Anicryde: Calculation of Elastic Properties in Silicon and Germanium Crystals

computer programs

AniCryDe: calculation of elastic properties in silicon and germanium crystals

ISSN 1600-5767 Riccardo Camattari,a* Luca Lanzoni,b Valerio Belluccia and Vincenzo Guidia

aDepartment of Physics and Earth Sciences, University of Ferrara, Via Saragat 1/c, 44122 Ferrara, and INFN, Section of Ferrara, Italy, and bDET – Dipertimento di Economia e Tecnologia, Universita` degli Studi della Repubblica di San Marino, Salita alla Rocca, 44, 47890 San Marino Citta` (Repubblica di San Marino), Italy. *Correspondence e-mail: Received 21 November 2014 [email protected] Accepted 12 March 2015

A code to calculate the anisotropic elastic properties in a silicon or germanium Edited by G. Kostorz, ETH Zurich, Switzerland crystal is introduced. The program, named AniCryDe, allows the user to select the crystallographic configuration of interest. For the selected crystallographic Keywords: anisotropic elastic properties; orientation, AniCryDe calculates several key mechanical parameters, such as silicon; germanium. Young’s modulus, Poisson’s ratio and the shear modulus. Furthermore, the program displays both the compliance and the stiffness tensors concerning the crystallographic orientation of interest. The code enables the user to set several parameters through a user-friendly control stage. As a result, the user obtains the complete displacement field of a deformed crystal and the curvature of any crystallographic plane. Manufacturing wafer defects such as miscut and misflat angle are also taken into account.

1. Introduction Since crystals are inherently anisotropic media, their elastic properties depend on the crystallographic orientation. Yet, to simplify Hooke’s equations, crystals are often regarded to be isotropic, making a zero-order approximation that renders the formalism far simpler than the general case. However, the knowledge of the anisotropic behaviour of crystals may be important for several applications, such as advanced engi- neering, electronics and microelectronics, physics, X-ray diffractometry, biology, and others. Among anisotropic materials, silicon and germanium are the most widely used, both as a substrate for compatibility with semiconductor processing equipment and as a structural material in device fabrication, such as micro- and nanoelec- tromechanical systems. Indeed, a large variety of miniaturized components like transducers, detectors, sensors, actuators and resonators are based on advanced Si and Ge micro- and nano- fabrication (Seacrist, 2005; Gevorgian, 2009). Furthermore, Si and Ge are extensively used for photovoltaic systems. During such usage, crystals are typically subjected to high stress and strain levels. The deformation resulting from stresses strongly depends on the crystal orientation. As a consequence, stressed crystals can be prone to a wide class of damaging phenomena that depend on the crystallographic orientation. In order to assess such anisotropy-based phenomena and to properly design crystalline systems, precise knowledge of the mechanical properties of crystals is mandatory. The anisotropic behaviour of crystals can also be directly exploited. For example, it is possible to use secondary (anisotropic) deformation of crystallographic planes for charged particle steering (Ivanov et al., 2006) or for focusing of # 2015 International Union of Crystallography hard X-rays (Guidi et al., 2011).

