9/18/2013. Symmetry Operations and Space Groups. Crystal Symmetry. Symmetry Elements. Center of Symmetry: 1. Rotation Axis: n. Center of Symmetry: 1

Size: px
Start display at page:

Download "9/18/2013. Symmetry Operations and Space Groups. Crystal Symmetry. Symmetry Elements. Center of Symmetry: 1. Rotation Axis: n. Center of Symmetry: 1"

Transcription

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 harles-victor Mauguin French there are 5 types in point symmetry. center of symmetry (or inversion) : point 2. rotation (or proper) axis : line 3. mirror : plane 5. identity Symmetry Elements 4. rotation inversion axis : line n : no element n m enter of Symmetry: a point in the molecule through which if another point on the molecule is taken, will meet an identical point on the molecule an equal distance away enter of Symmetry: Rotation Axis: n all points () ( x, y, z) if is placed at the origin y z + () n is an integer which gives the degrees of rotation: 2 or n 360 o n is the number of times molecule is rotated, each time stopping at an identical appearance, before returning to the starting point n x ( x, y, z) n is the foldness of the rotation axis only 2, 3, 4, and 6-fold axes allowed in crystal symmetry

2 Rotation Axis: 4 Mirror: m 360 o = 90 4 o plane within the molecule that, when acting as a mirror, reflects the molecule into itself o 4 Rotation-Inversion n Representation of Symmetry rotation followed by inversion this is a different definition than Schoenflies system Arthur Moritz Schönflies German 89 rotation followed by reflection point symmetry often represented symbolically in the form of points on a circle (projection of a sphere) a point above plane is a filled circle: a point below plane is an open circle: two points directly on top of each other: starting with one point, find other points generated by symmetry 32 point groups compatible with 7 crystal systems Triclinic Monoclinic 2 ( ) ( i ) highest symmetry in each crystal system is called: Laue Group 2 ( 2 ) m ( s ) 2/m ( 2h ) monoclinic convention: symmetry located wrt b axis 2: 2-fold axis along b m: mirror perpendicular to b (Schoenflies symbol) have center of symmetry 2/m: 2-fold axis along b, perpendicular to a mirror 2

3 Monoclinic Orthorhombic 2 2 m 2/m mm2 ( 2v ) 222 (D 2 ) mmm (D 2h ) xyz 3 symbols refer to: axes along a, b, or c mirrors perpendicular to a, b, or c Rhombohedral (Trigonal) Tetragonal 3 ( 3 ) 2 3 (S 6 ) 3m ( 3v ) 4 ( 4 ) 4 (S 4 ) 4/m ( 4h ) 32 (D 3 ) 3m (D 3d ) 4mm ( 4v ) 42m (D 2d ) 422 (D 4 ) 4/mmm (D 4h ) m 3m Tetragonal exagonal 4 4 4/m 6 ( 6 ) 6 ( 3h ) 6/m ( 6h ) 4mm 42m 422 6mm ( 6v ) 6m2 (D 3h ) 622 (D 6 ) 4/mmm 6/mmm (D 6h ) 3

4 exagonal ubic 6 6 6/m 23 (T) m3 (T h ) 43m (T d ) 6mm 6m (O) m3m (O h ) 6/mmm m3 m3m 43m Lattices 4 Bravais Lattices triclinic monoclinic 4 Bravais lattices have Laue symmetry P P2/m 2/m orthorhombic Pmmm mmm Immm Fmmm rhombohedral tetragonal R3m P4/mmm I4/mmm hexagonal cubic P6/mmm Pm3m Im3m Fm3m Lattices Translational Symmetry in repeating lattices, two additional symmetry elements 4 Bravais lattices have Laue symmetry all have a center of symmetry center of symmetry very important in crystallography: centrosymmetric or noncentrosymmetric translational elements. screw axis rotation and translation: n r rotation by 360 o /n; followed by translation of r/n along that axis (a, b or c) 2-fold screw axis most common: 2 2. glide plane reflection and translation: a, b, c, n or d reflection across plane; followed by translation of /2 (usually) unit cell parallel to plane along a, b, c, face diagonal (n), or body diagonal (d) 4

