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# Foundations of Crystallography with Computer Applications

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## Book Description

Taking a straightforward, logical approach that emphasizes symmetry and crystal relationships, **Foundations of Crystallography with Computer Applications, Second Edition** provides a thorough explanation of the topic for students studying the solid state in chemistry, physics, materials science, geological sciences, and engineering. It is also written for scientists who want to teach themselves. Computers are an essential part of crystallography, and computer-based exercises are integrated into this book. The material is presented with the goal of creating an understanding of how atoms are arranged in crystals and how crystal systems are related to each other.

See What’s New in the Second Edition:

- Eight new chapters that give detailed crystallographic analyses of one crystal chosen for each crystal system
- Numerous molecular examples and suggestions for student projects
- Coverage of special topics that naturally arise in the treatment of the crystals
- Suggestions for student projects with date that can be found in the free Teaching Subset of the Cambridge Structural Database
- Point group and space group diagrams have been color coded using a new scheme devised by the author to emphasize the change of handedness of the symmetry operations
- All the Starter Programs have been rewritten and improved, and a new one has been added in Chapter 6 on the graphing of intensity vs. 2θ for powder diffraction data
- New appendices contain detailed information about the 32 three-dimensional point groups and the 10 two-dimensional point groups

The book explains the individual entities, such as symmetry operations, and also explains how they fit together in a larger context. Coverage includes lattices, symmetry operations, metric matrices, point groups, space groups, reciprocal lattices, properties of x-rays, and electron density maps, all leading to a formal description of the crystal structures and an interpretation of the published crystallographic data. The author connects general properties such as the piezoelectric effect, compressibility, thermal expansion, and Mosely’s relationship in ordering the elements of the periodic table giving students a thorough foundation in the subject.

Print Versions of this book also include access to the ebook version.

## Table of Contents

**Lattices**Chapter Objectives

Introduction

Two-Dimensional Lattices

Two-Dimensional Basis Vectors and Unit Cells

Two-Dimensional Transformations between Sets of Basis Vectors

Three-Dimensional Basis Vectors, Unit Cells, and Lattice Transformations

Conversion into Cartesian Coordinates

A Crystal: Hexamethylbenzene

A Crystal: Anhydrous Alum

Effects of Temperature and Pressure on the Lattice Parameters

Definitions

Exercises

MATLAB Code: Starter Program for Chapter 1: Graphic of triclinic unit cell

**Unit Cell Calculations**

Chapter Objectives

Introduction

Fractional Coordinates

Plotting Atoms in the Unit Cell

Calculation of Interatomic Bond Distances

Calculation of Interatomic Bond Angles

Area and Volume of the Unit Cell

Summary of Metric Matrix Calculations

Quartz Example

Transformation Matrices

HMB Example

Crystallographic Directions

Crystallographic Planes and Miller Indices

Density

Revisiting Thermal Expansion and Isothermal Compressibility

Definitions

Exercises

MATLAB Code: Starter Program for Chapter 2: Graphic of HMB projection

**Point Groups**

Chapter Objectives

PART I: TWO DIMENSIONS

Introduction

Group Theory

Symmetry Operations

Crystallographic Rotations

Summary of the Two-Dimensional Crystallographic Operations

Two-Dimensional Crystallographic Point Groups

Two-Dimensional Crystal Systems

Two-Dimensional Point Group Tree

Part II: Three Dimensions

Three-Dimensional Point Groups

Three-Dimensional Crystal Systems

Examples of Three-Dimensional Point Groups with Multiple Generators

Three-Dimensional Point Group Trees

Point Group Symmetry and Some Physical Properties of Crystals

Definitions

Exercises

MATLAB Code: Starter Program for Chapter 3: Point Group

Multiplication Table 1

**Space Groups**

Chapter Objectives

Part I: Two Dimensions

Introduction

Two-Dimensional Bravais Lattices

Crystal Systems and the

**G**Matrices

Two-Dimensional Sp ace Groups

Color Coding and Overview of the Two-Dimensional Symbol Diagrams

Recipe for Analyzing a Periodic Pattern

Primitive Cells for

*cm*and

*c*2

*mm*

Two-Dimensional Space Group Tree

Summary of Two-Dimensional Space Groups

Part II: Three Dimensions

Three-Dimensional Bravais Lattices

Three-Dimensional Space Groups

HMB and Space Group No. 2,

*P*1

AA and Space Group No. 150,

*P*321

Caffeine Monohydrate and Two Effective Tools for Relating Symmetry and Structure

Definitions

Exercises

MATLAB Code: Starter Program for Chapter 4: Graphic of populated unit cell and projections

**The Reciprocal Lattice**

Chapter Objectives

Introduction

The Reciprocal Lattice

Relationships between Direct and Reciprocal Lattices

Reciprocal Lattice Calculations for Three Crystals

Relationships between Transformation Matrices

Diffraction Pattern and the Reciprocal Lattice

Three Applications of the Reciprocal Lattice

Definitions

Exercises

MATLAB Code: Starter Program for Chapter 5: Graphic of reciprocal cell superimposed on direct unit cell

