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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Authored by a university professor deeply involved in X-ray diffraction-related research, this textbook is based on his lectures given to graduate students for more than 20 years. It adopts a well-balanced approach, describing basic concepts and experimental techniques, which make X-ray diffraction an unsurpassed method for studying the structure of materials.
Both dynamical and kinematic X-ray diffraction is considered from a unified viewpoint, in which the dynamical diffraction in single-scattering approximation serves as a bridge between these two parts. The text emphasizes the fundamental laws that govern the interaction of X-rays with matter, but also covers in detail classical and modern applications, e.g., line broadening, texture and strain/stress analyses, X-ray mapping in reciprocal space, high-resolution X-ray diffraction in the spatial and wave vector domains, X-ray focusing, inelastic and time-resolved X-ray scattering. This unique scope, in combination with otherwise hard-to-find information on analytic expressions for simulating X-ray diffraction profiles in thin-film heterostructures, X-ray interaction with phonons, coherent scattering of Mossbauer radiation, and energy-variable X-ray diffraction, makes the book indispensable for any serious user of X-ray diffraction techniques.
Compact and self-contained, this textbook is suitable for students taking X-ray diffraction courses towards specialization in materials science, physics, chemistry, or biology. Numerous clear-cut illustrations, an easy-to-read style of writing, as well as rather short, easily digestible chapters all facilitate comprehension.
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Seitenzahl: 425
Veröffentlichungsjahr: 2014
Ähnliche
Table of Contents
Related Titles
Title Page
Copyright
Dedication
Preface
Introduction
1: Diffraction Phenomena in Optics
2: Wave Propagation in Periodic Media
3: Dynamical Diffraction of Particles and Fields: General Considerations
3.1 The Two-Beam Approximation
3.2 Diffraction Profile: The Laue Scattering Geometry
3.3 Diffraction Profile: The Bragg Scattering Geometry
4: Dynamical X-Ray Diffraction: The Ewald–Laue Approach
4.1 Dynamical X-Ray Diffraction: Two-Beam Approximation
5: Dynamical Diffraction: The Darwin Approach
5.1 Scattering by a Single Electron
5.2 Atomic Scattering Factor
5.3 Structure Factor
5.4 Scattering Amplitude from an Individual Atomic Plane
5.5 Diffraction Intensity in the Bragg Scattering Geometry
6: Dynamical Diffraction in Nonhomogeneous Media: The Takagi–Taupin Approach
6.1 Takagi Equations
6.2 Taupin Equation
7: X-Ray Absorption
8: Dynamical Diffraction in Single-Scattering Approximation: Simulation of High-Resolution X-Ray Diffraction in Heterostructures and Multilayers
8.1 Direct Wave Summation Method
9: Reciprocal Space Mapping and Strain Measurements in Heterostructures
10: X-Ray Diffraction in Kinematic Approximation
10.1 X-Ray Polarization Factor
10.2 Debye–Waller Factor
11: X-Ray Diffraction from Polycrystalline Materials
11.1 Ideal Mosaic Crystal
11.2 Powder Diffraction
12: Applications to Materials Science: Structure Analysis
13: Applications to Materials Science: Phase Analysis
13.1 Internal Standard Method
13.2 Rietveld Refinement
14: Applications to Materials Science: Preferred Orientation (Texture) Analysis
14.1 The March–Dollase Approach
15: Applications to Materials Science: Line Broadening Analysis
15.1 Line Broadening due to Finite Crystallite Size
15.2 Line Broadening due to Microstrain Fluctuations
15.3 Williamson–Hall Method
15.4 The Convolution Approach
15.5 Instrumental Broadening
15.6 Relation between Grain Size-Induced and Microstrain-Induced Broadenings of X-Ray Diffraction Profiles
16: Applications to Materials Science: Residual Strain/Stress Measurements
16.1 Strain Measurements in Single-Crystalline Systems
16.2 Residual Stress Measurements in Polycrystalline Materials
17: Impact of Lattice Defects on X-Ray Diffraction
18: X-Ray Diffraction Measurements in Polycrystals with High Spatial Resolution
18.1 The Theory of Energy-Variable Diffraction (EVD)
19: Inelastic Scattering
19.1 Inelastic Neutron Scattering
19.2 Inelastic X-Ray Scattering
20: Interaction of X-Rays with Acoustic Waves
20.1 Thermal Diffuse Scattering
20.2 Coherent Scattering by Externally Excited Phonons
21: Time-Resolved X-Ray Diffraction
22: X-Ray Sources
22.1 Synchrotron Radiation
23: X-Ray Focusing Optics
23.1 X-Ray Focusing: Geometrical Optics Approach
23.2 X-Ray Focusing: Diffraction Optics Approach
24: X-Ray Diffractometers
24.1 High-Resolution Diffractometers
24.2 Powder Diffractometers
References
Index
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The Author
Prof. Emil Zolotoyabko
Technion-Israel Institute of Technology
Department of Materials Science and Engineering
32000 Haifa
Israel
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In memory of my late parents: Galina Frenkel and Vulf Zolotoyabko
Preface
This book summarizes my more than 20 years' experience of teaching the graduate courses “X-Ray Diffraction” and “Dynamical X-Ray Diffraction” in the Department of Materials Science and Engineering, Technion-Israel Institute of Technology. These two courses based, respectively, on kinematic and dynamical diffraction theories, reflect the main trend in the field, that is, considering separately the X-ray diffraction in small and large crystals. The terms small and large, in this context, are used in comparison with some fundamental parameter of the theory called extinction length, which is inversely proportional to the strength of X-ray interaction with materials.