5 Screw Axis - 2 a c b,, +, Glide Plane - a ½ a c b, ½ a-glide c ab plane translational elements + point symmetry space groups in 2-D, referred to as plane groups there are 7 distinct ways of packing repeating object in 2-D p p2 wallpaper patterns pm pg p2mm p2mg cm p2gg c2mm 5

6 p4 p4mm p3 p3m p4gm p3m p pg p2gg pm cm c2mm pmg pmm p2 p4 p6 p6mm p4mm p4gm p3 p3m p3m p6 p6mm translational elements + 32 crystal point groups; 230 space groups 230 distinct ways of packing repeating object in 3-D triclinic P P entrosymmetric space groups monoclinic 2 P2 P2 2 m Pm Pc m c 2/m P2/m P2 /m 2/m P2/c P2 /c 2/c orthorhombic 222 P222 P222 P2 2 2 P F222 I222 I2 2 2 mm2 Pmm2 Pmc2 Pcc2 Pma2 Pca2 Pnc2 Pmn2 Pba2 Pna2 Pnn2 cc2 Amm2 Abm2 Ama2 Aba2 Fmm2 mm2 mc2 Fdd2 Imm2 Iba2 Ima2 mmm Pmmm Pnnn Pccm Pban Pmma Pnna Pmna Pcca Pbam Pccn Pbcm Pnnm Pmmn Pbcn Pbca Pnma mcm mca mmm ccm mma cca Fmmm Fddd Immm Ibam Ibca Imma 6

7 tetragonal 4 P4 P4 P4 2 P4 3 I4 I4 4 P4 I4 4/m P4/m P4 2 /m P4/n P4 2 /n I4/m I4 /a 422 P422 P42 2 P4 22 P4 2 2 P P P P I422 I4 22 4mm P4mm P4bm P4 2 cm P4 2 nm P4cc P4nc P4 2 mc P4 2 bc I4mm I4cm I4 md I4 cd 42m P42m P42c P42 m P42 c P4m2 P4c2 P42b P4n2 I4m2 I4c2 I42m I42d 4/mmm P4/mmm P4/mcc P4/nbm P4/nnc P4/mbm P4/mnc P4/nmm P4/nnc P4 2 /mmc P4 2 /mcm P4 2 /nbc P4 2 /nnm P4 2 /mbc P4 2 /mnm P4 2 /nmc P4 2 /ncm I4/mmm I4/mcm I4 /amd I4 /acd trigonal/rhombohedral 3 P3 P3 P3 2 R3 3 P3 R3 32 P32 P32 P3 2 P3 2 P3 2 2 P3 2 2 R32 3m P3m P3m P3c P3c R3m R3c 3m P3m P3c P3m P3c R3m R3c hexagonal 6 P6 P6 P6 5 P6 2 P6 4 P6 3 6 P6 6/m P6/m P6 3 /m 622 P622 P6 22 P P P P mm P6mm P6cc P6 3 cm P6 3 mc 6m2 P6m2 P6c2 P62m P62c 6m2 P6m2 P6c2 P62m P62c 6/mmm P6/mmm P6/mcc P6 3 /mcm P6 3 /mmc cubic 23 P23 F23 I23 P2 3 I2 3 m3 Pm3 Pn3 Fm3 Fd3 Im3 Pa3 432 P432 P F432 F4 32 I432 P P4 32 I m P43m F43m I43m P43n F43c I43d m3m Pm3m Pn3n Pm3n Pn3m Fm3m Fm3c Fd3m Fd3c Im3m Ia3d Symmetry 7 crystal systems: point symmetry of external lattice 4 Bravais lattices: translational symmetry of lattice points 32 point groups: point symmetry of external crystal 230 space groups: translational symmetry inside crystal molecules ambridge Structural Database (SD) all compounds crystallize in one or more of these space groups usually possible to find P, but always try to find the highest possible symmetry. structures observed in all 230 space groups ~95% of all structures: monoclinic, triclinic, orthorhombic ~83% of all structures: P2 /c, P, P2 2 2, 2/c, P2, Pbca Number of Structures structures Year 7

