PART 2:
Diane Holland P160 d.holland@warwick.ac.uk

2. Defects and disorder (10L)
 Lectures 1-2: crystal defects – point, line and planar defects; dislocations and mechanical behaviour
 Lectures 3-5: orientational disorder; point defects and nonstoichiometry; radiation induced defects; thermodynamics and stability of defects; elimination of defects
 Lectures 6-7: influence of defects on diffusion, ionic conductivity, optical and electronic properties
 Lectures 8-10:amorphous materials and glasses – formation and structure; structural theories; short and intermediate range order techniques for structural analysis – diffraction and the pair distribution function; total scattering; local probes
(NMR, EXAFS, Mössbauer, IR and Raman)

- ammonium salts
- linear chains
- Schottky
- Frenkel
- substitution

Convolution of Basis and lattice
 
Basis may be group of atoms which can adopt different orientations with respect to rest of lattice

Kermit is not symmetrical
 orientation is important a
0 b
0 defect

b
0 a
0 random
2b
0
2a
0 ordered
No repeat distance can cope with this disorder
Repeat distance has been doubled – extra peaks in diffraction pattern!
NB – not the same as the original structure!

- extent of disorder depends on T
NH
4
+
ND
4
+ e.g. ND
4
Br
< -104 o C
CsCl structure ordered orientations unit cell = a a
Br
D
N

-104 o C to -58 o C
CsCl structure
Ordered, alternating orientations unit cell = 2
 a
2a
2a
2a

-58 o C to +125 o C
CsCl structure random arrangement of orientations unit cell = a but disordered
> +125 o C
NaCl structure
(ND
4
) + ion rotating
 spherically symmetric
NB coordination number change from 8 to 6 i.e. rotating ion is ‘smaller’

e.g.organic polymers
• Carbon C 4-coordinated
Join two eclipsed
staggered


G eclipsed staggered
2

/3 4

/3


G eclipsed staggered

• Structural rearrangement requires activation energy
• Important in the formation of organic and polymeric glasses

vacancy interstitial small substitutional atom large substitutional atom
Frenkel defect Schottky defect
All of these defects disrupt the perfect arrangement of the surrounding atoms –relaxation effects

• Schottky and Frenkel normally v low conc since formation energy high e.g. NaCl at T
L
– 1 o C < 0.003% vacancies
• Frenkel high in some materials e.g. superionics
• substitution high in some materials e.g. alloys, spinels
• Stoichiometric Defects - stoichiometry of material not changed by introduction of defects Intrinsic defects

• vacancies - anion and cation vacancies balance such that charge neutrality is preserved e.g.
NaCl
 nV -
Na
+ nV +
Cl
MgCl
2
 nV 2-
Mg
+ 2nV +
Cl
• cation vacancy has net negative charge and vice versa because of non-neutralisation of nearest neighbour charges.
charges balance

• interstitial + vacancy e.g. AgCl
• atoms move from lattice site to interstitial position e.g. V i
+ Ag
Ag

Ag + i
+ V

Ag
• occurrence depends on - size of ion
- charge on ion
- electronegativity
• more common for small, monovalent cations which are not of low electronegativity

Ag + ( r = 1.15 Å; 
= 1.9) but not
Na + (r = 1.02 Å; 
= 0.9)
• can occur for small anions e.g. F in CaF
2

all defects are described in terms of charge on site and regular ion on site
(MX ionic compound with univalent ions)
SITES NOTATION SITES NOTATION
M
M
X on X site X
X
M + on M site
Vacancy on
M site
Interstitial M ion
Interstitial M atom
Foreign ion A + on M site
Free electron
V -
M
M + i
M i
A
M e -
Vacancy on X site
Interstitial X ion
Interstitial X atom
Foreign ion A 2+
M + site on
Free hole
V +
X
X i
X i
A +
M h +

in close-packed systems
TETRAHEDRAL
OCTAHEDRAL
For every sphere there is one octahedral and two tetrahedral interstitial sites
Can think of ionic compounds as one sublattice (usually anions) of close packed spheres with smaller
(cat)ions occupying suitable number of interstitial sites to give the correct stoichiometry.

c
0 .
732
 r r a c 
1
c
0 .
414
 r c r a

0 .
732
c
0 .
225
 r r a c 
0 .
414
N c
= 3
0 .
155
 r r a c 
0 .
225
N c
= 2
0
 r c r a

