Chapters 2 and 6 in Waseda. Lesson 8 Lattice Planes and Directions. Suggested Reading


 Marilynn Hill
 2 years ago
 Views:
Transcription
1 Analytical Methods for Materials Chapters 2 and 6 in Waseda Lesson 8 Lattice Planes and Directions Suggested Reading 192
2 Directions and Miller Indices Draw vector and define the tail as the origin. z Determine the length of the vector projection in unit cell dimensions a, b, and c. Remove fractions by multiplying by the smallest possible factor. [201] O [111] P a c y Enclose in square brackets Negative indices are written with a bar over the number.. b [110] x a b c [ 3 6 2] Point P Origin 193
3 Families of Directions (i.e., directions of a form) In cubic systems, directions that have the same indices are equivalent regardless of their order or sign. z y [010] [001] [100] x [010] [100] [001] We enclose indices in carats rather than brackets to indicate a family of directions The family of <100> directions is: [10 0], [100] [010], [0 10] [001], [001] All of these vectors have the same size and # lattice points/length 194
4 <100> CUBIC <aaa> [100] [010] [001] [100] [010] [001] TETRAGONAL <aac> [100] [010] [100] [010] In noncubic systems, directions with [100] [100] the same indices may not be equivalent. ORTHORHOMBIC <abc> 195
5 Directions in Crystals Directions and their multiples are identical [110] Ex.: z y [030] [020] [220] [220] 2 [110] x [010] Vectors and multiples of vectors have the same # lattice points/length 196
6 Miller Indices for Planes Specific crystallographic plane: (hkl) Family of crystallographic planes: {hkl} Ex.: (hkl), (lkh), (hlk) etc. In cubic systems, planes having the same indices are equivalent regardless of order or sign. In hexagonal crystals, we use a four index system (hkil) k i l). We can convert from three to four indices h+k = i 197
7 FAMILY OF PLANES ALL MEMBERS HAVE SAME ARRANGEMENT OF LATTICE POINTS {hkl} k We use Miller indices to denote planes 198
8 PROCEDURES FOR INDICES OF PLANES (Miller indices) 1. Identify the coordinate intercepts of the plane (i.e., the coordinates at which the plane intersects the x, y, and z axes). If plane is parallel to an axis, the intercept is taken as infinity (). If the plane passes through the origin, consider an equivalent plane in an adjacent unit cell or select a different origin for the same plane. 2. Take reciprocals of the intercepts. 3. Clear fractions to the lowest integers. 4. Cite specific planes in parentheses,(hkl), placing bars over negative indices. 199
9 MILLER INDICES FOR A SINGLE PLANE z x y z Intercept 1 1 Reciprocal 1/1 1/1 1/ Clear INDICES y (110) x Slide 200 The {110} family of planes (110), (011), (101), (110), (011), (101) (110), (1 10), (101), (10 1), (01 1), (0 11) 200
10 MILLER INDICES FOR A SINGLE PLANE cont d x y z z Intercept Reciprocal 1/1 1/1 1/1 Clear INDICES y x y z Intercept x Reciprocal 1/11/11/1 Clear ( 1 INDICES ( 111 ) 11 ) Slide
11 MILLER INDICES FOR A SINGLE PLANE cont d z x y z Intercept Reciprocal Clear INDICES 1/2 1/2 2/1 2/1 1/ (220) x y Planes and their multiples are not identical ( 220) (110) 202
12 Planes in Unit Cells Some important aspects of Miller indices for planes: 1. Planes and their negatives are identical. This was NOT the case for directions. 2. Planes and their multiples are NOT identical. This is opposite to the case for directions. 3. In cubic systems,, a direction that has the same indices as a plane is to that plane. This is not always true for noncubic systems. 203
