\r\n\tCreating decent living conditions for all people and at the same time decouple economic growth from the increasing use of virgin resources and environmental impacts will be the major challenge in this millennium. This is also the essence of the United Nations 2030 Sustainable Development Goals. There are many approaches suggested for solving these problems. One is to change consumption behaviour from material products to services. Another solution is to find technological solutions to create more closed loops for materials and use fewer virgin resources. All solutions will require more of the central resource, namely energy, and the hope is that energy will be obtained from clean renewable sources. A central question is if this complex equation has solutions or if there are barriers for the development which are unforeseen today. Another important question is how long a transition to a more sustainable use of resources will take.
\r\n\tResource efficiency thus involves using the Earth's limited virgin resources in a sustainable manner while at the same time creating liveable cities and minimising impacts on the environment. It allows us to create more with less and to deliver greater value with less input. An increasing consumption of virgin resources will inevitable create international conflicts as every nation will defend its own interests.
\r\n\tThe new situation for resource use is that material and product flows have become more and more globalized which make them difficult to control.
\r\n\tThe central aim of this book is to view resource efficiency from a more complex perspective looking at several resources and the causal links between them in order to point out more new pathways towards a more sustainable use of resources.
The Darwin theory  is based on the Bragg model for the crystal considered as a family of parallel crystal planes. The X-ray wave reflection is considered as a result of successive transmission and multiple reflections from planes. In this case determination of the diffracted wave amplitude is reduced to the solution of recurrent relations between the amplitudes of transmitted and scattered waves in passing through the specified atomic plane. In essence, the Darwin theory represents a direct extrapolation of the optical task of propagation of light in a layer continuum to the case of wavelengths of the X-ray range.\n
The Evald-Laue theory  was the next stage in the development of theoretical approximations about the character of X-ray wave propagation in the crystal under dynamic scattering. The model approximations of the X-ray crystal interaction were formulated in the framework of this theory, meaning that the X-ray wavelength is comparable to the interatomic distances.\n
Therefore, the standard continual approximation for electrodynamics of continua proves to be unacceptable, and the scattering from individual charges should be taken into consideration. As is known, taking this into account results in the formalism of 3D periodic dielectric permeability ε(
In spite of a series of unconditional advantages in the interpretation and theoretical prediction of experimental results on dynamic X-ray scattering in crystals, both the Evald-Laue theory and, substantially, the Darwin theory have the principal limitation that they describe the dynamical diffraction in perfect crystals only.\n
The necessity of taking into accounts the different deviations from ideal periodicity in the crystal and, first of all, deformations resulted in the creation of the generalized theory developed by Takagi and Taupin . This is based on the approximation of a wave field in the form of superposition of the transmitted and diffracted waves with slowly varied amplitudes depending on the coordinates that leads to the Takagi-Taupin equations to the system of differential equations relative to the field amplitudes. This formalism gives the possibility to describe the dynamical diffraction in a distorted crystal since the distortions of ideal periodicity can be taken into account in the explicit form in the wave field approximation. Correspondingly, the Takagi-Taupin equations become the system of differential equations with variable coefficients.\n
It is important that the Takagi-Taupin system is shortened and the coordinate second derivatives of field amplitudes are neglected in it. On the one hand, this significantly facilitates the theoretical consideration and makes observable the solution of a series of diffraction problems in standard diffraction geometries when this simplification proves to be justified. On the other hand, the Takagi-Taupin equations become inapplicable under the conditions, for example, of grazing diffraction geometry; then, it is necessary to solve the third- or even fourth-order differential equations .\n
The principal disadvantage of the procedure of equation shortening is related to the impossibility to state correctly the boundary conditions at the crystal-vacuum interface for field amplitudes. Instead of the known classical continuity conditions for tangential components of electric and magnetic fields, the boundary conditions of the type of setting of the normal components of field amplitudes on the crystal surface become vaguely clear but not in line with the Maxwell equations. Of cause the solutions of these boundary problems prove to be applicable only for rather large (more correctly significantly exceeding the angle of full external reflection) angles of radiation incidence and yield.\n
As a result the theory faces the difficulties related to the necessity to solve the third- or fourth-order equations which become virtually overwhelming for the case of the crystal with lattice deformation when the diffraction schemes of the type of sliding diffraction are considered.\n
At the same time, the Maxwell equations are the first-order equations or the second-order ones in the case of a single wave equation when passing, for example, to an electric field. Namely, this structure of equations agrees with the mentioned classical boundary conditions. This means that the requirement of taking correctly into account the boundary conditions in any theoretical diffraction scheme leads virtually unambiguously to the known structure of wave equation following from the Maxwell equation.\n
Thus, to overcome the aforementioned difficulties, it is necessary to create the theory directly based on the Maxwell equations using model approximations of crystal polarizability in the X-ray wavelength range.\n
The model approximations for crystal and X-rays propagating in it are reduced to the following.\n
The plane monochromatic wave falls from vacuum onto the crystal. The crystal-vacuum interface is considered as a geometrical one so that the classical boundary conditions for optics prove to be applicable. The uniform wave field in the crystal is described by the fundamental Maxwell equations supplemented by the constitutive equation
The Maxwell equations, as the equations of electromagnetic waves in dielectric, when dispersion is absent, can be written in the form (designations here and below are standard):\n
These equations are supplemented by the constitutive relations:\n
As usual, we will consider that the time dependence of E and H is harmonic:\n
We have the system\n
For economy we use here and below factor 2π into k, 2πk → k. We come from system (6) to the basic equation in the standard way:\n
In the present section, we will consider the case of a perfect crystal. This will allow us to compare the conclusions of the proposed formalism to the known theoretical results.\n
We choose the crystal model χ(
Below, we will bring 2π into
To take correctly into account the contribution of different terms into Eq. (9) when using the methods of perturbation theory, it is necessary to bring it to the dimensionless form. This procedure assumes the choice of a certain characteristic spatial scale of the problem. Apparently, this is the parameter determining the reciprocal lattice in our problem, namely, the modulus of the reciprocal lattice vector:\n
