Homogeneous self-standing curved monocrystals ... - IUCr Journals

research papers

ISSN 1600-5767

Received 1 October 2016 Accepted 23 November 2016

Homogeneous self-standing curved monocrystals, obtained using sandblasting, to be used as manipulators of hard X-rays and charged particle beams Riccardo Camattari,a Gianfranco Paterno`,a Marco Romagnoni,b Valerio Bellucci,b Andrea Mazzolaria,b and Vincenzo Guidia,b* a

INFN – Section of Ferrara, Italy, and bDepartment of Physics and Earth Sciences, University of Ferrara, Via Saragat 1/c, 44122 Ferrara, Italy. *Correspondence e-mail: [email protected]

Edited by G. Kostorz, ETH Zurich, Switzerland Keywords: bent crystals; surface sandblasting; X-ray focusing; crystalline undulators.

A technique to obtain self-standing curved crystals has been developed. The method is based on a sandblasting process capable of producing an amorphized layer on the substrate. It is demonstrated that the amorphized layer behaves as a thin compressive film, causing the curvature of the substrate. This procedure permits the fabrication of homogeneously curved crystals in a fast and economical way. It is shown that a sandblasted crystal can be used as an X-ray optical element for astrophysical or medical applications. A sandblasted bent crystal can also be used as an optical element for steering charged particles in accelerator beamlines. Several samples were manufactured and bent using the sandblasting method at the Sensor and Semiconductor Laboratory of Ferrara, Italy. Their curvature was verified using interferometric profilometry, showing a deformation in agreement with the Stoney formalism. The curvature of the machined samples was also tested using -ray diffraction at the Institut Laue– Langevin (ILL), Grenoble, France. A good agreement with the dynamical theory of diffraction was observed. In particular, the experiment showed that the crystalline quality of the bulk was preserved. Moreover, the method allowed curved samples to be obtained free of any additional material. Finally, a crystalline undulator was produced using sandblasting and tested using -ray diffraction at the ILL. The crystal showed a precise undulating pattern, so it will be suitable for hard X-ray production.

1. Introduction

# 2017 International Union of Crystallography

J. Appl. Cryst. (2017). 50, 145–151

The manufacture and development of bent crystals are progressing in different scientific fields. Such bent crystals can be used as optical elements for neutron beams and for X- and -rays; they can also be used to manipulate charged particle beams in accelerators. By exploiting the diffraction of highenergy radiation by bent crystals, many modern applications and tools have been developed, such as monochromators for X-ray beamlines (Schulze et al., 1998; Okuda et al., 2008), optics for instrumentation based on diffraction/scattering and spectroscopy (Okuda et al., 2006), and neutron beam controllers with wide angular acceptance (Mikula et al., 1990). Other applications are under development, such as a hard X-ray focusing system for astrophysical (Virgilli et al., 2013; Hudec, Marsikova et al., 2009; Hudec, Sik et al., 2009) and medical purposes (Roa et al., 2005). On the other hand, owing to the strong electric field generated by ordered atoms in a bent crystal, it is possible to manipulate charged particle trajectories via coherent effects such as channelling and volume reflection (Tsyganov, 1976; Taratin & Vorobiev, 1987). Bent crystals have already been proposed for use in https://doi.org/10.1107/S1600576716018768

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research papers collimation systems (Scandale et al., 2016) and for beam steering (Elishev et al., 1979) and extraction (Afonin et al., 2001). Finally, the emission of radiation due to the curved trajectories of charged particles in bent crystals has been studied; of particular interest are photon production through bremsstrahlung, channelling radiation, parametric X-ray radiation and undulation (Korol et al., 2013). In the light of the great interest shown by these scientific communities, several techniques for producing appropriately curved crystals have been developed, each of which has positive and negative features. One of the first methods proposed for bending consisted of the use of an external holder (Kawata et al., 2001; Carassiti et al., 2010). This method can be an optimal solution unless there are constraints such as encumbrance, weight or miniaturization. In such cases, selfstanding bent crystals are mandatory. A self-standing bent crystal can be obtained by applying a thermal gradient to a perfect crystal (Smither et al., 2006), but this method is energy consuming. A bent crystal can also be obtained by concentration gradient techniques, i.e. by growing a two-component crystal with graded composition along the growth axis (Keitel et al., 1999). These techniques have produced good results experimentally, but such bent crystals are not easy to manufacture, so they are not suitable for mass production. A selfstanding curved crystal can also be obtained by controlled surface damage due to a mechanical process performed on one side of the crystal. Within this category are the grooving method (Bellucci, Camattari, Guidi & Mazzolari, 2011) and the lapping process (Ferrari et al., 2013). These techniques are suitable for mass production, but cause non-negligible damage to the crystals. Another technique for obtaining thick selfstanding bent crystals consists of the deposition of a thick film made of carbon fibres (Camattari, Dolcini et al., 2014). However, this process does not permit the production of small bent crystals, and thus it is not suitable for applications where miniaturized samples are required. Finally, ion implantation has been proposed for crystal deformation (Bellucci et al., 2015). This technique is precise and suited to miniaturization, but it requires an ion implanter and does not permit the bending of crystals thicker than a few hundred micrometres. Here, we propose a technique for producing self-standing bent and periodically bent crystals. It is based on sandblasting one of the major sides of a crystal to produce an amorphized layer capable of keeping the sample bent (see Fig. 1). Sandblasting is a mechanical process consisting of driving a stream of abrasive material against a sample using a pressurized fluid,