J. Appl. Cryst. (2015). 48, 943–949 http://dx.doi.org/10.1107/S1600576715005087 943 computer programs

Several programs simulating the anisotropic behaviour of The characteristics of a crystal depend on its elastic prop- crystals are available, e.g. Ansys (http://www.ansys.com/), erties. Elasticity is the relationship between stress (force per Straus7 (http://www.straus7.com/), Dyna (http://www.lstc.com/), unit area) and strain " (ratio of deformation over initial Sap2000 (http://www.csiamerica.com/products/sap2000), ProSap length) through Hooke’s law. In the linear approximation, (http://www.2si.it/), ABAQUS (http://www.3ds.com/products- stress and strain are linked by first-order equations. In parti- services/simulia/products/abaqus/latest-release/), NASTRAN cular, for Si and Ge the relationships between stresses and (http://www.mscsoftware.com/product/msc-nastran) etc.All strains are linear up to the breaking point (Milman et al., these programs are based on finite element analysis and allow 2007). In tensor notation, these relationships are the user to simulate a wide range of applications. However, ¼ C"; " ¼ S; ð1Þ achieving results with these programs is usually not straightforward and requires specific expertise on the part of the user. where C is the tensor of stiffness and S the tensor of compli- In the present work, we propose a ready-to-use software ance, S ¼ C1. They contain all the useful crystallographic named AniCryDe (anisotropic crystal deformation). It is information. In general, they are fourth-rank tensors, and thus capable of assessing the elastic and mechanical behaviour of to deal with them fourth-rank 3 3 3 3 tensors are an anisotropic plate or beam with rectangular cross section of needed. However, owing to symmetry of the stress and strain a crystal with cubic symmetry. The program deals with such fields, the stress–strain relationship can be represented shapes because they are the typical geometries in which these through second-order tensors by using the Voigt compact crystals are manufactured. AniCryDe accurately calculates notation. Thus, without loss of generality, the stiffness and several quantities that can be very useful for both the crys- compliance tensors are completely defined by 6 6 matrices. tallographic community and the wider communities that use Si These are and Ge mono-crystals. The program has a user-friendly 0 1 0 10 1 interface that makes it readily operational and intuitive for 1 C11 C12 C13 C14 C15 C16 "1 B C B CB C any user without any specific expertise in crystallography. B 2 C B C22 C23 C24 C25 C26 CB "2 C B C B CB C Moreover, AniCryDe allows the user to visualize the results in B 3 C B C33 C34 C35 C36 CB "3 C B C ¼ B CB C; ð2Þ real time. B 4 C B C44 C45 C46 CB "4 C @ A @ A@ A 5 Sym C55 C56 "5 6 C66 "6 2. Program description and application where, for the Voigt notation, the indices that range from 1 to The program has been designed to simulate a mono-crystal 6 correspond to the following directions: with the shape of a plate or a beam in order to study its mechanical properties. The user specifies whether the material 1 ! xx; 2 ! yy; 3 ! zz; is Si or Ge, the crystallographic orientation of the plate, and 4 ! yz; 5 ! zx; 6 ! xy: the size of the plate. AniCryDe displays the crystal in a scaled representation. Then, the program allows the calculation of The Si and Ge C tensors for [100], [010], [001] as (xyz) axes Young’s modulus, the shear modulus and Poisson’s ratio of an are well known in the literature: arbitrary crystallographic direction. The code also calculates 0 1 165:64 63:94 63:94 0 0 0 the displacement field within the solid, as well as the radius of B C B 63:94 165:64 63:94 0 0 0 C curvature of any crystallographic direction. AniCryDe B C B 63:94 63:94 165:64 0 0 0 C displays the results through two- or three-dimensional C ¼B CGPa; Si B 00079:51 0 0 C diagrams. Finally, the program can take into account the @ 000079:51 0 A miscut and misflat angles. The next sub-sections describe the 0000079:51 program functionalities. ð3Þ 2.1. Selection of the crystal orientation 0 1 Since the purpose of AniCryDe is to simulate orientation- 129:247:947:9000 dependent properties, the first step is to set the crystal- B C B 47:9 129:247:9000C lographic orientation of the plate. In the program, crystal- B C B 47:947:9 129:2000C lographic planes and directions are described by Miller CGe ¼B CGPa: ð4Þ B 00067:00 0C indices, which are three-integer triples (hkl), corresponding to @ 000067:00A an (xyz) coordinate system. The (hkl) values are the reci- 0000067:0 procals of the coordinates of the intercepts on the (xyz) axes, multiplied by the lowest common denominator. The Miller indices correspond either to a direction or to a plane AniCryDe could be easily extended to simulate materials perpendicular to the direction. To represent a direction, other than silicon and germanium, by adding their stiffness square brackets are used, whereas round brackets describe a tensors. To calculate the anisotropic properties along an plane. arbitrary crystallographic orientation, the C and S tensors

944 Riccardo Camattari et al. AniCryDe J. Appl. Cryst. (2015). 48, 943–949 computer programs have to be rotated so that the axes are aligned with the directions of interest:

C0 ¼ C T; S0 ¼ S T: ð5Þ

It is not straightforward to calculate the orthogonal rotation tensor . The complete formulae needed for the transfor- mation are the following:

2 2 2 ð1;1Þ ¼ Qð1;1Þ; ð1;2Þ ¼ Qð1;2Þ; ð1;3Þ ¼ Qð1;3Þ; 2 2 2 ð2;1Þ ¼ Qð2;1Þ; ð2;2Þ ¼ Qð2;2Þ; ð2;3Þ ¼ Qð3;3Þ 2 2 2 ð3;1Þ ¼ Qð2;1Þ; ð3;2Þ ¼ Qð3;2Þ; ð3;3Þ ¼ Qð3;3Þ;

