Crystal Structure (21!10!2011)

8/3/2019 Crystal Structure (21!10!2011)

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CRYSTALSTRUCTURE



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 thearrangement is random.

Non crystalline substances are isotropic and they have nodirectional properties i.e., the magnitude of specific propertydoes not change with the change of direction.

Crystalline solids are anisotropic and they exhibit varyingphysical properties with directions.

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



Space latticeAn infinite array of points in three dimensions in which every An infinite array of points in three dimensions in which every

point has surroundings identical to that of every other point in point has surroundings identical to that of every other point in

the array is k nown as the array is k nown as Space latticeSpace lattice..

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

XX XX XX XX XX

Where a and b arecalled the repeatedtranslation vectors.

a

b



Lattice planes

Lattice lines

Lattice points

Three dimensional lattice



UNIT CELL

The minimum fundamental pattern or thesmallest block or minimum geometrical pattern

which exists in a crystal is called Unit cell.

Note: When unit cell is repeated indefinitely, in

all three dimensions a crystal is generated.

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 theany three faces of the unit cell which do not lie in thesame plane are calledsame plane are called CrystallographicCrystallographic axesaxes..

PR IMITIVES:

Intercepts made by the unit cell along thecrystallographic axes are called Primitives. These arerepresented by the notations a, b and c.

INTER FACIAL ANGLESINTER FACIAL ANGLES

The angles between three crystallographic axes areThe angles between three crystallographic axes areknown asknown as InteraxialInteraxial angles orangles or Interfacial angles and Interfacial angles and are denoted by the symbolsare denoted by the symbols , , and and ..



X

Y

Z

b

a

c

NOTE

1. Primitives decides the size of the unit cell.

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

X

X

X



PR IMITIVE CELLPR IMITIVE CELL

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

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

X

Y

Z

b

a

c

LATTICE PARAMETERSLATTICE PARAMETERS

The primitives and interfacialThe primitives and interfacial

angles together are calledangles together are called latticelattice parameters parameters..



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

Example for 2D lattic



BRAVIASBRAVIAS LATTICESLATTICES

ThereThere areare onlyonly fourteenfourteen distinguishabledistinguishable waysways of of arrangingarranging

thethe points points independentlyindependently inin threethree dimensionaldimensional spacespace andandthesethese spacespace latticeslattices areare knownknown asas BravaisBravais latticeslattices andand theythey

belong belong toto sevenseven crystalcrystal systemssystems



CRYSTAL TYPECRYSTAL TYPE BRAVAIS LATTICEBRAVAIS LATTICE

1. Cubic1. Cubic Simple Simple

Body centeredBody centered

Face centeredFace centered

2. Tetragonal2. Tetragonal SimpleSimple

Body centeredBody centered

3. Orthorhombic3. Orthorhombic Simple Simple

Base centeredBase centeredBody centeredBody centered

Face centeredFace centered

4. Monoclinic4. Monoclinic SimpleSimple

Base centeredBase centered

5. Triclinic 5. Triclinic SimpleSimple

6. Trigonal Simple6. Trigonal Simple

7. Hexgonal7. Hexgonal SimpleSimple

SYMBOLSSYMBOLS

SimpleSimple CubicCubic PP

Base CenteredBase Centered CC

Body CenteredBody Centered II

Face CenteredFace Centered FF



Crystal System Unit Vector Angles

CubicCubic a = b = ca = b = c = == = = 90= 90ÛÛ

TetragonalTetragonal a = b ca = b c = == = = 90= 90ÛÛ

Ortho rhombic a b cOrtho rhombic a b c = == = = 90= 90ÛÛ

R hombohedral or TrigonalR hombohedral or Trigonal a = b = ca = b = c == == 90 90ÛÛ

HexagonalHexagonal a = b ca = b c == = 90= 90Û,Û, =120=120ÛÛ

Mono clinicMono clinic a b ca b c = = 90 = = 90

TriclinicTriclinic a b ca b c 90 90ÛÛ

r



123

4

5

6

Cubic Crystal System

a = b = c &a = b = c & = == = =90Û=90Û



Tetragonal Crystal System

a = b c &a = b c & = == = =90Û=90Û



12

34

5

6

Ortho R hombic Crystal System

a b c &a b c & = == = =90Û=90Û



Monoclinic Crystal System

a b c &a b c & = = 90 = = 90



Triclinic clinic Crystal System

a b c &a b c & 90Û90Û



a = b = c &a = b = c & == == 90Û90Û

Trigonal Crystal System



Hexagonal Crystal System

a = b c &a = b c & == =90Û,=90Û, =120Û=120Û



NEAREST NEIGHBOUR DISTANCE

The distance between the centers of any two nearest

neighboring atoms is called nearest neighbor distance.