Eq. (10) cannot be exactly solved. Correspondingly, it is necessary to use some method of approximate solution. There are two main ways to analyze Eq. (10). In the first case, taking into account that χ(
The approximate solution is searched in the space beyond the scattering crystal at large distances from it in the form of the first term of a series of the Born expansion. The applicability of this approach is limited by the smallness of the scattering cross section as compared to the geometrical area of the crystal section. As is known this approach results in the kinematical theory .\n
In the second case, two variants of the dynamical theory are developed from Eq. (10). For the first variant, the solution of Eq. (10) is sought in the form of Bloch wave represented in the form of an infinite series of plane waves with wave vectors corresponding to the refracted wave and diffracted waves in the crystal. This Bloch wave is interpreted as a multiwave solution of the dynamical theory. As a result, the use of expansion of χ(
Since it is not possible to solve an infinite system, one has to be limited as a rule by two equations, that is, by the two-wave approximation.\n
For the second variant of the theory, the solution of Eq. (10) is presented in the form of plane waves with the slowly varying amplitude. As a result of the two-wave approximation, we obtain the Takagi-Taupin equations which can be interpreted as the recurrent Darwin relations written in the differential form .\n
We propose here to use a new approach to the analysis of Eq. (10). The physical justification of the proposed method indicates the choice of the perturbation parameter and is as follows. The propagation of an X-ray wave in the crystal unaccompanied by the appearance of diffracted beams is adequately described by the uniform wave equation with ε = 1 + χ0. This corresponds to the propagation of an X-ray wave in the crystal as a continuum with the refraction factor with respect to the usual laws of optics. This situation can be considered typical. On the contrary the appearance of diffracted beams requires that the definite geometrical conditions be fulfilled for the wave vectors and reciprocal lattice vector. Apparently, χ
Thus, in spite of the fact that all quantities χ0, χ
We use the simplest perturbation method, direct expansion, in χ
We will restrict ourselves to the first-order expansion. We have\n
The zeroth approximation corresponds to zero power of the perturbation parameter χ
We select the solution of this vector wave equation in the form of superposition of two plane waves:\n
The reasons for this selection are the following. The zeroth approximation corresponds to the propagation of two plane transverse waves in opposite directions in the continuum with χ = const. This is a singular analog of the total field in the crystal for the case of an empty lattice. The propagation directions and the wave amplitudes remain indeterminate and are specified below by the boundary conditions at the vacuum-crystal interface.\n
The first approximation is obtained when all terms in Eq. (11) proportional to the first power of χ
We obtained the inhomogeneous wave equation. According to the perturbation theory, it is necessary to find its particular solution. Since the continuum is uniform with an accuracy of the reciprocal lattice vector
takes the form\n
Eq. (13) is obtained taking into account the condition \n\n from which follows \n\n
Now, the solution in the first order of the perturbation theory can be written:\n
Finally, the direct expansion with an accuracy of \n\n takes the form\n
It follows from Eq. (15) that in addition to the direct (
However, this position radically changes when any denominator in Eq. (15) approaches zero. In this case \n\n, and we cannot consider a small correction to
This condition is well known: it is the Laue condition for X-rays, and therefore there is no need to detail its physical sense. Note only that all geometric constructions following from the Laue condition appear in this case as a natural consequence of validity violation of the direct field expansion in the parameter χ
Thus, the wave field structure principally changes for certain
Thus, it is necessary to modify the direct expansion near the
There are different methods to modify the direct expansion. All of them are directed to solve one problem: to obtain a so-called uniformly acceptable expansion near the values of parameters interesting for us. The method of multiple scales is most favorable for our investigation . However, method modification is necessary having in mind the vector character of the problem.\n
The main idea of the method for the considered problem is the following. The wave field singularities appear for different spatial scales determined by a small parameter of expansion of χ
Thus, we will search for the approximate solution of Eq. (10) for the most interesting case in the area of the Bragg maximum when the Laue condition
Hereinafter, we will restrict ourselves to the first order of expansion; correspondingly, we will consider two spatial scales
Then, for rot
that is, operator rot is linear relative to the carried-out substitution. The index of the operator signifies the space in which it operates. Using this property we obtain\n
As was indicated above, the interaction of the field with continuum has a parametric character. This means that along with the field expansion it is also necessary to expand the wave vector
We substitute all expansions into Eq. (10):\n
In this case, χ(
Namely, the initial approximation (unperturbed state) and subsequent ones are obtained when the constants of powers of the perturbation parameter χ
Let us demonstrate this procedure. As before the zeroth approximation, (the unperturbed equation) has the form of the standard vector wave equation for transverse waves propagating in continuum:\n
However, in contrast to the direct expansion, the operator rot acts here only on the single spatial scale
This structure of the wave field supposes strict fulfillment of the diffraction Laue condition
The condition of wave transversity (
The following first-order approximation to leads to the inhomogeneous equation:\n
Operator \n\n (gradient) acts here on r1. Eq. (21) is obtained taking into account \n\n and the additional condition \n\n. This means the following. The quantities \n\n are considered as the perturbed amplitudes of the corresponding plane waves. The two first terms in the right side of Eq. (21) generate the secular components in expansion which is seen from the abovementioned particular solution. In other words they provide parametric resonance in the system. Consequently, it is necessary to eliminate these terms in order to obtain the uniform approximation near the Laue condition. Then, we obtain the following system of vector equations:\n
In contrast to the usual scalar system, additional limitations are necessary to solve Eq. (22). The issue is that
The particular solution of the inhomogeneous equation\n
in the resonance case takes the form\n
It is seen from here that the infinite increase in the wave amplitude is associated with the wave vector \n\n in the direction coinciding with
Thus, multiplying scalarly the first equation of the system by
The obtained system (23) is virtually the dispersion relation written in the differential form for the transmitted and diffracted waves in the two-wave approximation. As follows from the presented conclusion, the possibility to obtain this system is dictated by the choice of the zeroth approximation. Namely, the wave vectors of transmitted and diffracted waves must have the identical component \n\n that leads to the scattering interpretation as a result of reflection from the atomic plane.\n
We pass from the differential form of system (23) to the algebraic one; for this purpose we make the following substitution \n\n, \n\n, where
