Domain wall solutions for EHM model of crystal:

Physica D 147 (2000) 122–134

Domain wall solutions for EHM model of crystal: structures with period multiple of four Sergey V. Dmitriev∗ , Kohji Abe, Takeshi Shigenari Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu-shi, Tokyo 182-8585, Japan Received 22 March 1999; received in revised form 6 June 1999; accepted 21 April 2000 Communicated by M. Sano

Abstract For a one-dimensional discrete model of a crystal the form of a moving domain wall in the structure with period multiple of four was derived in the constant amplitude approximation. The energy of the carrying commensurate structure, the width and the energy of the domain wall were expressed in terms of the amplitudes of Fourier harmonics of the carrying commensurate structure. With the use of the result by Ishibashi, the relation between different solutions was established. The applicability and the accuracy of the solutions were tested by comparing them with the numerically obtained exact solutions. © 2000 Elsevier Science B.V. All rights reserved. Keywords: One-dimensional model; Modulated phase; Domain wall; Sine-Gordon equation

1. Introduction Domain walls (DWs) play an important role in the physics of dielectric crystals [1–3]. DWs are an integral part of modulated phases [1,2,4]. Recently a new branch of ferroelectric technology and science named domain engineering has formed [3] and the interest in studying the processes of formation and kinetics of the domain structure has been increasing. Solutions of the form of DW in a commensurate structure have been obtained by Slot and Janssen [5] in the frame of the one-dimensional microscopic DIFFOUR model using the constant amplitude approximation (CAA) [6]. Their solutions were specified later by Hlinka et al. [7] for commensurate structures with κ = 13 , 15 and 17 . From the mathematical point of view, the DIFFOUR model is identical to the model with the next-nearest interaction and fourth-order polynomial background potential [8] and also to the elastically hinged molecule (EHM) model proposed by the present authors [9–15]. Static properties of the models mentioned above are identical to that of the model by Janssen and Tjon [8] with the acoustic mode. It means that, in spite of the different physical interpretations, any result obtained for one of these models, can be transformed to the notations of other models. ∗ Corresponding author. Permanent address: Department of General Physics, Barnaul State Technical University, 656099 Barnaul, Russia. Tel.: +81-424-83-2161; fax: +81-424-84-7013 E-mail address: [email protected] (S.V. Dmitriev).

0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 0 ) 0 0 1 5 8 - 5

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The DW solutions reported by Slot and Janssen [5] and Hlinka et al. [7] contain not only the usual error due to the continuum approximation, but also an error caused by poor estimation of the amplitudes of Fourier harmonics for the carrying commensurate structure. In our consideration [15], we used the same approach but in a more consistent way. As a result, the DW solutions, where the only source of the error is the continuum approximation, have been derived for any odd-periodic commensurate structure [15]. By means of the Ishibashi transformation [17] (see Section 3) the solutions for the odd-periodic structures can be transformed to the solutions in the (4l + 2)-periodic structures, where l is a positive integer. The problem for the 4l-periodic structures, however, has not been solved yet. In the present paper we give this solution, which completes the DW solutions for the commensurate phase in the EHM model. In Section 2, we describe the stable periodic structures of the EHM model. In Section 3 the Ishibashi transformation of the solutions is adopted to the notations of the EHM model. Using the CAA, we derive in Section 4 the continuum analog to the modulated structure with slowly varying phase. The DW solution is obtained in Section 5 as a kink solution to the sine-Gordon equation which plays the role of the continuum equation of motion. The DW solution is specified in Sections 6 and 7 for 4- and 8-periodic commensurate structures, respectively. In Section 8 the accuracy of the solutions is illustrated by comparison with the numerical results and with the other solutions. We conclude the paper in Section 9.

2. EHM model: equilibrium commensurate structures The kinetic and potential energy of the EHM model can be written as [9] T =

1X 2 y˙ , 2 n n

(1)

  1 4 1X 2 2 2 F (yn−1 − 2yn + yn+1 ) − P (yn+1 − yn ) + yn + yn , U= 2 n 2

(2)

respectively. The distance between two neighboring hinges is used as a unit of length. The model has two parameters: P ≥ 0 is the external pressure and F ≥ 0, which is temperature dependent, describes the elastic properties of the chain. Equation of motion of the EHM model has the form y¨n + F (yn−2 − 4yn−1 + 6yn − 4yn+1 + yn+2 ) + P (yn−1 − 2yn + yn+1 ) + yn + yn3 = 0.

