Crystal Structre

8/3/2019 Crystal Structre

1/86

1

AUTONOMOU

CRYSTALLOGRAPHY

&

CRYSTAL STRUCTURES


2/86

2

CRYSTALLINE NATURE OF MATTER:

Elements and their chemical compounds are found inthree states, i.e. Solids, liquids and Gases.

Some solids are brittle, some are ductile, some are strong,some are weak, some are good conductors of heat andelectricity.

Solids can be broadly classified as Crystalline or Non-Crystalline solids


3/86

3

If the atoms or molecules in a solid are arrangedin some regular fashion then it is known asCrystalline solids .

If the atoms or molecules in a solid are arrangedin irregular fashion ,then it is known as Non-Crystalline solids.

Non- Crystalline is also called as Amorphoussolids


4/86


5/86


6/86

6

All the bond in Crystalline solids are equallystrong due to their symmetry .When a crystallinesolids is heated all the bonds are ruptured at aunique temperature but amorphous solids are notequally strong .When it is heated, the weakestbonds are ruptured first while stronger ones laterat high temperature .

Crystalline solids have a very sharp melting pointbut in amorphous solids does not have the sharpmelting point


7/86

7

Space lattice

A crystal is a 3-Dimensional body.

Crystals are made up of a regular and periodic

3-D patterns of atoms in space is called theCrystal Structure

The Crystal Structure may be described in termsof an idealized geometrical concept called aSpace lattice.


8/86

8

Definition of Space lattice

A space lattice is defined as an infinite

array of points in three dimensions inwhich every point has surroundings

identical to that of every other point in

the array.


9/86

9

An array of points such that every point has identical

surroundings

In Euclidean space infinite array

We can have 1D, 2D or 3D arrays (lattices)

Space Lattice

Translationally periodic arrangement of points in space is

called a lattice

or
http://space.ppt/http://space.ppt/


10/86

10

a

b

2D Lattices

Two basis vectors generate the lattice

There are three lattice parameters which describe this lattice

One angle:

Two distances: a, b

= 90in the current example

ba


11/86

11

Lattice lines

Lattice points

Three dimensional lattice


12/86

12

Basis and Unit Cell:

Basis : A set of one or more atoms aboutthe lattice point is calledBasis .orA group

of atoms or molecule identical incomposition is called the basis

basis

Lattice point

The crystal structure is formation when a basis of atoms is attached identically to

each lattice point.

Lattice+basis = crystal structure


13/86

13

Basis and Unit Cell:

Space Lattice + Basis = Crystal Structure

=


14/86

14

UNIT CELL

UNIT CELL:A unit cell is the smallest geometric figurethe repetition which gives actual crystal structure

Or

The unit cell may also be defined as the fundamentalelementary pattern of minimum number of atoms,

molecules or group of molecules which represent fully

all the characteristics of crystal.


15/86

15

Square Lattice


16/86

16

X X X X X X

X X X X X X

X X X X X X

X X X X X X

X X X X X X

X X X X X X


17/86

17

Lattice Parameters of an unit cellZ

X

Y

a

b

c
http://../books/e-book/3D%20Unit%20Cell.png


18/86

18

Lattice Parameters of an unit cell

CRYSTALLOGRAPHIC AXES:

The lines drawn parallel to the lines of

intersection of any three faces of the unitcell which do not lie in the same plane are

called crystallographic axes.

PRIMITIVES:

a, b and c are the dimensions of an unit cell and are

known as Primitives.


19/86

19

Lattice Parameters of an unit cell

INTERFACIAL ANGLES:-

The angles between three crystallographic axesare known as Interfacial angles , and .

The primitives a,b and c and Interfacial angles , and .are the basic lattice parameters .

The unit cell formed by the primitives a,b and c is

called primitive cell A primitive cell will have only one lattice point


20/86

20

Crystal Systems:

on the basis of the lengths and direction of theaxis of the symmetry may be classified as Seven

Crystal Systems.

Cubic system:

* Lattice parameters

* All three sides are equal,

* All three angles , a b c

e.g :Nacl, diamond etc..