This system should be considered as the condition of field limitation in the specified direction of field propagation that is related to the structure parameters of the crystal and to the diffraction geometry. As for the physical meaning of the considered problem, this limitation should be in the direction of the normal inside crystal.\n
Here, \n\n are the direction cosines of the corresponding wave vectors, \n\n=\n\n.\n
The nontrivial solution of Eq. (25) requires the corresponding determinant to vanish:\n
The solution of quadratic equation relative to
Then, solving, for example, the first equation of system (25) relative to
Finally, going back to the initial variables, we represent the wave field in the crystal in the following form:\n
Here, the constants
The equation for the wave field is simplified for the case of semi-infinite crystal when the wave reflection from a lower crystal face is eliminated. In this case the choice of sign in
Finally, the wave field in the crystal takes the form\n
where the constant
Eq. (32) describes the uniform wave field in a perfect crystal near the Bragg maximum.\n
To compare the obtained expression to the known results and to the experiment, it is necessary to go to the angular variable which connects with the deviation from the exact Bragg angle Δθ. In this case, the parameter \n\n should be expressed by Δθ. As was mentioned the parametric resonance (diffraction Laue condition \n\n) is determined only by the component \n\n directed along
Here, we introduce the standard (with an accuracy of refraction) angular variable:\n
As is known, two principal schemes are considered in the diffraction theory, by Laue (\n\n) and by Bragg (\n\n). In the case of Bragg diffraction, the wave field structure will qualitatively differ depending on the considered angular range.\n
In particular in the range\n
the waves will exponentially attenuate along the normal surface into the crystal. In terms of the qualitative theory of differential equations, the solution will be unstable. The known interpretation [10, 11] leads to the conclusion of the expulsion of the wave from the crystal and the formation of a diffraction maximum. Thus, the indicated condition separates the stable solutions (oscillating type) from unstable ones (exponential type), that is, provides the equations of transition curves of the parametric plane (\n\n) [10, 11].\n
The width of the unstable region, the region of exponential wave attenuation, is determined by the expression\n
or proceeding to the angular variable\n
This is the known expression (with an accuracy of refraction) for the angular width of the Bragg table for the case of the semi-infinite perfect non-absorbing crystal. The extinction length \n\n is determined as a decrement of wave attenuation in the point Bragg position:\n
We now summarize the intermediate stage. The use of the generalized method of many scales allowed us to obtain the system of basic equations describing the behavior of wave field near the Bragg maximum in the two-wave approximation. This system is a direct analog of the dispersion relations of the Ewald-Laue theory and the Takagi-Taupin system of the generalized dynamic theory. The substantial difference of the developed variant of the theory consists in the cancelation of the shortening procedure of equations by neglecting the second derivatives. The principal moment here is the expansion in χ
Comparison of the obtained results to that known from the Takagi-Taupin theory shows the complete correspondence both in the qualitative interpretation of types of the solutions obtained in the different angular ranges in the case of Bragg diffraction and in the analytical expressions for the width of the Bragg maximum and the extinction length. This correspondence indicates that in spite of the formal representation of χ(
However, the value of the theory developed here shows itself to a large degree when the boundary conditions are taken into account that takes the explicit expressions for the reflection coefficient. Therefore, we consider now the boundary conditions and determine the principal difference between our approach and known variants of the dynamical diffraction theory.\n
According to Eq. (32), the expression above obtained for the wave field in the crystal depends on the constant
We find out now the differences appearing when the boundary conditions are strictly taken into account in our theory.\n
The reflection coefficient is determined as a ratio of the averaged values of normal components of the pointing vector for the diffracted and incident waves:\n
As is known, the boundary conditions require continuity of the tangential components of electric and magnetic fields which is a consequence of the uniformity of the problem along the surface. The boundary problem disintegrates into successive steps related to the determination of . In this case the elementary problem is considered at each stage, namely, the determination of the relation between the amplitudes of incident, transmitted, and secularly reflected waves. The solution of this problem leads to the known Fresnel formulas:\n
The obtained relations allow us to determine not only the diffracted wave but also the specularly reflected wave, which principally distinguishes our approach from the formalism of the Takagi-Taupin equations.\n
Eqs. (38)–(40) solve the problem of the determination of field amplitudes under the conditions of sliding noncoplanar diffraction when the incident and diffracted waves are near the critical angles of total external reflection (TER). They are analogous to the relations obtained in  where this problem was solved by the fourth-order dispersion equation.\n
The correspondence with the Takagi-Taupin theory must be undoubtedly fulfilled for the case of large angles of incidence and yield of the diffraction wave. Really in this case, the amplitude of specular wave tends to be zero, and the reflection coefficient takes the form\n
This is the known expression for the coefficient of reflection from a perfect half-infinite crystal, i.e. the Bragg table. In addition to this in the case of extremely asymmetric diffraction when the diffraction wave leaving the crystal is almost parallel to the surface, the amplitude is modulated by the factors taking into account the refraction of transmitted and diffracted waves at the crystal-vacuum interface and the diffraction wave interaction related to the vector
The proposed covariant (it may be named nonstandard) theory allows the generalization for the case of the crystal with lattice deformations. Therefore, a uniform approach to the account of deformations and other distortions in all diffraction schemes is realized.\n
It is known that Takagi-Taupin equations were obtained using the model concept of the character of lattice distortions which makes it possible to directly take into account the displacement of atomic planes in the three-dimensional periodic function of crystal polarizability. This concept allows for describing lattice displacements and strain using the methods of classical theory of elasticity formulated within the continuum approximation. To satisfy the condition of the dynamic character of scattering in the Takagi-Taupin theory, the lattice distortion is assumed to be rather weak; correspondingly, the strain is small. The character of variation in strain is implicitly taken into account only when the wave field is chosen in the form of a Bloch function with slowly varying amplitudes; thus, the question of applicability of this concept remains open.\n
There is another limitation of the Takagi-Taupin theory which is related to the correct statement of boundary conditions. Mathematically, the Takagi-Taupin equations form a first-order differential system with respect to the scalar amplitudes of transmitted and diffracted waves. The procedure of determining these amplitudes at the crystal-vacuum interface does not correspond to the classical boundary conditions.\n
