CRYSTAL STRUCTURE

Introduction A crystal is a solid composed of atoms or other microscopic

particles arranged in an orderly repetitive array.

Further Solids can be broadly classified into Crystalline and Non-crystalline or Amorphous.

In crystalline solids the atoms are arranged in a periodic manner in all three directions, where as in non crystalline the arrangement is random.

Non crystalline substances are isotropic and they have no directional properties.

Crystalline solids are anisotropic and they exhibit varying physical properties with directions.

Crystalline solids have sharp melting points where as amorphous solids melts over a range of temperature.

Space lattice A Space lattice is defined as an infinite array of points in three A Space lattice is defined as an infinite array of points in three dimensions in which every point has surroundings identical to dimensions in which every point has surroundings identical to that of every other point in the array.that of every other point in the array.

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

Where a and b are called the repeated translation vectors.

a

b

Lattice planes

Lattice lines

Lattice points

Three dimensional lattice

UNIT CELL

The unit cell is a smallest unit which is repeated in space indefinitely, that generates the space-lattice.

BASIS A group of atoms or molecules identical in composition is called the Basis.

Lattice + basis = Crystal structure

CRYSTALLOGRAPHIC AXESCRYSTALLOGRAPHIC AXES The lines drawn parallel to the lines of intersection of The lines drawn parallel to the lines of intersection of

any three faces of the unit cell which do not lie in the any three faces of the unit cell which do not lie in the same plane are called same plane are called CrystallographicCrystallographic axesaxes..

PRIMITIVES:The a, b and c are the dimensions of an unit cell and are known as Primitives.

INTERFACIAL ANGLESINTERFACIAL ANGLESThe angles between three crystallographic axes are The angles between three crystallographic axes are known as known as Interfacial anglesInterfacial angles α ,β α ,β and and γγ..

βα

γ X

Y

Z

a

b

c

NOTE 1. Primitives decides the size of the unit cell.

2. Interfacial angles decides the shape of the unit cell.

PRIMITIVE CELLPRIMITIVE CELL

The unit cell is formed by primitives is called The unit cell is formed by primitives is called primitive cellprimitive cell..

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

βα

γ X

Y

Z

a

b

cLATTICE PARAMETERSLATTICE PARAMETERS

The primitives and interfacial The primitives and interfacial angles together called as angles together called as lattice lattice parametersparameters..

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

BRAVIAS LATTICESBRAVIAS LATTICES

There are only fourteen distinguishable ways of arranging There are only fourteen distinguishable ways of arranging the points independently in three dimensional space and the points independently in three dimensional space and these space lattices are known as Bravais lattices and they these space lattices are known as Bravais lattices and they belong to seven crystal systemsbelong to seven crystal systems