0 .
155

SUBSTITUTIONAL DISORDER AND SPINELS
• general formula AB
2
X
4
X anions on fcc lattice
A,B cations in interstitial sites
•
• Normal spinels
Inverse spinels
A on tetrahedral sites
B on octahedral sites
A
T
(B
2
)
O
X
4 e.g. MgAl
2
O
4
( spinel )
½ B on tetrahedral sites
A and ½ B on octahedral sites e.g. Mg
2
TiO
4
; Fe
3
O
4
(magnetite) B
T
(AB)
O
X
4
• There are cases in between: degree of inversion
B
T
B
T

B
O
= 0 for normal;
= 0.5 for inverse;
= 0.33 for disordered

3
4

T
3+
2+
3+
O
4 e -
3+
2+ e -
3+
3+
2+ e -
Fe 2+ and Fe 3+ occupy adjacent, edge-sharing octahedra
-very easy for electrons to transfer from Fe 2+ to Fe 3+
 conduction
-would not occur if Fe
T
2+ [Fe
2
3+ ]
O
O
4
– no easy transfer oct  tet

Cation distribution depends on:
• Relative size of A and B radius ratio rules oct 0.414 – 0.732
tet 0.225 – 0.414
• charge - r i
+ usually decreases with higher charge
- affects Madelung const 2,3 usually normal
4,2 usually inverse
• crystal field stabilisation
• covalency

• superionics – gross vacancy/interstitial phenomenon
• f. rigid anion sublattice – sufficiently open that small cations can move through it
• AgI r(I ) = 2.15 Å ; r(Ag + ) = 1.15 Å

(wurtzite)

146 o C

(bcc)
• phase change accompanied by inc in  of 3-4 orders of magnitude
• 
-AgI I form close-packed lattice
21 roughly energetically
 nt sites available for each Ag + .
Hopping readily occurs between sites
 liquid sublattice

• e.g. 
- alumina NaAl
11
O
17
• Na + “liquid sublattice”
• 2D blocks of spinel structure linked by oxygens and mobile Na + ions
Na + oxygen ions interstitial sites
Na + migration sequence for sodium ions

• overall stoichiometry of material changes
• substitution interstitial vacancy
A

A
1-x
B x
AB

A
1+x
B
AB

A
1-x
B
• i.e. atom ratios change and foreign atoms may be present extrinsic defects
• Introduction of aliovalent foreign ions requires creation of vacancies or interstitials to maintain charge balance

Vacancy e.g.
NaCl + xCaCl
2

Na
1-2x
Ca x
(V
Na
) x
Cl
• normal anion lattice
• Ca 2+ substitutes for one Na + but another Na + must be removed to maintain charge balance creating a vacancy
• 2Na
Na
+ Ca

V -
Na
+ Ca +
Na
Interstitial e.g. CaF
2
+ YF
3

Ca
1-x
Y x
F
2
(F i
) x
• Normal cation lattice with 1 Y 3+ substituting for 1 Ca 2+ .
• Extra F required for charge balance goes on interstitial site.
• Ca
Ca
+ Y + F + V i

Y +
Ca
+ F i
• NB: F ( r i
= 1.33 Å ) much smaller than Cl ( r i
= 1.80 Å )

Variable valency
• e.g. reduction of TiO
2 by hydrogen
TiO
2
+ xH
2

TiO
2-x
+ xH
2
O

Ti 4+
1-2x
Ti 3+
2x
O
2-x complete cation lattice - oxygen vacancies
2Ti
Ti

2Ti
Ti
– + V
O
2+
• Materials with large non-stoichiometric regions usually contain elements which show variable valence transition metals e.g. Fe 2+
B metals e.g. Pb 2+
/Fe
/Pb
3+
4+
;

• External radiation or internally generated by radioactive decay of component atom
• Important in minerals containing radioactive elements
- metamict minerals
• Important in the storage of radioactive waste from nuclear programmes
- Chief sources of radiation damage are
 and

-decay
-

-decay responsible for most of heat generated in early history of waste but only produces 0.1 to 0.15 atomic displacements per event
-

-decay dominant after ~ 1000 yrs – produces ~ 1500 – 2000 atomic displacements per event

