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Emil Zolotoyabko

Basic Concepts of X-Ray Diffraction

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Emil Zolotoyabko

Basic Concepts of X-Ray Diffraction

The Author

Prof. Emil ZolotoyabkoTechnion-Israel Institute of TechnologyDepartment of Materials Science andEngineering32000 HaifaIsrael

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V

In memory of my late parents: Galina Frenkel and Vulf Zolotoyabko

VII

Contents

Preface XI

Introduction 1

1 Diffraction Phenomena in Optics 5

2 Wave Propagation in Periodic Media 11

3 Dynamical Diffraction of Particles and Fields: GeneralConsiderations 21

3.1 The Two-Beam Approximation 233.2 Diffraction Profile: The Laue Scattering Geometry 333.3 Diffraction Profile: The Bragg Scattering Geometry 38

4 Dynamical X-Ray Diffraction: The Ewald–Laue Approach 454.1 Dynamical X-Ray Diffraction: Two-Beam Approximation 494.1.1 The Role of X-Ray Polarization 504.1.2 The Two-Branch Isoenergetic Dispersion Surface for X-Rays 524.1.3 Isoenergetic Dispersion Surface for Asymmetric Reflection 56

5 Dynamical Diffraction: The Darwin Approach 615.1 Scattering by a Single Electron 615.2 Atomic Scattering Factor 645.3 Structure Factor 665.4 Scattering Amplitude from an Individual Atomic Plane 685.5 Diffraction Intensity in the Bragg Scattering Geometry 71

6 Dynamical Diffraction in Nonhomogeneous Media: The Takagi–TaupinApproach 77

6.1 Takagi Equations 776.2 Taupin Equation 846.2.1 Taupin Equation: The Symmetric Laue Case 84

VIII Contents

6.2.2 Taupin Equation: The Symmetric Bragg Case 866.2.3 Solution of the Taupin Equation for Multilayered Structures 88

7 X-Ray Absorption 91

8 Dynamical Diffraction in Single-Scattering Approximation: Simulation ofHigh-Resolution X-Ray Diffraction in Heterostructures andMultilayers 97

8.1 Direct Wave Summation Method 103

9 Reciprocal Space Mapping and Strain Measurements inHeterostructures 121

10 X-Ray Diffraction in Kinematic Approximation 13110.1 X-Ray Polarization Factor 13310.2 Debye–Waller Factor 135

11 X-Ray Diffraction from Polycrystalline Materials 13911.1 Ideal Mosaic Crystal 13911.2 Powder Diffraction 141

12 Applications to Materials Science: Structure Analysis 145

13 Applications to Materials Science: Phase Analysis 15513.1 Internal Standard Method 15813.2 Rietveld Refinement 159

14 Applications to Materials Science: Preferred Orientation (Texture)Analysis 161

14.1 The March–Dollase Approach 165

15 Applications to Materials Science: Line Broadening Analysis 17115.1 Line Broadening due to Finite Crystallite Size 17415.1.1 The Scherrer Equation 17515.1.2 Line Broadening in the Laue Scattering Geometry 17815.2 Line Broadening due to Microstrain Fluctuations 18015.3 Williamson–Hall Method 18115.4 The Convolution Approach 18315.5 Instrumental Broadening 18415.6 Relation between Grain Size-Induced and Microstrain-Induced

Broadenings of X-Ray Diffraction Profiles 186

16 Applications to Materials Science: Residual Strain/StressMeasurements 189

16.1 Strain Measurements in Single-Crystalline Systems 18916.2 Residual Stress Measurements in Polycrystalline Materials 190

Contents IX

17 Impact of Lattice Defects on X-Ray Diffraction 193

18 X-Ray Diffraction Measurements in Polycrystals with High SpatialResolution 203

18.1 The Theory of Energy-Variable Diffraction (EVD) 20618.1.1 Homogeneous Materials 21018.1.2 Inhomogeneous Materials 212

19 Inelastic Scattering 21719.1 Inelastic Neutron Scattering 21819.2 Inelastic X-Ray Scattering 221

20 Interaction of X-Rays with Acoustic Waves 22520.1 Thermal Diffuse Scattering 22820.2 Coherent Scattering by Externally Excited Phonons 230

21 Time-Resolved X-Ray Diffraction 237

22 X-Ray Sources 24122.1 Synchrotron Radiation 250

23 X-Ray Focusing Optics 25723.1 X-Ray Focusing: Geometrical Optics Approach 26123.2 X-Ray Focusing: Diffraction Optics Approach 26823.2.1 Bragg–Fresnel Lenses and Fresnel Zone Plates 26823.2.2 Using Asymmetric Reflections 272

