research papers
Geometrical prediction of cleavage planes in crystal structures
^{a}Department of Materials Science and Engineering, Tel Aviv University, Wolfson Building for Mechanical Engineering, Tel Aviv, 6997801, Israel, and ^{b}School of Mechanical Engineering, Tel Aviv University, Wolfson Building for Mechanical Engineering, Tel Aviv, 6997801, Israel
^{*}Correspondence email: gorfman@tauex.tau.ac.il
Cleavage is the ability of single crystals to split easily along specifically oriented planes. This phenomenon is of great interest for materials' scientists. Acquiring the data regarding cleavage is essential for the understanding of brittle fracture, plasticity and strength, as well as for the prevention of catastrophic device failures. Unfortunately, theoretical calculations of cleavage energy are demanding and often unsuitable for highthroughput searches of cleavage planes in arbitrary crystal structures. A simplified geometrical approach (GALOCS = gaps locations in crystal structures) is suggested for predicting the most promising cleavage planes. GALOCS enumerates all the possible reticular lattice planes and calculates the planeaverage electron density as a function of the position of the planes in the The assessment of the cleavage ability of the planes is based on the width and depth of planar gaps in crystal structures, which appear when observing the planes lengthwise. The method is demonstrated on twodimensional graphene and threedimensional silicon, quartz and LiNbO_{3} structures. A summary of planar gaps in a few more inorganic crystal structures is also presented.
Keywords: cleavage; transformations; lattice planes; GALOCS; computational modelling; inorganic materials; planar gaps.
1. Introduction
Cleavage is the ability of single crystals to split easily along specifically oriented planes (Hurlbut & Klein, 1977). Such planes are usually parallel to the reticular lattice planes with low Miller indices/large interplanar distances. The cleavage phenomenon was known long before the discovery of Xray diffraction by crystals; the observation of cleavage in crystals contributed greatly to the ideas about periodicity/longrange order of their structures (Tutton, 1922; Authier, 2013). Cleavage is the most striking example of anisotropy of physical properties (Nye, 1985). Finally, cleavage is the subject of many materials science oriented research (Gilman, 1960; Lawn, 1974): acquiring the information concerning cleavage in a given crystalline material is essential for the understanding of failure, brittle fracture, toughness, plasticity and strength (Lawn et al., 1993).
The prediction and discovery of cleavage planes is important for microelectronics and electrooptics where durability of crystalline materials and prevention of their catastrophic mechanical failures is critical (Spearing, 2000). Whether performed theoretically or experimentally, such prediction presents a challenging task.
The simpler approach to the prediction of cleavage planes involves counting the number of broken chemical bonds per unit area of a candidate plane. Such counting can be carried out analytically for the simplest crystal structures (e.g. of rock salt, diamond or sphalerite type) (Ramaseshan, 1946). However, it becomes impractical for larger (e.g. ternary) structures. Accurate calculations of cleavage energies are demanding (Zhang et al., 2007; Bitzek et al., 2015) and should involve e.g. density functional theory (DFT) calculation of the surface energies (Ong et al., 2013; Tran et al., 2016), which is half of the cleavage energy [for the case of slowly propagating cracks (Griffith, 1921)].
It is also possible to study cleavage experimentally (Cramer et al., 2000; Field, 1971; Jaccodine, 1963; Lawn et al., 1993; Michot, 1987; Sherman et al., 2008). While some cleavage planes may immediately appear because of a `mechanical impact', such impact `experiments' can hardly be helpful in the precise measurement of a cleavage energy. More complex arrangements are implemented for this purpose (Gilman, 1960; Gleizer & Sherman, 2014; Sherman & Gleizer, 2014; Hirsh et al., 2020). Specifically, a single crystal should be cut to a plate whose normal is parallel to a candidate cleavage plane while a uniaxial stress (socalled Mode I stress) must be applied in a controlled manner. In conclusion, existing experimental and theoretical methods are demanding and unsuitable for throughput studies of cleavage without preliminary suggestions of several likely possibilities.
Here, we suggest a simple geometrical algorithm and a computer program GALOCS (gaps' locations in crystal structures) which searches for promising cases of cleavage in single crystals of known structures. Apart from accepting the working hypothesis about the planar character of cleavage, we assume that such cleavage occurs along the planes, exposing the widest planar gaps (the intervals of empty space, appearing when observing the plane lengthwise). The manual variant of this approach is common in crystallography classes, where a lecturer exposes a threedimensional structural model and demonstrates prominent clearances/gaps in the structure to the audience. While such inspection can be performed both physically and graphically [e.g. by using the VESTA program (Momma & Izumi, 2011)], there are no computational methods for doing it automatically. This methodological gap is filled by the GALOCS package. In addition to its great illustrative potential, GALOCS can serve as a rough predictor of cleavage planes in crystals. If necessary, such planes can be analysed rigorously using DFT. Alternatively, the results may suggest a particular experimental geometry that allows measuring the cleavage energy of the plane.
2. The algorithm behind GALOCS
The proposed approach realizes an automatic (as opposed to visual) inspection of the `spacefilling function' (SFF). Specifically, we suggest inspecting the sections of the SFF by the planes that are parallel to the lattice planes with reasonable Miller indices/interplanar distances. The dependence of the average SFF on the depth of the plane in the
is produced in order to locate the most prominent planar gaps and measure their width and depth. The correlation between the width/depth of the gap and the cleavage ability of the candidate plane is the major hypothesis behind the algorithm.2.1. Spacefilling function
Calculation of the SFF requires knowledge of
type, lattice parameters, coordinates, occupancies of atomic/Wyckoff positions and types of atoms. The first definition of the SFF is the superposition of electron densities of individual pseudoatoms:Here, R_{m} = u_{mi}a_{i} (Einstein summation over the repeated index running from one to three is implemented everywhere throughout the article) are the positions of all the lattice nodes, u_{mi} are arbitrary integers and a_{i} are the lattice basis vectors i = 1, 2, 3. The vectors (0 ≤ x_{μi} < 1) list the positions of all the atoms in a are electron densities of a (pseudo)atom number μ, which can be approximated using e.g. electron density of an isolated atom (Su & Coppens, 1998) or atomic invarioms (Dittrich et al., 2004). All the atomic electron densities are normalized: [ is the number of electrons, associated with the (pseudo)atom number μ].