The first case (small crystals) is easy to treat analytically since one has to simply sum the amplitudes of X-ray waves scattered by each scattering center or atomic plane. X-ray diffraction in a large crystal is more difficult to analyze because the coherent interaction between transmitted and diffracted X-ray waves should be taken into account. The subdivision mentioned has always been supported by the facts that small individual crystals are much easier to grow and most engineering materials are polycrystalline in nature, that is, they comprise a number of small crystallites. Correspondingly, most classical applications of X-ray diffraction to chemistry and materials science, for example, structure determination or phase analysis, are theoretically based on the kinematic approximation.
However, the enormous progress in the microelectronics industry in the second half of twentieth century required the growth of large single crystals, mainly silicon, which challenged new developments in X-ray characterization techniques and, hence, the dynamical diffraction theory. Additional impetus to the field has been given by the advances in the growth of single-crystalline heterostructures and multilayers for optoelectronics and microelectronics, which stimulated the deployment of high-resolution X-ray diffraction as the main testing tool for the quality of the structures mentioned. Today, commercial computer programs that are in common use for simulating high-resolution X-ray diffraction profiles in multilayers are based on dynamical diffraction theory.
By permanently keeping in touch with graduate students taking my courses and involved in advanced X-ray diffraction measurements, I sensed the need for a textbook that unites kinematic and dynamical diffraction theories and gives a good introduction to modern characterization techniques. Besides, without a sound knowledge of the dynamical scattering theory, which forms the most comprehensive basis of X-ray diffraction, it is impossible to understand the limitations of the widely used kinematic approximation. I am confident that, especially for beginners, it is very important to provide a whole picture focused on the basic physical concepts that are distributed over numerous literature sources, rather than describing in detail the subsequent technical issues. Only after serious learning of the fundamentals of the field is it possible to follow more specialized literature and use sophisticated instruments for advanced materials characterization. The latter issue is of special importance because in last decades the progress in novel X-ray diffraction methods has been amazingly fast mainly due to new developments in synchrotron radiation sources and X-ray optics.
I kept these considerations in mind when working on this book. I believe that it will assist researchers in different disciplines who use X-ray diffraction in their studies and, especially, graduate students in materials science, physics, and chemistry.
The last remark relates to the literature sources that I have cited in this book. The list, generally, consists of other books on X-rays and complementary subjects as well as comprehensive reviews. I believe that student-oriented textbooks, in contrast to manuscripts focused on particular problems or describing rather narrow scientific fields, should not be overloaded by massive citations of technical papers published in specialized scientific journals. I have used a very limited number of the latter, whenever I felt that it was necessary.
2013
Emil Zolotoyabko
Haifa, Israel
Introduction
X-rays were discovered by Wilhelm Conrad Röntgen on 8 November 1895, that is, almost 120 years ago. Despite a very mature age, the global impact of this discovery on science, engineering and, generally, human life is only growing with time. We have no other example of a high-impact scientific discovery in modern era that has been so instrumental for groundbreaking developments in physics, chemistry, materials science, biology, and medicine. In fact, the list of Nobel Prize awards related to the field of X-rays alone is amazingly extensive; the most important examples with partial citations are given below:
A number of the X-ray crystallography works, that is, structure determination by X-ray diffraction, have been awarded the Nobel Prize. They include the following seminal discoveries:
I believe that this list will be further extended in the coming years. It clearly shows the 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 is frequently called X-ray diffraction.