This discrepancy is due to the fact that the Takagi-Taupin equations are obtained disregarding the second derivatives of the field amplitudes with respect to coordinates. Thus, the boundary conditions impose fundamental limitations on the applicability of the Takagi-Taupin equations (e.g., when analyzing extremely asymmetric diffraction schemes).\n
In this section we generalized the covariant theory of dynamic diffraction which was presented in the previous section, to a crystal with a distorted lattice. This approach makes it possible to formulate the limitation on the character of variation in strain for the applicability of the Takagi-Taupin equations. In addition, the equations obtained can be applied (as in the case of an ideal crystal) for arbitrary diffraction schemes.\n
The polarizability χ(
It can be seen that the general structure of χ(
As in the case of an ideal crystal, Eq. (43) should be reduced to a dimensionless form. To this end we will use the length
The approximate solution to Eq. (44) in the Takagi-Taupin theory is known to be sought after in the form of plane waves with slowly varying amplitudes. Finally, the Takagi-Taupin equations can be derived from Eq. (44) with allowance for the two-wave approximation .\n
The covariant dynamic theory in the case of an ideal crystal is due to the fact that the Fourier component of polarizability χ
Recall that in the case of an ideal crystal the direct expansion of the solution to Eq. (44) in the parameter χ
In this case, the scattered wave amplitude increases unlimitedly. This is due to the fact that the diffraction wave can be found as a particular solution to the inhomogeneous wave equation:\n
which according to Eq. (13) has the form\n
For a deformed crystal, the plane wave is replaced with
At the same time from the physical point of view, the displacement field changes significantly at distances much larger than the lattice parameter. This limitation is substantiated in particular by the fact that we describe lattice distortions within the continuum approximation. In this case the approximate solution to Eq. (46) can be obtained similarly to Eq. (47):\n
Correspondingly, the direct expansion of the solution to Eq. (44) for a deformed crystal is similar to that for an ideal crystal and is determined by the wave superposition in the form\n
The most general limitation on the strain in the crystal is imposed by the requirement for the smallness of the strain tensor
where ε0 is the strain amplitude in the structure. For sufficiently regular displacement fields, this requirement is reduced to the condition ε0 ≪ 1. It follows from Eq. (50) that in the case of a deformed crystal when the aforementioned conditions are satisfied the applicability of direct expansion remains limited because of the Laue resonant condition (45) where the amplitudes of waves excited in the crystal increase unlimitedly. Thus, the method for obtaining an approximate solution must also be modified in the case of a crystal with a distorted lattice. The choice of the modification technique depends on the character of the displacement field in the crystal and obviously cannot provide a universal solution for all physically possible cases. We will consider the most widespread situation where the displacement field changes at distances comparable with the extinction length. In this case the multiscale method which is the basis of the covariant theory of diffraction in an ideal crystal can directly be extended to a deformed structure.\n
The strain field in a crystal may have various forms depending on the nature of lattice distortions. These forms can mathematically be represented by setting different structural parameters (e.g., the thicknesses of epitaxial layers and transition regions between layers, sizes of lattice-strain regions caused by various defects, etc.). The values of these parameters are determined by not only the strain amplitudes ε0
Thus, the parameter χ
Thus, a consideration of different types of lattice distortions generally calls for taking into account different characteristic spatial regions; this approach completely corresponds to the main concept of the multiscale method . Obviously, different modifications of the method are required depending on the specific structure of Eq. (52). We will consider the simplest case of one scale which leads in a particular case to Takagi-Taupin equations.\n
Let us consider the atomic plane displacement occurring at some effective layer thickness
where \n\n is the displacement model and ε0 is the strain amplitude.\n
According to the multiscale method , the main equation (Eq. (44)) is analyzed near the Bragg maximum on different spatial scales (determined by χ
Then, according to Section 2, the field, the operator rot, and the wave vector
In contrast to the case of an ideal crystal, χ(
The parameter 1/
We will follow the scheme for solving Eq. (55) that was reported in Section 1. As for an ideal crystal, the initial approximation can be written in the form as in Eq. (18). Accordingly, the operator rot acts on only one spatial scale
The first-order approximation with respect to χ
Here, the gradient \n\n acts in the space
As can easily be found from Eq. (47), two first terms in the right-hand part of Eq. (60) generate the divergence of a particular solution. The following vector system can be obtained by excluding these terms from Eq. (60):\n
According to Section 2, to satisfy the field boundedness condition, the projections of Eq. (61) on the corresponding unit vectors
According to Section 2, the parameter \n\n can be expressed in terms of the deviation from the exact Bragg angle Δθ as follows:\n
Eq. (62) describes the changes in the wave amplitudes on the scale
Accordingly, Eq. (62) is reduced to the form\n
In a particular case of coplanar diffraction in the
Under the assumption that the displacement field in Eq. (64) depends on only
System (64) has a more general character because it was obtained without additional limitations on the Takagi-Taupin equations which is related to the rejection of the second derivatives of the field amplitudes with respect to coordinates. It is known that the boundary conditions at the crystal-vacuum interface cannot be correctly taken into account due to these limitations. Finally, the Takagi-Taupin equations cannot be applied to extremely asymmetric diffraction schemes.\n
Let us show how the consideration of the boundary conditions yields explicit expressions for the amplitudes of diffracted and specularly reflected waves for arbitrary angles of incidence. For simplicity we will consider diffraction in a semi-infinite crystal in the case of σ polarization where the vectors
where the constant
Finally, the wave field in the semi-infinite crystal near the Bragg angle of incidence can be described by an expression that formally corresponds to an ideal crystal:\n
The procedure of solving the boundary problem which can be reduced to a successive establishment of relations between wave amplitudes having common tangential components of the electric and magnetic fields at the crystal-vacuum interface remains the same. Therefore, one can use the expressions for amplitudes obtained in Section 2. As a result we arrive at formulas that are similar to the Fresnel formulas:\n
The following designations are introduced here:
Eqs. (68)–(70) can be used to find the field amplitudes for arbitrary angles of incidence including those in the vicinity of the critical total reflection angles. In particular the formula for reflectance can be found from Eq. (70) as follows:\n
In a particular case of large angles of incidence, Eq. (71) can be simplified, and Eq. (70) yields the following expression for the reflectance (which can be derived from the Takagi-Taupin equations):\n
