research papers
Permissible domain walls in monoclinic M_{C} phases
Part II. The case of^{a}Department of Materials Science and Engineering, Tel Aviv University, Wolfson Building for Mechanical Engineering, Tel Aviv, 6997801, Israel
^{*}Correspondence email: gorfman@tauex.tau.ac.il
This article is part of a collection of articles from the IUCr 2023 Congress in Melbourne, Australia, and commemorates the 75th anniversary of the IUCr.
Monoclinic ferroelectric phases are prevalent in various functional materials, most notably mixedion perovskite oxides. These phases can manifest as regularly ordered longrange crystallographic structures or as macroscopic averages of the selfassembled tetragonal/rhombohedral nanodomains. The structural and physical properties of monoclinic ferroelectric phases play a pivotal role when exploring the interplay between ferroelectricity, ferroelasticity, giant Acta Cryst. A80, 112–128] the geometrical principles governing the connection of domain microstructures formed by pairing M_{AB} type monoclinic domains were elucidated. Specifically, a catalog was established of `permissible domain walls', where `permissible', as originally introduced by Fousek & Janovec [J. Appl. Phys. (1969), 40, 135–142], denotes a mismatchfree connection between two monoclinic domains along the corresponding domain wall. The present article continues the prior work by elaborating on the formalisms of permissible domain walls to describe domain microstructures formed by pairing the M_{C} type monoclinic domains. Similarly to Paper I, 84 permissible domain walls are presented for M_{C} type domains. Each permissible domain wall is characterized by the transformation matrix between the vectors of the domains and, crucially, the expected separation of Bragg peaks diffracted from the matched pair of domains. All these parameters are provided in an analytical form for easy and intuitive interpretation of the results. Additionally, 2D illustrations are provided for selected instances of permissible domain walls. The findings can prove valuable for various domainrelated calculations, investigations involving Xray diffraction for domain analysis and the description of domainrelated physical properties.
and multiferroicity in crystals, ceramics and epitaxial thin films. However, the complex nature of this subject presents challenges, particularly in deciphering the microstructures of monoclinic domains. In Paper I [Biran & Gorfman (2024).Keywords: ferroelastic domains; monoclinic symmetry; Xray diffraction.
1. Introduction
The orientation and properties of permissible domain walls (PDWs) connecting domains of monoclinic (M_{A}/M_{B}) symmetry were thoroughly discussed in our previous paper, denoted as Paper I (Biran & Gorfman, 2024). In addition to motivation for the exploration of monoclinic ferroelectric phases, we systematically derived the catalog of 84 PDWs, which included their corresponding the orientation relationship between vectors and the separation between Bragg peaks diffracted from domains, connected along specific PDWs. Notably, we employed reasonable approximations to obtain analytical expressions for these quantities. We identified 48 PDWs of Wtype and 36 PDWs of Stype, signifying whether their are independent or dependent on the free lattice parameters. Moreover, we derived the specific combination of pseudocubic lattice parameters governing the orientation of the Stype domain walls as well as demonstrated how the change of a lattice parameter causes rotation of the domain wall.
According to Fu & Cohen (2000), Vanderbilt & Cohen (2001), Noheda et al. (1999, 2000), monoclinic ferroelectric phases (MFEP) can be categorized into M_{A}/M_{B} or M_{C} types. These phases are distinguished by the permissible crystallographic direction of spontaneous polarization, if present, and the set of independent pseudocubic lattice parameters. Both types of MFEP are prevalent in ferroelectric perovskites and are frequently employed to describe the fine details of their crystallographic structures (see e.g. Guo et al., 2003; Phelan et al., 2015; Wang et al., 2016; Gu et al., 2014; Zhang et al., 2011). Additionally, such phases are regularly reported in epitaxial thin films of (Luo et al., 2017; Bin Anooz et al., 2022; de Oliveira Guimarães et al., 2022; Wang et al., 2022; Gaal et al., 2023).
Paper I focused on MFEP of M_{A}/M_{B} type only. The current article extends the same formalism to encompass monoclinic phases of the M_{C} type. Because the framework of this paper aligns closely with that of Paper I, most of the mathematical derivations have been provided in the supporting information. For a comprehensive list of notations, please refer to the corresponding section of the paper Gorfman et al. (2022) and Appendix A of Paper I.
