content_image/443084.jpg content_image/443094.jpg content_image/443085.jpg content_image/443095.jpg
<img src='content_image/443084.jpg'>

erties in BaTiO$_{3}$ are the same as in our preceding work, 4 but for the sake of convenience, the definition and the GLD model and its parameters are resumed in Section III. Section IV. is devoted to the description of the com- putational scheme and approximations applied here to solve analytically the Euler-Lagrange equations. The main result of our study - systematic numerical evalu- ation of thicknesses, energies, polarization profiles and other properties for different domain walls, is presented in Section V. Sections VI. and VII. are devoted to the dis- cussion of validity of used approximations and the final conclusion, respectively. 

## II. MECHANICALLY COMPATIBLE DOMAIN WALLS IN BARIUM TITANATE 

The energy-degeneracy of different directions of spon- taneous polarization leads to the appearance of ferroelec- tric domain structure. Individual domains are separated by domain walls, where the polarization changes from one state to another. Here, only planar domain walls are considered. Orientations of mechanically compatible domain walls are determined by the equation for mechan- ically compatible interfaces separating two domains with the strain tensors e$_{ij}$ ( −∞ ) and e$_{ij}$ ( ∞ ) : 

<img src='content_image/443094.jpg'>

Systematic analysis of this equation using symmetry ar- guments has been done e.g. in Refs. 10–12. In general, the number N of mechanically compatible domain walls separating two particular domain states can have only one of the three values: N = 0, N = 2 or N = ∞ . In case of N = 2 there exist two mutually perpendic- ular domain walls. Each of them is either a crystallo- graphic ( W$_{f}$ -type) wall or non-crystallographic ( S -type) wall. Orientation of the W$_{f}$ -wall is fixed by symmetry of the crystal, while orientation of the S -wall is determined by components of the strain tensor in adjacent domains 

<img src='content_image/443085.jpg'>

(and its orientation can be therefore dependent on tem- perature). For N = ∞ there exists infinite number of wall orientations, some of them may be preferred ener- getically. 

Further, the electrically neutral domain walls will be considered. 10 It implies that the difference P ( ∞ ) − P ( −∞ ) between the spontaneous polarizations in the ad- jacent domains is perpendicular to the unit vector s , nor- mal to the domain wall: 

We also define a unit vector r ‖ ( P ( ∞ ) − P ( −∞ )), which identifies the component of the spontaneous polarization which reverses when crossing the wall. Then the charge neutrality condition (2) can be expressed as r · s = 0. Finally, let us introduce a third base vector t = r × s , 

<img src='content_image/443095.jpg'>