Another possible definition of the SFF involves an atomic probability density function (PDF). It describes the probability of finding an atom μ displaced by a vector u from its average position R_{μ}. Such displacements may originate from either a thermal motion or a static disorder. A PDF corresponds to the average of all the atomic positions over time or over different unit cells. A threedimensional Gaussian function is the simplest approximation of an atomic PDF (Coppens, 1997):
Here, is the secondrank tensor of atomic displacement. The components of this tensor are known from an Xray or neutron diffraction experiment and are represented as isotropic or anisotropic atomic displacement parameters (U_{ij} or β_{ij}) (Trueblood et al., 1996; Coppens, 1997; Tsirelson & Ozerov, 1996). If is represented by a 3 × 3 matrix then u^{T} and u are the rows or the column vectors, respectively. The SFF ρ(r) can now be calculated as
Here, describe the `weights' of every atom in the cumulative SFF. yields an SFF in the form of atomic density (it disregards the sort of atoms involved). Setting to the atomic masses defines the SFF as the
Finally, setting to will produce an SFF in the form of nuclei charge density.Notably, all possible definitions of the density function sustain the periodicity so that ρ(r) = ρ(r + R_{m}). Therefore, it is sufficient to calculate the values of ρ(r) inside the only. Because both and are negligible when r > r_{max} (), only the limited number of atoms whose centres are inside the or within r_{max} from its borders are included in the sums (1) or (3).
2.2. Forming a section of a by an arbitrary candidate cleavage plane
GALOCS inspects the sections of the SFF by the planes that are perpendicular to a unitary vector n (n = 1). Let us define the depth D of the plane inside the crystal and introduce the orientation and depthdependent average Γ(n, D) of the SFF ρ(r) as
Here the averaging of ρ(r) is performed over all the positions r, satisfying the condition rn = D (the equation of a plane, normal to the vector n and standing at the distance D from the origin). The periodicity of the means that D is nonnegative and below the corresponding interplanar distance d(n) = d_{hkl}. The latter is the inverse length of the primitive vector, {h_{i} or h, k, l are coprime integers/Miller indices of the plane [the are coprime if a primitive is used (Nespolo, 2015)] and are the basis vectors of the } where B_{hkl} ∥ n:
It is sufficient to average the SFF ρ(r)_{rn = D} over translationally independent locations only, rather than over the entire plane. It is therefore worth transforming the coordinates of r to the coordinate system A_{1}, A_{2}, A_{3} such that A_{i} and a_{i} are the bases of the same but the vectors A_{1} and A_{2} are parallel to the plane of interest (hkl) and A_{3} connects two adjacent lattice planes (A_{3i}h_{i} = 1). The numbertheoretical algorithms for such a transformation are described elsewhere (Gorfman, 2020) and the corresponding MATLABbased program MULDIN is deposited there. This step presents the core of the algorithm because it uses the periodicity of the Using the coordinates X_{1}, X_{2} and X_{3} such that r = X_{i}A_{i} = x_{i}a_{i} and expressing the SFF ρ(x_{1}, x_{2}, x_{3}) as ρ_{hkl}(X_{1}, X_{2}, X_{3}) yields
The cleavage planes are likely to appear where Γ(hkl, X_{3}) drops to the lowest possible values. Averaging of Γ(hkl, X_{3}) over X_{3} would yield the average of the SFF over the entire Let us introduce the constant threshold ρ_{S} = yρ_{0} (typically y = 0.75) such that all the X_{3} values where Γ(hkl, X_{3}) < ρ_{S} are considered as planar gaps. We define the maximum effective width of a planar gap as
Here ΔX_{3} is the length of the longest continuous range of X_{3} values where Γ(hkl, X_{3}) < ρ_{S} and is the average gap depth in this range. The denominator ρ_{0} is introduced in order to reduce the dimension of C_{hkl} to Å. Fig. 1 explains definition (7) graphically. It shows an arbitrary Γ(hkl, D) with the most prominent gap between two vertical dashed lines.
Definition (7) suggests that C_{hkl} must be centrosymmetric. Indeed, the sets of (hkl) and define the same lattice planes. The only difference between them concerns the direction of the plane movement with the increasing value of X_{3}. Specifically, , meaning that has the same minima, the same average value and the same width of the structural gap as Γ(hkl, X_{3}). Accordingly, , so the symmetry of the C_{hkl} is defined by one of the eleven Laue classes.
2.3. The flow chart of the GALOCS algorithm
GALOCS realizes the following steps:
(i) Calculating electron density ρ(x_{1}, x_{2}, x_{3}) (or any other SFF in future releases) inside a (0 ≤ x_{i} < 1) as a superposition of spherical atoms.
(ii) Generating the set of symmetryindependent (with respect to the relevant Laue class) lattice planes (hkl) such that d_{hkl} > d_{min}.
(iii) For each (hkl) transforming the basis vectors a_{i} → A_{i} using the MULDIN algorithm (Gorfman, 2020), and expressing ρ(x_{1}, x_{2}, x_{3}) as ρ_{hkl}(X_{1}, X_{2}, X_{3}) (0 ≤ X_{i} < 1).
(iv) Calculating C_{hkl} according to equation (7), and drawing the directional dependence of C_{hkl} on the stereographic projections and polar plots.
The MATLABbased software package includes several modules for:
(a) Automatic reading of relevant structural information from CIFs.
(b) Calculating ρ(x_{1}, x_{2}, x_{3}) in the on the predefined grid. The current version of the program uses pretabulated spherically symmetrical electron densities in isolated atoms (Su & Coppens, 1998).
(c) Drawing Γ(hkl, D) curves for any chosen hkl values.
(d) Calculation of C_{hkl} (according to equation 7 with userdefined clearance threshold, typically y = 0.75) for the set of lattice planes with interplanar distance above some userdefined value d_{min}.
The software package of GALOCS with corresponding manual is available in the supporting information and through the GitHUB platform. The functionality of the package will be extended (e.g. by introduction to the graphical user interface and by implementing various SFF models) subject to the interest of the user community.
2.4. Twodimensional illustration
Fig. 2 illustrates the GALOCS output for a twodimensional graphene structure. The conventional is based on the vectors a_{1} and a_{2} such that . The SFF was defined by the superposition of electron densities of isolated carbon atoms. Fig. 2(a) shows the transformed unit cells [based on the vectors A_{1} and A_{2} with A_{1} being parallel to (hk)] for the cases of (10), (11) and (12) planes. Fig. 2(b) illustrates the dependence of C_{hk} on the reciprocalspace directions using the polar plot. The spikes extend in the directions of the vectors B_{hk}, their lengths are proportional to the corresponding C_{hk} values. It shows that the longest gap in graphene is present for {10}_{H} and {11}_{H} families of planes [the notation stands for set of planes that are symmetry equivalent to (hk) with respect to the operations of the of twodimensional hexagonal lattice]. {21}_{H} planes have ∼15 times smaller C_{hk} (see Table 1 for the numerical values).

