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PX431 Structure and Dynamics of Solids. PART 2: Defects and Disorder 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

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px431 structure and dynamics of solids

PX431 Structure and Dynamics of Solids

PART 2:

Defects and Disorder

Diane Holland P160 d.holland@warwick.ac.uk

slide2

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 non- stoichiometry; 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)
slide3
Orientational disorder

groups of atoms

-ammonium salts

- linear chains

Point defects

vacancies, interstitials, incorrect atoms

-Schottky

- Frenkel

- substitution

orientational disorder
ORIENTATIONAL DISORDER

(conformational/rotational)

slide5

Crystal Structure

Convolution of Basis and lattice

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

slide6

a0

b0

defect

Kermit is not symmetrical  orientation is important

slide7

2a0

a0

b0

2b0

random

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!

example ammonium salts

Br

D

N

a

Example - ammonium salts

NH4+

ND4+

- extent of disorder depends on T

e.g. ND4Br

< -104oC

CsCl structure

ordered orientations

unit cell = a

slide9

2a

2a

2a

-104oC to -58oC

CsCl structure

Ordered, alternating orientations

unit cell = 2a

slide10

> +125oC

NaCl structure

(ND4)+ ion rotating  spherically symmetric

-58oC to +125oCCsCl structure

random arrangement of orientations

unit cell = a

but disordered

NB coordination number change from 8 to 6

i.e. rotating ion is ‘smaller’

chains
CHAINS

e.g.organic polymers

  • Carbon C 4-coordinated

Join two - eclipsed

- staggered

energetics of rotation

2/3

4/3

G

eclipsed

staggered

G

eclipsed

staggered

Energetics of rotation

  • Structural rearrangement requires activation energy
  • Important in the formation of organic and polymeric glasses
point defects
POINT DEFECTS

interstitial

vacancy

small substitutional atom

Schottky defect

Frenkel defect

large substitutional atom

All of these defects disrupt the perfect arrangement of the surrounding atoms –relaxation effects

slide14
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
slide15

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
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
kr ger vink notation simplified
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)

interstitial sites in close packed systems
INTERSTITIAL SITESin close-packed systems

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
RADIUS RATIO RULES

Nc = 8

Nc = 6

Nc = 4

Nc = 3

Nc = 2

substitutional disorder and spinels
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

BT(AB)OX4 e.g. Mg2TiO4; Fe3O4 (magnetite)

  • There are cases in between:

degree of inversion

= 0 for normal;

= 0.5 for inverse;

= 0.33 for disordered

slide21

e-

2+

3+

3+

2+

e-

e-

3+

Magnetite - Fe3O4  FeT3+[Fe2+Fe3+]OO4

  • 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
slide22
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
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)  (bcc)

146oC

  • 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
slide24

oxygen ions

interstitial sites

migration sequence for sodium ions

Na+

Na+

  • e.g.  - alumina NaAl11O17
  • Na+ “liquid sublattice”
  • 2D blocks of spinel structure linked by oxygens and mobile Na+ ions
non stoichiometric defects
Non-stoichiometric defects
  • overall stoichiometry of material changes
  • substitution A  A1-xBx

interstitial AB  A1+xB

vacancy 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
slide26
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)
slide27
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
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 – produces ~ 1500 – 2000 atomic displacements per event

slide29

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 equationamorphous fraction fa = 1 – exp(-NdD)
  • D number of -decays per atom
  • Nd number of permanently displaced atoms
thermodynamics
THERMODYNAMICS

Evidence for existence of non-stoichiometry:

  • continuous variation in composition
  • continuous change in structure e.g. lattice parameter
  • thermodynamic bivariance G = (T,x)
slide32

G

B

A

Y

X

AB

  • 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 non-stoichiometric phase determined by common tangent method.
  • High entropy S of non-stoichiometric phases stabilises them at high T. On cooling, form metastable phase or disproportionate.
  • e.g. “FeO”
slide33
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 G  -RTlnK

 (assumes S constant)

H  220 kJ mol-1 for NaCl

slide34
Frenkel

Take crystal of N molecules of AgCl

Vi + AgAg Ag+i + VAg

N N-Ni Ni Ni

H  130 kJ mol-1 for AgCl

why do defects occur

H

G = H - TS

G

[defect]

-TS

at this point a breakdown in structure will occur to form a new phase

WHY DO DEFECTS OCCUR?

requires energy to create them !

H inc butS also inc

G = H - TS

Temperature-TS incs with inc T  more defects at higher T

slide36
Probability

n number of defects

N-n normal species

N number of lattice sites

S = klnP

 k[NlnN – (N-n)ln(N-n) – nlnn]

 S depends on number of defectsNeglects lattice relaxation and defect interactions

slide37

‘FeO’

T

‘FeO’

+

Fe3O4

‘FeO’

+

Fe

570 oC

Fe + Fe3O4

0.85

0.97

1-x

Beyond a certain concentration,

defects will begin to interact and

even be eliminated.

‘FeO’ really Fe1-xO

Fe1-xO  Fe3O4 + Fe

elimination of disorder
ELIMINATION OF DISORDER

DEFECT INTERACTIONS- of increasing magnitude with defect conc

  • lattice relaxation
  • 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

slide39

Ca2+

Y3+

Fi-

Fi-

VF+

Cluster formation

2:2:2

slide40
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 - corner sharing  edge sharing

3D - (edge  face)

If shear planes regularly spaced then get new ‘stoichiometric’ phase TinO2n-1

slide41
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

nMgCl2 nV2–Mg + ?

Vi+ AgAg? + V–Ag

2NaNa + Ca V–Na + ?

(ii)Describe the effect of each of the above defect types on the density of a material