ABSTRACT
Above is an electron density map of hexamethylbenzene (HMB). e hydrogen atoms are omitted.
CONTENTS Chapter Objectives 318 7.1 Introduction 319 7.2 Scattering by an Electron 319 7.3 Scattering by an Atom 319
7.3.1 Atomic Scattering Factor 320 7.4 Scattering by a Crystal 322
7.4.1 Phase Angle 323 7.4.2 Amplitude 324 7.4.3 Structure Factor 326
7.5 Some Mathematical Identities 327
• Scattering by a single electron
• Scattering by a single atom and the atomic scattering factor
• Scattering by a crystal and the structure factor
• Structure factors and extinctions for cubic Bravais lattices
• Relationship of primitive, face-centered, and body-centered unit cells with their corresponding reciprocal unit cells
• Structure factors for unit cells with one or more dierent atoms
• Examples of crystals with real and complex structure factors
7.6 Structure Factors for Some Crystals 327 7.6.1 Simple Cubic, Polonium 328 7.6.2 Body-Centered Cubic, Chromium 332 7.6.3 Face-Centered Cubic, Copper 333 7.6.4 Crystal with Two Elements, NaCl 337 7.6.5 Crystal with Two Elements, Sphalerite 338 7.6.6 Structure Factors for Glide Planes 339 7.6.7 Structure Factors for Screw Axes 341 7.6.8 Structure Factor at the Origin 342
7.7 Structure Factors for Centrosymmetric and Noncentrosymmetric Crystals 342 7.7.1 Partitioning the Space Groups 342
7.7.1.1 Centrosymmetric Space Groups 343 7.7.1.2 Noncentrosymmetric Space Groups 343
7.7.2 Inversion Points and Structure Factors 343 7.7.3 Hexamethylbenzene (HMB): Example of a Centrosymmetric Crystal 344 7.7.4 Generalized Calculation of Structure Factors 345
7.8 Electron Density Maps 345 7.8.1 Friedel’s Law 345 7.8.2 Fourier Series 348
7.8.2.1 One-Dimensional Fourier Series 348 7.8.2.2 ree-Dimensional Fourier Series 351
7.9 Major Uses of Structure Factors 352 7.9.1 Structure Factors for the Crystal Model 352 7.9.2 Observed Data and the Phase Factor 354 7.9.3 Comparison of Experimental and Model Structure Factors 354 7.9.4 Calculation of Electron Density Maps from Experimental Data 354
Denitions 355 Exercises 355 MATLAB Code: Starter Program for Chapter 7: Graphic of Atomic Scattering Curve 357
• Structure factors and extinctions for glides and screws
• Structure factors for centrosymmetric crystals in general
• Example of structure factors using HMB
• Generalized calculation of structure factors
• Friedel’s law
• Fourier series and electron density maps in one dimension
• Electron density map for HMB
• Use the structure factors to compare the crystal model with the diraction data
• Construct electron density maps from diraction data
• Use R-value or the discrepancy index as a measure of the correctness of the structure
7.1 INTRODUCTION Diraction occurs when x-rays, neutrons, or electrons interact with matter. It is the result of both interference and scattering phenomena. is book emphasizes x-rays. In Chapter 6, the spatial distribution of the diracted x-rays was discussed in terms of Bragg’s law and the unit cell. e diraction spots vary in intensity. e scattering and interference due to the individual atoms or ions located within the unit cell contribute to the variation in intensity.
