Ion Doping Effects on the Lattice Distortion and Interlayer Mismatch of Aurivillius-Type Bismuth Titanate Compounds

Taking Bismuth Titanate (Bi4Ti3O12) as a Aurivillius-type compound with m = 3 for example, the ion (W6+/Cr3+) doping effect on the lattice distortion and interlayer mismatch of Bi4Ti3O12 structure were investigated by stress analysis, based on an elastic model. Since oxygen-octahedron rotates in the ab-plane, and inclines away from the c-axis, a lattice model for describing the status change of oxygen-octahedron was built according to the substituting mechanism of W6+/Cr3+ for Ti4+, which was used to investigate the variation of orthorhombic distortion degree (a/b) of Bi4Ti3O12 with the doping content. The analysis shows that the incorporation of W6+/Cr3+ into Bi4Ti3O12 tends to relieve the distortion of pseudo-perovskite layer, which also helps it to become more stiff. Since the bismuth-oxide layer expands while the pseudo-perovskite layer tightens, an analytic model for the plane stress distribution in the crystal lattice of Bi4Ti3O12 was developed from the constitutive relationship of alternating layer structure. The calculations reveal that the structural mismatch of Bi4Ti3O12 is constrained in the ab-plane of a unit cell, since both the interlayer mismatch degree and the total strain energy vary with the doping content in a similar trend to the lattice parameters of ab-plane.


Introduction
Fifty years ago, Aurivillius [1][2][3] discovered a family of bismuth compounds with a layer structure, and to date, more than 50 compounds are known to belong to this group. As most of them were found to be ferroelectric, this group was therefore called Bismuth Layer-Structured Ferroelectrics (BLSF), or Aurivillius compounds. In recent years, BLSFs as a lead-free piezoelectric systems have attracted much attention because of their high Curie temperatures (T c ), large spontaneous polarizations (P s ), and good antifatigue properties [4][5][6][7][8]. The crystal structure of BLSFs consist of [Bi 2 O 2 ] 2+ layers interleaved with pseudo-perovskite [A m−1 B m O 3m+1 ] 2− units stacked along the c-axis, which can be described by a general chemical formula of (Bi 2 O 2 ) 2+ (A m−1 B m O 3m+1 ) 2− , where A is mono-, di-, trivalent ions, or a mixture of them and B is tetra-, penta-, hexavalent ions, or a mixture of both, and the integer m is the number of BO 6 octahedral layer, taking any of the values from 1 to 5. The stability of

Phase Structure Analysis
The sintered BTWC ceramics were crushed into powder in an agate mortar and suspended in ethanol. The phase structures of powders were checked using an X-ray powder diffractometer (XRPD, Empyrean, PANalytical Co. Ltd., Almelo, The Netherlands) using CuKα radiation, in the 2θ range 10°-60°, with a step size of 0.02° (counting time was 15 s per step). The XRPD experimental diffraction patterns were analyzed via the Rietveld profile method using the MAUD program (Version: 2.7, written in Java by Luca Lutterotti, freely available his website).

Phase Structure of BTWC Ceramics
For a more quantitative assessment of the synthesis process on the phase structure of materials, XRPD pattern of some BTWC ceramics were registered at 295 K, as shown in Figure 3. In Figure 3a, all sets of reflections except for those specific ones noted by an asterisk could be indexed in the orthorhombic structure of the parent compound Bi4Ti3O12 (JCPDS card No. 72-1019), with a space group of B2cb(41). However, such impurity phases are most probably associated to the Bi2O3, TiO2 rutile and non-stoichiometric Ti2O3, which would be destroyed during longer milling time, without notable influence on the increased amount of the crystalline phase. As can be observed from Figure  3b-d, there seems to be very little difference in the patterns of BIT and W and/or Cr doped BIT. However, there is a slight shift of the reflections to lower 2θ values after the incorporation of W/Cr. According to the XRPD data, the Rietveld Refinement technique was used to calculate the parameters of the the pure Bi4Ti3O12 structure (Figure 3a) as follows, a = 5.4457 Å ; b = 5.4087 Å ; c = 32.8378 Å ; V = 967 Å 3 and a/b = 1.0068 (Refinement index: Rp = 5.67; Rwp = 7.42; RB = 4.74; and λ 2 = 1.37), which are basically in agreement with the values determined by Ivanov et al. [22]. Here, the factor with the form of a/b was used to indicate the degree of the lattice distortion of orthorhombic structure (ab.

