Line groups Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further,

Slides:

Advertisements

Similar presentations

Equivalent Positions in 3-D

Advertisements

Lesson 12—Working with Space Groups How to convert a space group to a point group Adding the translational elements Calculating the coordinates of the.

Why Study Solid State Physics?

Lecture 2: Crystal Symmetry

Wigner-Seitz Cell The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to.

1 Watkins/Fronczek - Space Groups Translational + Rotational Symmetry Introduction to Space Groups.

Crystals and Symmetry. Why Is Symmetry Important? Identification of Materials Prediction of Atomic Structure Relation to Physical Properties –Optical.

Today in Inorganic…. Symmetry elements and operations Properties of Groups Symmetry Groups, i.e., Point Groups Classes of Point Groups How to Assign Point.

Today in Inorganic…. Uses of Symmetry in Chemistry Symmetry elements and operations Properties of Groups Symmetry Groups, i.e., Point Groups Previously:

Mineralogy Carleton College Winter Lattice and its properties Lattice: An imaginary 3-D framework, that can be referenced to a network of regularly.

Lecture 12 (10/30/2006) Crystallography Part 5: Internal Order and 2-D Symmetry Plane Lattices Planar Point Groups Plane Groups.

17-plane groups When the three symmetry elements, mirrors, rotation axis and glide planes are shown on the five nets, 17-plane groups are derived.

CRYSTALLOGRAPHY TRIVIA FINAL ROUND!. Round 3 – Question 1 Twins are said to add another level of symmetry to a crystal. Why is this?

Crystallography Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern In 3-D, translation defines operations.

Symmetry of Single-walled Carbon Nanotubes. Outline Part I (November 29) Symmetry operations Line groups Part II (December 6) Irreducible representations.

Symmetry of Single-walled Carbon Nanotubes Part II.

PH0101 UNIT 4 LECTURE 3 CRYSTAL SYMMETRY CENTRE OF SYMMETRY

Elementary Crystallography for X-ray Diffraction

32 Crystallographic Point Groups. Point Groups The 32 crystallographic point groups (point groups consistent with translational symmetry) can be constructed.

CS 325 Introduction to Computer Graphics 02 / 17 / 2010 Instructor: Michael Eckmann.

Chem Lattices By definition, crystals are periodic in three dimensions and the X-ray diffraction experiment must be understood in the context of.

Today in Inorganic…. Symmetry elements and operations Properties of Groups Symmetry Groups, i.e., Point Groups Classes of Point Groups How to Assign Point.

Unit # 1 Vocabulary Review 1. coordinate plane 2.

General Introduction to Symmetry in Crystallography A. Daoud-Aladine (ISIS)

Crystallographic Points, Directions, and Planes. ISSUES TO ADDRESS... How to define points, directions, planes, as well as linear, planar, and volume densities.

Lecture 12 Crystallography

Chapter 4. Molecular Symmetry

Lesson 10-5: Transformations 1 Lesson 9 Transformations G.CO2, Represent transformation in plane describe as a function that take points in the plane as.

Homework Look at the Space Group P21/c (#14). This is the most common space group for small molecules. 1. No protein has ever been found to crystallize.

Key things to know to describe a crystal

1 Space groups Translations T = ut 1 (1-D) u is an integer The set of all lattice vectors is a group (the set of all integers (±) is a group)

Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

1 Algebra of operations Thm 1: Rotation about A thru angle  followed by translation T to axis = rotation thru same angle  about B on bisector of AA'

Symmetry in crystals. Infinitely repeating lattices.

Point & Space Group Notations

Lesson 11: Adding Translation Define pure translation within a cell—centering Combine translation with other symmetry operations to produce a new operations.

Crystal Structure and Crystallography of Materials

Parameters235 Formula MW cryst syst Space group a /Å b /Å c /Å α (°) β (°) γ (°) V /Å 3 Z T /K λ /Å ρ c (g/cm3) μ (mm-1) Goodness of fit θ range Total.

Crystal Structure and Crystallography of Materials

Periodic patterns.

Point Groups Roya Majidi 1393.

3. Transformation

SYMMETRY A A A DEFINITION:

Crystallographic Points, Directions, and Planes.

