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research papers Journal of Applied Crystallography Calculation of diffraction efficiency for curved crystals with arbitrary curvature radius ISSN 0021-8898 Valerio Bellucci, Vincenzo Guidi,* Riccardo Camattari and Ilaria Neri Received 24 October 2011 Accepted 2 January 2013 # 2013 International Union of Crystallography Printed in Singapore – all rights reserved Department of Physics, University of Ferrara, Via Saragat 1/c, 44122 Ferrara, and CNR-IDASC SENSOR Laboratory, Brescia, Italy. Correspondence e-mail: [email protected] A model is proposed to calculate the diffraction efficiency of X-rays in Laue geometry for curved crystals with an arbitrary value of the curvature radius. The model generalizes the results based on the dynamical theory of diffraction, which are valid only for crystals with a radius of curvature lower than the critical curvature. The model is proposed for any kind of crystal, and its efficiency tends to one-half in the limit of a thick flat crystal. On the basis of this model, it was possible to reconsider the results of recently observed diffraction efficiency for curved crystals. Finally, the model sets an upper limit for diffraction efficiency of low-curvature curved crystals, this latter being useful in applications such as the construction of a hard X-ray Laue lens. 1. Introduction Crystals with curved diffraction planes (CDPs) have recently become popular in X-ray optics because they allow easy manipulation of the trajectories of high-energy photons with efficiency near unity in a broad energy range. A monocrystal with zero-curvature planes, hereinafter referred to as a ‘flat crystal’, can diffract photons just within a very narrow energy range, and its reflectivity is physically limited to 50% at most (Bellucci, Camattari, Guidi, Neri & Barrière, 2011; Zachariasen, 1945). In fact, a photon has the same probability of undergoing an even or odd number of diffractions traversing the crystal thickness. Mosaic crystals, i.e. an ensemble of microscopic flat crystals slightly misaligned with respect to one another, can overcome the drawback regarding the energy range but the limitation to 50% in reflectivity still holds. In addition, they suffer from poor reproducibility in their fabrication. A method to circumvent these drawbacks is the use of CDP crystals, whose reflectivity can be close to 100%. In this case, the continuous curvature of lattice planes changes the Bragg condition along the crystal thickness, so that a diffracted photon has a low probability of undergoing re-diffraction inside the crystal (Fig. 1). The energy passband of the photons diffracted by these crystals is orders of magnitude broader Figure 1 X-ray diffraction in Laue geometry in the case of an unbent (a) and of a bent crystal (b). Multiple reflections in case (a) result in a maximum diffraction efficiency of 50%, while in case (b) the diffraction efficiency can be close to 100%. J. Appl. Cryst. (2013). 46, 415–420 than that for a flat crystal, featuring a uniform transfer function provided that the crystal curvature is homogeneous. The implementation of CDP crystals in particular geometries can even result in a focusing effect by the same crystal, making possible the construction of instruments with very high resolution and sensitivity (Guidi et al., 2011). An interesting application of crystals with CDPs is the construction of hard X-ray lenses for astrophysics and nuclear medicine (Frontera & Ballmoos, 2010). Up to now, the impossibility of efficiently focusing hard X-rays left the observation of the sky in this energy range to direct-view instruments, featuring low sensitivity and modest angular resolution. In fact, only the spectra of a few of the strongest sources are known above 70 keV (Frontera et al., 2005). Focusing optics with this purpose have already been realized with mosaic crystals, though some limitations were observed (von Halloin et al., 2004). The introduction of CDP crystals for the construction of high-efficiency focusing instruments for photons with energy up to 1 MeV would allow mapping of the distribution of antimatter and dark matter in our galaxy. High-efficiency instrumentation would also allow a detailed analysis of type 1A supernovae, as well as emissions from compact objects emitting X-rays. Nuclear medicine would also benefit from this technology. Imaging instruments in nuclear medicine are based on the detection of hard X-ray