Modern Diffraction Methods E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 965

Veröffentlichungsjahr: 2013

Beschreibung

The role of diffraction methods for the solid-state sciences has been pivotal to determining the (micro)structure of a material. Particularly,
the expanding activities in materials science have led to the development of new methods for analysis by diffraction. This book offers an authoritative overview of the new developments in the field of analysis of matter by (in particular X-ray, electron and neutron) diffraction. It is composed of chapters written by leading experts on 'modern diffraction methods'. The focus in the various chapters of this book is on the current forefront of research on and applications for diffraction methods. This unique book provides descriptions of the 'state of the art' and, at the same time, identifies avenues for future research.

The book assumes only a basic knowledge of solid-state physics and allows the application of the described methods by the readers of the
book (either graduate students or mature scientists).

Leseprobe

Table of Contents

Related Titles

Title Page

Copyright

Preface

About the Editors

Part I: Structure Determination

Chapter 1: Structure Determination of Single Crystals

1.1 Introduction

1.2 The Electron Density

1.3 Diffraction and the Phase Problem

1.4 Fourier Cycling and Difference Fourier Maps

1.5 Statistical Properties of Diffracted Intensities

1.6 The Patterson Function

1.7 Patterson Search Methods

1.8 Direct Methods

1.9 Charge Flipping and Low-Density Elimination

1.10 Outlook and Summary

References

Chapter 2: Modern Rietveld Refinement, a Practical Guide

2.1 The Peak Intensity

2.2 The Peak Position

2.3 The Peak Profile

2.4 The Background

2.5 The Mathematical Procedure

2.6 Agreement Factors

2.7 Global Optimization Method of Simulated Annealing

2.8 Rigid Bodies

2.9 Introduction of Penalty Functions

2.10 Parametric Rietveld Refinement

References

Chapter 3: Structure of Nanoparticles from Total Scattering

3.1 Introduction

3.2 Total Scattering Experiments

3.3 Structure Modeling and Refinement

3.4 Examples

3.5 Outlook

References

Part II: Analysis of the Microstructure

Chapter 4: Diffraction Line-Profile Analysis

4.1 Introduction

4.2 Instrumental Broadening

4.3 Structural, Specimen Broadening

4.4 Practical Application of Line-Profile Analysis

4.5 Conclusions

References

Chapter 5: Residual Stress Analysis by X-Ray Diffraction Methods

5.1 Introduction

5.2 Principles of Near-Surface X-Ray Residual Stress Analysis

5.3 Near-Surface X-Ray Residual Stress Analysis by Advanced and Complementary Methods

5.4 Final Remarks

References

Chapter 6: Stress Analysis by Neutron Diffraction

6.1 Introductory Remarks

6.2 Fundamentals of the Technique

6.3 Instrumentation

6.4 Capabilities

6.5 Examples

References

Chapter 7: Texture Analysis by Advanced Diffraction Methods

7.1 Introduction and Background

7.2 Synchrotron X-Rays

7.3 Neutron Diffraction

7.4 Electron Diffraction

7.5 Comparison of Methods

7.6 Conclusions

Acknowledgments

References

Chapter 8: Surface-Sensitive X-Ray Diffraction Methods

8.1 Introduction

8.2 X-Ray Reflectivity

8.3 Bragg Scattering in Reduced Dimensions (Crystal Truncation Rod Scattering)

8.4 Grazing Incidence X-Ray Diffraction

8.5 Experimental Geometries

8.6 Trends

Acknowledgments

References

Chapter 9: The Micro- and Nanostructure of Imperfect Oxide Epitaxial Films

9.1 The Diffracted Amplitude and Intensity

9.2 The Correlation Volume

9.3 Lattice Strain

9.4 Example

9.5 Strain Gradients

9.6 Conclusions

References

Part III: Phase Analysis and Phase Transformations

Chapter 10: Quantitative Phase Analysis Using the Rietveld Method

10.1 Introduction

10.2 Mathematical Basis

10.3 Applications in Minerals and Materials Research

10.4 Summary

Acknowledgments

References

Chapter 11: Kinetics of Phase Transformations and of Other Time-Dependent Processes in Solids Analyzed by Powder Diffraction

11.1 Introduction

11.2 Kinetic Concepts

11.3 Tracing the Process Kinetics by Powder Diffraction

11.4 Mode of Measurement:In Situ versus Ex Situ versus Methods

11.5 Types of Kinetic Processes and Examples

11.6 Concluding Remarks

References

Part IV: Diffraction Methods and Instrumentation

Chapter 12: Laboratory Instrumentation for X-Ray Powder Diffraction: Developments and Examples

12.1 Introduction: Historical Sketch

12.2 Laboratory X-Ray Powder Diffraction: Instrumentation

12.3 Examples

Acknowledgments

References

Chapter 13: The Calibration of Laboratory X-Ray Diffraction Equipment Using NIST Standard Reference Materials

13.1 Introduction

13.2 The Instrument Profile Function

13.3 SRMs, Instrumentation, and Data Collection Procedures

13.4 Data Analysis Methods

13.5 Instrument Qualification and Validation

13.6 Conclusions

References

Chapter 14: Synchrotron Diffraction: Capabilities, Instrumentation, and Examples

14.1 Introduction

14.2 The Underlying Physics of Synchrotron Sources

14.3 Diffraction Applications Exploiting High Source Brilliance

14.4 High Q-Resolution Measurements

14.5 Applications of Tunability: Resonant Scattering

14.6 Future: Ultrafast Science and Coherence

References

Chapter 15: High-Energy Electron Diffraction: Capabilities,Instrumentation, and Examples

15.1 Introduction

15.2 Instrumentation

15.3 Electron Diffraction Methods in the TEM

15.4 Summary and Outlook

Acknowledgment

References

Chapter 16: In Situ Diffraction Measurements: Challenges, Instrumentation, and Examples

16.1 Introduction

16.2 Instrumentation and Experimental Challenges

16.3 Examples

Acknowledgment

References

Index

Related Titles

Che, M., Vedrine, J. C. (eds.).

