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Order in crystals Symmetry, X-ray diffraction

Order in crystals Symmetry, X-ray diffraction. 2-dimensional square lattice. Translation. Translation. Rotation. Rotation. Point group symmetries : Identity (E) Reflection (s) Rotation (R n ) Rotation-reflection (S n ) Inversion (i). In periodic crystal lattice :

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Order in crystals Symmetry, X-ray diffraction

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  1. Order in crystalsSymmetry, X-ray diffraction

  2. 2-dimensional square lattice

  3. Translation

  4. Translation

  5. Rotation

  6. Rotation

  7. Point group symmetries : Identity (E) Reflection (s) Rotation (Rn) Rotation-reflection (Sn) Inversion (i) In periodic crystal lattice : (i) Additional symmetry - Translation (ii) Rotations – limited values of n

  8. a a q na ( n - 1 ) a / 2 Restriction on n-fold rotation symmetry in a periodic lattice cos (180-q) = - cos q = (n-1)/2 n 3 2 1 0 -1 qo180 120 90 60 0 Rotation 2 3 4 6 1

  9. Crystal Systems in 2-dimensions - 4 oblique square hexagonal rectangular

  10. Oblique a  b,   90o Rectangular a  b,  = 90o Square a = b,  = 90o Hexagonal a = b,  = 120o

  11. Crystal Systems in 3-dimensions - 7 Cubic Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Trigonal

  12. Bravais lattices in 2-dimensions - 5 square rectangular centred rectangular oblique hexagonal

  13. Bravais Lattices in 3-dimensions (in cubic system) Body centred cube (I) Primitive cube (P) Face centred cube (F)

  14. Bravais Lattices in 3-dimensions - 14 Cubic - P, F (fcc), I (bcc) Tetragonal - P, I Orthorhombic - P, C, I, F Monoclinic - P, C Triclinic - P Trigonal - R Hexagonal/Trigonal - P

  15. Point group operations 7 Crystal systems Point group operations + translation symmetries 14 Bravais lattices

  16. Lattice(o) + basis (x) = crystal structure

  17. C4 Spherical basis C4 Non-spherical basis

  18. Lattice+ SphericalBasis Lattice+ NonsphericalBasis Point group operations 7Crystalsystems 32 Crystallographic point groups Point group operations + translation symmetries 230 space groups 14 Bravais lattices Space Groups

  19. z y x Miller plane (100) Distance between planes = a a

  20. z y x (010) Distance between planes = a

  21. z y x (110) Distance between planes = a/2 = 0.7 a

  22. z y x (111) Distance between planes = a/3 = 0.58 a

  23. a h2+k2+l2 dhkl = Spacing between Miller planes for cubic crystal system

  24. Bragg’s law Wavelength =l q q d h k l h k l p l a n e q l 2 d s i n = n h k l

  25. von Laue’s condition for x-ray diffraction k k lattice point d d.i -d.i k = incident x-ray wave vector k = scattered x-ray wave vector d = lattice vector i = unit vector = (/2)k i = unit vector = (/2)k Constructive interference condition: d.(i-i) = m  (/2)d.(k-k) = m  d.k = 2m K = reciprocal lattice vector d.K = 2n  k = K

  26. Shkl = Sfne2pi(hx +ky +lz ) n n n Structure factor Atom position Relates to Atom type Intensity of x-ray scattered from an (hkl) plane Ihkl Shkl2

  27. Problem Set • Write down a set of primitive vectors for the following Bravais lattices : (a) simple cube, (b) body-centred cube, (c) face-centred cube, (d) simple tetragonal, (e) body-centred tetragonal. • Write down the reciprocal lattice vectors corresponding to the primitive direct lattice vectors in problem 1. • Prove with a simple geometric construction, that rotation symmetry operations of order 1, 2, 3, 4 and 6 only are compatible with a periodic lattice. • Determine the best packing efficiency among simple cube, bcc and fcc lattices. • In a system of close packed spheres, determine the ratio of the radius of tetrahedral interstitial sites to the radius of the octahedral interstitial sites. • List the Bravais lattices arising from the cubic, tetragonal and orthorhombic systems. Discuss their genesis and account for why cubic system has three, tetragonal system has two and orthorhombic system has four Bravais lattices. • Points on a cubic close packed structure form a Bravais lattice, but the points on a hexagonal close packed structure do not. Explain. • Write down the direct and reciprocal lattice vectors for diamond considering it as an fcc lattice with two atoms in the basis. Write down also the coordinates of the two atoms in the basis. Determine the systematic absences in its x-ray diffraction profile. • Discuss the spinel and inverse spinel structures. Give some examples of materials possessing such structures. • Draw schematic diagrams of the following structures : (a) rock salt, (b) cesium chloride, (c) fluorite, (d) rutile. • What is the difference in the structures of -graphite and -graphite ? • Contrast the zinc blende and wurtzite structures - give similarities and differences. • Diamond has a zinc blende structure - explain. • Draw schematic diagrams to illustrate the similarity between NiAs and CdI2 structures. • Illustrate with a diagram the perovskite structure for the general oxide formula ABO3. What is the oxygen coordination for A and B ? Indicate this on the diagram. • (Text books of Cotton & Wilkinson, Greenwood & Earnshaw, Wells etc. give details of various structural motifs). A more detailed presentation on x-ray diffractometry is also provided on the website

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