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Acta Crystallogr A. Jul 1, 2010; 66(Pt 4): 451–457.
Published online May 7, 2010. doi:  10.1107/S0108767310010433
PMCID: PMC2891002
High-resolution study of dynamical diffraction phenomena accompanying the Renninger (222/113) case of three-beam diffraction in silicon
A. Kazimirova* and V. G. Kohnb
aCornell High Energy Synchrotron Source (CHESS), Cornell University, Ithaca, 14853 NY, USA
bRussian Research Center ‘Kurchatov Institute’, 123182 Moscow, Russia
Correspondence e-mail: ayk7/at/cornell.edu
Received March 3, 2010; Accepted March 19, 2010.
X-ray optical schemes capable of producing a highly monochromatic beam with high angular collimation in both the vertical and horizontal planes have been evaluated and utilized to study high-resolution diffraction phenomena in the Renninger (222/113) case of three-beam diffraction in silicon. The effect of the total reflection of the incident beam into the nearly forbidden reflected beam was observed for the first time with the maximum 222 reflectivity at the 70% level. We have demonstrated that the width of the 222 reflection can be varied many times by tuning the azimuthal angle by only a few µrad in the vicinity of the three-beam diffraction region. This effect, predicted theoretically more than 20 years ago, is explained by the enhancement of the 222 scattering amplitude due to the virtual two-stage 000 An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi1.jpg 113 An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi1.jpg 222 process which depends on the azimuthal angle.
Keywords: dynamical diffraction, multiple diffraction, X-ray optics, plane wave, synchrotron radiation
The general theory of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi3.jpg-beam diffraction is based on the assumption that the incident beam is a perfect plane monochromatic wave and the theory describes the wavefield in a crystal as a superposition of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi3.jpg plane waves. To compare theory with experiment the resulting intensities are integrated over the finite angular spread on the incident beam (Colella, 1974 [triangle]). While this is sufficient for the analysis of the effects related to the phases of the reflections involved in multiple diffraction (which is one of the main established application areas for multiple diffraction), any high-resolution dynamical phen­om­ena taking place within a narrow angular range of a few Darwin widths for a particular reflection are washed out and lost from the analysis.
In non-coplanar multiple X-ray diffraction, the diffraction conditions are usually defined by two angles, a polar angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg and an azimuthal angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg. In a typical experiment at synchrotron sources the primary scattering plane is vertical, the polar angle is associated with the rotation around the horizontal axis perpendicular to this plane and the azimuthal angle is associated with the rotation around the axis lying in the vertical plane. Thus, for a well known Renninger scheme the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg angle corresponds to the Bragg angle for the forbidden 222 reflection and the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg angle describes the rotation around the normal to the crystal surface.
In spite of a great number of theoretical and experimental works on multiple diffraction, including the Renninger effect, published so far (see Chang, 2004 [triangle]; Authier, 2005 [triangle] and references therein), there are very few experimental works in which dynamical diffraction effects accompanying multiple diffraction have been studied with sufficiently high resolution. Indeed, to observe these effects one needs to condition the incident X-ray beam to a degree approaching a perfect plane wave. That requires a two-dimensional angular collimation and a high degree of monochromatization. The excitation of the X-ray standing wavefields in three-beam (111/220) diffraction was studied by Kazimirov et al. (1992 [triangle]) and Kazimirov, Kovalchuk, Kohn, Kharitonov et al. (1993 [triangle]). In these works, a two-dimensional angular collimation was achieved by using another interesting dynamical diffraction effect – the effect of the enhancement of the anomalous transmission (super-Borrmann effect) in a six-beam symmetrical Laue diffraction (Afanas’ev & Kohn, 1977 [triangle]). In the work by Kazimirov, Kovalchuk, Kohn, Ishikawa et al. (1993 [triangle]) this effect was experimentally measured and compared with theory. Pahl (1994 [triangle]) proposed a novel ultra-small-angle scattering camera based on a super-Borrmann effect, and its feasibility was experimentally verified. The idea of using the Renninger effect for a two-dimensional collimation was proposed by Colella (1974 [triangle]) and analyzed later in both Bragg and Laue cases by Stepanov et al. (1994 [triangle]), theoretically and experimentally.