Characterization of Solid Materials and Heterogeneous Catalysts

From Structure to Surface Reactivity

2012

ISBN: 978-3-527-32687-7

Friedbacher, G., Bubert, H. (eds.)

Surface and Thin Film Analysis

A Compendium of Principles, Instrumentation, and Applications

Second, Completely Revised and Enlarged Edition 2011

ISBN: 978-3-527-32047-9

Zolotoyabko, E.

Basic Concepts of Crystallography

2011

ISBN: 978-3-527-33009-6

Bennett, D. W.

Understanding Single-Crystal X-Ray Crystallography

2010

ISBN: 978-3-527-32677-8

The Editors

Prof. Dr. Eric J. Mittemeijer

Max Planck Institute for

Intelligent Systems

Heisenbergstraße 3

70569 Stuttgart

Germany

and

University of Stuttgart

Institute for Materials Science

Heisenbergstraße 3

D-70569 Stuttgart

Germany

Dr. Udo Welzel

Max Planck Institute for

Intelligent Systems

Heisenbergstraße 3

70569 Stuttgart

Germany

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Preface

Materials science and disciplines such as the physics and chemistry of solids owe a lot to the diffraction experiments performed, now precisely one century ago, by Friedrich, Knipping, and von Laue, demonstrating that crystals are characterized by a translationally periodic arrangement of atoms in three-dimensional space.1 The determination of the ideal(ized) arrangement of the atoms in a crystal, as usually represented by the filling of the unit cell, remained a cumbersome task for the scientist (crystallographer) until the last quarter of the twentieth century, which period of time is characterized by major methodological developments for which also Nobel prizes were given to emphasize the scientific and technological importance of diffraction methods.2, 3

It was realized rather soon after the discovery of the diffraction of X-rays that real materials are far from perfect regarding their atomic arrangement. The crystals can be small (i.e., not infinitely large) and can contain defects in the atomic arrangement. Scherrer (1918) and Dehlinger and Kochendörfer (1927, 1939) performed the first seminal works devoted to the determination of crystal(lite) size and mistakes in crystals, respectively, as revealed by the occurrence of the broadening of diffraction lines. Already in 1927, a textbook by Glocker was published, in which the essential elements of the (X-ray) diffraction method for (engineering) stress determination in materials, on the basis of the shift of the diffraction-line positions, were presented.

Through the years, the role of the diffraction methods for the solid state sciences has been of undiminished pivotal nature.2 There may have occurred a change of emphasis: it is obvious that the field of pure crystallography has been given less emphasis in the recent years (as illustrated by the decreased number of Chairs at universities devoted to this field of science), but this is overcompensated by the expanding activities in materials science in which diffraction analyses, in particular, on the basis of newly developed methods that are focused on in this book, play a cardinal role in the characterization of the so-called microstructure. “Modern diffraction methods” have a lot to do (but not only) with the investigation of the imperfect crystalline state: the “crystal imperfection.”

With this book, we have brought together contributions written by renown scientists, specialists in the field of diffraction analyses, invited by us, who are beyond any doubt capable of reviewing in an authoritative manner a certain part of the field of the diffraction analysis of materials. The focus in the various chapters of this book on the current front level of the research on and applications of diffraction methods has led to descriptions of the “state of the art” and at the same time has identified avenues for future research. As far as we know, a book like this is unique.

Part I of the book deals with the extraction of parameters of the (idealized) crystal structure from diffraction patterns. The classical and still important way to do this is based on the X-ray diffraction pattern of a single crystal. The current portfolio of methods utilized to this end is exposited in the first chapter and culminates with the presentation of recent developments for solving/circumventing the phase problem, as “charge flipping” in combination with “low-density elimination.”

The advent of the Rietveld-refinement procedure in 1967 for the refinement of the crystal-structure parameters of a material on the basis of a powder-diffraction pattern (i.e., not recorded from a single crystal but from a polycrystalline specimen (possibly but not necessarily a powder)) has led to a renaissance of the powder-diffraction method. Nowadays, in this way, even direct determination of the crystal structure has become feasible. A modern guide to the application of Rietveld-refinement procedures is provided in Chapter 2. In other chapters (notably Chapters 4 and 10), recent developments in the use of the Rietveld-refinement approach to determine the parameters characterizing the crystal imperfection (as the crystallite(domain) size and the microstrain, as, for example, pertaining to some dislocation density) and the relative amounts of phases in a material are outlined.

Total scattering analysis (i.e., considering both the Bragg and diffuse scattering contributions) allows characterization of the atomic arrangement in amorphous materials, in disordered crystalline materials, and in nanocrystalline particles via determination of the Pair Distribution Function. The possibilities of this method, originally applied to liquids and glassy materials, now for the structural characterization of nanocrystalline particles, are addressed in Chapter 3.

Part II is devoted to the analysis of the microstructure of materials by diffraction methods. Crystal imperfection can be assessed on the basis of the diffraction-line broadening induced by the structural defects. This is dealt with extensively in Chapter 4, in which the classical line-profile decomposition methods are reviewed and then compared with the more modern and promising line-profile synthesis methods, leading to a set of conclusions with distinct consequences for the current practice of often ill-considered application of line-profile analysis methods.