Since X-ray quanta have no electrical charge, their scattering by materials is rather weak, as compared to the electron scattering. For this reason, the development of powerful X-ray sources, which allow tremendous increase of the diffraction intensity and, correspondingly, shortening of the measurement time, is of enormous importance to the field. During the last 60 years, there has been continuous progress in the construction of the dedicated electron accelerators – synchrotrons, which produce intense X-ray beams with superior characteristics. The brilliance of synchrotron sources is many orders of magnitude higher than that of laboratory X-ray tubes. Nowadays, as a result of this progress, novel X-ray scattering experiments have become possible, which were considered only a dream a few decades ago: for example, inelastic X-ray scattering, ultrafast time-resolved measurements, diffraction measurements with small samples subjected to very high pressures and temperatures, X-ray diffraction with high spatial resolution, and magnetic X-ray scattering. Besides, there has been tremendous progress in X-ray focusing, which for a long time has been considered as hardly achievable because of the tiny differences between the refractive indices of materials and vacuum for electromagnetic waves in the X-ray range of wavelengths. This issue is of primary importance for continuous improvement of the spatial resolution of X-ray techniques toward the nanometer scale, which will allow us to compete in some aspects with electron microscopy.
In the nearest future, we expect fast development of electron accelerators of the next generation, the so-called free electron lasers. With the help of these machines, 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-ray imaging on a nanometer scale.
Bearing all this in mind, we are coming back to the content of the book, which provides a systematic description of X-ray diffraction using both dynamical and kinematic diffraction theories. A great deal of attention is given to the X-ray diffraction techniques developed for characterizing single-crystalline structures and polycrystalline materials. Certainly, the book reflects the scientific interests of 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 synchrotron beam lines.
The book starts with brief general description of diffraction phenomena in optics, with emphasis on the specific characteristics of X-rays considered in that context (Chapter 1). In Chapter 2, we discuss the fundamentals of X-ray diffraction 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 order to comprehensively analyze the diffraction conditions, the concept of the Ewald sphere is discussed. Chapter 3 is devoted to a general description of diffraction processes of particles and fields. Initially, we treat these processes in the framework of dynamical diffraction by using the Ewald–Laue approach applied to Schrödinger equation, that is, for scalar fields. Here we introduce the concept of two-beam approximation and the isoenergetic dispersion surface for quantum mechanical states within a crystal. This allows us to analyze the essential features of diffraction profiles in the Bragg and Laue scattering geometries, as well as the Pendellösung effect, which is also important for transmission electron microscopy.
In Chapter 4, dynamical diffraction is treated in the Ewald–Laue approach applied to Maxwell equations, that is, for vector fields, in order to take into account different X-ray polarizations. In this chapter, we derive the X-ray diffraction profiles for 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 kinematic diffraction theories.
Dynamical X-ray diffraction in nonhomogeneous structures is covered in Chapter 6 in the framework of the Takagi–Taupin approach. The obtained results are used in computer programs aimed at fitting the experimental diffraction profiles in multilayers.
Chapter 7 is devoted to the description of X-ray absorption, which does not directly relate to the formation of coherent X-ray scattering but has an important effect on it.
In Chapter 8, we develop a novel approach to dynamical diffraction, the so-called direct wave summation method, which takes into account the X-ray absorption 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 which can 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 space and related strain measurements in thin-film structures for microelectronics and optoelectronics.
Chapter 10 is devoted to the description of X-ray diffraction in the kinematic approximation, that is, when the crystal size is small compared to the extinction length. The intensity calculations use atomic and structure factors, as well as the scattering amplitude from an individual atomic plane which was introduced in Chapter 5.
In Chapter 11, the expressions for diffraction intensity are developed for polycrystalline materials and random powders.
Chapters 12–16 are devoted to classical applications of X-ray diffraction to materials science, that is, respectively, for structure analysis, phase analysis, preferred orientation, line broadening, and residual strain/stress analyses. Effects of preferred 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-ray diffraction.
Chapters 18–21 deal with specific subjects in which the recent progress and most spectacular achievements are directly related to the use of synchrotron radiation. These are X-ray diffraction measurements in polycrystalline materials with 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), and time-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), and X-ray diffractometers (Chapter 24). Without these technical developments, progress in the field would be impossible.
1
Diffraction Phenomena in Optics
The term diffraction in optics is usually used to explain the deviations of light propagation from the trajectories dictated by geometrical (ray) optics. One of the most famous examples is the so-called Fraunhofer diffraction, which explains the transmission of an initially parallel beam of light through a circular hole of radius D fabricated in a nontransparent screen. Within the framework of geometrical optics, behind the screen, the nonzero transmitted intensity will be detected just in front 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 reduction in 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 by the border of the hole can substantially modify this result and provide additional transmitted intensity in spatial directions that differ by angle Θ from the initial direction of light propagation before the screen (see Figure 1.2, upper panel). In other words, after passing through the screen, light propagates not only in one direction, which is defined by the initial wave vector , but also in many other directions defined by the vectors = + . Here, is a variable wave vector transfer to the screen during scattering events (see Figure 1.3). Note that, for elastic scattering processes
1.1
where λ is the wavelength of light. Taking into account Eq. (1.1) and the axial symmetry of the particular scattering problem (at a fixed scattering angle Θ, see Figure 1.3), we find that
1.2
Figure 1.1 Light transmission through a circular hole of radius D in the limit of geometrical optics.