photons emitted by tracing radionuclides injected into the patient (Roa et al., 2005). Today’s instruments use direct-view gamma cameras for this purpose, and complex algorithms to reconstruct the image of the patient’s body, reaching a resolution of a few millimetres at best (Adler et al., 2003). Highefficiency focalization instruments would improve the resolution and sensitivity of these analyses, allowing detection of physiological processes with submillimetre precision with no need for reconstruction algorithms, and thereby reducing the dose of radionuclide that needs to be injected into the patient. doi:10.1107/S0021889813000162 415 research papers Previous work (Bellucci, Camattari, Guidi, Neri & Barrière, 2011; Barrière et al., 2010) has demonstrated the possibility of producing good CDP crystals in a reproducible and controlled manner compatible with mass production. They also showed high reflectivity and a broad passband. It is the purpose of this article to provide a model for calculation of the ratio of diffracted to transmitted intensities in crystals with CDPs under general conditions and to compare this model with the experimental results obtained by Bellucci, Camattari, Guidi, Neri & Barrière (2011) and Barrière et al. (2010). 2. General background The ratio of diffracted to incident intensities is called reflectivity, and in crystals with CDPs it strongly depends on their curvature. The reflectivity is composed of two contributions. In a non-absorbing crystal, the reflectivity would be equal to the ratio of diffracted to transmitted intensities, called the diffraction efficiency, . A photon beam traversing a material undergoes absorption, and the contribution of absorption to the reflectivity can be expressed by the fraction of photons left by the absorption process. For a thick flat crystal, the diffraction efficiency in symmetrical Laue geometry is always equal to or lower than 50% (Authier, 2001; Zachariasen, 1945). As already mentioned, photons have the same probability of undergoing an even or odd number of diffractions inside the crystal. If the crystal is thick enough to guarantee that every photon interacts with the lattice, t0 being the crystal thickness traversed by the X-ray beam, the beam equally splits into diffracted and transmitted beams. With CDP crystals, the probability of a photon undergoing multiple reflections decreases because of the continuous change in the incidence angle with respect to the diffraction planes. The theory of diffraction in curved crystals was widely developed in the past half century in the context of the dynamical theory of diffraction, with particular contributions by Malgrange (2002). The thickness that leads to complete extinction of the diffracted photon beam in a flat crystal in the case of Laue symmetric diffraction is defined as extinction length ¼ ðV cos B Þ=ðre CjFh jÞ (Authier, 2001), where V is the volume of the unit crystalline cell, B the Bragg angle, re the classical electron radius, the wavelength of the incident photons, C their polarization factor and Fh the structure factor of the crystal diffracting planes. The quantity 2W, called the Darwin width, quantifies the broadening of the intensity profile while rocking a flat perfect crystal around the Bragg position. Such an intensity profile is referred to as the rocking curve (RC) in the literature. In the Laue symmetrical geometry, W is given by W ¼ d= (Authier, 2001), where d is the interplanar spacing between the diffraction planes. The physical quantity RC ¼ 2=ðW Þ is called the critical radius. If R RC, it is possible to find a simple quantitative expression for diffraction efficiency (Malgrange, 2002). In a CDP crystal with R RC, it is not possible to obtain complete extinction of the incident beam as for a flat crystal. In fact, for a flat crystal, or for a crystal where the variation of the angle of incidence on the diffracting planes induced by the 416 Valerio Bellucci et al. Diffraction efficiency for curved crystals distortion over its thickness is much smaller than the Darwin width, the entire thickness t0 is available for diffracting the photons of a particular energy. In the case of a highly curved crystal, only a fraction of the crystal thickness can diffract the photons with a particular energy, owing to the large change of incidence angle between photons and lattice planes (considering a parallel incident beam). As will be explained below, diffraction efficiency is influenced