In “materials engineering” and also in “materials science,” pronounced interest exists in the determination of the (macro)stress (components) acting on a component or specimen (e.g., thin films employed in microelectronic devices). For engineering applications, (residual) internal stresses can be detrimental or favorable; from a fundamental, scientific point of view, the development and relaxation of such stresses can teach us a lot of intrinsic material behavior. X-ray and neutron-diffraction methods have been developed for such (macro)stress analysis. X-ray diffraction is utilized for analyzing the state of stress in surface adjacent regions (at most, a few micrometers thick) of the specimen/component, whereas neutron diffraction is appropriate for analyzing the state of stress in the bulk of the component considered (i.e., the penetration depth of the neutrons is several centimeters). Chapters 5 and 6 provide an overview of the current possibilities in diffraction-stress analysis, including the analysis of stress-depth profiles.

The preferred orientation, called the texture, of a polycrystalline material can be of decisive importance for macroscopic properties of the specimen/component considered, for example, with a view to the macroscopic mechanical behavior, due to the intrinsic elastic anisotropy (of a single crystal of the same material). The classical and still important way to determine a pole figure and, from a number of pole figures, the orientation distribution function, by means of X-ray diffraction, utilizing an X-ray pole-figure goniometer, is well known, routinely applied, and not considered in this book. Instead, Chapter 7 focuses on the recently developed methods to determine orientation distributions in polycrystalline materials: synchrotron diffraction (for small volumes of material), neutron diffraction (also for the determination of magnetic pole figures), electron diffraction (in a transmission electron microscope (TEM)), and electron backscatter diffraction (from the surface of a specimen in a scanning electron microscope) (the last two techniques provide (i.e., keep) spatial information (misorientation with neighboring crystals).

Diffraction by atomic structures (developing) at and close to surfaces has been a topic of strongly growing importance in the recent years. These recent developments, in particular, involving surface X-ray diffraction, grazing-incidence X-ray diffraction for the analysis of surface adjacent regions, and also X-ray reflectivity, including their requirements/limitations, are dealt with in Chapter 8.

The final chapter of Part II, Chapter 9, deals with the analysis of epitaxial thin (oxide) films that contain defects. The chapter provides an example of the possibilities of “reciprocal space mapping,” including the determination of strain gradients.

Part III begins with a chapter dealing with a classical topic in powder-diffraction analysis: which phases and how much of these phases are present in the multiphase specimen investigated. As indicated above, here only the recent developments of Rietveld-refinement approaches, that is, the whole diffraction pattern is utilized, are presented and critically discussed (Chapter 10). In classical methods, not dealt with here, the quantitative phase analysis is based on individual peaks.

The investigation of processes in materials as function of time and temperature, “material dynamics,” belongs to the core of materials science. While traditional techniques, such as dilatometry and calorimetry, have been used frequently for quantitative analysis of the kinetics of phase transformations in materials, (X-ray) diffraction, to this end, has found only limited application until now. The inclusion of Chapter 11 in this book has been meant to provide an overview of the great, current methodological possibilities to analyze phase-transformation kinetics on the basis of diffraction measurements. As discussed in the chapter, both an acceptable kinetic description has to be adopted (this goes beyond simply, and often unvalidated, assuming Johnson-Mehl-Avrami-Kolmogorov kinetics) and an appropriate (experimental) approach has to be chosen (e.g., peak position, peak width, or peak area traced as function of time and temperature).

The final Part IV focuses on the modern techniques for diffraction analysis and, especially, their instrumental realization. The availability of laboratory facilities for (powder) diffraction is indispensable: X-ray sources such as storage rings and free electron lasers (synchrotron diffraction) and neutron sources such as reactors and spallation sources (neutron diffraction) are usually not directly deployable in research (usually access to these, relatively few, rather gigantic facilities is granted only after having “survived” in competition with applications from other scientists worldwide). There has been a continuous development in the laboratory instrumentation for X-ray diffraction, and indeed, as an example, one of the most modern possibilities for high brilliance and angular resolution, which can be installed “in-house,” is provided by the combination of a rotating anode and an X-ray mirror, leading to incident intensities of the same order of magnitude as offered by a synchrotron beamline of the second generation. This and other modern laboratory diffraction facilities, such as those provided by parallel-beam geometries, polycapillary collimators (X-ray lenses), and two-dimensional detectors, have been discussed in Chapter 12.

A topic of underestimated importance involves the calibration of the (laboratory) diffraction facilities. Although numerical calculation of the instrumental aberrations is a modern trend, it has to be recognized that a significant number of aberrations resist successful modeling (even more so if they occur in combination, as can happen in case of nonideal alignment of the instrument to be applied). As a remedy, and as the procedure to be advised to determine the diffraction consequences of the instrumental aberrations, calibration, and thus correction for the effect of instrumental aberrations in the recorded diffraction signal, can be achieved by application of “perfect” (but usually polycrystalline) materials. The high level of accuracy thus attainable is shown in Chapter 13, which thereby implies that low accuracy in diffraction measurements, as apparent in many published data of present day, is the result of work by careless experimentalists.

The still increasing use of synchrotron radiation for diffraction experiments to a large extent is due to its extraordinary brilliance and also advantages such as the possibility to study material-process dynamics, occurring on timescales of milliseconds and less (exploiting the pulsed nature of synchrotron radiation), and the strong beam collimation, which allows surface diffraction (as discussed in detail in Chapter 8). In the near future, certainly the application of coherent diffraction imaging (e.g., for the analysis of (the morphology and strains of) nanoparticles) will draw even much more attention. A survey of such and other applications of synchrotron radiation and the instrumental aspects is provided in Chapter 14.

