Permissible domain walls in monoclinic MAB ferroelectric phases

All the possibilities for permissible (mismatch-free) walls between monoclinic domains of pseudocubic ferroic perovskites are analyzed. Analytical expressions are derived for the orientation of such walls, the orientation relationship between the lattice vectors and the separation between Bragg peaks diffracted from matched domains.


Introduction
Monoclinic ferroelectric phases (MFEPs) have played an important role in understanding the structural mechanisms behind enhancement of properties in functional ferroelectric materials, particularly in mixed-ion perovskite oxides.The concept of 'monoclinic ferroelectrics' revolutionized the view on ferroelectricity by suggesting that spontaneous polarization can be rotated, rather than inverted or extended only (Davis et al., 2007;Damjanovic, 2010).Evidence of MFEPs was first reported by Noheda et al. (1999Noheda et al. ( , 2000)), Guo et al. (2000) and supported by the splitting of Bragg reflections in high-resolution X-ray diffraction patterns.MFEPs were later incorporated into the higher-order Devonshire theory (Vanderbilt & Cohen, 2001) and invoked to explain the enhancement of the giant piezoelectric effect in PbZr 1Àx Ti x O 3 at the so-called morphotropic phase boundary (MPB) (Fu & Cohen, 2000).The monoclinic space groups of ferroelectrics were used for many structural refinements based on X-ray and neutron scattering experiments (Gorfman & Thomas, 2010;Choe et al., 2018;Zhang et al., 2015;Zhang, Yokota et al., 2014;Aksel et al., 2011), and for the interpretation of the results of polarized light/birefringence experiments (Bokov et al., 2010;Gorfman et al., 2012).However, the true nature of the MFEPs is still debated: it is not clear if the MFEPs are truly long-range ordered or if the apparent long-range monoclinic order is 'mimicked' by the so-called adaptive state, consisting of assemblies of locally tetragonal or rhombohedral nanodomains (Jin et al., 2003;Viehland & Salje, 2014;Zhang, Xue et al., 2014).Regardless of the true character of MFEPs, the concept remains useful for the description of various phenomena in single-crystal ferroelectrics (Noheda et al., 2001;Choe et al., 2018;Gorfman et al., 2012), ferro/piezoceramics (Liu et al., 2017;Zhang, Yokota et al., 2014), epitaxial thin films (Wang et al., 2003;von Helden et al., 2018;Braun et al., 2018;Schmidbauer et al., 2017;de Oliveira Guimara ˜es et al., 2022) and shape memory alloys (Bhattacharya, 2003).
Besides the interesting intrinsic properties of MFEPs, rich microstructures of monoclinic domains (MDs) and domain walls (DWs) between them attract a great deal of interest (Nakajima et al., 2022;Mantri & Daniels, 2021).Any domain microstructures may underpin exotic physical properties such as giant electromechanical coupling (Hu et al., 2020), enhanced dielectric permittivity (Trolier-McKinstry et al., 2018), superelasticity (Viehland & Salje, 2014), the shape memory effect (Bhattacharya, 2003) and domain-wall superconductivity (Catalan et al., 2012).These microstructuredriven properties are particularly diverse when individual domains host several order parameters (e.g.electric, magnetic and elastic).Remarkably, such properties are often absent in a single domain.Their appearance and magnitude depend on the mobility of DWs.MFEPs should have rich and volatile domain microstructures.Therefore, the properties of DWs in MFEPs (such as crystallographic orientation and mobility) are relevant for the understanding of physical properties of materials.Although the algorithms for the prediction of DWs between domains of different symmetry are known (Fousek & Janovec, 1969;Sapriel, 1975;Authier, 2003), the underlying complexity of the subjects prevents any comprehensive understanding of domain microstructures of MFEPs.
The aim of this work is to describe the geometry of permissible DWs (PDWs) between domains of MFEPs.The term permissible [coined by Fousek & Janovec (1969), see also Sapriel (1975)] denotes a planar DW connecting two domains without any lattice mismatch.For example, tetragonal domains are permitted to connect along DWs of six different orientations (with the Miller indices belonging to the family {110}), rhombohedral domains are permitted to connect along DWs of 12 different orientations [with the Miller indices belonging to the families {110} and {100}, and exhibiting different physical properties (such as e.g.scattering of light) (Qiu et al., 2020)].
We demonstrate that, most generally, MDs are permitted to connect along 84 types of DW of 45 different orientations and five different orientational families.More specifically, we show that all the 84 DWs contain 48 prominent DWs (W-walls) which have fixed crystallographic orientation and 36 S-walls which change their orientation when free monoclinic lattice parameters change too.In addition, we present the analytical expressions for the matrices of transformation between the lattice basis vectors of matched domains and for the separation between Bragg peaks, diffracted from such domains.The presented equations create the direct path for the calculations of DW-related quantities, such as angles between polarization directions, the direction of DW motion under an electric field and so on.

