crystal structure sem 5

8/11/2019 Crystal Structure Sem 5

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Crystal Structure of Solids

Dr. Sukanta De

Books for Solid state physics Course: -

Introduction to Solid State Physics by Charles Kittel

Solid State Physics By A J Dekker

Solid State Physics By Ashcroft and Mermin


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How do atoms assemble into solid structures?

Crystalline Solids : Atoms are arranged in regular manner , form 3-D

pattern. (by 3-D repetition of a certain pattern unit.)

PERIODIC ARRANGEMENT OF ATOMS/IONS OVER LARGE ATOMI C

DI STANCES

Leads to structure displaying

LONG-RANGE ORDER that is

Measurable and Quantifiable

All metals, many ceramics, and some polymers


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When the periodicity of the pattern extends throughout a

certain piece of material Single Crystal

When the periodicity of the pattern interrupted at grain boundary

and grain size is at least several Angstroms

Polycrystalline materials

If the grain size is comparable to the size of pattern unit

Amorphous materials

Materials Lacking Long range order

Example: Ceramic GLASS and many plastics


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POLYCRYSTALLINE MATERIALS Nuclei form during solidification, each of which grows into crystals


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Ideal Crystal

An ideal crystal is a periodic arrayof structural units,such as atoms or molecules.

It can be constructed by the infinite repetition of

these identical structural units in space. Structure can be described in terms of a lattice, with

a group of atoms attached to each lattice point. The

group of atoms is the basis.


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Bravais Lattice

An infinite array of discrete points with anarrangement and orientation that appearsexactly the same, from any of the points the

array is viewed from. A three dimensional Bravais lattice consists of

all points with position vectors R that can bewritten as a linear combination ofprimitivevectors. The expansion coefficients must beintegers.


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Lattice-infinite,perfectly periodicarray ofpointsin a space


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Not a lattice:


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We abstracted points from the shape:

Now we abstract further:


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This is a UNIT CELL


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This is a UNIT CELL

Represented by two lengths and an angle

.or, alternatively, by two vectors


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Basis vectors and unit cells

T = ma + nb

a and b are the basis vectors for the lattice

a

b

T


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In 3-D:

a

b

a, b, and c are the basis vectors for the lattice

c


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In 3-D:

T = m1a + m2b + m3c

a

b

T

a, b, and c are the basis vectors for the lattice

c


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Crystal SystemsSome Definitional

information

7 crystal systems of varying

symmetry are known

These systems are built by changing

the lattice parameters:

a, b, and care the edge

lengths, , and are interaxial

angles

Fig. 3.4, Callister 7e.

Unit cell: smallest repetitive volume which contains

the complete lattice patternof a crystal.


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Crystal Systems

Crystal structures are divided

into groups according to unit

cell geometry (symmetry).


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CRYSTAL SYMMETRY

Symmetry defines the order resulting from how atoms

are arranged and oriented in a crystal

The definite ordered arrangement of the faces and edges

of a crystal known as Crystal Symmetry

The Symmetry Operation is one that leaves the crystal

and its environment invariant. i.e., actions which result inno change to the order of atoms in the crystal structure


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Imagine that this object will be rotated (maybe)


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Was it?


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The object is obviously symmetricit hassymmetry


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The object is obviously symmetricit has symmetry

Can be rotated 90 w/o detection


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so symmetry is really

doing nothing


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Symmetry is doing nothing -or at least doing

something so that it looks like nothing was done!


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What kind of symmetry does this object have?


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4


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4

m


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4

m


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4

m


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4

m

4mm


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Another example:


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Another example:

6

m

6mm


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Why Is Symmetry Important?

Identification of Materials

Prediction of Atomic Structure

Relation to Physical Properties

Optical

Mechanical

Electrical and Magnetic


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Symmetry operations performed about a point or a line are

calledpoint group symmetry operations

Point group symmetry elements exhibited by crystals

are

1. The inversion (centre of symmetry)

2. Reflection symmetry (The mirror reflection)3. Rotation Symmetry


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Inversion

11

A crystal will possess an inversioncentre if for every lattice point

given by a position vector rtherewill be a corresponding lattice point

at the position -r

Crystallographic symmetry

element : Centre of Symmetry


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In this operation, the reflection of a structure at a mirror plane m

passing through a lattice point leaves the crystal unchanged. The

mirror plane may or may not be composed of the atoms lying on

the concerned imaginary plane.