Electron diffraction as a tool for the analysis of the inner structure of materials has, at least at first sight, been of lesser importance than X-ray diffraction. The association induced with the term transmission electron microscopy undoubtedly in the first place concerns the diffraction-contrast images obtained with conventional TEM and the (flawed) images obtained by applying high resolution (TEM). However, the electron-diffraction patterns obtained in such imaging machines yield a wealth of structural information that has led to the development of a number of techniques, making in recent years such electron-diffraction analysis a very powerful tool for materials characterization. Therefore, it was a strong wish of the editors to include a chapter devoted to this topic in this book. A number of such electron-diffraction techniques and their instrumentation, and which techniques can generally be applied to very small specimen volumes and individual nanoparticles, are dealt with in Chapter 15, such as (variants of) convergent beam electron diffraction and large-angle rocking-beam electron diffraction, allowing for the determination of crystal structures, valence electron densities, and charge distributions and the analysis of crystal defects (such as dislocations, stacking faults), strain mapping, and, with a view to the remark made in the previous paragraph, even coherent diffraction imaging.

The final chapter of Part IV deals with the application of in situ diffraction measurements. This usually requires the availability of well-calibrated (temperature, diffraction angle) heating and cooling devices mounted on the diffractometer. In particular, if a transformation in the solid state runs fast, the instrumental and measurement conditions can be very demanding. Such considerations are part of Chapter 16.

Completing this book has required a lot of energy, time, and patience not only from us, as the editors, but also, in particular, from our authors. We are deeply grateful to all contributors. We understand, or better, we know on the basis of own experience, how difficult it can be to find the time to write an overview paper next to all daily duties. We also understand how frustrating it is, after having submitted your own chapter in final form, to wait until the last contribution has been put in final form, so that the book can be published. Now, looking at the result, we can only be satisfied: the contents and the quality of the book reflect rather well what we originally intended. Patience and perseverance have paid off.

Stuttgart

Eric J. Mittemeijer

February 2012

Udo Welzel

Notes

1Interestingly, at the time, the discoverers thought that the diffraction pattern produced by the crystal was the result of characteristic radiation emitted by the atoms upon being hit by the incident (primary) X-rays…(see also the account in M. Eckert, Z. Kristallogr. (2012), 27–35, also published in Acta Cryst. (2012), 30–39).

2It is important to note here that the, at the time of writing this preface, last Nobel Prize for chemistry, that is, for the year 2011, has been awarded to D. Schechtman for his discovery of the so-called quasicrystals, characterized by geometric ordering devoid of (long range) translational symmetry, on the basis of the analysis of electron-diffraction patterns. This is just another indication of the importance of diffraction analysis for solid-state science, until and beyond the present day (D. Schechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. (1984), 1951–1953).

3Also, the famous discovery of the double helix structure of DNA, thereby exposing the replication mechanism of genetic information, by Watson and Crick in 1953, was only possible on the basis of the X-ray diffraction experiments performed by (Rosalind) Franklin and Wilkens (see J.D. Watson, The double helix, Weidenfeld and Nicolson, 1968, London). This work was awarded with the Nobel Prize for medicine and physiology in 1962.

About the Editors

Welzel Udo Siegfried Welzel was born in 1972 in Selb, Germany. He studied physics, with specialization materials science, at the University of Bayreuth and acquired his diploma in 1998. After joining the Max Planck Institute for Metals Research in Stuttgart, he obtained his Ph.D. degree from the University of Stuttgart in 2002. From 2002 until 2012, he served as a scientific staff member in the Department of Prof. Eric Mittemeijer and, from 2005 until 2012, also as the head of the central scientific facility (i.e., a service laboratory) “X-ray diffraction” at the Max Planck Institute for Intelligent Systems (formerly Max Planck Institute for Metals Research). In February 2012, he joined the Robert Bosch GmbH in Schwieberdingen, where he now serves in the Automotive Electronics– Engineering Assembly and Interconnect Technology Department (AE/EAI3).

Part I

Structure Determination

Chapter 1

Structure Determination of Single Crystals

Sander van Smaalen

1.1 Introduction

Many crystalline materials possess translational symmetry. This property implies that the positions of the atoms can be obtained from the positions of a few atoms in a small volume (the unit cell), which is periodically repeated in space. Thus, the crystal structure is completely characterized by the translational symmetry — as given by the six lattice parameters — together with the positions of the atoms within one unit cell (Figure 1.1a) [1]. The goal of the procedure of structure determination is to obtain the atomic positions in the unit cell and the lattice parameters from a diffraction experiment.

Figure 1.1 (a) A unit cell with edge lengths a, b, and c and angles between edges with α, β, and γ. (b) The scattering vector S defines the difference between the directions of propagation of the primary and scattered waves.

X-rays are scattered by matter (Figure 1.1b). A consequence of translational symmetry is that scattered rays are only obtained for directions k corresponding to scattering angles 2θ as followed from Bragg's law. Furthermore, each Bragg reflection requires a specific orientation of the crystal with respect to the directions of the primary and scattered X-ray beams. Both properties depend on the lattice parameters, while different Bragg reflections are distinguished on the basis of a unique indexing with three integral numbers (h k l). The other way around, knowledge of the orientations of the crystal in conjunction with the directions of the diffracted beams for a sufficiently large number of Bragg reflections allows the determination of the lattice parameters and the indices of each Bragg reflection.