Phase Structure Analysis
The sintered BTWC ceramics were crushed into powder in an agate mortar and suspended in ethanol. The phase structures of powders were checked using an X-ray powder diffractometer (XRPD, Empyrean, PANalytical Co. Ltd., Almelo, The Netherlands) using CuKα radiation, in the 2θ range 10 • -60 • , with a step size of 0.02 • (counting time was 15 s per step). The XRPD experimental diffraction patterns were analyzed via the Rietveld profile method using the MAUD program (Version: 2.7, written in Java by Luca Lutterotti, freely available his website).

Phase Structure of BTWC Ceramics
For a more quantitative assessment of the synthesis process on the phase structure of materials, XRPD pattern of some BTWC ceramics were registered at 295 K, as shown in Figure 3. In Figure 3a, all sets of reflections except for those specific ones noted by an asterisk could be indexed in the orthorhombic structure of the parent compound Bi 4 Ti 3 O 12 (JCPDS card No. 72-1019), with a space group of B2cb(41). However, such impurity phases are most probably associated to the Bi 2 O 3 , TiO 2 rutile and non-stoichiometric Ti 2 O 3 , which would be destroyed during longer milling time, without notable influence on the increased amount of the crystalline phase. As can be observed from Figure 3b-d, there seems to be very little difference in the patterns of BIT and W and/or Cr doped BIT. However, there is a slight shift of the reflections to lower 2θ values after the incorporation of W/Cr. According to the XRPD data, the Rietveld Refinement technique was used to calculate the parameters of the the pure Bi 4 Ti 3 O 12 structure ( Figure 3a) as follows, a = 5.4457 Å; b = 5.4087 Å; c = 32.8378 Å; V = 967 Å 3 and a/b = 1.0068 (Refinement index: R p = 5.67; R wp = 7.42; R B = 4.74; and λ 2 = 1.37), which are basically in agreement with the values determined by Ivanov et al. [22]. Here, the factor with the form of a/b was used to indicate the degree of the lattice distortion of orthorhombic structure (ab. orthorhombic distortion), where a slight difference between the lattice constant of a and b was included.

Cell Parameters of BTWC Ceramics
The results from Rietveld Refinement for the crystal structures of BTWC ceramics are listed in Tables 2 and 3, respectively. For BTWx-0.2Cr, the clear decrease of the orthorhombic distortion with increase in W content (x ≥ 0.025) correlates with the cationic polar displacements from ideal positions and the tilting adjustment of oxygen-octahedron. Thus BTW0.05-0Cr (only doped with W) shows a remarkable decrease of the orthorhombic distortion, as compared to that of BIT. The lattice model of octahedral distortion and the corresponding regulatory mechanism of doping ions will be discussed for BIT in Section 4.1.3.

Cell Parameters of BTWC Ceramics
The results from Rietveld Refinement for the crystal structures of BTWC ceramics are listed in Tables 2 and 3, respectively. For BTW x -0.2Cr, the clear decrease of the orthorhombic distortion with increase in W content (x ≥ 0.025) correlates with the cationic polar displacements from ideal positions and the tilting adjustment of oxygen-octahedron. Thus BTW 0.05 -0Cr (only doped with W) shows a remarkable decrease of the orthorhombic distortion, as compared to that of BIT. The lattice model of octahedral distortion and the corresponding regulatory mechanism of doping ions will be discussed for BIT in Section 4.1.3. Table 2. Rietveld refinements for the crystal structure of BTW x -0.2Cr at 295 K.