8 Number of Published Structures Pī P2 /c P /c P2 Space Group Frequency Pbca Space group number Space Group Nomenclature space group name comes from Bravais lattice symbol, modified for translational symmetry easy to understand the components of many names, especially monoclinic and orthorhombic: P2 /c (P 2- on c) primitive unit cell ( lattice point) 2-fold screw axis along b (unique axis) c glide (translation along c axis) in ac plane ( to b) Pbca primitive unit cell ( lattice point) b glide (translation along b axis) in bc plane ( to a) c glide (translation along c axis) in ac plane ( to b) a glide (translation along a axis) in ab plane ( to c) Standard and Non-standard Settings sometimes a space group that is not on the list of 230 is given in a publication some space groups can be derived which are identical with another space group choice depends on convention P2 /a identical with P2 /c switching a and c label in monoclinic does not change the symmetry P2 /n alternate setting of P2 /c c c n closer to 90 o preferred a Pnam same as Pnma switch b and c label Equivalent Positions space groups used to locate symmetry related atoms in unit cell for example, if a benzene ring is located on a mirror: locate 3 and 3, asymmetric unit is the smallest part that generates the rest of the unit cell contents by all symmetry operations of space group others at symmetry equivalent positions Equivalent Positions, Asymmetric Unit and Z equivalent positions are divided into: general positions special positions asymmetric unit along with general and special positions allows an interpretation of Z (number of molecules in unit cell), and possible molecular symmetry Equivalent Positions of Symmetry Elements axis to position 2 a 2 b 2 c 2 a x + ½, y, z 2 b x, y + ½, z 2 c + ½ plane to n b x + ½, y, z + ½ m a n c x + ½, y + ½, z m b d a x, y + ¼, z + ¼ m c d b x + ¼, y, z + ¼ a b x + ½, y, z plane to position a c x + ½, y, z b a x, y + ½, z b c x, y + ½, z c a + ½ c b + ½ n a x, y + ½, z + ½ d c x + ¼, y + ¼, z 8

9 Equivalent Positions from entering Transforming oordinates for centered groups, add the following to each P general position: A x, y + ½, z + ½ x + ½, y + ½, z F x + ½, y + ½, z x + ½, y, z + ½ x, y + ½, z + ½ I x + ½, y + ½, z + ½ R x + 2/3, y + /3, z + /3 x + /3, y + 2/3, z + 2/3 x ¼ = (x + ¼) = (x ¼ + ½) = (x + ½) = x ½ = x + ½ (by adding ) y + ¼ = y ¼ + ½ = y + ½ Transforming oordinates Transforming oordinates x ¼ = (x + ¼) = (x ¼ + ½) = (x + ½) = x ½ = x + ½ (by adding ) y + ¼ = y ¼ + ½ = y + ½ x y, P2 /a x + ¼ = (x ¼) + ½ = x + ½ + ½ = x (by subtracting ) rename: x ¼as x y ¼as y z as z. x ¼, y ¼, z 2. x ¼, y +¼, z 3. x + ¼, y + ¼, z 4. x + ¼, y ¼, z x + ½, y + ½, z x + ½, y + ½, z Transforming oordinates Special Positions related by a change in sign related by a change in sign x + ½, y + ½, z x + ½, y + ½, z z y, 2 molecules/cell finally, change to preferred setting P2 /c; switch x and z x, y + ½, z + ½ x, y + ½, z + ½ general positions P2 /c if an object is located at = 0, 0, 0; only unique point generated by symmetry is at 0, ½, ½ also true for: 0, 0, ½ 0, ½, 0 ½, 0, ½ ½, ½, 0 ½, 0, 0 ½, ½, ½ 9