The amplitude and phase of the diffracted beam are the second unique property of each Bragg reflection. Their values depend on the structure of one unit cell of the crystal and the indices (h k l) of the Bragg reflection. Knowledge of the amplitudes and phases of many Bragg reflections allow the determination of the positions of the atoms in the crystal by the simple computational procedure of Fourier transform (Section 1.2).

The diffraction experiment provides for each Bragg reflection the orientation of the single crystal and the direction and intensity of the diffracted beam. This information is sufficient to compute the lattice parameters of the crystal and the indices of the Bragg reflections, but it does not allow computation of the crystal structure by the method of Fourier transform. The reason is that the intensity of an X-ray beam is proportional to the square of the amplitude of this electromagnetic wave, while it does not contain information about the phase. This is the crystallographic phase problem. Methods of structure solution aim at finding the crystal structure on the basis of the measured intensities Iobs(h, k, l) of the Bragg reflections. Solving the crystal structure implies solving the phase problem, as the phases of the Bragg reflections can be computed from the structure model and the structure model is obtained by Fourier transform of the Bragg reflections, if both amplitudes and phases of the Bragg reflections are known.

Methods of structure solution depend on a few fundamental properties of matter, which is introduced in Section 1.3. Furthermore, the methods of structure solution discussed in this chapter require the availability of the diffracted intensities of a sufficient number of Bragg reflections, as they can be measured, for example, by single-crystal X-ray diffraction. On the other hand, diffraction by microcrystalline powders results in diffracted intensity as a function of the scattering angle 2θ. Different Bragg reflections with equal or nearly equal scattering angles cannot be resolved, with the consequence that intensities of individual Bragg reflections cannot be obtained by powder diffraction for at least part of the reflections. Methods of structure solution have been developed, which account for the peculiarities of powder diffraction; these are discussed in Chapter 2.

Twinning is another feature, which prevents the measurement of intensities of individual Bragg reflections. Instead, apparent Bragg reflections of a twinned crystal may be the sum of two or more different reflections. For formal reasons, the methods of structure solution discussed in this chapter do not apply to this kind of diffraction data. Nevertheless, they may often give the correct solution (e.g., for inversion twins of a noncentrosymmetric crystal) or may lead to some average structure, which then provides the essential clue for the construction of the true structure model. However, twinning may also prohibit structure solution. Special problems and solutions related to twinning are not discussed in this chapter.

Apart from the crystal structure, intensities of Bragg reflections depend on a series of other effects, such as the geometry of the diffraction experiment, the polarization of the radiation, and absorption of X-rays by the sample. Different reflections may require different correction factors accounting for these effects, but these can be computed without knowledge of the crystal structure. It is understood that Iobs(h, k, l) has been obtained from the real measured intensity values by the application of these correction factors. Other, trivial dependencies are the proportionality of intensities of Bragg reflections to the intensity of the primary beam, to the volume of the crystal, and to the time of the measurement. Factors such as these affect all Bragg reflections in the same way, and they go into the scale factor (Section 1.5).

This chapter concentrates on structure determination by single-crystal X-ray diffraction. However, the same or similar methods can also be successful when applied to neutron diffraction or electron diffraction data.

1.2 The Electron Density

The elastic scattering of X-rays is determined by the electron density distribution in space. The periodicity of crystal structures determines that scattering is concentrated in directions represented by scattering vectors equal to lattice vectors of the reciprocal lattice of the crystal,

1.1

The integers (h k l) are used for indexing of the Bragg reflections. Non-Bragg scattering is experimentally eliminated by subtracting a background measured close to each Bragg reflection from the measured Bragg intensity, eventually leading to Iobs(h, k, l) (Section 1.1).

The amplitudes and phases of the scattered waves of the Bragg reflections are given by the Fourier coefficients of the periodic electron density ρ(x) as

1.2

The integration is performed over one unit cell. The structure factors F(h, k, l) are obtained in units of the amount of scattering by one electron, with F(0, 0, 0) equal to the number of electrons in the unit cell. The electron density can be computed by inverse Fourier transform of the structure factors,

1.3

The volume of the unit cell is Vcell and the summation runs over all Bragg reflections.

The electron density possesses an infinite number of Fourier coefficients F(h, k, l), obtained by enumerating h, k, and l over all integers. However, the values of the magnitudes |F(h, k, l)| tend to 0 for increasing length of the scattering vector (Eq. 1.1),

1.4

where θ is half the scattering angle (Figure 1.1b) and λ is the wavelength of the radiation used in the diffraction experiment. In practice, it thus appears sufficient to include in the summation of Eq. (1.3) the structure factors of all Bragg reflections with lengths of scattering vectors less than some upper limit. The resolution of diffraction data is often described by the maximum value of or by the minimum value reached for

1.5

Another popular designation for resolution is the maximum scattering angle, from which the actual resolution of the data can be obtained if the wavelength of the radiation is known (Eq. 1.5).

Table 1.1 Resolution of Diffraction Data in Dependence of the Maximum Scattering Angle for Mo–Kα and Cu–Kα Radiation

Figure 1.2 Molecular structure of paracetamol, C8H9NO2.

Figure 1.3 Fourier map of paracetamol in dependence on the resolution of the diffraction data. Sections are shown of 6 × 6 Å2 through the phenyl ring. Contour lines are at of the respective maximum densities, with dashed lines for negative values and long-dashed lines for the value 0. (a) 1.020 −1, (b) 0.704 −1, (c) 0.617 −1, (d) 0.481 −1, (e) 0.364 −1, and (f) 0.244 −1. See Table 1.1 for details.