Figure 1.2 Light transmission (upper panel) through a circular hole of radius D, taking into account diffraction phenomenon (Fraunhofer diffraction). Bottom panel: transmitted intensity as a function of angular deviation Θ.
Figure 1.3 Wave vector change in the course of elastic scattering of propagating light.
For each -value, the light scattering amplitude is given by the Fourier component of the wave field just after the screen [1]:
1.3
However, in the first approximation, we can set u = , that is, equal the amplitude of the homogeneous wave field before the screen, and then express the scattering amplitude as
1.4
where the integration proceeds over the entire area S of the hole. The diffraction intensity (relative to that in the incident beam) for a given -value within an element of solid angle Ω is expressed as follows [1]:
1.5
In order to find , let us introduce the polar coordinates and within the circular hole. In this coordinate system, Eq. (1.4) transforms into
1.6
where J0 is the Bessel function of zero order. Note that, in deriving Eq. (1.6), we used the fact that, for small scattering angles Θ, the vector is nearly situated in the plane of the hole. One can express the integral in (1.6) via a Bessel function of first order J1, as
1.7
and, finally
1.8
Substituting Eq. (1.8) into Eq. (1.5) and using Eq. (1.2), we obtain
1.9
The distribution of the transmitted intensity (Eq. (1.9)) as a function of the scattering angle Θ is shown in Figure 1.2 (bottom panel). With an increase in the absolute value of the angle Θ, the light intensity shows a fast overall reduction, on which the pronounced oscillating behavior is superimposed. The intensity oscillations are revealed as lateral maxima of diminishing height, separated by the zero-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:
1.10
That is,
1.11
It follows from Eq. (1.11) that diffraction is important when the wavelength λ is a significant part of the D-value. If , the angular deviations are subtle, which implies that diffraction effects (deviations from geometrical optics) are weak. For visible light with λ ≈ 0.5 μm, the diffraction phenomena are regularly observed for objects 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 optical instruments. For a telescope, the resolution is defined on an angular scale and is given by the so-called Rayleigh criterion. It states that two objects (stars) can be separately resolved if an angular distance ΔΘc between the maxima of their intensity distributions (Eq. (1.9)) exceeds the Θ0 value defined by Eq. (1.11). It implies that the angular resolution of a telescope is given by Eq. (1.11).
For a microscope, length limitations are most useful, helping us to evaluate the size of the smallest objects still visible with the aid of a particular optical device. In order to “translate” the Rayleigh criterion into the length-scale language, let us consider the simplified equivalent scheme of a microscope. The latter is represented by a circular lens of radius D and focal length f, and transforms an object 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
1.12
Figure 1.4 Illustration of the diffraction-limited spatial resolution of a microscope.
Applying the Rayleigh criterion means that Θ > Θ0 and hence
1.13
For focusing effect (see Figure 1.5), we illuminate our lens with a wide parallel beam and obtain a small spot in the focal plane (right side of the lens in Figure 1.5). Now
1.14
Figure 1.5 Illustration of the diffraction-limited focal spot size that is achievable by using a lens.
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,
1.15
Therefore, the spatial resolution , when using the circular focusing element, is completely defined by its radius D, focal length f, and radiation wavelength λ. We will use the obtained results in Chapter 23 when describing the focusing elements for X-ray optics. More information on diffraction optics of visible light and, in particular, on the Fraunhofer and Fresnel diffraction can be found in [2, 3].
When considering potential diffraction effects for X-rays, we stress that they have wavelengths of about 0.1 nm = 1 Å: 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 of Max von Laue, who had proposed in 1912 the diffraction experiment of X-rays in crystals, bearing in mind that crystals are built of periodic three-dimensional atomic networks; that is, they reveal translational symmetry. Fortunately, the characteristic distances between adjacent atomic unit cells (translation lengths) are comparable with X-ray wavelengths. Today, we can say that mainly translational symmetry together with appropriate lengths of the translation vectors is the origin of X-ray diffraction in crystals. This subject is comprehensively treated in Chapter 2.
2
Wave Propagation in Periodic Media
Let us consider, following the ideas of Brillouin [4], the propagation of plane waves within a medium. A plane wave is defined as
2.1
where Y stands for a physical parameter that oscillates in space (r) and time (t), while Y0, , and ω are the wave amplitude, wave vector, and angular frequency, respectively. The term in circular brackets in Eq. (2.1), that is,
2.2
is the phase of the plane wave. At any instant t, the surface of steady phase = const is defined by the condition = const. The latter is the equation of a geometrical plane perpendicular to the direction of wave propagation and, therefore, this type of wave has accordingly been so named (plane wave).