The variant of the dynamic X-ray diffraction presented in the present work is based on direct analysis of Maxwell equations for the definite model representations of the field-medium interaction taking into account the lattice presence which agree as a whole with the Ewald-Laue theory. This analysis proves to be available when the method of many scales adapted to the vector character of the problem is used. In this case the magnitude \n\n is the parameter of expansion that corresponds in full to the physical character of the problem. This correspondence is reflected in the mathematical structure of the analyzed field equation in the crystal under conditions of dynamic scattering.\n
The expressions obtained for the main field characteristics in the Bragg maximum region following from the qualitative singularities of the field propagation correspond to the known results of the dynamical theory. However, the correct use of the boundary conditions leads to an expression for the reflection coefficient that substantially differs from the classical one for the case of extremely asymmetric diffraction schemes. In addition the presented approach provides the amplitude of specularly reflected wave under conditions of dynamic diffraction, which cannot be apparently obtained in the framework of traditional approaches.\n
In the present work, we do not state the problem to analyze the features of dynamic scattering in the sliding diffraction geometry.\n
In conclusion, we note the most important in our opinion differences and advantages of the approach developed in the present work.\n
The second-order wave equation analyzed without any additional assumptions of the possibility of the interaction of refracted and scattered waves automatically results in dynamical scattering character; in this case the kinematical scattering can be considered to a certain extent as an artificial process having limited applicability. The diffraction Laue conditions appear as a result of natural limitations of the direct expansion of the solution in the resonance case.\n
In the framework of the developed theory, the total consideration of different geometrical diffraction schemes including sliding geometry and other surface variants proves to be possible. In this case the order of the dispersion equations does not change. This situation is related to the effective decomposition of the problem into the construction of the uniform wave field in the crystal and the determination of field amplitudes according to the boundary conditions.\n
Determination of the wave field as a whole without decomposition into refracted and scattered waves is the advantage of the theory. It is clear that this feature of the theory is most important for analysis of secondary diffraction processes.\n
We have generalized the covariant theory of dynamic X-ray diffraction to the case of a crystal with lattice deformation. In this case the displacement field is specified, starting from model representations used in Takagi-Taupin dynamic theory. In our case (in contrast to the formalism of the Takagi-Taupin equations), lattice distortions have been taken into account on various spatial scales that were different from the scale of the lattice period.\n
The displacement field was also a slowly varying function of coordinates. If the displacement field is considered on one spatial scale on the order of the extinction length, then the particular case of fundamental equations for the field amplitudes is obtained as a result.\n
In precisely this case, we have the same result as that of the Takagi-Taupin equations. By doing so we have shown possible restrictions on the applicability of the Takagi-Taupin equations to describing dynamic diffraction in crystals using various deformation models.\n
At the same time, the presented theory offers an opportunity for successively taking into account displacement fields of various types implemented on different spatial scales (that are larger or significantly smaller than the extinction length).\n
The possibility of the correct application of boundary conditions including cases of extremely asymmetric diffraction schemes in covariant theory for ideal crystals is also wholly retained for crystals with lattice distortions. Such a situation is due to the fact that the solution of the diffraction problem proper is not related to the boundary conditions; in particular the order of fundamental equations of the theory remains the same for arbitrary diffraction geometry.\n
Maize is an important crop plant and model organism. The maize karyotype was first characterized by the observation of pachytene chromosomes obtained from pollen mother cells, since the pioneering work by McClintock . The early cytological maps were constructed based on the identification of chromosome relative lengths, arm ratios, heterochromatin patterns, prominent chromomeres, and nuclear organizer region [2, 3, 4, 5]. Structures containing heterochromatin were described: heterochromatic knobs, centromeric heterochromatin, B chromosomes, abnormal chromosome 10, and nucleolus organizer region localized on chromosome 6 . Chromosome abnormalities were detected in several investigations, and collections were organized containing reciprocal translocations (A-A translocations), B-A translocations (interchanges between B chromosome and arms of the A set), inversions, and trisomics, available at the Maize Genetics Cooperation Stock Center  (www.maizegdb.org). These materials have been important tools for gene mapping.
The somatic chromosomes were identified by the C-banding procedure which was useful for the identification of chromosomal abnormalities in callus cultures [8, 9, 10]. The unequivocal identification of the somatic chromosomes is difficult due to their degree of condensation, and the use of the C-banding procedure was supplemented by an analysis of pachytene chromosomes of the lines from which callus cultures were derived. C-bands correspond to knobs visualized on meiotic chromosomes .
The characterization of meiotic and somatic chromosomes was improved by fluorescence in situ hybridization (FISH) using as probes repeated DNA sequences and genes, thus allowing the study of the molecular structure of chromosome components, such as centromere, neocentromere, B chromosome heterochromatic knobs, and gene mapping [12, 13, 14, 15, 16, 17, 18, 19, 20].
The maize chromosome structure has been extensively reviewed [21, 22, 23, 24, 25]. In the present review, we focus on the involvement of heterochromatic knobs on the occurrence of chromosome abnormalities in maize callus cultures. The size and number of knobs are variable, and they may be present in each of the 10 chromosomes of the complement at fixed positions in modern maize and its relatives, including species of
One genetic effect attributed to knobs is their influence on recombination [6, 19], and it was revealed that knobs in heterozygous condition can reduce local recombination . Another interesting genetic effect of knobs is their activity as neocentromeres resulting in meiotic drive. This meiotic event is a mechanism by which regions of the genome are preferentially transmitted to the progeny. In maize, meiotic drive is due to an uncommon form of chromosome 10, the abnormal chromosome 10 (Ab 10). In the presence of this chromosome, the knobs of other chromosomes are converted into motile neocentromeres. Thus the knobbed chromosomes preferentially segregate during female meiosis [30, 31]. The origin of maize polymorphism, including size and number, has been discussed in several reports, and it was proposed that meiotic drive was responsible for the evolution of knobs . Recently, a cluster of eight genes on Ab10 was identified, called
The effect of knobs on chromosome break and origin of abnormalities in maize callus culture is presented in this review.
Callus culture is an important step for genetic transformation in plants. The identification of maize genotypes showing high ability to form embryogenic callus type II (friable) and regenerate plants has progressed since the report by Green and Phillips . The genotypes identified since then were adapted to temperate regions , and maize genotypes of tropical and subtropical origin have also been shown to produce friable type II calli capable to develop somatic embryogenesis [36, 37, 38].
