# rotation,, axis of rotoinversion,, center of symmetry, and mirror planes can be

Save this PDF as:

Size: px
Start display at page:

Download "rotation,, axis of rotoinversion,, center of symmetry, and mirror planes can be"

## Transcription

1 Crystal Symmetry The external shape of a crystal reflects the presence or absence of translation-free symmetry y elements in its unit cell. While not always immediately obvious, in most well formed crystal shapes, axis of rotation,, axis of rotoinversion,, center of symmetry, and mirror planes can be spotted.

2 All discussed d operations may be combined, but the number of (i.e. unique) combinations is limited, to 32. Each of these is known as a point group, or crystal class. The crystal classes may be sub-divided into one of 6 crystal systems. Space groups are a combination of the 3D lattice Space groups are a combination of the 3D lattice types and the point groups (total of 65).

3 Each of the 32 crystal classes is unique to one of the 6 crystal systems: Triclinic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric (cubic) Interestingly, while all mirror planes and poles of rotation must intersect at one point, this point may not be a center of symmetry (i).

4 Crystallographic Axes The identification of specific symmetry operations enables one to orientate a crystal according to an imaginary set of reference lines known as the crystallographic axes. These are distinct and different from the classic Cartesian Axes, x, y and z, used in other common day usage, such as plotting graphs.

5 With the exception of the hexagonal system, the axes are designated a, a b, b and c. c The ends of each axes are designated + or -. This is important for the derivation of Miller Indices. The angles between the positive ends of the axes are designated α, β, and γ. α lies between b and c. β lies between a and c. γ lies between a and b.

6 Quantities can also be applied to further describe vectors and planes relative to a, b, and c These are u, v, w: u: projection along a v: projection along b w: projection along c

7 Quantities can also be applied to further describe vectors and planes relative to a, b, and c These are h, k, l: h: information relative to a axis v: information relative to b axis w: information relative to c axis [uvw] with (hkl) (hkl) faces on a cube

8 Axial Ratios With the exception of the cubic (isometric) system, there are crystallographic axes differing dff in length. Imagine one single unit cell and measuring the lengths of the a, b, and c axes. To obtain the axial ratios we normalise to the b axis. These ratios are relative.

9 Unique crystallographic axes of the 6 crystal systems Triclinic: Three unequal axes with oblique angles. Monoclinic: Three unequal axes, two are inclined to one another, the third is perpendicular. Orthorhombic: Three mutually perpendicular axes of different lengths. Tetragonal: Three mutually perpendicular axes, two are equal, the third (vertical) is shorter. Hexagonal: Three equal horizontal axes (a 1, a 2, a 3 ) and a 4 th perpendicular (vertical) of different length. Cubic: Three perpendicular axes of equal length.

10 Triclinic: Three unequal axes with oblique angles. To orientate a triclinic crystal the most pronounced nced zone should be vertical. c a and b are determined by the intersections of (010) and (100) with (001). b The b axis should be longer than the a axis. a

11 The unique symmetry operation in a triclinic system is a 1-fold axis of rotoinversion(equivalent to a center of symmetry or inversion, i). All forms are pinacoids therefore must consist of two identical and parallel faces. Common triclinic rock-forming minerals include microcline, some plagioclases, and wollastonite.

12 Monoclinic: Three unequal axes, two are inclined with oblique angles, the third is perpendicular. Orientation ti of a crystal has few constraints b is the only axis fixed by symmetry. c is typically chosen on the basis of habit and cleavage. α and γ = 90. There are some very rare cases where b equals 90 giving a pseudo- orthorhombic form. a c b

13 The unique symmetry operation in a monoclinic system is 2/m a twofold axis of rotation with a mirror plane. b is the rotation, while a and c lie in the mirror plane. Monoclinic crystals have two forms: pinacoids and prisms. Common monoclinic rock-forming minerals include clinopyroxene, mica, orthoclase and titanite.

14 Orthorhombic: Three mutually perpendicular axes of different lengths. Convention has it that a crystal is oriented such that c > b > a. c Crystals are oriented so that c is parallel to crystal elongation. In this case the length of the b axis is taken as unity and ratios are calculated thereafter. b a

15 The unique symmetry operation in an orthorhombic system is 2/m 2/m 2/m Three twofold axis of rotation coinciding with the three crystallographic axes. Perpendicular to each of the axes is a mirror plane. The general class for the orthorhombic system are rhombic dipyramid {hkl}. There are three types of form in the class: pinacoids, prisms, and dipyramids. Common orthorhombic rock-forming minerals include andalusite and sillimanite, orthopyroxene, olivine and topaz.