10 note: Special Positions 0, 0, 0 0, ½, ½ 0, 0, ½ 0, ½, 0 ½, 0, ½ ½, ½, 0 ½, 0, 0 ½, ½, ½ an object (molecule) at a special position has to have the same symmetry as the special position Special Positions an atom on a special position has at least one fixed coordinate; part of the atom generates the rest: one fixed position (axis to plane) for an atom on a mirror two fixed positions (other axes) for an atom on a rotation axis three fixed positions for an atom on an inversion center in P2 /c, a center of symmetry Z = 4 for an object on a general position in P2 /c ac mirror (x, 0, z) or (x, ½, z) Z = 2 for an object on a special position in P2 /c asymmetric unit is ½ of the molecule International Tables for rystallography International Tables for rystallography site symmetry general positions Z special positions 0

11 General Positions for Other P2 /c: Z = P2 /c: Z = 4 P2 /c: Z = 2 atom on special position no atom on special position

LMB Crystallography Course, 2013. Crystals, Symmetry and Space Groups Andrew Leslie

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

More information

Solid State Theory Physics 545

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

More information

Part 4-32 point groups

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

More information

12.524 2003 Lec 17: Dislocation Geometry and Fabric Production 1. Crystal Geometry

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

More information

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

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

More information

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

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

More information

Crystal Structure Determination I

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

More information

We shall first regard the dense sphere packing model. 1.1. Draw a two dimensional pattern of dense packing spheres. Identify the twodimensional

We shall first regard the dense sphere packing model. 1.1. Draw a two dimensional pattern of dense packing spheres. Identify the twodimensional Set 3: Task 1 and 2 considers many of the examples that are given in the compendium. Crystal structures derived from sphere packing models may be used to describe metals (see task 2), ionical compounds

More information

Chapter 2: Crystal Structures and Symmetry

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................

More information

Capacitance and Ferroelectrics

Capacitance and Ferroelectrics Ram Seshadri MRL 2031, x6129 seshadri@mrl.ucsb.edu; http://www.mrl.ucsb.edu/ seshadri/teach.html Capacitance and Ferroelectrics A voltage V applied across a capacitor of caacitance C allows a quantity

More information

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

rotation,, 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 translation-free symmetry y elements in its unit cell. While not always immediately obvious, in most well formed crystal

More information

Crystalline Structures Crystal Lattice Structures

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,

More information

Lecture 1 Symmetry in the solid state -

Lecture 1 Symmetry in the solid state - Lecture 1 Symmetry in the solid state - Part I: Simple patterns and groups 1 Introduction Concepts of symmetry are of capital importance in all branches of the physical sciences. In physics, continuous

More information

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-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

More information

Phase transitions with no group-subgroup relations between the phases

Phase transitions with no group-subgroup relations between the phases Phase transitions with no group-subgroup relations between the phases Michele Catti Dipartimento di Scienza dei Materiali, Universita di Milano Bicocca, Milano, Italy International School on the Use and

More information

Lecture 34: Symmetry Elements

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

More information

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

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

More information

Chapter 3: Structure of Metals and Ceramics. Chapter 3: Structure of Metals and Ceramics. Learning Objective

Chapter 3: Structure of Metals and Ceramics. Chapter 3: Structure of Metals and Ceramics. Learning Objective Chapter 3: Structure of Metals and Ceramics Chapter 3: Structure of Metals and Ceramics Goals Define basic terms and give examples of each: Lattice Basis Atoms (Decorations or Motifs) Crystal Structure

More information

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. 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

More information

1. Introduction: growth of the Powder Diffraction File over the past 60 years

1. Introduction: growth of the Powder Diffraction File over the past 60 years Acta Crystallographica Section B Structural Science ISSN 0108-7681 New Powder Diffraction File (PDF-4) in relational database format: advantages and data-mining capabilities Soorya N. Kabekkodu, John Faber*

More information

CRYSTALLINE SOLIDS IN 3D

CRYSTALLINE SOLIDS IN 3D CRYSTALLINE SOLIDS IN 3D Andrew Baczewski PHY 491, October 7th, 2011 OVERVIEW First - are there any questions from the previous lecture? Today, we will answer the following questions: Why should we care

More information

Chapter Outline. How do atoms arrange themselves to form solids?