Monoclinic ferroelectric phases: important definitions
This paper implements the list of notations and abbreviations introduced by Gorfman et al. (2022).Appendix A summarizes the most important ones.This section describes the definitions relevant for the description of the monoclinic phases of ferroelectric perovskites.
According to Fu & Cohen (2000), MFEPs can be of M A =M B or alternatively M C types.These types differ from one another by the set of independent pseudocubic lattice parameters and by the direction in which spontaneous polarization may develop.Note that while the spontaneous polarization vector, P (SP), is not mandatory in ferroelastic domains, it exists in practice for the case of ferroic perovskite oxides.Even if the magnitude of such polarization is zero, it is still useful to consider the potential SP direction(s) for domain referencing and numbering.3) and (4) correspond to the SPD along [111], [111], [111] and [111] directions, respectively.The SPDs within the 12 monoclinic domains are further explained in Table 1.
This paper focuses on the M A =M B case.The case of M C domains will be described in a follow-up paper.

The definition of M A /M B monoclinic domains
The crystallographic structures of the M A =M B phases of perovskite oxides belong to the space-group types Cm, Cc (Zhang, Yokota et al., 2014).These structures are obtained by the symmetry-lowering phase transitions from those described by the rhombohedral (R) space-group types R3m, R3c.The mirror (m)/glide (c) plane is parallel to two mutually perpendicular face-diagonals and the edge of the pseudocubic unit cell.These space-group types allow for rotation of the polar axis (e.g. the direction of the spontaneous polarization vector) within this mirror plane.Additionally, these space groups permit any distortion of the unit cell that maintains the mirror plane.Both the distortion of the pseudocubic unit cell (alongside the mirror plane) and the polar axis direction are shown in Fig. 1(a).
2.1.1.The numeration of the monoclinic domains and the potential spontaneous polarization.It is convenient to illustrate the monoclinic domains using stereographic projection and the corresponding potential spontaneous polarization direction (SPD).Since M A =M B domains arise from the transition from the rhombohedral R phase, we define the SPD by a small rotation angle from any of the four body-diagonal directions h111i towards any of the three adjacent unit-cell edges h001i.We mark the corresponding 12 monoclinic domains as M nm where the first index n lists the SPDs, R n in the 'parent' rhombohedral domain.In this case, The second index m ðm ¼ 1 . . .3Þ marks the pseudocubic axis T m so that T 1 ¼ AE½100, T 2 ¼ AE½010, T 3 ¼ AE½001 to which the polarization rotates.For example, the monoclinic domain M 13 has its SPD rotated from ½111 towards ½001, while M 21 has its SPD rotated from ½ " 1 111 towards ½ " 1 100.The SPDs in all 12 monoclinic domains are shown on the stereographic projections in Fig. 1  The definition of the 12 monoclinic (M A /M B -type) domain variants.
The first column contains the domain variant identifier [as also displayed in Fig. 1(b)].The second column contains the twinning matrix [the definition of this matrix is explained by Gorfman et al. (2022) but also presented in equation ( 66)].The third column contains the SPD for each domain, relative to the domain-related crystallographic coordinate system.The fourth column contains the pseudocubic lattice parameters expressed in terms of free parameters a, c, , .The notations !¼ À and !¼ À are used.The last column contains the reduced matrix ½G 0 mn ¼ ð½G mn =a 2 Þ À ½I.The calculations of the ½G 0 mn and corresponding lattice parameters are done using equation (67).

Twinning matrix [T]
[ In the following, we express the coordinates of the SPD relative to the axes of the Cartesian coordinate system, that are nearly parallel to the pseudocubic basis vectors.For the cases of domains M 13 we obtain Here, we introduced the notation with 0 ' 54:7 the angle between the body-diagonal and the edge of a cube, so that cos 0 ¼ 1= ffiffi ffi 3 p , sin 0 ¼ ffiffi ffi 2 p = ffiffi ffi 3 p .Assuming the SPD rotation angle is small and keeping the first term in the Taylor expansion with respect to , we can rewrite equation (2) as Note that the cases of > 0 and < 0 are referred to as M A and M B phases, correspondingly.

Pseudocubic lattice parameters of the monoclinic
M A /M B domains.Fig. 1(a) shows the M A /M B distortion of the pseudocubic unit cell.The corresponding pseudocubic lattice parameters a i ; i ði ¼ 1 . . .3Þ are described by four independent variables: a; c; ; (Gorfman & Thomas, 2010;Aksel et al., 2011;Choe et al., 2018): e.g. for the domain M 13 The corresponding matrix of the dot product is Here, [I] is the unitary matrix and Assuming that the monoclinic distortion is small and keeping the first power of ½ðc=aÞ À 1, ð=2Þ À , ð=2Þ À , we can write The resulting monoclinic crystal lattice is invariant with respect to N M ¼ 4 symmetry operations of the holohedry point group 2=m.The parent cubic crystal lattice is invariant with respect to N C ¼ 48 operations of the holohedry point group m3m.Because the monoclinic distortion may commence from any of these 48 equivalent variants, there are  The definitions of monoclinic M A =M B domain pair types.
The first two columns contain the domain pair name (full and short), the third column defines the pair, the fourth column lists the angle between the SPDs as a function of .This angle can be calculated by using   The same as Fig. 2 but for the case of the 'R-planar' type of DW pairs.cations, M nm , the ½G 0 metric tensors, the SPD and the lattice parameters a 1 , a 2 , a 3 , 1 , 2 , 3 .