16 Tetragonal: Three mutually perpendicular axes, two are equal, the third (vertical) is shorter. The two horizontal axis in a tetragronal mineral are oriented in the plane of the horizontal. Therefore, if a = b, c must be in the vertical. a 2 c There is no rule as to whether c is greater or less than a. a a 1

17 The unique symmetry operation in a tetragonal system is 4/m 2/m 2/m The vertical axis (c) is always a fourfold axis of rotation. There are 4 two-fold axis of rotation: 2 parallel to the crystallographic axes a and b, b the others at 45. There are 5 mirror planes. The general class for the orthorhombic system is known as the ditetragonal-dipyramidal class. There are four types of form in the class: basal pinacoids, tetragonal prisms, tetragonal dipyramids, and ditetragonal prisms. Common tetragonal rock-forming minerals include zircon, rutile and anatase, and apophyllite.

18 Hexagonal: Three equal horizontal axes (a 1, a 2, a 3 ) and a 4 th perpendicular vertical axis of different length. The three horizontal axis of a hexagonal mineral are oriented in the plane of the horizontal, with c in the vertical. c Unlike the other systems the Bravais-Miller nomenclature for crystal faces is given by 4 numbers (i.e. {0001}) The first three numbers are listed in order of a 1, a 2, a 3. a 3 a 2 a 1 90 = = 90 = 120

19 The unique symmetry operation in the hexagonal system is a sixfold axis of rotation, and the most common space group is 6/m 2/m 2/m. There vertical axis is the six-fold rotational operation, while there are a further 6 two-fold axis of rotation ti in the horizontal plane (3 coincide with the a n axes). There are 7 mirror planes. The general class for the orthorhombic system is known as the dihexagonal-dipyramidal class. There are five types of form in the class: pinacoids, hexagonal prisms, hexagonal dipyramids, dihexagonal prisms, and dihexagonal dipyramids. Common hexagonal minerals include beryl and apatite.

20 Isometric (cubic): Three equal length axes that intersecting at right-angles to one another. The axes are indistinguishable, as are the intersecting angles. As such all are interchangable. There are 15 isometric forms, but a 3 the most common are: a 2 Cube Octahedron Dodecahedron Tetrahexahedron Trapezohedron Trisoctahedron Hexoctahedron = = = 90 a 1

### Crystal 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 10-Jan-2011 Crystal Forms As stated at the end of the last lecture,

### ESS 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

### CHAPTER 4: THE 32 CRYSTAL CLASSES AND THE MILLER INDICES. Sarah Lambart

CHAPTER 4: THE 32 CRYSTAL CLASSES AND THE MILLER INDICES Sarah Lambart RECAP CHAP. 3 7 Crystal systems & 14 Bravais lattice RECAP CHAP. 3 Elements of symmetry Rotations: 1 fold, 2 fold, 3 fold, 4 fold,

### Repetitive 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

### CHAPTER 3: CRYSTAL STRUCTURES

CHAPTER 3: CRYSTAL STRUCTURES Crystal Structure: Basic Definitions - lecture Calculation of material density self-prep. Crystal Systems lecture + self-prep. Introduction to Crystallography lecture + self-prep.

### 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.

### Crystalline 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,

### Part 4-32 point groups

Part 4-32 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

### Earth and Planetary Materials

Earth and Planetary Materials Spring 2013 Lecture 11 2013.02.13 Midterm exam 2/25 (Monday) Office hours: 2/18 (M) 10-11am 2/20 (W) 10-11am 2/21 (Th) 11am-1pm No office hour 2/25 1 Point symmetry Symmetry

### Solid 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

### ACTIVITY 2: CRYSTAL FORM AND HABIT: MEASURING CRYSTAL FACES

WARD'S GEO-logic : Crystal Form Activity Set Name:. Group: Date: ACTIVITY 2: CRYSTAL FORM AND HABIT: MEASURING CRYSTAL FACES OBJECTIVE: To verify the relationship between interfacial angles and crystal

### Introduction and Symmetry Operations

Page 1 of 9 EENS 2110 Tulane University Mineralogy Prof. Stephen A. Nelson Introduction and Symmetry Operations This page last updated on 27-Aug-2013 Mineralogy Definition of a Mineral A mineral is a naturally

### 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

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

### 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

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

### 12.524 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

### Crystal 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

### International 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 three-dimensional crystallographic

### LMB 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

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

Analytical Methods for Materials Chapters 2 and 6 in Waseda Lesson 8 Lattice Planes and Directions Suggested Reading 192 Directions and Miller Indices Draw vector and define the tail as the origin. z Determine