13 204
14 Planes of a Zone A zone is a direction [uvw] z (1 10) ZONE AXIS [uvw] = [001] (220) Planes belonging to a particular zone are parallel to one direction known as the zone axis. hkl uvw 0 (010) lies on lies on 220 y 205
15 How to Determine the Zone Axis Take the cross product of the intersecting planes. z (1 10) ZONE AXIS [uvw] = [001] (220) (010) (h 1 k 1 l 1 ) (h 2 k 2 l 2 ) = [uvw] y u v w u v u 1 0 v 0 2 w 1 2 v 1 0 u 0 2 w 1 2 [0 04] [ 01] 0 206
16 Indexing in Hexagonal Systems The regular 3 index system is not suitable. c a Planes with the same a 1 indices do not necessarily look like. a 3 a 2 [1 10] [001] (0001) c 4 index system introduced. MillerBravais indices (1100) a 1 (10 11) [100] (1210) a 2 [010] 207
17 Indexing in Hexagonal Systems Planes: (hkl) becomes (hkil) i = (h+k) Directions: [UVW] becomes [uvtw] U = ut ; u = (2U V)/3 V = vt ; v = (2V U)/3 W=w ; t=(u+v) 208
18 PLANES Miller Indices MillerBravais Indices a 3 (hkl) a 3 (hkil) (100) (1010) (010) (110) (0 110) (1100) a 2 a 2 (110) (010) (1 100) (0110) a 1 (100) a 1 (10 10) DIRECTIONS (UVW) (uvtw) a 3 a 3 [120] [110] [1100] [0 110] a 2 a 2 [110] [100] [1120] [2110] a 1 [210] a 1 [1010] 209
19 c a 3 a a 2 a Some typical directions in an HCP unit cell using three and fouraxis systems. 210
20 Interplanar Spacings z y (100) (110) x Assuming no intercept on zaxis (210) d 210 The interplanar spacing in a particular direction is the distance between equivalent planes of atoms. Each material has a set of characteristic interplanar spacings. They are directly related to crystal size (i.e. lattice parameters) and atom location. a 211
21 Interplanar Spacing cont d 1 h k l CUBIC: 2 2 d a HEXAGONAL: TETRAGONAL: RHOMBOHEDRAL: ORTHORHOMBIC: h hk k l d 3 a c h k l d a c 1 d h hk k sin 2hk kl hlcos cos a 1 3cos 2cos h k l d a b c h k sin l 2hlcos a b c ac MONOCLINIC: 2 d sin TRICLINIC*: S h S22k S3l 2S12hk 2S23kl2S13hl d V S b c sin ; S a c sin ; S a b sin cos cos cos ; cos cos cos ; cos cos cos S abc S a bc S ab c V abc 1 cos cos cos 2 cos cos cos 212
Crystals are solids in which the atoms are regularly arranged with respect to one another.
Crystalline structures. Basic concepts Crystals are solids in which the atoms are regularly arranged with respect to one another. This regularity of arrangement can be described in terms of symmetry elements.
More informationCrystal Symmetries METE 327 Physical Metallurgy Copyright 2008 Loren A. Jacobson 5/16/08
Crystal Symmetries Why should we be interested? Important physical properties depend on crystal structure Conductivity Magnetic properties Stiffness Strength These properties also often depend on crystal
More informationBasic Concepts of Crystallography
Basic Concepts of Crystallography Language of Crystallography: Real Space Combination of local (point) symmetry elements, which include angular rotation, centersymmetric inversion, and reflection in mirror
More informationRelevant Reading for this Lecture... Pages 8387.
LECTURE #06 Chapter 3: Xray Diffraction and Crystal Structure Determination Learning Objectives To describe crystals in terms of the stacking of planes. How to use a dot product to solve for the angles
More informationCHAPTER 3 THE STRUCTURE OF CRYSTALLINE SOLIDS PROBLEM SOLUTIONS
CHAPTER THE STRUCTURE OF CRYSTALLINE SOLIDS PROBLEM SOLUTIONS Fundamental Concepts.6 Show that the atomic packing factor for HCP is 0.74. The APF is just the total sphere volumeunit cell volume ratio.
