Lecture2

S t r u ct u r e an d Sym m et r y

22.14 – In tro to N u c le a r M a te ria l s F e br ua r y 5, 2015

Scan n e d im ag es , u n les s cited , ar e f r o m Alle n & T h o m as , “ T h e Str u ctu r e o f M ater ials , ” 1999 .

Cr yst allo g r ap h y – T h e Co m m o n L an g u ag e o f M at er ials Scien ce

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Cr yst allin e vs. Am o r p h o u s

T he di f f er ence i s l ong - r an ge or de r , and sy m m e t r y

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ht t p: / / phy s i c s . a nu . e du. a u / e m e / r e s e a r c h/ a m o r pho us . php

S ym m et r y E vid en t in M at er ials

E tc h p its in s in g le c ry s ta l a lu m in u m

S o u r c e : J . H. Seo b, J . - H . R y u c , D . N . L e e . “F or m at ion of C r yst allog r ap h i c E t c h P it s d u r in g A C E t c h in g of A l u m i nu m . ” J. E le c troc h e m S oc . , 150( 9) : B 4 33 - B 4 3 8 ( 2003) .

S im p lest O p er at io n : T r an slat io n

M ove a poi nt by t w o b asi s v ect o r s, t 1 & t 2

t 2

t 1

Hig h er S ym m et r y

P l ace r es t r i ct i ons on t 1 and t 2 , and t he angl e bet w een t hem .

H ow m any com bi nat i on s can you t hi nk of ?

Ch o o sin g Un it Cells

D r aw a cel l t hat does t he f ol l ow i ng :

– C ont a i ns fe w e st num be r of a t om s

– H a s a n g le s c lo s e s t to 90 d eg r ees

– E x h ib its th e m o st sy m m e tr y

T r y w i t h di f f er ent pl ane gr oups i n cl as s

Ch o o sin g Un it Cells E xam p le

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Ch o o sin g Un it Cells

Ch o o sin g Un it Cells

Ch o o sin g Un it Cells

Ch o o sin g Un it Cells

M iller In d ices

D ire c tio n s w ritte n a s [ h k ] M u ltip le s o f t 1 a nd t 2

t 2

t 1

M iller In d ices

C a n you na m e t he s e c ry s ta l d ire c tio n s ?

t 2

t 1

S ym m et r y O p er at o r s in 2D

R ot a t i ona l

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S ym m et r y O p er at o r s in 2D

M irro r

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S ym m et r y O p er at o r s in 2D

G lid e

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S ym m et r y O p er at o r s in 2D

M irro r

S ym m et r y O p er at o r s in 2D

G lid e

S q u ar e L at t ice S ym m et r y

M o vin g t o 3D

F our new s ym m et r y oper at or s

– Inve rsi on

– R ot oi nve rsi on (rot a t i on & i nve rsi on)

– R ot ore fl e c t i on (rot a t i on & re fl e c t i on)

– Sc re w a xe s (rot a t i on & t ra ns l a t i on)

In ver sio n

N ew co o rd i n a t es

ℎ ′

𝑘 ′

𝑙 ′

=

− 1

𝟎𝟎

𝟎𝟎

𝟎𝟎

− 1

𝟎𝟎

𝟎𝟎

𝟎𝟎

− 1

ℎ − ℎ

𝑘 = − 𝑘

𝑙 − 𝑙

T ra n s f o rma t i o n O l d ma t ri x co o rd i n a t es

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Ro t o r ef lect io n & Ro t o in ver sio n

W h a t a r e t h e t ra n s f o rma t i o n ma t ri ces ?

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S cr ew Axes

R ot a t i on f ol l ow e d by t r a ns l a t i on

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S cr ew Axes

R ot a t i on f ol l ow e d by t r a ns l a t i on

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$R = \begin{bmatrix} \cos \theta +u_x^2 \left(1-\cos \theta\right) & u_x u_y \left(1-\cos \theta\right) - u_z \sin \theta & u_x u_z \left(1-\cos \theta\right) + u_y \sin \theta \\ u_y u_x \left(1-\cos \theta\right) + u_z \sin \theta & \cos \theta + u_y^2\left(1-\cos \theta\right) & u_y u_z \left(1-\cos \theta\right) - u_x \sin \theta \\ u_z u_x \left(1-\cos \theta\right) - u_y \sin \theta & u_z u_y \left(1-\cos \theta\right) + u_x \sin \theta & \cos \theta + u_z^2\left(1-\cos \theta\right) \end{bmatrix}.$

G en er aliz ed Ro t at io n M at r ix

O r m or e conci s el y:

$R = \cos\theta\mathbf I + \sin\theta[\mathbf u]_{\times} + (1-\cos\theta)\mathbf{u}\otimes\mathbf{u},$

W her e ( u x , u y , u z ) i s a uni t vect or

M iller In d ices in 3D

D i r ect i ons – [ hkl ]

F am i l i es of di r ect i ons – < hkl > P l anes – ( hkl )

