1. Ionic Conductors: Characterisation of Defect Structure Lectures 1-4 Introduction to Crystal Chemistry Dr. I. Abrahams Queen Mary University of London Lectures co-financed by the European Union in scope of the European Social Fund

2. Crystal Chemistry What is crystal chemistry? The study of the structures of crystals including: Description and classification of crystal structures Factors that govern structure types adopted Structure prediction Structure-property relationships What is a crystal? A solid that shows a regularly repeating structure that can be characterised by a basic repeating unit known as a unit cell.

3. Lectures co-financed by the European Union in scope of the European Social Fund Parallelepiped The unit cell is generally chosen as the smallest repeating unit with the highest symmetry. The unit cell, when repeated in 3D, must cover all the space in the crystal lattice. Different crystal structures have different unit cells. Unit cells are defined by six parameters in 3D. a, b, c are the unit cell edges and ,  and  are the inter-axial angles. (0 is the origin and its position is arbitrary). Unit Cells

4. Lectures co-financed by the European Union in scope of the European Social Fund Crystal Systems There are seven crystal systems. These can be distinguished by the different unit cell shapes and their minimum intrinsic symmetry. Crystal systemUnit cell shape Minimum symmetry Triclinica  b  c;  90  None Monoclinic 1 two fold axis or (standard setting)a  b  c;  =  =90 ,  90  mirror plane Orthorhombic a  b  c;  =  =  =90  3 two fold axes or mirror planes Tetragonal a=b  c;  =  =  =90  1 four fold axis Trigonal 1 three fold axis (rhombohedral setting) a=b=c;  =  =  90  (hexagonal setting) a=b  c;  =  =90  =120  Hexagonal a=b  c;  =  =90  =120  1 six fold axis Cubic a=b=c;  =  =  =90  4 three fold axes The symbol  used here refers to not necessarily equal to. In some cases there is accidental equivalence, but the minimum symmetry is not present.

5. Lectures co-financed by the European Union in scope of the European Social Fund Fractional Coordinates The location of the origin is arbitrary, but is usually chosen to correspond to a point of symmetry. It need not be an atom position. Atoms positions can be defined with respect to the unit cell using fractional coordinates x, y, z x = X/a where X is the distance parallel to the a-axis y = Y/b where Y is the distance parallel to the b-axis z = Z/c where Z is the distance parallel to the c-axis

6. Lectures co-financed by the European Union in scope of the European Social Fund Introduction to Crystal Chemistry There are many crystalline solids, but only a few basic structures. Many simple structures can be visualised in terms of close packing of identical spheres, in some case with smaller spheres in the spaces between the close packed spheres. Atoms or ions can be regarded as “squashy” spheres. The squashy character is a result of polarisation of the electronic cloud surrounding these atoms or ions. Different compounds with the same structure have the same geometry, but different size, i.e. different ionic radii and bond lengths.

7. Lectures co-financed by the European Union in scope of the European Social Fund e.g.NaCl, MgO, LiI, TiC all exhibit the rocksalt structure For compounds that adopt the rocksalt structure there is no direct correlation between structure and bonding, i.e. the rocksalt structure is adopted by ionic and covalent compounds.

8. Lectures co-financed by the European Union in scope of the European Social Fund Close Packing (cp) Identical spheres can pack in a number of ways. The closest way is known as close packing. Consider some arrays of identical spheres. 1-D 2-D cp, CN = 2 cp, CN = 6

9. Lectures co-financed by the European Union in scope of the European Social Fund 3-Dcp, CN = 12

10. Lectures co-financed by the European Union in scope of the European Social Fund Hexagonal and Cubic Close Packing There are two types of 3-D close packed arrays. Hexagonal close packing hcp ABA….. Cubic close packing ccp ABC… A A A B B C

11. Lectures co-financed by the European Union in scope of the European Social Fund hcp and ccp Unit Cells Like all crystalline solids hcp and ccp based solids can be described by unit cells. hcp ccp

12. Lectures co-financed by the European Union in scope of the European Social Fund Non-Close Packed Arrays Compare two similar 2-D arrays. 2-D cp, CN = 6 Non-cp, CN = 4

13. Lectures co-financed by the European Union in scope of the European Social Fund 3D Non-Close Packed Arrays Body centred cubic (bcc) packing is a non-close packed array. bcc CN = 8 Packing density Even in close packed arrays there are spaces between the spheres. A measure of how closely packed spheres are is the packing density e.g in ccp

