Order in crystals Symmetry, X-ray diffraction

2-dimensional square lattice

Translation

Rotation

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

Restriction on n-fold rotation symmetry in a periodic lattice a a na  (n-1)a/2 cos (180-  ) = - cos  = (n-1)/2 n  o Rotation

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

Obliquea  b,   90 o Rectangular a  b,  = 90 o Squarea = b,  = 90 o Hexagonala = b,  = 120 o

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

Bravais lattices in 2-dimensions - 5 square rectangular obliquehexagonal centred rectangular

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

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

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

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

C4C4 C4C4 Spherical basis Non-spherical basis

Lattice + Nonspherical Basis Point group operations Point group operations + translation symmetries 7 Crystal systems 32 Crystallographic point groups 14 Bravais lattices 230 space groups Lattice + Spherical Basis Space Groups

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

(010) Distance between planes = a x y z

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

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

a  h 2 +k 2 +l 2 d hkl = Spacing between Miller planes for cubic crystal system

 d hkl hkl plane 2d hkl sin  = n  Wavelength = Bragg’s law

von Laue’s condition for x-ray diffraction d k k lattice point k = incident x-ray wave vector k = scattered x-ray wave vector d = lattice vector d.i -d.i 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

S hkl =  f n e 2  i  hx +ky +lz ) nn n Relates to Atom type Atom position Structure factor Intensity of x-ray scattered from an (hkl) plane I hkl  S hkl 2

Problem Set 1.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. 2.Write down the reciprocal lattice vectors corresponding to the primitive direct lattice vectors in problem 1. 3.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. 4.Determine the best packing efficiency among simple cube, bcc and fcc lattices. 5.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. 6.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. 7.Points on a cubic close packed structure form a Bravais lattice, but the points on a hexagonal close packed structure do not. Explain. 8.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. 9.Discuss the spinel and inverse spinel structures. Give some examples of materials possessing such structures. 10.Draw schematic diagrams of the following structures : (a) rock salt, (b) cesium chloride, (c) fluorite, (d) rutile. 11.What is the difference in the structures of  -graphite and  -graphite ? 12.Contrast the zinc blende and wurtzite structures - give similarities and differences. 13.Diamond has a zinc blende structure - explain. 14.Draw schematic diagrams to illustrate the similarity between NiAs and CdI 2 structures. 15.Illustrate with a diagram the perovskite structure for the general oxide formula ABO 3. 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