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the O180 { 001 } wall has almost the same thickness as the O90 wall, while in the tetragonal phase, it is the 90 ◦ wall which is much thicker than 180 ◦ wall (3.59 nm compared to 0.63 nm). 

In general, the normal of the neutral 180 $^{◦}$-domain wall can take any direction perpendicular to the spontaneous polarization of the adjacent domains. Therefore also Σ depends on the orientation of the wall normal (in the s − t plane). We have checked in the orthorhombic phase that the 0180 { 001 } and 0180 { 1 ¯ 10 } correspond to the extremes in the angular dependence of Σ, which is monotonous between them. This strong angular dependence is corre- lated with the anisotropy of the tensor G$_{ijkl}$ . However, g is independent of the wall direction in the tetragonal and rhombohedral phase, and also the variation of coef- ficients a$_{1}$ , a$_{11}$ , and a$_{111}$ is insignificantly small, so that the effective directional dependence of 180 ◦ domain wall properties in these phases is negligible. 

The values of the shape factor A (see Table III) deter- mines the deviation of the P$_{r}$ ( s ) polarization profile of the domain walls from the simple tanh form. For all studied cases the values of A range between 1 and 2, where the correction factor A 5 / $^{2}$I ( A ) appearing in the Eqn. (25) is almost linear function of A , as it can be seen in Fig. 4. It means that the shape deviations are much smaller than those shown by broken line in Fig. 3. Unfortunately, in the case of broad walls T90 and O60, which are good candidates for study of the structure of the wall central part, A is almost one. 

Eqn. (14) can be also used to evaluate local strain vari- ation in the wall. In Fig. 5 the profiles of strain tensor components for T90 domain wall are shown for illustra- tion. Obviously, e$_{33}$, e$_{23}$ , and e$_{13}$ strain components are strictly constant and equal to their boundary values as it follows from mechanical compatibility conditions. The e$_{11}$ and e$_{22}$ components and polarization P$_{r}$ vary be- tween their spontaneous values. Let us stress that the ’re-entrant’ shear component e$_{12}$ (it has the same value in both adjacent domains) approaches the non-zero value of about 5 × 10 − 4 in the middle of the domain wall. 

The temperature dependence of the thickness and sur- face energy density of the 12 studied domain wall species is plotted in Fig. 6a and Fig. 6b, respectively. Although some properties do vary considerably, e.g. thickness of T90 wall in the vicinity of the paraelectric-ferroelectric phase transition, the sequence of the thickness values as well as the surface energy-density values of different do- main wall types remain conserved within each phase. 

The temperature dependence of domain wall thick- ness follows the trend given by Eqn. (23). It increases with increasing temperature due to the dependence on p$_{∞}$ (see dependence of U on p$_{∞}$ in Eqn. (24)). Such behavior is well known also from the experimental observations. 3,22,23 

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## VI. CURVED POLARIZATION PATH SOLUTIONS 

So far we have investigated domain walls within the SPP approximation, i.e. components P$_{s}$ and P$_{t}$ , which are the same in both domain states, were kept constant inside the whole wall. This is quite usual assumption made for a ferroelectric domain wall. Nevertheless, the full variational problem, where all three components of P could vary along s coordinate, leads in general to a lower energy solution corresponding to a curved polar- ization path (CPP) in the 3D primary order-parameter space. Appearance of the non-zero ’re-entrant’ compo- nents within a ferroelectric domain wall was considered 