(3)

Obviously, (3) has the trivial solution, yn = 0, and besides, it admits the stability of the J -periodic structures (yn = yn+J ) given below. Two-periodic structures (J = 2) are given by yn = ±A1 cos(πn) = ±(−1)n A1 .

(4)

Odd-periodic structures with J = 2l + 1, where l is a positive integer, can be expressed as (J −1)/2+1 X

Aj cos [j ∗ q(n + m)].

yn = ±

j =1

(5)

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Even-periodic structures are divided into two subsets, one with the period J = 4l + 2, yn =

(J −1)/2+1 X

Aj cos [j ∗ q(n + m)],

(6)

j =1

and another one with the period J = 4l, yn =

J /4 h  X π i Aj cos j ∗ q(n + m) + . J

(7)

j =1

In (5)–(7), we denote j ∗ = 2j − 1 and q = 2π κ, m can take any value from the set m = {0, 1, . . . , J − 1}, Aj are functions of parameters of the model P and F . We study the structures (5)–(7) with rational wave vectors I , 2I < J, (8) J where I and J are coprime positive integers. If J is not large, the structure is called a commensurate structure, otherwise, it can be regarded as an incommensurate structure. The numbering of harmonics in (5)–(7) differs from the traditional way of numbering of Fourier harmonics. The convenience of the present way of numbering is that one can take A1  A2  A3 . . . . Eq. (4) describes two different domains of two-periodic structure. Eq. (5) describes 2J different domains of J -periodic structure with an odd J , each domain being defined by m and by a specific choice of the sign. Eqs. (6) and (7) each describe J different domains of J -periodic structure with an even J , each domain being defined by m. Using the terminology introduced by Ishibashi [18], the structures given by (5)–(7) are the nonzero-structures which means that yn 6= 0 for any n. For the background potential with, for example, sixth-order anharmonicity, the stable zero-structures are possible [20,21]. For zero-structures (5)–(7) should be properly changed. However, in the case of fourth-order anharmonicity, discussed in the present paper, the zero-structures are always unstable. For the four-periodic structure (see (7) at J = 4), Ohfuti and Ono [19] considered a more general expression, yn = A cos(πn/2 + π/4) + B cos(πn), with the second term to describe a higher harmonics. Consequently, an additional term appears in their DW solution. However, for the particular anharmonicity (fourth-order anharmonicity) discussed here, for the stable four-periodic phase, B is equal to zero, as described in Ref. [10]. κ=

3. Relation between different structures Let yn (t) be a solution to the equation of motion (3). One can easily check that the function   √ (−1)n t yn , G = 16F − 4P + 1, yn∗ (t) = G G

(9)

is a solution to the following equation: y¨n + F ∗ (yn−2 − 4yn−1 + 6yn − 4yn+1 + yn+2 ) + P ∗ (yn−1 − 2yn + yn+1 ) + yn + yn3 = 0,

(10)

where F∗ =

F , G2

P∗ =

8F − P . G2

(11)

It means that knowing a solution yn (t) at a point of the phase space (F, P ), one can write a different solution yn∗ (t) at another point (F ∗ , P ∗ ).

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Factor (−1)n in (9) changes the sign of displacements of all even (or all odd) nodes for the solution yn (t). Factor (−1)n can be presented in the form cos(πn) and consequently, if the solution yn (t) is a periodic solution with wave vector κ, then the solution yn∗ (t) is also periodic and has the wave vector κ ∗ = 21 − κ [17]. Thus, if the solution yn (t) to (3) has the energy H , then the solution yn∗ (t) to (10) has the energy H ∗ = H /G4 . In the following we will refer to this transformation of EHM model solutions as to Ishibashi transformation. Ishibashi transformation transforms yn into yn∗ and vice versa, because from Eq. (11) it follows that 8F ∗ − P ∗ F∗ , P = , (G∗ )2 (G∗ )2 √ where G∗ = 16F ∗ − 4P ∗ + 1. As (9) suggests, Ishibashi transformation can be applied only if F =

16F − 4P + 1 > 0.