Reflection Symmetry

Crystallographic symmetry

element : Plane of Symmetry


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Rotational Symmetry

If a crystal left invariant after a rotation about an axis ,

is said to possess rotational symmetry.

The axis is called Axis of Symmetry

The axis is called `n-fold, axis if the angle of

rotation is 3600/n .

If equivalent configuration occurs after rotation of 180,

120 and 90, the axes of rotation are known as two-fold, three-

fold and four-fold axes of symmetry respectively.


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Symmetrythe rules behind the

shapes


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Symmetry elements For a simple Cubic Lattice

One CENTRE OF SYMMETRY


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Plane of Symmetry:

There are threeplanes of symmetry parallel to the faces ofthe cube and sixdiagonal planes of symmetry

9

planesofSymmetry

If 4 f 90 t ti i id i hi d d th i


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If n=4, for every 90 rotation, coincidence is achieved and the axis

is termed `tetrad axis.It is discussed already that a cube has `three

tetrad axes.

If n=3, the crystal has to be rotated through an angle = 120

about an axis to achieve self coincidence. Such an axis is called is`triad axis. In a cube, the axis passing through a solid diagonal

acts as a triad axis. Since there are 4 solid diagonals in a cube, the

number of triad axis is four.


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If n=2, the crystal has to be rotated through an angle = 180

about an axis to achieve self coincidence. Such an axis is called a`diad axis.Since there are 12 such edges in a cube, the number of

diad axes is six.

Total 13 axes of rotational symmetry for a Cube


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(a) Centre of symmetry 1

(b) Planes of symmetry 9

(Straight planes -3,Diagonal planes -6)

(c) Diad axes 6(d) Triad axes 4

(e) Tetrad axes 3

----

Total number of symmetry elements = 23----

Thus the total number of symmetry elements of a cubic structure is 23.

SYMMETRICAL ELEMENTS OF CUBE


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N-fold axes with n=5 or n>6 does not occur

in crystals

Adjacent spaces must be completely filled (no gaps, no overlaps).


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ABSENCE OF 5 FOLD SYMMETRY

We have seen earlier that the crystalline solids show only

1,2,3,4 and 6-fold axesof symmetry and not 5-fold axis of

symmetry or symmetry axis higher than 6.

The reason is that, a crystal is a one in which the atoms or

molecules are internally arranged in a very regular and

periodic fashion in a three dimensional pattern, and

identical repetition of an unit cell can take place only

when we consider 1,2,3,4 and 6-fold axes.


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P Q R S

a

MATHEMATICAL VERIFICATION

Let us consider a lattice P Q R Sas shown in figure

Let this lattice has n-fold axis of symmetryand the


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lattice parameter be equal to a.

Let us rotate the vectors Q Pand R Sthrough an

angle = 3600

/n , in the clockwise and anti clockwisedirections respectively.

After rotation the ends of the vectors be at x and y.

Since the lattice PQRS has n-fold axis of symmetry,

the points x and y should be the lattice points.

P Q R S

a

x y

Further the line xy should be parallel to the line PQRS.


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y p Q

Therefore the distance xy must equal to some integral

multiple of the lattice parameter a say, m a.

i.e., xy = a + 2a cos = ma (1)

Here, m = 0, 1, 2, 3, ..................

From equation (1),

2a cos = m aai.e., 2a cos = a (m - 1)

(or) cos = (2)

Here,

N = 0, 1, 2, 3, .....

22

1 Nm


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since (m-1) is also an integer, say N.

We can determine the values of which are allowed in a

lattice by solving the equation (2) for all values of N.

For example, if N = 0, cos = 0 i.e., = 90o

n = 4.

