Chapter 5

Unit 5.1 – Space Group P21/c and the Asymmetric Unit

Slides Unit 5.1

Transcript of Unit 5.1

Further material/links:

CSD Space group statistics
Space group frequency ranking for the 807,190 CSD structures.

Crystallographic Space Group Diagrams and Tables

Space Group Visualizer
Eckhard Hitzer and Christian Perwass worked on visualising point and space groups using the visualisation tool CLUCalc.This software implements Geometric Algebra, which allows for a direct implementation of symmetry operators in the algebra. Great tool!

The Space group list project
A complete collection of a list of examples containing at least one crystal structure for all of the 230 space groups.

Unit 5.2 – Ethylene, P21/n, Settings

Slides Unit 5.2

Transcript of Unit 5.2

Further material/links:

Mercury by CCDC
Free software for Crystal Structure Visualisation, Exploration and Analysis

CIF file of Ethylene

J. Haestier et al.: Poster P21/n
A poster about the transformation of P21/n nto P21/c

Unit 5.3 – Benzene and Polymorphism

Slides Unit 5.3

Transcript of Unit 5.3

Further material/links:

CIF file of polymorph I of benzene

CIF file of polymorph II of benzene

Paolo Raiteri et al.: Exploring Polymorphism: The Case of Benzene
(subscription required)

Sarah L. Price: Predicting crystal structures of organic compounds
(open access)

Unit 5.4 – NaCl – a simple-complicated Structure

Slides Unit 5.4

Transcript of Unit 5.4

Further material/links:

CIF file of NaCl

Unit 5.5 – Graphite and Diamond

Slides Unit 5.5

Transcript of Unit 5.5

Further material/links:

CIF file of Graphite

CIF file of Diamond

Unit 5.6 – Quasicrystals (I)

Slides Unit 5.6

Transcript of Unit 5.6

Further material/links:

D. Shechtman et al.: Metallic Phase with Long-Range Orientational Order and No Translational Symmetry
(open access)
First orginal article describing quasicrystals with 5-fold rotational symmetry.

Crystals of Golden Proportions
Background information for the Public about the Nobel Prize 2011 in Chemistry: Crystals of Golden Proportions

Interview with Prof. Dan Shechtman discussing his dicovery of quasicrystals
Prof. Shechtman shares his story of discovery of a new form of matter.
Film made by the American Technion Society.

Unit 5.7 – Quasicrystals (II)

Slides Unit 5.7

Transcript of Unit 5.7

Further material/links:

Steffen Weber: Introduction to Quasicrystals
This page is meant to be an introduction to the field of Quasicrystals in order to educate the interested reader on some basic concepts in this relatively new branch of Crystallography. The more advanced reader may proceed to other sites and sources on quasicrystals.

Eric Hwang: Penrose Tilings and Quasicrystals

Walter Steurer: The structure of quasicrystals
(subscription required)

The aim of the present article is to give a review of the state-of-the-art on quasicrystal structure analysis. After a short discussion of the term “crystal” in chapter 1, the geometrical generation of quasilattices is touched in chapter 2. In the following the higher-dimensional description of 1d, 2d and 3d quasi-crystals is demonstrated in detail as well as the derivation of structure factor equations and symmetry relationships in the higher-dimensional space. Chapter 4 shows the experimental techniques and structure determination methods for the study of quasicrystals. The experimental results of structural studies performed with different tech-niques are critically reviewed in chapter 5. Some of the results of the literature research are that five years after the detection of the first quasicrystal not a single quantitative (in terms of a regular structure determination) analysis of its structure has been carried out, and that the famous Mackay-icosahedra do not play the important role as the basic structural building elements as one supposed before.

A. Yamamoto: Crystallography of Quasiperiodic Crystals
(subscription required)

The aim of this review is to describe many approaches to modulated crystals and quasicrystals developed in two decades after the introduction of higher-dimensional crystallography in a unified way. Much attention is focused on higher-dimensional crystallography of quasi-crystals, which is under development. After discussions on symmetries of modulated crystals and methods of their structure analysis, many subjects on the analysis of quasicrystals are discussed, which include methods of generating quasiperiodic tilings, their diffraction patterns, similarity transformations, indexing problems, point density of quasicrystals, phason distortion, relations between quasicrystals and their crystalline approximants, model constructions, Patterson, refinement and maximum-entropy methods for quasicrystals, and superstructures in quasicrystals. In particular, the theories for octagonal, decagonal, dodecagonal and icosahedral quasicrystals are given in detail.

Donald L. D. Caspar and Eric Fontano: Five-fold symmetry in crystalline quasicrystal lattices
Colloquium Paper: To demonstrate that crystallographic methods can be applied to index and interpret diffraction patterns from well-ordered quasicrystals that display non-crystallographic 5-fold symmetry, we have characterized the properties of a series of periodic two-dimensional lattices built from pentagons, called Fibonacci pentilings, which resemble aperiodic Penrose tilings.

Paul J. Steinhardt: New perspectives on forbidden symmetries, quasicrystals, and Penrose tilings
Colloquium Paper: Quasicrystals are solids with quasiperiodic atomic structures and symmetries forbidden to ordinary periodic crystals—e.g., 5-fold symmetry axes. A powerful model for understanding their structure and properties has been the two-dimensional Penrose tiling. Recently discovered properties of Penrose tilings suggest a simple picture of the structure of quasicrystals and shed new light on why they form.

Forbidden crystal symmetry in mathematics and architecture
Sir Roger Penrose provides a unique insight into the “forbidden symmetry” of his famous penrose tiles and the use of non-repeating patterns in design and architecture.

Scientists Accidentally Create Improbable Two-Dimensional Quasicrystals
A strange new substance has unexpectedly emerged from a university lab in Germany: a two-dimensional quasicrystal, consisting of 12-sided, non-repeating atomic units.

+plus magazine: Penrose tilings
Some articles about the mathematical background of Penrose tilings at the ‘plus magazine’

Create and colour your very own Penrose tiling!