Domain pairs
Twelve ferroelastic domains (Table 1) can form 66 domain pairs.Some of these pairs can be connected via PDWs and some of them cannot.Before analyzing PDWs between various pairs of monoclinic domains, we will introduce five different pair types.These types are referred to as 'R-sibling', 'R-planar', 'R-semi-planar', 'R-semi-crossed' and 'R-crossed'.Each type has its own angle between the SPDs and its own expressions for the indices of PDWs.Accordingly, we expect different properties from various domain pair types, with respect to e.g.DW motion under an external electric field.Table 2 presents the information about all five domain pairs, including pair name, abbreviation, formal definition, the angles between SPDs and the reference figure.
2.2.1.Domain pairs of the type 'R-sibling'.We will use the term 'R-sibling' for 12 pairs of monoclinic domains M nk M nl such that the members of each pair originate from the same parent/rhombohedral domain R n .Three R-sibling pairs can be formed for each R n : M n1 M n2 , M n2 M n3 and M n3 M n1 .All such pairs are illustrated on the stereographic projections (viewed along [001] and [110] directions) in Fig. 2.
2.2.2.Domain pairs of the type 'R-planar'.We will use the term 'R-planar' for six pairs of monoclinic domains M mk M nk , originating from different rhombohedral domains R m and R n (m 6 ¼ n) but such that k ¼ Lðm; nÞ, where Lðm; nÞ marks the pseudocubic axis that is parallel to the R m R n plane so that All the R-planar domain pairs are illustrated in Fig. 3 on the same type of stereographic projection as in Fig. 2. 2.2.3.Domain pairs of the type 'R-semi-planar'.We use the term 'R-semi-planar' for 12 pairs of monoclinic domains Each R m R n pair produces two monoclinic domain pairs of this type, e.g.M 12 M 22 and M 13 M 23 for the case of R 1 R 2 .All the R-semiplanar domain pairs are illustrated in Fig. 4.
2.2.4.Domain walls of the type 'R-semi-crossed'.We will use the term 'R-semi-crossed' for 12 pairs of monoclinic domains M mk and M nl , originating from different rhombohedral domains R m and R n (m 6 ¼ n) and such that both k 6 ¼ Lðm; nÞ and l 6 ¼ Lðm; nÞ.In addition, k 6 ¼ l because the cases of k ¼ l are already included in the 'R-semi-planar' type of domain pairs.Each R m R n pair produces two pairs of monoclinic domains of this type, e.g.M 12 M 23 and M 13 M 22 for the case of R 1 R 2 .All the R-semi-crossed pairs of domains are illustrated in Fig. 5.
2.2.5.Domain pairs of the type 'R-crossed'.We will finally use the term 'R-crossed' for 24 pairs of monoclinic domains M mk and M nl such that m 6 ¼ n; k 6 ¼ l, while either k ¼ Lðm; nÞ or l ¼ Lðm; nÞ.Each R m R n pair produces four pairs of monoclinic domains of this type, for example M 11 M 23 , M 11 M 22 , M 13 M 21 and M 12 M 21 for the case of R 1 R 2 .All the R-crossed domains are illustrated in Fig. 6.We will see later that these pair types may generally not be connected via PDWs.

The orientation of PDWs between different pairs of domains
According to Fousek & Janovec (1969) The same as Fig. 2 but for the case of the 'R-semi-planar' type of DW pairs.

Figure 5
The same as Fig. 2 but for the case of the 'R-semi-crossed' twin domain pairs.