Electron densities are strictly positive maps because they give the number of electrons present at each point in the unit cell. On the other hand, the example of paracetamol shows that Fourier maps contain regions of negative “density” (Table 1.1 and Figure 1.3). These regions occur as the result of series termination effects. Taking the minimum value of the density as a measure for the magnitude of the noise in the map, it appears that the latter is not a simple function of resolution but that it is rather the ratio between ρmax and |ρmin| that increases on increasing resolution (Table 1.1). Better resolutions thus improve the likeliness to find atoms by way of local maxima in the map, but they still suffer from noise in the low-density regions. Strictly positive Fourier maps (Eq. 1.3) are obtained for resolutions dmin better than approximately 0.1  or with [sin(θ)/λ]max larger than 5 −1 [3].

1.3 Diffraction and the Phase Problem

Structure factors, F(h, k, l), are in general complex numbers, that can be represented by real and imaginary parts or by amplitudes and phases, according to Eq. (1.2)

1.6

The two notations of complex numbers are related by

1.7

The diffraction experiment measures the intensities of the Bragg reflections. The intensity of radiation is proportional to the square of the amplitude. Observed structure factor amplitudes are thus defined as

1.8

The diffraction experiment does not contain information on the phases ϕ(h, k, l). This is the crystallographic phase problem. It prevents calculation of the electron density of a crystal from its measured diffraction by Fourier inversion (Eq. 1.3).

Methods of structure solution aim at finding the phases of the structure factors from the knowledge of the measured diffraction intensities (Eq. 1.8). Since the amplitude does not contain information about the phase of a complex number, additional information must be taken into account for any method to work. This information is the following two properties of the electron density:

1. the electron density is everywhere positive;

2. the electron density has atomic character.

Randomly chosen values for the phases of the reflections lead to equal probabilities for any point of the Fourier map (Eq. 1.3) being positive or negative. Requiring a positive map thus provides a severe restriction on the acceptable combinations of phases of reflections. However, by itself it does not provide a solution to the phase problem, and finding the correct phases is the subject of the various methods of structure solution.

Atoms are at the basis of our understanding of matter. Translated to electron densities, it means that the latter can be described as the sum of atomic densities, ρμ(x), each centered on a different position in space: the position xμ of atom μ. This is the independent atom model (IAM) for the crystal structure, and the electron density

1.9

is an excellent approximation to the true density, requiring only small corrections because of the effects of chemical bonding [4]. The summation extends over all atoms N in the unit cell. The Fourier transform of the IAM can be evaluated term by term, resulting in the calculated structure factors, Fcal(h, k, l), of the structure model (Eqs. (1.2) and (1.9))

1.10

where fμ(|H|) is the atomic form factor. The isotropic temperature parameter Bμ is a positive quantity and should be replaced by the tensor of anisotropic ADPs in accurate models. Equation (1.10) shows that the intensities and phases of Bragg reflections are determined by the positions and ADPs of the atoms in the unit cell. Different atomic form factors are required for different chemical elements, but they do not depend on the positions of the atoms.

The correct structure model should give amplitudes of the calculated structure factors, |Fcal(h, k, l)| equal to the corresponding observed structure factors. Because the latter are proportional to the primary intensity of the diffraction experiment, equality is only obtained after the application of a, yet unknown, scale factor K (Eqs. (1.8) and (1.10))

1.11

Various methods exist, which solve the phase problem. Different methods rely in different ways on the two properties of the electron density as mentioned above. Here, we discuss the fundamental principles that are at the foundations of these methods. It should give an idea about the possibilities and pitfalls of each method and should provide a guide for selecting the most promising method for each problem.

The Patterson function can be computed from the diffracted intensities and thus is directly accessible from the experiment (Section 1.6). Structure models on the basis of the Patterson function can be determined by Search methods, whereby the orientations and positions of known fragments of molecules are determined on the basis of their matching with the experimental Patterson function (Section 1.7). Direct methods directly determine the phases of the reflections on the basis of the measured intensities (Section 1.8). They rely on the statistical properties of the structure factors, as these follow from the properties of positivity and atomic character of the density (Section 1.5). Charge flipping and low-density elimination are modern methods that involve manipulations of the density (Section 1.9), simultaneously solving for the reflection phases and the electron density. Other methods aim at finding a structure model (positions of the atoms in the unit cell) without solving the phase problem first. They are particularly important for solving crystal structures from powder diffraction and in the case of low-resolution data sets as are usually the only available data for protein crystals.

1.4 Fourier Cycling and Difference Fourier Maps

Methods of structure determination often result in approximate values for the phases of reflections. Alternatively, they may provide a partial structure model. The phases of the calculated structure factors of the partial model (Eq. 1.2) can then serve as an approximation to the phases of the true structure factors. Fourier maps based on measured amplitudes |Fobs(h, k, l)| and approximate phases of the reflections often allow an interpretation that leads to a structure model that is better than the (partial) structure model used for generating the phases. The new model can then be used to compute better phases (Eq. 1.8) and an improved Fourier map (Eq. 1.3). Repeating this procedure until convergence may eventually lead to a complete structure model of sufficient accuracy to initiate a successful refinement of the crystal structure. Structure refinements can often improve the atomic coordinates and ADPs of a partial structure model. A fruitful procedure for model completion can thus involve the alternate application of Fourier cycling and structure refinement.

Fourier maps may fail to disclose the positions of light atoms in the presence of heavy atoms. This failure may be due to the fact that the expected local maximum of the light atom is obliterated by the noisy features of the density of the heavy atom. In other cases, light atoms close to heavy atoms may not constitute local maxima in the density, and thus cannot be identified in Fourier maps by principle. This particularly applies to hydrogen atoms covalently bonded to atoms such as carbon, nitrogen, or oxygen [5]. The Fourier map of paracetamol is a good example of the latter feature, where the hydrogen atoms are apparently invisible at all resolutions (Figure 1.3).