3. Examples
Here, we demonstrate the implementation of the algorithm for the case of three inorganic structures (Si, LiNbO_{3} and SiO_{2}).
3.1. Silicon/diamond structural type
Fig. 3 projects the structure of silicon along the crystallographic [001] and [110] directions. The structure has one symmetryindependent atom sitting at the standard origin 1 of the . Each atom has four nearest neighbours at the corners of a regular tetrahedron. Table 2 summarizes the relevant structural information.

The calculation of C_{hkl} over the lattice planes with the interplanar distance above involved 17 330 primitive vectors enclosed in a corresponding reciprocalspace sphere of radius . This set included 446 symmetryindependent planes (with respect to the operations of the pointsymmetry group ). The cutoff d_{min} is justified by the fact that C_{hkl} drops to zero for those planes that have small interplanar distances. Fig. 4 illustrates this statement by showing C_{hkl} as a function of the length of the primitive vector (). Prominent gaps are present among low B_{hkl} (high d_{hkl}) planes only. The same figure indicates the of the planes with the highest cleavage ability. Note that the indices of the planes are given with respect to the conventional nonprimitive (face centred) and therefore some of the indices are not coprime. The indices are coprime if expressed using a primitive unitcell basis (Nespolo, 2015).
Figs. 5 and 6 illustrate the striking anisotropy of C_{hkl} in silicon using stereographic projections and polar plots. The stereographic projections contain a falsecolour map, which sets onetoone correspondence between the colour and the C_{hkl} values. The projection is viewed along the A_{uvw} = [uvw] = ua_{1} + va_{2} + wa_{3} {/[111] in Figs. 5(a)/6(a), respectively}. Figs. 5(b) and 6(b) show polar plots of C_{hkl} for all the directions B_{hkl} in the zone (i.e. such that B_{hkl} · A_{uvw} = 0). The individual spikes extend in the directions that are normal to the anticipated cleavage planes. Such polar plots provide specific guidelines for the cleavage experiment, in which a crystal is prepared in the form of a wafer whose surface is normal to A_{uvw}. Cleavage is initiated at a small precrack [see e.g. (Hirsh et al., 2020)] and at a uniaxial stress [also known as Mode I stress (Lawn et al., 1993)] along the direction n. Table 3 lists numerical values for the cleavage ability of the most prominent planes.