Miller indices/crystal forms/space groups

Crystal Structure and Crystallography of Materials

Moments of the forces Mo = F x d A moment is a turning force.

Textbook: Condensed matter physics, 2nd ed. By M. Marder

Honors Chemistry.

Crystals Crystal consist of the periodic arrangement of building blocks Each building block, called a basis, is an atom, a molecule, or a group of atoms.

Warm-Up Triangle RST has vertices R(0,4), S(3,3), and T(1, 1). Choose the graph that correctly displays a translation along – 4, 0, and a rotation of.

Crystallographic Points, Directions, and Planes.

SYMMETRY A A A DEFINITION:

NOTE: Symbolism In Schönflies notation, what does the symbol S2 mean?

Symmetry Elements II.

Symmetry “the correspondence in size, form and

WARM-UP 8 in. Perimeter = _____ 7 in. Area = _____ 12 in. 4 cm

9.5 : Symmetry I can identify line and rotational symmetries in two‐dimensional figures. I can identify plane and axis symmetries in three‐dimensional.

Crystals and Symmetry.

Translation (slide) TRANSLATION a slide without turning.

(xyz) (xyz) (xyz) (xyz)

Elementary Symmetry Operation Derivation of Plane Lattices

Crystallography.

Derivation of the 32 Point Groups

Properties of point groups:

Symmetry operations: Transformations mapping the nuclear frame of a molecule onto an indistinguishable version of itself  symmetry elements: symmetry.

Wallpaper Symmetries CS 39 Carlo H. Séquin

Crystallography lll.

Presentation transcript:

Line groups Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further, translations associated with glide planes & screw axes added by using homomorphism betwn isogonal pt. grps. & translation groups.

Line groups Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further, translations associated with glide planes & screw axes added by using homomorphism betwn isogonal pt. grps. & translation groups. The 17 2-D plane groups can be developed in a similar manner. Removal of a dimension severely restricts no. of symmetry combinations.

Line groups Review To enumerate the 230 3-D space groups: pt. grps. taken in turn & extended with conforming 3-D lattice translation groups. Further, translations associated with glide planes & screw axes added by using homomorphism betwn isogonal pt. grps. & translation groups. The 17 2-D plane groups can be developed in a similar manner. Removal of a dimension severely restricts no. of symmetry combinations. What about 1-D line groups? Some surprises(?)

Line groups What about 1-D line groups? Consider a 3-D object of arbitrary pt. symmetry repeated along 1-D lattice

Line groups What about 1-D line groups? Consider a 3-D object of arbitrary pt. symmetry repeated along 1-D lattice Only one type of translation - no possibility for centering translations Translations simply described by single lattice parameter

Line groups What about 1-D line groups? Consider a 3-D object of arbitrary pt. symmetry repeated along 1-D lattice Only one type of translation - no possibility for centering translations Translations simply described by single lattice parameter But lattice parameter not enough to describe structure of array

Line groups Need to also describe symmetry of object Can use pt. grps. for this All objects can be described by one of the 32 pt. grps.

Line groups Need to also describe symmetry of object Can use pt. grps. for this All objects can be described by one of the 32 pt. grps. Doesn't quite work Consider:

Line groups Note glide in this planar structure repeat unit

Line groups Note glide in this planar structure c-glides allowed repeat unit

Line groups Note glide in this planar structure 1 C2 C2' C2" m m' m" i repeat unit 1 C2 C2' C2" m m' m" i 2/m 2/m 2/m m m' C2 C2" m" C2'

Line groups G Note glide in this planar structure 1t 2t 3t zt xt yt it repeat unit G 1t 2t 3t zt xt yt it 2/m 2/m 2/m m,zt m'.xt C2,1t C2",3t m",yt C2',2t Rules: 1t 2t = 3t 1t zt = it 1t xt = yt (as before) 1t it = zt 1t yt = xt

Line groups G Note glide in this planar structure 1t 2t 3t zt xt yt it repeat unit G 1t 2t 3t zt xt yt it 2/m 2/m 2/m m,zt m'.xt C2,1t C2",3t m",yt C2',2t Rules: 1t 2t = 3t 1t zt = it 1t xt = yt Choose: 1t = c/2, 3t = 0 (as before) 1t it = zt 1t yt = xt it = 0, yt = 0 Then: 2t = zt = xt = c/2

