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Announcements Revised Lab timings: 1-3 PM (all groups) 2) Quiz 1, 28 th Jan 2014,

Announcements Revised Lab timings: 1-3 PM (all groups) 2) Quiz 1, 28 th Jan 2014, Tuesday 7:30 PM, WS 209, WS 213. Chapter 3 Crystal Geometry and Structure Determination. Recap. Lattice, Motif/Basis. Crystal = Lattice + Motif. e.g. Brass, diamond, ZnS.

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Announcements Revised Lab timings: 1-3 PM (all groups) 2) Quiz 1, 28 th Jan 2014,

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  1. Announcements • Revised Lab timings: 1-3 PM (all groups) • 2) Quiz 1, 28th Jan 2014, • Tuesday 7:30 PM, WS 209, WS 213

  2. Chapter 3Crystal Geometry and Structure Determination

  3. Recap Lattice, Motif/Basis Crystal = Lattice + Motif e.g. Brass, diamond, ZnS Miller indices of direction: components of vector w.r.t to basis vector a, b and c

  4. Miller Indices of directions and planes William Hallowes Miller(1801 – 1880) University of Cambridge

  5. Miller Indices for planes z 1. Select a crystallographic coordinate system with origin not on the plane 2. Find intercepts along axes in terms of respective lattice parameters 1 1 1 y 3. Take reciprocal 1 1 1 O 4. Convert to smallest integers in the same ratio x 1 1 1 5. Enclose in parenthesis (111)

  6. z z _ (1 1 0) O* y x x Miller Indices for planes (contd.) Plane ABCD OCBE origin O O* 1 ∞ ∞ 1 -1 ∞ intercepts E reciprocals 1 0 0 1 -1 0 Miller Indices A B (1 0 0) O Zero represents that the plane is parallel to the corresponding axis Bar represents a negative intercept D C

  7. Courtesy: H Bhadhesia

  8. Courtesy: H Bhadhesia

  9. Courtesy: H Bhadhesia

  10. Crystallographically equivalent planes

  11. Miller indices of a family of symmetry related planes = (hkl) and all other planes related to (hkl) by the symmetry of the crystal {hkl } All the faces of the cube are equivalent to each other by symmetry Front & back faces: (100) Left and right faces: (010) Top and bottom faces: (001) {100}= (100), (010), (001)

  12. Miller indices of a family of symmetry related planes z Tetragonal z Cubic y y x x {100}tetragonal = (100), (010) {100}cubic = (100), (010), (001) (001)

  13. CUBIC CRYSTALS [111] [hkl] (hkl) C (111) Angle between two directions [h1k1l1] and [h2k2l2]:

  14. Some IMPORTANT Results Weiss zone law Not in the textbook • If a direction [uvw] lies in a plane (hkl) then • uh+vk+wl = 0 [uvw] (hkl) True for ALL crystal systems

  15. dhkl Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell z E B O O (100) x x

  16. Summary of Notation convention for Indices [uvw] Miller indices of a direction (i.e. a set of parallel directions) (hkl) Miller Indices of a plane (i.e. a set of parallel planes) <uvw> Miller indices of a family of symmetry related directions {hkl} Miller indices of a family of symmetry related planes

  17. How do we determine the structure of a piece of crystalline solid? You can probe the atomic arrangements by X-ray diffraction (XRD)

  18. X-Ray Diffraction Diffracted Beam ≡ Bragg Reflection Sample Incident Beam Transmitted Beam Braggs Law (Part 1): For every diffracted beam there exists a set of crystal lattice planes such that the diffracted beam appears to be specularly reflected from this set of planes.

  19. r i   plane X-Ray Diffraction Braggs Law (Part 1): the diffracted beam appears to be specularly reflected from a set of crystal lattice planes. Specular reflection:Angle of incidence =Angle of reflection (both measured from the plane and not from the normal) The incident beam, the reflected beam and the plane normal lie in one plane

  20. r i   dhkl X-Ray Diffraction Bragg’s law (Part 2):

  21. dhkl r i     P R Q Path Difference =PQ+QR

  22. i r   P R Q Path Difference =PQ+QR Constructive inteference Bragg’s law

  23. Extinction Rules: Table 3.3

  24. + + 2 2 2 ( h k l ) q 2 sin µ Bragg’s Law: (1) Diffraction analysis of cubic crystals Cubic crystals (2) (2) in (1) => constant

  25. X Ray Diffractometer

  26. You do not get indices of plane!!

  27. Cu target, Wavelength = 1.5418 Angstrom • Unknown sample, cubic • Determine: • The crystal structure • Lattice parameter

  28. 5 step program for the determination of crystal structure • Start with 2θ values and generate a set of sin2θ values • Normalise the sin2θ values by dividing it with first entry • Clear fractions from normalised column: Multiply by • common number • 4) Speculate on the hkl values that, if expressed as • h2+k2+l2, would generate the sequence of the “clear • fractions” column • 5) Compute for each sin2θ /(h2+k2+l2) on the basis of the • assumed hkl values. If each entry in this column is identical, • then the entire process is validated.

  29. A father-son team that shared a Nobel Prize William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971) Nobel Prize (1915)

  30. Two equivalent ways of stating Bragg’s Law 1st Form 2nd Form

  31. Target Mo Cu Co Fe Cr Wavelength, Å 0.71 1.54 1.79 1.94 2.29 X-raysCharacteristic Radiation, K

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