14. Lectures co-financed by the European Union in scope of the European Social Fund Packing Density Look at a single unit cell face in ccp. Therefore the maximum packing density for identical spheres is 74% for a cp array Diameter of sphere = 2r  face diagonal = 4r Packing Density hcp 74% ccp 74% bcc 68%

15. Lectures co-financed by the European Union in scope of the European Social Fund Metals Metal atoms can be considered to be spherical and adopt structures that exhibit high coordination numbers in order to achieve maximum overlap of atomic orbitals. Metallic elements In metallic elements since all atoms are of the same type and size ccp, hcp and bcc packing are typically adopted. However, it should be noted that in some cases although a cp geometry is adopted the packing density may be lower than 74% i.e. not truly close packed. eg ccp Ag, Au, Fe, Pb hcp Be, Co,Mg bcc Ba, Cr, K

16. Lectures co-financed by the European Union in scope of the European Social Fund Alloys Metallic compounds with more than one atom type. If the atom sizes are similar then as with metallic elements ccp, hcp or bcc structures are adopted. e.g Cu:Au Alloy disordered ccp Note at certain compositions Cu and Au can order over the lattice.

17. Lectures co-financed by the European Union in scope of the European Social Fund Interstitial Sites In order to describe inorganic compounds using close packing it is first necessary to describe the interstitial sites present in a cp array. There are two important types of interstitial site 1. Tetrahedral sites Consider atoms from just two cp layers. Spheres in the top layer fit into dips between 3 spheres in the bottom layer and vice versa. This gives a tetrahedral interstitial site. A tetrahedron has 4 faces and 6 edges T+TT+T (pointing up) (pointing down) There are two types of tetrahedral interstitial site

18. Lectures co-financed by the European Union in scope of the European Social Fund Interstitial Sites - Tetrahedra Number of T + sites = Number of T  sites The tetrahedral sites do not lie strictly between the cp layers. T + in layer below The maximum radius r T of a sphere in a tetrahedral site is given by r T = r cp  0.225 Where r cp is the radius of the close packed sphere. T  in layer above

19. Lectures co-financed by the European Union in scope of the European Social Fund Interstitial Sites - Octahedra 2. Octahedral sites Where dips in the top and bottom layers coincide we get an octahedral site. An octahedron has 8 faces and 12 edges. The maximum radius r O for an atom to fit into an octahedral site is r O = r cp  0.414 i.e. much bigger than a tetrahedral site.

20. Lectures co-financed by the European Union in scope of the European Social Fund Location of interstitial sites in cp unit cells 1. hcp Tetrahedral Octahedral

21. Lectures co-financed by the European Union in scope of the European Social Fund 2. ccp TetrahedralOctahedral

22. Lectures co-financed by the European Union in scope of the European Social Fund Close packing described by polyhedra One can view cp structures as built up from polyhedra (representing the interstitial sites) that share faces, edges or corners. Using this type of representation (a) The centre of the polyhedron represents the interstitial site (b) The corner of the polyhedron represents the cp atom Polyhedral representations are very important as they emphasize the CN of the interstitial ions, their relative positions and linkage.

23. Lectures co-financed by the European Union in scope of the European Social Fund Interstitial sites in cp structures 1. hcp (a) Octahedra Octahedra share faces perpendicular (  ) to cp planes Octahedra share edges parallel (  ) to cp planes Results in columns of octahedra perpendicular to cp planes. Interstitial sites between cp layers 1 and 2, and 2 and 3 are identical and stacked one above the other resulting in mirror symmetry about B.

24. Lectures co-financed by the European Union in scope of the European Social Fund (b) Tetrahedra Tetrahedra share faces and corners  to cp layers Tetrahedra share edges  to cp layers T + shares a face with T  in layer below T  shares a face with T + in layer above T + and T  sharing faces gives a trigonal bipyramidal site CN = 5 Unique to hcp. T + shares edges with T  within cp layer T  shares edges with T + within cp layer (c) Inter-polyhedral linkages Octahedra and tetrahedra share faces within cp layer.

