HOOKE (1660) : canon ball

Crystal must owe its regular shape to the packing of spherical particles (balls) packed regularly, we get long-range order.

NEILS STEENSEN (1669) : quartz crystal

All crystals have the same angles between corresponding faces, regardless of their sizes he tried to make connection between macroscopic and atomic world.

If I have a regular cubic crystal, then if I divide it into smaller and smaller pieces down to an atomic dimension, will I get a cubic repeat unit?

RENĖ-JUST HAŪY (1781): cleavage of calcite

• Common shape to all shards: rhombohedral

• Mathematically proved that there are only 7 distinct space-filling volume elements

7 CRYSTAL SYSTEMS

CRYSTALLOGRAPHIC AXES

3 AXES 4 AXESyz = xz = xy =

yz = 90xy = yu = ux = 60

THE SEVEN CRYSTAL SYSTEMS

(rombhohedral)

SPACE FILLING

TILING

AUGUST BRAVAIS (1848): more math

How many different ways can we put atoms into these 7 crystal systems and get distinguishable point environment?

He mathematically proved that there are 14 distinct ways to arrange points in space

14 BRAVAIS LATTICES

The Fourteen Bravais Lattices

Simple cubic Body-centeredcubic

Face-centeredcubic

1 32

4

Simple tetragonal Body-centered tetragonal

5

Simple orthorhombic Body-centered orthorhombic

6 7

8 9

10 1211

1413

Hexagonal

A point lattice

Repeat unit

z

x

y

A unit cell

O

b

a

c

a, b, c

, , Lattice parameters

CRYSTAL STRUCTURE(Atomic arrangement in 3 space)

BRAVAIS LATTICE(Point environment)

BASIS(Atomic grouping at each lattice point)

EXAMPLE: properties of cubic system*)

BRAVAIS LATTICE

BASIS CRYSTAL STRUCTURE

EXAMPLE

FCC atom FCC Au, Al, Cu, Pt

molecule FCC CH4

ion pair(Na+ and Cl -)

Rock salt NaCl

Atom pair DC (diamond crystal)

Diamond, Si, Ge

C

C109

*) cubic system is the most simplemost of elements in periodic table have cubic crystal structure

CRYSTAL STRUCTURE OF NaCl

CHARACTERISTIC OF CUBIC LATTICES

SC BCC FCCUnit cell volume a3 a3 a3

Lattice point per cell 1 2 4Nearest neighbor distance a a3 / 2 a/2Number of nearest neighbors (coordination no.)

6 8 12

Second nearest neighbor distance

a2 a a

Number of second neighbor 12 6 6a = f(r) 2r 4/3 r 22 r

or 4r = a4 a3 a2Packing density 0.52 0.68 0.74

volumetotalatomsofvolume

densitypacking

3

3

344

a

r

EXAMPLE: FCC

3

3

223

44

r

r

74.02322

344

3

3

r

r

FCC74% matter (hard sphere model)

26% void

• In crystal structure, atom touch in one certain direction and far apart along other direction.

• There is correlation between atomic contact and bonding.

• Bonding is related to the whole properties, e.g. mechanical strength, electrical property, and optical property.

• If I look down on atom direction: high density of atoms direction of strength; low density/population of atom direction of weakness.

• If I want to cleave a crystal, I have to understand crystallography.

CRYSTALLOGRAPHIC NOTATION

POSITION: x, y, z, coordinate, separated by commas, no enclosure

O: 0,0,0

A: 0,1,1

B: 1,0,½

B

A

z

x

y

Unit cell

O

a

DIRECTION: move coordinate axes so that the line passes through origin

Define vector from O to point on the line Choose the smallest set of integers No commas, enclose in brackets, clear fractions

OB 1 0 ½ [2 0 1]

AO 0 -1 -1 110

B

A

z

x

y

Unit cell

O

Denote entire family of directions by carats < >

e.g.

all body diagonals: <1 1 1>

111 111 111 111

111 111 111 111

all face diagonals: <0 1 1>

110 110 110

101

110

101 101 101

011 011 011 011

all cube edges: <0 0 1>

100 100 010 001 010 001

MILLER INDICESFor describing planes.

Equation for plane: 1cz

by

ax

where a, b, and c are the intercepts of the plane with the x, y, and z axes, respectively.

Let:

so that

No commas, enclose in parenthesis (h k l) denote entirely family of planes by brace, e.g. all faces of unit

cell: {0 0 1}

ah

1

cl

1

bk

1

1 lzkyhx

100 100 001 etc.

MILLER INDICES

a

b

cIntercept at

Intercept at a/2

Intercept at b

Miller indices: (h k l)

121

(2 1 0)

Parallel to z axes

(h k l) [h k l]

[2 1 0]

(2 1 0)

Miller indices of planes in the cubic system

(0 1 0) (0 2 0)

011 111 210

011

Many of the geometric shapes that appear in the crystalline state are some degree symmetrical.

This fact can be used as a means of crystal classification.

The three elements of symmetry:

Symmetry about a point (a center of symmetry)

Symmetry about a line (an axis of symmetry)

Symmetry about a plane (a plane of symmetry)

CRYSTAL SYMMETRY

SYMMETRY ABOUT A POINT

A crystal possesses a center of symmetry when every point on the surface of the crystal has an identical point on the opposite side of the center, equidistant from it.