Most of the multiple diffraction experiments were performed at second-generation synchrotron radiation (SR) sources. The beams produced by modern third-generation SR sources, due to a very high intensity and a high degree of the ‘natural’ angular collimation of undulator radiation, can be more easily conditioned in terms of both a two-dimensional angular collimation and a monochromatization. Next-generation sources such as X-ray free electron lasers (XFELs) and energy-recovery linacs (ERLs) will be able to produce almost perfect plane-wave X-ray beams. These new experimental opportunities provide strong motivations to continue the study of multiple diffraction effects and their potential applications in X-ray optics.
In this article we present the high-resolution study of dynamical diffraction phenomena accompanying the Renninger Si (222/113) case of three-beam diffraction. First experimental rocking-curve measurements at the exact (222/113) excitation in Ge were performed by Colella (1974 [triangle]) using a double-crystal setup in the antiparallel arrangement. Much better collimation than that provided by a double-crystal setup is required in both directions to observe the details of the dynamical diffraction interaction. In particular, the goal of this work was to observe the effect of a total reflection of a parallel incident beam into the forbidden 222 reflected beam predicted by Kohn (1988 [triangle]). This effect takes place in a very narrow polar angular range and rather wide azimuthal angular range near the three-beam diffraction region where the 113 reflection is very small. It has never been observed experimentally because of the difficulty of preparing an incident beam approaching a perfect plane wave. This work presents such attempts. We studied five optical setups that provide various degrees of monochromatization and angular collimation in both the vertical and horizontal planes. For each of these optics we measured the 222 and 113 diffraction curves in the vicinity of the three-beam (222/113) diffraction region. For our best optics we recorded 68% reflectivity of the 222 reflected beam. We observed experimentally a remarkable phenomenon: the width of the 222 strong reflection region can be changed many times (about four times in our setup) by tuning the azimuthal angle by a few µrad while still at the 60% reflectivity level. In §2 the theoretical description of the excitation of the forbidden reflection is presented. Then we present the experimental results followed by the discussion and conclusions.
The general theory of plane-wave multiple diffraction in a single crystal is well developed (Colella, 1974 [triangle]; Kohn, 1979 [triangle]; Chang, 2004 [triangle]). 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
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd1.jpg
where the index An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi9.jpg is used for the beams (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi10.jpg for the incident beam), An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi11.jpg indicate the polarization state of the electric field vector of the beams, the index An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi12.jpg allows one to distinguish various Bloch waves or zones of dispersion surface, and the parameters An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi13.jpg determine the rate of excitation of the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi12.jpgth Bloch wave. The summation is performed over the values which correspond to the positive value of the absorption coefficent An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi15.jpg. The Bloch-wave amplitudes An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi16.jpg and the dispersion parameters An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi17.jpg are the solution of the eigenvalue problem
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd2.jpg
where the scattering matrix An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi18.jpg is determined as
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd3.jpg
Here An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi19.jpg are the unit polarization vectors, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi20.jpg is the wavelength of X-ray radiation, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi21.jpg is the cosine of the angle between the direction of the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi9.jpgth beam and the internal normal to the entrance surface, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi23.jpg is the Fourier image of the complex polarizability of the crystal with the reciprocal-lattice vector An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi24.jpg, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi25.jpg is the Kroneker’s symbol,
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd4.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi26.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi27.jpg mean the angular deviation of the incident-beam direction from the direction of kinematically exact multiple diffraction. The unit vectors An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi28.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi29.jpg are normal to the incident-beam direction An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi30.jpg. We choose An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi28.jpg to be in the scattering plane for the forbidden reflection (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi32.jpg), so An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi29.jpg is normal to this plane.