In a similar way, we can get four more rotation axes

in a lattice, i.e., n = 1, n = 2, n = 3, and n = 6.

Since the allowed values of cos have the limits1

to +1, the solutions of the equation (2) are notpossible for N > 2.

Therefore only 1, 2, 3, 4 and 6 fold symmetry axes

can exist in a lattice.


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Crystal Structure 53

Lattice Sites in Cubic Unit Cell


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Crystal Directions

Fig. Shows

[111] direction

We choose one lattice point on the line as anorigin, say the point O. Choice of origin iscompletely arbitrary, since every lattice point isidentical.

Then we choose the lattice vector joining O toany point on the line, say point T. This vectorcan be written as;

R = n1a + n2b + n3c

To distinguish a lattice direction from a latticepoint, the triple is enclosed in square brackets [

...] is used.[n1n2n3] [n1n2n3] is the smallest integer of the same

relative ratios.


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210

X = 1 , Y = , Z = 0

[1 0] [2 1 0]

X = , Y = , Z = 1

[ 1] [1 1 2]

Examples


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Negative directions

When we write the

direction [n1n2n3] dependon the origin, negative

directions can be writtenas

Y direction

(origin) O

- Y direction

X direction

- X direction

Z direction

- Z direction

][ 321 nnn

][ 321 nnn


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X = -1 , Y = -1 , Z = 0 [110]

Examples of crystal directions

X = 1 , Y = 0 , Z = 0 [1 0 0]


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Crystal Planes

Within a crystal lattice it is possible to identify sets ofequally spaced parallel planes. These are called

lattice planes.

b

a

b

a

The set of

planes in

2D lattice.


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Miller IndicesMiller Indices are a symbolic vector representation for the

orientation of an atomic plane in a crystal lattice and are definedas the reciprocals of the fractional intercepts which the planemakes with the crystallographic axes.

To determine Miller indices of a plane, take the following steps;

1) Determine the intercepts of the plane along each of the threecrystallographic directions

2) Take the reciprocals of the intercepts

3) If fractions result, multiply each by the denominator of thesmallest fraction


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Axis X Y Z

Intercept

points 1 Reciprocals 1/1 1/ 1/

Smallest

Ratio 1 0 0

Miller ndices (100)

Example-1

(1,0,0)


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Axis X Y Z

Intercept

points 1 1 Reciprocals 1/1 1/ 1 1/

Smallest

Ratio 1 1 0

Miller ndices (110)

Example-2

(1,0,0)

(0,1,0)


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Axis X Y Z

Intercept

points 1 1 1Reciprocals 1/1 1/ 1 1/ 1

Smallest

Ratio 1 1 1

Miller ndices (111)(1,0,0)

(0,1,0)

(0,0,1)

Example-3


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Axis X Y Z

Intercept

points 1/2 1 Reciprocals 1/() 1/ 1 1/

Smallest

Ratio 2 1 0

Miller ndices (210)(1/2, 0, 0)

(0,1,0)

Example-4


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Axis a b c

Intercept

points 1 Reciprocals 1/1 1/ 1/()

Smallest

Ratio 1 0 2

Miller ndices (102)

Example-5


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Axis a b c

Intercept

points -1 Reciprocals 1/-1 1/ 1/()

Smallest

Ratio -1 0 2

Miller ndices (102)

Example-6


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Miller Indices

Reciprocal numbers are:2

1,

2

1,

3

1

Plane intercepts axes at cba 2,2,3

Indices of the plane (Miller): (2,3,3)

(100)

(200)

(110)(111)

(100)

Indices of the direction: [2,3,3]a

3

2

2

bc

[2,3,3]


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Example-7


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Indices of a Family or Form

Sometimes when the unit cell has rotational symmetry, severalnonparallel planes may be equivalent by virtue of this symmetry, inwhich case it is convenient to lump all these planes in the sameMiller Indices, but with curly brackets.

Thus indices {h,k,l} represent all the planes equivalent to the

plane (hkl) through rotational symmetry.