Line groups G Note glide in this planar structure 1t 2t 3t zt xt yt it m'.xt C2",3t L 21/sm s2/c 2/m or L 21/mcm m,zt repeat unit m",yt C2',2t G 1t 2t 3t zt xt yt it 2/m 2/m 2/m m,zt m'.xt C2,1t C2",3t m",yt C2',2t Rules: 1t 2t = 3t 1t zt = it 1t xt = yt Choose: 1t = c/2, 3t = 0 (as before) 1t it = zt 1t yt = xt it = 0, yt = 0 Then: 2t = zt = xt = c/2

Line groups Note glide in this planar structure Also consider: repeat unit

Line groups Note glide in this planar structure Also consider: "Non-crystallographic" rotations can be crystallographic in 1-D lattices repeat unit

Line groups Rotation axes of any order, up to C∞, proper or improper, allowed C∞

Line groups Rotation axes of any order, up to C∞, proper or improper, allowed Multiple rotation axes (n/n) allowed (n even perpendicular mirror) C∞

Line groups Rotation axes of any order, up to C∞, proper or improper, allowed Multiple rotation axes (n/n) allowed (n even perpendicular mirror) Screw axes of any kind (21, 53, 149) allowed C∞

Line groups Summary: In line groups, pt. symmetry extended considerably Translations along lattice direction allowed No translations in any other direction

Line groups Procedure Combine lattice translations w/ 32 pt. grps. As in 2-D & 3-D, lattice group must be invariant under all pt. grp. operations Thus, while infinite no. of types of rotation axes allowed along lattice direction, only 2 & 2 axes allowed otherwise & must be perpendicular to lattice direction

Line groups Procedure 1. Combine lattice translations w/ 32 pt. grps. 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) Or remember: New axis appears at angle of 1/2 throw of main rotation axis new axis

Line groups Procedure 1. Combine lattice translations w/ 32 pt. grps. 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) 3. Determine line groups isogonal w/ all allowed point groups Infinite no. of line groups possible

Line groups Procedure 1. Combine lattice translations w/ 32 pt. grps. 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) 3. Determine line groups isogonal w/ all allowed point groups Infinite no. of line groups possible Examples for 1: L1, L4/m, L622, L2/m 2/m 2/m

Line groups Procedure 1. Combine lattice translations w/ 32 pt. grps. 2. Add rotation axes not allowed in 2-D & 3-D (use Euler construction) 3. Determine line groups isogonal w/ all allowed point groups Infinite no. of line groups possible Examples for 1: L1, L4/m, L622, L2/m 2/m 2/m Examples for 2: L5, L14/m, L8/m 2/m 2/m

Line groups Line groups (from Vujicic, Bozovic, & Herbut, J. Phys. A 10, 1271 (1977) Pt. grp. n odd n even (w/example) Cn Ln Lnp Cnv (4mm) Lnm Lnmm Lnc Lncc L(2q)qmc Cnh (4/m) Ln/m L(2q)q/m S2n (3) Ln L(2n) Dn (422) Ln2 Ln22 Lnp2 Lnp22 Dnd (42m) Lnm L(2n)2m Lnc L(2n)2c Dnh (4/m 2/m 2/m) L(2n)2m Ln/mmm L(2n)2c Ln/mcm L(2q)q/mcm n = 1, 2, …. p = 1, 2, …., n-1

Line groups Line groups - example - nanotubes 84 + 1/2 (4,0)

Line groups Line groups - example - nanotubes 84 84/m + 1/2 (4,0) ± 1/2

Line groups Line groups - example - nanotubes 84 84/m_m + 1/2 (4,0) ± 1/2

Line groups Line groups - example - nanotubes 84 84/mcm + 1/2 (4,0) 1/4 ± 1/2

Line groups Line groups - example - nanotubes 0° Ch = na1 + ma2 (n,m)

Line groups Line groups - example - nanotubes

Line groups Line groups - example - nanotubes armchair zigzag chiral