The absence of local maxima for light atoms in Fourier maps does not imply that the diffraction data would not contain information on the location of these atoms; it only demonstrates that Fourier maps are not the proper method to visualize light atoms next to heavy atoms. A solution to this problem is provided by the difference Fourier map. The latter is calculated with the difference between Fobs(h, k, l) and of a partial structure model replacing F(h, k, l) in Eq. (1.3):

1.12

Unlike the electron density, the electron difference density Δρ(x) should contain regions of both positive and negative values. Positive values indicate regions of density missing in the model and negative values indicate regions of too much density in the model. The latter usually occur because of inaccuracies in the positions of the atoms or yet incorrectly assigned values of ADPs.

As an example, we consider the structure of paracetamol (Section 1.2). Structure refinements have been performed against data at several resolutions of a partial structure model incorporating all nonhydrogen atoms. With the exception of the lowest resolution, Fourier maps are indistinguishable from those in Figure 1.3, and they have not been further analyzed. Minor differences have been found for the lowest resolution, which are due to variations of the structure model and the scale factor in the refinements when using a small selection of the data. These results show that a partial structure model can give phases of Bragg reflections, which are good approximations to the true phases.

Figure 1.4 Difference Fourier map of paracetamol in dependence on the resolution of the diffraction data, obtained after refinement of a model containing nonhydrogen atoms only. Sections are shown of 6 × 6 Å2 through the phenyl ring. Contour lines are at of the respective maximum densities, with dashed lines for negative values and long-dashed lines for the value 0. (a)1.020 −1, (b) 0.704 −1, (c) 0.617 −1, (d) 0.481 −1, (e) 0.364 −1, and (f) 0.244 −1. In panel (a), the structure model is superimposed as dots and sticks. Compare with Figure 1.3.

1.5 Statistical Properties of Diffracted Intensities

The atomic character of matter allows for a statistical analysis of the structure factors, eventually arriving at probability distribution functions (pdfs) for several quantities, such as the scattered intensity or the sum of phases of three matching reflections. These probability distributions form the foundation of direct methods (Section 1.8).

Consideration of the expression for the structure factor in Eq. 1.10 shows that each atom contributes one term that is the product of three factors: the atomic form factor, the Debije–Waller factor, and the phase factor. Both the atomic form factor and the Debije–Waller factor are positive real-valued functions, which gradually decrease on increasing length of the scattering vector. Because the phase factor is a complex number of magnitude 1, these properties imply that the magnitudes of structure factors are — on the average — smaller at high values of sin(θ)/λ than at low values. This property of structure factors has been formalized in the so-called Wilson plot that describes the averaged scattered intensity as a function of the length of the scattering vector. Realizing that the magnitudes of the observed and calculated structure factors are related by the scale factor K (Eq. 1.11), the Wilson plot is [6]

1.13

where B is the average or overall ADP. The symmetry enhancement factor pH accounts for different multiplicities of different reflections. The combined scattering power of the atoms in the unit cell is given by

1.14

Intensities of individual reflections are not expected to follow the linear dependence of Eq. (1.13) and Figure 1.5a. Instead, one expects that they cover a wide range about the average value. This property is used for the definition of normalized structure factors, E(H), which are given by the corresponding structure factor divided by the square root of the average intensity according to (Eq. 1.13)

1.15

Normalized structure factors bring the scattering of every crystalline material on the same scale. Unified probability relations exist for this quantity, which do not depend on the crystal structure. The chemical composition enters indirectly through the factor 2 in the definition of E(H) (Eqs. (1.14) and (1.15)).

Disregarding prior information about the crystal structure, all points in the unit cell have the same probability to be the site of an atom. For a sufficiently large structure, the positions of the atoms can be considered as being independent from each other. This feature has been used in statistical analyses of the diffraction, resulting in pdfs for the amplitudes and the phases of structure factors [6]. The pdf for the amplitudes |E| of normalized structure factors is defined as the probability of |E| to have values between |E| and |E| + d|E|. For an acentric structure (space group P1) this pdf is [6]

1.16

The distribution of normalized structure factors is found to be independent of the structural parameters and of the filling of the unit cell (Figure 1.5b). The integral of 1P|E|(|E|) over all possible values of its argument, that is, for |E|:0arrow∞, gives 1, because a reflection will have a value of its normalized structure factor with certainty.

Deviations from the acentric pdf will occur when correlations between structural parameters exist. An important type of correlation is the space group symmetry. In particular, the presence of an inversion center determines that the phase of each structure factor is restricted to one out of two possible values, instead of being a continuous variable in the acentric case. The centric pdf (space group ) for amplitudes of normalized structure factors is (Figure 1.5b) [6],

1.17

For other nontrivial symmetries, the form of the pdf will depend on the space group [8], even if the symmetry enhancement factor is taken into account.

The simple forms of the statistical functions (Eqs. (1.13), (1.16), and (1.17)) have been derived under the assumption of a random filling of the unit cell with equal atoms. Any violation of this assumption will lead to deviations of pdfs from those given above. Space group symmetry is one important source of correlations between the atoms. Other deviations from random filling may pertain to the presence in the structure of both heavy and light atoms, or to the presence of molecular groups, such as the phenyl group.

The second fundamental assumption of the statistical analyses of diffraction data is that expectation values can be computed as averages over many reflections. Accordingly, proper statistics require narrow intervals of [sin(θ)/λ]2 or |E|, each containing many reflections. These are contradictory requirements and their violation — as it is intrinsic to structures with small and intermediately sized unit cells — is the source of important deviations from the distributions in Eqs. (1.13), (1.16), and (1.17).