25. Lectures co-financed by the European Union in scope of the European Social Fund 2. ccp Octahedra share only edges  to cp planes Octahedra share only edges  to cp planes Tetrahedra share only edges  to cp planes Tetrahedra share only edges  to cp planes T + shares edges with T  only T  shares edges with T + only Comparison of oct and & tet linkages in hcp and ccp Oct shares face with oct in hcp only Tet shares faces with tet in hcp only Orientation of layers 1 and 2 and 2 and 3 now different. Octahedra are not above octahedra Tetrahedra are not above tetrahedra.

26. Lectures co-financed by the European Union in scope of the European Social Fund Interstitial Sites Summary cpcp atoms per cell Tet sites per cell Oct sites per cell Tet sites per cp atom Oct sites per cp atom hcp24321 ccp48421

27. Lectures co-financed by the European Union in scope of the European Social Fund Important Inorganic Structures Based on cp Many inorganic structures are based on close packing of spheres and can be described by close packing of one ion sublattice with counter ions in all or part of the interstitial sites. While these structures are not truly close packed (i.e. the ions do not touch each other), their geometry can be described as close packed. In the case of ionic conducting inorganic solids, many adopt ordered or disordered forms of the classic inorganic structural types.

28. Lectures co-financed by the European Union in scope of the European Social Fund 1. Cubic close packed structures (a) Li 3 Bi Li 3 Bi is an intermetallic compound and can be described as ccp Bi with Li in all the octahedral and tetrahedral sites. The Li 3 Bi structure therefore shows the complete filling of all interstitial sites. ccp Based Structures

29. Lectures co-financed by the European Union in scope of the European Social Fund (b) NaCl ccp Cl  with Na + in all the octahedral sites. ccp Cl  at corners and face centres of unit cell Na + in oct sites at centre of cube and mid point of each edge. NB Tet sites empty Cl  cp planes are  to body diagonal. NaCl 6 oct share all 12 edges with other NaCl 6 oct. NaCl 6 oct share faces with empty tet sites. Unit cell NaCl 6 oct Shared edge

30. Lectures co-financed by the European Union in scope of the European Social Fund Remember maximum ratio for octahedral coordination in cp system is 0.414 Therefore Cl  ions in NaCl are not close packed, but do have cp geometry with an fcc unit cell. Note each Cl  is surrounded by 6 Na + ions (and each Na + is surrounded by 6 Cl  ions). Many binary compounds exhibit the rocksalt structure. All are isostructural, but have different properties and bonding. Radius ratio

31. Lectures co-financed by the European Union in scope of the European Social Fund Compounda(Å)Compounda(Å) MgO4.213LiF4.0270 CaO4.8105NaF4.64 SrO5.160NaCl5.6402 BaO5.539AgF4.92 NiO4.1769AgCl5.549 TiO4.177AgBr5.7745 MnO4.445MgS5.200 FeO4.307CaS5.6948 UC4.955LaN5.30

32. Lectures co-financed by the European Union in scope of the European Social Fund (c) Zinc blende or sphalerite (ZnS) ccp S 2  with Zn 2+ in half the tet sites. Tet sites all T+(or T  ) avoiding edge sharing. Each S 2  is surrounded by 4 Zn 2+ and each Zn 2+ surrounded by 4 S 2 . Many other structures can be derived from zinc blende ZnSC (diamond)Si GaAs GaP

33. Lectures co-financed by the European Union in scope of the European Social Fund (d) Fluorite (CaF 2 ) ccp Ca 2+ with F  in all the tet sites. (Oct empty). Both T + and T  occupied. Therefore tet share edges and corners, but Ca 2+ large and not cp.  tet centres are far apart. Total 4 Ca 2+ per cell and 8 F  per cell  Ca:F = 1:2 i.e. CaF 2 CaF 8 = cubic coordination Antifluorite ccp anions with cations in all tet sites. e.g. Na 2 O

34. Lectures co-financed by the European Union in scope of the European Social Fund hcp based structures 2. Structures based on hcp (a) Nickel Arsenide (NiAs) hcp As with Ni in all the oct sites. (tet empty). Ni at 2/3, 1/3, 1/4 and 2/3, 1/3, 3/4

35. Lectures co-financed by the European Union in scope of the European Social Fund NiAs 6 octahedra AsNi 6 also 6 coordinate but not oct (trigonal prismatic) Each NiAs 6 oct shares 2 faces with other NiAs 6 oct resulting in columns of face sharing oct. NiAs is the hcp analogue of NaCl(ccp), but with face sharing. NaCl: Na +  Na + repulsions favour ccp. NiAs: Ni 2+  Ni 2+ repulsion reduced due to covalence and Ni-Ni bonding  hcp favoured. Structure adopted by FeS, NiS and CoS