Considering, first, a homogeneous medium, we can say that a plane wave having wave vector at a certain point in its trajectory will continue to propagate with the same wave vector because of the momentum conservation law. Note that the wave vector is linearly related to the momentum via the Planck constant : that is, = . We also remind the readers that the momentum conservation law is a direct consequence of the particular symmetry of a homogeneous medium, known as the homogeneity of space [5].
The situation drastically changes for a nonhomogeneous medium, in which the momentum conservation law, generally, is not valid because of the breaking of the above-mentioned symmetry. As a consequence, in such a medium, one can find wave vectors differing from the initial wave vector . The simplest case is realized when the medium comprises two homogeneous parts with dissimilar characteristics. Such breaking of symmetry is the origin of the refraction of waves at the interface between two parts. Refraction phenomena will also be touched upon later in this book (see Chapter 23). However, our focus in the current chapter is on the particular nonhomogeneous medium with translational symmetry, which comprises scattering centers in specific points rs only, that is,
2.3
the rest of the space being empty. Here, in Eq. (2.3), the vectors , , , are three noncoplanar translation vectors, while , , , are integer numbers (both positive, 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 infinite medium with no absorption, the magnitude of the plane wave Y should be the same in close proximity to any lattice node described by Eq. (2.3). It means that the amplitude Y0 is the same at all points , whereas the phase can differ by an integer number of (see Eq. (2.1)). Let us suppose that the plane wave has the wave vector at the starting point = 0 and = 0. Then, according to Eq. (2.2), = 0. If so, at point rs, the phase of the plane wave should be equal to . Note that the change of the wave vector from to physically means that the wave obeys scattering at point (see Figure 2.1). In this chapter, only elastic scattering (with no energy change) is considered. So
2.4
where stands for the radiation wavelength. Note also that Eq. (2.4) is equivalent to Eq. (1.1) introduced in Chapter 1.
Figure 2.1 Illustration of X-ray scattering in a periodic medium.
For further analysis, we recall the linear dispersion law for electromagnetic waves in vacuum, that is, the linear relationship between the absolute value of the wave vector and the angular frequency given by
2.5
where c is the speed of wave propagation. With the aid of Eq. (2.4) and Eq. (2.5), we can express the time interval for a wave traveling between points and as
2.6
By using Eq. (2.2), Eq. (2.4), Eq. (2.5), and Eq. (2.6), we calculate the phase of the plane wave, , after scattering at point as
2.7
Since the initial phase = 0, Eq. (2.7) determines the phase difference due to wave scattering. The difference vector between the wave vectors in the final and initial wave states is known as the wave vector transfer or the scattering vector:
2.8
Substituting Eq. (2.8) into Eq. (2.7) finally yields the phase difference as
2.9
According to Eq. (2.8), different values of are actually permitted, but only those that provide a scalar product in Eq. (2.9), that is, a scalar product of a certain scattering vector and different vectors from the lattice (Eq. (2.3)), equal to an integral multiple m of :
2.10
The vector is also called the diffraction vector. In order to avoid the usage of factor in Eq. (2.10), another vector is introduced as
2.11
for which Eq. (2.10) is rewritten as
2.12
By substituting Eq. (2.3) into Eq. (2.12), we finally obtain
2.13
In order to find the set of allowed vectors satisfying Eq. (2.13), the reciprocal space is introduced, which is based on three noncoplanar vectors , , and . Real space and reciprocal space are related to each other by the orthogonality conditions
2.14
where 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 following mathematical procedure:
2.15
where stands for the volume of the parallelepiped built in real space on vectors 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]/ = / = 1, whereas a2 · b1 = a2 · [a2 × a3]/ = 0. More information on the reciprocal space construction can be found, for example, in [6, 7].
In the reciprocal space, the allowed vectors are linear combinations of the basic vectors , , :
2.17
with integer projections (hkl), known as the Miller indices. The ends of vectors , being constructed from the common origin (000), form the nodes of a reciprocal lattice (see Figure 2.2). For all vectors , 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 may exist that are related to the initial wave vector as follows:
Figure 2.2 Reciprocal lattice (black spots) and the Ewald's sphere construction. Wave vectors of X-rays in the initial and final states are, respectively, indicated by and .
2.18
where the vectors 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 translational symmetry, which should be used instead of the momentum conservation law in a homogeneous medium. Note that the latter law means = = = 0, that is, = . Graphical representation of Eq. (2.18), which leads to the famous Bragg law, is given in Figure 2.3. This important point will be elaborated in more detail below.
Figure 2.3 Graphical representation of Eq. (2.18).