CRYSTAL TYPECRYSTAL TYPE BRAVAIS LATTICE BRAVAIS LATTICE

1. Cubic1. Cubic Simple Simple Body centeredBody centered Face centeredFace centered

2. Tetragonal2. Tetragonal Simple Simple Body centeredBody centered

3. Orthorhombic3. Orthorhombic Simple Simple Base centeredBase centered

Body centeredBody centered

Face centeredFace centered

4. Monoclinic4. Monoclinic Simple Simple Base centeredBase centered

5. Triclinic 5. Triclinic Simple Simple

6. Trigonal Simple6. Trigonal Simple

7. Hexgonal7. Hexgonal Simple Simple

SYMBOLSSYMBOLS Simple Simple CubicCubic

PP Base CenteredBase Centered C C

Body CenteredBody Centered

II Face CenteredFace Centered

FF

Crystal System Unit Vector Angles

Cubic Cubic a = b = c a = b = c αα = β = = β = γγ = 90 = 90˚̊

TetragonalTetragonal a = b ≠ c a = b ≠ c αα = β = = β = γγ = 90 = 90˚̊

Ortho rhombic a ≠ b ≠ cOrtho rhombic a ≠ b ≠ c αα = β = = β = γγ = 90 = 90˚̊

Mono clinicMono clinic a ≠ b ≠ c a ≠ b ≠ c αα = β = 90 ≠ = β = 90 ≠ γγ

TriclinicTriclinic a ≠ b ≠ c a ≠ b ≠ c αα ≠ β ≠ ≠ β ≠ γγ ≠ 90 ≠ 90˚̊

TrigonalTrigonal a = b = c a = b = c αα = = β β = = γγ ≠ 90≠ 90˚̊

HexagonalHexagonal a = b ≠ c a = b ≠ c αα = = β β = 90= 90˚,˚,γγ =120=120˚̊

12

34

5

6

Cubic Crystal System

a = b = c & a = b = c & αα = β = = β = γγ =90˚ =90˚

Tetragonal Crystal System

a = b ≠ c & a = b ≠ c & αα = β = = β = γγ =90˚ =90˚

12

34

5

6

Ortho Rhombic Crystal System

a ≠ b ≠ c & a ≠ b ≠ c & αα = β = = β = γγ =90˚ =90˚

Monoclinic Crystal System

a ≠ b ≠ c & a ≠ b ≠ c & αα = β = 90 ≠ = β = 90 ≠ γγ

Triclinic clinic Crystal System

a ≠ b ≠ c & a ≠ b ≠ c & αα ≠ β ≠ ≠ β ≠ γγ ≠90˚ ≠90˚

a = b = c & a = b = c & αα = = β β = = γγ ≠90˚≠90˚

Trigonal Crystal System

Hexagonal Crystal System

a = b ≠ c & a = b ≠ c & αα = = β β =90˚,=90˚,γγ =120˚ =120˚

NEAREST NEIGHBOUR DISTANCE The distance between the centers of two nearest neighboring atoms is called nearest neighbor distance.

CO – ORDINATION NUMBERCO – ORDINATION NUMBER Co-ordination number is defined as the number of Co-ordination number is defined as the number of equidistance nearest neighbors that an atom has in a given equidistance nearest neighbors that an atom has in a given structure.structure.

ATOMIC PACKING FACTORATOMIC PACKING FACTOR

Atomic packing factor is the ratio of volume occupied by the Atomic packing factor is the ratio of volume occupied by the atoms in an unit cell to the total volume of the unit cell. It is atoms in an unit cell to the total volume of the unit cell. It is

also called packing fractionalso called packing fraction..

cellunit a of volumeTotal

cellunit an in atoms by the occupied Volfactor Packing Atomic

Void Space

Vacant space left or unutilized space in unit cell , and more commonly known as interstitial space.

Void space = ( 1-APF ) X 100

SIMPLE CUBIC STRUCTURE - PACKING SIMPLE CUBIC STRUCTURE - PACKING FACTORFACTOR

1.1. Effective number of atoms per Effective number of atoms per unit cell (8 x 1/8) =1unit cell (8 x 1/8) =1

2.2. Atomic radius r = a / 2Atomic radius r = a / 2

3.3. Nearest neighbor distance Nearest neighbor distance 2r = a2r = a

4.4. Co-ordination number = 6Co-ordination number = 6

r r

a

6.Void space = (1-APF) X 1006.Void space = (1-APF) X 100

= (1-0.52)X 100= (1-0.52)X 100

= 48%= 48%

Example: PoloniumExample: Polonium..

3

3

3

3

41

3

2

41

3(2 )

0.52

(52%)

r

a

wherea r

r

r

5. Atomic packing factor

BCC STRUCURE – PACKING FACTOR

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

2. Atomic radius r = √3a /42. Atomic radius r = √3a /4

3. Nearest neighbor distance3. Nearest neighbor distance 2r =√3a/22r =√3a/2

4. Co-ordination number = 84. Co-ordination number = 8

a

a

aa2

a3

A B

C

D

r4

6.Void space = (1-APF) x 1006.Void space = (1-APF) x 100 = (1-0.68) x 100= (1-0.68) x 100 = 32%= 32%

Ex: Na, lithium and Chromium.Ex: Na, lithium and Chromium.

3

3

3

3

42

3

3

4

4 32 ( )

3 4( )

0.68

(68%)

r

a

wherer a

a

a

5.Atomic packing factor

FCC Crystal Structure – APF

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

2. Atomic radius r = a / 2√22. Atomic radius r = a / 2√2

3. Nearest neighbor distance3. Nearest neighbor distance 2r = a /√22r = a /√2

4. Co-ordination Number = 124. Co-ordination Number = 12

A a

C

B

a

r4

a2

12

34

6.Void space = (1-APF) X 1006.Void space = (1-APF) X 100 = (1-0.74) X 100= (1-0.74) X 100 = 26%= 26%

Ex: Cupper , Aluminum, Silver and LeadEx: Cupper , Aluminum, Silver and Lead

3

3

3

3

44

3

2 2

44 ( )

3 2 2( )

0.74

(74%)

r

a

ar

a

a

5.Atomic packing factor

Diamond Structure:Diamond is a combination of interpenetrating Fcc - Sub lattices along the body diagonal by 1/4th Cube edge.