Chapter 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 Face-centered cubic Body-centered cubic Hexagonal close-packed

More information

Group Theory and Chemistry

Group Theory and Chemistry Group Theory and Chemistry Outline: Raman and infra-red spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation

More information

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

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

More information

1. Human beings have a natural perception and appreciation for symmetry.

1. Human beings have a natural perception and appreciation for symmetry. I. SYMMETRY ELEMENTS AND OPERATIONS A. Introduction 1. Human beings have a natural perception and appreciation for symmetry. a. Most people tend to value symmetry in their visual perception of the world.

More information

A. The answer as per this document is No, it cannot exist keeping all distances rational.

A. The answer as per this document is No, it cannot exist keeping all distances rational. Rational Distance Conor.williams@gmail.com www.unsolvedproblems.org: Q. Given a unit square, can you find any point in the same plane, either inside or outside the square, that is a rational distance from

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

More information

2006 George Baloglou first draft: summer 2001 CHAPTER 7 COMPOSITIONS OF ISOMETRIES

2006 George Baloglou first draft: summer 2001 CHAPTER 7 COMPOSITIONS OF ISOMETRIES 2006 George Baloglou first draft: summer 2001 CHAPTER 7 COMPOSITIONS OF ISOMETRIES 7.0 Isometry hunting 7.0.1 Nothing totally new. Already in 4.0.4 we saw that the composition (combined effect) of a rotation

More information

An introduction to the tools hosted in the Bilbao Crystallographic Server

An introduction to the tools hosted in the Bilbao Crystallographic Server EPJ Web of Conferences 22, 00009 (202) DOI: 0.05/epjconf/2022200009 C Owned by the authors, published by EDP Sciences, 202 An introduction to the tools hosted in the Bilbao Crystallographic Server E.S.

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, June 20, 2012 9:15 a.m. to 12:15 p.m., only Student Name: School Name: Print your name and the name

More information

Basics of X-ray diffraction: From symmetry to structure determination

Basics of X-ray diffraction: From symmetry to structure determination Basics of X-ray diffraction: From symmetry to structure determination T. N. Guru Row Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore-560012 Email: ssctng@sscu.iisc.ernet.in

More information

X-ray Powder Diffraction Pattern Indexing for Pharmaceutical Applications

X-ray Powder Diffraction Pattern Indexing for Pharmaceutical Applications The published version of this manuscript may be found at the following webpage: http://www.pharmtech.com/pharmtech/peer-reviewed+research/x-ray-powder-diffraction-pattern-indexing-for- Phar/ArticleStandard/Article/detail/800851

More information

www.sakshieducation.com

www.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 information

Unit 8 Angles, 2D and 3D shapes, perimeter and area

Unit 8 Angles, 2D and 3D shapes, perimeter and area Unit 8 Angles, 2D and 3D shapes, perimeter and area Five daily lessons Year 6 Spring term Recognise and estimate angles. Use a protractor to measure and draw acute and obtuse angles to Page 111 the nearest

More information

Quick Guide for Data Collection on the NIU Bruker Smart CCD

Quick Guide for Data Collection on the NIU Bruker Smart CCD Quick Guide for Data Collection on the NIU Bruker Smart CCD 1. Create a new project 2. Optically align the crystal 3. Take rotation picture 4. Collect matrix to determine unit cell 5. Refine unit cell

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

More information

Geometry of Minerals

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

More information

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

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

More information

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

More information

Introduction to Powder X-Ray Diffraction History Basic Principles

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

More information

Phase determination methods in macromolecular X- ray Crystallography

Phase determination methods in macromolecular X- ray Crystallography Phase determination methods in macromolecular X- ray Crystallography Importance of protein structure determination: Proteins are the life machinery and are very essential for the various functions in the

More information

The Structure of solids.