2. Monoclinic ferroelectric phases: important definitions
2.1. The definition of the M_{C} monoclinic phase
The crystallographic structures of the M_{C} MFEP belong to the spacegroup types Pm, Pc. These structures are obtained through symmetrylowering phase transitions from those described by the tetragonal (T) spacegroup types P4mm, P4bm. The `monoclinic' mirror (m) /glide (c) plane aligns parallel to two edges of the pseudocubic The spacegroup types Pm, Pc allow for the rotation of the spontaneous polarization direction (SPD) within this mirror plane. Additionally, these space groups permit any distortions of the that maintain the mirror plane. Fig. 1(a) provides a visual representation of both the distortion of the pseudocubic and possible rotation of the SPD.
2.1.1. The directions of spontaneous polarization
The emergence of the M_{C} phase from the T phase prompts us to define the SPD through the slight rotation (by a small angle ρ) from one unitcell edge such as [001] towards another such as [100]. This definition gives rise to distinct orientational variants of the monoclinic domains, denoted as M_{nm}, where the first index n (n = 1–3) designates the SPD T_{n} in the `parent' tetragonal domain with T_{1} = [100], T_{2} = [010], T_{3} = [001]. The second index m lists four options of monoclinic distortion m = 1–3 such that m ≠ n. For instance, the monoclinic domain M_{12} has its SPD rotated from [100] towards [010], whereas features rotation from [100] towards . Fig. 1(b) represents the illustrating the potential SPDs in all 12 monoclinic domains.
In the following, we express the SPD relative to the axes of the Cartesian coordinate system, which are closely aligned with the pseudocubic basis vectors. For instance, in the case of the M_{31} domain we obtain
Herein, we introduce the notation
2.1.2. Pseudocubic lattice parameters
Fig. 1(a) shows the M_{C} distortion of the pseudocubic The corresponding pseudocubic lattice parameters a_{i}, α_{i} (i = 1–3) are defined in terms of four independent variables: a, b, c, β. For instance, in the case of the M_{31} domain, we have , , , = , . The matrix of dot products corresponding to these lattice parameters is
Here, [I] is the unitary matrix and
Assuming that the monoclinic distortions are small, i.e. keeping the first power of , , , we can write
The resulting monoclinic N_{M} = 4 symmetry operations of the 2/m. The original cubic is invariant with respect to N_{C} = 48 operations of the m3m. Given that the monoclinic distortion can originate from any of these 48 equivalent variants, there are a total of monoclinic domain variants. These domains are listed in Table 1, which includes domain identifiers, M_{nm}, the [G′] metric tensors, the corresponding SPD as well as the lattice parameters a_{1}, a_{2}, a_{3}, α_{1}, α_{2}, α_{3}.
is invariant with respect to2.2. Domain pairs
Here we introduce different types of domain pairs, denoted as `Tsiblingplanar' (TSBP), `Tsiblingcrossed' (TSBC), `Tplanar1' (TP1), `Tplanar2' (TP2), `Tsemiplanar' (TSP), `Tsemicrossed' (TSC) and `Tcrossed' (TC). The angles between the SPDs within each pair can be calculated using equation (2) and the third column of Table 1. A comprehensive summary of information pertaining to the domain pair types can be found in Table 2 (see Figs. 2–8).

3. The orientation of PDWs between different pairs of domains
The key steps for determining the orientation of the PDWs between two arbitrary domains (Table 1) have already been elucidated in the corresponding section of Paper I. These steps involve calculating the difference [G′]_{n} − [G′]_{m} and evaluating their respective eigenvalues and eigenvectors. As was done in Paper I, we present the detailed derivation for representatives of each domain pair type. However, for the sake of brevity, most technical details are provided in the supporting information.
