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“There are two things to aim at in life: first, to get what you want; and, after that, to enjoy it. Only the wisest of mankind achieve the second.” Logan Pearsall Smith, Afterthought (1931), “Life and Human Nature” Chapter 3 Crystal Geometry and Structure Determination Contents Crystal Crystal, Lattice and Motif Symmetry Crystal systems Bravais lattices Miller Indices Structure Determination Crystal ? A 3D translationaly periodic arrangement of atoms in space is called a crystal. Unit cell description : 1 Translational Periodicity One can select a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps) Unit Cell UNIT CELL: Unit cell description : 2 The most common shape of a unit cell is a parallelopiped. The description of a unit cell requires: 1. Its Size and shape (lattice parameters) Unit cell description : 3 2. Its atomic content (fractional coordinates) Size and shape of the unit cell: Unit cell description : 4 1. A corner as origin c 2. Three edge vectors {a, b, c} from the origin define a CRSYTALLOGRAPHIC a COORDINATE SYSTEM b 3. The three lengths a, b, c and the three interaxial angles , , are called the LATTICE PARAMETERS Lattice? A 3D translationally periodic arrangement of points in space is called a lattice. Lattice A 3D translationally periodic arrangement of points Each lattice point in a lattice has identical neighbourhood of other lattice points. Classification of lattice The Seven Crystal System And The Fourteen Bravais Lattices 7 Crystal Systems and 14 Bravais Lattices Crystal System Bravais Lattices 1. Cubic P I 2. Tetragonal P I 3. Orthorhombic P I 4. Hexagonal P 5. Trigonal P 6. Monoclinic P 7. Triclinic P P: Simple; I: body-centred; F: Face-centred; C: End-centred F F C C The three cubic Bravais lattices Crystal system Bravais lattices 1. Cubic P Simple cubic Primitive cubic Cubic P I F Body-centred cubic Face-centred cubic Cubic I Cubic F Orthorhombic C End-centred orthorhombic Base-centred orthorhombic 15/87 Cubic Crystals? a=b=c; ===90 7 crystal Systems Unit Cell Shape Crystal System 1. a=b=c, ===90 Cubic 2. a=bc, ===90 Tetragonal 3. abc, ===90 Orthorhombic 4. a=bc, == 90, =120 Hexagonal 5. a=b=c, ==90 Rhombohedral OR Trigonal 6. abc, ==90 Monoclinic 7. abc, Triclinic Why half the boxes are empty? Crystal System Bravais Lattices 1. Cubic P I 2. Tetragonal P I 3. Orthorhombic P I 4. Hexagonal P 5. Trigonal P 6. Monoclinic P 7. Triclinic P F ? F C E.g. Why cubic C is absent? C End-centred cubic not in the Bravais list ? a 2 a 2 End-centred cubic = Simple Tetragonal 14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices 1. Cubic P I 2. Tetragonal P I 3. Orthorhombic P I 4. Hexagonal P 5. Trigonal P 6. Monoclinic P 7. Triclinic P F C F C C Now apply the same procedure to the FCC lattice Cubic F = Tetragonal I ?!!! 14 Bravais lattices divided into seven crystal systems Crystal system Bravais lattices 1. Cubic P I 2. Tetragonal P I 3. Orthorhombic P I 4. Hexagonal P 5. Trigonal P 6. Monoclinic P 7. Triclinic P F C F C C History: ML Frankenheim 1801-1869 1835: IIT-Delhi Auguste Bravais 1811-1863 15 X lattices 1850: 14 lattices 1856: 14 lattices Couldn’t find his photo on the net 26th July 2013: AML120 Class Sem I 20132014 13 lattices !!! UNIT CELLS OF A LATTICE Nonprimitive cell If the lattice points are only at the corners, the unit cell is primitive otherwise non- primitive Primitive cell A unit cell of a lattice is NOT unique. Primitive cell Unit cell shape CANNOT be the basis for classification of Lattices End of Lec 3 (Lec 1 on crystallography) UNIT CELLS OF A LATTICE A good after-class question from the last class: If we are selecting smallest possible region as a unit cell, why can’t we select a triangular unit cell? Primitive cell Unit cell is a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps) Why can’t the FaceCentred Cubic lattice (Cubic F) be