(12)

This restriction comes from the fact that, in order to reduce the number of parameters of the model from three to two, we consider only the case of the single-well background potential. A more general case of a double-well potential would not give new important physical effects in comparison to our simplified consideration. It is obvious that the Ishibashi transformation transforms the odd-periodic structures given by (5) into the even-periodic structures with J = 4l + 2 given by (6) and vice versa. Each structure of the form (7) with J = 4l will be transformed to another structure of the same form. Two-periodic structure (4) cannot be transformed to another solution, because it exists in the region 16F − 4P + 1 ≤ 0 [9], where the transformation (9) cannot be applied. 4. Modulated structures with period J = 4l 4l: continuum approximation Using the CAA, let us seek the solution to (3) of the following form: J /4 h  i X π yn (t) = ± Aj cos j ∗ q(n + m) + + ϕn (t) . J

(13)

j =1

Eq. (13) differs from (7) by the presence of the unknown slowly varying (with n) function ϕn (t). One can say that the function ϕn (t) modulates the J -periodic structure. Particularly, the function ϕn (t) can describe the DW in the J -periodic structure as the variation of the phase between two adjacent domains given by (7). Let us substitute the ansatz (13) into (1) and (2). The fast varying terms can be averaged over J neighboring nodes assuming that ϕn is almost constant within one period. It means that one can substitute, for example, cos(ϕn ) cos2 (nq)

by

J X 1 1 cos(ϕn ) cos2 (jq) = cos(ϕn ). J 2

(14)

j =1

Averaging the harmonic terms of (1) and (2) gives J /4

y˙n2 =

ϕ˙n2 X ∗ (j Aj )2 , 2

yn2 =

1X 2 Aj , 2

(15)

j =1

J /4

j =1

(16)

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(yn+1 − yn )2 =

J /4 X A2j {1 − cos [j ∗ (q + ϕn+1 − ϕn )]},

(17)

j =1

(yn−1 − 2yn + yn+1 )2 =

J /4 X A2j {3 + cos(2j ∗ q) cos [j ∗ (ϕn+1 − ϕn−1 )]} j =1

−2

J /4 X A2j cos(j ∗ q) sin(j ∗ q) sin [j ∗ (ϕn+1 − ϕn−1 )] j =1

−4

    J /4 X ϕn−1 − 2ϕn + ϕn+1 ϕn+1 − ϕn−1 A2j cos j ∗ cos j ∗ + j ∗q . 2 2

(18)

j =1

The averaged anharmonic term of (2) for a given J can be written in the form yn4 = XJ − ZJ cos(J ϕ),

(19)

where XJ , ZJ are functions of the amplitudes of harmonics Aj , j = 1, . . . , 41 J . Let us write the functions XJ , ZJ in an explicit form for J = 4, 8, 12: X4 = 38 A41 ,

Z4 = 18 A41 .

(20)

X8 = 38 (A41 + A42 ) + 23 A21 A22 + 21 A31 A2 ,

Z8 = 43 A21 A22 + 21 A1 A32 .

(21)

X12 = 38 (A41 + A42 + A43 ) + 23 (A21 A22 + A21 A23 + A22 A23 + A21 A2 A3 + A1 A22 A3 ) + 21 A31 A2 , Z12 = 18 A42 + 43 A21 A23 + 21 A2 A33 + 23 (A1 A22 A3 + A1 A2 A23 ).