•Most damage produced by recoil of atom
M m

M d
+

E(

) ~ 4.5 – 5.5 MeV E(nucleus recoil) ~ 70 – 100 keV
•recoiling nucleus produces ionisation and displacement of surrounding atoms (Frenkel defects) cascade of collisions = metamictisation
•Produces amorphous regions and bloating
•direct damage equation amorphous fraction f a
= 1 – exp(-N d
D

)
D

N d number of

-decays per atom number of permanently displaced atoms

actinide atoms substituted for some Zr atoms in zircon, ZrSiO
4

Evidence for existence of non-stoichiometry:
- continuous variation in composition
- continuous change in structure e.g. lattice parameter
- thermodynamic bivariance G =

(T,x)

• Stability region
• G v x curve for non-stoichiometric phase
(AB) very broad for stoichiometric phases X and
Y narrow (line phase).
• Stability region of nonstoichiometric phase determined by common tangent method.
G
• High entropy S of nonstoichiometric phases stabilises them at high T.
On cooling, form metastable phase or disproportionate.
• e.g. “FeO”
A X
AB
Y B

Schottky
Take crystal of N molecules of NaCl
N
V vacancies on both lattices
Na
Na
+ Cl
Cl

V -
Na
N-N
V
N-N
V
N
V
+ V +
Cl
N
V
Equilibrium constant

N
V

NK 0.5
K 

 V 
Na
V 
   
Na
Na
 


N
V
2

V
 2

N
V
2
N 2

G

-RTlnK Energy

G required to form defects


K
N
 exp
V

RT
G
 const exp
 
H
RT
Nconst exp
 
H
2 RT
(assumes S constant)

H

220 kJ mol -1 for NaCl

Frenkel
V i
+ Ag
Ag

N N-N i

Ag + i
N i
+ V

Ag
N i
K


N

N i
2
N i
 
N

N

N i
2
2
N i
 constN exp
 
H
2 RT

H

130 kJ mol-1 for AgCl

requires energy to create them !

G

H

G =

H - T

S

H inc but

S also inc

G =

H - T

S
[defect]
Temperature
-T

S incs with inc T
 more defects at higher T
-T

S at this point a breakdown in structure will occur to form a new phase

P

N !

N
 n

!
n !
n number of defects
N-n normal species
N number of lattice sites

S = klnP
S k ln


N !
( N

)! !


 k[NlnN – (N-n)ln(N-n) – nlnn]

S depends on number of defects
Neglects lattice relaxation and defect interactions

Beyond a certain concentration, defects will begin to interact and even be eliminated.
‘FeO’ really Fe
Fe
1-x
O

Fe
3
O
1-x
4
O
+ Fe
‘FeO’
T
570 o C
‘FeO’
+
Fe
Fe + Fe
3
O
4
‘FeO’
+
Fe
3
O
4
0.97
1-x
0.85

DEFECT INTERACTIONS
- of increasing magnitude with defect conc
1.
lattice relaxation
2.
short-range order clustering e.g. Ca
1-x
Y x
F
2+x
Y 3+ substitutes for Ca x small – xs F goes into interstitial sites
2+ inc x – clusters of F , Y, and vacancies form e.g. 2:2:2 higher x – increasingly large clusters

Cluster formation
Ca 2+
Y 3+
F i
-
F i
-
V
F
+
2:2:2

3.
long-range order
(a) superlattice formation – defects assimilated by ordering to form a new structure type – often gives new unit cell where one or more parameters are multiples of the original.
(b) crystallographic shear - vacancies eliminated by cooperative movement over long distances to give change in linkage of coordination polyhedra e.g. TiO
2-x
2D
3D
-
corner sharing
 edge sharing
(edge
 face)
If shear planes regularly spaced then get new ‘stoichiometric’ phase Ti n
O
2n-1

(i) Complete the following equations (i.e. replace the question marks), using Kroger-Vink notation, and state which type of defect is being formed in each case.
nNaCl

?
+ nV
+Cl nMgCl
V i
+ Ag
Ag
2Na
Na
2
 nV 2 –
Mg
+ ?

?
+ V
–
Ag
+ Ca

V
–
Na
+ ?
(ii) Describe the effect of each of the above defect types on the density of a material