Parameters
Tungsten Content in BTW x -0.2Cr (Space Group of B2cb)   . In this pattern, a lattice distortion is observed for the pseudo-perovskite layer of BIT which is constructed by the cluster of oxygen-octahedron. Here, a shear force induced by the shorter and stronger Bi-O bond between bismuth atoms at A-site (Bi1) and oxygen atoms in octahedron (O 3 ), drives the central octahedron to rotate along the polarization axis (i.e., a-axis), thus stabilizing the pseudo-perovskite layer. Moreover, those three horizontal octahedra tend to rotate with the same angle (about 4 • ), leading to the same displacement between octahedra and bismuth atoms [23]. Even this octahedral distortion contributes to the principal component of the spontaneous polarization of BIT (i.e., P s,a ).
Further, Martin Kunz [24] suggested a length matching relationship (a A /a B ) between A-O bond and B-O bond as the criterion for the rotation of octahedron in the perovskite structure, as follows, a lattice stress will be induced due to the mismatch of bond length, causing the spontaneous distortion of octahedra to relieve such a mismatch.
Based on this theory, if a A /a B < 1, the A-O bond will be stretched, while the B-O bond will be compressed, because A-site atoms look smaller. Conversely, if a A /a B > 1, the A-O bond is compressed while the B-O bond will be stretched, because A-site atoms look bigger. Here, the relationship of a A /a B < 1 for BIT according to the length of chemical bonds [25] can be obtained. As a result, the octahedron will rotate around the bismuth atom at the A-site along the a-axis, for the sake of relieving the mismatch of bond length by stretching Bi-O bond. However, it is worty noting that the rotation angle must meet the principle for the valence bond sum of cations in the structure unit.
cluster of oxygen-octahedron. Here, a shear force induced by the shorter and stronger Bi-O bond between bismuth atoms at A-site (Bi1) and oxygen atoms in octahedron (O3), drives the central octahedron to rotate along the polarization axis (i.e., a-axis), thus stabilizing the pseudo-perovskite layer. Moreover, those three horizontal octahedra tend to rotate with the same angle (about 4°), leading to the same displacement between octahedra and bismuth atoms [23]. Even this octahedral distortion contributes to the principal component of the spontaneous polarization of BIT (i.e., Ps,a).

Tilting of Oxygen-Octahedron
When the structure of BIT undergoes the phase transition from the paraelectric phase to the ferroelectric phase, a stronger Bi-O bond between the bismuth atom in the bottom of the bismuth-oxide layer and the oxygen atom in the top of pseudo-perovskite layer will be produced to ensure the stability of the structure. Such a strong chemical bond tends to pull the two apical oxygen-octahedra, leading them to tilt along the c-axis. According to the description in [26], if choosing to define the octahedral tilts explicitly with respect to the unit cell edges, octahedral tilts can be calculated from the locations of the apical oxygen ions in the direction of interest.
However, the tilt angles are absolute tilts calculated from the apical oxygen ion positions along the c-axis with respect to the ab-plane. The tilt angles are within~1 • for the central and outer BO 6 oxygen-octahedra across the series, but the octahedra are also strongly distorted.
In addition, according to the global parameterization method (GPM) proposed by Thomas [27], for the perovskite structure with any symmetry system, the lattice strain with the symbol of 90 • -α pc (α pc : pseudo-cubic angle) will cause the tilting of octahedron, and the corresponding tilting angle can be evaluated by the following formula, where, θ m and θ z are the tilting angle related to the pseudo-cubic axes. V A is the volume of AO 12 polyhedron and V B is the volume of BO 6  , respectively. It is believed that the tilting of octahedron stems forms the finite rotation of octahedron. A-site ions still need to deviate from the centre to form a displacement to coordinate the rotation, which can ensure that B-site ions are located in the centre of octahedron. If V A /V B < 5, the octahedron will tilt. It was reported that V Ca /V Ti = 4.61 in Ref. [28]. For the BIT with the orthorhombic structure, the value of V Bi /V Ti may be smaller than 4.61, due to the stronger combining ability between Bi 3+ and O 2− leading to the shorter Bi-O bond. This is because they have the similar ionic radius, but the electronegativity of Bi (2.02) is twice that of Ca (1.00). Therefore, the oxygen-octahedron of BIT is about to tilt, except for rotation. In addition, the first-principle calculations have shown a large cooperative coupling of Jahn-Teller (JT) distortion to oxygen-octahedron rotations in perovskite LaMnO 3 . The combination of the two distortions is responsible for stabilizing the strongly orthorhombic A-AFM insulating (I) Pbnm ground state, relative to a metallic ferromagnetic (FM-M) phase.

