ch04_The structure of crystalline solids

7/26/2019 ch04_The structure of crystalline solids

http://slidepdf.com/reader/full/ch04the-structure-of-crystalline-solids 1/45

Chapter 4 - 1

ISSUES TO ADDRESS...

• What are common crystal structures for

metals and ceramics?

• What features of a metal’s/ceramic’s atomic

structure determine its density?

• How do the crystal structures of ceramicmaterials differ from those for metals?

Chapter 4: The Structure of Crystalline Solids



Chapter 4 - 2

etallic Crystal Structures

• !o" can "e stac# metal atoms to minimi$e

empty space?

2-dimensions

vs.

%o" stac# these 2-& layers to ma#e '-& structures



Chapter 4 - '

• Tend to (e densely pac#ed)• *easons for dense pac#in+:

- Typically, only one element is present, so all atomic

radii are the same)

- etallic (ondin+ is not directional)- %earest nei+h(or distances tend to (e small in

order to lo"er (ond ener+y)

- lectron cloud shields cores from each other)

• etals ha.e the simplest crystal structures)

We "ill eamine three such structures)))

etallic Crystal Structures



Chapter 4 - 4

• *are due to lo" pac#in+ density 0only o has this structure

• Close-pac#ed directions are cu(e ed+es)

• Coordination 3 5

03 nearest nei+h(ors

Simple Cu(ic Structure 0SC

6i+) 4)2, Callister & Rethwisch 9e.



Chapter 4 - 7

• 86 for a simple cu(ic structure 9)72

86 a'

4

'π 09)7a '1

atoms

unit cellatom

.olume

unit cell

.olume

8tomic ac#in+ 6actor 086

86 olume of atoms in unit cell;

olume of unit cell

;assume hard spheres

8dapted from 6i+) 4)2 0a,

Callister & Rethwisch 9e.

close-pac#ed directions

a

R

9)7a

contains < 1/< 1 atom/unit cell



Chapter 4 - 5

• Coordination 3 <

8dapted from 6i+) 4)1,


• 8toms touch each other alon+ cu(e dia+onals)--%ote: 8ll atoms are identical= the center atom is shaded

differently only for ease of .ie"in+)

1) >ody Centered Cu(ic Structure 0>CC

e: Cr, W, 6e 0α, Tantalum, oly(denum

2 atoms/unit cell: 1 center < corners 1/<


http://slidepdf.com/reader/full/ch04the-structure-of-crystalline-solids 7/45Chapter 4 - @

8tomic ac#in+ 6actor: >CC

86

4

'π 0 'a/4 '2

atoms

unit cell atom

.olume

a'

unit cell

.olume

len+th 4R

Close-pac#ed directions:

' a

• 86 for a (ody-centered cu(ic structure 9)5<

aR8dapted from

6i+) 4)10a, Callister &

Rethwisch 9e.

a

a2

a'


http://slidepdf.com/reader/full/ch04the-structure-of-crystalline-solids 8/45Chapter 4 - <


8dapted from 6i+) ')1, Callister & Rethwisch

9e.

• 8toms touch each other alon+ face dia+onals)--%ote: 8ll atoms are identical= the face-centered atoms are shaded

differently only for ease of .ie"in+)

2) 6ace Centered Cu(ic Structure 06CC

e: 8l, Cu, 8u, (, %i, t, 8+

4 atoms/unit cell: 5 face 1/2 < corners 1/<


http://slidepdf.com/reader/full/ch04the-structure-of-crystalline-solids 9/45Chapter 4 - A

• 86 for a face-centered cu(ic structure 9)@4

8tomic ac#in+ 6actor: 6CC

maimum achie.a(le 86

86

4

'π 0 2a/4 '4

atoms

unit cell atom.olume

a'

unit cell

.olume

Close-pac#ed directions:

len+th 4R 2 a

Bnit cell contains: 5 1/2 < 1/<

4 atoms/unit cella

2 a

8dapted from

6i+) ')10a,

Callister &

Rethwisch 9e.


http://slidepdf.com/reader/full/ch04the-structure-of-crystalline-solids 10/45Chapter 4 - 19