The calculations predict {111}_{C} and {110}_{C} as the most prominent cleavage planes in silicon [the notation stands for the set of planes that are symmetry equivalent to (hkl) with respect to the operations of the cubic ]. These planes are well known from real cleavage experiments on single crystals of silicon (Gleizer et al., 2014).
The output is further illustrated in Figs. 7 and 8. Fig. 7 shows the corresponding sections of electron density by (111) planes and the Γ(n, D) function where the gap is seen clearly. Fig. 8 shows the view of the structure along the predicted cleavage plane. This figure was produced using the VESTA program, where (111) plane was added artificially and the structural model was rotated in a way that the normal to the plane is parallel to the horizontal axis of the screen. The easy ability to produce the images, exposing the structural gaps in the crystal clearly, is one of the goals of the suggested algorithm.
3.2. Quartz (SiO_{2})
Fig. 9 projects the structure of quartz along [100] and [001] directions, while Table 4 summarizes the relevant structural information (Levien et al., 1980). crystallizes in the type P3_{2}21 (No. 154) and corresponds to the . The structure has two symmetryindependent atoms (Si and O). Enumeration of the lattice planes with the interplanar distance above results in 49 358 vectors, 4393 of them are symmetry independent (with respect to the pointsymmetry operations of the Laue class). Figs. 10–12 and Table 5 show the results of the calculations (as in chapter 3.1). Fig. 10 illustrates the dependence of C_{hkl} on the length of the vectors. Figs. 11 and 12 illustrate the anisotropy of C_{hkl} using stereographic projections along [100] and [001] zone axes [defining the normal to X and Z cuts of quartz wafers (Brainerd et al., 1949)]. Table 5 lists the numerical values of C_{hkl} of seven planes with the most prominent gaps. Figs. 13 and 14 (organized in the same way as Figs. 7 and 8) illustrate the output for the case of (011) planes in quartz.


3.3. Lithium niobate (LiNbO_{3})
LiNbO_{3} crystallizes in the type R3c (Weis & Gaylord, 1985; Weigel et al., 2020), . Table 6 summarizes the relevant structural information. The structure has three symmetryindependent atoms (Li, Nb and O).