F am i l i es of pl anes – { hkl }

E xp lo r e S o m e E xam p les

D o n e i n cl a ss, u si n g C r ys t al m ak e r

M iller In d ices – L at t ice P ar am et er

H er e, a=b =c

a

– N o t al w ay s t h e cas e c

b

M iller In d ices – Dir ect io n s

(1 , 2 , 1 )

c

(½ , 1 , ½ )

b

a

( 1, 1, 1/ 3)

O rig in

D r aw i ng di r ect i ons i ns i de uni t cel l :

– [ 121 ]

– 𝟎𝟎 1 1 � ( 1 m ean s

ne ga t i ve – [ 331 ]

• D i v i d e so l ar g est

i nde x = 1 t o ge t i n t er cep t s

M iller In d ices – Dir ect io n E xam p les

D r aw t he f ol l ow i ng di r ect i ons :

– [ 001]

– 00 1 � c

– [ 250]

– 1 1 � 1

b

– [ 441] a

– [ 632]

– [ 633]

M iller In d ices – P lan es

c

a

O rig in

E xam pl e:

– (234)

• T ak e r eci p r o cal s o f

i ndi c e s ( ½, 1/ 3, ¼) b

• M ul t i pl y s o l a r ge s t i nde x i s one ( 1, 2/ 3, ½)

• T h ese ar e t h e p l an e i n t er cep t s o n l at t i ce ax es

M iller In d ices – Dir ect io n s an d P lan es

O rig in

E xam pl e:

– (234) c

– [234]

b

a

M iller In d ices – P lan e E xam p les

D r aw t he f ol l ow i ng pl anes :

– (001) c

– (001)

– (251)

b

– ( 1 1 1 )

a

– (441)

– (632)

– (633)

F am ilies o f Dir ect io n s & P lan es

F a m i l y o f [1 1 1 ] di r e c t i ons

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M iller In d ices – Dir ect io n s an d P lan es

O rig in

c

I n a cubi c l at t i ce di r ect i ons ar e nor m al t o pl anes . E xam pl e:

– (234)

– [234] b

a

M iller In d ices – An g le Bet w een P lan es in a Cu b ic L at t ice

a = b = c = 3Å

E xam pl e:

– (234)

– ( 1 10)

• 97.55 d e gr e e s

c

b

a

M iller In d ices – An g le Bet w een P lan es in a No n - Cu b ic L at t ice

M ul t i pl y vect or s b y l at t i ce

a = c = 3Å , b = 5Å

cons t ant s

E xam pl e:

– (234)

– ( 1 10)

c

b

a

• 108.44 d e gr e e s

M iller In d ices – Dir ect io n s Co m m o n t o P lan es

D i r e c t i on [ uvw ] c om m on t o pl a ne s (h 1 k 1 l 1 ) a nd h 2 k 2 l 2 ):

C he c k t he W ei ss Z o n e L a w :

a = b = c = 3Å

c

E xam pl e:

– (234) and ( 1 10)

b

• [ 4,4,5] a

Br avais L at t ices

1 ) C h ar act er i z e t h ese sy st em s i n t e r m s of a , b, c , a nd a ngl e s

2) W hy i s body - cen t er ed m onoc l i ni c e qui va l e nt t o ba s e - c e nt e r e d m onoc l i ni c ?

P ackin g F r act io n

T hi s s l i de i nt ent i onal l y l ef t bl ank. . . done i n cl as s !

S p ace G r o u p s

U ni que com bi nat i o n s of s ym m et r y , denot ed by cer t ai n s ym bol s

F i nd t hem i n:

T h e Int ’ l T a bl e s fo r C ryst a l l ogra phy

h ttp ://it. iu c r . o r g /

O r fo r fre e a t t h e U ni ve rsi t y C ol l e g e o f L ondon:

h ttp :// i m g. c hem . u cl . ac. uk / s gp / l ar ge / s gp . h t m

E xam p le: T r iclin ic ( P 1)

h t t p : // im g. c h e m . u c l. a c . u k /sgp / l ar g e / sgp . h t m

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E xam p le: T r iclin ic ( P 1 � )

h t t p : // im g. c h e m . u c l. a c . u k /sgp / l ar g e / sgp . h t m

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E xam p le S p ace G r o u p s

h t t p : // im g. c h e m . u c l. a c . u k /sgp / l ar g e / sgp . h t m

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P6 3 / mmc

h t t p : // im g. c h e m . u c l. a c . u k /sgp / l ar g e / sgp . h t m

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E xam p le S p ace G r o u p s

h t t p : // im g. c h e m . u c l. a c . u k /sgp / l ar g e / sgp . h t m

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information, see http://ocw.mit.edu/help/faq-fair-use/ .

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E xp lo r e S o m e E xam p les

D o n e i n cl a ss, u si n g C r ys t al m ak e r

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2 2.14 Materials in Nuclear Engineering

Spring 20 1 5

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