90


21/86

21

Crystal Systems

Tetragonal System:

Lattice Parameters

* two sides equal

* all three right angles equal

a b c

90

e.g: NiSo4 etc..


22/86

22

Crystal Systems

Orthorhombic system:

Lattice parameters

* All three sides different

* All three right angles equal

a b c

90

Eg; KNO3


23/86

23

Monoclinic System :

Lattice parameters

* All three sides

different

Two right angles ,

third arbitrary,

a b c

90

E.g; FeSo4


24/86

24

Crystal Systems

Triclinic System:

Lattice parameters

*All three sides

different

* All three angles

different

a b c

Eg. CuSO4


25/86

25

Crystal Systems

Trigonal(rhombohedral):

Lattice parameters

*All three sides equal

All three angles ,

of arbitrary value ,

a b c

90

E.g CaSO4,Quartz,etc.


26/86

26

Some times an alternate hexagonal cell

is used instead of the Trigonal Cell


27/86

27

Crystal Systems

Hexagonal System:

Lattice parameters

* Two sides equal,

third arbitrary

* Two right angles ,

third angle

a b c

90 , 120

a b c

E.g SiO2 ,etc..


28/86

28

Crystal Systems

* Cubic a = b=c = = =90

* Tetragonal a = b c = = =90

* Orthorhombic a b c = = =90

* Monoclinic a b c = = 90

* Triclinic a b c 90

* Trigonal a = b = c = = 90

* Hexagonal a=b=c

=

=90,=120


29/86


30/86

30

Simple Cubic or (Primitive) Or Cubic P

type Lattice:

There is one lattice point at each of eight

corners of the unit cell. There is no latticepoint inside the unit cell.


31/86

31


32/86

32

Lattice Point per Unit cell:

Simple Cubic lattice :one lattice point at eachof the eight corners of the unit cell. Each latticepoint forming the simple cubic lattice is amember of 8 surrounding cell.

the share of each lattice point to the cubiclattice=1/8.

Number of lattice points in unit cell=(8*1/8)=1

Hence it is Primitive.


33/86

33

Body Centred Cubic (bcc) I Lattice:

* There is one lattice point at each of the

eight corners and one lattice point at centreof each cell. So there is a lattice point at

the centre of each unit cell


34/86

34


35/86

35

lattice Points Per Unit Cell:

There are eight lattice points at the eight corners and eachof them is a member of 8 surrounding cells.

Also there in one lattice point at the centre of the body.

** total number of body centre of lattice point per unitcell =1

** total number of corner lattice point attached to unitcell=(8*1/8)=1

** total number of lattice point in unit cell 1+1=2

*** hence its not primitive.


36/86

36

Face Centred Cubic or Cubic F Lattice:

* There is one lattice Point at each of the

eight corners and one lattice point at thecentres of each of the six faces of the unit

cell.

* Thus there is an extra point point in at thecentre of each face.


37/86

37


38/86

38

LatticePoints Per Unit Cell:

unit cell consists of 8 corner lattice point ,eachbeing a member of 8 surrounding cells.

*** Six face lattice points,each being a memberof two cells.

** total no. of corner lattice point concerned withunit cell=8*1/8=1.

** total no.of face centred lattice atomsconcerned with unit cell=6*1/2=3

*** total no.of lattice points concerned with unitcell=1+3=4


39/86

39

Position of lattice

points

Effective number of Lattice points /

cell

1 P 8 Corners = [8 (1/8)] = 1

2 I8 Corners+

1 body centre

= [1 (for corners)] + [1 (BC)] = 2

3 F

8 Corners

+

6 face centres

= [1 (for corners)] + [6 (1/2)] = 4

4

A/

B/

C

8 corners

+

2 centres of opposite

faces

= [1 (for corners)] + [2 (1/2)] = 2


40/86

40

1 Cubic Cube

P I F C

P

I

F90 a b c

BRAVAIS LATTICES


41/86

41

P I F C

2 Tetragonal Square Prism (general height)

IP

a b c 90


42/86

42

P I F C

3Orthorhomb

icRectangular Prism (general height)