)111(),111(),111(),111(),111(),111(),111(),111(}111{

)001(),100(),010(),001(),010(),100(}100{

Coo din t on N mbe


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Coordinaton Number

Coordinaton Number (CN) : The Bravais lattice pointsclosest to a given point are the nearest neighbours.

Because the Bravais lattice is periodic, all points have

the same number of nearest neighbours or coordinationnumber. It is a property of the lattice.

A simple cubic has coordination number 6; a body-centered cubic lattice, 8; and a face-centered cubiclattice,12.


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Atomic Packing Factor

Atomic Packing Factor (APF) is defined as thevolume of atoms within the unit cell divided by

the volume of the unit cell.


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1-CUBIC CRYSTAL SYSTEM

Simple Cubic has one lattice point so its primitive cell.

In the unit cell on the left, the atoms at the corners are cut

because only a portion (in this case 1/8) belongs to that cell.The rest of the atom belongs to neighboring cells.

Coordinatination number of simple cubic is 6.

a- Simple Cubic (SC)

a

bc
http://www.kings.edu/~chemlab/vrml/simcubun.wrlhttp://www.kings.edu/~chemlab/vrml/simcubun.wrl


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APF for a simple cubic structure = 0.52

APF =

a3

4

3p (0.5a) 31

atoms

unit cellatom

volume

unit cell

volume

Atomic Packing Factor (APF)

APF =

Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

Adapted from Fig. 3.23,

Callister 7e.

close-packed directions

a

R=0.5a

contains (8 x 1/8) =

1 atom/unit cell Here: a = Rat*2

Where Rat is the handbook atomic

radius


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b-Body Centered Cubic (BCC)

BCC has two lattice points so BCC

is a non-primitive cell.

BCC has eight nearest neighbors.

Each atom is in contact with its

neighbors only along the body-

diagonal directions.

Many metals (Fe,Li,Na..etc),

including the alkalis and several

transition elements choose the

BCC structure.a

b c


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Atomic Packing Factor: BCC

a

APF =

4

3 p( 3a/4 )32

atoms

unit cell atom

volume

a3

unit cell

volume

length = 4R=

Close-packed directions:

3 a

APF for a body-centered cubic structure = 0.68

aR

Adapted from

Fig. 3.2(a), Callister 7e.

a2

a3


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c- Face Centered Cubic (FCC)

There are atoms at the corners of the unit cell and at the center ofeach face.

Face centered cubic has 4 atoms so its non primitive cell. Many of common metals (Cu,Ni,Pb..etc) crystallize in FCC

structure.

3 F C d C b


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3 - Face Centered Cubc

Atoms are all same.

A i P ki F FCC


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APF for a face-centered cubic structure = 0.74

Atomic Packing Factor: FCC

The maximum achievable APF!

APF =

4

3p ( 2a/4 )34

atomsunit cell atom

volume

a3

unit cell

volume

Close-packed directions:

length = 4R= 2 a

Unit cell contains:

6 x1/2 + 8 x1/8

= 4 atoms/unit cella

2 a

Adapted from

Fig. 3.1(a),

Callister 7e.

(a = 22*R)


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Atoms Shared Between: Each atom counts:

corner 8 cells 1/8

face centre 2 cells 1/2

body centre 1 cell 1

edge centre 2 cells 1/2

lattice type cell contents

P 1 [=8 x 1/8]I 2 [=(8 x 1/8) + (1 x 1)]

F 4 [=(8 x 1/8) + (6 x 1/2)]C 2 [=(8 x 1/8) + (2 x 1/2)]

Unit cell contents

Counting the number of atoms within the unit cell

h l


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Theoretical Density, r

where n= number of atoms/unit cell

A=atomic weight

VC= Volume of unit cell = a3for cubicNA= Avogadros number

= 6.023 x 1023atoms/mol

Density = r =

VCNA

nAr =

CellUnitofVolumeTotal

CellUnitinAtomsofMass

Th ti l D it


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Ex: Cr (BCC)A=52.00 g/mol

R= 0.125 nm

n= 2

a= 4R/3 = 0.2887 nma

R

r=a3

52.002

atoms

unit cell mol

g

unit cell

volume atoms

mol

6.023x1023

Theoretical Density, r

rtheoretical

ractual

= 7.18 g/cm3

= 7.19 g/cm3


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The Concept of the reciprocal lattice devised to tabulate two

i i f l l Th i l d h i i


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important properties of crystal planes: Their slopes and their inter

planer distance.