Enumeration of the lattice planes with the interplanar distance exceeding results in the generation of 46 340 –19 and Table 7, organized as in the previous sections. The analysis predicts the most prominent cleavage plane with the (012). This plane is also known from the measurements of cleavage energies in LiNbO_{3} crystals (Hirsh et al., 2020).
vectors, where 4108 of them are symmetry independent (with respect to the pointsymmetry group ). The calculation results are presented in Figs. 15

4. Further examples
Tables 8–12 present a brief summary of the C_{hkl} calculations for the structures of wurzite (AlN), fluorite (CaF_{2}), diamond (C), pyrite (FeS_{2}) and corundum (Al_{2}O_{3}). They are organized in the same way as Tables 3, 5 and 7. All the examples are included in the GALOCS user manual in the supporting information.





5. Discussion
GALOCS is an easy and illustrative way to find planar gaps in arbitrary crystal structures. Although we do not claim a onetoone correspondence between the size of these gaps and the cleavage energies, some correlation between them exists. Specifically, the calculations in Section 3.1 suggest that the most prominent gap is seen along (111) and the next most prominent is seen along (110) planes. This result is proven by both experiments (Gleizer et al., 2014) and calculations (Pérez & Gumbsch, 2000), suggesting that single crystals of silicon will indeed break most readily along (111) and next most readily along (110) planes. Specifically, these literature results imply that cleavage along (110) planes requires ∼20% more energy input than cleavage along (111) planes. Table 13 shows the numerical values of cleavage energies against the calculated parameters of the gaps.

Cleavage of quartz single crystals is debated in the literature [see e.g. (White, 2006; Bloss & Gibbs, 1963)]. We are unaware of any accurate calculations or precise measurements of the cleavage energies in quartz. Nonetheless, according to Bloss & Gibbs (1963), when crushed, quartz cleaves most readily along (101)/(011) planes and next most readily along (112) planes. All these planes appear at the top of the list calculated by our algorithm.
Cleavage of LiNbO_{3} crystals was recently investigated by Hirsh et al. (2020), featuring (012), (010) and (116) cleavage planes [the cleavage energies of (012) and (010) were measured]. While all these planes appear on the most prominent planargap list in Table 7, there is some disagreement between the measured cleavage energy and the gap width. Specifically (see Table 14), while the measured cleavage energy of (010) is ∼40% smaller than that of (012), the gap of (010) is four times narrower than (012). It is important to reiterate that this type of disagreement is expected, while the presence of these planes on the list is the simplified GALOCS scheme main goal (since most of the planes do not exhibit any gaps at all).

Still, in order to investigate the matter deeper, we inspected the Γ(hkl, D) for both (012) and (010) planes. Fig. 20 shows that the Γ(012, D) curve has a well, which separates the gap into two `valleys'. This suggests that the chosen threshold value y = 0.75 results in overestimation of the actual gap size for (012) planes. Such close inspection of Γ(hkl, D) is therefore recommended for all planes that are selected as candidate cleavage planes, and can be carried out by using the GALOCS package. Additionally, it is possible to recalculate all the C_{hkl} values using different y thresholds if necessary.
The algorithm can be improved by adding dummy atoms into the structures (e.g. to mimic chemical bonds). Alternatively, it may implement advanced models of electron density, which, in turn, is obtained by DFT calculations or results from a multipole model of Xray diffraction intensities (Hansen & Coppens, 1978). Additionally, atomic electron densities may be convoluted with their PDFs; this can be particularly valuable for disordered materials with high atomic displacement parameters. In general, the SFF ρ(x_{1}, x_{2}, x_{3}) can be customized, e.g. by adding `sticks' (additional electron densities) along the bond lines and removing the atoms themselves. This way the algorithm and the program will be capable of counting the number of chemical bonds intersected by the planes.
6. Conclusions
We developed geometrical algorithm GALOCS to locate the planes that expose the most prominent planar gaps in crystal structures. Such planes are listed as candidate cleavage planes, to be explored experimentally or by using densityfunctionaltheory calculations. GALOCS implements some known generalized spacefilling function (e.g. superposition of atomic electron densities or the probability density function). It calculates the average values of this function within specific planes and as a function of plane depth with respect to the unitcell origin. We also provided detailed illustration of the algorithm for silicon, quartz and LiNbO_{3} where a clear correlation between our calculations and existing experimental studies of cleavage energy is present.
Supporting information
GALOCS package. DOI: https://doi.org/10.1107/S2052252521007272/lt5039sup1.zip
User manual for the GALOCS package. DOI: https://doi.org/10.1107/S2052252521007272/lt5039sup2.pdf
Funding information
The following funding is acknowledged: Israel Science Foundation (grant No. 1561/18).
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