24 X-Ray Diffractometers 27524.1 High-Resolution Diffractometers 27524.2 Powder Diffractometers 280

References 285

Index 291

XI

Preface

This book summarizes my more than 20 years’ experience of teaching the graduatecourses ‘‘X-Ray Diffraction’’ and ‘‘Dynamical X-Ray Diffraction’’ in the Departmentof Materials Science and Engineering, Technion-Israel Institute of Technology.These two courses based, respectively, on kinematic and dynamical diffractiontheories, reflect the main trend in the field, that is, considering separately the X-raydiffraction in small and large crystals. The terms small and large, in this context,are used in comparison with some fundamental parameter of the theory calledextinction length, which is inversely proportional to the strength of X-ray interactionwith materials.

The first case (small crystals) is easy to treat analytically since one has to simplysum the amplitudes of X-ray waves scattered by each scattering center or atomicplane. X-ray diffraction in a large crystal is more difficult to analyze because thecoherent interaction between transmitted and diffracted X-ray waves should betaken into account. The subdivision mentioned has always been supported by thefacts that small individual crystals are much easier to grow and most engineeringmaterials are polycrystalline in nature, that is, they comprise a number of smallcrystallites. Correspondingly, most classical applications of X-ray diffraction tochemistry and materials science, for example, structure determination or phaseanalysis, are theoretically based on the kinematic approximation.

However, the enormous progress in the microelectronics industry in the secondhalf of twentieth century required the growth of large single crystals, mainlysilicon, which challenged new developments in X-ray characterization techniquesand, hence, the dynamical diffraction theory. Additional impetus to the field hasbeen given by the advances in the growth of single-crystalline heterostructuresand multilayers for optoelectronics and microelectronics, which stimulated thedeployment of high-resolution X-ray diffraction as the main testing tool for thequality of the structures mentioned. Today, commercial computer programs thatare in common use for simulating high-resolution X-ray diffraction profiles inmultilayers are based on dynamical diffraction theory.

By permanently keeping in touch with graduate students taking my coursesand involved in advanced X-ray diffraction measurements, I sensed the need fora textbook that unites kinematic and dynamical diffraction theories and gives agood introduction to modern characterization techniques. Besides, without a sound

XII Preface

knowledge of the dynamical scattering theory, which forms the most comprehensivebasis of X-ray diffraction, it is impossible to understand the limitations of the widelyused kinematic approximation. I am confident that, especially for beginners, it isvery important to provide a whole picture focused on the basic physical concepts thatare distributed over numerous literature sources, rather than describing in detailthe subsequent technical issues. Only after serious learning of the fundamentalsof the field is it possible to follow more specialized literature and use sophisticatedinstruments for advanced materials characterization. The latter issue is of specialimportance because in last decades the progress in novel X-ray diffraction methodshas been amazingly fast mainly due to new developments in synchrotron radiationsources and X-ray optics.

I kept these considerations in mind when working on this book. I believethat it will assist researchers in different disciplines who use X-ray diffraction intheir studies and, especially, graduate students in materials science, physics, andchemistry.

The last remark relates to the literature sources that I have cited in this book. Thelist, generally, consists of other books on X-rays and complementary subjects as wellas comprehensive reviews. I believe that student-oriented textbooks, in contrast tomanuscripts focused on particular problems or describing rather narrow scientificfields, should not be overloaded by massive citations of technical papers publishedin specialized scientific journals. I have used a very limited number of the latter,whenever I felt that it was necessary.

2013 Emil ZolotoyabkoHaifa, Israel

1

Introduction

X-rays were discovered by Wilhelm Conrad Rontgen on 8 November 1895, that is,almost 120 years ago. Despite a very mature age, the global impact of this discoveryon science, engineering and, generally, human life is only growing with time. Wehave no other example of a high-impact scientific discovery in modern era thathas been so instrumental for groundbreaking developments in physics, chemistry,materials science, biology, and medicine. In fact, the list of Nobel Prize awardsrelated to the field of X-rays alone is amazingly extensive; the most importantexamples with partial citations are given below:

1901 – Wilhelm Conrad Rontgen – the first Nobel Prize in physics: ‘‘for thediscovery with which his name is linked for all time: the … so-called Rontgenrays or, as he himself called them, X-rays.’’

1914 – Max Theodor Felix von Laue – Nobel Prize in physics: ‘‘for his discoveryof the diffraction of X-rays by crystals.’’

1915 – William Henry Bragg and William Lawrence Bragg – Nobel Prize inphysics: ‘‘for their services in the analysis of crystal structure by means ofX-rays.’’