Regulation of Doping-Ions for the Lattice Distortion
As analyzed above, the rotation and tilting of oxygen-octahedron within the pseudo-perovskite layer cause a slight difference in the length between a-axis and b-axis for Bi 4 Ti 3 O 12. Based on the deformation mechanism of oxygen-octahedron, the status change of oxygen-octahedron in Bi 4 Ti 3 O 12 structure after W/Cr co-doping can be described by the lattice model built in Figure 5. Here, the introduction of W 6+ was performed in the calcining process of starting materials including WO 3 , while the incorporation of Cr 3+ was addressed during sintering of the green bodies with the addition of Cr 2 O 3 . In view of their similar ionic radius (W 6+ : 0.600 Å, Cr 3+ : 0.615 Å and Ti 4+ : 0.605 Å) and same coordination number (6), both W 6+ and Cr 3+ are considered to substitute for Ti 4+ in the core of oxygen-octahedron (i.e., B-site of perovskite structure). The difference in their bonding properties involved with oxygen atoms is likely responsible for the status change of oxygen-octahedron, which will be analyzed as follows.
Materials 2018, 11, x 9 of 15 As analyzed above, the rotation and tilting of oxygen-octahedron within the pseudo-perovskite layer cause a slight difference in the length between a-axis and b-axis for Bi4Ti3O12. Based on the deformation mechanism of oxygen-octahedron, the status change of oxygen-octahedron in Bi4Ti3O12 structure after W/Cr co-doping can be described by the lattice model built in Figure 5. Here, the introduction of W 6+ was performed in the calcining process of starting materials including WO3, while the incorporation of Cr 3+ was addressed during sintering of the green bodies with the addition of Cr2O3. In view of their similar ionic radius (W 6+ : 0.600 Å , Cr 3+ : 0.615 Å and Ti 4+ : 0.605 Å ) and same coordination number (6), both W 6+ and Cr 3+ are considered to substitute for Ti 4+ in the core of oxygenoctahedron (i.e., B-site of perovskite structure). The difference in their bonding properties involved with oxygen atoms is likely responsible for the status change of oxygen-octahedron, which will be analyzed as follows. Rietveld refinements of XRD have been used to calculate the value of orthorhombic distortion (a/b) for these BIT-based compounds. For BTWx-0.2Cr (Table 2), the orthorhombic distortion seems to linearly decrease with an increasing W content (x) from 0.025. Since the electronegativity of W (2.36) is larger than that of Ti (1.54), the length of W-O bond may be shorter than that of Ti-O bond within the octahedron. When W 6+ occupies the B-site instead of partial Ti 4+ , the average value of aA/aB will have an increase getting closer to 1, which tends to relieve the rotation of oxygen-octahedron according to Equation (2). Further evidence for this mechanism is derived from the composition of BTW0.05-0Cr, which was only doped by W 6+ , with its a/b value (1.0023) being much lower than that of BIT (1.0068). On the other hand, for BTW0.05-yCr (Table 3), Cr as a substitute has an approximatively equal electronegativity (1.66) as compared with Ti (1.54), which will not change the length of B-O bond significantly. Thus, the rotation status of oxygen-octahedron could remain constant, according to Equation 2. However, the substitution of Cr 3+ for Ti 4+ tends to create oxygen vacancy (VO •• ) in the oxygen-octahedron (as represented by the white circle in Figure 5), based on the principle of charge compensation [29], which has a significant influence on the volume of BO6 octahedron. As the result, this effect will change the tilting status of oxygen-octahedron, according to Equation 3. Finally, the value of a/b first increases, and then decreases with an increase of the amount of Cr2O3 (y), which can be considered as the synergistic reaction of these two regulatory mechanisms. It has been reported Rietveld refinements of XRD have been used to calculate the value of orthorhombic distortion (a/b) for these BIT-based compounds. For BTW x -0.2Cr (Table 2), the orthorhombic distortion seems to linearly decrease with an increasing W content (x) from 0.025. Since the electronegativity of W (2.36) is larger than that of Ti (1.54), the length of W-O bond may be shorter than that of Ti-O bond within the octahedron. When W 6+ occupies the B-site instead of partial Ti 4+ , the average value of a A /a B will have an increase getting closer to 1, which tends to relieve the rotation of oxygen-octahedron according to Equation (2). Further evidence for this mechanism is derived from the composition of BTW 0.05 -0Cr, which was only doped by W 6+ , with its a/b value (1.0023) being much lower than that of BIT (1.0068). On the other hand, for BTW 0.05 -yCr (Table 3), Cr as a substitute has an approximatively equal electronegativity (1.66) as compared with Ti (1.54), which will not change the length of B-O bond significantly. Thus, the rotation status of oxygen-octahedron could remain constant, according to Equation 2. However, the substitution of Cr 3+ for Ti 4+ tends to create oxygen vacancy (V O •• ) in the oxygen-octahedron (as represented by the white circle in Figure 5), based on the principle of charge compensation [29], which has a significant influence on the volume of BO 6 octahedron. As the result, this effect will change the tilting status of oxygen-octahedron, according to Equation 3. Finally, the value of a/b first increases, and then decreases with an increase of the amount of Cr 2 O 3 (y), which can be considered as the synergistic reaction of these two regulatory mechanisms. It has been reported that the ferroelectric pseudo-perovskite of BLSF, with a higher degree of lattice distortion, usually possesses a larger spontaneous polarization in single ferroelectric domain [30]. As for W/Cr co-doped Bi 4 Ti 3 O 12 , the compositions with a higher orthorhombic distortion have also been shown to possess a larger spspontaneous polarization in polycrystalline ceramics, which was reported in one of our previous works [14,15].