1 23

4

5

6

Diamond - APF

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

2. Atomic radius r = √3a / 8.2. Atomic radius r = √3a / 8.

3. Nearest neighbor distance3. Nearest neighbor distance 2r = √3a / 4.2r = √3a / 4.

4. Co-ordination number = 4.4. Co-ordination number = 4.

a/4

a/4

a/4

x p

z

y

2r

x p

y

z

a/4

a/4

a/2

a

a

6. Void space = (1-APF) x 1006. Void space = (1-APF) x 100

= (1-0.34) x 100= (1-0.34) x 100

= 66%= 66%

Ge, Si and Carbon atoms are Ge, Si and Carbon atoms are possess this structurepossess this structure

3

3

3

3

48

3

3

8

4 38 ( )

3 8( )

0.34

(34%)

r

a

r a

a

a

5. Atomic packing factor

Hexagonal Close Packed Structure

1.1. Effective number of atoms per Effective number of atoms per unit cellunit cell2 x (6x 1/6) + 2 x 1/2 + 3 = 6.2 x (6x 1/6) + 2 x 1/2 + 3 = 6.

2. Atomic radius r = a / 2.2. Atomic radius r = a / 2.

3. Nearest neighbor distance 2r = a3. Nearest neighbor distance 2r = a

4. Co-ordination number = 12.4. Co-ordination number = 12.

5.Volume of the HCP unit cell

The volume of the unit cell determined by computing the area of the base of the unit cell and then by multiplying it by the unit cell height.

Volume = (Area of the hexagon) x (height of the cell)

0

2 0

2

6 ( )

16 ( )( sin 60 )

2

3 sin 60

3 3

2

ABC

a a

a

a

Area of the hexagon

If c is the height of the unit cell23 3

2V a c

A B

C

a60°

c/a ratio:The three body atoms lie in a horizontal plane at a height c/2 from the base or at top of the Hexagonal cell.

3

8

3

8

34

)2

()3

(

)2

()3

()2(

3

30cos

)2

()2(

2

2

22

2

222

222

20

222

a

c

a

c

aa

c

caa

car

ax

xAPQ

cxr

a

A

B

p

o

a

30°

c/22r

x

Nq

6. Void space = (1-APF) x 1006. Void space = (1-APF) x 100

= (1-0.74) x 100= (1-0.74) x 100

= 26%= 26%

Ex: Mg, Cd and Zn.Ex: Mg, Cd and Zn.

3

3

3

3

46

33 2

2

46 ( )

3 23 2( )

0.74

(74%)

r

a

ar

a

a

5. Atomic packing factor

Sodium Chloride Structure

Nacl Crystal is an ionic crystal. It consists of two Nacl Crystal is an ionic crystal. It consists of two FCC sub lattices.FCC sub lattices.

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

Each ion in a NaCl lattice has six nearest neighbor Each ion in a NaCl lattice has six nearest neighbor ions at a distance a/2. i,e its Co-ordination number is ions at a distance a/2. i,e its Co-ordination number is 6.6.

Na

Sodium Chloride structureSodium Chloride structure

Cl

Each unit cell of a sodium chloride as four sodium ions Each unit cell of a sodium chloride as four sodium ions and four chlorine ions. Thus there are four molecules in and four chlorine ions. Thus there are four molecules in each unit cell.each unit cell.

Cl : (0,0,0) (1/2,1/2,0) (1/2,0,1/2) (0,1/2,1/2)Cl : (0,0,0) (1/2,1/2,0) (1/2,0,1/2) (0,1/2,1/2)

Na : (1/2,1/2,1/2),(0,0,1/2) (0,1/2,0)(1/2,0,0)Na : (1/2,1/2,1/2),(0,0,1/2) (0,1/2,0)(1/2,0,0)

Structure of Cesium chloride:Structure of Cesium chloride:

• Cscl is an ionic Compound.Cscl is an ionic Compound.

• The lattice points of CsCl are two The lattice points of CsCl are two interpenetrating simple cubic lattices.interpenetrating simple cubic lattices.

• One sub lattice occupied by cesium One sub lattice occupied by cesium ions and another occupied by Cl ions.ions and another occupied by Cl ions.

• The co-ordinates of the ions are The co-ordinates of the ions are Cs : (000),(100),(010),(001),(110), Cs : (000),(100),(010),(001),(110),

(110),(011),(111). (110),(011),(111).

Cl : (1/2,1/2,1/2).Cl : (1/2,1/2,1/2).

Cs

Cl

Some important directions in Cubic Crystal

Square brackets [ ] are used to indicate the directions

The digits in a square bracket indicate the indices of that direction.

A negative index is indicated by a ‘bar’ over the digit .

Ex: for positive x-axes→[ 100 ]

for negative x-axes→[ 100 ]

zz

xx

yy[100][100]

[000][000]

[001][001]

[010][010]

Fundamental directions in crystals

Crystal planes & Miller indices

Reciprocals of intercepts made by the plane which are simplified into the smallest possible numbers or integers and represented by (h k l ) are known as Miller Indices.

(or)

The miller indices are the three smallest integers which have the same ratio as the reciprocals of the intercepts having on the three axes.