Figure 1 A schematic representation of a sandblasted sample.

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Table 1 Features of the sandblasting process. Sample material Sample size (mm) Sandblaster Compressed air consumption Blasting medium Blasting size Blasting density Blasting hardness Nozzle-to-sample distance Process duration

Si 10 10 SAMAC 560 l s1 at 6 bar (1 bar = 100 kPa) Natron glass 1–50 mm 2.3 0.3 g cm3 6 Mohs 10 cm 300 s

usually compressed air. The advantages of this technique are that it is suitable for mass production, it is fast and economical, it does not add any materials to the crystal, and it permits the bending of thin or thick crystals, up to a few millimetres. The drawback is that a thin layer of the material becomes damaged by the process itself. Several Si samples were bent using the sandblasting method and tested at the DIGRA facility at Institut Laue–Langevin (ILL, Grenoble, France). Here, we show that an application of the sandblasting method is the realization of self-standing bent crystals to be used as optical elements in a Laue lens for medical purposes (Paterno` et al., 2015, 2016). A sandblasted crystal could also be used as an optical element to manipulate the trajectories of charged particles in cases where additional materials within the crystal are not allowed, such as in the vacuum pipe of an accelerator. Finally, we propose a particular application of the sandblasting method, i.e. the fabrication of a crystalline undulator, a device suitable for generating hard Xrays via coherent interaction with charged particle beams (Korol et al., 2004). We manufactured a crystalline undulator using the sandblasting method and tested the undulation of the crystallographic planes at the DIGRA facility.

2. Experimental method Several bent Si crystals were prepared and machined at the Sensor and Semiconductor Laboratory of Ferrara, Italy. The samples were shaped using a high-precision dicing saw (DISCO, DAD3220) into squared tiles, 10 10 mm wide, with different thicknesses. The samples were deformed by sandblasting one of the larger surfaces of the tiles. The features of the manufacturing process are listed in Table 1. The sandblasting process produced an amorphized layer on the machined surface of the samples, the characteristics of the layer being dependent on the parameters of the sandblasting process. This amorphized layer resulted in a thin film capable of bending the crystals. We used crystals made of silicon because it is a hard material and shows high crystallographic perfection. Hardness is necessary because a soft material may be consumed by the sandblasting process. On the other hand, the crystallographic perfection of Si allows evaluation of the extent of damage to the crystal bulk via X-ray diffraction. In particular, the diffraction efficiency only attains the theoretical value if the crystalline order is preserved. J. Appl. Cryst. (2017). 50, 145–151

research papers Even though it is not proved in this paper, in principle the sandblasting method would allow a stress field to be applied to any material. For each case it would be important to select appropriate process parameters. As in other contexts where an amorphized layer lies on the crystal surface, it is possible to model the layer as a compressive thin film. Indeed, the amorphized layer is capable of transferring coactive forces to the crystal bulk, thus producing an elastic strain field within the crystal. Therefore, the Stoney formalism for an equi-biaxial planar stress regime can be applied (Janssen et al., 2009): f ¼

h2s

Es 1 ; 6ð1 s Þ hf R

ð1Þ

where hs and hf are the thickness of the substrate and of the compressive film, respectively, Es and s are Young’s modulus and Poisson’s ratio of the substrate, respectively, f is the film stress, and R is the radius of curvature of the sample. In order to take into account the anisotropic behaviour of Si crystals, the Stoney formula can be written (Camattari et al., 2013) as f ¼

1 h2s 1 ; 6ðS11 þ S12 Þ hf R

ð2Þ

where Sij are the components of the compliance tensor for an anisotropic material with reference to the (x, y, z) Cartesian system (Lekhnitskii et al., 1956). It can be noted that the radius of curvature depends on the square of the sample thickness. Thus, it would be possible to obtain a sample with an arbitrary radius of curvature. An increase in the sample thickness leads to an arbitrarily large radius of curvature, i.e. to a small curvature, while reducing the sample thickness leads to high curvatures. However, a sample smaller than few hundred micrometres would be broken by