Various studies have shown the occurrence of cytogenetic and genetic variability in plants regenerated from maize callus cultures [39, 40]. This so-called somaclonal variation  is undesirable when genetic stability is required, but interesting for the study of mechanisms that give rise to chromosome abnormalities. Chromosome breakage associated with heterochromatin was shown in several plant callus cultures [8, 9, 10, 42, 43, 44, 45, 46, 47, 48].
Breakpoints involved in chromosome abnormalities associated with heterochromatin were previously detected in maize regenerated plants. The analysis of pachytene chromosomes of these plants revealed that most breakpoints were localized in chromosomes bearing a knob. The authors hypothesized that late-replicating heterochromatin would replicate later in tissue culture, giving origin to bridges in anaphases and occurrence of breakage between the knob and centromere . This would explain the presence of knobs in chromosomes involved in abnormalities observed in regenerated plants. The authors identified in meiotic cells alterations in chromosome structure, such as translocations, intercalary deficiencies, and heteromorphic pairs in 91 of 189 plants regenerated from callus cultures originated from an Oh43-A188 genetic background .
The first reports on breakage-fusion-bridge (BFB) cycles were made by McClintock [49, 50]. In investigations on the behavior in successive nuclear divisions of a chromosome broken at meiosis, it was shown that the chromatid type of BFB cycle initiated by broken chromosome ends occurs in gametophyte mitoses and in the endosperm. In the zygote the broken chromosome ends heal. BFB cycles have also been observed in other species including wheat (
Investigations of mitotic cells in maize callus cultures detected anaphase bridges resulting in delayed separation at knob regions and typical bridges originated from dicentric chromatids. The observation of C-banded anaphases showed that the chromatids were held together at C-band sites (corresponding to knob) . Typical bridges with and without C-bands were observed. These events were interpreted as derived from a chromatid type of BFB cycle initiated by chromatids that were broken during the primary event.
The analysis of abnormalities in chromosomes 7 and 9 of maize callus cultures gave evidence of their origin from BFB cycles [8, 9, 10]. As illustration, we show here the mechanism that would have originated these abnormalities. The callus culture was induced from a hybrid between two sister inbred lines derived from a tropical maize variety (Jac Duro [JD]). These JD lines possessed the same knob composition: K6 L2, K6 L3, K7S, K7L, K8 L1, K8 L2, and K9S [8, 20]. K refers to knob, the number identifies the chromosome, and S refers to short arm and L to long arm. Numbers 1, 2, and 3 refers to knob positions, according to the literature . Thus, chromosome 7 possessed large knobs on both arms, and chromosome 9 had a very large knob on the short arm. Therefore, these chromosomes were more prone to suffer alterations.
An abnormal chromosome 7 carrying two knobs on the short arm was observed in metaphases of a callus culture designed 3–57. This abnormality was interpreted as being a deficiency-duplication (Df-Dp) derived from a BFB cycle and healing of the broken arm, for it was observed in various cells of the culture . Figure 1 [9, 25] shows the mechanism that would have originated this aberration. The two knobs on the short arm would bear a deficiency in the terminal region (RTD). Therefore this abnormal chromosome 7 possessed reverse tandem duplications of these knobs and of a segment designated “b.”
This chromosome 7 carrying a deficiency on K7S and duplications of the knob and of a “b” segment (Df-Dp7) was stable in culture and was transmitted to regenerated plants. Thus, R0 plants regenerated from the 3–57 culture were heterozygous for this chromosome alteration. R1 and R2 plants were recovered and analyzed. Homozygotes for normal chromosome 7 and heterozygotes for the Df-Dp7 were detected. Plants homozygous for the Df-Dp7 were not recovered. Presumably, seeds carrying homozygotes were not viable. Figure 2A [9, 25] shows a metaphase of a regenerated plant homozygous for normal chromosome 7, and Figure 2B shows a metaphase of a plant heterozygous for the aberration. The distal knob (K7S) is subterminal, for there is a tiny terminal euchromatic segment on the short arm.
Fluorescent in situ hybridization (FISH) using the telomeric sequence (TTTAGGG)6 showed signals in all the somatic chromosomes of the regenerated plants, including the Df-Dp7 chromosome (Figure 2C–E) [9, 25]. This result gives evidence of telomere healing at the end of the broken short arm. In these DAPI-stained metaphases, bands corresponding to the knobs could be clearly visualized. In the less condensed metaphases, the telomeric signals could be detected at the end of the euchromatic segment of the duplicated short arm.
The healing of chromosome ends, i.e., the addition of telomere sequences to the broken chromosome ends, has been observed in diverse plant species. In wheat, FISH telomeric signals were detected at the broken ends of deleted chromosomes and at the centromeric regions of telocentric chromosomes [52, 53]. The expression of telomerase has been reported for diverse plant tissues, such as the meristematic tissue and suspension cell cultures . In barley, there was a decrease in the number of telomeric sequences in differentiated cells, and the number of telomeric sequences increased in callus cultures . High telomerase activity was observed in calli derived from tobacco plants, while in leaves the activity was very low . In wheat, during the divisions of the gametophyte, dicentric chromosomes undergo BFB cycles. De novo addition of telomere sequences occurs gradually during the early mitotic divisions in the sporophyte .
The present study showing the telomere healing of the broken short arm of chromosome 7 gives evidence of telomerase expression in maize callus culture. The addition of telomeric repeats occurred on a euchromatic region, which was certainly non-telomeric.
The meiosis of the regenerated plants heterozygous for the Df-Dp chromosome 7 was normal. The terminal euchromatic segment was clearly observed at pachytene stage on the duplicated short arm. In the diplotene and diakinesis stages, a heteromorphic pair corresponding to chromosome 7 was observed, as expected for heterozygotes bearing a duplication .
C-banded metaphases of subcultures prepared after 18 months of the initiation of the 3–57 callus culture were analyzed during a cultivation period from the 18-month-old original culture to 42-month-old cell lines. The subcultures were designated as cell lines 1-MS2, 2-MS-2, 1-MS1, 2-MS1, 1 N6, and 2 N6. Feulgen-stained anaphases were also observed.