by two phenomena: the variation of the fraction of the incident beam that undergoes extinction (prevalent for R RC), and re-diffraction inside the traversed crystal thickness (prevalent for R RC). In the case R RC, the first is the only relevant phenomenon, and the formula below [elaborated by Barrière et al. (2010)] provides a good approximation for diffraction efficiency: ¼ 1 expð2 dR=2 Þ: ð1Þ Thus, diffraction efficiency strongly depends on the curvature radius of the CDPs. Of course, equation (1) cannot work for crystals with low curvature, as it would approach unity as the curvature radius tends to infinity instead of approaching 0.5 as it does for a flat crystal. To date, there exists no analytical theory that quantitatively calculates the diffraction efficiency for crystals with low curvature. In this case, a complete resolution of the Takagi–Taupin equations would be needed, which is not simple and is often not possible (Authier, 2001). Indeed, the applications of CDP crystals sometimes require a curvature radius in the range where equation (1) is not valid, and for that reason we developed the model described in the next section. 3. Modeling X-ray propagation in a crystal is considered without absorption and the crystal is assumed to be at the Bragg angle for the first interaction. The basic approach is to divide the crystal into finite elements, which is similar to the initial assumption of the lamellar model of White (1950). However, multiple diffraction is considered in this article, with the consequent rediffraction effects produced by the misalignment of neighboring elements. One can regard a CDP crystal as a series of very thin flat crystals (lamellae) slightly misaligned with respect to one another. The thickness of these crystals is set equal to half the extinction length (Fig. 2), because within this length, 96% of the photons in the X-ray beam undergo a single interaction with the crystal (Authier, 2001). Thus, the total crystal thickness t0 is divided into N lamellae of thickness =2, i.e. t0 ¼ N=2. The relative misalignment between two subsequent crystals is 0 ¼ =2R and depends on the curvature, which is considered to be perfectly cylindrical. The beam is assumed to be monochromatic and arbitrarily narrow, interacting with the lattice every half extinction length, as explained above. In this case, the totality of the photons in the beam can be considered to interact with the lamella, but the interaction is free of re-diffraction processes. In some intuitive sense, it would be possible to define a lamella as a crystal thickness within which the totality of the photons interact only J. Appl. Cryst. (2013). 46, 415–420 research papers once with the crystal. Naturally, the thickness of a lamella is dependent on photon energy and the nature of the crystal, because the extinction length is a function of energy too. In other words, propagation of an X-ray beam in a curved crystal can be studied once the relative misalignments between the lamellae are known. Diffraction efficiency for a flat and very thin crystal in Laue geometry is an oscillating function of the crystal thickness. It approaches unity when the thickness is equal to half of the extinction length and the angle between the beam and the diffraction planes is exactly the Bragg angle. If the incoming beam is slightly misaligned from the Bragg condition, diffraction efficiency falls off. Following Zachariasen (1945), the dependence of diffraction efficiency on misalignment can be described by a Gaussian distribution, whose maximum attains unity at zero misalignment and whose FWHM equals the Darwin width, i.e. the standard deviation of the Gaussian is ¼ 2W =2:35. The effects of misalignment with respect to the Bragg angle and the variation of the fraction of the incident beam undergoing extinction can be regarded as independent physical quantities. Thereby, the expression for the diffraction efficiency profile of a single lamella is the product of the two individual distributions, namely it is equation (1) multiplied by the misalignment function exp½ðÞ2 =2 2 : f ð; RÞ ¼ exp ðÞ =ð2 Þ 1 exp dR= : 2 2 2 2 0 pathð3Þ i;j 1 B 1 B B 1 B B 1 ¼B B þ1 B B þ1 B @ þ1 þ1 1 1 þ1 C C 1 C C þ1 C C; 1 C C þ1 C C 1 A þ1 i ¼ 1; . . . ; 2n ; j ¼ 1; . . . ; n: ð3Þ At the nth order, the fraction of the initial intensity streaming into the ith branch is 2 2 i;n ¼ ð1 pathðnÞ Þ=2 exp dR= i;1 n1 Q ðnÞ ðnÞ pathi;j þ pathi;jþ1 =2 j¼1 # 2 j P ðnÞ exp 0 pathi;k =ð2 2 