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

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Chapter 3 Crystal Geometry and Structure Determination

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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

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

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

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

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Courtesy: H Bhadhesia

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Crystallographically equivalent planes

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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)

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

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CUBIC CRYSTALS [hkl] (hkl) Angle between two directions [h 1 k 1 l 1 ] and [h 2 k 2 l 2 ]: C [111] (111)

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

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

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[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) Miller indices of a family of symmetry related directions {hkl}Miller indices of a family of symmetry related planes Summary of Notation convention for Indices

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

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Incident Beam X-Ray Diffraction Transmitted Beam Diffracted Beam Sample 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. ≡ Bragg Reflection

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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 X-Ray Diffraction i plane r

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X-Ray Diffraction i r d hkl Bragg’s law (Part 2):

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i r Path Difference =PQ+QR P Q R d hkl

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

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Extinction Rules: Table 3.3 Bravais LatticeAllowed Reflections SCAll BCC(h + k + l) even FCCh, k and l unmixed DC h, k and l are all odd Or if all are even then (h + k + l) divisible by 4

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Diffraction analysis of cubic crystals 2 sin 222 )lkh( constant Bragg’s Law: Cubic crystals (1) (2) (2) in (1) =>

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X Ray Diffractometer

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You do not get indices of plane!!

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Cu target, Wavelength = 1.5418 Angstrom 2θ2θ 44.48 51.83 76.35 92.90 98.40 121.87 144.54 Unknown sample, cubic Determine: 1)The crystal structure 2)Lattice parameter

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5 step program for the determination of crystal structure 1)Start with 2θ values and generate a set of sin 2 θ values 2)Normalise the sin 2 θ values by dividing it with first entry 3)Clear fractions from normalised column: Multiply by common number 4) Speculate on the hkl values that, if expressed as h 2 +k 2 +l 2, would generate the sequence of the “clear fractions” column 5) Compute for each sin 2 θ /(h 2 +k 2 +l 2 ) on the basis of the assumed hkl values. If each entry in this column is identical, then the entire process is validated.

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2θ2θSin 2 θSin 2 θ/Sin 2 θ 1 Clear fractions (hkl)?sin 2 θ /(h 2 +k 2 +l 2 ) 44.480.1431.0031110.0477 51.830.1911.3442000.0478 76.350.3822.6782200.0478 92.900.5253.67113110.0477 98.400.5734.01122220.0478 121.870.7645.34164000.0477 144.540.9076.34194200.0477

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

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h 2 + k 2 + l 2 SCFCCBCCDC 1100 2110 3111 4200 5210 6211 7 8220 9300, 221 10310 11311 12222 13320 14321 15 16400 17410, 322 18411, 330 19331

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

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X-rays Characteristic Radiation, K Target Mo Cu Co Fe Cr Wavelength, Å 0.71 1.54 1.79 1.94 2.29

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