Figure 6
The same as Fig. 2 but for the case of the 'R-crossed' type of domain pair.
dot products ½G n and ½G m , such a plane should satisfy the equations The key steps (see Gorfman et al., 2022) for finding the orientation of the PDWs between two arbitrary domains (Table 1) are: (i) Finding the eigenvalues ( 1 , 2 and 3 ) of ½ÁG or, equivalently, ½ÁG 0 ¼ ½G 0 n À ½G 0 m .
(ii) Checking if these domains have PDWs.This is the case if at least one eigenvalue is zero (e.g. 2 = 0).This condition is fulfilled if and only if jÁG 0 j ¼ 0.
(iii) Rearranging the eigenvalues so that 2 = 0, 3 > 0. Importantly, for all the cases considered in this paper (iv) Forming the orthogonal matrix [V] (½V À1 ¼ ½V T ) whose columns are the corresponding normalized eigenvectors of ½ÁG 0 .
(v) Finding the PDW indices (the coordinates of the PDW normal with respect to the reciprocal basis vector a Ã im ) according to i can be extended to the nearest allinteger values to get the Miller indices of the corresponding DW hkl.
Besides the ability to calculate the Miller indices of the PDW, this approach provides the basis for the calculation of the orientation relationship between the domain's basis vectors and separation of Bragg peaks, diffracted from a matched pair of domains.This possibility is the main advantage of this approach over those already existing (Fousek & Janovec, 1969).The relevant information for calculating these quantities is given further in Section 5.

PDWs connecting domain pairs of the type 'R-sibling'
We will demonstrate the derivation of the PDWs connecting the representative domain pair M 12 M 13 and obtain similar results for all the other pairs of this type analogously.Using the last column of Table 1 and equations ( 6) we obtain The following notation is introduced here: and The eigenvalues of the ½ÁG RSB can be found trivially as The corresponding eigenvalues of the matrix ½G 0 13 À ½G 0 12 are À 3 RSB ; 0; 3 RSB with The orthogonal matrix of eigenvectors of ½ÁG RSB (as well as ½G 0 13 À ½G 0 12 ) can be expressed as Accordingly, two PDWs normal to the vectors RSB ð1;2Þ The ½RSB ð1Þ normal has fixed coordinates that do not depend on the free lattice parameters.According to Fousek & Janovec (1969), such a wall can therefore be referred to as a W-wall.In contrast, the ½RSB ð2Þ depends on the monoclinic distortion parameter r and according to Fousek & Janovec (1969) it can be referred to as an S-wall ('strange' DW).Although the monoclinic distortion parameters C, A, B are small, the value of r (as a ratio of C and A À B) is not.This means that even a small change of monoclinic distortion may cause significant reorientation of the PDW.Table 3 highlights several favorable cases of the monoclinic distortion parameter r which sets the S-wall to have rational 'Miller' indices.For example, r = 2 [when 1 À ðc=aÞ ¼ À ] creates a PDW along the (111) plane.Approaching r ¼ 1 (e.g.= ) would mean the appearance of a PDW parallel to (011).

PDWs connecting domain pairs of the type 'R-planar'
We will demonstrate the derivation of the PDWs connecting the representative domain pair M 11 M 21 and obtain similar results for the other pairs of this type analogously.Using the last column of Table 1 and equations (6), Here, we introduce the following notation: It is straightforward to see that the eigenvalues of ½ÁG RP are À ffiffi ffi 2 p ; 0; ffiffi ffi 2 p .Similarly, the eigenvalues of the matrix ½G 0 21 À ½G 0 11 are À 3 RP ; 0; 3 RP with Table 3 The special cases of monoclinic distortion, leading to the appearance of Swalls with rational Miller indices.
The first column contains the relevant condition for the lattice parameters, the second column contains the corresponding value of r.The third column contains the eigenvalue 3 RSB of the matrix ½G 0 13 À ½G 0 12 .The condition of mismatch-free connection is only relevant for the case if 3 RSB 6 ¼ 0 (otherwise the domains may connect along any plane).The last column contains the Miller indices of the DW.

Lattice parameters
The orthogonal matrix of eigenvectors of both ½ÁG RP and ½G 0 21 À ½G 0 11 is Accordingly, two PDWs normal to the vectors RP ð1;2Þ i $ ðV RPi1 Ç V RPi3 ) exist: Both are W-walls, i.e. the crystallographic orientation of these walls does not depend on the values of the lattice parameters.

PDWs connecting domain pairs of the type 'R-semiplanar'
We will demonstrate the derivation of the PDWs connecting the representative domain pair M 12 M 22 and obtain similar results for the other pairs of this type analogously.According to the last column of Table 1 and equations ( 6), Here, we introduce the following notation: The eigenvalues and eigenvectors of ½ÁG RSP can be written as Accordingly, the corresponding eigenvalues of ½G 0 22 À ½G 0 12 are À 3 RSP ; 0; 3 RSP with It is straightforward to see that the orthogonal matrix of eigenvectors of ½ÁG 0 RSP (as well as ½G 0 22 À ½G 0 12 ) can be expressed as Accordingly, two PDWs normal to the vectors RSP ð1;2Þ i $ ðV RSPi1 Ç V RSPi3 ) exist: As in the case of PDWs connecting domain pairs of the type R-sibling, both W-and S-type DWs are present.Notably, it is shown in equation ( 22) that the orientation of the DW depends on the ratio of the angles Á and Á rather than the lengths of the pseudocubic cell edges.The corresponding Swall becomes parallel to the lattice plane with rational Miller indices for the special case such as Á ¼ Á, Á ¼ 0 or Á ¼ 0. Table 4 lists these favorable cases.