The investigation of mitotic instability by the analysis of Feulgen-stained anaphases showed abnormalities similar to those previously described  and shown in Figure 3: (i) bridges resulting from delayed chromatids held together at knob sites (Figure 3A), (ii) broken bridges (Figure 3B), (iii) typical bridges (Figure 3C, D), and (iv) fragments (not shown) . The analysis of the frequency of these abnormalities showed a tendency of decreasing frequency with time in culture. Three samples of each cell line were harvested in different periods of cultivation, except 1-MS1 from which seven samples were analyzed. The frequency of anaphase abnormalities observed varied from 4 to 10% in the first sample and from 0.67 to 5.33% in the last sample. This tendency of decreasing frequency was a consequence of the healing of the broken chromosomes, therefore, avoiding an accumulation of BFB cycles , as discussed below. Interestingly, the total frequency of abnormalities varied from 0.67 to 10%, and this result was quite similar to the ones observed in a previous study of 5-month-old cultures derived from related inbred lines .
The analysis of the cell line pedigree showing the types of chromosomes 7 and 9 observed in C-banded metaphases in different subcultures of the six cell lines is displayed in Figure 4 [9, 25]. A karyotype diversity among cell lines was detected in this analysis, but homogeneity within some of them was observed in samples harvested at different age transfers. Then, new abnormal chromosomes were stable in different subcultures. Gross aberrations were not observed in chromosomes 6 and 8 that possess knobs smaller than those found in chromosomes 7 and 9.
Different types of abnormal chromosomes 7 and 9 were observed in the cell lines. In the original 18-month-old callus culture 3–57, two types of chromosome 7 were detected. One of the chromosomes possessed two knobs on the short arm (K7S) corresponding to the Df-Dp chromosome 7 described above, and the other type possessed K7S on an interstitial position of a duplicated short arm (Figure 4). This chromosome would have originated from a mechanism similar to that shown in Figure 1 [9, 25]. The following types of chromosome 7 were distinguished in the cell lines (Figure 5A): 7A, normal type, with a terminal K7S and a subterminal K7L; 7B, with a duplicated short arm and a subterminal K7S; 7C, with two knobs on the short arm and a terminal euchromatic segment (similar to Figure 2B); 7D, similar to 7C, but without the terminal euchromatic segment; 7E, similar to 7D, with a smaller deficient and terminal K7S; 7F, with a larger short arm, a very large interstitial K7S and without the K7L on the long arm. The 7A, 7B, and 7C types were found in the original 18-month-old culture (Figure 4). Figure 5B, C illustrates the 7D and 7B types, respectively, and the 7E type can be seen in Figure 5D. Figure 5E illustrates the 7C chromosome. The 7F type can be seen in the pedigree of the 1-MS1 cell line (42-month-old culture, Figure 6) .
Different types of altered chromosome 9 were also observed in the samples of cell lines (Figure 5A) : the normal type corresponds to 9A; a smaller terminal K9S corresponds to 9B; a smaller subterminal K9S corresponds to 9C; 9D is a chromosome without the knob; and 9E is a minichromosome derived from chromosome 9. The 9A, 9C, and 9E types can be seen in Figure 5B, and the 9B type is shown in Figure 5D, E. Figure 6 shows the 9D type, which appears in the 31-month-old subculture of the 1-MS1 cell line. This figure illustrates the different types of chromosomes 7 and 9 detected in 1-MS1 cell line [9, 25].
Therefore, the analysis of metaphases of the cell lines showed new abnormalities in chromosomes 7 and 9. The occurrence of delayed chromatid separation and bridges in anaphases provided evidence of BFB cycle events, and healing of the broken chromosomes could be inferred by the stability of the same abnormal chromosome in different subcultures of the same cell line [9, 25].
In most cell lines, the original abnormal chromosomes 7 (7B and 9C types) were maintained. The 7E type (with a smaller distal K7S) was found in the 42-month-old subculture of the 2-N6 cell line (see Figure 4) [9, 25]. The 7D type (chromosome 7 without the terminal euchromatic segment) was observed in the 1-MS1 and 2-MS1. These data suggest that cells bearing the original Df-Dp chromosome 7 (7B or 7C types) were highly adapted in culture and that the new types (7D and 7E) found in some subcultures were derived from the original altered chromosome 7 (7C type) through new events of delayed chromatid separation at the knob region and breakage. The 7F type would be a new alteration of the normal chromosome and was detected in the 42-month-old subculture of the 1-MS1 cell line (Figures 4,6) [9, 25]. Its origin would be through a delay in sister chromatids on K7S at anaphase, and an amplified subterminal knob would appear if the duplicate knob did not separate and a breakage occurred at an adjacent euchromatic region. A delayed separation of chromatids on K7L at anaphase, followed by breakage, would explain the absence of this knob in the 7F type.
Chromosome 9 suffered alterations in most cell lines, except for the 1-MS2 and 1-N6 cell lines. The 9D type (K9S deleted) was detected in the 2-MS2 and 2-MS1 cell lines, and the 9B type (partial deletion of the knob) was observed in the 2-N6 cell line (see Figure 4 [9, 25]). A total or partial deletion of K9S would have occurred after a delay of separation of the chromatids on this knob region and breakage totally or partially eliminating the knob or a segment of it. Interestingly, in the cell line 1-MS1, two types of chromosome 9 appeared, the 9C displaying a subterminal smaller K9S (9C type) and a chromosome without the knob (9D type). In addition, a minichromosome (9E type) appeared in the subcultures possessing one of these abnormal types. These abnormalities could have resulted from the mechanism suggested in Figure 7 [9, 25]. The primary event would be a delay in the separation of chromatids at K9S region followed by breakage originating a deficient knob. Then, two types of BFB cycles, the chromosome  and the chromatid types, would have originated the 9C, 9D, and 9E chromosome types. The 9C and 9E chromosome types were observed in several subcultures, thus providing evidence of healing of the broken chromosome ends. In the cell lines analyzed, abnormalities were detected only in chromosomes 7 and 9. These alterations were derived from a primary event of chromatid delayed separation at knob sites in anaphases, followed by breakage and BFB cycle. The presence of large knobs in these chromosomes would lead to this kind of primary event. A case of elimination of chromosome segments from knobbed chromosomes was reported by Rhoades and Dempsey . In the presence of B chromosomes, a bridge formation would occur due to delayed replication of the knob at the second microspore division. Chromosomes containing large knobs would be involved more frequently in this kind of event.