Table 2 Experimental and simulated results for the radius of curvature of the samples along the [111] direction. Sample thickness (mm)

Interferometric measurements (m)

Analytical calculation (m)

Simulation with Straus7 (m)

Simulation with AniCryDe (m)

0.5 1.0 2.0

6.2 0.7 21 2 80 6

5.1 0.2 20.4 0.6 81 3

5.3 0.1 20.8 0.4 82 2

5.1 0.2 20.3 0.7 81 3

the sandblasting process, placing the limit for the radius of curvature of Si crystals at a few metres. Since, for the applications reported here, the optimal radius of curvature is tens of metres, sandblasting would represent an optimal method for crystal bending. 2.1. Film stress calculation

The manufactured samples were characterized by three different thicknesses, 0.5, 1.0 and 2.0 mm, and three samples were produced for each thickness. The crystallographic orientations of the nine samples are the same as those of the samples depicted in Fig. 2. The radii of curvature of the samples were measured using an optical profilometer (VEECO, NT1100) with 1 mm lateral and 1 nm vertical resolution. Since the machined surface is damaged, the profilometric characterization was carried out on the back face of the samples. The results of the profilometric measurements are listed in Table 2. The values concerning the interferometric measurements are the average of the radii of curvature for the three samples of each group with the same thickness. Their uncertainties take into account both the experimental uncertainty of each measurement and the dispersion of the experimental data for the three measurements. We studied the dependence of the sample curvature on the process duration. In particular, Fig. 3 shows this trend for the 2 mm thick samples. Here, every point represents the average of the radii of curvature for the set of samples, while the error bars take into account both the experimental uncertainty of each measurement and the dispersion of the experimental data for the three measurements.

Figure 2 The two geometries used in this paper. Red arrows represent an X-ray beam, while the tiles represent the crystals. The crystallographic orientations used in the experiment are reported. (a) Geometry 1, aimed at measuring the principal curvature. (b) Geometry 2, aimed at measuring the secondary (QM) curvature. J. Appl. Cryst. (2017). 50, 145–151

Figure 3 The radius of curvature of the 2 mm thick sample as a function of process duration. The error bars take into account the instrumental uncertainty and the dispersion of the experimental data. Riccardo Camattari et al.


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research papers It can be hypothesized that the sandblasting process produces an amorphized layer on the sample surface, with the stress of the machined surface increasing with time. It can be seen that the gradient of the graph decreases with time and the radius of curvature attains a stable value after 150 s. In this study, we decided to blow the sand for 300 s to achieve a high reproducibility of the method within a relatively short time. The film stress f was calculated using equation (2) under the assumption that the film thickness hf was of the order of 5 mm (Gogotsi et al., 1999). By fitting the experimental data with equation (2), we obtained f = 374 8 MPa. The effect of sandblasting was then simulated using the Straus7 finite element package (http://www.straus7.com/). An equivalent amorphized Si layer, 5 mm thick and with a compressive stress of 374 MPa, bonded to Si crystal tiles with the same size and crystallographic orientation as the manufactured samples was simulated. The output of the simulations is reported in Table 2. Finally, the effect of sandblasting was evaluated using the AniCryDe software (Camattari, Lanzoni et al., 2015), imposing a couple of perpendicular moments per unit length on the crystal plate Mx = My = f hf [(hs + hf)/2]. The simulation results are again reported in Table 2. Here, the uncertainties related to the simulations are due to the uncertainty in the calculated stress field. All the simulations are in good agreement with the experimental data, except for the 0.5 mm thick samples, which showed a radius of curvature slightly higher than the simulated values, maybe because the approximation of a thin film is more suitable for describing the thicker samples. The layer of amorphized material can be appropriately considered as a compressive thin film.