3.1. PDWs connecting domain pairs of the type Tsiblingplanar (TSBP)
Supporting information section S1.1 demonstrates the derivation of PDW orientation for the representative TSBP domain pair . It reveals that this pair of domains can be connected via two PDWs, each normal to the directions :
As in Paper I, the components of these vectors have the meaning of the ) these are Wwalls.
of the PDW plane. The of both PDWs are independent of the lattice parameters. According to Fousek & Janovec (19693.2. PDWs connecting domain pairs of the type Tsiblingcrossed (TSBC)
Supporting information section S1.2 demonstrates the derivation of PDW orientation for the representative TSBC domain pair M_{12}M_{13}. It shows that this pair of domains may be connected along PDWs normal to the vectors :
The wall, normal to [TSBC^{(1)}], can be referred to as a Wwall. In contrast, the of an Swall [TSBC^{(2)}] depend on the monoclinic distortion parameter u, as defined by
It is worth noting that, while both the numerator and denominator in equation (8) involve small monoclinic distortions, their ratio is generally not small. Specifically, u is strongly dependent on and Δβ. Remarkably, the orientation of this domain wall remains independent of the monoclinic lattice parameter c. Table 3 highlights cases in which the of the Swall are rational numbers.

3.3. PDWs connecting domain pairs of the type Tplanar1 (TP1)
Supporting information section S1.3 demonstrates the derivation of PDW orientation between the representative TP1 pair of domains M_{12}M_{21}. It reveals the existence of two Wwalls normal to the vectors :
3.4. PDWs connecting domain pairs of the type Tplanar2 (TP2)
Supporting information section S1.4 demonstrates somewhat more intricate derivation of the PDW orientation between the representative TP2 types of domain pairs M_{12} and . The analysis reveals the existence of two PDWs, each normal to the vectors :
Here we introduced the notation
and
Both PDWs are Swalls. Their orientation appears to be significantly influenced by the lattice parameters c, a and Δβ but independent of b. Table 4 showcases two special/favorable cases in which these walls are perpendicular to the directions with rational Miller indices.

3.5. PDWs connecting domain pairs of the type Tsemiplanar (TSP)
Supporting information section S1.5 presents the derivation of PDW orientation for the representative TSP type of domains M_{31}M_{21}. As a result, two PDWs normal to the vectors are identified:
In this context, the following notation is employed:
Table 5 lists special cases where the Swall, normal to [TSP^{(2)}], exhibits rational Notably, the wall's orientation generally remains independent of the lattice parameter a.

3.6. PDWs connecting domain pairs of the type Tsemicrossed (TSC)
Supporting information section S1.6 provides the derivation of PDW orientation for the representative pair of TSPtype . The analysis reveals that this pair may connect via two PDWs, normal to the vectors :
Similar to some cases discussed above, both W and Stype domain walls (DWs) are present here. Favorable cases in which the [TSC^{(1)}] Swall exhibits rational are contained in Table 5.
4. The transformation matrices and the separation between Bragg peaks
Supporting information section S2 provides the derivation of the `Delta' transformation matrices between the pseudocubic basis vectors of two domains m and n. These matrices [ΔS] are defined as follows:
The methodology for deriving these matrices aligns with the procedure described in Paper I. These transformation matrices enable various domainrelated calculations, such as precise calculation of the angles between the SPDs in the corresponding pair of domains connected along the relevant PDW. Most notably, the formalism provides expressions for the separation of Bragg peaks HKL diffracted from these domain pairs. This separation can be calculated using the matrix [ΔS*] between the corresponding reciprocallattice vectors:
This leads to the expression for the splitting of the Bragg peaks, relative to the reciprocal coordinate system of domain m:
Such splitting is routinely measured in highresolution singlecrystal diffraction experiments (Gorfman & Thomas, 2010; Vergentev et al., 2016; Zhang et al., 2018; Choe et al., 2018; Gorfman et al., 2011, 2020, 2021, 2022). Therefore, expression (18) finds direct application in recognizing connected domain pairs within 3D diffraction patterns. Remarkably, when monoclinic distortion parameters (5) are small, both the elements of these transformation matrices as well as the components of the Bragg peak separation can be obtained analytically (see corresponding expressions in sections S2.1–S2.6).
5. Numerical examples
In this section we illustrate the principles underlying PDWs and the separation among the associated Bragg peaks, focusing on the domain pairs of TP1 and TP2 type. Given that these pairs have a common monoclinic twofold axis, we can illustrate the connection between such domains on the 2D drawings within the monoclinic mirror plane which is perpendicular to this axis [this plane is highlighted in Fig. 1(a)].