ð4;1Þ ¼ Qð2;1ÞQð3;1Þ; ð5;1Þ ¼ Qð3;1ÞQð1;1Þ; ð6;1Þ ¼ Qð1;1ÞQð2;1Þ;

ð4;2Þ ¼ Qð2;2ÞQð3;2Þ; ð5;2Þ ¼ Qð3;2ÞQð1;2Þ; ð6;2Þ ¼ Qð1;2ÞQð2;2Þ;

ð4;3Þ ¼ Qð2;3ÞQð3;3Þ; ð5;3Þ ¼ Qð3;3ÞQð1;3Þ; ð6;3Þ ¼ Qð1;3ÞQð2;3Þ;

ð1;4Þ ¼ 2Qð1;2ÞQð1;3Þ; ð2;4Þ ¼ 2Qð2;2ÞQð2;3Þ;

ð3;4Þ ¼ 2Qð3;2ÞQð3;3Þ; ð4;4Þ ¼ Qð2;2ÞQð3;3Þ þ Qð2;3ÞQð3;2Þ;

ð5;4Þ ¼ Qð3;2ÞQð1;3Þ þ Qð3;3ÞQð1;2Þ;

ð6;4Þ ¼ Qð1;2ÞQð2;3Þ þ Qð1;3ÞQð2;2Þ; ð1;5Þ ¼ 2Qð1;3ÞQð1;1Þ;

ð2;5Þ ¼ 2Qð2;3ÞQð2;1Þ; ð3;5Þ ¼ 2Qð3;3ÞQð3;1Þ;

ð4;5Þ ¼ Qð2;3ÞQð1;3Þ þ Qð2;1ÞQð3;3Þ;

ð5;5Þ ¼ Qð3;3ÞQð1;1Þ þ Qð3;1ÞQð1;3Þ;

ð6;5Þ ¼ Qð1;3ÞQð2;1Þ þ Qð1;1ÞQð2;3Þ; ð1;6Þ ¼ 2Qð1;1ÞQð1;2Þ;

ð2;6Þ ¼ 2Qð2;1ÞQð2;2Þ; ð3;6Þ ¼ 2Qð3;1ÞQð3;2Þ;

ð4;6Þ ¼ Qð2;1ÞQð3;2Þ þ Qð2;2ÞQð3;1Þ;

ð5;6Þ ¼ Qð3;1ÞQð1;2Þ þ Qð3;2ÞQð1;1Þ;

ð6;6Þ ¼ Qð1;1ÞQð2;2Þ þ Qð1;2ÞQð2;1Þ; ð6Þ where 0 1 h k l @ x x x A Q ¼ hy ky ly : ð7Þ hz kz lz

Through the proposed code, the user only has to specify the Miller indices of the x and y axes of the reference system of interest. The z axis, C0 and S0 are automatically calculated and displayed by the program.

2.2. Poisson’s ratio, Young’s modulus and shear modulus We consider the case of a crystal plate under a point load or bent by external moments, where there is linearity between stress and strain (the Hooke regime). The value E, i.e. Young’s modulus or the elastic modulus, is needed to quantify the elastic behaviour of the material in question (Hopcroft et al., 2010). It can be very important to take into account the anisotropic nature of a crystal to evaluate Young’s modulus because the crystal response can vary by as much as 45% as a Figure 1 function of the crystallographic direction. The most important Young’s modulus, Poisson’s ratio and the shear modulus calculated by the program for a silicon plate. Left column: isotropic (111) surface. Right quantities in addition to Young’s modulus are Poisson’s ratio column: anisotropic (110) surface. It can be noticed that for the isotropic and the shear modulus. They are described as follows: (111) plane none of the plots depend on the angle.

