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361: Crystallography and Diffraction 361: Crystallography and Diffraction Michael Bedzyk Department of Materials Science and Engineering Northwestern University February 2, 2021 Contents 1 Catalog Description3 2 Course Outcomes3 3 361: Crystallography and Diffraction3 4 Symmetry of Crystals4 4.1 Types of Symmetry.......................... 4 4.2 Projections of Symmetry Elements and Point Groups...... 9 4.3 Translational Symmetry ....................... 17 5 Crystal Lattices 18 5.1 Indexing within a crystal lattice................... 18 5.2 Lattices................................. 22 6 Stereographic Projections 31 6.1 Projection plane............................ 33 6.2 Point of projection........................... 33 6.3 Representing Atomic Planes with Vectors............. 36 6.4 Concept of Reciprocal Lattice .................... 38 6.5 Reciprocal Lattice Vector....................... 41 7 Representative Crystal Structures 48 7.1 Crystal Structure Examples ..................... 48 7.2 The hexagonal close packed structure, unlike the examples in Figures 7.1 and 7.2, has two atoms per lattice point. These two atoms are considered to be the motif, or repeating object within the HCP lattice. ............................ 49 1 7.3 Voids in FCC.............................. 54 7.4 Atom Sizes and Coordination.................... 54 8 Introduction to Diffraction 57 8.1 X-ray .................................. 57 8.2 Interference .............................. 57 8.3 X-ray Diffraction History ...................... 59 8.4 How does X-ray diffraction work? ................. 59 8.5 Absent (or forbidden) Reflections.................. 63 9 Absorption/Emission 64 9.1 Spectrometers............................. 64 9.2 Bremsstrahlung Radiation...................... 65 9.3 Synchrotron Radiation........................ 66 9.4 Characteristic Spectrum ....................... 69 9.5 Auger Effect.............................. 71 9.6 Radioactive Source .......................... 73 9.7 X-ray Absorption........................... 74 9.8 X-ray Transmission Filters...................... 76 9.9 Photoelectric effect .......................... 79 9.10 X ray scattered by an electron.................... 80 9.11 Theory for Radiation Generating by Accelerated Charge . 81 10 Scattering 82 10.1 X-ray Scattering & Polarization................... 82 10.2 Thomson Scattering by two electrons................ 85 10.3 Compton Scattering.......................... 93 11 Kinematic Theory 98 12 Diffraction 107 12.1 Laue condition for diffraction.................... 107 12.2 Single Crystal Diffractometer .................... 109 12.3 Diffraction Methods ......................... 110 12.4 Structure Factor Examples...................... 113 12.5 Width of Diffraction Peaks (single crystal) . 119 12.6 Transverse scan ¹l − B20=º ≡ rock crystal through peak with 2\ fixed .................................. 121 12.7 Polycrystalline Aggregate (Powder sample) . 123 12.8 Rotating Crystal Method....................... 134 13 Temperature Effects 141 13.1 Describe a crystal in terms of primitive unit crystal. 142 13.2 Debye Theory of Lattice Dynamics/heat capacity — Corrected by Waller................................ 146 13.3 Symmetrical Reflection Powder Diffraction . 148 2 14 Thin Film X-ray Scattering 149 14.1 1-D interference function....................... 150 14.2 HRXRD of single crystal Ferro-Electric Ultra-Thin Film . 150 14.3 Friedel’s Law ............................. 152 14.4 Experimental Reflectivity along the (00L) Crystal Truncation Rod (CTR) for ferro-electric PZT thin-film capacitor heterostructure . 153 15 Order-Disorder 166 15.1 Order-disorder geometry....................... 166 15.2 Long-range order (W.L. Bragg and E.J. Williams) . 167 16 Special Topics in X-ray Physics: Extended X-ray Absorption Fine Structure (EXAFS) 170 16.1 X-ray Absorption of Fine Structure (XAFS): an overview . 170 16.2 Outgoing photoelectron: electrons as spherical waves . 172 16.3 XAFS spectra of mono and diatomic gases . 172 17 361 Problems 194 18 361 Laboratories 202 18.1 Laboratory 1: Symmetry in Two Dimensions . 202 18.2 Laboratory 2: Symmetry in Three Dimensions . 208 18.3 Laboratory 3: Laue Diffraction Patterns . 211 18.4 Laboratory 4: X-Ray Diffractometer Part I . 213 18.5 Laboratory 5: X-Ray Diffractometer Part 2 . 216 18.6 Laboratory 6: Quantitative Analysis of a Mixture . 220 18.7 Laboratory 7: Rotating Crystal Method (et al.) . 223 18.8 Laboratory 8: Single Crystal Epitaxial Thin Film . 226 1 Catalog Description Elementary crystallography. Basic diffraction theory; reciprocal space. Ap- plications to structure analysis, preferred orientation. Point and 2D Detector techniques. Lectures, laboratory. Prerequisites: GEN ENG 205 4; PHYSICS 135 2,3. Mathematics including Calculus 1-3 and Linear Algebra will be required. 