36. Lectures co-financed by the European Union in scope of the European Social Fund (b) Wurtzite (ZnS) hcp S 2  with Zn 2+ in half the tet sites. (Oct empty). Zn 2+ on edges 0,0,5/8 Zn 2+ in cell at 1/3, 2/3, 1/8

37. Lectures co-financed by the European Union in scope of the European Social Fund Only T + or T  occupied. Avoids tet sharing faces which is energetically unfavourable. Also avoids tet sharing edges. ZnS 4 tet corner sharing only SZn 4 also tet ZnS either wurtzite or zinc blende Both tet ZnS 4 Both corner share Wurtzite more ionic

38. Lectures co-financed by the European Union in scope of the European Social Fund Layered structures (a) CdCl 2 and CdI 2 The structures of CdCl 2 and CdI 2 can be described as being based on ccp and hcp halide lattices respectively with Cd 2+ filling octahedral sites in alternate layers. This results in layered compounds with alternate layers held together by van der Waals forces. In both structures CdX 6 octahedra share edges with other octahedra in same layer Cdl 2 CdCl 2

39. Lectures co-financed by the European Union in scope of the European Social Fund (b) CrCl 3 and BiI 3 The structures of CrCl 3 and BiI 3 can be described as being based on ccp and hcp halide lattices respectively. In both structures 1/3 of the available oct sites are occupied. 2/3 of the oct sites in alternate layers are filled by cations resulting in layered structures. Each octahedron shares edges with 3 other octahedra within a layer.

40. Lectures co-financed by the European Union in scope of the European Social Fund Other Important Structures (a) Rutile (TiO 2 ) Essentially distorted hcp O 2  with Ti 4+ in half the oct sites. Every alternate octahedron is filled resulting in chains of edge sharing TiO 6 octahedra. Columns of octahedra with alternate columns empty. Columns corner share with neighbouring columns. The columns run parallel to the cp layers OTi 3 trigonal planar O 2  coordination. Other examples MnO 2, SnO 2, CrO 2, MnF 2.

41. Lectures co-financed by the European Union in scope of the European Social Fund (b) Corundum (  -Al 2 O 3 ) hcp O 2  with Al 3+ in 2/3 of the oct sites. Al 3+ displaced resulting in distorted tet coordination for O 2 . Corundum is noted for its hardness. Doping with Cr or Ti results in the gemstones ruby and sapphire. Other examples Ti 2 O 3 V 2 O 3 Cr 2 O 3 Ga 2 O 3.

42. Lectures co-financed by the European Union in scope of the European Social Fund (c) ReO 3 ccp O 2  with ¼ of the O 2  ions missing. Re 6+ locate in ¼ of the the octahedral sites. The resulting structure is a 3-dimensional array of corner sharing ReO 6 octahedra. Each ReO 6 octahedron shares all six corners with other ReO 6 octahedra and linear Re-O-Re linkages. Other examples ScF 3 NbF 3 TaF 3 MoF 3

43. Lectures co-financed by the European Union in scope of the European Social Fund (d) Perovskite (CaTiO 3 ) Closely related to ReO 3.. A ccp array of O 2  with ¼ of the O 2  ions missing. Ti 4+ located in ¼ of the the octahedral sites. Ca 2+ is located in the oxide ion vacancy. TiO 6 octahedra share corners to give the 3-D framework, with Ca 2+ in essentially a 12 CN site. However distortion lowers the coordination number to 8.

44. Lectures co-financed by the European Union in scope of the European Social Fund (e) Spinel (MgAl 2 O 4 ) ccp array of O 2  with Al 3+ located in 1/2 of the the octahedral sites and Mg 2+ in 1/8 of the tetrahedral sites. The structure consists of columns of edge sharing octahedra which share edges with parallel columns. The tetrahedra share corners with the octahedra. Inverse spinel Fe 2 MgO 4 adopts an inverse spinel structure. With half the Fe ions (Fe 3+ ) in tetrahedral sites and the other half in octahedral sites with Mg 2+.

45. Lectures co-financed by the European Union in scope of the European Social Fund