Þ k¼1 1 exp 2 dR=2 : ð4Þ The diffraction efficiency of the entire crystal after n interactions is equal to the sum of the intensities held by the branches oriented in the diffraction direction: n ¼ ð2Þ Multiple diffraction splits the initial beam into several branches (Fig. 3). The branches continue propagating and interacting with the lattice every half extinction length, so that after n interactions with the lattice, the initial beam is divided into 2n branches. During this process, the misalignment between the branches propagating in the crystal and the diffraction planes varies because of the crystal curvature. The information about the misalignment between branches and diffraction planes (Fig. 3) is contained in a matrix. The entries of the matrix represent misalignment expressed in units of 0, i.e. misalignment divided by 0. The rows of such a matrix represent the 2n branches and the n columns represent the interactions undergone by the beam. As an example, for n = 3, this matrix holds: 1 1 þ1 þ1 1 1 þ1 þ1 2n P i;n ð1 pathðnÞ i;n ½i; nÞ=2: ð5Þ i¼1 According to the model, in the case of a flat crystal, the intensity of the initial beam is a single branch, which is either the diffracted beam if n is an odd number or the transmitted beam if n is even. Hence, the diffraction efficiency is either 1 or 0, respectively. The elaboration of the model has called for the discretization at ‘deterministic’ locations of the interaction between X-rays and the lattice, but for real situations the interaction Figure 3 Figure 2 Schematic representation of the diffraction process in a Laue crystal. Dotted lines represent the diffraction planes. The X-ray beam interacts with the crystal by traversing the entire crystal thickness t0. The crystal can be divided into elements with thickness =2. J. Appl. Cryst. (2013). 46, 415–420 Subdivision of the initial beam into branches inside the crystal for subsequent steps n (Borrmann triangle). The number beside each branch represents the misalignment with respect to the Bragg angle at the next interaction point of the beam with the CDPs. Misalignment is expressed in units of 0, i.e. misalignment divided by 0. The angles between the lines of different layers are physically the same, but in the figure they are plotted differently to avoid line overlap. Valerio Bellucci et al. Diffraction efficiency for curved crystals 417 research papers occurs continuously. For a thick crystal, the effect of continuous interaction, ruled by the laws of probability, changes the effective thickness within which the beam is completely diffracted. This stochastic effect is equivalent to a crystal with deterministic diffraction locations but with random thickness. Thus, for a given thickness, the diffraction efficiency of the outgoing diffracted beam is the average of the two extreme values 0 and 1, which is 0.5 for a flat crystal. The same holds for a curved crystal, the real diffraction efficiency being the average between the extreme values as calculated by the model. Hence, the diffraction efficiency of the entire crystal at the nth order is the average between the intensities held by the branches in the diffraction directions at the order n and at the order (n + 1): ntot ¼ ðn þ nþ1 Þ=2: ð6Þ Since it is necessary to consider the contribution to diffraction efficiency up to the order n + 1, the corresponding number of lamellae is N = n + 1. Figure 4 Diffraction efficiency versus curvature radius in a Ge crystal. Photons with energy E = 150 keV are diffracted by (111) CDPs. The dashed curve represents the prediction of dynamical diffraction theory through equation (1). The vertical line indicates the critical radius RC = 48.5 m for Ge(111). The dotted horizontal line is the theoretical limit to diffraction efficiency for a flat crystal ( = 0.5). The set of curves are diffraction efficiency ntot at n = 1, 3, 5, 7, 9, 12 and 124, corresponding to a number of lamellae N = n + 1 = 2, 4, 6, 8, 10, 13 and 125. With =2 = 78.8 mm, the corresponding crystal thicknesses are t0 = 0.158, 315, 0.473, 0.630, 0.788, 1.024 and 9.85 mm. The latter thickness is for the Ge crystal that was experimentally tested in Fig. 6(b). An increase in the number of lamellae tends to be even less effective. Figure 6 Figure 5 The same physical quantities as in Fig. 4 as a function of normalized radius. In addition, the simulation was repeated within the interval of photon energy 150–5000 keV with steps of 150 keV in the cases of both Si and Ge. For each number of lamellae all the curves overlap each other irrespective of the energy and of the material. 