PDWs connecting domain pairs of the type 'R-semicrossed'
We will demonstrate the derivation of the PDWs connecting the representative domain pair M 12 M 23 and obtain similar results for all the other pairs of this type analogously.Using the last column of Table 1 and equations ( 6), Here we introduce the following notation: The eigenvectors and eigenvalues of the ½ÁG RSC can be found as À RSC , 0, RSC , Accordingly, the corresponding eigenvalues of ½G 0 23 À ½G 0 12 are À 3 RSC ; 0; 3 RSC with It is straightforward to see that the orthogonal matrix of eigenvectors of ½ÁG RSC (as well as ½G 0 23 À ½G 0 12 ) can be expressed as Accordingly, two PDWs normal to the vectors RSC ð1;2Þ i $ ðV RSCi1 Ç V RSCi3 ) exist: As The same as Table 3 just for the case of S-walls separating the domain pairs of the 'R-semi-planar' types.
Lattice parameters f 3 RSP S-wall orientation present here.In addition, some favorable cases (Table 5) of the lattice parameters turn the S-type of PDW into the PDW with rational Miller indices.

PDWs connecting domain pairs of the type 'R-crossed'
We will show that the corresponding domain pairs of this type do not generally have any PDWs.Indeed, we can attempt to find such for the case of the representative pair of domains M 32 M 23 .According to the last column of Table 1 and equations (6) we get The determinant of ½ÁG RC can be calculated as Accordingly, this pair of domains may connect along the PDW if one of the following conditions is fulfilled: or These conditions are generally not fulfilled and therefore we can consider domain pairs of the type 'crossed' not compatible.The special conditions under which domain pairs may connect could be the subject of future work.

The change of polarization direction across the domain walls
The SPD changes across any DW.This section demonstrates the calculation of the change of the SPD projection on the DW normal.Such a change is numerically equal to the surface density of electric charge at the wall (Jackson, 2007).We will consider that each ferroelastic domain mn may host spontaneous polarization þP mn or ÀP mn (the coordinates of the vectors P mn are defined in Table 1).Accordingly, the specific pair of domains m 1 n 1 and m 2 n 2 may meet along a DW that switches SPD according to the configuration P m 1 n 1 !P m 2 n 2 (+) or P m 1 n 1 !ÀP m 2 n 2 (À).We will see which of these configurations ensures zero (or minimal) charge at the corresponding DW using equations ( 14), ( 19), ( 26), (33) for the normal to the DW of each type.Table 6 summarizes the results.It shows that uncharged DWs occur in the following cases: (i) W-and S-type 'R-sibling' PDWs change SPD by nearly 180 or 0 , respectively.
These results have significant implications, particularly in the context of describing the DW motion under external electric fields and assessing the role of the specifically connected domain pair in the extrinsic contribution to the electromechanical coupling (Pramanick et al., 2011;Jones et al., 2006;Tutuncu et al., 2016;Gorfman et al., 2020).Indeed, this contribution hinges on the orientation of the SPD with respect to the electric field: domains with positive/negative projection of the SPD to the applied electric field would expand/contract, respectively.Consequently, comprehending the SPD's orientation and its change across the DW is pivotal.

General expressions
After calculating the indices of PDWs, connecting the specific pair of domains, it is also possible to calculate the orientation relationship between domains and separation of Bragg peaks diffracted from them.Full details of these calculations are presented by Gorfman et al. (2022) and briefly summarized here.The matrix of transformation ½ÁS between the basis vectors of the domains m and n is defined as ða 1n a 2n a 3n Þ ¼ ða 1m a 2m a 3m Þð½I þ ½ÁS).This matrix can be calculated according to research papers Acta Cryst.(2024).A80, 112-128 Biran and Gorfman Permissible domain walls in ferroelectric phases 119 Table 5 The same as Table 3 just for the case of S-walls separating the domain pairs of the 'R-semi-crossed' type.
Lattice parameters p 3 RSC S-wall orientation Table 6 The change of the SPD across each of the DWs described above.
The first column contains the type of domain pair.The second column contains the PDW Miller indices.The third column contains the domain numbers m 1 n 1 | m 2 n 2 meeting along the wall.The fourth column contains the sign involved in the connection: the sign + means e.g.P m 1 n 1 !P m 2 n 2 , the sign À stands for e.g.
The fifth column contains the projection of the P m 1 n 1 to the DW normal.The last column shows the angles between the corresponding SPDs as defined in Here ½Z ¼ The sign AE before Z 31 is used for the cases when the PDW normal is V i1 Ç V i3 , respectively.The coefficients y 1 and y 2 can be calculated according to Similarly, the matrix of transformation ½ÁS Ã between the reciprocal basis vectors of the domains m and n is defined as and can be calculated according to We can use (42) to calculate the separation between the Bragg peaks H, K, L so that