J. Appl. Cryst. (2015). 48, 943–949 Riccardo Camattari et al. AniCryDe 945 computer programs

(a) Young’s modulus or the tensile modulus or the elastic superconductive 1000 T magnet would do. Such crystals can be modulus (E) is a measure of the stiffness of an elastic material, employed in particle accelerators as highly efﬁcient colli- namely the ratio of along an axis over " along that axis. mating systems for charged particle beams (Scandale et al.,

(b) Poisson’s ratio () is defined as "trans="axial. It describes 2010). Another example is the following: in order to focus a the deformation of a body that occurs along a direction polychromatic hard X-ray beam, a bent sample is an order of orthogonal to the direction of applied tension. Usually, it magnitude more efficient with respect to a flat crystal and at assumes a positive value. Thus, if a body is stretched, exhibits least twice more efficient than a traditional mosaic crystal a transverse contraction. (Authier & Malgrange, 1998). Properly manufactured Si and (c) The shear modulus or modulus of rigidity (G) describes Ge crystals constitute the optical elements in X-ray analyses the material’s response to shear stress. and in telescope prototypes to focalize cosmic hard X-rays These quantities can be calculated in terms of the compli- (Virgilli et al., 2013). A monitoring system with low power ance tensor as follows: consumption can be realized by applying distributed strip-like piezoceramic sensors to structural frames in order to control E ¼ 1=S ; ð8Þ j jj the frame integrity and reliability. Piezoceramics bonded to a flexible structure can also be used to convert the strain energy v ¼S =S ; ð9Þ ij ji ii of the host structure into usable electrical energy (Anton & Sodano, 2007). Gij ¼ ijk =Skþ3;kþ3; ð10Þ A crystal for such applications is deformed because of with i, j = 1, 2, 3 and the Levi-Civita tensor. external forces. The deformation can be induced in a crystal The variation of these values with respect to a rotation of through several methods, such as mechanical holders (Caras- the coordinate system around a given direction can be easily siti et al., 2010), controlled surface damage (Ferrari et al., obtained through relations (6) by varying the corresponding 2013), superficial grooves (Camattari, Guidi, Lanzoni & Neri, angle of rotation and calculating the compliance tensor S0. 2013), film deposition (Camattari et al., 2014) or the piezo- The program calculates and shows the variation of the electric effect (Anton & Sodano, 2007). However, regardless Young and shear moduli, and the in-plane and out-of-plane of the cause that bends the crystal, a precise assessment of the Poisson ratios for the crystallographic orientations set by the stress and strain fields is mandatory in order to foresee the user. Fig. 1 shows an example of the code output with an Si crystal deformation. crystal oriented in two different configurations: one taking In order to evaluate the mechanical behaviour of a bent into account the isotropic (111) plane and the other with the crystal, the Euler–Bernoulli beam model (e.g. Guidi et al., anisotropic (110) plane. 2010) or thin Kirchhoff plate model (e.g. Bellucci et al., 2013) The compliance matrices for the two cases are reported can be adopted. A complete description of the deformation of here: anisotropic bodies is given by Lekhnitskii (1981) and Nye 0 1 (1985). Loads and stresses acting on a bent crystal may be 0:592 0:155 0:096 0 0:167 0 complex. Nonetheless, the mechanical behaviour of complex B C B 0:155 0:592 0:096 0 0:167 0 C B C loads can be reduced to a system of bending moments acting B C B 0:096 0:096 0:533 0 0 0 C only at the edges of the crystal (Lanzoni et al., 2008). The Sð111Þ ¼ B C B 0001:73 0 0:334 C curvature of the crystallographic planes can be calculated B C @ 0:167 0:167 0 0 1:73 0 A through the displacement field as a function of u(r), v(r) and 0000:334 0 1:494 w(r), which are the displacements along the x, y and z axes, respectively. The normal () and tangential () components of 1011 Pa1; ð11Þ the stress tensor are bound to the mechanical moments Mx 0 1 and M applied to the crystal via 0:533 0:096 0:096 0 0 0 y B C B 0:096 0:592 0:155 0 0 0:167 C M M B C ¼ x z;¼ y z;¼ 0; B 0:096 0:155 0:592 0 0 0:167 C x y z B C Ix Iy ð13Þ Sð110Þ ¼ B C B 0001:494 0:334 0 C B C yz ¼ 0;xz ¼ 0;xy ¼ 0; @ 0000:334 1:73 0 A

00:167 0:167 0 0 1:73 where Ix and Iy are the moments of inertia. As known from the 1011 Pa1: ð12Þ theory of homogeneous anisotropic thin plates subjected to bending, from the following boundary conditions, dw dw dv du 2.3. Calculation of the crystal deformation ¼ ¼ 0; ¼ 0; dx 0 dy 0 dx dy ð14Þ Several applications exploit the anisotropic properties of uð0Þ¼vð0Þ¼wð0Þ¼0; bent crystals to obtain surprising effects. As an example, a bent Si or Ge strip of a few tens of millimetres is capable of the displacement ﬁeld arising from the deformation of a deﬂecting a charged particle beam (Tsyganov, 1976) as a crystal plate is