For the TP1 case we illustrate the connection between domains M_{12}M_{21}. These domains have the lattice parameters c a b β and a c b β, respectively (Table 1). For this numerical example, we assumed that and β = 88°. According to (9) these domains connect along (110) or PDWs, both walls are normal to the mirror plane. Fig. 9(a) illustrates (110)connection of these domains. Notably, these domains can selforganize into a lamellatype microstructure pattern, wherein M_{12} and M_{21} domains alternate periodically along the PDW normal. This arrangement introduces the concept of `adaptive' phase as discussed by Jin et al. (2003), Viehland & Salje (2014). In this concept, the alternation and miniaturization of domains create states with macroscopic longrange periodicity and symmetry controlled by the volume ratios of the domains, rather than their lattice parameters only. Fig. 9(a) illustrates the possibility of such alternation while avoiding, however, the effects of domain miniaturization. Furthermore, Fig. 9(b) depicts the reciprocal lattices of these domains, clearly indicating that the separation between the Bragg peak diffracted from the corresponding matched domains occurs in the direction parallel to the PDW normal. It is worth noting that additional diffraction effects may emerge due to domain miniaturization and periodicity, as described by Wang (2006, 2007) in the case of tetragonal and rhombohedral nanodomains.
For the TP2 case we elucidate the connection between M_{12} and domains. These domains have the corresponding lattice parameters c a b β and a c b , respectively (Table 1). According to (10) these domains can form a connection along Swalls (g10) and . In this specific numerical example, with and β = 88°, we obtain that, according to (11) and (12), g ≃ 0.52. Fig. 10(a) illustrates the pairing of the domains along the (g10) plane, like Fig. 9(a), while Fig. 10(b) provides visual representation of their corresponding reciprocal lattices. It is crucial to note that the orientation of this wall can vary with changes in lattice parameters.
6. Summarizing tables
The previous paragraphs and the supporting information outline the derivation of the equation for the PDWs'
orientation relationship between the lattice basis vectors, and the separation of Bragg peaks diffracted from the representative domain pairs. Similar equations can be derived for all the other pairs of domains. Tables and figures presented here list the corresponding quantities for all 84 existing PDWs. The full list includes:12 PDWs connecting domain pairs of the type `Tsiblingplanar'. All of them are Wwalls.
24 PDWs connecting domain pairs of the type `Tsiblingcrossed'. 12 of them are Wwalls and another 12 of them are Swalls.
12 PDWs connecting domain pairs of the type `Tplanar1'. All of them are Wwalls.
12 PDWs connecting domain pairs of the type `Tplanar2'. All of them are Swalls.
12 PDWs connecting domain pairs of the type `Tsemiplanar'. Six of them are Wwalls and another six of them are Swalls.
12 PDWs connecting domain pairs of the type `Tsemicrossed'. Six of them are Wwalls and another six of them are Swalls.
Tables 6–11 contain the list of 84 PDW including 36 S and 48 Wwalls. Each table includes PDW number, the identifiers of the connected domains, the of the corresponding PDW, the orientation relationship and the reciprocalspace separation between the Bragg peaks diffracted from this domain pair. In addition, the fifth column of these tables contains the approximate angle between the SPDs, providing zero or minimal domain wall charge, meaning that the SPDs (listed in Table 1) on both sides of the domain wall should have the same signs as the projection on the domain wall normal. For example, Table 6 shows that TSBP pair M_{12} and may connect along the PDWs parallel to either (100) or (010) lattice planes. According to Table 1 these domains contain polarization vectors parallel (or antiparallel) to the directions [1x0] or . The projection of these directions to the (100) normal is equal to 1 while the projection of these directions to the (010) plane normal is . Accordingly, the (100) DW should separate domains with the polarization vectors with the angle between them close to 0. In contrast, the (010) DW should separate domains with the polarization vectors with the angle between them close to 180°.






Tables 6–11 reveal that certain Wwalls have the same orientations. Table 12 presents all the distinct PDW orientations and their relevant details. It reveals that all the PDWs belong to five orientation families {100}, {110}, {2uu}, {g01}, {2tt}, so that PDWs of 45 distinct orientations are present. Furthermore, the table demonstrates the distribution of PDWs based on the pair type and the angle between the polarization directions.

Fig. 11 displays the orientation of all the PDWs for various choices of lattice parameters. The normals to these walls are depicted using the poles on the Wwalls are marked by poles with solidline edges, and the color of the pole reflects the angle between SPDs, which is close to 0, 90 and 180° (as specified in Table 4).