It is known (Kohn, 1979 [triangle]) that in a thick crystal the parameters An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi34.jpg if An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi12.jpg indicates the Bloch wave with An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi36.jpg. The remaining values may be found from the linear set of equations
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd5.jpg
where the index An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi9.jpg runs only over the Laue beams with An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi38.jpg. If all the diffracted beams are the Bragg-diffracted beams with An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi39.jpg the situation becomes simpler and we have An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi40.jpg. 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 (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi32.jpg) 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 An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi42.jpg vectors are approximately parallel to each other, all An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi43.jpg vectors are also approximately parallel to each other, but all An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi42.jpg vectors are normal to the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi43.jpg vectors. Then, the general sixfold system can be divided into two threefold systems of equations and the index An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi46.jpg can be omitted.
Taking into account that An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi47.jpg we have for the amplitude of the forbidden beam in the three-beam case
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd6.jpg
As we see from this equation, the forbidden beam can be excited by the beam An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi48.jpg and there are two mechanisms for this excitation. The first one corresponds to the large value of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi48.jpg in the angular region where the value of (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi50.jpg) 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 (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi51.jpg). This excitation is realized in the angular region of the multiple diffraction which satisfies the two-beam diffraction condition for beam An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi52.jpg. The second type of excitation takes place if the denominator is small, i.e. An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi53.jpg. This corresponds to the angular region of the two-beam Bragg condition for the forbidden beam. Now the significant value of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi54.jpg can be obtained even for a small value of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi48.jpg. This case can be called a virtual Bragg diffraction, or a resonance diffraction. Indeed, the value An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi48.jpg 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 An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi57.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi54.jpg amplitudes but a small An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi48.jpg amplitude. Also we have small An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi60.jpg and rather large An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi61.jpg. The amplitude An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi48.jpg can be calculated by means of the perturbation method from the system [equation (2)] as
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd7.jpg
Substitution of this equation into equations for B 0 and B 1 leads to the system
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd8.jpg
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd9.jpg
This system of equations describes a two-beam diffraction with the reflection of the incident beam An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi57.jpg into the forbidden beam An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi54.jpg. The matrix of this system is
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd10.jpg
Since these equations are written for the case of a large value of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi61.jpg and we consider small values of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi66.jpg, we can replace the denominator by the value An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi67.jpg. 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 [triangle]) 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 (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi68.jpg) and a pure forbidden first reflection (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi69.jpg),
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd11.jpg
where
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd12.jpg
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd13.jpg
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd14.jpg
The angular dependence of the reflection power to the forbidden beam is determined as
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd15.jpg
where
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd16.jpg
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd17.jpg
Here the square root should be taken with the positive imaginary part. It can be verified straightforwardly that
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd18.jpg
where
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd19.jpg
The imaginary part of the parameter An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi70.jpg is determined by the absorption. If the absorption is negligible then the total reflection takes place in the region of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi71.jpg. According to equation (13) the angular width of the total reflection region is determined by the equation
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd20.jpg
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 An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi52.jpg, which is proportional to An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi73.jpg.
In the center of the angular region of total reflection we have An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi74.jpg, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi75.jpg, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi76.jpg for small values of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi77.jpg. For large values of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi78.jpg we have an approximate expression for the parameter An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi77.jpg as An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi80.jpg, where An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi81.jpg is a linear absorption coefficient, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi82.jpg is an extinction length. Therefore An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi77.jpg describes the absorption on the extinction length. Since An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi84.jpg becomes very large for the large azimuthal angles the total reflection cannot be realized for all azimuthal angles. For example, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi85.jpg for An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi86.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi87.jpg.
Inside the total reflection region the maximum value of the reflection power corresponds to the angle where An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi77.jpg is minimal. From equation (13) we can write approximately
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd21.jpg
In the normal two-beam case An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi89.jpg is comparable with An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi90.jpg, and in the left part of the total reflection region at An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi91.jpg a significant decrease in the value of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi77.jpg takes place and this is the reason for the Borrmann anomalous transmission phenomenon. However, in our case An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi89.jpg is much smaller than An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi90.jpg; 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
A mathematical equation, expression, or formula.
