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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
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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.
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MILLER INDICES
d
DIFFERENT LATTICE PLANES
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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.
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MILLER INDICES
!iller indices is defined as thereciprocalsof
the interceptsmade by the plane on the three
aes.
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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.
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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
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ILLUSTRATION
PLANES IN A CRYSTAL
*lane +- has intercepts of units along $/ais, 0
units along %/ais and units along &/ais.
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DETERMINATION OF MILLER INDICES
S&e' 1(The intercepts are ,0 and on the three aes.
S&e' 2(The reciprocals are 12, 120 and 12.
S&e' )(The least common denominator is 3.
!ultiplying each reciprocal by lcd, we get, 0, and 0.
S&e' 4("ence !iller indices for the plane +- is '0 0(
ILLUSTRATION
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IMPORTANT FEATURES OF MILLER INDICES
For the cubic crystal especially, the important features of !illerindices are,
+ plane which is parallel to any one of the co/ordinate aes
has an intercept of infinity '(. Therefore the !iller inde forthat ais is 4ero5 i.e. for an intercept at infinity, the
corresponding inde is 4ero.
MILLER INDICES
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EXAMPLE
( 1 0 0 ) plane
Plane parallel t Y an! " a#e$
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MILLER INDICES
IMPORTANT FEATURES OF 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.
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MILLER INDICES
IMPORTANT FEATURES OF 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.
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MILLER INDICES OF SOME IMPORTANT PLANES
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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
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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
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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
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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 = >.
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IMPORTANT DIRECTIONS IN CRYSTAL
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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
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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 .
+ = =
+ + + +
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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 (
+ + =
+ + + +
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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
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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.
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SEPARATION %ET&EEN LATTICE PLANES
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SEPARATION %ET&EEN LATTICE PLANES
Aet 1, 1and 1'different from the interfacial
angles, and ( be the angles between co/ordinate aes $,%,& and B7 respectively.
The intercepts of the plane on the three aes are,
'1(a a aB+ , B and B-
h k l= = =
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SEPARATION %ET&EEN LATTICE PLANES
From the figure, ;.1;'a(, we have,
'(
From the property of direction of cosines,
'0(
Csing equation 1 in , we get,
1 1 11 1 1d d dcos ,cos and cosB+ B B-
= = =
1 1 1cos cos cos 1 + + =
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PH 0101 UNIT 4 LECTURE 2 29
SEPARATION %ET&EEN LATTICE PLANES
Csing equation 1 in , we get,
';(
6ubstituting equation ';( in '0(, we get,
1 1 11 1 1d h d k d l
cos ,cos , and cosa a a = = =
. . .
1 1 1d h d k d l 1
a a a
+ + =
1 1 1
d h d k d l1
a a a+ + =
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PH 0101 UNIT 4 LECTURE 2 30
i.e.
'@(
i.e. the perpendicular distance between the origin
and the 1st plane +- is,
1
d
'h k l ( 1a + + =
1
ad
'h k l (=
+ +
1. . .
ad B7
h k l= =
+ +
1. . .
ad
h k l=
+ +
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PH 0101 UNIT 4 LECTURE 2 31
7ow, let us consider the net parallel plane.
Aet B!8dbe the perpendicular distance of this
plane from the origin.
The intercepts of this plane along the three aes are
1 1 1a a aB+ ,B ,B- ,h k l
= = =
.. . .
.aB! d
h k l = =
+ +
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PH 0101 UNIT 4 LECTURE 2 32
=Therefore, the interplanar spacing between twoad
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PH 0101 UNIT 4 LECTURE 2 34
PRO%LEMS
&r'e! E#aple.*ind the perpendic&lar distance )et'een the t'o planes indicated )ythe "iller indices 1 2 1/ and 2 1 2/ in a &nit cell of a c&)ic lattice
'ith a lattice constant parameter a.
;e
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PRO%LEMS
T(e perpe%&ic-ar &i!ta%ce etee% t(e p-a%e! 1 2 1 a%& 2 1 2 are$
& " &1> &2 "
)r ! 3 0209: a2
0a 3a a'0 3(a a03 0 3 0 3
= =
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