More informationChapter 2 Crystal Lattices and Reciprocal Lattices
Chapter 2 Crystal Lattices and Reciprocal Lattices Abstract In this chapter, the basic unit vectors in real space and the basic unit vectors in reciprocal space, as well as their reciprocal relationships,
More informationCrystallographic Directions, and Planes
Crstallographic Directions, and Planes Now that we know how atoms arrange themselves to form crstals, we need a wa to identif directions and planes of atoms. Wh? Deformation under loading (slip) occurs
More informationSolid State Theory Physics 545
Solid State Theory Physics 545 CRYSTAL STRUCTURES Describing periodic structures Terminology Basic Structures Symmetry Operations Ionic crystals often have a definite habit which gives rise to particular
More information12.524 2003 Lec 17: Dislocation Geometry and Fabric Production 1. Crystal Geometry
12.524 2003 Lec 17: Dislocation Geometry and Fabric Production 1. Bibliography: Crystal Geometry Assigned Reading: [Poirier, 1985]Chapter 2, 4. General References: [Kelly and Groves, 1970] Chapter 1. [Hirth
More informationElementary Crystallography for XRay Diffraction
Introduction Crystallography originated as the science of the study of crystal forms. With the advent of the xray diffraction, the science has become primarily concerned with the study of atomic arrangements
More informationMAE 20 Winter 2011 Assignment 2 solutions
MAE 0 Winter 0 Assignment solutions. List the point coordinates of the titanium, barium, and oxygen ions for a unit cell of the perovskite crystal structure (Figure.6). In Figure.6, the barium ions are
More informationA crystalline solid is one which has a crystal structure in which atoms or ions are arranged in a pattern that repeats itself in three dimensions.
CHAPTER ATOMIC STRUCTURE AND BONDING. Define a crstalline solid. A crstalline solid is one which has a crstal structure in which atoms or ions are arranged in a pattern that repeats itself in three dimensions..2
More informationXRay Diffraction HOW IT WORKS WHAT IT CAN AND WHAT IT CANNOT TELL US. Hanno zur Loye
XRay Diffraction HOW IT WORKS WHAT IT CAN AND WHAT IT CANNOT TELL US Hanno zur Loye Xrays are electromagnetic radiation of wavelength about 1 Å (1010 m), which is about the same size as an atom. The
More informationChapter Outline. How do atoms arrange themselves to form solids?
Chapter Outline How do atoms arrange themselves to form solids? Fundamental concepts and language Unit cells Crystal structures Simple cubic Facecentered cubic Bodycentered cubic Hexagonal closepacked
More informationCrystal Structure Determination I
Crystal Structure Determination I Dr. Falak Sher Pakistan Institute of Engineering and Applied Sciences National Workshop on Crystal Structure Determination using Powder XRD, organized by the Khwarzimic
More informationESS 212: Laboratory 2 and 3. For each of your paper models, assign crystallographic axes that will serve to orient the model.
ESS 212: Laboratory 2 and 3 Lab Topics: Point groups of symmetry Crystal systems and Crystallographic axes Miller Indices of crystal faces; crystal forms Exercises to be handed in: In these two labs you
More informationCrystallographic Points, Directions, and Planes. ISSUES TO ADDRESS... a v. Points, Directions, and Planes in Terms of Unit Cell Vectors
Crstallographic Points, Directions, and Planes. Points, Directions, and Planes in Terms of Unit Cell Vectors ISSUES TO ADDRESS... How to define points, directions, planes, as well as linear, planar, and
More informationLMB Crystallography Course, 2013. Crystals, Symmetry and Space Groups Andrew Leslie
LMB Crystallography Course, 2013 Crystals, Symmetry and Space Groups Andrew Leslie Many of the slides were kindly provided by Erhard Hohenester (Imperial College), several other illustrations are from
More informationCrystal Form, Zones, Crystal Habit. Crystal Forms
Page 1 of 15 EENS 2110 Tulane University Mineralogy Prof. Stephen A. Nelson Crystal Form, Zones, Crystal Habit This page last updated on 10Jan2011 Crystal Forms As stated at the end of the last lecture,
More informationCrystal symmetry X nd setting X Xm m mm2 4mm 3m 6mm 2 or. 2m m2 3m m even X2 + centre Xm +centre.