(22)

The averaged kinetic energy (1) in view of (15) in the continuum approximation reads J /4

1X ∗ (j Aj )2 T = 4

Z

j =1

∞ −∞

ϕt2 dx,

(23)

and the potential energy (2) in view of (16)–(19) reads     Z ∞ Z ∞ X J /4 J /4 X P 1 XJ − ZJ cos(J ϕ) +  A2j [1 − cos(j ∗ q + j ∗ ϕx )] dx A2j  dx − U= 4 −∞ 2 −∞ j =1 j =1   Z J /4 F ∞ X 2 Aj [3 + cos(2j ∗ q) cos(2j ∗ ϕx )] dx + 2 −∞ j =1    ∗  Z ∞ X J /4  A2j [ cos(j ∗ q) sin(j ∗ q) sin(2j ∗ ϕx ) + 2 cos j ϕxx cos(j ∗ q + j ∗ ϕx )] dx. −F 2 −∞

(24)

j =1

Using the assumption ϕx  1, we put cos(j ∗ ϕx ) ≈ 1 − 21 (j ∗ ϕx )2 ,

sin(j ∗ ϕx ) ≈ j ∗ ϕx ,

cos(j ∗ ϕxx ) ≈ 1.

(25)

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Eq. (24), in view of (25) can be presented as   Z ∞ J /4 X 1 1  [XJ − ZJ cos(J ϕ)] + A2j [Rj (q) + Rj∗ (q)j ∗ ϕx + Rj∗∗ (q)(j ∗ ϕx )2 ] dx, U= 2 −∞ 4

127

(26)

j =1

where Rj (q) =

1 2

− P + 3F − (4F − P ) cos(j ∗ q) + F cos(2j ∗ q),

(27)

Rj∗ (q) = (4F − P ) sin(j ∗ q) − 2F sin(2j ∗ q),

(28)

Rj∗∗ (q) = 21 (4F − P ) cos(j ∗ q) − 2F cos(2j ∗ q).

(29)

Let us find the equations of motion by minimizing the integral Z t2 (T − U ) dt

(30)

t1

(Hamilton’s principle), where the kinetic and potential energy are given by (23) and (26), respectively. The Euler–Lagrange equation for the function ϕ(x, t) is the sine-Gordon equation ρϕtt − σ ϕxx + η sin(J ϕ) = 0,

(31)

where J /4

ρ=

1X ∗ (j Aj )2 , 2 j =1

σ =

J /4 X (j ∗ Aj )2 Rj∗∗ (q), j =1

η=

J ZJ . 4

(32)

The sine-Gordon equation is a completely integrable equation, one of the most extensively studied nonlinear equations. In the following subsections we will study only the kink solution to the sine-Gordon equation, or the solution of the form of a moving DW. 5. DW solution The DW solution to (31) is ϕ(x, t) =

4 arctan exp[SJ (x − x0 − ct)], J

(33)

or in the discrete form ϕn (t) =

4 arctan exp[SJ (n − x0 − ct)], J

where x0 is the position of the DW at t = 0, c the velocity of DW and SJ the inverse DW width: s Jη . SJ = ± σ − ρc2

(34)

(35)

Eq. (13) together with (34) and (35) describe all possible J DWs in the J -periodic structure. Note that SJ can be positive or negative. Positive SJ corresponds to an extended DW (kink) and negative SJ to a compressed one (antikink). Energy of the extended DW, generally speaking, is not equal to that of the compressed one. √ As (35) suggests, the velocity of DW can vary from 0 to the limiting value σ/ρ.

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Substituting (33) into (23) one can find the kinetic energy of DW: J /4

TDWJ

2c2 SJ X ∗ =± 2 (j Aj )2 , J

(36)

j =1

where the upper sign is for SJ > 0 and the lower sign for SJ < 0. Eq. (26), in view of (33) and in view of the integrals Z ∞ Z ∞ Z ∞ 4 2π cos(J ϕ) dx = − + dx, ϕx dx = , S J J −∞ −∞ −∞

Z

∞

−∞

ϕx2 dx =

8SJ , J2

can be written in the form Z ∞ UJ dx. U = UDWJ + −∞

(37)

(38)

The first term in the right-hand side of (38) is the potential energy of DW in J -periodic structure: UDWJ = ±

  ∗ J /4 X ZJ 4SJ (j ∗ )2 ∗∗ πj ∗ Rj (q) + ± A2j R (q) , j SJ J J2

(39)

j =1

where the upper sign is for SJ > 0 and the lower sign for SJ < 0. The second term in the right-hand side of (38) is the potential energy of the ideal J -periodic structure with the energy density J /4 1 1X 2 Aj Rj (q). UJ = (XJ − ZJ ) + 4 2