The 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 information

Number Sense and Operations

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

More information

http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4

http://www.castlelearning.com/review/teacher/assignmentprinting.aspx 5. 2 6. 2 1. 10 3. 70 2. 55 4. 180 7. 2 8. 4 of 9 1/28/2013 8:32 PM Teacher: Mr. Sime Name: 2 What is the slope of the graph of the equation y = 2x? 5. 2 If the ratio of the measures of corresponding sides of two similar triangles is 4:9, then the

More information

Theory of X-Ray Diffraction. Kingshuk Majumdar

Theory of X-Ray Diffraction. Kingshuk Majumdar Theory of X-Ray Diffraction Kingshuk Majumdar Contents Introduction to X-Rays Crystal Structures: Introduction to Lattices Different types of lattices Reciprocal Lattice Index Planes X-Ray Diffraction:

More information

Crystal Optics of Visible Light

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

More information

EXPERIMENT O-6. Michelson Interferometer. Abstract. References. Pre-Lab

EXPERIMENT O-6. Michelson Interferometer. Abstract. References. Pre-Lab EXPERIMENT O-6 Michelson Interferometer Abstract A Michelson interferometer, constructed by the student, is used to measure the wavelength of He-Ne laser light and the index of refraction of a flat transparent

More information

Chapter 3. 1. 3 types of materials- amorphous, crystalline, and polycrystalline. 5. Same as #3 for the ceramic and diamond crystal structures.

Chapter 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 information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

Chem 106 Thursday Feb. 3, 2011

Chem 106 Thursday Feb. 3, 2011 Chem 106 Thursday Feb. 3, 2011 Chapter 13: -The Chemistry of Solids -Phase Diagrams - (no Born-Haber cycle) 2/3/2011 1 Approx surface area (Å 2 ) 253 258 Which C 5 H 12 alkane do you think has the highest

More information

Symmetry Relations between Crystal Structures

Symmetry Relations between Crystal Structures Summer School on Mathematical and Theoretical Crystallography 27 April May 2008 Gargnano, Italy Symmetry Relations between Crystal Structures Ulrich Müller Fachbereich Chemie, Philipps-Universität Marburg,

More information

Vocabulary Words and Definitions for Algebra

Vocabulary 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 information

Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item 2) (MAT 360) Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

More information

COMPOSITION. SPACE GROUP AND UNIT CELL OF HANKSITE LBwrs S. RRusonr-r., Uni.aersity of Michigan, Ann Arbor, Michigan' Assrnecr

COMPOSITION. SPACE GROUP AND UNIT CELL OF HANKSITE LBwrs S. RRusonr-r., Uni.aersity of Michigan, Ann Arbor, Michigan' Assrnecr COMPOSITION. SPACE GROUP AND UNIT CELL OF HANKSITE LBwrs S. RRusonr-r., Uni.aersity of Michigan, Ann Arbor, Michigan' Assrnecr The composition of gnazsor'narcog'kcl for hanksite is verified. There are

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A. Monday, January 27, 2003 1:15 to 4:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A. Monday, January 27, 2003 1:15 to 4:15 p.m. The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS A Monday, January 27, 2003 1:15 to 4:15 p.m., only Print Your Name: Print Your School s Name: Print your name and the

More information

Proton Nuclear Magnetic Resonance ( 1 H-NMR) Spectroscopy

Proton Nuclear Magnetic Resonance ( 1 H-NMR) Spectroscopy Proton Nuclear Magnetic Resonance ( 1 H-NMR) Spectroscopy Theory behind NMR: In the late 1940 s, physical chemists originally developed NMR spectroscopy to study different properties of atomic nuclei,

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, June 17, 2010 1:15 to 4:15 p.m., only Student Name: School Name: Print your name and the name of your

More information

Chapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold

Chapter 18 Symmetry. Symmetry of Shapes in a Plane 18.1. then unfold Chapter 18 Symmetry Symmetry is of interest in many areas, for example, art, design in general, and even the study of molecules. This chapter begins with a look at two types of symmetry of two-dimensional

More information

Three-Dimensional Figures or Space Figures. Rectangular Prism Cylinder Cone Sphere. Two-Dimensional Figures or Plane Figures