418 Valerio Bellucci et al. Diffraction efficiency for curved crystals (a) Rocking curve of an Si(111) curved crystal with size 25.5 25.5 1.0 mm analyzed through one of its longest sides by 150 keV photons. At this energy, the crystal thickness traversed by X-rays is t0 = 25.5 mm, corresponding to N = 139 lamellae. (1) Open circles: the intensity of the diffracted beam divided by the intensity of the transmitted beam, i.e. diffraction efficiency. (2) Filled circles: the intensity of the transmitted beam over the intensity of the transmitted beam when no diffraction occurs, i.e. transmission efficiency. The experimental diffraction efficiency is 94.0 (30)%, while the FWHM of the distribution is 15.4 (11)00 . (3) Dashed lines: prediction of the model about the theoretical diffraction efficiency [93.1 (42)%]. The simulation accounts for the uncertainties of experimental parameters; dashed lines define the tolerance range of theoretical predictions. (b) Rocking curve of a Ge(111) curved crystal with size 9.8 9.8 1.0 mm analyzed through one of its longest sides by 150 keV photons. At this energy, the traversed crystal thickness t0 = 9.8 mm, corresponding to N = 125 lamellae. (4), (5), (6) Same meanings as for curves (1), (2), (3), respectively. The experimental diffraction efficiency is 58.1 (19)%, while the FWHM of the distribution is 22.0 (17)00 . The theoretical diffraction efficiency is 67.6 (44)%, a value slightly larger than the experimental diffraction efficiency owing to partial mosaicity in the crystalline structure (Bellucci, Camattari, Guidi, Neri & Barrière, 2011). J. Appl. Cryst. (2013). 46, 415–420 research papers 4. Discussion and results Fig. 4 shows the calculated diffraction efficiency ntot for an Si crystal as a function of the curvature radius for several values of n. The diffraction efficiency tends to 0.5 as R ! 1, i.e. for a flat crystal. Indeed, the diffraction efficiency perfectly overlaps with the expectation of the dynamical theory [equation (1)] in the range R RC. As the crystal becomes thicker owing to an increase of n, the change in the efficiency curve becomes even less important. The critical radius is the condition at which the change in the orientation of lattice planes over half the extinction length, 0 , is equal to the Darwin width. Since 0 depends on curvature, as the curvature radius exceeds the critical radius, the effect of misalignment between near points of the lattice dominates with respect to the effect of variation of the extinction length, producing an increase in diffraction efficiency with respect to the case of a flat crystal. As the curvature radius becomes smaller than the critical radius, the effect of variation of the extinction length dominates, thus reducing the diffraction efficiency. The analysis has been repeated for Ge, which has also been used to fabricate curved crystals for X-ray diffraction and presents a similar crystalline structure but higher atomic number and electronic density than Si. These characteristics lead to a smaller extinction length for Ge than for Si. As an example, for diffraction of photons with energy E = 150 keV by a Ge(111) crystal RC = 48.5 m, while for Si(111) RC = 267 m. For a given n, the efficiency can be expressed in terms of normalized curvature radius R/RC as in Fig. 5. Here, the efficiency is the same for both materials, reflecting the fact that the information regarding X-ray diffraction in a bent crystal is conveyed by the critical radius. The achievements of this paper are useful for revisiting some results of a measurement campaign run at ESRF in 2010 (Bellucci, Camattari, Guidi, Neri & Barrière, 2011). The aim of the experiment was the realization of high-efficiency optics for hard X-rays by diffraction in crystals curved by the method of grooving (Bellucci, Camattari, Guidi & Mazzolari, 2011). A series of Si and Ge crystals was analyzed by X-ray diffraction of their CDPs. A typical diffraction plot for an Si crystal, Figure 7 Theoretical rocking curves (the intensity of the diffracted beam divided by the intensity of the transmitted beam, i.e. diffraction efficiency th) of an Si(111) curved crystal with size 25.5 25.5 1.0 mm analyzed through one of its longest sides by 150 keV photons, at four curvature radii, R. The FWHMs of the distributions are always 15.400 . The theoretical prediction fits the experimental data (open circles) well for (c); a curve joining the experimental data points has been drawn to guide the eye. The curvature radii are R = RC / 4 = 68.75 m (a), R = RC = 275 m (b), R = 1.28RC = 341 m (c), R = 1.89RC = 519.75 m (d). The diffraction efficiency predicted by dynamical theory would, respectively, be 0.793, 1, 1 and 1; the diffraction efficiency predicted by the model th = 0.793, 0.998, 0.931 and 0.676. J. Appl. Cryst. (2013). 