(40)

j =1

Note that (40) is not an approximation, it is an exact expression of the energy density of the J -periodic structure in terms of Aj . In this section the inverse DW width (35), kinetic (36) and potential (39) energies of DW and the energy density of the ideal J -periodic structure (40) were expressed in terms of the amplitudes of Fourier harmonics Aj of the ideal J -periodic commensurate structure. The form of equations is correct for any J which is a multiple of four. In the following sections the analytical estimations for Aj in terms of parameters of the EHM model P , F will be given and the DW solutions for the 4-, 8- and 12-periodic structures will be specified. The same results can be obtained for any J if the functions XJ , ZJ in (19) are known. 6. Four-periodic structure (J = 4) There exists only one four-periodic structure with κ = 41 (q = π/2). Substituting J = 4 into (40) gives the following expression for energy density of the ideal four-periodic commensurate structure in terms of A1 : U4 =

1 4 16 A1

+ (1 − 2P + 4F ) 41 A21 .

(41)

The minimum energy condition, dU4 /dA1 = 0, has the following exact solution: A21 = 2(2P − 4F − 1). Four-periodic structure exists if 2P > 4F + 1.

(42)

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With the use of (42) one can specify (35), (36) and (39) for q = π/2 as follows: A1 . S4 = ± √ 4F − c2

(43)

TDW4 = ± 18 c2 S4 A21 ,

(44)

√ The limiting value of the velocity of DW is 2 F ,

UDW4 = ±

A41 A2 ± [π(4F − P ) + 2FS4 ] 1 . 8S4 4

(45)

One can see that the potential energy of DW contains a term which does not depend on the inverse width S4 of DW. This term is equal to zero on the line P = 4F . In view of (13) and (34) the DW in the four-periodic structure is given by h π i π (46) (n + m) + + arctan exp[S4 (n − x0 − ct)] , yn (t) = ±A1 cos 2 4 where A1 and S4 are given by (42) and (43). 7. Eight-periodic structures (J = 8) There exist two different eight-periodic structures, with κ = 18 (q = π/4) and κ = structures are related to each other by means of transformation (9). One can rewrite (40) for J = 8 as U8 =

4 3 32 (A1

+ A42 ) +

3 2 2 16 A1 A2

3 8

(q = 3π/4). These two

+ 18 A31 A2 − 18 A1 A32 + 21 A21 R1 (q) + 21 A22 R2 (q).

The energy of the eight-periodic structure (47) has a minimum at, approximately, r A31 8 , A2 = − A1 = − R1 (q), 3 8R2 (q) or more precisely r A1 = 2 R2 (q) −

q R22 (q) + 43 R1 (q)R2 (q),

A2 = −

A31 8R2 (q) + 3A21

.

(47)

(48)

(49)

Estimation (48) can be used for small displacements (A1  1), whereas (49) is valid in a wider range of displacements. More precise estimation of the amplitudes A1 , A2 and A3 is derivable from (47) by solving the cubic algebraic equations. The exact magnitudes of the amplitudes can be found numerically. In view of (48) or (49) one can express the parameters of DW S8 (35), TDW8 (36) and UDW8 (39) in terms of parameters P , F, κ for either of the two eight-periodic commensurate structures. Note that for a not very wide DW the estimation (48) cannot be used.

8. Discussion For the four-periodic structure another form of DW solution has been given [10]. The solution reads

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A1 S4 y4n = ± √ − √ (P − 4F ) cosh−2 2 2 A1 y4n+2 = −y4n , y4n+3 = −y4n+1 ,



 S4 (n + n0 − ct) , 2

  A1 S4 (n + n0 − ct) , y4n+1 = √ tanh 2 2 (50)

where A1 and S4 are given by (42) and (43). There exist eight different DWs. To describe the four compressed walls, it is necessary to choose sign (−) in the first equation of (50) and to shift the indices by unity sequentially. Four extended walls correspond to sign +. The solution exists in the whole region where the four-periodic structure exists. For |n| → ∞ (50) are reduced to the solution (7) with κ = 41 . Let us compare the two different DW solutions in the four-periodic structure. First, we describe the region of the phase plane (P , F ), where the carrying commensurate four-periodic structure is stable and then we discuss the DW solution in this structure. The parabola F = 41 P 2 ,

(51)

and the smoothly joined straight line F = 41 P −

1 16

(52)

define the boundary of the region where the trivial solution exists. The four-periodic structure is stable in the area bounded by the hyperbola F2 +

1 2 48 P

− 21 FP + 16 F < 0,

(53)

and the smoothly joined straight line F > − 21 P + 21 .