 Object name is a-66-00451-efd22.jpg
The Bragg condition for the first reflection reads An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi95.jpg. One can verify straightforwardly that in this case the system has a solution with An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi96.jpg, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi97.jpg, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi98.jpg. 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 An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi99.jpg if An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi100.jpg. 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.
The experiment was performed at the Cornell High Energy Synchrotron Source (CHESS) at the A2 beamline. The X-ray beam from the 49-pole wiggler was monochromated to an energy of 24.982 keV by a double-crystal Si 111 upstream water-cooled monochromator. Post-monochromator optics for additional monochromatization and angular collimation were assembled on the optical table in the experimental hutch. The sample, an Si (111)-oriented thick perfect crystal, was mounted on a four-circle diffractometer. The diffraction plane for the forbidden 222 reflection was vertical. The 222 and 113 diffracted intensities were recorded as a function of the polar angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg for various values of the azimuthal angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg. The 222 intensity was normalized to the intensity of the incident beam recorded by the intensity monitor installed in front of the sample.
The optical arrangements are shown schematically in Fig. 1 [triangle]. Two diffraction planes, vertical (V) and horizontal (H), are shown for each setup. The values for the energy bandwidth and the angular collimation characteristic for each setup are summarized in Table 1 [triangle]. Experimental diffraction curves measured by using each setup are compiled in Fig. 2 [triangle] (note different angular and intensity scales). Each column in Fig. 2 [triangle] contains diffraction data obtained by using a particular optical setup.
Figure 1
Figure 1
Experimental setups are indicated by the numbers shown on the left. Each setup is shown by two panels: one, marked by ‘V’, is in the vertical plane (diffraction plane for the 222 reflection) and the other one, marked by ‘H’, (more ...)
Table 1
Table 1
The estimated values of the energy and angular resolution for the experimental setups in Fig. 1 [triangle]
Figure 2
Figure 2
The 222 (thick black lines) and 113 (thin red lines) diffraction curves measured using the experimental setups shown in Fig. 1 [triangle]. The 222 curve is normalized on the incident intensity. Setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi121.jpg (first column): the azimuthal angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg, from top to bottom: (more ...)
We start with a ‘standard’ beamline setup in which the energy bandwidth and the angular collimation are determined by the double-crystal Si 111 monochromator, the source size and the slits. With a wiggler source size of 0.38 (vertical) × 3.38 mm (horizontal), a size of the S3 slit in front of the sample of 0.5 × 1 mm and a distance between the source and the slit of 38 m, we estimate the angular collimation in the vertical plane as 23 µrad and in the horizontal plane as about 115 µrad. (These values are smaller than the angular spread of the wiggler radiation of 150 µrad in the vertical and 285 µrad in the horizontal plane.) The estimated energy bandwidth An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi103.jpg is An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi104.jpg, close to the experimentally measured value of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi105.jpg. The experimental curves (first column in Fig. 2 [triangle]) are much narrower for the 222 reflection than for the 113 reflection because the 222 reflection does not depend on the azimuthal angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg whereas for the 113 reflection the region of the strong reflection projected on the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg axis shifts with changing An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg. The enhancement of the 222 peak in the center of the three-beam region is clearly seen with the maximum 222 reflectivity of about 2%. Any dynamical effects described in the previous sections are washed out by the angular spread of the incident beam.
In all other optical schemes two double-bounce channel-cut crystals in the antiparallel (+/+) setting were used to obtain a higher degree of monochromatization and improve the angular collimation in the vertical plane: Si 004 in setups An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi52.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi110.jpg, and Si 008 in setups An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi111.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg (Fig. 1 [triangle]). For the antiparallel setting both the energy bandwidth and the angular spread are determined by the intrinsic rocking-curve width An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi113.jpg of the crystals: An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi114.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi115.jpg. The calculated values are An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi116.jpg and 4.7 µrad for setups 2 and 4, and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi117.jpg and 1.4 µrad for setups 3 and 5. As one can see by comparing the results shown in the first three columns in Fig. 2 [triangle], the additional collimation/monochromatization has a strong effect on the 222 reflection: the reflectivity of 36% was measured using setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi111.jpg.