### Earth Sciences: Course Presenters

Earth Sciences: Course Presenters 625-102 Geology Dr Stephen Gallagher Prof Andrew Gleadow Dr Malcolm Wallace Prof Ian Plimer Course Coordinator Room 214 Series - Lecture Topics 1. Mineralogy and Crystals

### Basic Concepts of Crystallography

Basic Concepts of Crystallography Language of Crystallography: Real Space Combination of local (point) symmetry elements, which include angular rotation, center-symmetric inversion, and reflection in mirror

### Elementary Crystallography for X-Ray Diffraction

Introduction Crystallography originated as the science of the study of crystal forms. With the advent of the x-ray diffraction, the science has become primarily concerned with the study of atomic arrangements

### Crystal Optics of Visible Light

Crystal Optics of Visible Light This can be a very helpful aspect of minerals in understanding the petrographic history of a rock. The manner by which light is transferred through a mineral is a means

### Crystal 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 3-3 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

### George 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

### Basics of Mineralogy. Geology 200 Geology for Environmental Scientists

Basics of Mineralogy Geology 200 Geology for Environmental Scientists Terms to Know: Atom Molecule Proton Neutron Electron Isotope Ion Bonding ionic covalent metallic Fig. 3.3 Periodic Table of the Elements

### X-Ray Diffraction HOW IT WORKS WHAT IT CAN AND WHAT IT CANNOT TELL US. Hanno zur Loye

X-Ray Diffraction HOW IT WORKS WHAT IT CAN AND WHAT IT CANNOT TELL US Hanno zur Loye X-rays are electromagnetic radiation of wavelength about 1 Å (10-10 m), which is about the same size as an atom. The

### X-ray diffraction: theory and applications to materials science and engineering. Luca Lutterotti

X-ray 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

### Geometry of Minerals

Geometry of Minerals Objectives Students will connect geometry and science Students will study 2 and 3 dimensional shapes Students will recognize numerical relationships and write algebraic expressions

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Chapter 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,

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Lecture 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,

### Chapter 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

### Precession 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

### Crystal 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

### ENGINEERING DRAWING. UNIT I - Part A

ENGINEERING DRAWING UNIT I - Part A 1. Solid Geometry is the study of graphic representation of solids of --------- dimensions on plane surfaces of ------- dimensions. 2. In the orthographic projection,

### Crystal 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

### 15. PRISMS AND CYLINDERS

15. PRISMS AND CYLINDERS 15-1 Drawing prisms 2 15-2 Modelling prisms 4 15-3 Cross-sections of prisms 6 15-4 Nets of prisms 7 15-5 Euler's formula 8 15-6 Stacking prisms 9 15-7 Cylinders 10 15-8 Why is

### Elements and Operations. A symmetry element is an imaginary geometrical construct about which a symmetry operation is performed.

Elements and Operations A symmetry element is an imaginary geometrical construct about which a symmetry operation is performed. A symmetry operation is a movement of an object about a symmetry element

### 2 Identifying Minerals

CHAPTER 5 2 Identifying Minerals SECTION Minerals of Earth s Crust KEY IDEAS As you read this section, keep these questions in mind: What are seven physical properties scientists use to identify minerals?

### acute angle acute triangle Cartesian coordinate system concave polygon congruent figures

acute angle acute triangle Cartesian coordinate system concave polygon congruent figures convex polygon coordinate grid coordinates dilatation equilateral triangle horizontal axis intersecting lines isosceles

### Symmetry and Molecular Spectroscopy

PG510 Symmetry and Molecular Spectroscopy Lecture no. 2 Group Theory: Molecular Symmetry Giuseppe Pileio 1 Learning Outcomes By the end of this lecture you will be able to:!! Understand the concepts of

### Chapter 1 Symmetry of Molecules p. 1 - Symmetry operation: Operation that transforms a molecule to an equivalent position

Chapter 1 Symmetry of Molecules p. 1-1. Symmetry of Molecules 1.1 Symmetry Elements Symmetry operation: Operation that transforms a molecule to an equivalent position and orientation, i.e. after the operation

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Engineering Drawing. Anup Ghosh. September 12, Department of Aerospace Engineering Indian Institute of Technology Kharagpur

Department of Aerospace Engineering Indian Institute of Technology Kharagpur September 12, 2011 Example -1 1 A vertical square prism (50mm side) 2 A horizontal square prism (35mm side) with axis to VP

### Chapter 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 Two-Dimensional Lattices................. 3 1.2 Three-Dimensional Lattices................