8 sites

> >

>

>>

> >

C sites

C C

C8

>

> sites

• 8>C8>C))) Stac#in+ Seuence

• 2& roDection

• 6CC Bnit Cell

6CC Stac#in+ Seuence

> >

>

>>

> >

> sites

C C

C8

C C

C8

8

>C




• 8>8>))) Stac#in+ Seuence

• 86 9)@4

• '& roDection • 2& roDection

8dapted from 6i+) 4)'0a,


') !ea+onal Close-ac#ed Structure

0!C

5 atoms/unit cell

e: Cd, +, Ti, En

• c/a 1)5''

c

a

8 sites

> sites

8 sites >ottom layer

iddle layer

Top layer



Theoretical &ensity, ρ

"here n num(er of atoms/unit cell

A atomic "ei+ht

V C olume of unit cell a' for cu(ic

N8 8.o+adro’s num(er

5)922 192' atoms/mol

&ensity ρ

V C N8

n A ρ

CellBnitofolumeTotal

CellBnitin8tomsofass


http://slidepdf.com/reader/full/ch04the-structure-of-crystalline-solids 13/45Chapter 4 - 1'

• : Cr 0>CC

A 72)99 +/mol

R 9)127 nm

n 2 atoms/unit cell

ρtheoretical

a 4R/ ' 9)2<<@ nm

ρactual

aR

ρ

a'

72)992

atoms

unit cell mol

+

unit cell

.olume atoms

mol

5)922 192'

Theoretical &ensity, ρ

@)1< +/cm'

@)1A +/cm'

8dapted from

6i+) 4)10a, Callister &

Rethwisch 9e.



Chapter 4 - 14

• >ondin+:

-- Can (e ionic and/or co.alent in character) -- F ionic character increases "ith difference in

electrone+ati.ity of atoms)

• &e+ree of ionic character may (e lar+e or small:

8tomic >ondin+ in Ceramics

SiC: small

Ca62: lar+e



Chapter 4 - 17

Ceramic Crystal Structures

Gide structuresH oy+en anions lar+er than metal cations

H close pac#ed oy+en in a lattice 0usually 6CC

H cations fit into interstitial sites amon+ oy+en ions



Chapter 4 - 15

6actors that &etermine Crystal Structure1) *elati.e si$es of ions H 6ormation of sta(le structures:

--maimi$e the 3 of oppositely char+ed ion nei+h(ors)



- -

- -

unsta(le

- -

- -

sta(le

- -

- -

sta(le

2) aintenance of

Char+e %eutrality : --%et char+e in ceramic

should (e $ero) --*eflected in chemical

formula:

Ca62: Ca2

cation

6-

6-

anions

8mIp

m, p .alues to achie.e

char+e neutrality



Chapter 4 - 1@

• Coordination %um(er increases "ith

Coordination %um(er and Jonic *adii

8dapted from Ta(le 4)',


2

r cationr anion

Coord)

%um(er

K 9)177

9)177 - 9)227

9)227 - 9)414

9)414 - 9)@'2

9)@'2 - 1)9

'

4

5

<

linear

trian+ular

tetrahedral

octahedral

cu(ic





8dapted from 6i+) 4)@,Callister & Rethwisch 9e.

EnS

0$inc (lende

%aCl

0sodium

chloride

CsCl0cesium

chloride

r cation

r anionTo form a sta(le structure, ho" many anions can

surround around a cation?



Chapter 4 - 1<

>ond !y(ridi$ation

>ond !y(ridi$ation is possi(le "hen there is si+nificant

co.alent (ondin+

H hy(rid electron or(itals form

H 6or eample for SiC

• X Si 1)< and X C 2)7

• L <AF co.alent (ondin+

• >oth Si and C prefer sp' hy(ridi$ation

• Therefore, for SiC, Si atoms occupy tetrahedral sites



Chapter 4 - 1A

• Gn the (asis of ionic radii, "hat crystal structure

"ould you predict for 6eG?

• 8ns"er:

5500

1400

0770

anion

cation

.

.