Actually, Eq. (2.18) describes the kinematics of the diffraction process in an infinite periodic medium, since the presence of waves propagating along different directions , in addition to the incident wave with wave vector , is the essence of the diffraction phenomenon. According to Eq. (2.18), the necessary condition for the diffraction process is the quasi-momentum (or the quasi-wave vector) conservation law, which defines the specific angles 2 between wave vectors and , at which diffraction intensity could, in principle, be observed (see Figure 2.3). Solving the wave vector triangle in Figure 2.3, together with Eq. (2.4), yields
2.19
Note that each vector of reciprocal lattice, that is, = hb1 + kb2 + lb3, is perpendicular to a specific crystallographic plane in real space. This connection is directly given by Eq. (2.12), which defines the geometric plane for the ends of certain vectors rs, the plane being perpendicular to the specific vector (see Figure 2.4). Using Eq. (2.19) and introducing a set of parallel planes of this type, which are separated by the d-spacing
2.20
we finally obtain the so-called Bragg law:
2.21
which provides the relation between the possible directions for the diffracted wave propagation (via Bragg angles ) and interplanar spacings (d-spacings) d in crystals. By using Eq. (2.15), Eq. (2.16), Eq. (2.17), and Eq. (2.20), one can calculate the d-spacings in crystals, as functions of their lattice parameters and Miller indices, for all possible symmetry systems in real space (see Chapter 12). Therefore, measuring the diffraction peak positions 2 and calculating lattice d-spacings via the Bragg law (Eq. (2.21)) provides an important tool for solving crystal structures by diffraction methods. This line is elaborated in more detail in Chapter 12.
Figure 2.4 Schematic illustration of Eq. (2.12).
It is worth further analyzing Bragg's law in terms of the phases of propagating and scattered waves. Let us consider the fate of an incident X-ray wave with wave vector between two parallel atomic planes separated by an interplanar spacing d (see Figure 2.5). The wave crosses the first atomic plane at point I (with radius vector rI), where it is scattered, and then is scattered again by the second atomic plane at point II (with radius vector rII). After each scattering, the wave vector is changed from to . According to Eq. (2.8) and Eq. (2.9), the phase difference between these two scattered waves is
2.22
where r = rII − rI is a vector connecting points rI and rII. For specular reflection, the vector is perpendicular to the chosen atomic planes (see Figure 2.5) and then the phase difference is simply
2.23
By using Eq. (2.19) and Eq. (2.21), we find that the phase difference between two neighboring diffracted waves in the exact Bragg position is
2.24
that is, the diffracted waves are all in phase. This is the essence of diffraction physics in crystals when using the wave language.
Figure 2.5 X-ray scattering by a system of parallel atomic planes.
Note that Bragg's law, expressed in the form of Eq. (2.21) or Eq. (2.24), in fact, reflects only a necessary condition for the diffraction process and in that way helps us to calculate the angles between the incident and diffracted waves. However, it does not say much about the required orientation of a single crystal with respect to an incident beam in order to realize diffraction conditions. This information is contained in the initial Eq. (2.18), expressed in the vector form. In order to extract this information, the so-called Ewald construction within the reciprocal lattice is used (see Figure 2.2).
In this construction, the wave vector of the incident wave (in fact, reduced in length by a factor 2π) is placed within the reciprocal lattice of the investigated crystal. Since the wave vector can be moved in space without changing its direction (i.e., remaining parallel to itself), we choose the vector /2π to be ended at some node of the reciprocal lattice, which is taken now as the 0-node. After that, the starting point A of vector /2π is also well defined (see Figure 2.2). Since we are interested in elastic scattering (see Eq. (2.4)), the ends of all possible wave vectors (divided by 2π) should be located on the surface of the sphere of radius /2π = /2π = /2π = 1/λ, drawn from common center located at point A (see Figure 2.2). This sphere is called the Ewald sphere. With the aid of Figure 2.2, we can say that the diffraction condition (Eq. (2.18)) simply means that the Ewald sphere intersects at least one additional node besides the 0-node, the latter being always located on the Ewald sphere, according to the chosen construction procedure. In the light of this, the alignment procedure of the crystal to fit the diffraction conditions means proper crystal rotation (and correspondingly the rotation of its reciprocal lattice) until at least one additional node will touch the surface of the Ewald sphere.
The latter claim requires further clarification. In fact, if the length of the wave vector /2π is much smaller than the length of the smallest vector of the reciprocal lattice, , the Ewald sphere does not intersect any node of the reciprocal lattice (except a trivial intersection at the 0-node, see Figure 2.6). In other words, X-ray diffraction within atomic network, in this case, cannot be realized. Using Eq. (2.4) and Eq. (2.20), we can formulate a quantitative criterion for such a situation:
2.25
where dmax is the maximal d-spacing within unit cell of a crystal. By using Eq. (2.21), one can find the stronger constraint
2.26
Figure 2.6 Ewald's construction in the case described by Eq. (2.25).
For example, the wavelength range for visible light fits this criterion (Eq. (2.25)). As was explained in Chapter 1, visible light does obey diffraction when it meets obstacles (holes, slits, screens, etc.) with geometrical sizes comparable with λ. However, according to Eq. (2.25), visible light does not “feel” the periodicity of the atomic network.