III Crystal symmetry 33 Point group and space group A. Point group 1. Symbols of the 32 three dimensional point groups General Triclinic Monoclinic Tetragonal Trigonal Hexagonal Cubic symbol 1 st setting
More informationChapter 3: The Structure of Crystalline Solids
Sapphire: cryst. Al 2 O 3 Insulin : The Structure of Crystalline Solids Crystal: a solid composed of atoms, ions, or molecules arranged in a pattern that is repeated in three dimensions A material in which
More informationrotation,, axis of rotoinversion,, center of symmetry, and mirror planes can be
Crystal Symmetry The external shape of a crystal reflects the presence or absence of translationfree symmetry y elements in its unit cell. While not always immediately obvious, in most well formed crystal
More informationXray diffraction: theory and applications to materials science and engineering. Luca Lutterotti
Xray diffraction: theory and applications to materials science and engineering Luca Lutterotti luca.lutterotti@unitn.it Program Part 1, theory and methodologies: General principles of crystallography
More informationRepetitive arrangement of features (faces, corners and edges) of a crystal around
Geology 284  Mineralogy, Fall 2008 Dr. Helen Lang, West Virginia University Symmetry External Shape of Crystals reflects Internal Structure External Shape is best described by Symmetry Symmetry Repetitive
More informationLECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 6483.
More informationChapter 3. 1. 3 types of materials amorphous, crystalline, and polycrystalline. 5. Same as #3 for the ceramic and diamond crystal structures.
Chapter Highlights: Notes: 1. types of materials amorphous, crystalline, and polycrystalline.. Understand the meaning of crystallinity, which refers to a regular lattice based on a repeating unit cell..
More informationReading. Chapter 12 in DeGraef and McHenry Chapter 3i in Engler and Randle
Class 17 Xray Diffraction Chapter 1 in DeGraef and McHenry Chapter 3i in Engler and Randle Reading Chapter 6 in DeGraef and McHenry (I WILL ASSUME THAT YOU KNOW ABOUT THIS CONCEPT ALREADY! IF NOT, READ
More informationPart 5 Space groups. 5.1 Glide planes. 5.2 Screw axes. 5.3 The 230 space groups. 5.4 Properties of space groups. 5.5 Space group and crystal structure
Part 5 Space groups 5.1 Glide planes 5.2 Screw axes 5.3 The 230 space groups 5.4 Properties of space groups 5.5 Space group and crystal structure Glide planes and screw axes 32 point groups are symmetry
More informationPAPER CRYSTAL STURCTURE Put your name and period on structure near the name of the crystal. Cut along SOLID lines. Fold inward along DASH lines so
PAPER CRYSTAL STURCTURE Put your name and period on structure near the name of the crystal. Cut along SOLID lines. Fold inward along DASH lines so that your name and crystal s name is visible. Put your
More informationPrecession photograph
Precession photograph Spacing of spots is used to get unit cell dimensions. Note symmetrical pattern. Crystal symmetry leads to diffraction pattern symmetry. Groups Total For proteins Laue group 230 space
More informationCHAPTER 3: CRYSTAL STRUCTURES
CHAPTER 3: CRYSTAL STRUCTURES Crystal Structure: Basic Definitions  lecture Calculation of material density selfprep. Crystal Systems lecture + selfprep. Introduction to Crystallography lecture + selfprep.