Example: cube

If a crystal is rotated 360 about any given axis, it obviously returns to its original position.

If the crystal appears to have reached its original more than once during its complete rotation, the chosen axis is an axis of symmetry.

SYMMETRY ABOUT A LINE

DIAD AXIS

TRIAD AXIS

TETRAD AXIS

HEXAD AXIS

AXIS OF SYMMETRY

• Rotated 180• Twofold rotation axis

• Rotated 120• Threefold rotation axis

• Rotated 90• Fourfold rotation axis

• Rotated 60• Sixfold rotation axis

THE 13 AXES OF SYMMETRY IN A CUBE

A plane of symmetry bisects a solid object in a such manner that one half becomes the mirror image of the other half in the given plane.

A cube has 9 planes of symmetry:

SYMMETRY ABOUT A PLANE

THE 9 PLANES OF SYMMETRY IN A CUBE

Cube (hexahedron) is a highly symmetrical body as it has 23 elements of symmetry (a center, 9 planes, and 13 axis).

Octahedron also has the same 23 elements of symmetry.

ELEMENTS OF SYMMETRY

Combination forms of cube and octahedron

IONIC

COVALENT

MOLECULAR

METALLIC

SOLID STATE

BONDING

• Composed of ions• Held by electrostatic force• Eg.: NaCl

• Composed of neutral atoms• Held by covalent bonding• Eg.: diamond

• Composed of molecules• Held by weak attractive force• Eg.: organic compounds

SOLID STATE BONDING

• Comprise ordered arrays of identical cations

• Held by metallic bond• Eg.: Cu, Fe

ISOMORPH Two or more substances that crystallize in almost

identical forms are said to be isomorphous.

Isomorphs are often chemically similar.

Example: chrome alum K2SO4.Cr2(SO4)3.24H2O (purple) and potash alum K2SO4.Al2(SO4)3.24H2O (colorless) crystallize from their respective aqueous solutions as regular octahedral. When an aqueous solution containing both salts are crystallized, regular octahedral are again formed, but the color of the crystals (which are now homogeneous solid solutions) can vary from almost colorless to deep purple, depending on the proportions of the two alums in the solution.

CHROME ALUM CRYSTAL

A substance capable of crystallizing into different, but chemically identical, crystalline forms is said to exhibit polymorphism.

Different polymorphs of a given substance are chemically identical but will exhibit different physical properties, such as density, heat capacity, melting point, thermal conductivity, and optical activity.

Example:

POLYMORPH

ARAGONITE

CRISTOBALITE

Polymorphic Forms of Some Common Substances

Material that exhibit polymorphism present an interesting problem:

1. It is necessary to control conditions to obtain the desired polymorph.

2. Once the desired polymorph is obtained, it is necessary to prevent the transformation of the material to another polymorph.

Polymorph 1Poly-

morph 2

Polymorphic transition

In many cases, a particular polymorph is metastable

Transform into more stable state

Relatively rapid infinitely slow

Carbon at room temperature

Diamond(metastable)

Graphite(stable)

POLYMORPH

MONOTROPIC ENANTIOTROPIC

One of the polymorphs is the stable form at all

temperature

Different polymorphs are stable at different

temperature

The most stable is the one having lowest

solubility

CRYSTAL HABIT

In nature perfect crystals are rare. 

The faces that develop on a crystal depend on the space available for the crystals to grow. 

If crystals grow into one another or in a restricted environment, it is possible that no well-formed crystal faces will be developed. 

However, crystals sometimes develop certain forms more commonly than others, although the symmetry may not be readily apparent from these common forms. 

The term used to describe general shape of a crystal is habit.

Crystal habit refers to external appearance of the crystal.

A quantitative description of a crystal means knowing the crystal faces present, their relative areas, the length of the axes in the three directions, the angles between the faces, and the shape factor of the crystal.

Shape factors are a convenient mathematical way of describing the geometry of a crystal.

If a size of a crystal is defined in terms of a characteriza-tion dimension L, two shape factors can be defined:

Volume shape factor : V = L3

Area shape factor : A = L2

Some common crystal habits are as follows.

Cubic - cube shapes

Octahedral - shaped like octahedrons, as described above.

Tabular -  rectangular shapes.

Equant - a term used to describe minerals that have all of their boundaries of approximately equal length.

Fibrous -  elongated clusters of fibers.

Acicular -  long, slender crystals.

Prismatic -  abundance of prism faces.

Bladed -  like a wedge or knife blade

Dendritic - tree-like growths

Botryoidal - smooth bulbous shapes

Internal structure External habit ?=

Tabular Prismatic Acicular

External shape of hexagonal crystal displaying the same faces

Crystal habit is controlled by:

1. Internal structure

2. The conditions at which the crystal grows (the rate of growth, the solvent used, the impurities present)

Variation of sodium chlorate crystal shape grown: (a) rapidly; (b) slowly

(a) (b)

Sodium chloride grown from: (a) pure solution; (b) Solution containing 10% urea