Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14...

8
crystallograp hy lll

description

Space group diagrams Example: P222 Space group diagrams Example: P222 a b

Transcript of Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14...

Page 1: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

crystallography lll

Page 2: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

230 space groups

Combine 32 point groups (rotational symmetry) witha. 14 Bravais lattices (translational symmetry)b. glide planes (rotational + translational symmetry)

- a, b, c, n, d

• screw axes (rotational + translational symmetry) - 21, 31, 32, 41, 42,43, 61, 62, 63, 64, 65

lattice centering screw axis glide plane

Examples:

Ccc2, P1, I 41/a 2/c 2/d, F 4/m 3 2/m, P 63/m 2/m 2/c

Page 3: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

Space group diagrams

Example: P222

a

b

Page 4: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

Space group diagrams

Example: P222

a

bx,y,z

x,y,z

x,y,z

x,y,z

General equipoint: 4u (x,y,z) (x,y,z) (x,y,z) (x,y,z)Special equipoints: 1a (0,0,0); 1b (1/2,0,0); … 2i (x,0,0), (x,0,0); 2j (x,0,1/2), (x,0,1/2); … 2m (0,y,z) (0,y,0); …

Page 5: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

Space group tables

International Tables for Crystallography, Vol. A

Page 6: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

Describing crystal structures

Need:

lattice parametersspace groupequipoints occupiedpositional parameters

Example:

LaB6 a = 4.1566 ÅP 4/m 3 2/mLa in 1a, B in 6fx = 0.207

Page 7: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

Describing crystal structures

LaB6 a = 4.1566 ÅP 4/m 3 2/mLa in 1a, B in 6fx = 0.207

In P 4/m 3 2/m

1a - (0,0,0)6f - (x,1/2,1/2)6f - (x,1/2,1/2)

1/2

1/2

6f - (1/2,x,1/2) 1/2

6f - (1/2,x,1/2)1/2

6f - (1/2,1/2,x)

0.207

6f - (1/2,1/2,x)

0.793

Page 8: Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

Exercises

1. C a = 3.500Å F 41/d 3 2/m C in 8a

F (0,0,0)+ (1/2,1/2,0)+ (1/2 ,0,1/2)+ (0,1/2,1/2)+

8a: (0,0,0) (3/4,1/4,3/4) projection down [001] 2. CaCu5 a = 5.082, c = 4.078Å P6/mmm Ca in 1a, Cu in 2c, 3g

2c: (1/3,2/3,0) (2/3,1/3,0)3g: (1/2,0,1/2) (0,1/2,1/2) (1/2,1/2,1/2)

projection down [001]


Geodesic spaces : momentum :: Groups : symmetry

Geodesic spaces : momentum :: Groups : symmetry

INDRASHIL UNIVERSITYindrashiluniversity.edu.in/brochures/pdf/M.Sc_Chemistry_syllabus.pdf · symmetry elements and optical isomerism, symmetry point groups, classes of symmetry operations,

INDRASHIL UNIVERSITYindrashiluniversity.edu.in/brochures/pdf/M.Sc_Chemistry_syllabus.pdf · symmetry elements and optical isomerism, symmetry point groups, classes of symmetry operations,

Red de Bravais

Red de Bravais

Space Group = {Essential Symmetry Operations} {Bravais Lattice} N = # of translation operations in the Bravais lattice (N is a very large number) h =

Space Group = {Essential Symmetry Operations} {Bravais Lattice} N = # of translation operations in the Bravais lattice (N is a very large number) h =

COMPUTING CHARACTER TABLES OF FINITE GROUPS · COMPUTING CHARACTER TABLES OF FINITE GROUPS Jay Taylor (Università degli Studi di Padova) Symmetry. Symmetry Chemistry. ... Groups

COMPUTING CHARACTER TABLES OF FINITE GROUPS · COMPUTING CHARACTER TABLES OF FINITE GROUPS Jay Taylor (Università degli Studi di Padova) Symmetry. Symmetry Chemistry. ... Groups

Contents What are Plane Symmetry Groups?may/VIGRE/VIGRE2008/REUPapers/Levine.pdf · PLANE SYMMETRY GROUPS MAXWELL LEVINE Abstract. This paper discusses plane symmetry groups, also

Contents What are Plane Symmetry Groups?may/VIGRE/VIGRE2008/REUPapers/Levine.pdf · PLANE SYMMETRY GROUPS MAXWELL LEVINE Abstract. This paper discusses plane symmetry groups, also

Introduction and point groups Stereographic projections Low symmetry systems Space groups

Introduction and point groups Stereographic projections Low symmetry systems Space groups

Symmetry Groups of Boolean Functions and Constructions of ... · Symmetry Groups of Boolean Functions and Constructions of Permutation Groups Andrzej Kisielewicz* Institute of Mathematics,

Symmetry Groups of Boolean Functions and Constructions of ... · Symmetry Groups of Boolean Functions and Constructions of Permutation Groups Andrzej Kisielewicz* Institute of Mathematics,

Groups, symmetry and fractals

Groups, symmetry and fractals

Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Part 2.2: Symmetry and Point Groups 1. VSEPR Define Symmetry Symmetry elements Symmetry operations Point groups Assigning point groups Matrix math Outline.

Symmetry-operations, point groups, space groups and ...folk.uio.no/ravi/cutn/scm/symmetry_and_crystallography/Compendium... · 1 Symmetry-operations, point groups, space groups and

Symmetry-operations, point groups, space groups and ...folk.uio.no/ravi/cutn/scm/symmetry_and_crystallography/Compendium... · 1 Symmetry-operations, point groups, space groups and

FINITE SYMMETRY GROUPS IN COMPLEX GEOMETRY

FINITE SYMMETRY GROUPS IN COMPLEX GEOMETRY

INDRASHIL UNIVERSITYsymmetry elements and optical isomerism, symmetry point groups, classes of symmetry operations, classification of molecular point groups. Representations of groups.

INDRASHIL UNIVERSITYsymmetry elements and optical isomerism, symmetry point groups, classes of symmetry operations, classification of molecular point groups. Representations of groups.

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

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

Bravais lattices

Bravais lattices

GROUPS AND SYMMETRY CHAPTER 2. Contents 1.Groups and subgroups 2.Group actions 3.Symmetry groups 4.Covering graphs 5.Symmetry in graphs 6.Symmetry in.

GROUPS AND SYMMETRY CHAPTER 2. Contents 1.Groups and subgroups 2.Group actions 3.Symmetry groups 4.Covering graphs 5.Symmetry in graphs 6.Symmetry in.

Translation symmetry, Space groups, Bloch functions, · PDF file · 2014-07-29Translation symmetry, Space groups, Bloch functions, Fermi energy Roberto Orlando ... • point symmetry

Translation symmetry, Space groups, Bloch functions, · PDF file · 2014-07-29Translation symmetry, Space groups, Bloch functions, Fermi energy Roberto Orlando ... • point symmetry

CH6. Symmetry Symmetry elements and operations Point groups Character tables Some applications

CH6. Symmetry Symmetry elements and operations Point groups Character tables Some applications

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

Group Theory and Symmetry - California State … Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ... •PtCl 4 2-5 symmetry …

Group Theory and Symmetry - California State … Theory and Symmetry point groups, symmetry elements, matrix representations and properties of groups 2 ... •PtCl 4 2-5 symmetry …