u4lecture 2(Crystal Physics)

7/23/2019 u4lecture 2(Crystal Physics)

1/36

PH 0101 UNIT 4 LECTURE 2

1

PH0101 UNIT 4 LECTURE 2

MILLER INDICES

PROCEDURE FOR FINDING MILLER INDICES

DETERMINATION OF MILLER INDICES

IMPORTANT FEATURES OF MILLER INDICES

CRYSTAL DIRECTIONS

SEPARATION BETWEEN LATTICE PLANES


2/36


2

MILLER INDICES

The crystal lattice may be regarded as made

up of an infinite set of parallel equidistant

planes passing through the lattice points

which are known as lattice planes.

In simple terms, the planes passing through

lattice points are called lattice planes.

For a given lattice, the lattice planes can be

chosen in a different number of ways.


3/36


3

MILLER INDICES

d

DIFFERENT LATTICE PLANES


4/36


4

MILLER INDICES

The orientation of planes or faces in a crystal can be

described in terms of their intercepts on the three

aes.

!iller introduced a system to designate a plane in a

crystal.

"e introduced a set of three numbersto specify a

plane in a crystal.

This set of three numbers is known as Miller Indices

of the concerned plane.


5/36


5

MILLER INDICES

!iller indices is defined as thereciprocalsof

the interceptsmade by the plane on the three

aes.


6/36


6

MILLER INDICES

Procedure or !"d!"# M!$$er I"d!ce%

S&e' 1( #etermine the interceptsof the planealong the aes $,% and & in terms of

the lattice constants a,b and c.

S&e' 2( #etermine the reciprocalsof these

numbers.


7/36


7

S&e' )(Find the least common denominator 'lcd(and multiply each by this lcd.

S&e' 4(The result is written in paranthesis.This is

called the )!iller Indices of the plane in

the form 'h k l(.

This is called the )!iller Indices of the plane in the form

'h k l(.

MILLER INDICES


8/36


8

ILLUSTRATION

PLANES IN A CRYSTAL

*lane +- has intercepts of units along $/ais, 0

units along %/ais and units along &/ais.


9/36


9

DETERMINATION OF MILLER INDICES

S&e' 1(The intercepts are ,0 and on the three aes.

S&e' 2(The reciprocals are 12, 120 and 12.

S&e' )(The least common denominator is 3.

!ultiplying each reciprocal by lcd, we get, 0, and 0.

S&e' 4("ence !iller indices for the plane +- is '0 0(

ILLUSTRATION


10/36


10


For the cubic crystal especially, the important features of !illerindices are,

+ plane which is parallel to any one of the co/ordinate aes

has an intercept of infinity '(. Therefore the !iller inde forthat ais is 4ero5 i.e. for an intercept at infinity, the

corresponding inde is 4ero.

MILLER INDICES


11/36


11

EXAMPLE

( 1 0 0 ) plane

Plane parallel t Y an! " a#e$


12/36


13/36


13

MILLER INDICES


+ plane passing through the origin is defined in terms of aparallel plane having non 4ero intercepts.

+ll equally spaced parallel planes have same !iller

indicesi.e. The !iller indices do not only define a particular

plane but also a set of parallel planes. Thus the planeswhose intercepts are 1, 1,15 ,,5 /0,/0,/0 etc., are all

represented by the same set of !iller indices.


14/36


14

MILLER INDICES


It is only the ratio of the indices which is important in thisnotation. The '3 ( planes are the same as '0 1 1( planes.

If a plane cuts an ais on the negative side of the origin,

corresponding inde is negative. It is represented by a bar,

like '1 9 9(. i.e. !iller indices '1 9 9( indicates that theplane has an intercept in the :ve $ :ais.


15/36


15

MILLER INDICES OF SOME IMPORTANT PLANES


16/36


16

PRO%LEMS

&r'e! E#apleA certain crystal has lattice parameters of 4.24, 10 and 3.66 on X,

Y, Z axes respectiely. !etermine the "iller indices of a plane hain#

intercepts of 2.12, 10 and 1.$3 on the X, Y and Z axes.Lattice paraeter! are " 4#24$ 10 a%& 3#66 'T(e i%tercept! )* t(e +i,e% p-a%e " 2#12$ 10 a%& 1#83 'i#e# T(e i%tercept! are$ 0#5$ 1 a%& 0#5#Step 1. T(e I%tercept! are 1/2$ 1 a%& 1/2#Step 2. T(e recipr)ca-! are 2$ 1 a%& 2#

Step 3. T(e -ea!t c))% &e%)i%at)r i! 2#Step 4. M-tip-i%+ t(e -c& eac( recipr)ca- e +et$ 4$ 2 a%& 4#Step 5. riti%+ t(e i% pare%t(e!i! e +et 4 2 4

T*ere+re t*e M,ller ,n!,-e$ + t*e .,/en plane ,$ ( ) r ( 1 )2


17/36


17

PRO%LEMS

&r'e! E#aple%alc&late the miller indices for the plane 'ith intercepts 2a,( 3) and 4c the alon# the crystallo#raphic axes.