More informationPart 432 point groups
Part 432 point groups 4.1 Subgroups 4.2 32 point groups 4.2 Crystal forms The 32 point groups The point groups are made up from point symmetry operations and their combinations. A point group is defined
More informationLecture 2. Surface Structure
Lecture 2 Surface Structure Quantitative Description of Surface Structure clean metal surfaces adsorbated covered and reconstructed surfaces electronic and geometrical structure References: 1) Zangwill,
More informationEarth and Planetary Materials
Earth and Planetary Materials Spring 2013 Lecture 11 2013.02.13 Midterm exam 2/25 (Monday) Office hours: 2/18 (M) 1011am 2/20 (W) 1011am 2/21 (Th) 11am1pm No office hour 2/25 1 Point symmetry Symmetry
More informationReciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following
Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following form in two and three dimensions: k (r + R) = e 2 ik
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationSolid State Device Fundamentals
Solid State Device Fundamentals ENS 345 Lecture Course by Alexander M. Zaitsev alexander.zaitsev@csi.cuny.edu Tel: 718 982 2812 Office 4N101b 1 Solids Three types of solids, classified according to atomic
More informationThe University of Western Ontario Department of Physics and Astronomy P2800 Fall 2008
P2800 Fall 2008 Questions (Total  20 points): 1. Of the noble gases Ne, Ar, Kr and Xe, which should be the most chemically reactive and why? (0.5 point) Xenon should be most reactive since its outermost
More informationExplain the ionic bonds, covalent bonds and metallic bonds and give one example for each type of bonds.
Problem 1 Explain the ionic bonds, covalent bonds and metallic bonds and give one example for each type of bonds. Ionic Bonds Two neutral atoms close to each can undergo an ionization process in order
More information9/18/2013. Symmetry Operations and Space Groups. Crystal Symmetry. Symmetry Elements. Center of Symmetry: 1. Rotation Axis: n. Center of Symmetry: 1
Symmetry Operations and rystal Symmetry 32 point groups of crystals compatible with 7 crystal systems crystallographers use ermannmauguin symmetry symbols arl ermann German 89896 harlesvictor Mauguin
More informationCrystalline Structures Crystal Lattice Structures
Jewelry Home Page Crystalline Structures Crystal Lattice Structures Crystal Habit Refractive Index Crystal Forms Mohs Scale Mineral Classification Crystal Healing Extensive information on healing crystals,
More information1. By how much does 1 3 of 5 2 exceed 1 2 of 1 3? 2. What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3.
1 By how much does 1 3 of 5 exceed 1 of 1 3? What fraction of the area of a circle of radius 5 lies between radius 3 and radius 4? 3 A ticket fee was $10, but then it was reduced The number of customers
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More informationSpecimen paper MATHEMATICS HIGHER TIER. Time allowed: 2 hours. GCSE BITESIZE examinations. General Certificate of Secondary Education
GCSE BITESIZE examinations General Certificate of Secondary Education Specimen paper MATHEMATICS HIGHER TIER 2005 Paper 1 Noncalculator Time allowed: 2 hours You must not use a calculator. Answer all
More informationMaths for Computer Graphics
Analytic Geometry Review of geometry Euclid laid the foundations of geometry that have been taught in schools for centuries. In the last century, mathematicians such as Bernhard Riemann (1809 1900) and
More informationCrystal Structure. A(r) = A(r + T), (1)
Crystal Structure In general, by solid we mean an equilibrium state with broken translation symmetry. That is a state for which there exist observables say, densities of particles with spatially dependent
More informationLecture Outline Crystallography
Lecture Outline Crystallography Short and long range Order Poly and single crystals, anisotropy, polymorphy Allotropic and Polymorphic Transitions Lattice, Unit Cells, Basis, Packing, Density, and Crystal
More informationSTRAND B: Number Theory. UNIT B2 Number Classification and Bases: Text * * * * * Contents. Section. B2.1 Number Classification. B2.