Simplifications
Equations ( 38) and ( 42) can be used to obtain the elements of ½ÁS and ½ÁS Ã numerically.However, we will show that reasonable approximation leads to more visually appealing analytical expressions.Let us notice that the right-hand side of (40) can be derived from the elements 13 and 23 of the matrix ½ÁG ðWÞ ¼ ½Z T ½V T ½ÁG 0 ½V½Z.Considering that the columns of the matrix [V] are the eigenvectors of ½ÁG 0 with the eigenvectors of À 3 ; 0 and 3 , respectively, we can write Here, the same sign as in equation ( 39 We will now consider the second term in the right side of equation ( 41) ½Z T ½V T ½G 0 m;n ½V½Z is proportional to the parameters of monoclinic distortion A, B, C [see equation ( 6)] and therefore it is much smaller than the first term ½Z T ½Z.Accordingly, we can rewrite (41) in the form Considering (46) we can rewrite (45) as Substituting ( 47) into ( 38) and ( 42) we get and Using the notations ½ÁS þ ; ½ÁS À for the case of signs + and À in front of 3 , respectively, we can see that ( 48), (49) lead to 5.3.The case of domain pairs of the type 'R-sibling' We will now apply ( 48) and ( 49) for PDWs connecting domain pairs of the type R-sibling.The corresponding transformation matrices ½ÁS þ ; ½ÁS À are marked explicitly as ½ÁS RSB ð0 " 1 11Þ and ½ÁS RSB ð2rrÞ .According to (12) we obtain 13) into (48) and using (50) we get Using ( 43) and ( 50) we can obtain the separation between the Bragg peaks diffracted from the corresponding pair of domains as and As mentioned by Gorfman et al. (2022), the three-dimensional separation between the Bragg peaks diffracted from a pair of connected domains is parallel to the DW normal.

The case of domain pairs of the type 'R-planar'
Similarly, for the case of domain pairs of the type 'Rplanar', the corresponding transformation matrices ½ÁS þ , ½ÁS À are marked explicitly as ½ÁS RP ð100Þ and ½ÁS RP ð011Þ .According to (17) 3 ¼ À2 ffiffi ffi 2 p Á. Substituting ( 18) into (48) and using ( 50) we obtain Equivalently we will obtain the following expression for the separation of Bragg peaks: and 5.5.The case of domain pairs of the type 'R-semi-planar' The corresponding transformation matrices ½ÁS þ ; ½ÁS À are then marked explicitly as ½ÁS RSP ð100Þ and ½ÁS RSP ð01f Þ .According to (24 . Substituting ( 25) into (48) and using ( 50): The separation of Bragg peaks diffracted from the correspondingly connected domain pairs is and 5.6.The case of domain pairs of the type 'R-semi-crossed' The corresponding transformation matrices ½ÁS þ ; ½ÁS À are marked explicitly as ½ÁS RSC ð2p " p pÞ and ½ÁS RSC ð011Þ .According to (31 . Substituting ( 32) into (48) and using (50): The separation of the Bragg peaks diffracted from the domains, meeting along the DW normal to ½RSC ð1Þ is research papers Table 7 Summary of 24 PDWs connecting domain pairs of the R-sibling type.
The first column contains the PDW number, while the second and third columns contain the domain identifiers based on Fig. 1 and Table 1.The fourth column displays the Miller indices of the PDW.The fifth column contains the transformation matrix between the basis vectors of the domain m 1 n 1 and the basis vectors of the domain m 2 n 2 .The last column contains the separation between the Bragg peaks with the indices H, K, L diffracted from these domains.

Summarizing tables
The previous paragraphs demonstrated how to derive key quantities such as Miller indices, the orientation relationship between the lattice basis vectors, and the separation of Bragg peaks for representative domain pairs only.Similar equations can be derived for all the other domain pairs.The tables and figures below list the corresponding quantities for all 84 existing PDWs.The full list includes: (i) 24 PDWs connecting domain pairs of the type 'R-sibling', including 12 W-and 12 S-walls.
(ii) 12 PDWs connecting domain pairs of the type 'Rplanar'.All of them are W-walls.
The list of 84 PDWs contains 36 S-walls and 48 W-walls as listed in Tables 7, 8, 9, 10.Each row of these tables contains domain pair number, the Miller indices of the PDW, the matrix of transformation ½ÁS between the corresponding basis vectors and the separation of Bragg peaks H, K, L diffracted from this pair of domains.
Tables 7, 8, 9, 10 reveal that certain W-walls have the same orientations.For instance, all domain pairs of the type 'Rplanar' M 11 M 21 ; M 31 M 41 and all domain pairs of the type 'Rsemi-planar', M 12 ; M 22 , M 13 ; M 23 have (100)-oriented PDWs.Table 11 presents all the distinct PDW orientations and their relevant details.It reveals that all the PDWs belong to five orientation families {100}, {110}, {2rr}, {10f}, {2pp}, 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.It indicates that 84 PDWs are classified into 12 DWs, 30 DWs, 30 DWs, 12 DWs with the angles between SPDs close to 0, 71, 109 and 180 , respectively.Fig. 7 displays the orientation of all the PDWs for different choices of lattice parameters.The normal vectors to these walls are shown using the poles on the stereographic projection.The W-walls are marked by the poles with a solid line edge and the color of the pole reflects the angle between the SPDs being close to 0, 71, 109 and 180 (as specified in the last column of Table 5).Each stereographic projection, from left to right, shows DWs between the domain pairs of the types 'Rsibling', 'R-planar', 'R-semi-planar' and 'R-semi-crossed'.The supporting information includes the animated version of this figure showing how the orientation of these DWs changes with the lattice parameters.