considered as a Body-Centred Tetragonal lattice (Tetragonal I) ? What is the basis for classification of lattices into 7 crystal systems and 14 Bravais lattices? Lattices are classified on the basis of their symmetry Symmetry? If an object is brought into selfcoincidence after some operation it said to possess symmetry with respect to that operation. Translational symmetry Lattices also have translational symmetry In fact this is the defining symmetry of a lattice Rotation Axis If an object come into self-coincidence through smallest non-zero rotation angle of then it is said to have an nfold rotation axis where 0 n 360 =180 n=2 2-fold rotation axis =90 n=4 4-fold rotation axis Rotational Symmetries Z Angles: 180 120 90 72 60 45 3 4 5 6 8 Fold: 2 Graphic symbols Crsytallographic Restriction 5-fold symmetry or Pentagonal symmetry is not possible for Periodic Tilings Symmetries higher than 6-fold also not possible Only possible rotational symmetries for lattices 2 3 4 5 6 7 8 9… Symmetry of lattices Lattices have Translational symmetry Rotational symmetry Reflection symmetry Point Group and Space Group The group of all symmetry elements of a crystal except translations (e.g. rotation, reflection etc.) is called its POINT GROUP. The complete group of all symmetry elements including translations of a crystal is called its SPACE GROUP Classification of Lattices Crystal systems and Bravais Lattices Classification of lattices Based on the point group symmetry alone (i.e. excluding translational symmetry 7 types of lattices 7 crystal systems Based on the space group symmetry, i.e., rotational, reflection and translational symmetry 14 types of lattices 14 Bravais lattices 7 crystal Systems 37/87 Defining symmetry 4 A single 3 1 Crystal system Conventional unit cell Cubic a=b=c, ===90 Tetragonal a=bc, ===90 Orthorhombic abc, ===90 Hexagonal a=bc, == 90, =120 A single Rhombohedral a=b=c, ==90 A single Monoclinic abc, ==90 Triclinic abc, none 38/87 Tetragonal symmetry Cubic C = Tetragonal P Cubic symmetry Cubic F Tetragonal I End of Lec4 on 30.07.2013 Lec2 on crystallography Crystal Lattice A 3D translationally periodic arrangement of atoms A 3D translationally periodic arrangement of points What is the relation between the two? Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated with each lattice point Crystal=lattice+basis Lattice: the underlying periodicity of the crystal, Basis: atom or group of atoms associated with each lattice points Lattice: how to repeat Motif: what to repeat Lattice + Motif = Crystal + Love Lattice + Heart = = Love Pattern Air, Water and Earth by M.C. Esher Every periodic pattern (and hence a crystal) has a unique lattice associated with it The six lattice parameters a, b, c, , , The cell of the lattice lattice + Motif crystal Richard P. Feynman Nobel Prize in Physics, 1965 Feynman’s Lectures on Physics Vol 1 Chap 1 Fig. 1-4 Hexagonal symmetry 360 60 6 “Fig. 1-4 is an invented arrangement for ice, and although it contains many of the correct features of the ice, it is not the true arrangement. One of the correct features is that there is a part of the symmetry that is hexagonal. You can see that if we turn the picture around an axis by 120°, the picture returns to itself.” o Correction: Shift the box One suggested correction: But gives H:O = 1.5 : 1 http://www.youtube.com/watc h?v=kUuDG6VJYgA The errata has been accepted by Michael Gottlieb of Caltech and the corrections will appear in future editions Website www.feynmanlectures.info QUESTIONS? Miller Indices 1 Miller Indices of directions and planes William Hallowes Miller (1801 – 1880) University of Cambridge Miller Indices 2 Miller Indices of Directions z 1. Choose a point on the direction as the origin. 2. Choose a coordinate system with axes parallel to the unit cell edges. y 3. Find the coordinates of another point on the direction in terms of a, b and c x 1a+0b+0c 1, 0, 0 4. Reduce the coordinates to smallest integers. 