946 Riccardo Camattari et al. AniCryDe J. Appl. Cryst. (2015). 48, 943–949 computer programs

1 u ¼ ½M ðS z2 þ S yz þ 2S xzÞ 2I x 51 61 11 2 þ MyðS52z þ S62yz þ 2S12xzÞ; 1 v ¼ ½M ðS z2 þ 2S yz þ S xzÞ 2I x 41 21 61 2 þ MyðS42z þ 2S22yz þ S62xzÞ; 1 w ¼ ½M ðS z2 S x2 S y2 2I x 31 11 12 2 2 S16xy þ MyðS32z S12x 2 S22y S26xyÞ: ð15Þ In an anisotropic material, this displacement field may manifest itself in a non-intuitive way. As an example, Fig. 2 shows the deformation of the largest surface of a 10 10 1mmSi plate subjected to two couples of perpendicular and identical moments. Two cases are displayed. In the first case, the crystal has the largest surface oriented as a (111) plane. Since the elastic properties along any direction in the (111) plane are constant, the plate behaves as an isotropic medium and Figure 2 Crystal plate deformed by two couples of perpendicular and identical moments. Two cases are shown. takes the shape of a spherical cup. In Upper part: isotropic configuration. Bottom part: anisotropic configuration. Left panel: two- the second case, the crystal has the dimensional view. Right panel: three-dimensional view. largest surface oriented as a (110) plane. In this plane, the elastic constants depend on the direction. Thus, the deformation is anisotropic and not homogeneous. Some similar experiments are described by Bellucci et al. (2011). Other particular deformations that can be calculated through the code are the secondary curvatures, namely the deformations that arise along unex- pected (and often unwanted) directions. They are a direct consequence of off-diagonal terms in the S0 tensor. The most famous secondary deformation is the anticlastic curvature, which is the transverse curvature developed in the direction parallel to the bending axis (Wang et al., 2005). For an elastic deformation, this happens by the differential lateral contraction caused by the Poisson effect. Fig. 3 shows an example obtained through the program for a 10 10 1 mm Si crystal on which two couples of moments are applied. If the crystal is oriented with Figure 3 the largest surfaces as (111), the Anticlastic deformation for a silicon crystal in the case of isotropic orientation (top) and in the case of anisotropic orientation (bottom). In the former case the couples of moments act about the [110] behaviour is isotropic and the crystal axis, while in the latter they act about the [111] axis. Left panel: two-dimensional view. Right panel: deforms as a saddle. For the (110) three-dimensional view.

