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Metallic Structures: Unit CellsMetallic Structures: Unit Cells

Simplifed 2D DiscussionSimplifed 2D Discussion

Unit Cell: Smallest Repeating unit that represents symmetry/order ostructure.

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Metallic Structures: Unit CellsMetallic Structures: Unit Cells

z

x

y

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7 systems o crystal symmetry

!"amples o the Unit Cells#

Source: R.E. Smallman, Ch1, F1.7

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Source: R.E. Smallman, Ch1, F1.15

7 Systems o Crystal Symmetry$% Crystal Structures

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&oint Coordinates: Determining'here atoms sit.

&oint coordinates or unit cellcenter are

a/2( b/2( c/2 ) ( )( )#

&oint coordinates or unit cellcorner are $( $( $#

*ranslation: integer multiple olattice constants identical

position in another unit cell

z

x

y

a b

c

000

111

y

z

2c

b

b

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Crystallographic Directions: Mo+ing,et-een tomic &ositions

$. 0ector repositioned i necessary# topass

through origin.

2. Read o1 proections in terms ounit cell dimensions a( b( and c3. dust to smallest integer +alues%. !nclose in s4uare 5rac6ets( nocommas

uvw

8ex:1, 0, => 2, 0, 1 => [201]

-1, 1, 1

amilies o directions

z

x

l!or"#hm

-here o+er5ar represents a negati+einde"

[111]=>

y

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ex: l"near $en%"#& o' l "n [110]$"rec#"on

a= 0.(05 nm

;inear Density;inear Density

;inear Density o toms ;D &roperties +ary -ith direction: anisotropic.

>!"ample: the modulus o elasticity !# in ,CC iron:

?mportance o ;inear/ &lanar Densities?mportance o ;inear/ &lanar Densities

! diagonal# < 273 @&a

! edge# < $2A @&a

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Metallic Structure:Metallic Structure:

Crystallographic &lanesCrystallographic &lanes

$a#e$ 'rom F"!. *., Callister 7e.

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Metallic Structure:Metallic Structure:

Crystallographic &lanesCrystallographic &lanes z

x

ya b

c

(. /"ller n$"ce% 110

examle a b cz

x

ya b

c

(. /"ller n$"ce% 100

1. n#erce#% 1 1 2. Rec"rocal% 131 131 13

1 1 0*. Re$uc#"on 1 1 0

1. n#erce#% 132 2. Rec"rocal% 13 13 13

2 0 0*. Re$uc#"on 2 0 0

examle a b c

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Re+ie- o Determining &lanes CoordinatesRe+ie- o Determining &lanes Coordinates

examle a b cz

x

y

a b

c

(. /"ller n$"ce% 100

1. n#erce#% 1 2. Rec"rocal% 1 13 13

*. Re$uc#"on 1 0 0

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Crystallographic &lanes

z

x

y

a b

c

(. /"ller n$"ce% 4*(

examle

1. n#erce#% 132 1 *3(

a b c

2. Rec"rocal% 13 131 13

2 1 (3*

*. Re$uc#"on 4 * (

001010,

Fam"l& o' 6lane% hkl8

100, 010,001,Ex: 1008 = 100,

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Spacing 5et-een e4ui+alent planes

dhkl

=a

h2+ k

2+ l

2

'hat is the spacing 5et-een $$$#

planes in BCC =ic6el *he atomicradius is .$2% nm.

a# .2 nm5# .3A nmc# .3 nm

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&lanar Density o $# ?ron

t * 9 E$2C iron has the ,CC structure.

100

Radius o iron R< .$2%$ nm

R*

*(a=

$a#e$ 'rom F"!. *.2c, Callister 7e.

2D repeat unit

6lanar en%"#& =a2

1

a#om%

2 reea# un"#

=

m2a#om%= 1.2 x 101

12

R*

*(area

2 reea# un"#

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&lanar Density o $$$# ?ron

Solution cont#: $$$# plane 1 a#om "n lane3 un"# %ur'acecell

***

2

2

R*

14R

*

(

2a*ah2area =

===

atoms in plane

atoms a5o+eplane

atoms 5elo-

plane ah2*=

a2

2,re

-ea#un"#

1

= =

nm2a#om%

7.0m2

a#om%0.70 x 101

* 2R*

146lanar en%"#& =

a#om%2 reea# un"#

area

2 reea# un"#

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F Coordination G < $2

F ,,... Stac6ing Se4uence

F &B < .7%

F 3D &roection F 2D &roection

$a#e$ 'rom F"!. *.*a,

Callister 7e.

He"agonal Close>&ac6ed StructureHe"agonal Close>&ac6ed Structure

HCHC

I atoms/unit cell

e": Cd( Mg( *i( Jn

F c/a< $.I33

c

a

%"#e%

9 %"#e%

%"#e% 9o##om la&er

/"$$le la&er

o la&er

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F Coordination G < $2

F ,,... Stac6ing Se4uence

F &B < .7%

F 3D &roection

$a#e$ 'rom F"!. *.*a,

Callister 7e.

He"agonal Close>&ac6ed StructureHe"agonal Close>&ac6ed Structure

HCHC

F c/a< $.I33

c

a

%"#e%

9 %"#e%

%"#e%

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+1;

HC& Crystallographic Directions

1.

$a#e$ 'rom F"!. *.;a, Callister 7e.

$a%he$ re$ l"ne% "n$"ca#e

roec#"on% on#o a1an$ a2axe%a1

a2

a*

-a*

2

a2

2

a1

-a*

a1

a2

zl!or"#hm

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%"#e%

9 9

9

99

9 9

C %"#e%

C C

C

9

9 %"#e%

F ,C,C... Stac6ing Se4uence

F 2D &roection

BCC Stac6ing Se4uence

9 9

9

99

9 9

C C

C

C C

C

9o##om la&er

/"$$le la&er

o la&er

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He"agonal Close Pac6ed

For "$eall& acAe$ hc, c3a=1.4**

c3aBn=1.;54, c3a"=1.5;7Source: R.E. Smallman, Ch1, F1.15

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Stac6ing: Di1erence 5et-eenHC& .7%# and BCC .7%#

FCC: 9C9C9C

C6: 999

Source: R.E. Smallman, Ch1, F1.14

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BCC +s. HC&

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Source: R.E. Smallman