Violations of both assumptions explain the deviations from linearity of the Wilson plot for paracetamol (Figure 1.5a). In general, the discrepancy between the measured diffraction data and their expected statistical properties is one reason why direct methods may fail to solve the phase problem, as it happens to be the case for a small percentage of compounds.

1.6 The Patterson Function

The crystallographic phase problem prevents the calculation of the electron density by the inverse Fourier transform (Section 1.3). Nevertheless, the inverse Fourier transform of the intensities of Bragg reflections can be computed as (Eq. (1.3))

1.18

The Patterson function can alternatively be defined on the basis of the electron density,

1.19

This second definition has the interpretation of a “sum” (as approximation to the integral) over the unit cell of the product of the electron density at one point with the electron density at a point displaced over u. Large contributions to the integral occur if both ρ(r) and ρ(r + u) have large values. Large values of the electron density are concentrated at the positions of the atoms, while most of the unit cell has only a small density. As a result, the Patterson function possesses local maxima for u equal to interatomic vectors, that is, for u connecting one atom with another (Figure 1.6a). Equation (1.19) shows that the Patterson function is periodic with the same lattice as the crystal structure. In all analyses, it is therefore sufficient to consider P(u) over one unit cell.

Figure 1.6 Patterson function of paracetamol in dependence on the resolution of the diffraction data (Eq. 1.18) and calculated with |Fobs|2 from [2]. Sections are shown of 6 × 6 Å2 parallel to the plane of phenyl ring and centered on the origin. (a) 1.020 −1, (b) 0.704 −1, (c) 0.617 −1, (d) 0.481 −1, (e) 0.364 −1, and (f) 0.244 −1. Contour lines are at of the maximum densities except the origin peak.

A rotation or screw axis leads to the concentration of Patterson peaks in Harker planes perpendicular to the direction of the axis. For paracetamol, the larger number of Patterson peaks in the plane than in (2x, 0, 2z) is in agreement with the presence of a 21 screw axis in its space group (Figure 1.8). Harker planes would allow the determination of the x- and z-coordinates of all atoms, if all peaks could be resolved. This is not the case for paracetamol, and the use of the Patterson function as ab initio method of structure solution fails again. Even if Patterson peaks are not resolved, Harker lines and Harker planes allow the determination of the point symmetry of the structure and the intrinsic translations of the symmetry elements. Since the Patterson function is based on all diffraction information, this is a much more robust method than the analysis of reflection conditions for the determination of the intrinsic translations.

1.7 Patterson Search Methods

Once partial information is available about the structure, Patterson search methods can successfully be used for completing the structure. One type of information is the molecular formula of the compound. For the example of paracetamol, this includes the phenyl ring, which together with the N and O atoms, forms a planar rigid group of which the structure is well known from the crystal structures of other compounds containing phenyl rings. While the Patterson function of this group suffers from many overlaps of peaks (Section 1.6), it also displays a typical pattern of local maxima close to the origin and arranged within a plane that is parallel to the plane of the phenyl ring (Figure 1.6a). The determination of the plane of this typical pattern within the Patterson function then provides the orientation of the phenyl ring in the crystal structure.

An automated search for the orientation of a typical pattern in the Patterson function can be achieved by the rotation function [10]. The theoretical Patterson pattern of the rigid group, Prg(u) is rotated by the rotation R until the integral

1.20

attains its maximum value. The integration is restricted to a sphere that just covers all peaks of Prg(u). A maximum value of RI is obtained if all local maxima of Prg(u) are matched with local maxima of the Patterson function. The rotation bringing this coincidence then defines the sought orientation of the rigid group. The translation function can be used in a similar way to determine the position of a rigid group within the unit cell, after its orientation has been found through the rotation function [10].

Of more general applicability is the superposition minimum function, SMF(u), defined as [11]

1.21

At each point in space the minimum value is taken from the values of the shifted Patterson function, P(u − u1), and a Harker line or Harker section, P(u − Ru), where R is one of the rotational symmetry operators of the crystal. More than one term of each type can be included in the definition of the SMF, and the SMF can be used to compare different functions. The integral over u of the SMF of P(u) and Prg(Ru) can be used instead of the rotation function (Eq. 1.20) in a rotation search of a known fragment. For u1 equal to an interatomic vector between heavy atoms, the SMF can be used as a method of phase determination.

Modern computer programs for ab initio phase determination include both Patterson search methods and Direct methods (Section 1.8), as it is the case, for example, for SHELXS and SHELXD [9], SIR [11], and SnB [12].

1.8 Direct Methods

Direct methods aim at the determination of phases of structure factors from the knowledge of their (measured) amplitudes. The first problem is the feature that phases of structure factors are not a given quantity but depend on the choice of the origin of the coordinate system. For example, the rumor that reflection phases are equal to 0 or π for centrosymmetric crystals is only true if the origin has been chosen on an inversion center. Therefore, care should be taken that an admissible origin is chosen, which conforms the space group symmetry [1].

A second result of the above observation is that general relations do not exist concerning the phases of single reflections. Instead, properties need to be considered, which are independent from the choice of the origin. The most important of these relations exists for groups of three reflections, called triplets, related by

1.22

1.23

1.24

The Cochran distribution is illustrated in Figure 1.9 for several values of GHK. If more than one triplet contributes to a reflection H, the phase of its structure factor can be estimated with the tangent formula [6].

Figure 1.9 (a) The Cochran distribution describing the probability distribution for the value of the triplet phase ΦHK