Three-Dimensional Figures or Space Figures. Rectangular Prism Cylinder Cone Sphere. Two-Dimensional Figures or Plane Figures SHAPE NAMES Three-Dimensional Figures or Space Figures Rectangular Prism Cylinder Cone Sphere Two-Dimensional Figures or Plane Figures Square Rectangle Triangle Circle Name each shape. [triangle] [cone]

More information

Structure Factors 59-553 78

Structure Factors 59-553 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 information

SPRING UNIT 14. second half. Line symmetry and reflection. Measuring angles. Naming and estimating angles. Drawing angles

SPRING UNIT 14. second half. Line symmetry and reflection. Measuring angles. Naming and estimating angles. Drawing angles PART SPRING second half SHAPE AND SPACE SECTION Line symmetry and reflection SECTION Measuring angles SECTION Naming and estimating angles SECTION Drawing angles SECTION 5 Calculations involving angles

More information

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.

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

More information

INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY. jueves 6 de diciembre de 12

INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY. jueves 6 de diciembre de 12 INTERNATIONAL SCHOOL ON FUNDAMENTAL CRYSTALLOGRAPHY CRYSTAL-STRUCTURE TOOLS BILBAO CRYSTALLOGRAPHIC SERVER PRACTICAL EXERCISES Mois I. Aroyo Universidad del Pais Vasco, Bilbao, Spain Bilbao Crystallographic

More information

Geometry Regents Review

Geometry Regents Review Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles that are between parallel lines, but on opposite sides of a transversal. GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

More information

Geometry Progress Ladder

Geometry Progress Ladder 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

More information

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 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

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, January 26, 2016 1:15 to 4:15 p.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, January 26, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The possession or use of any communications

More information

1 of 9 2/9/2010 3:38 PM

1 of 9 2/9/2010 3:38 PM 1 of 9 2/9/2010 3:38 PM Chapter 23 Homework Due: 8:00am on Monday, February 8, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]

More information

Hybrid Molecular Orbitals

Hybrid Molecular Orbitals Hybrid Molecular Orbitals Last time you learned how to construct molecule orbital diagrams for simple molecules based on the symmetry of the atomic orbitals. Molecular orbitals extend over the entire molecule

More information

SAT Subject Math Level 2 Facts & Formulas

SAT Subject Math Level 2 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

More information

Questions: Does it always take the same amount of force to lift a load? Where should you press to lift a load with the least amount of force?

Questions: Does it always take the same amount of force to lift a load? Where should you press to lift a load with the least amount of force? Lifting A Load 1 NAME LIFTING A LOAD Questions: Does it always take the same amount of force to lift a load? Where should you press to lift a load with the least amount of force? Background Information:

More information

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

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.

More information

Numeracy Targets. I can count at least 20 objects

Numeracy Targets. I can count at least 20 objects Targets 1c I can read numbers up to 10 I can count up to 10 objects I can say the number names in order up to 20 I can write at least 4 numbers up to 10. When someone gives me a small number of objects

More information

Infrared Spectroscopy: Theory

Infrared Spectroscopy: Theory u Chapter 15 Infrared Spectroscopy: Theory An important tool of the organic chemist is Infrared Spectroscopy, or IR. IR spectra are acquired on a special instrument, called an IR spectrometer. IR is used

More information

Section 16: Neutral Axis and Parallel Axis Theorem 16-1

Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Geometry of deformation We will consider the deformation of an ideal, isotropic prismatic beam the cross section is symmetric about y-axis All parts

More information

CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

More information

Cabri Geometry Application User Guide

Cabri Geometry Application User Guide Cabri Geometry Application User Guide Preview of Geometry... 2 Learning the Basics... 3 Managing File Operations... 12 Setting Application Preferences... 14 Selecting and Moving Objects... 17 Deleting

More information

Common Lay-up Terms and Conditions

Common Lay-up Terms and Conditions Mid-Plane: Centerline of the lay-up. Plane forming the mid-line of the laminate. Symmetry: A laminate is symmetric when the plies above the mid-plane are a mirror image of those below the mid-plane. Symmetrical

More information

A reinterpretation of the phase transitions in Na 2 CO 3

A reinterpretation of the phase transitions in Na 2 CO 3 Acta Crystallographica Section B Structural Science ISSN 0108-7681 Editor: Carolyn P. Brock A reinterpretation of the phase transitions in Na 2 CO 3 Alla Arakcheeva and Gervais Chapuis Copyright International

More information

MATHS LEVEL DESCRIPTORS

MATHS LEVEL DESCRIPTORS MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and

More information

Molecular Graphics What?