3. Sandblasted crystals as optical elements for hard X-ray concentrators

Figure 4 A schematic representation of a Laue lens. The red arrows represent an X-ray beam that is diffracted towards a target placed at the focal point of the lens.

was E/E ’ 106. The beam flux was produced by neutron capture in a gadolinium target (157 64 Gd) inserted close to the nuclear reactor of ILL at a temperature of about 673 K. The beam divergence after an Si(220) monochromator was 3.500 , as measured by recording a rocking curve (RC) of the monochromator itself. The collimated beam size was 1 2 mm. A standard electrode coaxial Ge detector with 25% relative efficiency was used. Sample characterization was carried out by measuring RCs, i.e. by recording either the transmitted or diffracted beam intensity while the crystal was being rotated around the position where the Bragg condition was satisfied. A 2 mm thick crystal was analysed using two different geometries. In geometry 1 (Fig. 2a), the beam passes through the sample along the [111] direction, traversing 10 mm of material. In this geometry, the curvature induced directly by the sandblasting process was measured. Since ð112Þ diffraction is forbidden in silicon, we used the ð224Þ planes for the experiment. In geometry 2 (Fig. 2b), the beam passes through the shortest side of the crystal, i.e. along the ½112 direction. In this configuration, it was possible to record the RC of the crystallographic planes bent by the quasi-mosaic (QM) effect (Camattari, Guidi et al., 2015). Indeed, the (111) planes proved to be bent by the QM effect, as shown in Fig. 2(b). QM

A hard X-ray concentrator can be exploited for astrophysical (Von Ballmoos, 2013) or medical purposes (Paterno` et al., 2016). Indeed, for both applications there is the aim of concentrating hard X-rays for which the energy ranges from about 100 to 1000 keV. Such hard X-rays can be focused through diffraction via a Laue lens (Lund, 1992), which is conceived as an ensemble of many crystals arranged in such a way that as much radiation as possible is diffracted onto the lens focus over a selected energy band (see Fig. 4). Curved crystals are optimal candidates for focusing a hard X-ray beam with high efficiency (Bellucci, Camattari, Guidi, Neri & Barrie`re, 2011; Camattari, 2016). We tested the sandblasted samples using -ray diffraction on the DIGRA facility at the ILL, a facility specifically built for characterFigure 5 izing instrumentation for astrophysics. Rocking curves for the two geometries. Black dots represent the diffracted beam and red dots the The -ray beam energy was transmitted beam. (a) Sample in geometry 1 (primary curvature). (b) Sample in geometry 2 (QM curvature). Grey areas represent the theoretical expectations. 181.931 keV and its monochromaticity

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research papers curvature is due to the anisotropic behaviour of silicon and can be exploited to obtain high focusing of a parallel X-ray beam (Bellucci et al., 2013; Camattari, Paterno` et al., 2014). The results of the measurements are shown in Fig. 5. By fitting the RCs with the dynamical theory of diffraction (Authier & Malgrange, 1998), it is possible to derive the sample curvature, the homogeneity of the curvature and the quality of the crystal bulk. If the measured RCs are compatible with theoretical expectations, we can deduce that the crystal is bent homogeneously and the lattice quality is preserved. The diffraction efficiency is given by the formula ¼ 1 exp

2 T0 dhkl ;

20

experimental setup, which is represented by a normal distribution with a standard deviation equal to 3.500 . As a result, the RC is a symmetric function of height equal to 0.234 in units of normalized counts and a standard deviation of 3.6500 . In summary, Fig. 5 highlights the very good agreement between theoretical expectations and experimental results. Since the RCs show a flat-top profile, it can be deduced that the curvature of the sandblasted sample is homogeneous and the quality of the crystal bulk is not compromised by the manufacturing process. Therefore, samples produced using the sandblasting method can profitably be used as X-ray optical elements and as charged particle beam steerers.

ð3Þ

where T0 is the crystal thickness traversed by the radiation, dhkl the d spacing of the diffracting planes (hkl), 0 the extinction length as defined by Authier & Malgrange (1998) for the Laue symmetric case and the bending angle of the curved diffracting planes. The grey area in Fig. 5 represents the theoretical expectation, taking into account the uncertainty on the primary curvature of the sample. Fig. 5(a) shows the RC for the sample in geometry 1. The height of the RC corresponds to the diffraction efficiency , while the FWHM corresponds to the bending angle of the sample. The result reported in Fig. 5(b) is here explained. From the theory of elasticity, the ratio between the QM radius of curvature (RQM) and the superficial radius of curvature (the primary curvature, RP) is RQM/RP = 2.61 for the (111) planes. For this sample, RP = 74 4 m, and thus RQM = 2.61RP = 192 m (Camattari, Guidi et al., 2015). Here, the QM curvature corresponds to an angular spread of 2.1400 for the (111) planes. The diffraction efficiency was expected to be 95.0%. The RC reported in Fig. 5(b) shows a far lower peak. However, there are some factors that have to be taken into consideration, chiefly that the measured RCs are the convolution of three functions. The first function is a uniform distribution due to the diffracting QM planes, 2.1400 wide and 0.95000 in height. The second function represents the spread owing to the primary curvature, which is a uniform distribution 2.8000 wide and 100 in height. The beam has a finite size of 1 mm along the x direction; thus, the primary curvature results in a rotation of the diffracting planes by 2.8000 . The third function takes into account the resolution of the