1, 0, 0 5. Put in square brackets [100] Miller Indices 3 z y x Miller indices of a direction represents only the orientation of the line corresponding to the direction and not its position or sense [100] All parallel directions have the same Miller indices Miller Indices of Directions (contd.) z OA=1/2 a + 1/2 b + 1 c Q A z 1/2, 1/2, 1 [1 1 2] y y O P x PQ = -1 a -1 b + 1 c -1, -1, 1 __ [111] x -ve steps are shown as bar over the number Miller Indices 4 Miller indices of a family of symmetry related directions uvw = [uvw] and all other directions related to [uvw] by the symmetry of the crystal [001] Tetragonal Cubic [010] [010] [100] 100 cubic = [100], [010], [001] [100] 100 = [100], [010] tetragonal End of Lec 05 31/07/2013 Lec 3 on crystallography Lec 6 Lec 4 on crystallography Miller Indices Continues z Miller Indices for planes 1. Select a crystallographic coordinate system with origin not on the plane 2. Find intercepts along axes in terms of respective 1 1 1 lattice parameters O x y 3. Take reciprocal 1 1 1 4. Convert to smallest integers in the same ratio 1 1 1 5. Enclose in parenthesis (111) Miller Indices for planes (contd.) z z E Plane ABCD OCBE origin O O* intercepts 1 ∞ ∞ reciprocals 1 0 0 A B O O* y D x C x Miller Indices (1 0 0) Zero represents that the plane is parallel to the corresponding axis 1 -1 ∞ 1 -1 0 _ (1 1 0) Bar represents a negative intercept Miller indices of a plane specifies only its orientation in space not its position z E A All parallel planes have the same Miller Indices _ _ _ (h k l ) (h k l ) B O y D C x (100) _ (100) (100) Z Y X Miller indices of a family of symmetry related planes {hkl } = (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal 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 Cubic z Tetragonal z y y x {100}cubic = (100), (010), (001) x {100}tetragonal = (100), (010) (001) Some IMPORTANT Results Weiss zone law Not in the textbook Condition for a direction [uvw] to be parallel to a plane or lie in the plane (hkl): hu+kv+lw=0 True for ALL crystal systems CUBIC CRYSTALS [111] [hkl] (hkl) C (111) Angle between two directions [h1k1l1] and [h2k2l2]: h1 h 2 k 1 k 2 l1l 2 cos h1 k 1 l1 2 2 2 h2 k 2 l2 2 2 2 dhkl Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell cubic d hkl a 2 2 d1 1 0 a h k l z E B O O d 100 a x (100) x 2 2 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 In the fell clutch of circumstance I have not winced nor cried aloud. Under the bludgeonings of chance My head is bloody, but unbowed. From "Invictus" by William Ernest Henley (1849–1903). End of Lecture 6 Lec 4 on crystallography Some crystal structures Crystal Lattice Motif Lattice parameter a=3.61 Å Cu FCC Cu 000 Zn Simple Hex Zn 000, a=2.66 Zn 1/3, 2/3, 1/2 c=4.95 Q1: How do we determine the crystal structure? X-Ray Diffraction Sample Incident Beam Transmitted Beam Diffracted Beam X-Ray Diffraction ≡ 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. X-Ray Diffraction Braggs’ recipe for Nobel prize? Call the diffraction a reflection!!! “The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them”. W.L. Bragg 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 i =Angle of reflection (both measured from the plane plane and not from the normal) r The incident beam, the reflected beam and the plane normal lie in one plane X-Ray Diffraction r i dhkl Bragg’s law (Part 2): n 2 d hkl sin r i P R dhkl Q Path Difference =PQ+QR 2 d hkl sin i r P R Q Path Difference =PQ+QR 2 d hkl sin Constructive inteference n 2 d hkl sin Bragg’s law Two equivalent ways of stating Bragg’s Law 1st Form n 2 d hkl sin 2 d hkl sin n d nh , nk , nl a ( nh ) ( nk ) ( nl ) 2 2 d nh nk nl sin 2 2 d hkl n 2nd Form Two equivalent ways of stating Bragg’s Law n 2 d hkl sin nth order reflection from (hkl) plane 2 d nh nk nl sin 1st order reflection from (nh nk nl) plane