Crystal Slides

8/12/2019 Crystal Slides

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METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson !"#!08

Crystal $y%%etries


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&hy shoul' (e be intereste')

*%portant physical properties 'epen' on crystalstructure

Con'ucti+ity

Magnetic properties $ti,,ness

$trength

These properties also o,ten 'epen' on crystalorientation


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$o%e *%portant Ter%s ,or Crystals

Crystal $tructure -ra+ais lattices

$y%%etry operations

-asis o, Close Pace'$tructures

Miller *n'ices Planes /irections

$tereographicProection

$tan'ar' Proection Principal Metal $tructures

1-CC CC 4CP

*onic Crystals

/ia%on' $tructure

T(inne' Crystals

*so%orphis%

Poly%orphis%


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-ra+ais Lattices 5"6 Total

Cubic1abc all angles 90o Tetragonal1ab !c all angles90o :rthorho%bic1a!b!c all angles90o ;ho%bohe'ral1abc three e> a!b!c no angles nor 90o


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Cubic -ra+ais Lattices


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Tetragonal an' :rthorho%bic


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:rthorho%bic ;ho%bohe'ral

4e=agonal


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Monoclinic Triclinic


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Miller *n'icesA Con+enient &ay to *'enti,y

Crystal Planes an' /irections or Planes the in'e= is the

reciprocal o, the +alue o, the

intersection o, the plane (ith aparticular a=is con+erte' to (holenu%bers.

or /irections the in'e= is the a=is

coor'inate o, the en' point o, the+ectorcon+erte' to nearest (holenu%bers.


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Miller *n'ices 5'irections

h,k,an'l re,er to principal a=esx,yan' z. /irections are in'icate' by s"00@?0>"0@ an' ?00>"@ The ,irst three are principal a=es

=y an' D.


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Miller *n'ices 5planes

Planes are in'icate' byparentheses 5hl an' ,a%ilies o,planes by curly bracets hlF.

E=a%ple o, a plane ,a%ily is as,ollo(s "00F 5"00 50"0 500"5>"00 50>"0 an' 500>" These are all si= ,aces o, a cube.


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Calculating Miller *n'e= ,or planes

x

z

Intercepts= 0.5, 1, 1/3

Index = (213)

Intercepts= -1, 1, 0.5

Index = (-112)


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Calculating Miller *n'e= 'irections

End coordinates= 0.5,0.5,1

Index = [112]

End=0,1,0.5

Index=[021]


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4e=agonal Miller>-ra+ais *n'ices

a1

a2

a3

c!ree a directions,as s!o"n, p#$s c

Indices are%

(!,&,-(!'&), #)

!e p#ane s!o"n is%

(11-20)


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:ther Crystal CharacteriDation

*t is o,ten i%portant to 'eter%ine crystalorientation.

$ingle crystals.

*n'i+i'ual grains in a polycrystal *, there is a pre,erre' grain orientation

this is re,erre' to as Gte=tureH.

:ne %etho' is to e%ploy the $tereographic

Proection.


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$tereographic Proection


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$o%e Cubic Crystal Planes

001 #ane

110 #ane

111 #ane


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Crystal $y%%etries

Translational $y%%etry1a %o+e o, one cellin each o, 3 a=is 'irections restores thestructure.

;otational $y%%etry1rotation o, speci,icangle 590o "20o "80o restores the structure.

Mirror $y%%etry1re,lection across a planerestores the structure.


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A si%ple Cubic $tructure5illustrating translational sy%%etry

The cubic unit cell

Eight unit cells the starto, a crystal structure.


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;otational $y%%etry Cubic

o$r-*o#d

+otation

"o-o#d

+otation

!ree-o#d+otation


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"00F Poles o, a Cubic Crystal


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*nterplanar Angles

oints on t!e sp!ere are intersectionso* p#ane nora#s.


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Calculating *nterplanar Angles5Cubic Crystals

os = h1h

2+ k

1k

2+ l

1l2

-------------------------- +((h

1

2 + k1

2+ l1

2)(h2

2

+ k

2

2+ l2

2))


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Calculating *nterplanar $pacing5Cubic Crystals

1/d2 = (h2+ k2+ l2)/ a2


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$tan'ar' 500" Cubic Proection


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The $tan'ar' $tereographic Triangle

se*$# *or s!o"in crysta# axis orientations.


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Metallic -on'ing

Ialence or outer electrons o, %etallic ato%s are 'istribute' throughoutthe structure. Positi+ely charge' %etal ions are 'istribute' (ithin thisGseaH o, electrons.

This allo(s %etals to be electrical con'uctors.

There are secon' nearest neighbor interactions that in,luence thecrystal structure


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-o'y Centere' Cubic 5-CC Metals

-e, r,o, , -i,

-4r


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ace>Centere' Cubic 5CC Metals

$, #, 6i, 7,

-e

4 l Cl P ' 54CP


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4e=agonal Close>Pace' 54CPMetals

8e, , 4n, d, -i, -4r


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;elation -et(een CC an' 4CP

8

8..

.

88

8..

.

-CC t 4CP T , ti


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-CC to 4CP Trans,or%ation

#ose ac&ed 8 p#ane 9110: 7ecoes#ose ac&ed ; p#ane (0002).


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-CC to 4CP Trans,or%ation 52

1.ariants o* ;.)


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Types o, Trans,or%ations

/isplaci+e (hich %eans that ato%s 'onot ha+e to %o+e +ery ,ar an' o,ten ashear 'isplace%ent can lea' ,ro% onecrystal structure to another.

;eplaci+e (hich %eans that ato%s (ill%o+e so%e 'istance to their ne(locations an' so longer range 'i,,usionis re


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a

7

a

7

c = a x 7

a 1

7 3

7 3= a1x a2 a3 (a1x a2)

a3a 2

c ab sin a

c a = b a b ab cos a

Iector Multiplication E=a%ples


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