1917 – Charles Glover Barkla – Nobel Prize in physics: ‘‘for his discovery of thecharacteristic Rontgen radiation of the elements.’’

1924 – Karl Manne Georg Siegbahn – Nobel Prize in physics: ‘‘for his discoveriesand research in the field of X-ray spectroscopy.’’

1927 – Arthur Holly Compton – Nobel Prize in physics: ‘‘for his discovery of theeffect named after him.’’

1936 – Petrus (Peter) Josephus Wilhelmus Debye – Nobel Prize in chemistry:‘‘for his contributions to our knowledge of molecular structure throughhis investigations on dipole moments and on the diffraction of X-rays andelectrons in gases.’’

1946 – Hermann Joseph Muller – Nobel Prize in physiology and medicine: ‘‘forthe discovery of the production of mutations by means of X-ray irradiation.’’

1979 – Allan Cormack and Godfrey Hounsfield – Nobel Prize in physiology ormedicine: ‘‘for the development of computer assisted tomography.’’

2002 – Riccardo Giacconi – Nobel Prize in physics: ‘‘for pioneering contribu-tions to astrophysics, which have led to the discovery of cosmic X-raysources.’’

Basic Concepts of X-Ray Diffraction, First Edition. Emil Zolotoyabko.c© 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 Introduction

A number of the X-ray crystallography works, that is, structure determination byX-ray diffraction, have been awarded the Nobel Prize. They include the followingseminal discoveries:

1962 – James Watson, Francis Crick, and Maurice Wilkins – Nobel Prize inmedicine or physiology: ‘‘for their determination … of the structure ofdeoxyribonucleic acid (DNA).’’

1962 – Max Ferdinand Perutz and John Cowdery Kendrew – Nobel Prize inchemistry: ‘‘for their studies of the structures of globular proteins (mainly bymeans of a heavy atom method in X-ray diffraction).’’

1964 – Dorothy Crowfoot Hodgkin – Nobel Prize in chemistry: ‘‘for her deter-minations by X-ray techniques of the structures of important biochemicalsubstances (for example, vitamin B12).’’

1985 – Herbert Hauptman and Jerome Karle – Nobel prize in chemistry: ‘‘fortheir outstanding achievements in the development of direct methods forthe determination of crystal structures.’’

2009 – Venkatraman Ramakrishnan, Thomas Steitz, and Ada Yonath – NobelPrize in chemistry: ‘‘for studies of the structure and function of the ribo-some.’’

I believe that this list will be further extended in the coming years. It clearly showsthe uniqueness of X-rays: they can equally well be used for imaging, spectroscopy,and scattering measurements. Only the last domain is considered in this book,which is devoted to coherent X-ray scattering in crystals – the field which isfrequently called X-ray diffraction.

Since X-ray quanta have no electrical charge, their scattering by materials israther weak, as compared to the electron scattering. For this reason, the develop-ment of powerful X-ray sources, which allow tremendous increase of the diffractionintensity and, correspondingly, shortening of the measurement time, is of enor-mous importance to the field. During the last 60 years, there has been continuousprogress in the construction of the dedicated electron accelerators – synchrotrons,which produce intense X-ray beams with superior characteristics. The brilliance ofsynchrotron sources is many orders of magnitude higher than that of laboratoryX-ray tubes. Nowadays, as a result of this progress, novel X-ray scattering experi-ments have become possible, which were considered only a dream a few decadesago: for example, inelastic X-ray scattering, ultrafast time-resolved measurements,diffraction measurements with small samples subjected to very high pressures andtemperatures, X-ray diffraction with high spatial resolution, and magnetic X-rayscattering. Besides, there has been tremendous progress in X-ray focusing, whichfor a long time has been considered as hardly achievable because of the tiny differ-ences between the refractive indices of materials and vacuum for electromagneticwaves in the X-ray range of wavelengths. This issue is of primary importance forcontinuous improvement of the spatial resolution of X-ray techniques toward thenanometer scale, which will allow us to compete in some aspects with electronmicroscopy.

Introduction 3

In the nearest future, we expect fast development of electron accelerators ofthe next generation, the so-called free electron lasers. With the help of thesemachines, the field of X-ray diffraction and scattering will be further expanded.One already speaks of scattering experiments with individual molecules, time-resolved diffraction measurements in the femtosecond range, and coherent X-rayimaging on a nanometer scale.

Bearing all this in mind, we are coming back to the content of the book, whichprovides a systematic description of X-ray diffraction using both dynamical andkinematic diffraction theories. A great deal of attention is given to the X-raydiffraction techniques developed for characterizing single-crystalline structuresand polycrystalline materials. Certainly, the book reflects the scientific interestsof the author, for example, the field of X-ray interaction with acoustic waves.Other examples include direct wave summation method in high-resolution X-ray diffraction and energy-variable depth-resolved X-ray diffraction at synchrotronbeam lines.