T(e i%tercept! are 2$ 3 a%& 4

Step 1. T(e i%tercept! are 2$ 3 a%& 4 a-)%+ t(e 3 ae!

Step 2. T(e recipr)ca-! are

Step 3. T(e -ea!t c))% &e%)i%at)r i! 12#

M-tip-i%+ eac( recipr)ca- -c&$ e +et 6 4 a%& 3

Step 4. He%ce t(e Mi--er i%&ice! *)r t(e p-a%e i!

1 1 1, and

0 ;

( )3 ; 0


18/36


18

CRYSTAL DIRECTIONS

In crystal analysis, it is essential to indicate certain

directions inside the crystal.

+ direction, in general may be represented in terms of

three aes with reference to the origin.In crystal system,

the line


19/36


19

CRYSTAL DIRECTIONS

To find the !iller indices of a direction,

-hoose a perpendicular plane to that direction.

Find the !iller indices of that perpendicular plane.

The perpendicular plane and the direction have

the same !iller indices value.

Therefore, the !iller indices of the perpendicular

plane is written within a square bracket to

represent the !iller indices of the direction like = >.


20/36


20

IMPORTANT DIRECTIONS IN CRYSTAL


21/36


21

PRO%LEMS

&r'e! E#aple*ind the an#le )et'een the directions +2 1 1 and +1 1 2 in a

c&)ic crystal.

T(e t) &irecti)%! are 2 1 1: a%& 1 1 2:

;e


22/36


22

PRO%LEMS

I% t(i! ca!e$ 1" 2$ ,1" 1$ 1" 1$ 2" 1$ ,2" 1$ 2" 2

)r c)! " 0#833

3 456 45

40

2

. . . . . .

'. 1( '1 1( '1 .( @cos

3. 1 l 1 1 .

+ = =

+ + + +


23/36


23

DESIRA%LE FEATURES OF MILLER INDICES

The angle between any two crystallographic directions=u1v1w1> and =uvw> can be calculated easily. The

angle is given by,

The direction =h k l> is perpendicular to the plane 'h k l(

1 . 1 . 1 .

. . . 12 . . . . 12 .

1 1 1 . . .

u u v v w wcos

'u v w ( 'u v w (

+ + =

+ + + +


24/36


24

DESIRA%LE FEATURES OF MILLER INDICES

The relation between the interplanar distance and the

interatomic distance is given by,

for cubic crystal.

If 'h k l( is the !iller indices of a crystal plane then the

intercepts made by the plane with the crystallographicaes are given as

(ere a$ a%& c are t(e priiti,e!#

. . .

adh k l

=+ +

a b c, and

h k l


25/36


25

SEPARATION %ET&EEN LATTICE PLANES

-onsider a cubic crystalof side a, and a

plane +-.

This plane belongs to a family of planes

whose !iller indices are 'h k l( because

!iller indices represent a set of planes.

Aet B7 8d, be the perpendicular distance of

the plane + - from the origin.


26/36


26



27/36


27


Aet 1, 1and 1'different from the interfacial

angles, and ( be the angles between co/ordinate aes $,%,& and B7 respectively.

The intercepts of the plane on the three aes are,

'1(a a aB+ , B and B-

h k l= = =


28/36

PH 0101 UNIT 4 LECTURE 2 28


From the figure, ;.1;'a(, we have,

'(

From the property of direction of cosines,

'0(

Csing equation 1 in , we get,

1 1 11 1 1d d dcos ,cos and cosB+ B B-

= = =

1 1 1cos cos cos 1 + + =


29/36



Csing equation 1 in , we get,

';(

6ubstituting equation ';( in '0(, we get,

1 1 11 1 1d h d k d l

cos ,cos , and cosa a a = = =

. . .

1 1 1d h d k d l 1

a a a

+ + =

1 1 1

d h d k d l1

a a a+ + =


30/36


i.e.

'@(

i.e. the perpendicular distance between the origin

and the 1st plane +- is,

1

d

'h k l ( 1a + + =

1

ad

'h k l (=

+ +

1. . .

ad B7

h k l= =

+ +

1. . .

ad

h k l=

+ +


31/36


7ow, let us consider the net parallel plane.

Aet B!8dbe the perpendicular distance of this

plane from the origin.

The intercepts of this plane along the three aes are

1 1 1a a aB+ ,B ,B- ,h k l

= = =

.. . .

.aB! d

h k l = =

+ +


32/36


=Therefore, the interplanar spacing between twoad


33/36


34/36


PRO%LEMS

&r'e! E#aple.*ind the perpendic&lar distance )et'een the t'o planes indicated )ythe "iller indices 1 2 1/ and 2 1 2/ in a &nit cell of a c&)ic lattice

'ith a lattice constant parameter a.

;e


35/36


PRO%LEMS

T(e perpe%&ic-ar &i!ta%ce etee% t(e p-a%e! 1 2 1 a%& 2 1 2 are$

& " &1> &2 "

)r ! 3 0209: a2

0a 3a a'0 3(a a03 0 3 0 3

= =


36/36

PH 0101 UNIT 4 LECTU 36

Documents

u4lecture 2(Crystal Physics)