4. Sandblasted crystals as crystalline undulators As an application of sandblasting for obtaining a deformed crystal, here we propose the fabrication of a crystalline undulator (CU). A CU consists of a crystal whose planes are periodically bent, to be used as a generator of hard electromagnetic radiation (see Fig. 6). It is possible to confine ultrarelativistic positrons within the atomic planes by the strong electrostatic field generated by the aligned atoms of the lattice, i.e. the channelling phenomenon (Biryukov et al., 1997). Similar to a magnetic undulator, the charged particles are forced to follow an undulating trajectory within the CU, emitting very hard X-ray radiation (Korol et al., 2004). We exploited sandblasting to produce a precisely undulating Si crystal, leaving the bulk substantially defect free. We manufactured an Si CU, 8 mm long, 0.5 mm thick and 10 mm wide; the undulation period was 2 mm and the channelling planes were (111). A similar CU was realized using the

Figure 6 A schematic representation of a crystalline undulator. If a charged particle beam is confined in an undulating crystal, coherent emission of hard -rays can be achieved. The undulating rectangle is the crystalline undulator, the particle trajectory is represented with a red dashed line and the resulting -rays are represented by pale-green arrows. J. Appl. Cryst. (2017). 50, 145–151

Figure 7 (a) The angular shift in crystallographic planes due to deformation along the crystal length, measured using hard monochromatic X-rays. (b) The deformation of crystallographic planes obtained using numerical integration of the experimental data. Riccardo Camattari et al.


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research papers grooving method and successfully tested using a proton beam (Bagli et al., 2014). These parameters fulfil the condition for an optimal undulator in the case of 15 GeV positrons (Korol et al., 2004). The CU was tested using -ray diffraction at the DIGRA facility of ILL. The -ray beam energy was 181.931 keV, its monochromaticity was E/E ’ 106 and the beam size was 0.5 0.5 mm. The sample was characterized by recording the position of the diffraction peak for the (220) lattice plane for 39 consecutive points along the [111] direction. In order to subtract a possible shift due to instrumental tolerances, a flat reference crystal with identical lattice orientation was set behind the undulator, and the (220) diffraction peak of the reference crystal was recorded together with the peak of the undulator. The shift in the angular position of the CU diffraction peak with respect to the reference as a function of the impact position of the -ray beam is shown in Fig. 7. The measured average amplitude of undulation was 23.1 0.5 nm with an average period of 2.01 0.03 mm. As can be seen, a precise and homogeneous undulating pattern was imparted to the sandblasted sample.

5. Conclusions A method for manufacturing self-standing curved crystals has been shown. The manufactured samples were characterized using interferometric profilometry and -ray diffraction, showing good agreement with theoretical expectations. This means that the layer of material amorphized by the sandblasting process behaves as a compressive thin film and the substrate of the crystal is not damaged by the manufacturing process. Indeed, sandblasting leaves the crystal bulk substantially free from defects, since the superficial amorphized layer is very thin compared with the entire crystal volume, less than 0.25% for a 2 mm thick crystal. Moreover, contamination of the machined samples is not expected because the sand microspheres used for sandblasting are chemically inert. The sandblasting method results are reproducible. The relatively small uncertainty in the radius of curvature for each group with the same thickness is representative of the reproducibility of the technique, as can be seen in Table 2. Moreover, by observing Fig. 7, it can be seen that the amplitude of each period of the manufactured undulator shows a very high homogeneity, highlighting that each strip of sandblasted material produces the same stress field and thus the same deformation on the sample. Indeed, if the technique were not reproducible, it would be impossible to obtain such a regular periodic deformation pattern. Other parameters which have not been investigated in this paper, such as nozzle-to-sample distance, blasting medium and flux characteristics, may affect the sample curvature. However, since we observed a relatively good reproducibility, we have probably identified the key parameters behind the method. Nevertheless, a deeper analysis of the parameters we have not studied would lead to better comprehension and in turn to better reproducibility.

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Sandblasted curved crystals may be optimal candidates for optical elements of a Laue lens for medical or astrophysical applications, where a large number of homogeneous and small crystals are mandatory to focus the radiation towards the focal plane. Moreover, the resulting sample deformation is suitable for patterning. The sandblasting method has been demonstrated to be a possible method for producing self-standing periodically bent crystals, i.e. a crystalline undulator (Korol et al., 2013).

Acknowledgements The authors are grateful to the INFN for financial support through the Lauper project and to the ILL for the allocation of beam time.

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