COCO ± ± ORDINATIONORDINATION NUMBER NUMBER

TheThe number number of of nearestnearest equidistantequidistant neighborsneighbors thatthat anan atomatomhashas inin aa givengiven structurestructure isis knownknown asas CoCo--ordinationordination

NumberNumber..

ATOMIC RADIUSATOMIC RADIUS

Half Half thethe dist ancedist ance betweenbetween anyany twotwo nearest nearest neighborsneighbors isis calledcalledatomicatomic radiusradius



ATOMIC PACK ING FACTOR ATOMIC PACK ING FACTOR

Atomic packing factor is the ratio of volume occupied by theAtomic packing factor is the ratio of volume occupied by theeffective no. of atoms in a unit cell to that of the total volumeeffective no. of atoms in a unit cell to that of the total volume

of the unit cell. It is also called packing fractionof the unit cell. It is also called packing fraction..

cellunitaof volumeTotal

cellunitaninatoms by theoccupiedVolfactor PackingAtomic !

PF = nv / V



Void Space

Vacant space left or unutilized space in unitcell is called void space and more commonlyknown as interstitial space.

Void space = ( 1-APF ) X 100



SIMPLE CUBIC STR UCTUR ESIMPLE CUBIC STR UCTUR E -- PACKING FACTOR PACKING FACTOR

1.1. Each corner is shared by eightEach corner is shared by eightidentical unit cells. Thusidentical unit cells. Thusevery unit cell contributesevery unit cell contributes1/8th of atom to the corner.1/8th of atom to the corner.Hence effective number of Hence effective number of atoms per unit cell (8 x 1/8)atoms per unit cell (8 x 1/8)=1=1

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

3.3. Nearest neighbor distance Nearest neighbor distance2r = a2r = a

4.4.CoCo--ordination number = 6;ordination number = 6;i.e., four will lie in the samei.e., four will lie in the same

plane, 1 above the plane and 1 plane, 1 above the plane and 1



r r

a



6.Void space = (16.Void space = (1--APF) X 100APF) X 100

= (1= (1--0.52)X 1000.52)X 100

= 48%= 48%

Example: PoloniumExample: Polonium..

3

3

3

3

41

3

2

413

(2 )

0.52

(52%)

r

a

wherea r

r

r

T

T

v

!

!

v

!

!

!

5. Atomic packing factor



BCC STRUCURE ± PACK ING FACTOR

1.1. Effective number of atoms per Effective number of atoms per unit cell (8 x 1/8) + 1 =2 i.e., 1unit cell (8 x 1/8) + 1 =2 i.e., 1from corners and one fromfrom corners and one fromcenter of the cubecenter of the cube

2.A

tomic radius r = ¥3a /42.A

tomic radius r = ¥3a /43. Nearest neighbor distance3. Nearest neighbor distance2r =¥3a/22r =¥3a/2

4. Co4. Co--ordination number = 8 i.e.,ordination number = 8 i.e.,

all corner atoms are at theall corner atoms are at theequidistance from center atomequidistance from center atomor 8 body centered atoms are ator 8 body centered atoms are atthe equidistance from thethe equidistance from the

corner atomcorner atom



a

a

aa2

a3

A B

C

D

r 4



6.Void space = (16.Void space = (1--APF) x 100APF) x 100= (1= (1--0.68) x 1000.68) x 100= 32%= 32%

Ex: Na, lithium andEx: Na, lithium andChromium.Chromium.

3

3

3

3

42

3

3

4

4 32 ( )

3 4

( )

0.68

(68%)

r

a

wherer a

a

a

T

T

v

!

!

v

!

!

!

5.Atomic packing factor



FCC Crystal Structure ± APF

1.1. Effective number of atoms per unitEffective number of atoms per unitcell (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. Co4. Co--ordination Number = 12ordination Number = 12i.e., 4 in the same plane, 4 above thei.e., 4 in the same plane, 4 above the

plane and 4 below the plane. All plane and 4 below the plane. All

these are represented at face centersthese are represented at face centers



6.Void space = (16.Void space = (1--APF) X 100APF) X 100

= (1= (1--0.74) X 1000.74) 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

T

T

v

!

!

v

!

!

!

5.Atomic packing factor



Diamond Structure:

Diamond is a combinationof two interpenetratingFcc - sub lattices along the

body diagonal by 1/4th

Cube edge.