The book starts with brief general description of diffraction phenomena inoptics, with emphasis on the specific characteristics of X-rays considered inthat context (Chapter 1). In Chapter 2, we discuss the fundamentals of X-raydiffraction due to wave propagation in periodic media. We introduce the quasi-wave vector conservation law and show how the Bragg law is related to it. In orderto comprehensively analyze the diffraction conditions, the concept of the Ewaldsphere is discussed. Chapter 3 is devoted to a general description of diffractionprocesses of particles and fields. Initially, we treat these processes in the frameworkof dynamical diffraction by using the Ewald–Laue approach applied to Schrodingerequation, that is, for scalar fields. Here we introduce the concept of two-beamapproximation and the isoenergetic dispersion surface for quantum mechanicalstates within a crystal. This allows us to analyze the essential features of diffractionprofiles in the Bragg and Laue scattering geometries, as well as the Pendellosungeffect, which is also important for transmission electron microscopy.

In Chapter 4, dynamical diffraction is treated in the Ewald–Laue approachapplied to Maxwell equations, that is, for vector fields, in order to take into accountdifferent X-ray polarizations. In this chapter, we derive the X-ray diffraction profilesfor asymmetric reflections and introduce the X-ray extinction length.

In Chapter 5, we show the Darwin approach to the dynamical X-ray diffraction,which helps us later on to bridge the gap between dynamical and kinematicdiffraction theories.

Dynamical X-ray diffraction in nonhomogeneous structures is covered inChapter 6 in the framework of the Takagi–Taupin approach. The obtained resultsare used in computer programs aimed at fitting the experimental diffraction profilesin multilayers.

Chapter 7 is devoted to the description of X-ray absorption, which does notdirectly relate to the formation of coherent X-ray scattering but has an importanteffect on it.

In Chapter 8, we develop a novel approach to dynamical diffraction, the so-calleddirect wave summation method, which takes into account the X-ray absorption

4 Introduction

and attenuation of the transmitted X-ray beam due to the diffraction process itself,but in the single scattering approximation. We show that, in many practical cases,this method allows us to obtain analytic expressions for diffraction profiles whichcan successfully be applied for fitting experimental data taken from thin-film,single-crystalline structures.

In Chapter 9, we describe the X-ray mapping method in the reciprocal spaceand related strain measurements in thin-film structures for microelectronics andoptoelectronics.

Chapter 10 is devoted to the description of X-ray diffraction in the kinematicapproximation, that is, when the crystal size is small compared to the extinctionlength. The intensity calculations use atomic and structure factors, as well as thescattering amplitude from an individual atomic plane which was introduced inChapter 5.

In Chapter 11, the expressions for diffraction intensity are developed for poly-crystalline materials and random powders.

Chapters 12–16 are devoted to classical applications of X-ray diffraction tomaterials science, that is, respectively, for structure analysis, phase analysis,preferred orientation, line broadening, and residual strain/stress analyses. Effects ofpreferred orientation are analytically described within the March–Dollase approach,that is, for uniaxial texture.

In Chapter 17, we analyze the fundamental effect of lattice defects on X-raydiffraction.

Chapters 18–21 deal with specific subjects in which the recent progress andmost spectacular achievements are directly related to the use of synchrotronradiation. These are X-ray diffraction measurements in polycrystalline materialswith high spatial resolution (Chapter 18), inelastic X-ray scattering (Chapter 19)and the related field of the X-ray interaction with acoustic waves (Chapter 20), andtime-resolved X-ray scattering (Chapter 21).

We end the book with three chapters devoted to the essential technical issues,which include X-ray sources (Chapter 22), X-ray optical elements (Chapter 23), andX-ray diffractometers (Chapter 24). Without these technical developments, prog-ress in the field would be impossible.