2 Course Outcomes 3 361: Crystallography and Diffraction At the conclusion of the course students will be able to: 3 1. Identify different types of crystal structures and symmetry elements such as space groups, point symmetry and glide planes that occur in metals, ceramics, and polymers. 2. Perform standard x-ray diffraction measurements on metals, ceramics and polymers and quantitatively determine lattice constants, grain size, texture, and strain in bulk crystals and epitaxial films. 3. Describe the basic particle and wave physical processes underlying x-ray emission, elastic and inelastic scattering, absorption, and interference of coherent waves. 4. Identify symmetry elements in (point, translation) in 2D and 3D patterns and crystals. 5. Describe basic principles underlying synchrotron x-ray sources, x-ray fluorescence spectroscopy, and electron and neutron diffraction. 6. Use reciprocal space graphical constructions and vector algebra to inter- pret diffraction from 3D and 2D single crystals, and random and textured polycrystalline samples. 4 Symmetry of Crystals A crystal can be defined as a solid which is composed of an atomic array that is periodic in three dimensions. Crystal systems may exhibit various kinds of symmetry, or types of repetitiveness. The specific type of symmetry is de- termined by a number of operations, which when performed on the crystal, bring it into self-coincidence. In other words, a crystal will be symmetric with regard to a certain operation, if it is impossible to distinguish the crystal before or after said operation. 4.1 Types of Symmetry Symmetry elements can be of the following types: • Rotational axis • Mirror plane • Inversion center • Translation symmetry (or a combination of these): 4 • Rotoinversion axis • Glide plane • Screw axis 4.1.1 Mirror Planes and Chirality Probably the most familar form of symmetry from daily life is reflection sym- metry. We refer to some object as being symmetric with respect to reflection if putting a mirror crossing through the object would result in no apparrent change to the object (we call this lack of change self coincidence). The lines (or planes in 3D) on which these hypothetical mirrors could be placed are ap- propriately called mirror planes. Observe Figure 4.1, noticing that the image could be replicated by holding half of it up to an appropriately placed mirror. In a sense, symmetry thus allows us to define structures or patterns in terms of unique, fundamental units. Mirror planes will typically be denoted in figures by bolded lines. It is also worth noting that the two shapes in Figure 4.1 have an opposite handedness, or chirality. The points of the two objects proceed outward in either a clockwise or counterclockwise manner; as a result, the ob- ject on the right can be identified as distinct from the one on the left. Thus, we see that a reflection changes the chirality of an object. We encounter this phenomenon in daily life; it is possible to tell a person’s right hand from their left because of the arrangement of their fingers, but an image of a left hand reflected from a mirror will have the same arrangement as that of a right hand (similar conclusions follow for the reflection of the right hand). Figure 4.1: A figure containing a mirror plane. Note that the objects on either side of the plane have opposite chirality 4.1.2 Proper Rotation Axes A rotational axis refers to some line which can be utilized as the center of some specific rotation that will leave the system (such as a pattern or crystal) seemingly unchanged (invariant). Rotational axes can occur in both three dimensions, or two dimensions, with the “line” being represented as a point in the latter case, indicating lines perpendicular to the plane of view. The 5 plane object pictured in Figure 4.2 demonstrates rotational symmetry, as a 120◦ turn about its center brings it into self coincidence. In addition, the object has three mirror planes, and the object’s reflection across these planes also brings it into self coincidence. Figure 4.2: A figure containing 3-fold rotational symmetry, as well as 3 mirror planes Figure 4.3 shows the types of proper rotation axes that we’re concerned with, as well as the symbols which will be used to represent each type. Referring back to Figure 4.2, the object pictured has 3-fold rotational symmetry about an axis directed out of the page, as a 360◦/3 or 120◦rotation allows for self coincidence. In other words, an object has =-fold symmetry about an axis if a 360◦/= rotation about that axis brings it into self coincidence. All objects exhibit 1-fold symmetry, or are brought into self coincidence by a full 360◦ turn. This case is trivial (hence why there is no associated symbol) but important nonetheless. Proper Improper Figure 4.3: Symbols for
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