1 23

4

5

6



Diamond - APF1.1. It is the combination of interpenetrationIt is the combination of interpenetration

of two FCC sub lattices along the bodyof two FCC sub lattices along the bodydiagonal by ¼diagonal by ¼ thth cube edge.cube edge.One sub lattice say µOne sub lattice say µ x x¶(0,0,0), has its origin¶(0,0,0), has its originand the other sub lattice µand the other sub lattice µ y y¶(¶(aa/4,/4, aa/4,/4, aa/4)./4).

2.Effective number of atoms per unit cell2.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.

3. Atomic radius r = ¥3a / 8.3. Atomic radius r = ¥3a / 8.

4. Nearest neighbor distance4. Nearest neighbor distance

2r = ¥3a / 4.2r = ¥3a / 4.

5. Co5. Co--ordination number = 4.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 = (16. Void space = (1--APF) x 100APF) x 100

= (1= (1--0.34) x 1000.34) x 100

= 66%= 66%GeGe, Si and Carbon atoms are, Si and Carbon atoms are

possess this structure possess this structure

3

3

3

3

48

3

3

8

4 38 ( )3 8

( )

0.34

(34%)

r

a

r a

a

a

T

T

v

!

!

v

!

!

!




Structure of a Zinc Sulphide:

Zn

sZnS structure is interpenetration of two FCCsub lattices where One of zinc and other isof Sulphur atoms. This structure is identicalto that of the Diamond structure except thatthe two interpenetrating sub lattices are

occupied by two different elements. i.e.,Zinc and sulphur.



Zinc sulphide structure results when ZnZinc sulphide structure results when Zn

atoms are placed on one FCC lattice andatoms are placed on one FCC lattice andS atoms on the other FCC lattice asS atoms on the other FCC lattice asshown in figure below.shown in figure below.

1 23

4

5

6 Zn

S



Effective no . of atoms per unit cell are 8.

i.e., 4 Zinc atoms and 4 Sulphur atoms or

Effective no . of molecules is 4 .

The co-ordination Number is 4.

Packing fraction is 34 %

The other examples are InSb2, GaAs, andCuCl.



Hexagonal Close Pack ed 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 =2. Atomic radius r = aa / 2./ 2.

3. Nearest neighbor distance 2r =3. Nearest neighbor distance 2r = aa

4. Co4. Co--ordination number = 12.ordination number = 12.

i.e., 6 in the same plane 3 above thei.e., 6 in the same plane 3 above the plane and 3 below the plane. plane and 3 below the plane.



5.Volume of the HCP unit cell

The volume of the unit cell determined by computing the area of the

base of unit cell and then by multiplying it with height of the unitcell.

Volume = (Area of the base) x (height of the Unit cell)

0

2 0

2

6 ( )

16 ( )( sin 60 )

2

3 sin 60

3 3

2

ABC

a a

a

a

! v (

! v

!

!

Area of the hexagon

If ³c´ is the height of the unit cell23 3

2

V a c!

A B

C

a60°

c/a ratio:



c/a ratio:

The three middle layer atoms lie in ahorizontal plane at a height c/2 from

the base or at top of the Hexagonalcell.

2 2 2

2 2 2

2 22

2

2

(2 ) ( )2

2

3

2 3( )

3 2 3

( ) ( )23

4 3

8

3

8

3

cr x

x AN

a x a

a ca

c aa

c

a

c

a

!

!

! !

!

!

!

!

A

B

a

o

a

30°

c/2

2r

x

N



6. Void space = (16. Void space = (1--APF) x 100APF) x 100

= (1= (1--0.74) x 1000.74) x 100

= 26%= 26%

Ex: Mg,Ex: Mg, CdCd and Zn.and Zn.

3

3

3

3

46

33 2

2

46 ( )

3 2

3 2 ( )

0.74

(74%)

r

a

ar

a

a

T

T

v

!

!

v

!

!

!




Sodium Chloride Structure

Nacl Nacl CrystalCrystal isis anan ionicionic crystalcrystal.. ItIt consistsconsists of of twotwoFCCFCC subsub latticeslattices..

OneOne of of thethe chlorinechlorine ionion havinghaving itsits originorigin atat thethe((00 00 00)) point point andand other other of of thethe sodiumsodium ionsions havinghaving itsitsoriginorigin atat (a/(a/22 00 00))..

EachEach ionion inin aa NaCl NaCl latticelattice hashas sixsix nearestnearest equidistantequidistantoppositeopposite ionsions atat aa distancedistance a/a/22.. i,ei,e itsits CoCo--ordinationordinationnumber number isis 66..