5

1Diffraction Phenomena in Optics

The term diffraction in optics is usually used to explain the deviations of lightpropagation from the trajectories dictated by geometrical (ray) optics. One of themost famous examples is the so-called Fraunhofer diffraction, which explains thetransmission of an initially parallel beam of light through a circular hole of radiusD fabricated in a nontransparent screen. Within the framework of geometricaloptics, behind the screen, the nonzero transmitted intensity will be detected just infront of the hole (see Figure 1.1). It means that, after passing through the screen,the direction of light propagation does not change; the only effect is a reductionin the total light intensity in a proportion dictated by the area of the hole S=𝜋D2

with respect to the cross section of the incident beam. However, light scattering bythe border of the hole can substantially modify this result and provide additionaltransmitted intensity in spatial directions that differ by angle Θ from the initialdirection of light propagation before the screen (see Figure 1.2, upper panel). Inother words, after passing through the screen, light propagates not only in onedirection, which is defined by the initial wave vector k i, but also in many otherdirections defined by the vectors k s = k i + q. Here, q is a variable wave vectortransfer to the screen during scattering events (see Figure 1.3). Note that, for elasticscattering processes

|k s| = |k i| = 2𝜋𝜆

(1.1)

where 𝜆 is the wavelength of light. Taking into account Eq. (1.1) and the axialsymmetry of the particular scattering problem (at a fixed scattering angle Θ, seeFigure 1.3), we find that

|q| = q ≈ 2𝜋𝜆Θ (1.2)

For each q-value, the light scattering amplitude is given by the Fourier componentu𝐪 of the wave field u(r) just after the screen [1]:

u𝐪 = ∫ ∫ u(r)e−iqr𝑑𝑥𝑑𝑦 (1.3)

However, in the first approximation, we can set u= u0, that is, equal the amplitudeof the homogeneous wave field before the screen, and then express the scattering


6 1 Diffraction Phenomena in Optics

2D

Figure 1.1 Light transmission through a circular hole of radius D in the limit ofgeometrical optics.

−0.61λ/D 0.61λ/D

2D

Θ-axis

Θ Θ

Figure 1.2 Light transmission (upper panel) through a circular hole of radius D, takinginto account diffraction phenomenon (Fraunhofer diffraction). Bottom panel: transmittedintensity as a function of angular deviation Θ.

k i

ks

q

Θ

Figure 1.3 Wave vector change q in the course of elasticscattering of propagating light.


amplitude u𝐪 as

u𝐪 = ∫ ∫ u0e−iqr𝑑𝑥𝑑𝑦 (1.4)

where the integration proceeds over the entire area S of the hole. The diffractionintensity (relative to that in the incident beam) for a given q-value within an elementof solid angle Ω is expressed as follows [1]:

dIrel = 𝜆−2|||||u𝐪

uo

|||||2

dΩ (1.5)

In order to find u𝐪, let us introduce the polar coordinates r and 𝜑 within the circularhole. In this coordinate system, Eq. (1.4) transforms into

u𝐪 = u0∫D

0 ∫2𝜋

0e−𝑖𝑞𝑟 cos𝜑 r 𝑑𝜑 𝑑𝑟 = 2𝜋u0∫

D

0J0(𝑞𝑟)r 𝑑𝑟 (1.6)

where J0 is the Bessel function of zero order. Note that, in deriving Eq. (1.6), weused the fact that, for small scattering angles Θ, the vector q is nearly situated inthe plane of the hole. One can express the integral in (1.6) via a Bessel function offirst order J1, as

∫D

0J0 (𝑞𝑟)r 𝑑𝑟 =

Dq

J1(𝐷𝑞) (1.7)

and, finally

u𝐪 =2𝜋u0D

qJ1(𝐷𝑞) (1.8)

Substituting Eq. (1.8) into Eq. (1.5) and using Eq. (1.2), we obtain

dIrel =D2

Θ2J2

1

(2𝜋𝐷𝜆

Θ)𝑑𝛺 (1.9)

The distribution of the transmitted intensity (Eq. (1.9)) as a function of thescattering angle Θ is shown in Figure 1.2 (bottom panel). With an increase in theabsolute value of the angle Θ, the light intensity shows a fast overall reduction,on which the pronounced oscillating behavior is superimposed. The intensityoscillations are revealed as lateral maxima of diminishing height, separated by thezero-intensity points. The latter are determined by the zeros of the J1 function.Most of the diffraction intensity (about 84%) is confined within the angular interval−Θ0 ≤Θ≤Θ0, which is defined by the first zero of the Bessel function J1:

2𝜋𝜆

D Θ0 = 3.832 (1.10)

That is,

Θ0 = 0.61𝜆

D(1.11)

It follows from Eq. (1.11) that diffraction is important when the wavelength 𝜆 is asignificant part of the D-value. If 𝜆∕D ≪ 1, the angular deviations are subtle, whichimplies that diffraction effects (deviations from geometrical optics) are weak. For

8 1 Diffraction Phenomena in Optics

visible light with 𝜆≈ 0.5 μm, the diffraction phenomena are regularly observed forobjects with the characteristic size D ranging from few micrometers and up to∼103 μm.