The reciprocal space lattice is a set of imaginary points

constructed in such a way that the direction of vector from one

point to another coincides with the direction of a normal to the

real space planes and the separation of those points (absolute

value of the vector) is equal to the reciprocal of the real inter-planer distance.


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Reciprocal Lattice Vectors

The electronic number density is a periodic function inspace with a period equal to the lattice translationvector T, i.e.

This means that one can use a Fourier seriesexpansion to represent in 1D n(x) as

where:

)()( rTr nn

apxip

pp

pp enapxSapxCnxn /2

00 )/2sin()/2cos()(

p

pp

p

aapxi

ap exdxnn

0

/21 )(

In 3D, we have a

ii 1 GG


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The set of reciprocal lattice vectorsthat lead to electrondensity invariant under lattice translations is found from thecondition:

The reciprocal lattice vectors that satisfy the aboverequirement are of the form

where v1, v2and v3are integers and

i

Vi edVnnenn

c0

1 )()( rGGrG

G

G rr

1)()( )( TGTGrG

G

GTrG

G

G rTr iiii ewhenneenenn

332211 bbbG vvv

ijjikjikj

i zyxi p

p 2,,,2 ab

aaa

aab


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HW:

Reciprocal of a reciprocal lattice is the direct lattice

Wigner Seitz Method


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Wigner-Seitz Method

A simply way to find the primitive

cellwhich is called Wigner-Seitz

cell can be done as follows;

1. Choose a lattice point.2. Draw lines to connect these

lattice point to its neighbours.

3. At the mid-point and normal tothese lines draw new lines.

The volume enclosed is called as a

Wigner-Seitz cell.

Wigner Seitz Cell 3D


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Wigner-Seitz Cell - 3D


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X-ray Diffraction

/hcE

Typical interatomic distances in solid are of the order of an angstrom.

Thus the typical wavelength of an electromagnetic probe of such distances

Must be of the order of an angstrom.

Upon substituting this value for the wavelength into the energy equation,

We find that E is of the order of 12 thousand eV, which is a typical X-ray

Energy. Thus X-ray diffraction of crystals is a standard probe.

X-Rays to Determine Crystal Structure


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y y

X-rayintensity(fromdetector)

c

d

n

2sinc

Measurement of

critical angle, c,allows computation of

planar spacing, d.

Incoming X-rays diffractfrom crystal planes.

Adapted from Fig. 3.19,

Callister 7e.

reflections mustbe in phase fora detectable signal!

spacingbetweenplanes

d

extradistancetraveledby wave 2

2 2 2hkl

ad

h k l

For Cubic Crystals:

h, k, l are Miller Indices

Figure 3.34 (a) An x-ray diffractometer. (Courtesy of Scintag,


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Inc.) (b) A schematic of the experiment.

X-Ray Diffraction Pattern


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y

Adapted from Fig. 3.20, Callister 5e.

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2

Diffraction pattern for polycrystalline -iron (BCC)

Intensity

(relative)

z

x

ya b

c

z

x

ya b

c


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INTER TOMIC FORCES

Energies of Interactions Between Atoms

Ionic bonding

NaCl

Covalent bonding

Comparison of ionic and covalent bonding

Metallic bonding

Van der waals bonding

Hydrogen bonding

What kind of forces hold the atoms together in

a sol id?


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Energies of Interactions Between toms

The energy of the crystal is lower than that of the freeatoms by an amount equal to the energy required to pullthe crystal apart into a set of free atoms. This is called thebinding (cohesive) energy of the crystal.

NaCl is more stable than a collection of free Na and Cl. Ge crystal is more stable than a collection of free Ge.