### Symmetry-operations, point groups, space groups and crystal structure

1 Symmetry-operations, 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

### Twinning 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:

### Chapter Draw sketches of C, axes and a planes? (a) NH3? (b) The PtC1 2-4 ion? 4.2 S4 or i : (a) C02? (b) C2H2? (c) BF3? (d) SO 2-4?

Chapter 4 4.1 Draw sketches of C, axes and a planes? (a) NH 3? In the drawings below, the circle represents the nitrogen atom of ammonia and the diamonds represent the hydrogen atoms. The mirror plane

### Minerals. L3: Minerals and Rocks. Common Minerals to Learn. Crystallography

L3: Minerals and Rocks Identification Classification Formation } of common minerals & rocks Feedback on Lab 1: Rock Identification Minerals Definition: Highly-ordered crystalline atomic structure Definite

### Mineral Properties. Hand sample description

Mineral Properties Hand sample description How do we characterize minerals? 1) Color and streak 2) Luster 3) Hardness 4) Fracture and tenacity 5) Crystal form and system 6) Crystal shape and habit 7) Cleavage

### SHAPE, SPACE AND MEASURES

SHPE, SPCE ND MESURES Pupils should be taught to: Understand and use the language and notation associated with reflections, translations and rotations s outcomes, Year 7 pupils should, for example: Use,

### Geometry Vocabulary Booklet

Geometry Vocabulary Booklet Geometry Vocabulary Word Everyday Expression Example Acute An angle less than 90 degrees. Adjacent Lying next to each other. Array Numbers, letter or shapes arranged in a rectangular

### Silicate Structures. The building blocks of the common rock-forming minerals

Silicate Structures The building blocks of the common rock-forming minerals Mineral classes and the silicates There are a total of 78 mineral classes 27 of these are the silicates which constitute ~92%

### Mineral List Observed in lab: Name Formula Crystal System Hand specimen Thin section

Igneous Minerals (Labs 3 & 9) Framework Silicates (Tectosilicates) The SiO 4 tetrahedrons are linked together in three-dimensional forming a framework silicate where the Si:O ratio is 1:2. SiO 2 Group

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### Lecture 34: Symmetry Elements

Lecture 34: Symmetry Elements The material in this lecture covers the following in Atkins. 15 Molecular Symmetry The symmetry elements of objects 15.1 Operations and symmetry elements 15.2 Symmetry classification

### Reading. Chapter 12 in DeGraef and McHenry Chapter 3i in Engler and Randle

Class 17 X-ray 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

### Coordination and Pauling's Rules

Page 1 of 8 EENS 2110 Tulane University Mineralogy Prof. Stephen A. Nelson Coordination and Pauling's Rules This document last updated on 24-Sep-2013 The arrangement of atoms in a crystal structure not

### All 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.

### The ash heap of crystallography: restoring forgotten basic knowledge

ISSN: 1600-5767 journals.iucr.org/j The ash heap of crystallography: restoring forgotten basic knowledge Massimo Nespolo J. Appl. Cryst. (2015). 48, 1290 1298 IUCr Journals CRYSTALLOGRAPHY JOURNALS ONLINE

### A 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

### Stereographic projections

Stereographic projections 1. Introduction The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. The projection is defined as shown in Fig. 1. If

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Space group symmetry

Space group symmetry Screw axes Sometimes also referred to as rototranslation axes Screw axes, notation nr, involve a rotation by 360º/n about an unit cell axis followed by translation parallel to that

### 9/18/2013. 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 ermann-mauguin symmetry symbols arl ermann German 898-96 harles-victor Mauguin

### Symmetry. The primitive lattice vectors are:

Syetry Previously, we noted all crystal structures could be specified by a set of Bravais lattice vectors, when describing a lattice you ust either use the priitive vectors or add a set of basis vectors

### Year 8 - Maths Autumn Term

Year 8 - Maths Autumn Term Whole Numbers and Decimals Order, add and subtract negative numbers. Recognise and use multiples and factors. Use divisibility tests. Recognise prime numbers. Find square numbers

### Symmetry and Molecular Structures

Symmetry and Molecular Structures Some Readings Chemical Application of Group Theory F. A. Cotton Symmetry through the Eyes of a Chemist I. Hargittai and M. Hargittai The Most Beautiful Molecule - an Adventure

### Begin recognition in EYFS Age related expectation at Y1 (secure use of language)

For more information - http://www.mathsisfun.com/geometry Begin recognition in EYFS Age related expectation at Y1 (secure use of language) shape, flat, curved, straight, round, hollow, solid, vertexvertices

### 2.2 Symmetry of single-walled carbon nanotubes

2.2 Symmetry of single-walled carbon nanotubes 13 Figure 2.5: Scanning tunneling microscopy images of an isolated semiconducting (a) and metallic (b) singlewalled carbon nanotube on a gold substrate. The

### PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT

UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,

### Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (2005), Chap. 3, and Atkins and Friedman, Chap. 5.