Elastic Model Considering Short Range Forces
Due to the rotation and tilting of oxygen octahedron, the pseudo-perovskite layer is distorted, which interferes the configuration of the bismuth-oxide layer. The structural mismatch between the pseudo-perovskite layer and the bismuth-oxide layer can be explained by the elastic strain energy [19]. Elastic model is a general method to deal with the lattice mismatch in epitaxial growth. Based on this model, the interlayer mismatch problem in the structure of Bi 4 Ti 3 O 12. can be addressed.
It is supposed that the tetragonal structure of BIT has a lattice parameter of a T . Thus the relationship of lattice parameter between orthorhombic structure and tetragonal structure can be expressed as follows: If half a unit cell is taken into consideration, the cell volume, V, can be expressed as follows: where V B is the volume of bismuth-oxide layer unit and V P is that of pseudo-perovskite layer unit. Assuming that both the two units are elastomer, and then the internal stresses induced in the structure are expressed as: where K B and K p are the bulk modulus, V B and V P are the volume of the constrained bismuth-oxide layer unit and pseudo-perovskite layer unit, respectively, V B ' and V p ' are the volume of the unconstrained units. (V' − V)/V is defined as the volume strain (ε). When the two units are settled in the space lattice, F B and F P can be affected by not only the short range forces within the units but also the long range forces outside of the units, i.e., Madelung forces. In order to simplify the problem, only the short range forces are considered in this model. Here, the unit volume of the tetragonal structure can be expressed as follows: For the tetragonal structure, Equations (6a) and (6b) can be rewritten as follows: Here, both (a B ', c B ') and (a P ', c P ') are the lattice parameters of the unconstrained bismuth oxide layer and pseudo-perovskite layer, respectively. For half an unit cell of BIT, a relationship between lattice parameters is expressed as: The mismatch theory proposed by Armstrong and Newnham [13] pointed out the relationship that for a B ' < a < a P ' there is a lattice strain existing in the ab-plane, which is normal to the c-axis. Especially, under the biaxial compression or tensile load, the elastic responses of the bismuth oxide layer and pseudo-perovskite layer are similar in both tetragonal crystal and cubic crystal. Since the lattice strain tends to be distributed in the ab-plane because of the interval and alternation of two layers along the c-axis, the volume change in a unit cell can be considered without taking into account the length change of c-axis. If we consider the change of a-axis only, Equations (8a) and (8b) can be further simplified as follows: For a true crystal lattice of Bi 4 Ti 3 O 12 , the structure cannot be stable unless F B and F P not only act in the opposite manner, but also remain balanced with each other. Thus, an analytical model for the plane stress distribution in the crystal lattice of Bi 4 Ti 3 O 12 could be constructed using the constitutive relationship of the alternating layer structure, as shown in Figure 6.
For a true crystal lattice of Bi4Ti3O12, the structure cannot be stable unless FB and FP not only act in the opposite manner, but also remain balanced with each other. Thus, an analytical model for the plane stress distribution in the crystal lattice of Bi4Ti3O12 could be constructed using the constitutive relationship of the alternating layer structure, as shown in Figure 6. According to the balance of forces, it yields: Substituting Equations (10a) and (10b) into Equation (11), we can obtain: where K is defined as the interlayer mismatch degree, which is mainly determined by KP since KB is irrelevant to the composition of pseudo-perovskite layer. This is a confirmed structure and uniform composition for the bismuth-oxide layer of all BLSFs. aB' is about 3.78 Å [13]. aP' is assumed to be equal to the average lattice parameter of perovskite unit, which can be calculated as follows: (13) where rA and rB are the radius of the ions at A-site (Bi 3+ : 1.34 Å ) and B-site of the perovskite unit. Finally, the total strain energy (E) generated by the expansion of the bismuth-oxide layer and the compression of pseudo-perovskite layer can be calculated by the following formula: According to the balance of forces, it yields: Substituting Equations (10a) and (10b) into Equation (11), we can obtain: where K is defined as the interlayer mismatch degree, which is mainly determined by K P since K B is irrelevant to the composition of pseudo-perovskite layer. This is a confirmed structure and uniform composition for the bismuth-oxide layer of all BLSFs. a B ' is about 3.78 Å [13]. a P ' is assumed to be equal to the average lattice parameter of perovskite unit, which can be calculated as follows: where r A and r B are the radius of the ions at A-site (Bi 3+ : 1.34 Å) and B-site of the perovskite unit. Finally, the total strain energy (E) generated by the expansion of the bismuth-oxide layer and the compression of pseudo-perovskite layer can be calculated by the following formula: where V B ' = a B ' 2 c B ≈ 65.4 Å 3 , V P ' = a P ' 3 . According to the principle of solid physics, the bulk modulus of ionic crystal is determined by the following formula [30]: Here, α is Madelung constant, which is 24.76 for A II B IV O 3 [19]. r 0 is the shortest distance between cations and anions, which is 1.73 Å for Bi 4 Ti 3 O 12 (B = Ti) [23], and e = 4.8 × 10 −10 esu. Hereby, K P can be approximatively calculated as 44.26 GPa, and K B can be also determined as 98.36 GPa according to the value of K (0.45) for Bi 4 Ti 3 O 12 reported in Ref. [19].