**Properties of X-Rays**

Chapter Objectives

Introduction

The Discovery of X-rays

Properties of Waves

X-Ray Spectrum

The X-Ray Tube

X -Ray Diffraction

Synchrotron X-Rays

Definitions

Exercises

MATLAB Code: Starter Program for Chapter 6: Graphic of powder diffraction file

**Electron Density Maps**

Chapter Objectives

Introduction

Scattering by an Electron

Scattering by an Atom

Scattering by a Crystal

Some Mathematical Identities

Structure Factors for Some Crystals

Structure Factors for Centrosymmetric and Noncentrosymmetric Crystals

Electron Density Maps

Major Uses of Structure Factors

Definitions

Exercises

MATLAB Code: Starter Program for Chapter 7: Graphic of atomic scattering curve

**Introduction to the Seven Crystals Exemplifying the Seven Crystal Systems**

Chapter Objectives

Introduction

Crystallographic Data for the Seven Crystal Examples

Presentation of Crystals in Chapters 9 through 15

Color-Coding Point Group and Space Group Diagrams

Crystal Selection Criteria

Student Projects

Distribution of Crystal Structures among Space Groups and Crystal Systems

**Triclinic Crystal System: DL-Leucine**

Chapter Objectives

Introduction

DL-Leucine: Point Group Properties

DL-Leucine: Sp ace Group Properties

DL-Leucine: Direct and Reciprocal Lattices

DL-Leucine: Fractional Coordinates and Other Data for the Crystal Structure

DL-Leucine: Crystal Structure

DL-Leucine: Reciprocal Lattice and

*d*-Spacings

DL-Leucine: Atomic Scattering Curves

DL-Leucine: Structure Factor

Definitions

Exercises

**Monoclinic System: Sucrose**

Chapter Objectives

Introduction

Sucrose: Point Group Properties

Sucrose: Sp ace Group Properties

Sucrose: Direct and Reciprocal Lattices

Sucrose: Fractional Coordinates and Other Data for the Crystal Structure

Sucrose: Crystal Structure

Sucrose: Atomic Scattering Curves

Sucrose: Structure Factor

Definitions

Exercises

**Orthorhombic Crystal System: Polyethylene**

Chapter Objectives

Introduction

Polyethylene: Point Group Properties

Polyethylene: Space Group Properties

Polyethylene: Direct and Reciprocal Lattices

Polyethylene: Fractional Coordinates and Other Data for the Crystal Structure

Polyethylene: Crystal Structure

Polyethylene: Reciprocal Lattice and

*d*-Spacings

Polyethylene: Atomic Scattering Curves

Polyethylene: Structure Factor

Definitions

Exercises

**Tetragonal System: α-Cristobalite**

Chapter Objectives

Introduction

α-Cristobalite: Point Group Properties

α-Cristobalite: Sp ace Group Properties

α-Cristobalite: Direct and Reciprocal Lattices

α-Cristobalite: Fractional Coordinates and Other Data for the Crystal Structure

α-Cristobalite: Crystal Structure

α-Cristobalite: Reciprocal Lattice and

*d*-Spacings

α-Cristobalite: Atomic Scattering Curves

α-Cristobalite: Structure Factor

Definitions

Exercises

**Trigonal Crystal System: H12B12**

^{-2}, 3K^{+}, Br^{-}Chapter Objectives

Introduction

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Point Group Properties

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Space Group Properties

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Direct and Reciprocal Lattices

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Fractional Coordinates and Other Data for the Crystal Structure

Special Topic: Boron Icosahedron

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Crystal Structure

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Reciprocal Lattice and

*d*-Sp acings

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Atomic Scattering Curves

H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{-}: Structure Factor

Special Topic: H

_{12}B

_{12}

^{-2},3K

^{+},Br

^{- }Isotypic Crystal Structures

Definitions

Exercises

**Hexagonal System: Magnesium**

Chapter Objectives

Introduction

Magnesium: Point Group Properties

Magnesium: Space Group Properties

Magnesium: Direct and Reciprocal Lattices

Magnesium: Fractional Coordinates and Other Data for the Crystal Structure

Magnesium: Crystal Structure

Magnesium: Reciprocal Lattice and

*d*-Spacings

Magnesium: Atomic Scattering Curve

Magnesium: Structure Factor

Definitions

Exercises

**Cubic System: Acetylene**

Chapter Objectives

Introduction

Acetylene: Point Group Properties

Acetylene: Space Group Properties

Acetylene: Direct and Reciprocal Lattices

Acetylene: Fractional Coordinates and Other Data for the Crystal Structure

Acetylene: Crystal Structure

Acetylene: Reciprocal Lattice and

*d*-Spacings

Acetylene: Atomic Scattering Curves

Acetylene: Structure Factor

Definitions

Exercises

References,

Appendix 1: Definitions,

Appendix 2: The Ten Two-Dimensional Point Groups,

Appendix 3: The Thirty-Two Three-Dimensional Point Groups,

Index