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Spatial scaling in the theory of X-ray scattering. Non-standard dynamic theory.

1. Introduction

This study is concerned with the fundamentals of the non-standard dynamic theory of X-ray scattering based on the spatial scaling of the total wave field. The need for developing such as theory has been caused by the following circumstances.

The currently available theory of dynamic x-ray diffraction is based on Takagi's equations. The main principle of the formalism of the Takagi equations is the representation of the wave field in the form of a superposition of the transmitted and diffracted waves with slowly changing amplitudes.

The important factor is that the Takagi system is 'shortened', i.e., does not consider the second derivatives of the amplitude of the fields with respect to the coordinate. On the one side, this greatly facilitates the theoretical investigations and enables solution of a number of diffraction problems in standard diffraction geometry when such a simplification is justified. On the other side, the Takagi equations cannot be used in the conditions of, for example, sliding diffraction geometry.

This restriction follows from the fact that it is not possible to formulate correctly the boundary conditions at the crystal-vacuum interface for the field amplitudes. Instead of the well-known classic conditions of continuity of the tangential components of the electrical and magnetic fields, there are new intuitive boundary conditions of the definition of the normal components of the amplitude

of the fields on the crystal surface which are clear but do not agree with the Maxwell equations. Of course, the solutions of these boundary problems are applicable only for relatively high (more accurately, greatly exceeding the PVO angle) angles of incidence and the exit of radiation.

The currently available theoretical approaches, generalising the Takagi theory for the case of maximally asymmetric diffraction schemes, requires the solution of the equations of the third or fourth order [1].

At the centre, the Maxwell equations at the wave equations of the second order with respect to, for example, the electrical field. In particular, a similar structure of the equations is in agreement with the previously mentioned classic bounty conditions. This means that the requirement for taking into account correctly the boundary conditions in any theoretical diffraction scheme leads almost always the well-known structure of the wave equation following from the Maxwell equations.

Thus, to overcome these difficulties, it is necessary to develop a theory based directly on the Maxwell equations using modelling considerations of the polarisability of the crystal in the x-ray wavelength range.

2. The main equation and direct expansion

From the system of the Maxwell equations using the standard procedure we obtain the main equation for the amplitude of the electrical field in the crystal:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

Equation (1) is reduced to the dimensionless form using the reciprocal lattice vector H

H/H = h, r = Hr', [kappa]k/H.

The quantities [[chi].sub.0], [[chi].sub.H], [[chi].sub.[bar.H]] ~ [10.sup.5]-[10.sup.6]so that the equation (1) can be solved by the perturbation method. [[chi].sub.H] should be selected as the perturbation parameter. Initially, we considere the direct expansion of the first order with respect to [[chi].sub.H]:

E(r) = E0 (r) + [[chi].sub.H] [E.sub.1] (r) + [[chi].sup.2.sub.H] [E.sub.2] (r) + ...

We restrict our considerations to the first order expansion. Consequently:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

Direct expansion with the accuracy to XH as the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where 20= 2(1 +X0).

Expansion (3) ceases to be valid at the well-known Laue diffraction condition for the x-rays:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here we restrict ourselves to the case in which the transmitted and diffracted waves--two-wave approximation--satisfy the Laue equation.

Thus, in the vicinity of the values K0 at which diffraction is detected, it is necessary to modify the direct expansion. The principal moment here is the parametric nature of the interaction of the medium and the wave field.

3. Many scales method

The main concept of the method applied to the given task is as follows [2]. Special features of the wave field are demonstrated on different spatial scales, determined by [[chi].sub.H], Correspondingly, these special features can be investigated independently in a specific approximation. For this purpose, we transfer from a single variable r to several variables, reflecting different scales of the task. The fixed number of the scales determines the order of expansion of the solution.

Thus, we determine the approximate solution (1) in the region of the Bragg maximum when the Laue condition [[kappa].sub.0] [+ or -] h = [[kappa].sub.h] is satisfied. After substituting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], in equation (1)

E(r) = E([r.sub.0], [r.sub.1], ...).