7. Conclusion
In this study, we have applied the geometrical theory of permissible domains walls (PDWs) to compile a comprehensive list of 84 PDWs connecting ferroelastic domains of monoclinic M_{C} symmetry. Our list not only includes analytical expressions for the of the PDWs but also matrices for transforming the corresponding pseudocubic basis vectors and formulas for calculating the reciprocalspace separation between corresponding Bragg peak pairs. These 84 PDWs encompass 45 different orientations and are organized into five distinct orientational families.
Our derivation of this extensive list is predicated on the assumption that the twostep transition from the cubic ( phase to the monoclinic (Pm/Pc) phase leads to the formation of 12 ferroelastic monoclinic domains. The first step of this transition, from the cubic to the tetragonal P4mm/P4bm phase, results in the creation of three ferroelastic domains. In the second step, from the tetragonal P4mm/P4bm to the monoclinic Pm/Pc phase, each of these three domains divides into a group of four monoclinic domains. We have identified six distinct types of domain pairs, referred to as `Tsiblingplanar', `Tsiblingcrossed', `Tplanar1', `Tplanar2', `Tsemiplanar' and `Tsemicrossed', each characterized by their own expression for the PDW orientation. As previously shown (Fousek & Janovec, 1969; Sapriel, 1975), we obtained that the of PDWs can either remain fixed (Wwalls) or depend on the values of the monoclinic lattice parameters (Swalls). Our investigation has revealed that the orientation of Swalls can be determined by three straightforward parameters, u, g, t with , {here } and .
The results of our work (both the present one and the preceding one) can be useful in several ways. Firstly, the availability of simple analytical expressions for domain wall orientation aids in describing the domain switching through domain wall rotation or domain wall motion. Such a process can be initiated by a change in the temperature or the application of an external electric field, for example. Secondly, the formulas for calculating the separation between Bragg peaks (as found in Tables 6–11) can facilitate the study of monoclinic domain patterns, using singlecrystal Xray diffraction. Lastly, we provide expressions that may prove valuable for precisely calculating the angles between the spontaneous polarization directions of various domains.
Supporting information
Supporting information. DOI: https://doi.org/10.1107/S2053273324002419/lu5034sup1.pdf
Funding information
The following funding is acknowledged: Israel Science Foundation (grant Nos. 1561/18, 1365/21 to Semën Gorfman; grant No. 1365/23); United States–Israel Binational Science Foundation (award No. 2018161).
References
Bin Anooz, S., Wang, Y., Petrik, P., de Oliveira Guimaraes, M., Schmidbauer, M. & Schwarzkopf, J. (2022). Appl. Phys. Lett. 120, 202901. Google Scholar
Biran, I. & Gorfman, S. (2024). Acta Cryst. A80, 112–128. CrossRef IUCr Journals Google Scholar
Choe, H., Bieker, J., Zhang, N., Glazer, A. M., Thomas, P. A. & Gorfman, S. (2018). IUCrJ, 5, 417–427. Web of Science CrossRef CAS PubMed IUCr Journals Google Scholar
Fousek, J. & Janovec, V. (1969). J. Appl. Phys. 40, 135–142. CrossRef CAS Web of Science Google Scholar
Fu, H. & Cohen, R. E. (2000). Nature, 403, 281–283. Web of Science CrossRef PubMed CAS Google Scholar
Gaal, P., Schmidt, D., Khosla, M., Richter, C., Boesecke, P., Novikov, D., Schmidbauer, M. & Schwarzkopf, J. (2023). Appl. Surf. Sci. 613, 155891. Web of Science CrossRef Google Scholar
