crystal systems&millerindices

7/24/2019 Crystal Systems&MillerIndices

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Assignment # 1

Subject: CHARACTERIZATION OF MATERIALS

Topic: Crystal systems, Planes of Intersection, and

Miller indices

Submitted to: r! Israr "adir

Submitted by: Faroo# S$a$

Re%! & '()*' +SMME

SCME e-artment, N.ST

ated/ 01t$No2em3er, 1)04


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Crystal:A solid com-osed of atoms, ions, or molec5les arran%ed in a -attern t$at is re-eated in

t$ree dimensions! It is a material in 6$ic$ atoms are sit5ated in a re-eatin% or -eriodicarray o2er lar%e atomic distances!

T$ere are se2en different -ossi3le com3inations of a, b, and c, and and

eac$ of 6$ic$ re-resents a distinct crystal system.

Geometrical characteristics of the seven crystal systems


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Bravais Lattices

simple cubic body-centered face-centeredcubic cubic

simple body-centeredtetragonal tetragonal

simple body-centered base-centered face-centeredorthorhombic orthorhombic orthorhombic orthorhombic

rhombohedral hexagonal

sim le base-centered triclinicmonoclinic monoclinic


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iller !ndices and corresponding "lane of !nterceptions:

T$e orientation of a s5rface or a crystal -lane may 3e defined 3y considerin% $o6 t$e -lane +or

indeed any -arallel -lane intersects t$e main crystallo%ra-$ic a7es of t$e solid! T$e a--licationof a set of r5les leads to t$e assi%nment of t$e Miller Indices , + hkl 8 a set of n5m3ers 6$ic$

#5antify t$e interce-ts and t$5s may 3e 5sed to 5ni#5ely identify t$e -lane or s5rface!

T$e follo6in% treatment of t$e -roced5re 5sed to assi%n t$e Miller Indices is a sim-lified one +it

may 3e 3est if yo5 sim-ly re%ard it as a 9reci-e9 and only a cubiccrystal system +one $a2in% ac53ic 5nit cell 6it$ dimensions a7 a7 a 6ill 3e considered!

T$e -roced5re is most easily ill5strated 5sin% an e7am-le so 6e 6ill first consider t$e follo6in%

s5rface:-lane/


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Step 1/ Identify the intercepts on the x- , y- and z- axes.

In t$is case t$e interce-t on t$e 7;a7is is at x< a+ at t$e -oint +a,),) , 35t t$e s5rface is -arallel

to t$e y; andz;a7es ; strictly t$erefore t$ere is no interce-t on t$ese t6o a7es 35t 6e s$all

consider t$e interce-t to 3e at infinity + = for t$e s-ecial case 6$ere t$e -lane is -arallel to ana7is! T$e interce-ts on t$ex; ,y; andz;a7es are t$5s

Interce-ts / a $ $

Step %/ S-ecify t$e interce-ts in fractional co;ordinates

Co;ordinates are con2erted to fractional co;ordinates 3y di2idin% 3y t$e res-ecti2e cell;

dimension ; for e7am-le, a -oint +x,y,z in a 5nit cell of dimensions a7 b7 c $as fractional co;ordinates of + x:a , y:b , z:c! In t$e case of a c53ic 5nit cell eac$ co;ordinate 6ill sim-ly 3e

di2ided 3y t$e c53ic cell constant , a! T$is %i2es

Fractional Interce-ts / a&a $&a $&a i!e! 1 $ $

Step '/ Ta>e t$e reci-rocals of t$e fractional interce-ts

T$is final mani-5lation %enerates t$e Miller Indices 6$ic$ +3y con2ention s$o5ld t$en 3es-ecified 6it$o5t 3ein% se-arated 3y any commas or ot$er sym3ols! T$e Miller Indices are also

enclosed 6it$in standard 3rac>ets +?! 6$en one is s-ecifyin% a 5ni#5e s5rface s5c$ as t$at

3ein% considered $ere!