e.g. a 2nd order reflection from (111) plane can be described as 1st order reflection from (222) plane X-rays Characteristic Radiation, K Target Mo Cu Co Fe Cr Wavelength, Å 0.71 1.54 1.79 1.94 2.29 Powder Method is fixed (K radiation) is variable – specimen consists of millions of powder particles – each being a crystallite and these are randomly oriented in space – amounting to the rotation of a crystal about all possible axes Powder diffractometer geometry Strong intensity Diffracted beam 2 i plane t r Transmitted sample beam Intensity Incident beam 21 Zero Diffracted intensity beam 1 X-ray detector 21 22 2 Crystal monochromator detector X-ray tube X-ray powder diffractometer The diffraction pattern of austenite Austenite = fcc Fe Bcc crystal x /2 d100 = a z 100 reflection= rays reflected from adjacent (100) planes spaced at d100 have a path difference y No 100 reflection for bcc No bcc reflection for h+k+l=odd Extinction Rules: Table 3.3 Bravais Lattice Allowed Reflections SC All BCC (h + k + l) even FCC h, k and l unmixed h, k and l are all odd DC Or if all are even then (h + k + l) divisible by 4 End of Lec 7 06/08/13 Lec 5 on crystallography (Last lecture) Diffraction analysis of cubic crystals Bragg’s Law: (1) 2 d hkl sin Cubic crystals d hkl (2) a h k l 2 2 2 (2) in (1) => sin 2 2 4a (h k l ) 2 2 2 2 constant 2 2 2 sin (h k l ) 2 h2 + k2 + l2 SC 1 100 2 110 3 111 111 4 200 200 5 210 6 211 FCC BCC DC 110 111 200 211 7 8 220 220 9 300, 221 10 310 11 311 311 12 222 222 13 320 14 321 220 220 310 311 222 321 15 16 400 17 410, 322 18 411, 330 19 331 400 400 400 411, 330 331 331 Crystal Structure Allowed ratios of Sin2 (theta) SC 1: 2: 3: 4: 5: 6: 8: 9… BCC 1: 2: 3: 4: 5: 6: 7: 8… FCC 3: 4: 8: 11: 12… DC 3: 8: 11:16… Ananlysis of a cubic diffraction pattern 19.0 22.5 33.0 39.0 41.5 49.5 56.5 59.0 69.5 84.0 sin2 0.11 0.15 0.30 0.40 0.45 0.58 0.70 0.73 0.88 0.99 p sin2 h2+k2+l2 p sin2 h2+k2+l2 p=18.87 p=9.43 1.0 1.4 2.8 3.8 4.1 5.4 6.6 6.9 8.3 9.3 1 2 3 4 5 6 8 9 10 11 sc 2 2.8 5.6 7.4 8.3 10.9 13.1 13.6 16.6 18.7 2 4 6 8 10 12 14 16 18 20 bcc p sin2 h2+k2+l2 p=27.3 2.8 4.0 8.1 10.8 12.0 15.8 19.0 20.1 23.9 27.0 3 4 8 11 12 16 19 20 24 27 fcc This is an fcc crystal Ananlysis of a cubic diffraction pattern contd. 19.0 22.5 33.0 39.0 41.5 49.5 56.5 59.0 69.5 84.0 h2+k2+l2 3 4 8 11 12 16 19 20 24 27 hkl a 111 200 220 311 222 400 331 420 422 511 4.05 4.02 4.02 4.04 4.02 4.04 4.03 4.04 4.01 4.03 Indexing of diffraction patterns 2 4 a (h k l ) 2 sin2 2 2 2 The diffraction pattern is from an fcc crystal of lattice parameter 4.03 Å Education is an admirable thing, but it is well to remember from time to time that nothing that is worth knowing can be taught. -Oscar Wilde A father-son team that shared a Nobel Prize William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971) Nobel Prize (1915) One of the greatest scientific discoveries of twentieth century Max von Laue, 1879-1960 Nobel 1914 Two Questions Q1: X-rays waves or particles? Father Bragg: Particles Son Bragg: Waves “Even after they shared a Nobel Prize in 1915, … this tension persisted…” – Ioan James in Remarkable Physicists Q2:Crystals: Perodic arrangement of atoms? X-RAY DIFFRACTION: X-rays are waves and crystals are periodic arrangement of atoms If it is permissible to evaluate a human discovery according to the fruits which it bears then there are not many discoveries ranking on par with that made by von Laue. -from Nobel Presentation Talk Watson, Crick and Wilkins Nobel Prize, 1962 Rosalind Elsie Franklin Rosy, of course, did not directly give us her data. For that matter, no one at King's realized they were in our hands. J.D. Watson "However, the data which really helped us to obtain the structure was mainly obtained by Rosalind Franklin". -Francis Crick Google Doodle on 25th July, 2013 Franklins birth centenary 25 July 2020 Rosalind Elsie Franklin 25 July 1920 – 16 April 1958