Chapter 5. Geometrical ymmetry Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (005), Chap., and Atkins and Friedman, Chap. 5. 5.1 ymmetry Operations We have already

### Experiment 5: Magnetic Fields of a Bar Magnet and of the Earth

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2005 Experiment 5: Magnetic Fields of a Bar Magnet and of the Earth OBJECTIVES 1. To examine the magnetic field associated with a

### Mathematics 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

### Introduction to Powder X-Ray Diffraction History Basic Principles

Introduction to Powder X-Ray Diffraction History Basic Principles Folie.1 History: Wilhelm Conrad Röntgen Wilhelm Conrad Röntgen discovered 1895 the X-rays. 1901 he was honoured by the Noble prize for

### Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

### Notes pertinent to lecture on Feb. 10 and 12

Notes pertinent to lecture on eb. 10 and 12 MOLEULAR SYMMETRY Know intuitively what "symmetry" means - how to make it quantitative? Will stick to isolated, finite molecules (not crystals). SYMMETRY OPERATION

Geometry Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes

### GENERATED CRYSTALS USING SHAPE Kenneth J. Brock

COMPUTER GENERATED CRYSTALS USING SHAPE Kenneth J. Brock Indiana University Northwest Gary, IN 46408-1197 kebrock@ucs.indiana.edu The software package known as SHAPE (Shape Software 521 Hidden Valley Road,

### F B = ilbsin(f), L x B because we take current i to be a positive quantity. The force FB. L and. B as shown in the Figure below.

PHYSICS 176 UNIVERSITY PHYSICS LAB II Experiment 9 Magnetic Force on a Current Carrying Wire Equipment: Supplies: Unit. Electronic balance, Power supply, Ammeter, Lab stand Current Loop PC Boards, Magnet

### Relevant Reading for this Lecture... Pages 83-87.

LECTURE #06 Chapter 3: X-ray 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

### Nonlinear Optics. University of Osnabrück Summer Term 2003, rev. 2005

Nonlinear Optics Manfred Wöhlecke Klaus Betzler Mirco Imlau University of Osnabrück Summer Term 2003, rev. 2005 a Physics would be dull and life most unfulfilling if all physical phenomena around us were

### CHAPTER 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 volume-unit cell volume ratio.

### Lab 3: Stereonets. Fall 2005

Lab 3: Stereonets Fall 2005 1 Introduction In structural geology it is important to determine the orientations of planes and lines and their intersections. Working out these relationships as we have in

### Lab 6: Fabrics and folds

Lab 6: Fabrics and folds Introduction Any rock that has properties that vary with direction is said to have fabric. Like most structures, fabrics can be primary or secondary. Examples of primary fabric

### 17 PLANE SYMMETRY GROUPS

17 PLANE SYMMETRY GROUPS ANNA NELSON, HOLLI NEWMAN, MOLLY SHIPLEY 1. Introduction Patterns are everywhere. They are deeply embedded all around us. We observe patterns in nature, art, architecture, mathematics,

### Wallpaper Patterns. The isometries of E 2 are all of one of the following forms: 1

Wallpaper Patterns Symmetry. We define the symmetry of objects and patterns in terms of isometries. Since this project is about wallpaper patterns, we will only consider isometries in E 2, and not in higher

### 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

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

### Rectangular Prisms Dimensions

Rectangular Prisms Dimensions 5 Rectangular prisms are (3-D) three-dimensional figures, which means they have three dimensions: a length, a width, and a height. The length of this rectangular prism is

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### Digital Image Processing. Prof. P.K. Biswas. Department of Electronics & Electrical Communication Engineering

Digital Image Processing Prof. P.K. Biswas Department of Electronics & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture - 27 Colour Image Processing II Hello, welcome

### State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. esolutions Manual - Powered by Cognero Page 1 1. A figure

### Conic Sections in Cartesian and Polar Coordinates

Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane

### BCM 6200 - Protein crystallography - I. Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS

BCM 6200 - Protein crystallography - I Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS Elastic X-ray Scattering From classical electrodynamics, the electric field of the electromagnetic

### Crystal Structures and Symmetry

PHYS 624: Crystal Structures and Symmetry 1 Crystal Structures and Symmetry Introduction to Solid State Physics http://www.physics.udel.edu/ bnikolic/teaching/phys624/phys624.html PHYS 624: Crystal Structures