PI

F

C

a b c 90


43/86

43

6 Monoclinic Parallogramic Prism

I F

P I F C

a b c

P C

90 a b c


44/86

44

P I F C

7 Triclinic Parallelepiped (general)

a b c


45/86

45

P I F C

5 TrigonalParallelepiped (Equilateral,

Equiangular)

90 a b c

Rhombohedral


46/86

46

P I F C

4 Hexagonal 120 Rhombic Prism

Note: there is only one type of hexagonal

lattice (the primitive one)

a b c 90 , 120


47/86

47

BRAVAIS LATTICES

**Cubic Simple p

Body centred I

Face centred F

**Tetragonal simple p

Body centered I

**Orthorhombic simple pBase centred C

Body centred I

Face centred F

**Monoclinic sim ple p

Base centred c

**Triclinic simple p

**Trigonal simple p

**Hexagonal simple p

`


48/86

48

The distance between the centers of two nearest

neighboring atoms is called nearest neighbordistance.

Atomis radius (r). Half the distnce between the NND

i.e 2r/2=r

NEAREST NEIGHBOUUR DISTANCE

(2r):


49/86

49

CO-ORDINATION NUMBER:

Co-ordination number is defined as thenumber of equidistance nearest neighbors

that an atom has in a given structure.


50/86

50

ATOMIC PACKING FACTOR:

Atomic packing factor is the ratio of volumeoccupied by the atoms in an unit cell to the total

volume of the unit cell. It is also called packingfraction.

V

vA.P.F =


51/86

51

Void space:

Void space in the unit cell is the vacantspace left or unutilized space in unit cell ,and more commonly known as interstitialspace.

Void space = ( 1-APF ) X 100


52/86

52

SIMPLE CUBIC STRUCTURE-

Effective number of atoms per unit cell

(8 x 1/8) =1

Nearest neighbour distance 2r = a

. Atomic radius r = a / 2

co-ordination number = 6 a


53/86

53

SIMPLE CUBIC STRUCTURE


54/86

54

SIMPLE CUBIC STRUCTURE

5. Atomic packingfactor

Vol. of all atoms in the unit cell

Vol.of the unit cell

6.Void space = (1-APF) X 100

= (1-0.52)X 100

=48%

Ex: polonium.

3

3

3

3

41

3

2

41

3

(2 )0.52

(52%)

r

a

wherea r

r

r


55/86

55

BCC STRUCTURE

Close packed directions are cube diagonals.


56/86

56

BCC STRUCTURE

Effective number of atoms per unit cell (8 x 1/8) + 1 =2

Atomic radius r = 3a /4

Nearest neighbour distance

2r =3a/2

Co-ordination number = 8

C

Aa

aB

a2

a3

D

D

C

a


57/86

57

BCC STRUCTURE

5.Atomic packing factor

6.Void space = (1-APF) x 100

= (1-0.68) x 100

= 32%

Ex:Na, lithium andChromium.

3

3

3

3

42

3

34

4 32 ( )

3 4

( )

0.68

(68%)

r

a

wherer a

a

a


58/86

58

FCC Crystal Structure

Effective number of atoms per unit cell (8 x 1/8) + 1/2 X 6 = 4

2.Atomic radius r = a / 22

3.Nearest neighbour distance

2r = a /2

4.Co-ordination number = 12

A B

C

4r


59/86

59


Close packed directions are face diagonals.


60/86

60


How the co-ordination number is 12 in FCCStructure ?

Ans. Centre of each face as one atom. this centreatom touches four corner atoms in its plane, four

body centered atoms in each of the two planes oneither side of its plane. thus the co-ordinationnumber is 12.


61/86

61

FCC coordination number

4

C

1

2

35

6

7

89

10

11

12

a

a /2

a /2

a


62/86

62

5.Atomic packing factor

6.Void space = (1-APF) X 100

= (1-0.74) X 100

= 26%

Ex: Copper , Aluminum, silver and Led

3

3

3

3

443

2 2

44 ( )

3 2 2

( )

0.74

(74%)

r

a

a

r

a

a


63/86

63

Hexagonal (Unit cell)

H l l k d


64/86

64

Hexagonal close packed

structure: Effective number of atoms per unit cell

2 x (6x 1/6) + 2 x 1/2 + 3 = 6.