Further, as in the direct expansion, we consider two scales [r.sub.0], [r.sub.1]. Together with the expansion of the field, we expand the operator rot and also the wave vector [[kappa].sub.0] with respect to the degrees [[chi square].sub.H], as a result of the parametric nature of the interaction of the field with the medium:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

All the expansions are substituted into the main equation (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Further, the approximate solution is obtained by consecutive equating of the coefficients and the degrees [[chi].sub.H]. The uniform suitable expansion is obtained by excluding the diverging terms of the expansion.

The solution in the zero approximation is determining the former superposition of the transmitted and diffracted waves (two-wave approximation):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

The quantities [c.sub.i] ([r.sub.1]) are regarded as 'slow' variables. In this case, we consider the case of a semi-infinite crystal because equation (4) does not take into account the waves reflected from the lower face of the crystal.

The condition of exclusion of the diverging terms in the first approximation generates the system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

The system (5) is the dispersion relationship, written in the differential form, for the transmitted and diffracted waves in the two-wave approximation.

In (5) we substitute [c.sub.j] ([r.sub.1]) [right arrow] [c.sub.j] exp (iPr), [[DELTA].sub.1][c.sub.j] = iPc., where P is some constant vector:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

This system is regarded as the condition of restriction of the field in the given direction of propagation of the waves, connected with the structural parameters of the crystal and diffraction geometry. According to the physical meaning of the investigated problem, this restriction should be applied in the direction of the normal into the thickness of the crystal. Consequently, the system (6) acquires the new form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

Here [[gamma].sub.0], [[gamma].sub.h] are the directing cosines of the appropriate wave vectors. Consequently, we obtain the expression 4P:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

Consequently, for the system (7) we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

The selection of the sign in [P.sub.1,2] and, correspondingly, in [[alpha].sub.1,2] is determined by the physical considerations. In particular, it is necessary to ensure that in the deviations from the exact Bragg condition the wave field transfers to the conventional form of the refracted wave, propagating in the crystal as in a continuum. Consequently, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In the final analysis, the wave field in the crystal has to form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where constant c is determined from the boundary conditions.

The relationship (10) is the unique wave field in the ideal crystal in the vicinity of the Bragg maximum. In order to compare the resultant equation with the results, it is necessary to transfer to the angular variable - deviation from the exact Bragg angle [DELTA][theta]. Taking into account the fact that the angular scanning leads to the variation [[kappa].sub.0n] = ([[kappa].sub.0n]), and also that [[kappa].sub.01] = [[kappa].sub.01]h, we obtain the relationship for X:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In the case of Bragg diffraction (y0 > 0, yh <0) in the range

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

waves will be subjected to exponential attenuation along the normal into the thickness of the crystal; the width of the range is determined by the value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

This is the well-known expression (with the accuracy to diffraction) for the angular width of the Bragg table in the case of the semi-infinite ideal nonabsorbing crystal.

The extinction length [[LAMBDA].sub.ext] is determined as the decrement of attenuation of the wave in the exact Bragg position. This also leads to the well-known equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

Thus, the application of the method of many scales yielding the system of fundamental equations describing the behaviour of the wave field in the vicinity of the Bragg maximum in the two-wave approximation. This system is the direct analogue of the dispersion relationships of the Ewald-Laue theory and the TakagiTopen system of the generalised dynamic theory. Comparison of the resultant relationships with the available theoretical results shows complete agreement for the width of the Bragg maximum and the extension length.

The distinguishing feature of the investigated variant of the theory is that it does not use the procedure of shortening the equations by rejecting second derivatives. The most important moment here is the expansion with respect to [[chi].sub.H] which makes it possible to preserve to the maximum extent the structure of the Maxwell equations for the wave field in the crystal in the dynamic diffraction conditions.

