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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. 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

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  1. PX431 Structure and Dynamics of Solids PART 2: Defects and Disorder Diane Holland P160 d.holland@warwick.ac.uk

  2. 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)

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

  4. ORIENTATIONAL DISORDER (conformational/rotational)

  5. Crystal Structure Convolution of Basis and lattice Basis may be group of atoms which can adopt different orientations with respect to rest of lattice

  6. a0 b0 defect Kermit is not symmetrical  orientation is important

  7. 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!

  8. 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

  9. 2a 2a 2a -104oC to -58oC CsCl structure Ordered, alternating orientations unit cell = 2a

  10. > +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’

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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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)

  18. 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.

  19. RADIUS RATIO RULES Nc = 8 Nc = 6 Nc = 4 Nc = 3 Nc = 2

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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)

  27. 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+

  28. 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

  29. 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

  30. actinide atoms substituted for some Zr atoms in zircon, ZrSiO4

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

  32. 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”

  33. 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

  34. 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

  35. 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

  36. 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

  37. ‘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

  38. 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

  39. Ca2+ Y3+ Fi- Fi- VF+ Cluster formation 2:2:2

  40. 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

  41. 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

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