These improvements do not affect the width of the 113 curves but lead to a sharp intensity drop in the angular range of the 222 reflection. To improve the angular collimation in the other direction an additional Si 022 channel-cut crystal diffracting in the horizontal plane was installed (setups An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi110.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg). The width of the 113 reflection changes from about 80 µrad for setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi121.jpg to 6.8 µrad for setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi110.jpg. The best two-dimensional collimation and monochromatization were achieved with setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg, which resulted in the maximum 222 reflectivity of 68%. The experimental width of the 222 curve off the three-beam condition was 1.2 µrad. The deep minimum in the 113 intensity is observed in the central three-beam diffraction region for the angular interval of the 222 reflection. According to the theory presented in the previous section, for a perfect incident plane wave the 113 intensity should be zero for the exact 222 diffraction condition. This effect may serve as an indicator of the quality of the plane-wave optics. The azimuthal dependence of the maximum reflectivity (thick black line) and the integral intensity (thin red line) of the 222 reflection measured using setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg are shown in Fig. 3 [triangle]. Both curves show a strong An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg asymmetry.
Figure 3
Figure 3
Azimuthal (An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg) dependences of the absolute reflectivity (thick black line, left scale) and the intensity integrated over the polar angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg (thin red line, arbitrary units) of the 222 reflection.
To quantify independently setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg, after the three-beam measurements were complete the sample was turned off the three-beam condition and was used as an analyzer crystal. The Si 111, 333 and 555 analyzer curves are shown in Fig. 4 [triangle] (lower panel). The 111 curve shows a perfect Darwin curve with the reflectivity very close to 100%. The 333 reflectivity is 86% and the full width at half maximum is 2.9 µrad versus 99% and 1.97 µrad for the theoretical ‘intrinsic’ curve. The 555 analyzer curve gives 40% reflectivity and 1.68 µrad width versus 97.5% and 0.84 µrad for the ‘intrinsic’ curve. The later results confirm that the beam produced with our best setup is far from being a perfect plane wave. Two 222 curves are shown in the upper panel of Fig. 4 [triangle]. The thick black line is the 222 curve in the center of the three-beam diffraction region; its width is close to the width of the 113 reflection. The 222 curve measured at the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi127.jpg = 10 µrad off the exact three-beam condition is shown as a thin (red) line. Note that the width of this curve is close to the width of the 555 analyzer curve, indicating that they are both limited by the resolution of our setup. A remarkable effect which has never been observed before is clearly demon­strated: the width of the 222 curve can be changed several times (about four times with these optics) by tuning the azimuthal angle by only a few µrad. This effect can be used for tuning the energy resolution of monochromators based on multiple diffraction.
Figure 4
Figure 4
Top: the 222 reflectivity (thick black line) corresponds to the center of the three-beam diffraction region, the thin red line corresponds to the angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi127.jpg = 10 µrad off the exact three-beam condition. Bottom: the analyzer 111 (black), 333 (more ...)
The analysis of the experimental results assembled in Fig. 2 [triangle] led us to the conclusion that improving the angular collimation in the horizontal plane results in narrowing the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg width of the 113 diffraction curves (compare setups An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi52.jpg, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi110.jpg, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi111.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg), and it also affects the 222 reflection inside the ‘intrinsic’ three-beam region. On the other hand, better monochromatization and angular collimation in the vertical plane lead to narrower 222 curves outside the three-beam region, and, finally, to the observation of the fine structure in both 222 and 113 reflections in setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg. This is remarkably different from the two-beam diffraction curves usually measured in a non-dispersive setup when the spreads in energy and in the azimuthal angle do not affect the curves. The 222 reflection curves outside the three-beam case still depend on the azimuthal angle because of the virtual three-beam diffraction, though this dependence is slow. As for the 113 reflection, as one can see in §2, both angles, An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg and An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg, as well as the energy, enter the formula in equation (4), which describes the deviation from the exact three-beam condition, in a very similar way.