Conclusion
We have applied the theory of PDWs to create a list of 84 PDWs connecting ferroelastic domains of monoclinic (Cm/Cc) The same as Table 7 but for the case of PDWs connecting domain pairs of the type 'R-planar'.
The same as Table 7 but for the case of PDWs connecting domain pairs of the type 'R-semi-planar'.
symmetry.Our list includes analytical expressions for the Miller indices of the PDWs, matrices of transformation between the corresponding pseudocubic basis vectors and expressions for the reciprocal-space separation between the corresponding Bragg peak pairs.The 84 PDWs can have 45 different orientations and are grouped into five orientational families.
Our derivation of this list assumed that the two-step transition from the cubic (Pm " 3 3mÞ phase to the monoclinic (Cm/ Cc) phase results in the formation of 12 ferroelastic monoclinic domains.The first step of this transition (from the cubic to the rhombohedral R3m=R3c phase) results in the formation of four ferroelastic domains, while the second step (from the rhombohedral to the monoclinic phase) splits each of them into groups of three monoclinic domains.We identified five different types of domain pairs (referred to here as 'R-sibling', 'R-planar', 'R-semi-planar', 'R-semi-crossed' and 'Rcrossed'), each with its own expression for the PDW orientation.As shown in previous works (Fousek & Janovec, 1969;Sapriel, 1975), we found that the crystallographic orientation/ Miller indices of PDWs can be fixed (for the so-called W-walls) or depend on the values of the monoclinic lattice parameters (for the so-called S-walls).We found that the orientation of such walls can be controlled by the three simple parameters r ¼ ½2=ðÁ À ÁÞ½ðc=aÞ À 1, f ¼ ðÁ=ÁÞ and p ¼ ½2=ðÁ þ ÁÞ½ðc=aÞ À 1.
We have demonstrated that the rotatable domain walls can be described by the Miller indices {2rr}, {10f}, {2pp}.Even a small change in the monoclinic distortion (such as c=a; Á; Á) can cause a significant rotation of the PDW.This process is often referred to as 'thermal switching'.Furthermore, we have predicted the angles between polarization directions for the cases when DWs are not charged.
The results of this work can be useful in several different ways.First, the availability of simple analytical expressions (Tables 7-10) for the orientation of DWs can help in describing the domain switching through DW rotation or DW motion.Such a process can be induced by the change of the temperature or external electric field, for example.Second, the expressions for the separation between Bragg peaks (Tables 7-10) can help investigate monoclinic domain patterns, using 'single-crystal' X-ray diffraction.Third, the expressions may be useful for the precise calculation of the angles between SPDs of various domains.Such angles can be easily evaluated using the corresponding matrices of transformation between the domain basis vectors in Tables 7-10.
The results have significant importance in the analysis of domains within crystals and epitaxial thin films.Indeed, the observation of monoclinic domains in epitaxial thin films is common (see e.g.Schmidbauer et al., 2017;Gaal et al., 2023) where one or another type of monoclinic distortion is stabilized by the substrate-film lattice mismatch.Modulating this mismatch can influence the monoclinic lattice parameters and, consequently, the orientation of PDWs between them.It is worth highlighting that certain distinctions may arise due to variations in the number of monoclinic domains present.In the case of 'free-standing' single crystals, the phase transition sequence from cubic to rhombohedral to monoclinic ideally results in the presence of 12 equivalent domains.However, introducing bias at any of these transitional stages can alter this configuration.For instance, the application of an electric field along the pseudocubic [111] direction during the cubicto-rhombohedral phase transition may lead to the formation of just one rhombohedral domain instead of the expected four.Subsequently, the rhombohedral-to-monoclinic transition further divides this domain into three monoclinic domains.Consequently, in such scenarios, only 'R-sibling' domain pairs, connected by six PDWs, must be considered.The presence of the substrate can bias or suppress the formation of specific domains, such as favoring the presence of domain pairs of the R-sibling type exclusively, and this, in turn, can impact the number of PDWs.A detailed characterization of PDWs in relation to the origin of these domains can prove useful for cataloging the potential PDWs existing between thin film domains or in other cases when formation of domains is biased or engineered.
Finally, this article describes the PDWs between monoclinic domains of M A /M B type.A similar formalism for the monoclinic M C symmetry case will be presented in a follow-up publication.
Basis vectors: a im ði ¼ 1 . . .3Þ are the basis vectors of a crystal lattice.The second index refers to the ferroelastic domain variant m. m ¼ 0 corresponds to the crystal lattice of the higher-symmetry (e.g.cubic) 'parent' phase (Fig. 1).The parallelepiped based on the vectors a im forms a unit cell.
Unit-cell settings: many unit-cell settings exist for the same lattice (Gorfman, 2020).Here, we prefer the cell settings a im (m > 0Þ obtained by the smallest possible distortion/rotation of the parent-phase basis vectors a i0 . Metric tensor/matrix of dot products: G ij ¼ a i a j is the metric tensor (Giacovazzo, 1992;Hahn, 2005).The corresponding 3 Â 3 matrix ½G m is the matrix of dot products for the domain variant m.Their determinants are jGj ¼ V 2 A (V A is the unitcell volume).
The transformation matrix: the transformation e.g. from the basis vectors a im to the basis vectors a in is defined by the 3 Â 3 transformation matrix [S] Transformation of the metric tensor: the transformation of the basis vectors (63) leads to the following transformation of the corresponding metric tensors: This relationship can be extended to any cases of transformation between coordinate systems.The difference transformation matrix is defined as the difference between [S] and the unitary matrix [I]:

½ÁS ¼ ½S À ½I: ð65Þ
Twinning matrix: [T] represents a symmetry operation of the parent-phase lattice (i.e. the one built using the basis vectors a i0 ) that is no longer the symmetry operation of a ferroelastic phase lattice.We define [T] as a 3 Â 3 matrix, which describes the transformation to the coordinate system a i0 from its symmetry equivalent a 0 i0 using the following formal matrix equation: The number of symmetry-equivalent coordinate systems is equal to the order of the holohedry point symmetry group (e.g.48 for a cubic lattice).The transition from a paraelastic to a ferroelastic phase is associated with the distortion of the basis vectors a i0 ! a im .Such a distortion, however, can commence from any of the symmetry-equivalent a 0 i0 .Let us assume that a i0 and a 0 i0 serve as the starting points for domain variants m and n, correspondingly.The following relationship between ½G n and ½G m exists: Reciprocal basis vectors: the superscript * refers to the reciprocal bases, e.g. a Ã i are such that a i a Ã j ¼ ij .The reciprocal metric tensor is G Ã ij ¼ a Ã i a Ã j .The relationship ½G Ã ¼ ½G À1 holds.
Transformation between the reciprocal basis vectors: if the direct basis vectors (e.g. a im and a in ) are related by the matrix [S] [according to equation ( 63)], then the corresponding reciprocal-lattice vectors (a Ã im and a Ã in ) are related by the matrix ½S Ã .The following relationship between [S] and ½S Ã holds: The difference transformation matrix between the reciprocal basis vectors is defined according to the equation ½ÁS Ã ¼ ½S Ã À ½I: ð69Þ

Figure 2
Figure 2 Schematic illustration of the 'R-sibling' type of monoclinic domain pairs.The term 'R-sibling' refers to the case when both pair members originate from the same R domain.The figure includes: (a) stereographic projection viewed along the [001] direction, showing the SPDs in the 12 monoclinic domains.(b) Stereographic direction viewed along the direction [110], highlighting the sibling pair types, originating from R 1 and R 2 .

Figure 3
Figure3 Figure 4 for the cases of DWs connecting domain pairs of the type 'R-sibling' and 'R-semi-planar', W-type and S-type DWs are 118 Biran and Gorfman Permissible domain walls in ferroelectric phases Acta Cryst.(2024).A80, 112-128 research papers Table 4 ) is implemented instead of AE.Using (44) we can rewrite (40

Figure 7
Figure 7The orientation of the DW is normal for all the PDWs which connect domain pairs of the R-sibling, R-planar, R-semi-planar and R-semi-crossed types, with 45 different orientations in total.These orientations are distributed among five different orientation families.The normals are shown using the poles on the stereographic projection viewed along the direction [001] with the poles corresponding to the W-walls framed by a solid line.The lattice parameters are chosen arbitrarily.The supporting information includes animated versions of the same figure for different values of the monoclinic lattice parameters.
Permissible domain walls in ferroelectric phases 123Table 8

Table 2
Table 1, equation (2) and keeping the first power of in the Taylor series expansion.The fifth column contains the number of the corresponding domain pairs, the last column refers to the corresponding figure.
. The columns of the matrix [S] are the coordinates of a in with respect to a im : ð a 1n a 2n a 3n Þ ¼ ð a 1m a 2m a 3m Þ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33