Transcript Defects and Disorder PX431 Structure and Dynamics of Solids PART 2: Diane Holland

PX431 Structure and Dynamics of Solids
PART 2:
Defects and Disorder
Diane Holland
P160
[email protected]
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)
Orientational disorder
groups of atoms
- ammonium salts
- linear chains
Point defects
vacancies, interstitials, incorrect atoms
- Schottky
- Frenkel
- substitution
ORIENTATIONAL DISORDER
(conformational/rotational)
Crystal Structure
Convolution of Basis and lattice
Basis  Lattice  Crystal
Basis may be group of atoms which can adopt different
orientations with respect to rest of lattice
Kermit is not symmetrical  orientation is important
a0
b0
defect
random
ordered
2a0
a0
b0
2b0
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!
Example - ammonium salts
NH4+
- extent of disorder depends on T
ND4+
Br
e.g. ND4Br
< -104oC
CsCl structure
ordered orientations
unit cell = a
D
N
a
-104oC to -58oC
CsCl structure
Ordered, alternating
orientations
unit cell = 2a
2a
2a
2a
-58oC to +125oC
> +125oC
CsCl structure
NaCl structure
random
arrangement of
orientations
(ND4)+ ion rotating
 spherically
symmetric
unit cell = a
but disordered
NB coordination number change
from 8 to 6
i.e. rotating ion is ‘smaller’
CHAINS
e.g.organic polymers
• Carbon C 4-coordinated
Join two
-
eclipsed
-
staggered
Energetics of rotation
G

2/3
4/3
eclipsed
staggered

G

eclipsed
staggered

• Structural rearrangement requires activation energy
• Important in the formation of organic and polymeric glasses
POINT DEFECTS
vacancy
large substitutional
atom
interstitial
Frenkel defect
small substitutional atom
Schottky defect
All of these defects disrupt the perfect arrangement of the
surrounding atoms –relaxation effects
Schottky, Frenkel, substitution
• Schottky and Frenkel normally v low conc since
formation energy high
e.g. NaCl at TL – 1 oC < 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
Schottky defects
• vacancies - anion and cation vacancies balance such
that charge neutrality is preserved
e.g.
NaCl 
nV-Na + nV+Cl
MgCl2 
nV2-Mg + 2nV+Cl
• cation vacancy has net negative charge and vice versa
because of non-neutralisation of nearest neighbour
charges.
charges balance
Frenkel defect
• interstitial + vacancy e.g. AgCl
• atoms move from lattice site to interstitial position
e.g. Vi + AgAg  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 CaF2
Kröger-Vink Notation (simplified)
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+ on M site
MM
X- on X site
XX
Vacancy on
M site
V-M
Vacancy on X site
V+X
Interstitial M ion
M+i
Interstitial X ion
X-i
Interstitial M
atom
Mi
Interstitial X atom
Xi
Foreign ion A+
on M site
AM
Foreign ion A2+ on
M+ site
A+M
Free electron
e-
Free hole
h+
INTERSTITIAL SITES
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.
RADIUS RATIO RULES
Nc = 8
0.732 
rc
1
ra
Nc = 3
0.155 
Nc = 6
0.414 
rc
 0.732
ra
Nc = 4
0.225 
rc
 0.414
ra
rc
 0.225
ra
Nc = 2
0
rc
 0.155
ra
SUBSTITUTIONAL DISORDER AND SPINELS
• general formula
AB2X4
X anions on fcc lattice
A,B cations in interstitial sites
• Normal spinels
A on tetrahedral sites
B on octahedral sites
AT(B2)OX4
e.g. MgAl2O4 (spinel)
• Inverse spinels
½ B on tetrahedral sites
A and ½ B on octahedral sites
e.g. Mg2TiO4; Fe3O4 (magnetite)
BT(AB)OX4
• There are cases in between:
degree of inversion
BT
BT  BO
= 0 for normal;
= 0.5 for inverse;
= 0.33 for disordered
Magnetite - Fe3O4  FeT3+[Fe2+Fe3+]OO4
e2+
3+
3+
2+
e-
e-
3+
Fe2+ and Fe3+ occupy adjacent, edge-sharing octahedra
-very easy for electrons to transfer from Fe2+ to Fe3+  conduction
-would not occur if FeT2+[Fe23+]OO4 – 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
- ri+ usually decreases with higher charge
- affects Madelung const 2,3 usually normal
4,2 usually inverse
• crystal field stabilisation
• covalency
FRENKEL DISORDER AND SUPERIONICS
• 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)