(54)

The hyperbola (53) is tangent to the parabola (51) at the point (P , F ) = (1, 41 ). In Fig. 1, we schematically show the regions where the trivial solution and the four-periodic structure are stable (left and right shaded regions, respectively). The trivial solution is stable in the region bounded by the parabola (1a)

Fig. 1. Schematic representation of the phase diagram. Parabola (1a) (51) and smoothly joined line (1b) (52) make the boundary of the trivial solution stability region (left shaded region). Hyperbola (2a) (53) and smoothly joined line (2b) (54) make the boundary of the stability region for the four-periodic structure (right shaded region). Lines (1a) and (2a) are tangent to each other. The coordinates of the tangency point are (P , F ) = (1, 41 ). The DW solution obtained in the frame of CAA, (46), and the solution given by (50) are compared with the exact solution obtained numerically at the dashed line (3) given by P = 2F + 0.502. The line (3) is parallel to the tangent to the parabola (1a) at the point (1, 41 ).

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Fig. 2. The energy of the extended DW at c = 0 as the function of F with P = 2F + 0.502, solid line: the exact solution, open circles: (46), solid circles: (50).

(51) and smoothly joined line (1b) (52). The four-periodic structure stability region is bounded by hyperbola (2a) (53) and smoothly joined line (2b) (54). The comparison of the solution obtained in the frame of CAA, (46), and the solution given by (50) with the exact solution obtained numerically, is given in Figs. 2–5. In Fig. 2, we compare the energy of the extended DW at c = 0 as the function of F with P = 2F + 0.502 (dashed line (3) in Fig. 1). The estimation of DW energy was found by substituting the approximate solutions given by (46) (open circles) and (50) (solid circles) into the Hamiltonian (2) and subtracting the energy of the four-periodic commensurate structure given by (41). The exact solution is shown by solid line. Below F = 0.212 and above F = 0.265 the DW is unstable. CAA solution (46) gives a better result in the region where the DW has negative energy, while the solution (50) is very accurate for the DW with positive energy. Similar results are shown in Fig. 3, but for the compressed DW. The DW is stable in the range 0.237 < F < 0.297. The DW with negative energy is better described by (46) and that with positive energy is better described by (50). In order to answer the question why the accuracy of the solutions (46) and (50) is different at low and high limits of DW stability, in Figs. 4 and 5 we compare, for the compressed DW, the profile of DW close to both the limits of stability, namely, at F = 0.238 (Fig. 4) and at F = 0.296 (Fig. 5). In (a) the exact solution is shown and (b) and (c) show the approximate solutions (46) and (50), respectively. The displacements yn are shown by four envelops for the nodes 4n, 4n + 1, 4n + 2 and 4n + 3.

Fig. 3. The energy of the compressed DW at c = 0 as the function of F with P = 2F + 0.502, solid line: the exact solution, open circles: (46), solid circles: (50).

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Fig. 4. The profile of the compressed DW at F = 0.238, P = 2F + 0.502: (a) is the exact solution; (b) and (c) are the approximate solutions (46) and (50), respectively. The displacements yn are shown by four envelops for the nodes 4n, 4n + 1, 4n + 2 and 4n + 3.