Diffraction of light imposes the main limitation on the resolving power of opticalinstruments. For a telescope, the resolution is defined on an angular scale andis given by the so-called Rayleigh criterion. It states that two objects (stars) canbe separately resolved if an angular distance ΔΘc between the maxima of theirintensity distributions (Eq. (1.9)) exceeds the Θ0 value defined by Eq. (1.11). Itimplies that the angular resolution of a telescope is given by Eq. (1.11).

For a microscope, length limitations are most useful, helping us to evaluatethe size of the smallest objects still visible with the aid of a particular opticaldevice. In order to ‘‘translate’’ the Rayleigh criterion into the length-scale language,let us consider the simplified equivalent scheme of a microscope. The latter isrepresented by a circular lens of radius D and focal length f , and transforms anobject of size Y into its image of size Y ′ (see Figure 1.4). For high magnification,an object is placed close to the focus (left side of the lens in Figure 1.4). Then

𝜃 ≈ Yf

(1.12)

Applying the Rayleigh criterion means that Θ>Θ0 and hence

Y > Δ = f 𝜃0 = 0.61𝜆

Df (1.13)

For focusing effect (see Figure 1.5), we illuminate our lens with a wide parallelbeam and obtain a small spot Y ′ in the focal plane (right side of the lens inFigure 1.5). Now

𝜃 = Y ′

f(1.14)

Applying again the Rayleigh criterion and Eq. (1.11), we find that the spot size Y ′

cannot be smaller than parameter Δ given by Eq. (1.13), that is,

Y ′ > Δ = f 𝜃0 = 0.61𝜆

Df (1.15)

Therefore, the spatial resolution Δ, when using the circular focusing element, iscompletely defined by its radius D, focal length f , and radiation wavelength 𝜆. Wewill use the obtained results in Chapter 23 when describing the focusing elementsfor X-ray optics. More information on diffraction optics of visible light and, inparticular, on the Fraunhofer and Fresnel diffraction can be found in [2, 3].

D

f

Y ′

Y Θ

Figure 1.4 Illustration of the diffraction-limited spatial resolution of a microscope.


f

DΘ

Θ

Y ′Y

Figure 1.5 Illustration of the diffraction-limited focal spot size that is achievable by using alens.

When considering potential diffraction effects for X-rays, we stress that they havewavelengths of about 0.1 nm= 1 A: that is, 5000 times shorter than for visible light.If so, what kind of objects could potentially cause the diffraction of X-rays? Clearly,characteristic sizes in these objects should be very small. It was the great idea ofMax von Laue, who had proposed in 1912 the diffraction experiment of X-rays incrystals, bearing in mind that crystals are built of periodic three-dimensional atomicnetworks; that is, they reveal translational symmetry. Fortunately, the characteristicdistances between adjacent atomic unit cells (translation lengths) are comparablewith X-ray wavelengths. Today, we can say that mainly translational symmetrytogether with appropriate lengths of the translation vectors is the origin of X-raydiffraction in crystals. This subject is comprehensively treated in Chapter 2.

11

2Wave Propagation in Periodic Media

Let us consider, following the ideas of Brillouin [4], the propagation of plane waveswithin a medium. A plane wave is defined as

Y = Y0 exp[i(kr − 𝜔𝑡)] (2.1)

where Y stands for a physical parameter that oscillates in space (r) and time (t),while Y0, k , and 𝜔 are the wave amplitude, wave vector, and angular frequency,respectively. The term in circular brackets in Eq. (2.1), that is,

𝜑 = kr − 𝜔𝑡 (2.2)

is the phase of the plane wave. At any instant t, the surface of steady phase𝜑= constis defined by the condition kr = const. The latter is the equation of a geometricalplane perpendicular to the direction of wave propagation k and, therefore, this typeof wave has accordingly been so named (plane wave).

Considering, first, a homogeneous medium, we can say that a plane wave havingwave vector k i at a certain point in its trajectory will continue to propagate withthe same wave vector because of the momentum conservation law. Note that thewave vector k is linearly related to the momentum P via the Planck constant ℏ: thatis, P =ℏk . We also remind the readers that the momentum conservation law is adirect consequence of the particular symmetry of a homogeneous medium, knownas the homogeneity of space [5].

The situation drastically changes for a nonhomogeneous medium, in which themomentum conservation law, generally, is not valid because of the breaking ofthe above-mentioned symmetry. As a consequence, in such a medium, one canfind wave vectors k f differing from the initial wave vector k i. The simplest caseis realized when the medium comprises two homogeneous parts with dissimilarcharacteristics. Such breaking of symmetry is the origin of the refraction of wavesat the interface between two parts. Refraction phenomena will also be touchedupon later in this book (see Chapter 23). However, our focus in the current chapteris on the particular nonhomogeneous medium with translational symmetry, whichcomprises scattering centers in specific points rs only, that is,

rs = n1a1 + n2a2 + n3a3 (2.3)


12 2 Wave Propagation in Periodic Media

the rest of the space being empty. Here, in Eq. (2.3), the vectors a1, a2, a3, arethree noncoplanar translation vectors, while n1, n2, n3, are integer numbers (bothpositive, negative, and zero). Currently, this is our model of a crystal.