Molecular Graphics What? Molecular Graphics Molecular Graphics What? PDF 00-057-0248 ULM-8 A microporous florinated gallium phosphate with N 4 C 6 H 19 in the pores PDF-4 products contain data sets with atomic coordinates. Two

More information

Geometric Optics Converging Lenses and Mirrors Physics Lab IV

Geometric Optics Converging Lenses and Mirrors Physics Lab IV Objective Geometric Optics Converging Lenses and Mirrors Physics Lab IV In this set of lab exercises, the basic properties geometric optics concerning converging lenses and mirrors will be explored. The

More information

2014 Chapter Competition Solutions

2014 Chapter Competition Solutions 2014 Chapter Competition Solutions Are you wondering how we could have possibly thought that a Mathlete would be able to answer a particular Sprint Round problem without a calculator? Are you wondering

More information

ME 111: Engineering Drawing

ME 111: Engineering Drawing ME 111: Engineering Drawing Lecture 4 08-08-2011 Engineering Curves and Theory of Projection Indian Institute of Technology Guwahati Guwahati 781039 Eccentrici ty = Distance of the point from the focus

More information

Principles & Practice of Electron Diffraction

Principles & Practice of Electron Diffraction Principles & Practice of Electron Diffraction Duncan Alexander EPFL-CIME 1 Contents Introduction to electron diffraction Elastic scattering theory Basic crystallography & symmetry Electron diffraction

More information

Chapter 8 Geometry We will discuss following concepts in this chapter.

Chapter 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 information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 16, 2012 8:30 to 11:30 a.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

More information

MENSURATION. Definition

MENSURATION. Definition MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters

More information

Chapter 9. Chemical reactivity of molecules depends on the nature of the bonds between the atoms as well on its 3D structure

Chapter 9. Chemical reactivity of molecules depends on the nature of the bonds between the atoms as well on its 3D structure Chapter 9 Molecular Geometry & Bonding Theories I) Molecular Geometry (Shapes) Chemical reactivity of molecules depends on the nature of the bonds between the atoms as well on its 3D structure Molecular

More information

2 Session Two - Complex Numbers and Vectors

2 Session Two - Complex Numbers and Vectors PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

More information

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions:

Math 10C. Course: Polynomial Products and Factors. Unit of Study: Step 1: Identify the Outcomes to Address. Guiding Questions: Course: Unit of Study: Math 10C Polynomial Products and Factors Step 1: Identify the Outcomes to Address Guiding Questions: What do I want my students to learn? What can they currently understand and do?

More information

12-1 Representations of Three-Dimensional Figures

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

More information

North Carolina Math 2

North Carolina Math 2 Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.

More information

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

More information

9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS 9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

More information

SURFACE TENSION. Definition

SURFACE TENSION. Definition SURFACE TENSION Definition In the fall a fisherman s boat is often surrounded by fallen leaves that are lying on the water. The boat floats, because it is partially immersed in the water and the resulting

More information

Level 1 - Maths Targets TARGETS. With support, I can show my work using objects or pictures 12. I can order numbers to 10 3

Level 1 - Maths Targets TARGETS. With support, I can show my work using objects or pictures 12. I can order numbers to 10 3 Ma Data Hling: Interpreting Processing representing Ma Shape, space measures: position shape Written Mental method s Operations relationship s between them Fractio ns Number s the Ma1 Using Str Levels

More information

Group Theory and Molecular Symmetry

Group Theory and Molecular Symmetry Group Theory and Molecular Symmetry Molecular Symmetry Symmetry Elements and perations Identity element E - Apply E to object and nothing happens. bject is unmoed. Rotation axis C n - Rotation of object

More information