It can be seen from Fig. 4(a) that the DW at F = 0.238 has a concave shape. The solution (50) reproduces the concave shape of the DW (see Fig. 4(c)), whereas the CAA approximation does not (see Fig. 4(b)). That is why the solution given by (50) is more accurate in this case. The solution (50) is more flexible because it assumes that there are two unknown functions instead of one, as it is in the CAA approach. As Fig. 5(a) suggests, the DW at F = 0.296 has a convex shape. This shape is reproduced by both the approximate solutions, (46) and (50), shown in Fig. 5(b) and (c), respectively, but (46) gives a better result. The CAA solution is in good agreement with the numerical results in the parameter region where the DW energy is negative. Let us briefly discuss the role of the negative-energy DWs in dielectric materials. The DWs with negative energy can appear as a result of evolution of the incommensurate phase. The incommensurate phase appears in dielectric crystals usually on cooling. The transition to the incommensurate phase from

Fig. 5. The profile of the compressed DW at F = 0.296, P = 2F + 0.502: (a) is the exact solution; (b) and (c) are the approximate solutions (46) and (50), respectively. The displacements yn are shown by four envelops for the nodes 4n, 4n + 1, 4n + 2 and 4n + 3.

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the high-symmetry commensurate phase (trivial solution in Fig. 1) is of the second order due to the softening of a phonon mode inside the Brillouin zone. If the softening occurs, for example, at the point κ = 41 +ε, where ε is small, the incommensurate phase has the form of a modulated four-periodic structure. The stability region of the structure with κ = 41 + ε mostly overlaps with that of the four-periodic commensurate phase shown in Fig. 1. Close to the transition from the high-symmetry commensurate to the incommensurate phase, the modulation has a sinusoidal form. Note that at this stage the modulated phase has negative energy with respect to the low-symmetry commensurate (ideal four-periodic) structure. Further cooling leads to the transformation of the sinusoidal modulation to the so-called DW regime with periodically arranged DWs separating the domains of commensurate four-periodic phase. Therefore, depending on the parameters of the system, DWs can have either positive or negative energy [10]. As a rule, on further cooling, one more phase transition, called the lock-in transition, takes place. As a result of this first-order transition the DWs disappear from the crystal. The behavior of crystal on heating differs from that on cooling. On heating, the phase transition from the low-symmetry commensurate (let us say, four-periodic) phase to an incommensurate phase can take place. Recently we have proposed a possible mechanism of this transition [16], where the existence of at least one DW in the crystal is postulated. In some region of the external parameters, the DW can have negative energy. In this region the DW plays a role of nucleus of a lower energy phase in a metastable phase. Such a DW can initialize a transition to the lower energy phase with an interesting kinetics. The negative-energy DW splits into two autowaves moving in opposite directions and transforming the metastable phase to an incommensurate phase with lower energy [10,16]. 9. Conclusion In the frame of the CAA, continuum analogs to the potential and kinetic energies of the 4l-periodic structure modulated by spatially slowly varying phase ϕ(x, t) were given. It was shown that the equation of motion for the unknown function ϕ(x, t) is the sine-Gordon equation. The kink solution of the sine-Gordon equation was employed in order to write the DW solution for any 4l-periodic commensurate structure. The solution was specified for the 4-, 8- and 12-periodic commensurate structures. The expressions of the energy of the commensurate structure, the width of DW, the kinetic and potential energies of DW were given in terms of amplitudes of Fourier harmonics Aj of the carrying commensurate structure. By means of Ishibashi transformation, the relation between DW solutions in different commensurate structures was established. The obtained solutions were tested numerically. The DW solution for four-periodic structure obtained within the CAA was compared with the solution obtained by another method [10] (50). The CAA gives a better result in the region where the DW has negative energy, while the solution (50) is very accurate for the DW with positive energy. The CAA assumes only one unknown function and that is why it cannot reproduce the concave shape of the DW, whereas the solution (50), which assumes two unknown functions, can reproduce it. Acknowledgements The authors would like to thank Dr. Hlinka of Nagoya University for the fruitful discussions. One of us, SVD, wishes to thank the Ministry of Education, Science, Sports and Culture of Japan for their financial support. References [1] R. Blinc, A.P. Levanyuk (Eds.), Incommensurate Phases in Dielectrics, Vols. 14.1 and 14.2, North-Holland, Amsterdam, 1986. [2] K. Hamano, K. Abe, T. Mitsui, J. Phys. Soc. Jpn. 67 (1998) 1037.

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