Influence of Doping-Ions on the Interlayer Mismatch
In this experiment, the chemical formula of BTWC ceramics is: Bi 4 Ti 3−x W x O 12+x + y wt % Cr 2 O 3 (0 ≤ x ≤ 0.1; 0 ≤ y ≤ 0.4). The part of titanium at the B-site of perovskite structure was co-substituted by the tungsten and chromium introduced by dopants. If the mass fraction of Cr 2 O 3 is changed into the molar fraction relative to the chemical composition, the chemical formula of BTWC ceramics can be written as Bi 4 Ti 3−x−0.154y W x Cr 0.154y O 12+x−0.077y . As the B-site is occupied by Ti 4+ /W 6+ /Cr 3+ together, in order to simply the calculation, r B is given by the average value of radius of all local ions (W 6+ 0.600 Å, Cr 3+~0 .615 Å and Ti 4+~0 .605 Å), in relation to their molar ratio: Finally, according to Equations (13)- (15) obtained from the elastic model and lattice parameters derived from XRD, the interlayer mismatch degree (K) and the total strain energy (E) can be calculated for BTWC. Figure 7 shows the variation of interlayer mismatch degree and total strain energy of BTWC with the composition. As can be seen, the interlayer mismatch degree shows a similar varying trend to the total strain energy. That is to say, a more mismatched layer structure stores a larger strain energy for a crystal. In addition, the calculated value of K is 0.48 for BIT in this paper, while BTW x -0.2Cr and BTW 0.05 -yCr take the values of K in the range of 0.97-3.45 and 0.60-0.97, respectively. It seems to be that either W 6+ or Cr 3+ can aggravate the interlayer mismatch of Bi 4 Ti 3 O 12 structure. This may be partly due to the mechanism whereby doping of W 6+ /Cr 3+ could relieve the distortion of pseudo-perovskite layer by depressing the rotation and tilting of oxygen octahedron (see the values of a/b in Tables 2 and 3), which helps the pseudo-perovskite layer to be more stiff. energy for a crystal. In addition, the calculated value of K is 0.48 for BIT in this paper, while BTWx-0.2Cr and BTW0.05-yCr take the values of K in the range of 0.97-3.45 and 0.60-0.97, respectively. It seems to be that either W 6+ or Cr 3+ can aggravate the interlayer mismatch of Bi4Ti3O12 structure. This may be partly due to the mechanism whereby doping of W 6+ /Cr 3+ could relieve the distortion of pseudo-perovskite layer by depressing the rotation and tilting of oxygen octahedron (see the values of a/b in Tables 2 and 3), which helps the pseudo-perovskite layer to be more stiff. However, in fact, the interlayer mismatch degree should be more dependent on the size in abplane for the Bi4Ti3O12 structure, which is illuminated by Figure 8. Also, the variation of the size in the ab-plane with the dopant content is consistent with that of the bulk modulus of the perovskite unit, which indicates that a larger perovskite unit tends to possess a higher bulk modulus. Above all, for all the compositions of BTWC, the size in ab-plane keeps the same varying tendency with the However, in fact, the interlayer mismatch degree should be more dependent on the size in ab-plane for the Bi 4 Ti 3 O 12 structure, which is illuminated by Figure 8. Also, the variation of the size in the ab-plane with the dopant content is consistent with that of the bulk modulus of the perovskite unit, which indicates that a larger perovskite unit tends to possess a higher bulk modulus. Above all, for all the compositions of BTWC, the size in ab-plane keeps the same varying tendency with the interlayer mismatch degree. The results can strongly prove that the interlayer mismatch in Bi 4 Ti 3 O 12 structure has to be constrained in the ab-plane of the unit cell. In general, with larger size of ab-plane, a higher bulk modulus of perovskite unit is achieved; then, a larger interlayer mismatch of Bi 4 Ti 3 O 12 structure, as well as a higher strain energy stored in the crystal occur. interlayer mismatch degree. The results can strongly prove that the interlayer mismatch in Bi4Ti3O12 structure has to be constrained in the ab-plane of the unit cell. In general, with larger size of ab-plane, a higher bulk modulus of perovskite unit is achieved; then, a larger interlayer mismatch of Bi4Ti3O12