Na

Sodium Chloride structureSodium Chloride structure

Cl



EachEach unitunit cellcell of of aa sodiumsodium chloridechloride hashas four four sodiumsodiumionsions andand four four associatedassociated chlorinechlorine ionsions.. ThusThus therethere areare

fourfour moleculesmolecules inin eacheach unitunit cellcell..

TheThe ionicionic radiusradius of of chlorinechlorine isis aboutabout 11..8181 AUAU andand for for sodium,sodium, itit isis aboutabout 00..9898 AUAU..

TheThe other other examplesexamples of of thisthis structurestructure areare KClKCl,, K Br K Br,,

MgOMgO,,A

gB

r A

gB

r etcetc..



Structure of Cesium chloride:Structure of Cesium chloride:

It is an ionic Crystal.It is an ionic Crystal.

It is the result of interpenetration of It is the result of interpenetration of two simple cubic sub lattices.two simple cubic sub lattices.

One sub lattice is occupied by cesiumOne sub lattice is occupied by cesiumions and the another one is occupiedions and the another one is occupied

by by Cl Cl ions.ions. Both ions have almost same sizeBoth ions have almost same size The coThe co--ordinates of the ions areordinates of the ions are

CsCs : (000),(100),(010),(001),(110),: (000),(100),(010),(001),(110),(110),(011),(111).(110),(011),(111).

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

The other examples areThe other examples are RbCl RbCl andand LiHg LiHg ..

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 thatdirection.

A negative index is indicated by a µbar ¶ 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

R eciprocals of intercepts made by the plane which aresimplified into the smallest possible numbers or integersand represented by (h k l ) are known as Miller Indices.

(or)The miller indices are the three smallest integers whichhave the same ratio as the reciprocals of the interceptshaving 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 eachwith their L.C.M , to get the smallest whole number.

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



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 termsof a parallel plane having non-zero intercept.



The numerical parameters of

the planeAB

C are (2,2,1). The reciprocal of these

values are given by(1/2,1/2,1).

LCM is equal to 2.

Multiplying the reciprocalswith 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(areinterceptsof lsR`eciproca

),,1(arePlanetheof Intercepts

gg!

gg!



z

x

y Set of [100] parallel planes



x

y

z

[010] plane.


)1

,1

1,


),1,(arePlanetheof Intercepts

gg!

gg!



x

y

z

Set of ( 0 1 0 ) parallel planes



x

y

z

[ 001 ] plane


)

1

1,

1,


)1,,(arePlanetheof Intercepts

gg

!

gg!



x

y

z

[ 001 ]

Set of ( 0 0 1 ) parallel planes



y

x

[110]

z

Construction of [110] plane


)1

,1

1,

1


),1,1(arePlanetheof Intercepts

g!

g!



x

y

z

[110]

Set of [110] parallel planes



z

x

y

Construction of ( 0 0) Planes


)1

,1

,1


),,1(arePlanetheof Intercepts

g g

!

g g!



Intercepts of the planes are 1,1,1

R eciprocals 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 planeLet ( h ,k, l ) be the miller indices of the plane ABC.ABC.

Let ON=d be a normal to the plane passing through theLet ON=d be a normal to the plane passing through the

origin µ0¶.origin µ0¶. Let this ON make anglesLet this ON make angles ,, andand with x, y and zwith x, y and z

axes respectively.axes respectively.

Imagine the reference plane passing through the originImagine the reference plane passing through the origin³o´ and the next plane cutting the intercepts a/h, b/k ³o´ and the next plane cutting the intercepts a/h, b/k and c/l on x, y and z axes.and c/l on x, y and z axes.



x

y

Z

A

B

C

N

ha k

b

l

c

E

FK o

o

d

OA /h OB b/k OC /lOA /h OB b/k OC /l



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 theA normal ON is drawn to the plane ABC from theorigin ³o´. the length ³d´ of this normal from theorigin ³o´. the length ³d´ 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

WhereWhere coscos,, coscos ,,coscos are directional cosines of are directional cosines of ,,,,angles.angles.

)(

cos

)(

cos

)(

cos

h

c

d

OC

ON

h

b

d

OB

ON

h

a

d

OA

ON

!!

!!

!!

K

F

E



According to law of directional cosinesAccording to law of directional cosines

2

2

2

2

2

2

2

2

2

2

2

22

222

222

1

1}{

1]

)(

[]

)(

[]

)(

[

1coscoscos

c

l

b

k

a

hd

c

l

b

k

a

hd

l

c

d

k

b

d

h

a

d

!

!

!

! KFE



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 theIn cubic system as we know that a = b = c, so theexpression becomesexpression becomes

222 l k h

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!

Crystal Structure (21!10!2011)

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