46, 415–420 Valerio Bellucci et al. Diffraction efficiency for curved crystals 419 research papers 5. Conclusions Table 1 Comparison of experimental results with theory for Si and Ge crystal plates. An Si crystal plate with dimensions 25.5 25.5 1.0 mm was analyzed through one of its longest sides; the error on geometrical dimensions is 25 mm. Photons with energy E = 150 keV were diffracted by (111) CDPs. In these conditions, the critical radius was RC = 275 m, while the curvature radius of the diffraction planes was R = 341 (25) m. The analysis was repeated with a Ge crystal of dimensions 9.8 9.8 1.0 mm analyzed through one of its longest sides with the same photon energy and diffraction planes. In these conditions the critical radius was RC = 48.5 m while the curvature radius of the diffraction planes was R = 92 (7) m. Values in parentheses are standard uncertainties on the least significant digits. exp and dyn stand for the experimental diffraction efficiency and the diffraction efficiency calculated by the dynamical theory, respectively. The errors on the theoretical diffraction efficiency th are calculated by propagating the error bounds of the experimentally measured curvature radius and thickness, as half the difference between the extreme values of efficiency. Si Ge E (keV) R/RC exp (%) dyn (%) th (%) 150 150 1.28 (9) 1.89 (15) 94.0 (30) 58.1 (19) 100 100 93.1 (42) 67.6 (44) A model of Laue diffraction in a curved crystal has been developed, whose results agree very well with those of the dynamical theory (Barrière et al., 2010) for R RC. The model also produces the same results for a flat crystal when the curvature radius is infinitely large (R ! +1) and provides a quantitative description of diffraction efficiency when neither of these cases are applicable. The model allows a refinement in the interpretation of previously achieved experimental data for curved crystals in the range R > RC. The model sets an upper limit of efficiency for CDP crystals, even in the region R > RC. This knowledge is important when designing new schemes for a Laue lens in next-generation satellite-borne experiments in astrophysics. We recognize financial support by ASI through the LAUE project. References obtained by rocking the crystal with respect to the incident X-ray beam, is shown in Fig. 6(a). The energy passband of the photons that a curved crystal can diffract is determined by its curvature. Indeed, most of the samples characterized at ESRF showed a diffraction efficiency consistent with equation (1), but for the samples with moderate curvature the experiment highlighted a discrepancy between measured efficiency and the theoretical expectations relying on the dynamical theory of diffraction (Table 1). Indeed, under some conditions of application, the curvature radius of crystals exceeded the critical radius. The measured diffraction efficiency for Si (Fig. 6a) is consistent with the prediction of the model developed in this paper in all cases. In fact, the predicted efficiency is 93.1 (42)%, while a diffraction efficiency of 94.0 (30)% was recorded experimentally. For Ge, the diffraction efficiency was somewhat lower than the theoretical prediction, inasmuch as the predicted efficiency is 67.6 (44)% while the experimental diffraction efficiency was 58.1 (19)%. However, for the samples of that run, grooving of Ge samples for bending was found to be too aggressive, resulting in lower diffraction efficiency (Fig. 6b). In fact, as determined by the slope of this curve in Fig. 6(b), the crystal exhibited a mosaicity = 3.5 (4)00 . The theoretical diffraction efficiency of a silicon sample versus its orientation with respect to the beam is plotted in Fig. 7 at four values of the curvature radius. In particular, Fig. 7(c) is the case of the sample examined in Fig. 6(a). The rocking curve calculated by the model fits the experimental data quite well. 420 Valerio Bellucci et al. Diffraction efficiency for curved crystals Adler, L. 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