2.Atomic radius r = a / 2.

3.Nearest neighbor distance

2r = a

4.Co-ordination number = 12.

CC/2

a


65/86

65

HCP Co ordination Number


66/86

66

Calculation of c/a ratio

Let c be the hight of the unit cell and abe its edge, the three body atoms lie in a

horizontal plane at height c/2 from the

bottom layer

a

a

cc

B

A

O

B

Ox

A30

0

C

XA

/

O/

a2

cCX


67/86

67

A/AB=300

From the figure, Triangle ABA/

Cos

=A

300

A/

AB

A

=

A/

But AX=2/3 A=A/

23a

3

a

=


68/86

683

8

32

4

2

3

2

3

2

2

22

2

222

222

a

c

actherefore

cbutCX

aa

a

AXACCX

CXAXACIn Triangle AXC,


69/86

69

Atomic packing factor

Area of the base=6* Area of the triangle ABO

6*1/2*AB*OO/

2/332

a

2/332

a* C

aa

2

3**3

VOL. OF UNIT CELL IS=Area of the base * hight


70/86

7000

2

3

2

3

74

23

8

3

33

2

2.

33

16

2

33

3

4*6

c

a

Since

c

a

arSub

ca

r

ca

r A.P.F=

Eg; Mg,Zn,Cd


71/86

71

Diamond structure:

Diamond structure:

Diamond is a combination of twointerpenetrating FCC - sub lattices alongthe body diagonal by 1/4thCube edge.


72/86

72


73/86

73

1 23

4

5

6


74/86

74

x p

y

z

a/4

a/4

a/2

a

a


75/86

75

Diamond - APF

Effective number of atoms perunit cell (8 x 1/8) + 1/2 X 6+ 4 = 8.

2.Atomic radius r = 3a / 8.

3.Nearest neighbor distance 2r = 3a / 4.

4.Co-ordination number = 4.

a/4

a/4

a/4

x p

z

y

2r

8

a


76/86

76

Diamond Structure


77/86

77

Diamond Structure

* The fraction such as 1/4,1/2,3/4 denote

the height above the base in units of cubeedge.

1/2

1/2

0 1/2 0

00

1/4

0

3/4

1/2

1/43/4


78/86

78

5. Atomic packing

factor

3

3

3

3

48

3

38

4 38 ( )

3 8

( )

0.34

(34%)

r

a

r a

a

a

6. Void space = (1-APF) x 100

= (1-0.34) x 100

= 66%Ge, Si and Carbon atoms are

possess this structure


79/86

79

Structure of a zinc sulphide:

ZnS structure is interpenetration of two fcclattices where One of zinc and other is of

sulpher atoms. This structure is similar to

Diamond crystal structure


80/86

80

Zn

S


81/86


82/86

82

Sodium chloride structure:

Nacl crystal is an ionic crystal.

In Nacl structure Na+ and Cl- are situated side by side

NaCl Structure is consists of two fcc sub lattices.

One of the chlorine ion having its origin at the(o, o, o) point. and other of the sodium ions having itsorigin at (a/2,0,0).

Each ion in a Nacl lattice has six nearest neighbor ions ata distance a/2.i,e its co-ordination number is 6.


83/86

83

2a

2

a


84/86

84

2

a

a

a2

a

2

a


85/86

85

Structure of cesium chloride

CsCl is an ionic compound. Ithas simple cubic structure.

The Unit cell of CsCl as shownIn Fig.

From the figure Cs+ ions aresituated at the corners of thecubic cell and Cl- ions at thebody centre or vice versa

So the crystal may be consideredto be a combination of twosimple cubic Sublattice one is

cs+ ions and other of Cl- ion.

The co-ordination number is 8 Cs+

Cl


86/86

Crystal Structre

Documents

Transcript of Crystal Structre