To compare experimental results with theory, the 222 and 113 diffraction curves were calculated for approximately the same values of the azimuthal angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg as for the experimental curves in Fig. 2 [triangle] (last column, setup An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi112.jpg). They are shown in Fig. 5 [triangle]. Calculations for the perfect plane wave are in the left panel. The 222 curves show 100% reflectivity and their shape clearly demonstrates two excitation mechanisms as described in §2. The first one corresponds to the ‘pumping’ of the strong 113 intensity into the 222 beam; it results in the peak with the angular position which corresponds to the 113 reflection. The second, a narrower peak, is excited resonantly and its center angular position corresponds to the Bragg condition for the 222 reflection. At this angle the 113 beam has zero intensity; therefore the total region of reflection splits into two peaks.
Figure 5
Figure 5
Theoretical diffraction curves 222 (thick black line) and 113 (thin red line) calculated for the perfect plane wave (left) and taking into account the energy and the angular resolution of the experimental setup (right). The values of the azimuthal angle (more ...)
The effect of the finite angular and energy resolution on the diffraction curves can be taken into account based on equation (4). The change in energy is equivalent to the angular shift in An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg of both 222 and 113 curves. It was found that for the experimental value of An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi139.jpg the effect of the finite monochromaticity is negligibly small. To account for the finite angular spread the 222 and 113 curves were summed over the angular interval An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi127.jpg = 8 µrad and the result was convoluted with a Gaussian function with width An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi141.jpg = 1 µrad. The resulting curves are shown in the right panel in Fig. 5 [triangle]. As one can see, they reproduce the experimental curves remarkably well. We may conclude that although the achieved resolution was sufficient to observe the effects predicted by theory, a better angular resolution in both vertical and horizontal planes is required for a detailed quantitative comparison with theory.
The theoretical treatment (§2) assumed a purely forbidden reflection. This is not the case for Si 222. It is well known that the 222 reflection is very weak but not completely forbidden (Bragg, 1921 [triangle]) due (mostly) to the non-spherical valence charge density (Roberto & Batterman, 1970 [triangle]; Colella, 1977 [triangle]). In our experiment it can be seen in two ways (see Fig. 3 [triangle]): first, the intensity of the 222 beam is approaching a nonzero value as the crystal is rotated far from the three-beam condition; secondly, the strong asymmetry of the An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi6.jpg dependence clearly indicates the phase of the structure factor. Most of the experimental measurements of the structure factors for forbidden reflections were performed by measuring the intensity integrated over the polar angle An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi5.jpg (see e.g. Roberto & Batterman, 1970 [triangle]). Accurate rocking-curve measurements in a double-crystal 222 nondispersive (+/An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi144.jpg) setup were performed by Entin & Smirnova (1989 [triangle]). They reported a width of the double-crystal rocking curve measured with Mo An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi145.jpg radiation of 0.339 µrad, which was consistent with the structure factor An external file that holds a picture, illustration, etc.
Object name is a-66-00451-efi146.jpg reported by others. This value gives us an estimate of the intrinsic Si 222 width for our energy as 0.13 µrad. The experimental data presented in Fig. 3 [triangle] can be used for the determination of both the modulus and the phase of the structure factor.
A high degree of two-dimensional collimation was achieved in this work by using perfect crystal optics. This resulted in a loss of intensity by many orders of magnitude. Next-generation synchrotron sources such as the ERLs (Gruner & Bilderback, 2003 [triangle]) and the XFELs (Pellegrini & Stöhr, 2003 [triangle]) are characterized by extremely high brilliance and produce highly parallel X-ray beams. For such sources the requirements for two-dimensional collimation can be easily fulfilled without significant loss in flux, thus making X-ray optics based on multi-beam diffraction effects extremely attractive.
Acknowledgments
This work is based upon research conducted at the Cornell High Energy Synchrotron Source (CHESS), which is supported by the National Science Foundation and the National Institutes of Health/National Institute of General Medical Sciences under NSF award No. DMR-0225180. The work of V. G. Kohn was supported by RFBR grant Nos. 09-02-12164-Ofi_m and RS-4110.2008.2.
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