The general theory of plane-wave multiple diffraction in a single crystal is well developed (Colella, 1974

; Kohn, 1979

; Chang, 2004

). In this section, we formulate the theoretical approach which has been effectively utilized for computer simulations. The normalized reflection powers for the beams which leave the crystal through the entrance surface can be calculated by means of the formula

where the index

is used for the beams (

for the incident beam),

indicate the polarization state of the electric field vector of the beams, the index

allows one to distinguish various Bloch waves or zones of dispersion surface, and the parameters

determine the rate of excitation of the

th Bloch wave. The summation is performed over the values which correspond to the positive value of the absorption coefficent

. The Bloch-wave amplitudes

and the dispersion parameters

are the solution of the eigenvalue problem

where the scattering matrix

is determined as

Here

are the unit polarization vectors,

is the wavelength of X-ray radiation,

is the cosine of the angle between the direction of the

th beam and the internal normal to the entrance surface,

is the Fourier image of the complex polarizability of the crystal with the reciprocal-lattice vector

,

is the Kroneker’s symbol,

where

and

mean the angular deviation of the incident-beam direction from the direction of kinematically exact multiple diffraction. The unit vectors

and

are normal to the incident-beam direction

. We choose

to be in the scattering plane for the forbidden reflection (

), so

is normal to this plane.

It is known (Kohn, 1979

) that in a thick crystal the parameters

if

indicates the Bloch wave with

. The remaining values may be found from the linear set of equations

where the index

runs only over the Laue beams with

. If all the diffracted beams are the Bragg-diffracted beams with

the situation becomes simpler and we have

. Equations (1)

, (2)

, (3)

, (4)

and (5)

allow one to calculate numerically the angular dependence of the reflection power for the diffracted beams. We are interested mainly in the first beam (

) which we will treat as a pure forbidden. To illustrate the phenomena analytically we simplify the problem and consider hard enough radiation so that the Bragg angles are small which is, in fact, close to our experimental conditions. In this case, one can choose the polarization vectors in such a way that all

vectors are approximately parallel to each other, all

vectors are also approximately parallel to each other, but all

vectors are normal to the

vectors. Then, the general sixfold system can be divided into two threefold systems of equations and the index

can be omitted.

Taking into account that

we have for the amplitude of the forbidden beam in the three-beam case

As we see from this equation, the forbidden beam can be excited by the beam

and there are two mechanisms for this excitation. The first one corresponds to the large value of

in the angular region where the value of (

) is not too large. In this region the forbidden beam is pumped by another strong reflection, so this is the case of a double reflection (

). This excitation is realized in the angular region of the multiple diffraction which satisfies the two-beam diffraction condition for beam

. The second type of excitation takes place if the denominator is small,

*i.e.*
. This corresponds to the angular region of the two-beam Bragg condition for the forbidden beam. Now the significant value of

can be obtained even for a small value of

. This case can be called a virtual Bragg diffraction, or a resonance diffraction. Indeed, the value

may be small but the amplitude is strongly enhanced by a small value of the resonance denominator.

Let us consider this case in more detail. We have strong

and

amplitudes but a small

amplitude. Also we have small

and rather large

. The amplitude

can be calculated by means of the perturbation method from the system [equation (2)

] as

Substitution of this equation into equations for

*B*
_{0} and

*B*
_{1} leads to the system

This system of equations describes a two-beam diffraction with the reflection of the incident beam

into the forbidden beam

. The matrix of this system is

Since these equations are written for the case of a large value of

and we consider small values of

, we can replace the denominator by the value

. Equations (9)

and (10)

, in a more general case and with taking into account polarizations, were obtained for the first time by Høier & Marthinsen (1983

) as a method of approximate numerical solution of the multiple diffraction problem. However, we can perform the numerical solution accurately.

Thus, we obtained the case of a two-beam diffraction with a nonzero diffraction parameter for the forbidden beam. It is easy to calculate the solution analytically assuming a symmetrical case (

) and a pure forbidden first reflection (

),

where

The angular dependence of the reflection power to the forbidden beam is determined as

where

Here the square root should be taken with the positive imaginary part. It can be verified straightforwardly that

where

The imaginary part of the parameter

is determined by the absorption. If the absorption is negligible then the total reflection takes place in the region of

. According to equation (13)

the angular width of the total reflection region is determined by the equation

It depends on the deviation of the azimuthal angle from the exact multiple diffraction condition. Interestingly, this dependence is slower compared to the reflection power of beam

, which is proportional to

.

In the center of the angular region of total reflection we have

,

,

for small values of

. For large values of

we have an approximate expression for the parameter

as

, where

is a linear absorption coefficient,

is an extinction length. Therefore

describes the absorption on the extinction length. Since

becomes very large for the large azimuthal angles the total reflection cannot be realized for all azimuthal angles. For example,

for

and

.

Inside the total reflection region the maximum value of the reflection power corresponds to the angle where

is minimal. From equation (13)

we can write approximately

In the normal two-beam case

is comparable with

, and in the left part of the total reflection region at

a significant decrease in the value of

takes place and this is the reason for the Borrmann anomalous transmission phenomenon. However, in our case

is much smaller than

; therefore the Borrmann effect is not observed.

There is an interesting peculiarity of the reflection into the second reflection if the Bragg condition (including refraction) is fulfilled completely for the first (forbidden) reflection. Indeed, the set of equations (2)

without polarizations can be written as

The Bragg condition for the first reflection reads

. One can verify straightforwardly that in this case the system has a solution with

,

,

. This solution is independent of the azimuthal angle,

*i.e.* even in the exact multiple diffraction region the reflection into a second beam vanishes. In reality we can satisfy the condition only for the real parts

if

. The imaginary parts do not satisfy this condition; therefore the reflection into the forbidden beam vanishes for very large values of the azimuthal angle. Nevertheless, the phenomenon of vanishing of the reflection into the second beam inside the multiple diffraction region seems to be rather interesting.