Gorfman, S., Choe, H., Zhang, G., Zhang, N., Yokota, H., Glazer, A. M., Xie, Y., Dyadkin, V., Chernyshov, D. & Ye, Z.G. (2020). J. Appl. Cryst. 53, 1039–1050. Web of Science CrossRef CAS IUCr Journals Google Scholar
Gorfman, S., Keeble, D. S., Glazer, A. M., Long, X., Xie, Y., Ye, Z.G., Collins, S. & Thomas, P. A. (2011). Phys. Rev. B, 84, 020102. Web of Science CrossRef Google Scholar
Gorfman, S., Spirito, D., Cohen, N., Siffalovic, P., Nadazdy, P. & Li, Y. (2021). J. Appl. Cryst. 54, 914–923. Web of Science CrossRef CAS IUCr Journals Google Scholar
Gorfman, S., Spirito, D., Zhang, G., Detlefs, C. & Zhang, N. (2022). Acta Cryst. A78, 158–171. Web of Science CrossRef IUCr Journals Google Scholar
Gorfman, S. & Thomas, P. A. (2010). J. Appl. Cryst. 43, 1409–1414. Web of Science CrossRef CAS IUCr Journals Google Scholar
Gu, Y., Xue, F., Lei, S., Lummen, T. T. A., Wang, J., Gopalan, V. & Chen, L.Q. (2014). Phys. Rev. B, 90, 024104. CrossRef Google Scholar
Guo, Y., Luo, H., Ling, D., Xu, H., He, T. & Yin, Z. (2003). J. Phys. Condens. Matter, 15, L77–L82. CrossRef CAS Google Scholar
Jin, Y. M., Wang, Y. U., Khachaturyan, A. G., Li, J. F. & Viehland, D. (2003). J. Appl. Phys. 94, 3629–3640. Web of Science CrossRef CAS Google Scholar
Luo, J., Sun, W., Zhou, Z., Lee, H.Y., Wang, K., Zhu, F., Bai, Y., Wang, Z. J. & Li, J.F. (2017). Adv. Elect Mater. 3, 1700226. CrossRef Google Scholar
Noheda, B., Cox, D. E., Shirane, G., Gonzalo, J. A., Cross, L. E. & Park, S.E. (1999). Appl. Phys. Lett. 74, 2059–2061. Web of Science CrossRef CAS Google Scholar
Noheda, B., Cox, D. E., Shirane, G., Guo, R., Jones, B. & Cross, L. E. (2000). Phys. Rev. B, 63, 014103. Web of Science CrossRef Google Scholar
Oliveira Guimarães, M. de, Richter, C., Hanke, M., Bin Anooz, S., Wang, Y., Schwarzkopf, J. & Schmidbauer, M. (2022). J. Appl. Phys. 132, 154102. Google Scholar
Phelan, D., Rodriguez, E. E., Gao, J., Bing, Y., Ye, Z.G., Huang, Q., Wen, J., Xu, G., Stock, C., Matsuura, M. & Gehring, P. M. (2015). Phase Transit. 88, 283–305. CrossRef CAS Google Scholar
Sapriel, J. (1975). Phys. Rev. B, 12, 5128–5140. CrossRef CAS Web of Science Google Scholar
Vanderbilt, D. & Cohen, M. H. (2001). Phys. Rev. B, 63, 094108. Web of Science CrossRef Google Scholar
Vergentev, T., Bronwald, I., Chernyshov, D., Gorfman, S., Ryding, S. H. M., Thompson, P. & Cernik, R. J. (2016). J. Appl. Cryst. 49, 1501–1507. Web of Science CrossRef CAS IUCr Journals Google Scholar
Viehland, D. D. & Salje, E. K. H. (2014). Adv. Phys. 63, 267–326. Web of Science CrossRef CAS Google Scholar
Wang, R., Yang, B., Luo, Z., Sun, E., Sun, Y., Xu, H., Zhao, J., Zheng, L., Zhou, H., Gao, C. & Cao, W. (2016). Phys. Rev. B, 94, 054115. Google Scholar
Wang, Y., Bin Anooz, S., Niu, G., Schmidbauer, M., Wang, L., Ren, W. & Schwarzkopf, J. (2022). Phys. Rev. Mater. 6, 084413. CrossRef Google Scholar
Wang, Y. U. (2006). Phys. Rev. B, 74, 104109. CrossRef Google Scholar
Wang, Y. U. (2007). Phys. Rev. B, 76, 024108. Google Scholar
Zhang, N., Gorfman, S., Choe, H., Vergentev, T., Dyadkin, V., Yokota, H., Chernyshov, D., Wang, B., Glazer, A. M., Ren, W. & Ye, Z.G. (2018). J. Appl. Cryst. 51, 1396–1403. Web of Science CrossRef CAS IUCr Journals Google Scholar
Zhang, N., Yokota, H., Glazer, A. M. & Thomas, P. A. (2011). Acta Cryst. B67, 386–398. Web of Science CrossRef CAS IUCr Journals Google Scholar
This is an openaccess article distributed under the terms of the Creative Commons Attribution (CCBY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.