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

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

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**Announcements**• Revised Lab timings: 1-3 PM (all groups) • 2) Quiz 1, 28th Jan 2014, • Tuesday 7:30 PM, WS 209, WS 213**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**Miller Indices of directions and planes**William Hallowes Miller(1801 – 1880) University of Cambridge**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)**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**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)**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)**CUBIC CRYSTALS**[111] [hkl] (hkl) C (111) Angle between two directions [h1k1l1] and [h2k2l2]:**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**dhkl**Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell z E B O O (100) x x**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**How do we determine the structure of a piece of crystalline**solid? You can probe the atomic arrangements by X-ray diffraction (XRD)**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.**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**r**i dhkl X-Ray Diffraction Bragg’s law (Part 2):**dhkl**r i P R Q Path Difference =PQ+QR**i**r P R Q Path Difference =PQ+QR Constructive inteference Bragg’s law**+**+ 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**Cu target, Wavelength = 1.5418 Angstrom**• Unknown sample, cubic • Determine: • The crystal structure • Lattice parameter**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.**A father-son team that shared a Nobel Prize**William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971) Nobel Prize (1915)**Two equivalent ways of stating Bragg’s Law**1st Form 2nd Form**Target**Mo Cu Co Fe Cr Wavelength, Å 0.71 1.54 1.79 1.94 2.29 X-raysCharacteristic Radiation, K