PX431 Structure and Dynamics of Solids

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PX431 Structure and Dynamics of Solids

• PART 2
• Defects and Disorder
• Diane Holland P160 d.holland_at_warwick.ac.uk

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• 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)

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• Orientational disorder
• groups of atoms
• - ammonium salts
• - linear chains
• Point defects
• vacancies, interstitials, incorrect atoms
• - Schottky
• - Frenkel
• - substitution

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ORIENTATIONAL DISORDER
(conformational/rotational)
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Crystal Structure
Convolution of Basis and lattice
Basis may be group of atoms which can adopt
different orientations with respect to rest of
lattice
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Kermit is not symmetrical ? orientation is
important
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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!
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Example - ammonium salts
NH4 ND4

• - extent of disorder depends on T

e.g. ND4Br lt -104oC CsCl structure ordered
orientations unit cell a
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-104oC to -58oC CsCl structure Ordered,
alternating orientations unit cell 2?a
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-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
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CHAINS

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

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Energetics of rotation
?
?

• Structural rearrangement requires activation
  energy
• Important in the formation of organic and
  polymeric glasses

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POINT DEFECTS
interstitial
small substitutional atom
vacancy
Schottky defect
Frenkel defect
large substitutional atom
All of these defects disrupt the perfect
arrangement of the surrounding atoms relaxation
effects
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• Schottky, Frenkel, substitution
• Schottky and Frenkel normally v low conc since
  formation energy high e.g. NaCl at TL 1 oC lt
  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

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Schottky defects

• vacancies - anion and cation vacancies balance
  such that charge neutrality is preserved
• e.g. NaCl ? nV-Na nVCl
• MgCl2 ? nV2-Mg 2nVCl
• cation vacancy has net negative charge and vice
  versa because of non-neutralisation of nearest
  neighbour charges.
• charges balance

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Frenkel defect

• interstitial vacancy e.g. AgCl
• atoms move from lattice site to interstitial
  position
• e.g. Vi AgAg ? Agi 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

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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 VX
Interstitial M ion Mi 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 AM
Free electron e- Free hole h
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INTERSTITIAL SITESin 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.
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RADIUS RATIO RULES

• Nc 8
• Nc 6
• Nc 4

Nc 3 Nc 2
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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

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Magnetite - Fe3O4 ? FeT3Fe2Fe3OO4

• Fe2 and Fe3 occupy adjacent, edge-sharing
  octahedra
• very easy for electrons to transfer from Fe2 to
  Fe3 ? conduction
• would not occur if FeT2Fe23OO4 no easy
  transfer oct ? tet

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• 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

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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

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• e.g. ? - alumina NaAl11O17
• Na liquid sublattice
• 2D blocks of spinel structure linked by oxygens
  and mobile Na ions

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Non-stoichiometric defects

• overall stoichiometry of material changes
• substitution A ? A1-xBx
• interstitial AB ? A1xB
• 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

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• 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 CaNa
• 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 ? YCa F-i
• NB F- ( ri 1.33 Å) much smaller than Cl- (
  ri 1.80 Å)

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• Variable valency
• e.g. reduction of TiO2 by hydrogen
• TiO2 xH2 ? TiO2-x xH2O
• ? Ti41-2xTi32xO2-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

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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

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• 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

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actinide atoms substituted for some Zr atoms in
zircon, ZrSiO4
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THERMODYNAMICS

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

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• 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

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• Schottky
• Take crystal of N molecules of NaCl
• NV vacancies on both lattices
• NaNa ClCl ? V-Na VCl
• 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

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• Frenkel
• Take crystal of N molecules of AgCl
• Vi AgAg ? Agi V?Ag
• ?N N-Ni Ni Ni
• ?H ? 130 kJ mol-1 for AgCl

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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

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• Probability
• n number of defects
• N-n normal species
• N number of lattice sites
• ?S klnP
• ? kNlnN (N-n)ln(N-n) nlnn
• ? S depends on number of defectsNeglects
  lattice relaxation and defect interactions

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• Beyond a certain concentration,
• defects will begin to interact and
• even be eliminated.
• FeO really Fe1-xO
• Fe1-xO ? Fe3O4 Fe

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ELIMINATION OF DISORDER

• DEFECT INTERACTIONS - of increasing magnitude
  with defect conc
• lattice relaxation
• short-range order - clustering
• e.g. Ca1-xYxF2x Y3 substitutes for Ca2
• x small xs F- goes into interstitial sites
• inc x clusters of F- , Y, and vacancies form
  
• e.g. 222
• higher x increasingly large clusters

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Cluster formation
222
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• 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

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• 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 ? ? nVCl
• nMgCl2 ? nV2Mg ?
• Vi AgAg ? ? VAg
• 2NaNa Ca ? VNa ?
• (ii) Describe the effect of each of the above
  defect types on the density of a material