structure 3

8/9/2019 Structure 3

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Chapter 3:

The Structure of Crystalline Solids

ISSUES TO ADDRESS...

• How do atoms assemble into solid structures?

(for now, focus on metals)

• How does the density of a material depend on

its structure?

• When do material properties vary with the

sample (i.e., part) orientation?

MM !"#1 $ n%ineerin% Materials


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!

Materials and Packing

• atoms pac& in periodic, ' arraysCrystalline materials...

$metals

$many ceramics

$some polymers

• atoms have no periodic pac&in%

Noncrystalline materials...

$comple structures$rapid coolin%

crystalline *i+!

noncrystalline *i+!Amorphous - oncrystalline

Si Oxygen

• typical of/

• occurs for/



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Crystal Systems'

crystal systems

12 crystal lattices

3nit cell/ smallest repetitive volume whichcontains the complete lattice pattern of a crystal.

a, b, and c are the lattice constants



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Crystal Systems2



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"

Metallic Crystal Structures

• 4end to be densely pac&ed.• 5easons for dense pac&in%/

$ 4ypically, only one element is present, so all atomic

radii are the same.

$ Metallic bondin% is not directional.$ earest nei%hbor distances tend to be small in

order to lower bond ener%y.

$ lectron cloud shields cores from each other.

• Have the simplest crystal structures.



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6

Simple Cubic Structure (SC)• 5are due to low pac&in% density (only 7o has this structure)

• 0lose$pac&ed directions are cube ed%es.

• 0oordination 8 - 6

(8 nearest nei%hbors)



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Atomic Packing actor (AP)

• 97: for a simple cubic structure - #."!

97: -a'

2

'π (#."a) '1

atoms

unit cellatom

volume

unit cell

volume

97: -;olume of atoms in unit cell<

;olume of unit cell


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=

!ody Centered Cubic Structure (!CC)

• 0oordination 8 - =

• 9toms touch each other alon% cube dia%onals.$$ote/ 9ll atoms are identical the center atom is shaded

e/ 0r, W, :e (α), 4antalum, Molybdenum

! atoms>unit cell/ 1 center @ = corners 1>=



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A

Atomic Packing actor: !CC

a

97: -

2

'π ( 'a>2)'!

atoms

unit cell atom

volume

a'

unit cell

volume

len%th - 2R -

0lose$pac&ed directions/

' a

• 97: for a body$centered cubic structure - #.6=

aR

a2

a3



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1#

ace Centered Cubic Structure (CC)

• 0oordination 8 - 1!

• 9toms touch each other alon% face dia%onals.$$ote/ 9ll atoms are identical the face$centered atoms are shaded

e/ 9l, 0u, 9u, 7b, i, 7t, 9%

2 atoms>unit cell/ 6 face 1>! @ = corners 1>=



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11

Atomic Packing actor: CC• 97: for a face$centered cubic structure - #.2

maimum achievable 97:

97: -

2

'π ( !a>2)'2

atoms

unit cell atomvolume

a'

unit cell

volume

0lose$pac&ed directions/

len%th - 2R - ! a

3nit cell contains/ 6 1>! @ = 1>=

- 2 atoms>unit cella

! a



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1!

9 sites

B B

B

BB

B B

0 sites

0 0

0 9

B

B sites

• 9B09B0... *tac&in% *eCuence

• ! 7roDection

• :00 3nit 0ell

:00 *tac&in% *eCuence

B B

B

BB

B B

B sites

0 0

0 9

0 0

0 9

A

BC



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1'

"e#agonal Close$Packed Structure ("CP)

• 0oordination 8 - 1!

• 9B9B... *tac&in% *eCuence

• 97: - #.2

• ' 7roDection • ! 7roDection

6 atoms>unit cell

e/ 0d, M%, 4i, En• c >a - 1.6''

c

a

9 sites

B sites

9 sites Bottom layer

Middle layer

4op layer



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Theoretical %ensity& ρ12

where n - number of atoms>unit cell

A - atomic wei%ht

V C - ;olume of unit cell - a' for cubic

N 9 - 9vo%adroFs number

- 6.#!' 1#!' atoms>mol

ensity - ρ -

V C N 9

n Aρ -

0ell3nitof ;olume4otal

0ell3nitin 9tomsof Mass



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Theoretical %ensity& ρ1"

'#: Cr (!CC) A *+,, g-mol

R ,+.* nm

n *

ρtheoretical

a - 2R > ' - #.!== nm

ρactual

aR

ρ -a'

"!.##!

atoms

unit cell mol

%

unit cell

volume atomsmol

6.#!' 1#!'

- .1= %>cm'

- .1A %>cm'



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%ensities of Material Classes16ρmetals

Gρceramics

Gρpolymers

Why?

ρ

( % > c m

) '

raphite>0eramics>

*emiconductor

Metals> 9lloys

0omposites>fibers

7olymers

1

!

!#

'#


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1

Crystals as !uilding !locks

• Some en%ineerin% applications reCuire sin%le crystals/

• 7roperties of crystalline materials

often related to crystal structure.

$$/ uartN fractures more easily

alon% some crystal planes thanothers.

$$diamond sin%le

crystals for abrasives

$$turbine blades



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1=

Polycrystals• Most en%ineerin% materials are polycrystals.

• b$Hf$W plate with an electron beam weld.

• ach %rain is a sin%le crystal.• Lf %rains are randomly oriented, overall component properties are not directional.