Conclusions
For some W/Cr co-doped Bi4Ti3O12 (ab. BTWC) Aurivillius compounds, XRD analysis shows that all compositions of BTWC belong to the orthorhombic structure, and Rietveld refinements for the crystal structure are used to calculate their lattice constants. Since the oxygen-octahedron rotates in the ab-plane, as well as inclines away from the c-axis, W 6+ could occupy the B-site instead of partial Ti 4+ , which tends to relieve the rotation of oxygen-octahedron. The substitution of Cr 3+ for Ti 4+ tends to create an oxygen vacancy (VO •• ) in the oxygen-octahedron, which has an influence on the tilting of oxygen-octahedron. Due to the strong combining power between the pseudo-perovskite layer and the bismuth oxide layer along the c-axis, a lattice misfit tends to be constrained to the ab-plane, which causes a structural mismatch between the two layers. Both the degree of interlayer mismatch and the total strain energy vary with the compositions of BTWC in a similar trend to the lattice parameters in the ab-plane. The doping of W 6+ /Cr 3+ could relieve the distortion of pseudo-perovskite layer of BIT, which helps it to be more stiff.

Conclusions
For some W/Cr co-doped Bi 4 Ti 3 O 12 (ab. BTWC) Aurivillius compounds, XRD analysis shows that all compositions of BTWC belong to the orthorhombic structure, and Rietveld refinements for the crystal structure are used to calculate their lattice constants. Since the oxygen-octahedron rotates in the ab-plane, as well as inclines away from the c-axis, W 6+ could occupy the B-site instead of partial Ti 4+ , which tends to relieve the rotation of oxygen-octahedron. The substitution of Cr 3+ for Ti 4+ tends to create an oxygen vacancy (V O •• ) in the oxygen-octahedron, which has an influence on the tilting of oxygen-octahedron. Due to the strong combining power between the pseudo-perovskite layer and the bismuth oxide layer along the c-axis, a lattice misfit tends to be constrained to the ab-plane, which causes a structural mismatch between the two layers. Both the degree of interlayer mismatch and the total strain energy vary with the compositions of BTWC in a similar trend to the lattice parameters in the ab-plane. The doping of W 6+ /Cr 3+ could relieve the distortion of pseudo-perovskite layer of BIT, which helps it to be more stiff.