These indices are used to indicate the different sets of parallel planes in a crystal.

Procedure for finding Miller indices

Find the intercepts of desired plane on the three Co-ordinate axes. Let they be (pa, qb, rc).

Express the intercepts as multiples of the unit cell dimensions i.e. p, q, r. (which are coefficients of primitives a, b and c)

Take the ratio of reciprocals of these numbers i.e. a/pa : b/qb : c/rc. which is equal to 1/p:1/q:1/r.

Convert these reciprocals into whole numbers by multiplying each with their L.C.M , to get the smallest whole number.

These smallest whole numbers are Miller indices (h, k, l) of the crystal.

Important features of miller indices

When a plane is parallel to any axis, the intercept of the plane on that axis is infinity. Hence its miller index for that axis is zero.

When the intercept of a plane on any axis is negative a bar is put on the corresponding miller index.

All equally spaced parallel planes have the same index number (h, k, l).

If a plane passes through origin, it is defined in terms of a parallel plane having non-zero intercept.

The numerical parameters of the plane ABC are (2,2,1).

The reciprocal of these values are given by (1/2,1/2,1).

LCM is equal to 2.

Multiplying the reciprocals with LCM we get Miller indices [1,1,2].

x

y

z

2

2

1

y

z

x

[1 0 0] plane

Construction of [100] plane

)0,0,1(:indicesMiller

)1

,1

,1

1( are intercepts of lsR`eciproca

),,1( are Plane theof Intercepts

z

x

y

Set of [100] parallel planes

x

y

z

[010] plane.

)0,1,0(:indicesMiller

)1

,1

1,

1( are intercepts of lsR`eciproca

),1,( are Plane theof Intercepts

x

y

z

Set of ( 0 1 0 ) parallel planes

x

y

z

[ 001 ] plane

)1,0,0(:indicesMiller

)1

1,

1,

1( are intercepts of lsR`eciproca

)1,,( are Plane theof Intercepts

x

y

z

[ 001 ]

Set of ( 0 0 1 ) parallel planes

y

x

[110]

z

Construction of [110] plane

)0,1,1(:indicesMiller

)1

,1

1,

1

1( are intercepts of lsR`eciproca

),1,1( are Plane theof Intercepts

x

y

z

[110]

Set of [110] parallel planes

z

x

y

Construction of ( ī 0 0) Planes

)0,0,1(:indicesMiller

)1

,1

,1

1( are intercepts of lsR`eciproca

),,1( are Plane theof Intercepts

Intercepts of the planes are 1,1,1

Reciprocals of interceptsare 1/1,1/1,1/1

Miller indices:(111)

x

y

z

( 1 1 1 ) plane

Inter planner spacing of orthogonal crystal system:

Let ( h ,k, l ) be the miller indices of the plane Let ( h ,k, l ) be the miller indices of the plane ABC.ABC. Let ON=d be a normal to the plane passing through the Let ON=d be a normal to the plane passing through the

origin ‘0’.origin ‘0’. Let this ON make angles Let this ON make angles αα, , ββ and and γγ with x, y and z axes with x, y and z axes

respectively.respectively. Imagine the reference plane passing through the origin Imagine the reference plane passing through the origin

“o” and the next plane cutting the intercepts a/h, b/k and “o” and the next plane cutting the intercepts a/h, b/k and c/l on x, y and z axes. c/l on x, y and z axes.

x

y

Z

A

B

C

N

h

a k

b

l

c

o

od

OA = a/h, OB = b/k, OC = c/lOA = a/h, OB = b/k, OC = c/lA normal ON is drawn to the plane ABC from the A normal ON is drawn to the plane ABC from the origin “o”. the length “d” of this normal from the origin “o”. the length “d” of this normal from the origin to the plane will be the inter planar separation.origin to the plane will be the inter planar separation.

from∆ ONAfrom∆ ONA

from∆ ONBfrom∆ ONB

from∆ ONCfrom∆ ONC

Where cosWhere cosαα, cos, cosββ ,cos ,cosγγ are directional cosines of are directional cosines of αα,,ββ,,γγ angles.angles.

)(cos

)(cos

)(cos

lcd

OC

ONkbd

OB

ONhad

OA

ON

According to law of directional cosines According to law of directional cosines

2

2

2

2

2

2

2

2

2

2

2

22

222

222

1

1}{

1])(

[])(

[])(

[

1coscoscos

cl

bk

ah

d

c

l

b

k

a

hd

lcd

kbd

had

This is the general expression for inter planar separation for This is the general expression for inter planar separation for any set of planes.any set of planes.

In cubic system as we know that a = b = c, so the In cubic system as we know that a = b = c, so the expression becomes expression becomes

222 lkh

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