The observation of mitotic and meiotic aspects of an amplification of the knob localized on the long arm of chromosome 7 (K7L), in plants regenerated from a long-term callus culture designated 12-F, was carried out aiming to investigate the origin of this amplification. The 12-F original culture was 28 months old when the R1 plants were obtained . The original callus 12-F was heterozygous for the amplified K7L. Therefore, segregation was expected in R1 progenies derived by selfing R0 plants. Plants homozygous for the normal and amplified K7L, and plants heterozygous for the amplified K7L, were recovered (Figure 8) . The frequency of plants homozygous for the amplification was lower than expected according to Mendelian segregation, while the frequency of plants homozygous for the normal K7L was higher than expected. The frequency of heterozygotes was according to the expected value.
Some plants whose karyotype was investigated were selected for meiotic analysis. The homologous chromosomes were completely synapsed on knobs and terminal euchromatic segment on the long arm at pachytene in plants homologous for normal and amplified K7L (Figure 9A, B) . In plants heterozygous for the amplification, the knobs and the terminal euchromatic segments were completely synapsed in some cells (Figure 9C) , but synapsis failure was also detected in these regions (Figure 9D) . In chromosomes bearing the K7L amplification and in normal chromosomes, the size of the distal euchromatic segment was similar, but the size of the amplified knob was significantly larger than the normal knob. In a possible case of delay of chromatid separation at this knob site followed by breakage and a BFB cycle, the distal euchromatic segment would be lost as discussed below .
Other abnormalities such as translocations, inversions, duplications, and deletions were not found in the chromosomes of these plants derived from a long-term callus culture .
The analysis of microsporocytes at the diakinesis stage showed the presence of two types of univalents: one larger with two C-bands, thus corresponding to chromosome 7, and a small one. The frequency of univalents was low for both types. The frequency of the large univalents in heterozygous plants was higher (2.88%) than control plants (0.55%). Differences were not observed in the frequency of small univalents in the heterozygotes (1.14%) in comparison with control plants (1.92%) . Therefore, the meiosis was normal in most microsporocytes, and R2 progenies were also obtained.
The investigation of short-term cultures derived from inbred lines and hybrids related to the inbred line donor of culture 12-F showed interesting alterations on the long arm of chromosome 7 bearing K7L. The cytogenetic analysis of these cultures detected abnormalities in chromosomes 7 and 9 and other chromosomes with and without knobs. Here we focus only on alterations in the long arm of chromosome 7 to infer the origin of these abnormalities, aiming to understand the origin of the K7L amplification observed in plants derived from culture 12-F. A total of 5223 cells of the callus cultures from 6 genotypes were examined. In three cells from different cultures, chromosome 7 bearing with asymmetric C-bands (corresponding to K7L) was observed: one band was amplified, and the other was reduced in sister chromatids (Figure 10A,B). These band alterations would have appeared due to the occurrence of unequal sister chromatid recombination .
Unequal crossing over in regions containing duplicate genes or repetitive DNA has been demonstrated in several organisms, such as yeast , apes , and humans . Two reciprocal products, a directly amplified tandem duplication and a deletion, can result from unequal crossing over between homologous chromosomes. In patients with chromosome duplications involving some types of Charcot–Marie–Tooth disease, the occurrence of unequal recombination between homologous segments (interchromosomal) and sister chromatids (intrachromosomal) has been shown . Thus, the generation of deletions and duplications by unequal recombination can affect the copy number of repeated genes and noncoding repeated DNA sequences.
From this scenario, we can assume that the asymmetric chromosome 7 observed in callus cultures resulted from unequal chromatid recombination at the K7L site. Therefore, the amplification of K7L detected in R1 plants derived from the 12F culture would have originated from an unequal recombination at K7L. This event would not alter the size of the distal euchromatic segment, as observed here .
Other alterations were observed on the long arm of chromosome 7 in the callus cultures analyzed: duplication of K7L (Figure 10C), amplification of K7L (Figure 10D), reduction of K7L (Figure 10E), K7L localized on telomeric position (Figure 10F), and the absence of K7L (Figure 10G). Figure 10H shows a diagrammatic representation of these abnormalities. The frequency of these aberrations varied from 0.98 to 4.82% in the six genotypes evaluated .
These abnormalities could have originated from a delay of sister chromatid separation at the K7L region, followed by breakage and BFB cycles as suggested in Figure 11 . After the primary event of delayed chromatid separation, breakage could occur in three different positions at the knob region, terminal (1), proximal (2), and middle (3), giving rise to the different types of abnormalities shown in Figure 10C–G. Note that in all possible events, the terminal euchromatic “b” segment would be lost. Regenerated plants homozygous for these aberrations probably would not survive with the deletion of the terminal segment. The recovery of regenerated plants homozygous for the K7L amplification gives support to the hypothesis that this amplification was originated from an unequal chromatid amplification.
Therefore, the results show that knob amplification or reduction can appear as a result of BFB cycles or unequal crossing over, but if they are originated from BFB cycles, they would not survive in homozygous regenerated plants.
The presence of some chromosome abnormalities in maize callus cultures can be explained by the occurrence of delay of chromatid separation in mitotic anaphases. This primary event gives origin to a bridge followed by a breakage-fusion-bridge cycle and chromosome healing. FISH using telomere sequences as probes gave evidence of de novo telomere formation at broken chromosome ends. Amplifications and deficiencies in the knobs may also occur via unequal chromosome crossing over evidenced in culture by the presence of chromosome 7 showing differences in the size of K7L (C-band) in sister chromatids.
The data suggest interesting questions for further investigations such as the mechanism underlying the delay in chromatid separation at knob sites and that of de novo telomere formation at the broken chromosome ends in callus culture. Changes in DNA methylation could be the cause of unusual later replication of knobs (see ).
The observations on the chromosome healing of chromosomes 7 and 9 showed that this event occurred in euchromatic and heterochromatic regions, certainly non-telomeric sites. Mechanistic information on telomere formation is available through studies on
In conclusion, mechanisms of chromosomal evolution like the related here might occur in plants. It has been suggested that structural chromosomal rearrangements frequently appear in euchromatin-heterochromatin borders .
The authors acknowledge: Canadian Science Publishing for the permission to use figures and data of the article . S. Karger AG for the permission to use figures and data of the article . Nova Science Publishers Inc. for the permission to use figures and data of the review . Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for research support.
The authors declare no conflict of interest.