On the basis of the translational symmetry only, we can say that, in an infinitemedium with no absorption, the magnitude of the plane wave Y should be thesame in close proximity to any lattice node described by Eq. (2.3). It means thatthe amplitude Y0 is the same at all points rs, whereas the phase 𝜑 can differ byan integer number m of 2𝜋 (see Eq. (2.1)). Let us suppose that the plane wave hasthe wave vector k i at the starting point r0 = 0 and t0 = 0. Then, according to Eq.(2.2), 𝜑(0) = 0. If so, at point rs, the phase 𝜑(rs) of the plane wave should be equalto 𝜑(rs) = k f rs − 𝜔𝑡 = 2𝜋𝑚. Note that the change of the wave vector from k i to k f

physically means that the wave obeys scattering at point rs (see Figure 2.1). In thischapter, only elastic scattering (with no energy change) is considered. So

|k f | = |k i| = |k | = 2𝜋𝜆

(2.4)

where 𝜆 stands for the radiation wavelength. Note also that Eq. (2.4) is equivalentto Eq. (1.1) introduced in Chapter 1.

For further analysis, we recall the linear dispersion law for electromagnetic wavesin vacuum, that is, the linear relationship between the absolute value of the wavevector |k | and the angular frequency 𝜔 given by

𝜔 = c|k | (2.5)

where c is the speed of wave propagation. With the aid of Eq. (2.4) and Eq. (2.5), wecan express the time interval t for a wave traveling between points r0 = 0 and rs as

t =k irs|k i|c (2.6)

r =0

r = rs

ki

kf

Figure 2.1 Illustration of X-ray scattering in a periodic medium.


By using Eq. (2.2), Eq. (2.4), Eq. (2.5), and Eq. (2.6), we calculate the phase of theplane wave, 𝜑(rs), after scattering at point rs as

𝜑(rs) = k f rs − 𝜔𝑡 = (k f − k i)rs (2.7)

Since the initial phase 𝜑(0) = 0, Eq. (2.7) determines the phase difference 𝜑 dueto wave scattering. The difference vector Q between the wave vectors in the final(k f ) and initial (k i) wave states is known as the wave vector transfer or the scatteringvector:

Q = k f − k i (2.8)

Substituting Eq. (2.8) into Eq. (2.7) finally yields the phase difference 𝜑 as

𝜑 = 𝜑(rs) = Qrs (2.9)

According to Eq. (2.8), different values of k f are actually permitted, but only thosethat provide a scalar product in Eq. (2.9), that is, a scalar product of a certainscattering vector QB and different vectors rs from the lattice (Eq. (2.3)), equal to anintegral multiple m of 2𝜋:

𝜑 = QBrs = 2𝜋𝑚 (2.10)

The vector QB is also called the diffraction vector. In order to avoid the usage offactor 2𝜋 in Eq. (2.10), another vector H is introduced as

H =QB

2𝜋(2.11)

for which Eq. (2.10) is rewritten as

H ⋅ rs = m (2.12)

By substituting Eq. (2.3) into Eq. (2.12), we finally obtain

H ⋅ (n1a1 + n2a2 + n3a3) = m (2.13)

In order to find the set of allowed vectors H satisfying Eq. (2.13), the reciprocalspace is introduced, which is based on three noncoplanar vectors b1, b2, and b3.Real space and reciprocal space are related to each other by the orthogonalityconditions

aib j = 𝛿𝑖𝑗 (2.14)

where 𝛿𝑖𝑗 is the Kronecker symbol, equal to 1 for i= j or 0 for i≠ j (i, j= 1, 2,3). In order to build the reciprocal space from real space, we use the followingmathematical procedure:

b1 =[a2 × a3]

Vc

b2 =[a3 × a1]

Vc

b3 =[a1 × a2]

Vc(2.15)


where Vc stands for the volume of the parallelepiped built in real space on vectorsa1, a2, a3:

Vc = a1 ⋅ [a2 × a3] (2.16)

By using Eq. (2.16), it is easy to directly check that the procedure (Eq.(2.15)) provides the orthogonality conditions (Eq. (2.14)). For example,a1 ⋅ b1 = a1 ⋅ [a2 × a3]/ Vc = Vc/ Vc = 1, whereas a2 ⋅ b1 = a2 ⋅ [a2 × a3]/ Vc = 0. Moreinformation on the reciprocal space construction can be found, for example, in[6, 7].