• rain siNes typically ran%e from 1 nm to ! cm

(i.e., from a few to millions of atomic layers).

1 mm

Lsotropic

9nisotropic



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1A

Single /s Polycrystals• *in%le 0rystals

$7roperties vary with direction/ anisotropic.

$ample/ the modulus

of elasticity () in B00 iron/

• 7olycrystals

$7roperties may>may not

vary with direction.

$Lf %rains are randomly

oriented/ isotropic. (poly iron - !1# 7a)

$Lf %rains are tetured,

anisotropic.

!## µm

(dia%onal) - !' 7a

(ed%e) - 1!" 7a



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Polymorphism!#

T0o or more distinct crystal structures for the samematerial (allotropy-polymorphism)

titanium

α& β$Ti

carbon

diamond& graphite

B00

:00

B00

1"'=O0

1'A2O0

A1!O0

-:e

γ-:e

-:e

liCuid

iron system



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Point Coordinates!1

Point coordinates for unit cellcenter are

a-*& b-*& c-* 1 1 1

Point coordinates for unit cellcorner are ...

Translation: integer multiple oflattice constants identicalposition in another unit cell

z

x

y

a b

c

###

111

y

z

•

2c

•

•

•

b

b



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Crystallographic %irections!!

1. ;ector repositioned (if necessary) to pass

throu%h ori%in.

!. 5ead off proDections in terms of

unit cell dimensions a, b, and c

'. 9dDust to smallest inte%er values

2. nclose in sCuare brac&ets, no commas

Puvw Q

e/ 1, #, R -G !, #, 1 -G P !#1 Q

$1, 1, 1

z

x

9l%orithm

where overbar represents a

ne%ative inde

P 111 Q-G

y



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Crystallographic Planes!"



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Crystallographic Planes!6

Miller 2ndices: eciprocals of the (three) a#ialintercepts for a plane& cleared of fractions 4common multiples+ All parallel planes ha/esame Miller indices+

Algorithm.+ ead off intercepts of plane 0ith a#es in

terms of a& b& c

*+ Take reciprocals of intercepts3+ educe to smallest integer /alues5+ 'nclose in parentheses& no

commas i+e+& (hkl )



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Crystallographic Planes!

z

x

y a b

c

2. Miller Lndices (11#)

eample a b c z

x

y a b

c

2. Miller Lndices (!##)

1. Lntercepts 1 1 ∞!. 5eciprocals 1>1 1>1 1>∞

1 1 #'. 5eduction 1 1 #

1. Lntercepts 1>! ∞ ∞!. 5eciprocals 1>R 1>∞ 1>∞

! # #'. 5eduction ! # #

eample a b c



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Crystallographic Planes!=

z

x

y a b

c •

•

•

2. Miller Lndices (6'2)

eample1. Lntercepts 1>! 1 '>2

a b c

!. 5eciprocals 1>R 1>1 1>S

! 1 2>'

'. 5eduction 6 ' 2



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6inear %ensity '#

6inear %ensity of Atoms≡

6%

e/ linear density of 9l in P11#Q

direction

a - #.2#" nm

a

P11#Q

3nit len%th of direction vector

umber of atoms

8 atoms

len%th

1'." nma!

!K −==



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Crystallographic Planes'1

7e 0ant to e#amine the atomic packing ofcrystallographic planes

2ron foil can be used as a catalyst+ Theatomic packing of the e#posed planes is

important+a) %ra0 (.,,) and (...) crystallographic planes

for e+

b) Calculate the planar density for each of theseplanes+



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Planar %ensity of (.,,) 2ron'!Solution: At T 8 9.*°C iron has the !CC structure+

(1##)

5adius of iron R - #.1!21 nm

R '

'2a =

9dapted from :i%. '.!(c), Callister 7e.

! repeat unit

-7lanar ensity - a!

1

atoms

! repeat unit

-nm!

atoms1!.1

m!atoms

- 1.! 1#1A 1

!

R '

'2area

! repeat unit



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Planar %ensity of (...) 2ron

''Solution (cont): (...) plane 1 atom in plane> unit surface cell

'''

!

!

R '

16R

'

2!a'ah!area =÷÷

===

atoms in plane

atoms above plane

atoms below plane

ah23=

a2

! (

r e p e a t u n i t

1

- -nm!

atoms.#

m!atoms

#.# 1#1A

' !R '

167lanar ensity -

atoms

! repeat unit

area

! repeat unit



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$ay %iffraction'2

%iffraction gratings must ha/e spacings comparable tothe 0a/elength of diffracted radiation+

Can;t resol/e spacings


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$ays to %etermine Crystal Structure'"

T$rayintensity(fromdetector)

θ

θc

d =n λ

! sinθc

Measurement ofcritical an%le, θc,allows computation of

planar spacin%, d .

• Lncomin% T$rays diffract from crystal planes.

9dapted from :i%. '.1A,

Callister 7e.

reflections mustbe in phase fora detectable si%nal

spacin%betweenplanes

d

i n c o m i n %

T $ r a y s

o u t %

o i n %

T $ r a

y s

d e t e c t o r

θλ

θetra

distancetravelledby wave U!V

U 1 V U ! V

U 1 V

U ! V



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$ay %iffraction Pattern'6

(11#)

(!##)

(!11)

z

x

y a b

c

iffraction an%le !θ

iffraction pattern for polycrystalline α$iron (B00)

L n t e n s i t y ( r

e l a t i v e )

z

x

y a b

c

z

x

y a b

c



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S


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structure 3

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