.

r

r

=

=

(ased on this ratio,

-- coord 3 5 (ecause 9)414 K 9)779 K 9)@'2

-- crystal structure is %aCl

&ata from Ta(le 4)4,


ample ro(lem: redictin+ the Crystal

Structure of 6eG

Jonic radius 0nm

9)97'

9)9@@9)95A

9)199

9)149

9)1<1

9)1''

Cation

8nion

8l'

6e26e'

Ca2

G2-

Cl-

6-



Chapter 4 - 29

*oc# Salt Structure

Same concepts can (e applied to ionic solids in +eneral)

ample: %aCl 0roc# salt structure

r %a 9)192 nm

r %a/r Cl 9)754

∴ cations 0%a prefer octahedral sites



r Cl 9)1<1 nm



Chapter 4 - 21

+G and 6eG

G2- r G 9)149 nm

+2 r + 9)9@2 nm

r +/r G 9)714

∴ cations prefer octahedral sites

So each +2 0or 6e2 has 5 nei+h(or oy+en atoms



+G and 6eG also ha.e the %aCl structure



Chapter 4 - 22

8I Crystal Structures

6i+) 4)5, Callister & Rethwisch 9e.

Cesium Chloride structure:

∴ Since 9)@'2 K 9)A'A K 1)9,

cu(ic sites preferredSo each Cs has < nei+h(or Cl-

8IHType Crystal Structures include %aCl, CsCl, and $inc (lende



Chapter 4 - 2'

8I2 Crystal Structures

• Calcium 6luorite 0Ca62

• Cations in cu(ic sites

• BG2, ThG2, ErG2, CeG2

• 8ntifluorite structure H

positions of cations and

anions re.ersed

6i+) 4)<, Callister & Rethwisch 9e.

6luorite structure



Chapter 4 - 24

8>I' Crystal Structures

6i+) 4)A, Callister &

Rethwisch 9e.

• ero.s#ite structure

: comple oide

>aTiG'



Chapter 4 - 27

&ensity Computations for Ceramics

%um(er of formula units/unit cell

olume of unit cell

8.o+adro’s num(er

sum of atomic "ei+hts of all anions in formula unit

sum of atomic "ei+hts of all cations in formula unit



Chapter 4 - 25

&ensities of aterial Classes

ρmetals M ρceramics M ρpolymers

Why?

&ata from Ta(le >)1, Callister & Rethwisch, 9e.

ρ

0 + / c m

2 '

Nraphite/Ceramics/

Semicond

etals/8lloys

Composites/fi(ers

olymers

1

2

29

'9>ased on data in Ta(le >1, Callister

;N6*, C6*, O 86* are Nlass,Car(on, O 8ramid 6i(er-*einforcedpoy composites 0.alues (ased on59F .olume fraction of ali+ned fi(ers

in an epoy matri)19

'

4

7

9)'

9)4

9)7

a+nesium

8luminum

Steels

Titanium

Cu,%i

Tin, Einc

Sil.er, o

TantalumNold, Wlatinum

Nraphite

Silicon

Nlass -sodaConcrete

Si nitride&iamond8l oide

Eirconia

!&, S, P&

C

T6

T

CSilicone

Wood

86*;

C6*;

N6*;

Nlass fi(ers

Car(on fi(ers

8ramid fi(ers

etals ha.e))) • close-pac#in+

0metallic (ondin+

• often lar+e atomic masses Ceramics ha.e))) • less dense pac#in+

• often li+hter elements

olymers ha.e)))

• lo" pac#in+ density 0often amorphous

• li+hter elements 0C,!,G

Composites ha.e))) • intermediate .alues

Jn +eneral



Chapter 4 - 2@

Silicate Ceramicsost common elements on earth are Si O G

• SiG2 0silica polymorphic forms are uart$,crysto(alite, O tridymite

• The stron+ Si-G (onds lead to a hi+h meltin+

temperature 01@19QC for this material

Si4

G2-

6i+s) 4)19 O 4)11, Callister & Rethwisch

9e crysto(alite



Chapter 4 - 2<

>ondin+ of adDacent SiG4

4-

accomplished (y the sharin+of common corners, ed+es, or faces

Silicates

+2SiG4 Ca2+Si2G@

8dapted from 6i+)

4)1', Callister &

Rethwisch 9e.

resence of cations such as Ca2, +2, O 8l'

1) maintain char+e neutrality, and

2) ionically (ond SiG44- to one another



Chapter 4 - 2A

• Ruart$ is crystalline

SiG2:

• >asic Bnit: Nlass is noncrystalline 0amorphous

• 6used silica is SiG2 to "hich noimpurities ha.e (een added

• Gther common +lasses contain

impurity ions such as %a, Ca2,

8l', and >'

0soda +lass


Callister & Rethwisch 9e)

Nlass Structure

Si94 tetrahedron4-

Si4

G2-

Si4

%a

G2-



Chapter 4 - '9

Payered Silicates• Payered silicates 0e)+), clays, mica,

talcH SiG4 tetrahedra connected

to+ether to form 2-& plane

• 8 net ne+ati.e char+e is associated

"ith each 0Si2G72-

unit• %e+ati.e char+e (alanced (y

adDacent plane rich in positi.elychar+ed cations

6i+) 4)14, Callister &

Rethwisch 9e.



Chapter 4 - '1

• aolinite clay alternates 0Si2G72- layer "ith 8l20G!42?

layer

Payered Silicates 0cont

%ote: 8dDacent sheets of this type are loosely (ound to

one another (y .an der Waal’s forces)

6i+) 4)17, Callister &

Rethwisch 9e.



Chapter 4 - '2

olymorphic 6orms of Car(on

DiamondH tetrahedral (ondin+ of

car(on• hardest material #no"n

• .ery hi+h thermal

conducti.ity H lar+e sin+le crystals H +em

stones

H small crystals H used to+rind/cut other materials

H diamond thin films• hard surface coatin+s H used

for cuttin+ tools, medicalde.ices, etc)

6i+) 4)1@, Callister &

Rethwisch 9e.



Chapter 4 - ''

olymorphic 6orms of Car(on 0cont

GraphiteH layered structure H parallel hea+onal arrays ofcar(on atoms

H "ea# .an der Waal’s forces (et"een layersH planes slide easily o.er one another -- +ood lu(ricant

6i+) 4)1<, Callister &

Rethwisch 9e.



Chapter 4 - '4

Crystallinity in olymers

• Grdered atomic

arran+ements in.ol.in+

molecular chains

• Crystal structures in terms

of unit cells

• ample sho"n

H polyethylene unit cell

6i+) 4)1A, Callister &

Rethwisch 9e.



Chapter 4 - '7

• Some en+ineerin+ applications reuire sin+le crystals:

• roperties of crystalline materials

often related to crystal structure)

0Courtesy )) 8nderson

-- : Ruart$ fractures more easily

alon+ some crystal planes than

others)

-- diamond sin+le crystals for a(rasi.es

-- tur(ine (lades

0Courtesy artin &ea#ins,

N Supera(rasi.es, Worthin+ton,

G!) Bsed "ith permission)

Crystals as >uildin+ >loc#s



Chapter 4 - '5

• Most en+ineerin+ materials are polycrystals)

• %(-!f-W plate "ith an electron (eam "eld)

• ach +rain is a sin+le crystal)• Jf +rains are randomly oriented, o.erall component properties are not directional)

• Nrain si$es typically ran+e from 1 nm to 2 cm

0i)e), from a fe" to millions of atomic layers)

6i+) , color inset pa+es

of Callister 5e)0Courtesy of aul )

&anielson, Teledyne Wah

Chan+ 8l(any

1 mm

olycrystals

Jsotropic

8nisotropic



Chapter 4 - '@

• Sin+le Crystals

-roperties .ary "ith direction: anisotropic)

-ample: the modulus

of elasticity 0E in >CC iron:

&ata from Ta(le ')',

Callister & Rethwisch 9e)0Source of data is *)W)

!ert$(er+, Deformation an

!ract"re Mechanics of

En#ineerin# Materials, 'rd ed),

Uohn Wiley and Sons, 1A<A)

• olycrystals

-roperties may/may not

.ary "ith direction)

-Jf +rains are randomly

oriented: isotropic) 0E poly iron 219 Na

-Jf +rains are tetured,

anisotropic)

299 μm 8dapted from 6i+)

5)1A0(, Callister &

Rethwisch 9e)V6i+) 5)1A0( is courtesy of

P)C) Smith and C) >rady, the

%ational >ureau of

Standards, Washin+ton, &C0no" the %ational Jnstitute of

Standards and Technolo+y,

Naithers(ur+, &)