J. Appl. Cryst. (2015). 48, 943–949 Riccardo Camattari et al. AniCryDe 947 computer programs orientation, the response is anisotropic and the crystal 3. Program availability deforms as a twisted saddle. The program AniCryDe was developed on the basis of the Another secondary curvature, less known but potentially Mathematica language (Wolfram, Champaign, IL, USA) and very useful for certain applications, is the quasi-mosaic works on Windows, Mac and Linux operating systems. curvature. This curvature concerns the inner planes of a AniCryDe can be found at http://web.fe.infn.it/u/camattari in crystal. It can be used, for instance, to diffract hard X-rays with the ‘Software’ section. AniCryDe can also be found and used high focusing power (Camattari, Guidi, Bellucci et al., 2013) or online as a demonstration project at http://demonstrations. to steer charged particles under particular conditions (Scan- wolfram.com/ElasticPropertiesOfDiamondLikeCrystals/. dale et al., 2007). This curvature manifests itself only in anisotropic media. The program calculates the curvature radii through the following procedure. An array of points is generated along a 4. Conclusions direction of interest. Then, the displacement field is calculated The program named AniCryDe has been conceived to calcu- at these points, which are moved from their initial position. late the elastic properties of the most used semiconductor The final configuration corresponds to the deformation of the mono-crystals, i.e. silicon and germanium. AniCryDe has been crystal along the considered direction. Then, AniCryDe fits a worked out as a ready-to-use software with a user-friendly function on these displaced points. The radius of curvature of interface. In particular, this code allows the user to calculate the fitted function turns out to be the radius of curvature that several physical quantities, such as Poisson’s ratio, the Young the user is interested in. Since the initial array of points can be and shear moduli, and the curvature along any crystal- oriented by the user along an arbitrary direction, AniCryDe lographic direction. Finally, the effects of miscut and misflat calculates the curvature radius for every orientation. angles are accounted for. For every application where mono-crystals are required, the knowledge of their anisotropic properties can be important. In particular, the code has been implemented and used by the authors for their work in particle steering and X-ray focali- 2.4. Calculation of the misflat and miscut effects zation. Usually, a producer specifies the crystal orientation of wafers in two ways. First, the orientation of the plane of the Acknowledgements wafer surface is shown; for example, a (110) wafer has the top surface oriented as a (110) crystallographic plane. The second The authors are thankful to INFN for financial support part of the orientation information is given by the location of through the LOGOS project. the wafer primary flat. The primary flat is a cut perpendicular to the largest surfaces that indicates another crystallographic References orientation, for example the [111]. The accuracy of the Anton, S. R. & Sodano, H. A. (2007). Smart Mater. Struct. 16, R1. alignment of the primary flat with the real crystal lattice is Authier, A. & Malgrange, C. (1998). Acta Cryst. A54, 806–819. named misflat. As known, crystalline Si or Ge wafers are Bellucci, V., Camattari, R. & Guidi, V. (2013). Astron. Astrophys. 560, produced with a typical misflat of about 0.5. Furthermore, a 1–8. miscut can arise during the crystal production, i.e. a wafer can Bellucci, V., Camattari, R., Guidi, V. & Mazzolari, A. (2011). Thin be affected by a crystalline misorientation of the surface with Solid Films, 520, 1069–1073. Bellucci, V., Paterno`, G., Camattari, R., Guidi, V., Jentschel, M. & respect to the real crystallographic planes. Though the miscut Bastie, P. (2015). J. Appl. Cryst. 48, 297–300. and misflat do not have much effect on the crystal behaviour, Camattari, R., Dolcini, E., Bellucci, V., Mazzolari, A. & Guidi, V. they can significantly affect the performance of a crystal in (2014). J. Appl. Cryst. 47, 1762–1764. various applications. As an example, the miscut can affect the Camattari, R., Guidi, V., Bellucci, V., Neri, I. & Jentschel, M. (2013). film deposition and its growth on a crystalline substrate. It is Rev. Sci. Instrum. 84, 053110. Camattari, R., Guidi, V., Lanzoni, L. & Neri, I. (2013). Meccanica, 48, also important to know the actual alignment of the crystal- 1875–1882. lographic planes to perform precise anisotropic etching Carassiti, V., Dalpiaz, P., Guidi, V., Mazzolari, A. & Melchiorri, M. (Vangbo & Ba¨cklund, 1996). (2010). Rev. Sci. Instrum. 81, 066106. The program calculates the effects of miscut and misflat Ferrari, C., Buffagni, E., Bonnini, E. & Korytar, D. (2013). J. Appl. angles on the elastic properties of a crystal. These angles are Cryst. 46, 1576–1581. 0 Gevorgian, S. (2009). Ferroelectrics in Microwave Devices, Circuits taken into account by the code during the calculation of C and and Systems. London: Springer. 0 S as described by Bellucci et al. (2015). Thus, all the other Guidi, V., Bellucci, V., Camattari, R. & Neri, I. (2011). J. Appl. Cryst. outputs are modified accordingly. In particular, Young’s 44, 1255–1258. modulus, Poisson’s ratio and the shear modulus change Guidi, V., Lanzoni, L. & Mazzolari, A. (2010). J. Appl. Phys. 107, because they actually correspond to the morphological surface 113534. Hopcroft, M., Nix, W. & Kenny, T. (2010). J. Microelectromech. Syst. of the solid. The displacement field also changes accordingly 19, 229–238. because it takes into account the real crystallographic orien- Ivanov, Y., Bondar, N., Gavrikov, Y., Denisov, A., Zhelamkov, A., tation of the crystal. AniCryDe displays all this information. Ivochkin, V., Kos’yanenko, S., Lapina, L., Petrunin, A., Skorobo-

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J. Appl. Cryst. (2015). 48, 943–949 Riccardo Camattari et al. AniCryDe 949