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T$e reci-rocals of 0 and = are 0 and ) res-ecti2ely, t$5s yieldin%

Miller Indices / (1))*

So t$e s5rface:-lane ill5strated is t$e +0)) -lane of t$e c53ic crystal!

+ther ,-amples

0! T$e +00) s5rface

Interce-ts / a, a, =

Fractional interce-ts / 0 , 0 , =

Miller Indices / (11)*

1! T$e +000 s5rface

Interce-ts / a, a, a

Fractional interce-ts / 0 , 0 , 0

Miller Indices / (111*


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T$e +0)), +00) and +000 s5rfaces considered a3o2e are t$e so;called lo6 inde7 s5rfacesof a

c53ic crystal system +t$e 9lo69 refers to t$e Miller indices 3ein% small n5m3ers ; ) or 0 in t$is

case! T$ese s5rfaces $a2e a -artic5lar im-ortance 35t t$ere an infinite n5m3er of ot$er -lanest$at may 3e defined 5sin% Miller inde7 notation! @e s$all 5st loo> at one more ?

(! T$e +10) s5rface

Interce-ts / B a, a, =

Fractional interce-ts / B , 0 , =

Miller Indices / (%1)*

+i In some instances t$e Miller indices are 3est m5lti-lied or di2ided t$ro5%$ 3y a common

n5m3er in order to sim-lify t$em 3y, for e7am-le, remo2in% a common factor! T$is o-eration of

m5lti-lication sim-ly %enerates a -arallel -lane 6$ic$ is at a different distance from t$e ori%in oft$e -artic5lar 5nit cell 3ein% considered! e!%! +1)) is transformed to +0)) 3y di2idin% t$ro5%$

3y 1!

+ii If any of t$e interce-ts are at ne%ati2e 2al5es on t$e a7es t$en t$e ne%ati2e si%n 6ill carryt$ro5%$ into t$e Miller indices8 in s5c$ cases t$e ne%ati2e si%n is act5ally denoted 3y

o2erstri>in% t$e rele2ant n5m3er! e!%! +)) ;0 is instead denoted 3y

+iii in t$e hcpcrystal system t$ere are fo5r -rinci-al a7es8 t$is leads to fo5r Miller Indices e!%!

yo5 may see articles referrin% to an hcp+)))0 s5rface! It is 6ort$ notin%, $o6e2er, t$at t$einterce-ts on t$e first t$ree a7es are necessarily related and not com-letely inde-endent8


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conse#5ently t$e 2al5es of t$e first t$ree Miller indices are also lin>ed 3y a sim-le mat$ematical

relations$i-!

eferences:

0! 1))* Callister ; Materials Science and En%ineerin% ; An Introd5ction *t$

1! dave.ucsc.edu/myrtreia/crystal.html

(! $tt-/::666!crystala%e!com:crystalinformation:se2encrystalsystems:D! https://en.wikipedia.org/wiki/Miller_index

4! https://web.iit.edu/.../Miller_Indices.pdf

'! $tt-/::reference!i5cr!or%:dictionary:Millerindices
http://www.crystalage.com/crystal_information/seven_crystal_systems/https://en.wikipedia.org/wiki/Miller_indexhttps://en.wikipedia.org/wiki/Miller_indexhttps://en.wikipedia.org/wiki/Miller_indexhttps://en.wikipedia.org/wiki/Miller_indexhttps://web.iit.edu/.../Miller_Indices.pdfhttps://web.iit.edu/.../Miller_Indices.pdfhttps://web.iit.edu/.../Miller_Indices.pdfhttps://web.iit.edu/.../Miller_Indices.pdfhttps://web.iit.edu/.../Miller_Indices.pdfhttp://reference.iucr.org/dictionary/Miller_indiceshttps://en.wikipedia.org/wiki/Miller_indexhttps://web.iit.edu/.../Miller_Indices.pdfhttp://reference.iucr.org/dictionary/Miller_indiceshttp://www.crystalage.com/crystal_information/seven_crystal_systems/

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