In the reciprocal space, the allowed vectors H are linear combinations of thebasic vectors b1, b2, b3:

H = hb1 + kb2 + lb3 (2.17)

with integer projections (hkl), known as the Miller indices. The ends of vectors H,being constructed from the common origin (000), form the nodes of a reciprocallattice (see Figure 2.2). For all vectors H, which are called vectors of reciprocal lattice,Eq. (2.13) is automatically valid because of the orthogonality conditions (Eq. (2.14)).So, in a medium with translational symmetry, only those wave vectors k f may existthat are related to the initial wave vector k i as follows:

k f − k i = QB = 2𝜋H (2.18)

where the vectors H are given by Eq. (2.17). Sometimes, Eq. (2.18) is called the quasi-momentum (or quasi-wave vector) conservation law in the medium with translationalsymmetry, which should be used instead of the momentum conservation law ina homogeneous medium. Note that the latter law means QB = 2𝜋H = k f − k i = 0,that is, k f = k i. Graphical representation of Eq. (2.18), which leads to the famousBragg law, is given in Figure 2.3. This important point will be elaborated in moredetail below.

0

H

ki /2π

kf/2π

Trace of the Ewald sphere

A

Figure 2.2 Reciprocal lattice (black spots) and the Ewald’s sphere construction. Wave vec-tors of X-rays in the initial and final states are, respectively, indicated by ki and kf.


kf

k i

2πH2ΘB

Figure 2.3 Graphical representation of Eq. (2.18).

Actually, Eq. (2.18) describes the kinematics of the diffraction process in aninfinite periodic medium, since the presence of waves propagating along differentdirections k f , in addition to the incident wave with wave vector k i, is the essence ofthe diffraction phenomenon. According to Eq. (2.18), the necessary condition for thediffraction process is the quasi-momentum (or the quasi-wave vector) conservationlaw, which defines the specific angles 2ΘB between wave vectors k f and k i, at whichdiffraction intensity could, in principle, be observed (see Figure 2.3). Solving thewave vector triangle in Figure 2.3, together with Eq. (2.4), yields

2|k | sinΘB =4𝜋 sinΘB

𝜆= 2𝜋|H| = |QB| = QB (2.19)

Note that each vector of reciprocal lattice, that is, H = hb1 + kb2 + lb3, is perpen-dicular to a specific crystallographic plane in real space. This connection is directlygiven by Eq. (2.12), which defines the geometric plane for the ends of certainvectors rs, the plane being perpendicular to the specific vector H (see Figure 2.4).Using Eq. (2.19) and introducing a set of parallel planes of this type, which areseparated by the d-spacing

d = 1|H| (2.20)

we finally obtain the so-called Bragg law:

2d sinΘB = 𝜆 (2.21)

H

rs rs

rs

Figure 2.4 Schematic illustration of Eq. (2.12).


which provides the relation between the possible directions for the diffractedwave propagation (via Bragg angles ΘB) and interplanar spacings (d-spacings) din crystals. By using Eq. (2.15), Eq. (2.16), Eq. (2.17), and Eq. (2.20), one cancalculate the d-spacings in crystals, as functions of their lattice parameters andMiller indices, for all possible symmetry systems in real space (see Chapter 12).Therefore, measuring the diffraction peak positions 2ΘB and calculating latticed-spacings via the Bragg law (Eq. (2.21)) provides an important tool for solvingcrystal structures by diffraction methods. This line is elaborated in more detail inChapter 12.

It is worth further analyzing Bragg’s law in terms of the phases of propagatingand scattered waves. Let us consider the fate of an incident X-ray wave with wavevector k i between two parallel atomic planes separated by an interplanar spacingd (see Figure 2.5). The wave crosses the first atomic plane at point I (with radiusvector r I), where it is scattered, and then is scattered again by the second atomicplane at point II (with radius vector r II). After each scattering, the wave vector ischanged from k i to k f . According to Eq. (2.8) and Eq. (2.9), the phase difference 𝜑between these two scattered waves is

𝜑 = (k f − k i)(r II − r I) = Qr (2.22)

where r = r II − r I is a vector connecting points r I and r II. For specular reflection,the vector Q is perpendicular to the chosen atomic planes (see Figure 2.5) and thenthe phase difference is simply

𝜑 = 𝑄𝑑 (2.23)

I

II

d

rII

r I

Q kf

kf

ki

Figure 2.5 X-ray scattering by a system of parallel atomic planes.

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