Cl Na NaCl

Types of Bonding Mechanisms


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Types of Bonding Mechanisms

It is conventional to classify the bonds betweenatoms into different types as

Ionic Covalent

Metallic Van der Waals Hydrogen

All bonding is a consequence of the electrostatic interaction betweenthe nuclei and electrons.


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IONIC BONDING

Ionic bonding is the electrostatic force of attraction between

positively and negatively charged ions (between non-metals

and metals).

All ionic compounds are crystalline solids at room

temperature.

NaCl is a typical example of ionic bonding.

Metallic elements have only up to the valence electrons in their outer


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Metallic elements have only up to the valence electrons in their outershell.

When losing their electrons they become positive ions.

Electronegative elements tend to acquire additional electrons tobecome negative ions or anions.

Na Cl

When the Na+ and Cl- ions approach each other closely


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When the Na and Cl ions approach each other closelyenough so that the orbits of the electron in the ions beginto overlap with each other, then the electron begins to

repel each other by virtue of the repulsive electrostaticcoulomb force. Of course the closer together the ions are,the greater the repulsive force.

Pauli exclusion principle has an important role in repulsiveforce. To prevent a violation of the exclusion principle, thepotential energy of the system increases very rapidly.


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COV LENT BONDING

Covalent bonding takes place between atoms with small differencesin electronegativity which are close to each other in the periodictable (between non-metals and non-metals).

The covalent bonding is formed when the atoms share the outershell electrons (i.e., s and p electrons) rather than by electrontransfer.

Noble gas electron configuration can be attained.

Each electron in a shared pair is attracted to both nuclei


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Each electron in a shared pair is attracted to both nucleiinvolved in the bond. The approach, electron overlap, and

attraction can be visualized as shown in the following figure

representing the nuclei and electrons in a hydrogen molecule.

e

e

Comparison of Ionic and Covalent

d


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Bonding


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MET LLIC BONDING

Metallic bonding is found in metalelements. This is the electrostaticforce of attraction between positivelycharged ions and delocalized outerelectrons.

The metallic bond is weaker than theionic and the covalent bonds.

A metal may be described as a low-density cloud of free electrons.

Therefore, metals have high electricaland thermal conductivity.

+

+

+

+

+

+

+

+

+


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V N DER W LS BONDING

These are weak bonds with a typical strength of 0.2 eV/atom.

Van Der Waals bonds occur between neutral atoms andmolecules.

Weak forces of attraction result from the natural fluctuations inthe electron density of all molecules that cause smalltemporary dipoles to appear within the molecules.

It is these temporary dipoles that attract one molecule toanother. They are called van der Waals' forces.

The shapeof a molecule influences its ability to form temporarydipoles. Long thin molecules can pack closer to each other thanmolecules that are more spherical The bigger the 'surface area' of


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molecules that are more spherical. The bigger the 'surface area' ofa molecule, the greater the van der Waal's forces will be and thehigher the melting and boiling points of the compound will be.

Van der Waal's forces are of the order of 1% of the strength of acovalent bond.

Homonuclear molecules,such as iodine, develop

temporary dipoles due to

natural fluctuations of electron

density within the molecule

Heteronuclear molecules,

such as H-Cl have permanent

dipoles that attract the opposite

pole in other molecules.

These forces are due to the electrostatic attraction between


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the nucleus of one atom and the electrons of the other.

Van der waals interaction occurs generally betweenatoms which have noble gas configuration.

van der waals

bonding


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HYDROGEN BONDING

A hydrogen atom, having one electron, can be covalently bonded toonly one atom. However, the hydrogen atom can involve itself in an

additional electrostatic bond with a second atom of highly

electronegative character such as fluorine or oxygen. This second

bond permits a hydrogen bond between two atoms or strucures.

The strength of hydrogen bonding varies from 0.1 to 0.5 ev/atom.

Hydrogen bonds connect water

molecules in ordinary ice.Hydrogen bonding is also very

important in proteins and

nucleic acids and therefore in

life processes.


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crystal structure sem 5

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