STRAND B: Number Theory B2 Number Classification and Bases Text Contents * * * * * Section B2. Number Classification B2.2 Binary Numbers B2.3 Adding and Subtracting Binary Numbers B2.4 Multiplying Binary
More informationIt is possible to map the surfaces of conducting solids at the atomic level using an
3 C H A P T E R Crstal Structures and Crstal Geometr It is possible to map the surfaces of conducting solids at the atomic level using an instrument called the scanning tunneling microscope (STM). The
More informationIntroduction to Powder XRay Diffraction History Basic Principles
Introduction to Powder XRay Diffraction History Basic Principles Folie.1 History: Wilhelm Conrad Röntgen Wilhelm Conrad Röntgen discovered 1895 the Xrays. 1901 he was honoured by the Noble prize for
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationSTRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2
STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of xaxis in anticlockwise direction, then the value of tan θ is called the slope
More informationwww.sakshieducation.com
LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c
More informationBCM 6200  Protein crystallography  I. Crystal symmetry Xray diffraction Protein crystallization Xray sources SAXS
BCM 6200  Protein crystallography  I Crystal symmetry Xray diffraction Protein crystallization Xray sources SAXS Elastic Xray Scattering From classical electrodynamics, the electric field of the electromagnetic
More informationREVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example.
REVIEW OVER VECTORS I. Scalars & Vectors: A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example mass = 5 kg A vector is a quantity that can be described
More informationInternational Tables for Crystallography (2006). Vol. A, Section 10.1.2, pp. 763 795.
International Tables for Crystallography (2006). Vol. A, Section 10.1.2, pp. 763 795. 10.1. CRYSTALLOGRAPHIC AND NONCRYSTALLOGRAPHIC POINT GROUPS Table 10.1.1.2. The 32 threedimensional crystallographic
More informationSolution: 2. Sketch the graph of 2 given the vectors and shown below.
7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit
More informationGeorge R. McCormick Department of Geology The University of Iowa Iowa City, Iowa
CRYSTAL MEASUREMENT AND AXIAL RATIO LABORATORY George R. McCormick Department of Geology The University of Iowa Iowa City Iowa 52242 george_mccormick@uiowa.edu Goals of the Exercise This exercise is designed
More informationREVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012
REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationEarth Sciences: Course Presenters
Earth Sciences: Course Presenters 625102 Geology Dr Stephen Gallagher Prof Andrew Gleadow Dr Malcolm Wallace Prof Ian Plimer Course Coordinator Room 214 Series  Lecture Topics 1. Mineralogy and Crystals
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationPOWDER XRAY DIFFRACTION: STRUCTURAL DETERMINATION OF ALKALI HALIDE SALTS
EXPERIMENT 4 POWDER XRAY DIFFRACTION: STRUCTURAL DETERMINATION OF ALKALI HALIDE SALTS I. Introduction The determination of the chemical structure of molecules is indispensable to chemists in their effort
More informationThe Structure of solids.
Chapter S. The Structure of solids. After having studied this chapter, the student will be able to: 1. Distinguish between a crystal structure and an amorphous structure. 2. Describe the concept of a unit
More informationRemember that the information below is always provided on the formula sheet at the start of your exam paper
Maths GCSE Linear HIGHER Things to Remember Remember that the information below is always provided on the formula sheet at the start of your exam paper In addition to these formulae, you also need to learn
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationMathematics 1. Lecture 5. Pattarawit Polpinit
Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance
More informationTwinning and absolute structure. Bill Clegg Newcastle University, UK. SHELX Workshop Denver 2016
Twinning and absolute structure Bill Clegg Newcastle University, UK SHELX Workshop Denver 2016 Outline Twinning Definition, characterisation, illustration Classification: different types of twin Recognition:
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More informationGeometry A Solutions. Written by Ante Qu
Geometry A Solutions Written by Ante Qu 1. [3] Three circles, with radii of 1, 1, and, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to
More informationAll points on the coordinate plane are described with reference to the origin. What is the origin, and what are its coordinates?
Classwork Example 1: Extending the Axes Beyond Zero The point below represents zero on the number line. Draw a number line to the right starting at zero. Then, follow directions as provided by the teacher.
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is socalled because when the scalar product of
More informationOne advantage of this algebraic approach is that we can write down
. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the xaxis points out
More informationIf a question asks you to find all or list all and you think there are none, write None.