This difference is manifested most clearly when taking into account the boundary conditions which provide explicit expressions for the reflection coefficients.

4. Reflection coefficient and boundary conditions

We now explain the differences resulting from strict consideration of the boundary conditions in our theory.

The reflection coefficient is determined as the ratio of the average values of the normal components of the Pointing vector of the diffracted and incident waves:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

Here n is the only director of the normal directed into the thickness of the crystal, [kappa], [c.sup.0] and [[kappa].sup.0.sub.h], [c.sup.0.sub.h], and the wave vectors and the amplitude of the incident and diffracted waves, respectively. Index 0 shows that the values of these quantities relate to the external medium--vacuum. Thus, the determination of the reflection coefficient is associated with the fact that the amplitude of the diffraction wave is located in vacuum. This problem is solved using boundary conditions.

It is well-known that the boundary conditions require continuous tangential components of the electrical and magnetic fields. The boundary problem breaks down into individual stages, associated with the determination of [c.sup.0.sub.h]. In each stage we examine the elementary task of determination of the relationship between the amplitude of the incident, transmitted and mirror-reflected waves. The solution of these problems it used to the well-known Frenel equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

Here [c.sub.R] is the amplitude of the mirror-reflected wave, [[kappa].sub.hR] is the wave vector of the diffraction wave, mirror-reflected from the lower side of the crystal-vacuum interface.

Equations (14)-(16) solve the problem of determination of the amplitude of the fields in the conditions of sliding non-complanar diffraction in which the incident and diffracted waves are located in the vicinity of the critical angles. They are identical with the relationships obtained in [3], where this problem is solved using the dispersion equation of the fourth order.

In the case of the high incidence angles and the angle of exit of the diffracted waves the amplitude of the mirror wave may tend to 0, and the reflection coefficient has the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

This is the well-known equation for the coefficient of reflection from the ideal semi-infinite crystal. At the same time, in the case of sliding non-complanar diffraction the amplitude of the diffracted wave is modulated by the factors which take into account the refraction of the diffraction wave at the crystal-vacuum interface and mirror reflection.

5. Conclusions

The variant of the theory of dynamic x-ray diffraction, presented here, is based on the direct analysis of the Maxwell equations for the given modelling considerations of the interaction of the field with the medium, taking the presence of the lattice into account. This analysis can be carried out using the method of many scales, adapted for the vector nature of the problem. The expansion parameter is the quantity [[chi].sub.H] so that the structure of the analysed equation of the field can be fully retained.

The resultant equations for the main characteristics of the field in the region of the Bragg maximum, derived from the quantitative special features of propagation of the waves, correspond to the well-known results of dynamic theory. However, the correct use of the boundary conditions leads to the expression for the reflection coefficient which greatly differs from the classic coefficient in the case of the maximally asymmetric diffractive schemes. In addition to this, the approach gives the amplitude of the mirror-reflected wave in the conditions of dynamic diffraction which, evidently, can be obtained using the conventional approaches.

A detailed analysis of the resultant special features of diffraction in the sliding diffraction geometries is outside the framework of this study.

References

[1.] M.A. Andreeva and R.N. Kuz'min, Mossbauer and x-ray surface optics, Izd. Obshchenats. Akad. Znanii, Moscow, 1996.

[2.] A. Naiphe. Introduction to pertubation methods, Mir, Moscow, 1984.

[3.] V.A. Bushuev and A.P. Oreshkom, FTT, 43, 906 (2001).

Dyshekov A. A. Faculty of Physics, Kabardino-Balkarsk State University, Chernyshevky Street 173, 360004 Nal'chik, Russia
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Title Annotation:INTERACTION OF RADIATION AND PARTICLES WITH CONDENSED MATTER
Author:Dyshekov, A.A.
Publication:Physics of Metals and Advanced Technologies
Article Type:Report
Geographic Code:4EXRU
Date:Jan 1, 2010
Words:2321
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