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Crystal Structures

 Types of crystal structures

Face centered cubic (FCC)

Body centered cubic (BCC) Hexagonal close packed (HCP)

Close Packed Structures

Dierent Packing of HCP and FCC

Crystallograpic Directions and Planes

cubic syste!s

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Face Centered Cubic (FCC)

"to!s are arranged at te cornersand center of eac cube face of te

cell# "to!s are assu!ed to touc along face

diagonals

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Face Centered Cubic (FCC)

 Te lattice para!eter$ a$ is related to teradius of te ato! in te cell troug%

Coordination nu!ber% te nu!ber ofnearest neigbors to any ato!# For FCCsyste!s$ te coordination nu!ber is &'#

22 Ra   =

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Face Centered Cubic (FCC)

"to!ic Packing Factor% te ratio ofato!ic spere olu!e to unit cell

olu!e$ assu!ing a ard spere!odel#

FCC syste!s ae an "PF of #*+$ te

!axi!u! packing for a syste! in ,icall speres ae e-ual dia!eter#

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Body Centered Cubic

"to!s are arranged at te corners ofte cube ,it anoter ato! at te

cube center#

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Body Centered Cubic

Since ato!s are assu!ed to toucalong te cube diagonal in BCC$ te

lattice para!eter is related to ato!icradius troug%

3

4 R

a   =

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Body Centered Cubic

Coordination nu!ber for BCC is .#/ac center ato! is surrounded by

te eigt corner ato!s#  Te lo,er coordination nu!ber also

results in a sligtly lo,er "PF for BCC

structures# BCC as an "PF of #0.$rater tan #*+ in FCC

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Hexagonal Close Packed

Cell of an HCP lattice is isuali1ed asa top and botto! plane of * ato!s$

for!ing a regular exagon around acentral ato!# 2n bet,een teseplanes is a alf3exagon of 4 ato!s#

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Hexagonal Close Packed

 Tere are t,o lattice para!eters in HCP$ a

and c$ representing te basal and eigtpara!eters respectiely# 2n te ideal case$

te c/a ratio is ,$ o,eer$ deiations dooccur#

Coordination nu!ber and "PF for HCP areexactly te sa!e as tose for FCC% &' and#*+ respectiely#

 Tis is because tey are bot considered closepacked structures#

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Close Packed Structures

/en toug FCC and HCP are closepacked structures$ tey are -uite dierentin te !anner of stacking teir closepacked planes#

Close packed stacking in HCP takes placealong te c direction ( te (&) plane)# FCCclose packed planes are along te (&&&)#

First plane is isuali1ed as an ato! surroundedby 0 nearest neigbors in bot HCP and FCC#

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Close Packed Structures

 Te second plane in bot HCP and FCC issituated in te 5oles6 aboe te 7rst plane ofato!s#

 T,o possible place!ents for te tird plane ofato!s

 Tird plane is placed directly aboe te 7rst planeof ato!s

8 "B" stacking 33 HCP structure  Tird plane is placed aboe te 5oles6 of te 7rst

plane not coered by te second plane8 "BC stacking 33 FCC structure

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Close Packed Structures

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Crystallographic

Directions

Cubic syste!s

directions are na!ed based upon te

pro9ection of a ector fro! te origin of tecrystal to anoter point in te cell#

Conentionally$ a rigt and Cartesiancoordinate syste! is used#

 Te cosen origin is arbitrary$ but isal,ays selected for te easiest solution tote proble!#

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Crystallographic

Directions

Points ,itin te lattice are ,ritten inte for! $k$l$ ,ere te tree

indices correspond to te fraction ofte lattice para!eters in te x$y$1direction#

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Miller Indices

Procedure for ,riting directions in:iller 2ndices

Deter!ine te coordinates of te t,opoints in te direction# (Si!pli7ed if oneof te points is te origin)#

Subtract te coordinates of te second

point fro! tose of te 7rst# Clear fractions to gie lo,est integer

alues for all coordinates

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Miller Indices

2ndices are ,ritten in s-uare brackets,itout co!!as (ex% ;kl ten te direction is

 

][   kl h

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Miller Indices

Crystallograpic Planes

2dentify te coordinate intercepts of te

plane te coordinates at ,ic te plane intercepts

te x$ y and 1 axes#

2f a plane is parallel to an axis$ its intercept istaken as ∞#

2f a plane passes troug te origin$ coose ane-uialent plane$ or !oe te origin

 Take te reciprocal of te intercepts

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Miller Indices

Clear fractions due to te reciprocal$ but do notreduce to lo,est integer alues#

Planes are ,ritten in parenteses$ ,it bars

oer te negatie indices# /x% (kl) or if > ten it beco!es

ex% plane " is parallel to x$ and intercepts yand 1 at &$ and terefore is te (&&)#

Plane B passes troug te origin$ so teorigin is !oed to ?@$ tereby !aking teplane te

)(   kl h

)121(

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Miller Indices