If a question asks you to find all or list all and you think there are none, write None 1 Simplify 1/( 1 3 1 4 ) 2 The price of an item increases by 10% and then by another 10% What is the overall price
More informationMidterm Exam I, Calculus III, Sample A
Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of
More informationLAB #1 THE CRYSTALLOGRAPHY AND MILLER INDICES
LAB #1 THE CRYSTALLOGRAPHY AND MILLER INDICES THE OBJECTIVES: 1. To study simple crystal lattices using a simulation packages. THE THEORY: Solid state semiconductor technology has brought valuable systems
More informationAlgebra 1 Chapter 3 Vocabulary. equivalent  Equations with the same solutions as the original equation are called.
Chapter 3 Vocabulary equivalent  Equations with the same solutions as the original equation are called. formula  An algebraic equation that relates two or more reallife quantities. unit rate  A rate
More informationSelfDirected Course: Transitional Math Module 2: Fractions
Lesson #1: Comparing Fractions Comparing fractions means finding out which fraction is larger or smaller than the other. To compare fractions, use the following inequality and equal signs:  greater than
More informationSymmetryoperations, point groups, space groups and crystal structure
1 Symmetryoperations, point groups, space groups and crystal structure KJ/MV 210 Helmer Fjellvåg, Department of Chemistry, University of Oslo 1994 This compendium replaces chapter 5.3 and 6 in West. Sections
More informationBryn Mawr College Department of Physics Undergraduate Teaching Laboratories Xray Crystallography
Bryn Mawr College Department of Physics Undergraduate Teaching Laboratories Xray Crystallography Introduction In this lab you will explore the Xray spectrum produced by copper in an Xray tube and investigate
More informationStructure Factors 59553 78
78 Structure Factors Until now, we have only typically considered reflections arising from planes in a hypothetical lattice containing one atom in the asymmetric unit. In practice we will generally deal
More informationAQA Level 2 Certificate FURTHER MATHEMATICS
AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the
More informationExperiment: Crystal Structure Analysis in Engineering Materials
Experiment: Crystal Structure Analysis in Engineering Materials Objective The purpose of this experiment is to introduce students to the use of Xray diffraction techniques for investigating various types
More informationChapter 2: Crystal Structures and Symmetry
Chapter 2: Crystal Structures and Symmetry Laue, ravais December 28, 2001 Contents 1 Lattice Types and Symmetry 3 1.1 TwoDimensional Lattices................. 3 1.2 ThreeDimensional Lattices................
More informationGeometry Unit 1. Basics of Geometry
Geometry Unit 1 Basics of Geometry Using inductive reasoning  Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture an unproven statement that is based
More informationChapter 12. The Straight Line
302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,
More informationTriangle Definition of sin and cos
Triangle Definition of sin and cos Then Consider the triangle ABC below. Let A be called. A HYP (hpotenuse) ADJ (side adjacent to the angle ) B C OPP (side opposite to the angle ) sin OPP HYP BC AB ADJ
More informationLECTURE SUMMARY September 30th 2009
LECTURE SUMMARY September 30 th 2009 Key Lecture Topics Crystal Structures in Relation to Slip Systems Resolved Shear Stress Using a Stereographic Projection to Determine the Active Slip System Slip Planes
More informationEach grain is a single crystal with a specific orientation. Imperfections
Crystal Structure / Imperfections Almost all materials crystallize when they solidify; i.e., the atoms are arranged in an ordered, repeating, 3dimensional pattern. These structures are called crystals
More information7.3 Volumes Calculus
7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are
More informationUnit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook
Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest.
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationENGR1100 Introduction to Engineering Analysis. Lecture 3
ENGR1100 Introduction to Engineering Analysis Lecture 3 POSITION VECTORS & FORCE VECTORS Today s Objectives: Students will be able to : a) Represent a position vector in Cartesian coordinate form, from
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationUse order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
More informationHigher. Functions and Graphs. Functions and Graphs 18
hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms
More information