Another extreme is realized when or
2.27
In this case, many nodes of the reciprocal lattice can simultaneously be intersected by the Ewald sphere (see Figure 2.7). It means that many diffracted waves, differing in wave vectors , can simultaneously propagate within the crystal and, if so, the diffraction process should be treated in the framework of the so-called multiwave approximation. This is typical for electron diffraction in crystals (see e.g., [8]).
Figure 2.7 Ewald's construction in the case described by Eq. (2.27).
For X-rays, as a rule, the following condition is fulfilled:
2.28
which allows the propagation of a single diffracted wave in a crystal, in addition to the incident wave, that is, the location of two nodes, 0 and , on the Ewald sphere, as displayed in Figure 2.2. In this case, the diffraction process, generally, should be handled in the two-beam approximation, which is described in detail in Chapter 3. In specially designed experiments, diffraction of a few waves can also be achieved with X-rays and then treated appropriately within a multibeam approximation (see e.g., [9]).
It is important to stress that, under the Bragg condition (Eq. (2.21)), the incident and diffracted waves in an infinite crystal describe identical quantum mechanical states since they are related to each other by the quasi-wave vector conservation law (Eq. (2.18)), which is a direct consequence of translational symmetry. These two waves strongly interact via the periodic lattice potential (in case of electrons) or the periodic dielectric permittivity (in case of X-rays) and such interaction is the subject of dynamical diffraction theory, which is considered in Chapters 3–6. If the crystal is small, the accumulated diffracted wave is rather weak compared to the incident wave and its effect on the incident beam is negligible. X-ray diffraction in small crystals is described in the so-called kinematic approximation (see Chapters 10 and 11), abandoning the interaction mentioned.
The last remark in this chapter relates to the X-ray coherence. This concept is very important to the entire field since noncoherent waves do not interfere. Note that an infinite monochromatic plane wave, by definition, is fully coherent since it has a fixed wave vector (both in its length and direction). Correspondingly, the coherence length tends to infinity since the spread of the wave vector distribution tends to zero. In reality, we have no perfect plane waves. X-rays always emanate from a source and then are restricted in space and time. For this reason, they are better described by wave packets with nonzero spread . The vector has two components: one with magnitude Δk along vector , and the other with magnitude (Δα) perpendicular to it (see Figure 2.8). Here, Δα is the angular spread of X-ray wave vectors. Correspondingly, it is acceptable to introduce two different coherence lengths, that is, longitudinal and transverse (Δα). Recalling that = 2π/λ (Eq. (2.4)) and then, , we find
2.29
2.30
Figure 2.8 Illustration of calculating the coherence length Lc.
According to Eq. (2.29) and Eq. (2.30), at a steady monochromatization degree Δλ/λ and angular divergence Δα, the coherence length is linearly proportional to the wavelength λ. It means that, generally, the Lc values will be much smaller for X-rays than for visible light. For example, for λ = 1 Å and excellent monochromatization, Δλ/λ = 2 × 10−5, Lcl = 5 μm. For comparison, even five times worse monochromatization, Δλ/λ = 10−4 for visible light with λ = 500 nm, leads to the longitudinal coherence length Lcl = 5000 μm. At long synchrotron beam lines, the angular divergence can be reduced to the level of Δα ≈ 10−6 = 1 μrad, so the transverse coherence length for X-rays in best cases reaches Lct ≈ 100 μm. Note that only crystal regions with linear sizes comparable with the X-ray coherence lengths will effectively participate in the diffraction of individual X-ray waves.
3
Dynamical Diffraction of Particles and Fields: General Considerations
Dynamical diffraction of X-rays in crystals is comprehensively treated in [10, 11]. Here we briefly describe the basic concepts and main results that are necessary for further reading through this book. The theory of dynamical diffraction is fundamentally very similar for particles (electrons, neutrons) and electromagnetic waves (X-rays) since the analyses are performed in the framework of the perturbation theory. Nevertheless, one important exception still exists: for classical particles, we use scalar fields (wave functions) and Schrödinger equation, whereas for X-rays, vector fields and Maxwell equations are used. Correspondingly, in the case of X-ray diffraction, theoretical description should take into account different X-ray polarizations (see Chapter 4). Before doing this, let us start with a mathematically easier case of electron (neutron) dynamical diffraction, which illustrates well the most essential aspects of this area of diffraction physics.
Motion of electrons in crystals obeys the nonstationary Schrödinger equation
3.1
where is the electron wave function, is periodic lattice potential, m is the mass of electron, is the reduced Planck constant, and
3.2
is the Laplace operator. Let us find first the solution of Eq. (3.1) in free space, that is, when = 0. It is natural to assume that in free space an electron exists in the form of a plane wave with amplitude :
3.3
which propagates with constant wave vector and angular frequency . Substituting Eq. (3.3) into Eq. (3.1) yields the relationship between the magnitude of the wave vector = and frequency :
3.4
which is the dispersion law for an electron wave in vacuum.