Sin+le .s olycrystalsE 0dia+onal 2@' Na

E 0ed+e 127 Na



Chapter 4 - '<

olymorphism

• T"o or more distinct crystal structures for the same

material 0allotropy/polymorphism

titanium

$ , β -Ti

car(on

diamond, +raphite

>CC

6CC

>CC

17'<XC

1'A4XC

A12XC

δ -6e

γ -6e

$ -6e

liuid

iron system



Chapter 4 - 'A

e: linear density of 8l in V119

direction

a 9)497 nm

Pinear &ensity

• Pinear &ensity of 8toms ≡ P&

a

V119

8dapted from

6i+) ')10a,

Callister &

Rethwisch 9e.

Bnit len+th of direction .ector

%um(er of atoms

3 atoms

len+th

1')7 nma2

2P& −==



Chapter 4 - 49

lanar &ensity of 0199 Jron

Solution: 8t T K A12XC iron has the >CC structure)

0199

*adius of iron R 9)1241 nm

R'

'4a =

2& repeat unit

lanar &ensity a 2

1

atoms

2& repeat unit

nm2

atoms12)1

m2

atoms 1)2 191A

1

2

R'

'4area

2& repeat unit

6i+) 4)20c, Callister & Rethwisch 9e Vfrom W) N) offatt, N) W)

earsall, and U) Wulff, %he Str"ct"re an roperties of Materials, ol) J,

Str"ct"re, p) 71) Copyri+ht Y 1A54 (y Uohn Wiley O Sons, %e" Zor#)

*eprinted (y permission of Uohn Wiley O Sons, Jnc)



Chapter 4 - 41

lanar &ensity of 0111 JronSolution 0cont: 0111 plane 1 atom in plane/ unit surface cell

atoms in plane

atoms a(o.e plane

atoms (elo" plane

ah2'=

a2

2 &

r e

p e a t u n i t

1

nm2

atoms@)9

m2

atoms9)@9 191A

' 2R'

15lanar &ensity

atoms

2& repeat unit

area

2& repeat unit

'''

2

2

R'

15R

'

4

2a'ah2area ====



Chapter 4 - 42

I-*ay &iffraction

• &iffraction +ratin+s must ha.e spacin+s compara(le tothe "a.elen+th of diffracted radiation)

• Can’t resol.e spacin+s < '

• Spacin+ is the distance (et"een parallel planes ofatoms)



Chapter 4 - 4'

I-*ays to &etermine Crystal Structure

I-ray

intensity0fromdetector

(

(c

=n '

2 sin(c

easurement of

critical an+le, θ c,allo"s computation of

planar spacin+, )

• Jncomin+ I-rays diffract from crystal planes)

8dapted from 6i+) 4)2A,

Callister & Rethwisch 9e)

reflections must(e in phase fora detecta(le si+nal

spacin+(et"eenplanes

i n c o m i n +

I - r a y s

o u t +

o i n +

I - r a

y s

d e t e c t o r

( '

(

etradistancetra.elled

(y "a.e [2\

[ 1 \ [ 2 \

[ 1 \

[ 2 \



Chapter 4 - 44

I-*ay &iffraction attern

8dapted from 6i+) ')22, Callister )e.

0119

0299

0211

*

+

a -

c

&iffraction an+le 2]

&iffraction pattern for polycrystalline ^-iron 0>CC

J n t e n s i t y 0 r e l a t i . e 2

*

+

a -

c

*

+

a -

c



Summary

• I-ray diffraction is used for crystal structure and

interplanar spacin+ determinations)

• We can predict the density of a material, pro.ided "e

#no" the atomic "ei+ht, atomic radius, and crystal

+eometry 0e)+), 6CC, >CC, !C)

• Common metallic crystal structures are 6CC, >CC, and!C) Coordination num(er and atomic pac#in+ factor

are the same for (oth 6CC and !C crystal structures)

• Some materials can ha.e more than one crystal structure)

This is referred to as polymorphism 0or allotropy)

• Ceramic crystal structures are based on:

-- maintaining charge neutrality

-- cation-anion radii ratios.

• Jnteratomic (ondin+ in ceramics is ionic and/or co.alent)

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ch04_The structure of crystalline solids