146oC
 (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
NaAl11O17
• Na+ “liquid sublattice”
Na+
Na+
• 2D blocks of spinel structure linked
by oxygens and mobile Na+ ions
oxygen ions
interstitial
sites
migration sequence for
sodium ions
Non-stoichiometric defects
• overall stoichiometry of material changes
•
substitution
interstitial
vacancy
A  A1-xBx
AB  A1+xB
AB  A1-xB
• 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 + xCaCl2  Na1-2xCax(VNa)xCl
• normal anion lattice
• Ca2+ substitutes for one Na+ but another Na+ must be removed to
maintain charge balance creating a vacancy
• 2NaNa + Ca  V-Na + Ca+Na
Interstitial
e.g.
CaF2 + YF3  Ca1-xYxF2(Fi)x
• Normal cation lattice with 1 Y3+ substituting for 1 Ca2+.
• Extra F- required for charge balance goes on interstitial site.
• CaCa + Y + F + Vi  Y+Ca + F-i
• NB: F- ( ri = 1.33 Å) much smaller than Cl- ( ri = 1.80 Å)
Variable valency
• e.g. reduction of TiO2 by hydrogen
TiO2 + xH2  TiO2-x + xH2O
 Ti4+1-2xTi3+2xO2-x
complete cation lattice - oxygen vacancies
2TiTi
 2TiTi– + VO2+
• Materials with large non-stoichiometric regions usually contain
elements which show variable valence
transition metals
e.g. Fe2+/Fe3+;
B metals
e.g. Pb2+/Pb4+
Radiation damage
•
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
atomic displacements per event
– produces ~ 1500 – 2000
•Most damage produced by recoil of atom
Mm  Md + 
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
fa = 1 – exp(-NdD)
D number of -decays per atom
Nd number of permanently displaced atoms
actinide atoms substituted for some Zr atoms
in zircon, ZrSiO4
THERMODYNAMICS
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.
• High entropy S of nonstoichiometric phases stabilises
them at high T.
On cooling, form metastable
phase or disproportionate.
• e.g. “FeO”
G
A
X
AB
Y
B
Schottky
Take crystal of N molecules of NaCl
NV vacancies on both lattices
NaNa + ClCl  V-Na + V+Cl
N-NV N-NV NV
NV
Equilibrium constant
 NV  NK0.5
Energy G required to form defects
VNa  VCl 
NV2
NV2
K

 2
2
Na
Cl
 Na  Cl   N  NV  N
G  -RTlnK
 K  exp  G   const exp  H  (assumes S constant)
 RT 
 RT 

  H 
NV  Nconst exp

2
RT


H  220 kJ mol-1 for NaCl
Frenkel
Take crystal of N molecules of AgCl
Vi + AgAg  Ag+i + VAg
N N-Ni
Ni
Ni
N i2
N i2
K

 N  N i N N 2
  H 
N i  constN exp 

 2 RT 
H  130 kJ mol-1 for AgCl
WHY DO DEFECTS OCCUR?
requires energy to create
them !
H
G
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
Probability
N!
P
N  n!n!
n number of defects
N-n normal species
N number of lattice sites

S = klnP
S  k ln

N!
 (N  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’
T
‘FeO’ really Fe1-xO
Fe1-xO  Fe3O4 + Fe
‘FeO’
+
Fe3O4
‘FeO’
+
Fe
570 oC
Fe + Fe3O4
0.97
1-x
0.85
ELIMINATION OF DISORDER
DEFECT INTERACTIONS
- of increasing magnitude with defect conc
1.
lattice relaxation
2.
short-range order
clustering
e.g. Ca1-xYxF2+x
Y3+ substitutes for Ca2+
x small – xs F- goes into interstitial sites
inc x – clusters of F- , Y, and vacancies form
e.g. 2:2:2
higher x – increasingly large clusters
Cluster formation
Ca2+
Y3+
FiFiVF+
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. TiO2-x
2D
3D
-
corner sharing  edge sharing
(edge  face)
If shear planes regularly spaced then get new ‘stoichiometric’
phase TinO2n-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 
nMgCl2 
Vi + AgAg 
2NaNa + Ca
? + nV+Cl
nV2–Mg + ?
? + V–Ag
 V–Na + ?
(ii) Describe the effect of each of the above defect types
on the density of a material