In the presence of a periodic lattice potential , we are trying to find the solution of Schrödinger equation (Eq. (3.1)) as linear combination of plane waves with different wave vectors and amplitudes :
3.5
In turn, the periodic lattice potential can be expanded in the Fourier series using the set of vectors of reciprocal lattice or, more exactly, the set of diffraction vectors = 2π (see Eq. (2.11)):
3.6
The Fourier coefficients are given by the expression
3.7
in which integration is performed over the volume of unit cell . Substituting the plane wave from the set (Eq. (3.5)) into the Schrödinger equation (Eq. (3.1)) yields
3.8
Introducing the amplitude of scattered wave as
3.9
we finally obtain the system of algebraic equations
3.10
which connect the amplitude of the incident wave with amplitudes of an infinite number of scattered waves, each corresponding to a specific scattering vector . Note that across this book (except Chapters 19 and 20), we will consider elastic scattering only (see Eq. (2.4)) and, hence, the frequency does not change as the result of the wave scattering processes.
Further analysis depends on how many nodes of reciprocal lattice, that is, vectors = /2π, are simultaneously located in close proximity to the Ewald sphere. Let us start with a simple case where there are no additional nodes (except the 0-node) at the surface of the Ewald sphere. It means that only one strong wave is propagating through the crystal, that is, the incident wave with wave vector . Correspondingly, there is no diffraction, that is, = 0. Applying this consideration to Eq. (3.10), we find
3.11
which represents the dispersion law in a homogeneous medium, differing from vacuum. The magnitude of wave vector in a medium is related to that in vacuum (Eq. (3.4)) by the following expression:
3.12
where is the electron energy in vacuum. By means of Eq. (3.12), we immediately obtain the refractive index for electron waves in a homogeneous medium with respect to vacuum as
3.13
For electron waves, > 0 and then > 1; that is, materials are optically denser than vacuum (as for visible light).
Note that for neutrons the refractive index is described by the same expression (3.13). However, because of the peculiar features of nuclear scattering, the Fourier coefficient may be positive or negative, depending on the internal structure of a particular nucleus. It means that some materials (with negative values of ) will be optically less dense than vacuum, which leads to the possibility of total external reflection for neutrons propagating from the vacuum side toward such a medium. Moreover, by cooling neutrons, that is, reducing their kinetic energy E, we can arrive at the situation that the refractive index becomes an imaginary number. It means that such ultracold neutrons cannot penetrate into particular material from the vacuum side at any angle of incidence and, hence, will stay for a while (limited only by the mean neutron lifetime of about 15 min) within the evacuated volume surrounded by the walls built of this material. This interesting effect was theoretically predicted by Yaakov Zel'dovich in 1959 and is used to study the quantum characteristics of neutrons (see, e.g., [12]). Comprehensive information on the dynamical neutron diffraction can be found in [13].
For X-rays, because of the specific features of their scattering amplitude (see Chapter 4), the refractive index is slightly smaller than 1 for all materials only if the X-ray energy is not very close to the so-called X-ray absorption edge (see Chapter 7). Therefore, for X-ray waves, all materials are regularly less optically dense than vacuum. It means that, at small incident angles, all materials will reveal total external reflection when X-rays are entering from the vacuum side at small incident angles (a few tenths of degree) with respect to the sample surface (see Chapter 23). This phenomenon allows us to concentrate the X-ray energy in ultrathin layers beneath the sample surface and is used in grazing incidence diffraction (GID) and X-ray reflectivity measurements (for details see [14, 15]).
3.1 The Two-Beam Approximation
If additional nodes of the reciprocal lattice are located in close proximity to the surface of the Ewald sphere, then diffraction phenomena occur. If the total number of such nodes is N, then dynamical diffraction is described by the system of N algebraic equations (3.10), relating the amplitudes of one incident and (N − 1) diffracted waves. Regularly, in X-ray diffraction, we have two nodes (the 0-node and the -node) being intersected by the Ewald sphere (see Figure 2.2). It implies that the system of N equations (Eq. (3.10)) is converted to only two equations, which is the basis of the so-called two-beam approximation.
The first equation is obtained by attributing the wave vector to the incident wave and the wave vector to the diffracted wave:
3.14
Since in the two-beam approximation, the quantum states of the incident and diffracted waves are related to each other by the vector of the reciprocal lattice and in that sense are equivalent, we can attribute the wave vector () to the incident wave and the wave vector to the diffracted wave. In this case, these two vectors are connected by the vector (−) (see Figure 3.1), which gives us the second equation
3.15
Figure 3.1 Wave vector construction used for deriving Eq. (3.15).
