Solitons motion

The concept of the soliton has been introduced in the theoretical treatment by Su et al. (1979, 1980), in which they have employed a model Hamiltonian within the framework of the Hiickel approximation, including both cr-bond compressibility and the kinetic energy term of the CH units. The soliton is an elementary excitation and, in the case of the transient in Fig. 12a, is expected to satisfy a wave equation akin to the if/4 field theory (Krumhansl and Schrieffer, 1975). The estimated energy of the creation of a soliton is 0.4 eV and the periodic-lattice-induced activation energy for the soliton motion is 0.002 eV, the latter being consistent with the result of the ESR observation. The energy of a soliton is most stabilized when its tail (that is, the spatial halfwidth of the kink) extends over seven carbon sites. 

Carter [18] proposed that soliton motion in polyacetylenes could be harnessed in various switches, gates, and logic circuits. However, apart from the difficulties of synthesizing or addressing such molecules, soliton switches would be relatively slow. 

Nechtschein et al. [118] have investigated the frequency dependence of the relaxation of the proton NMR signal in fran -polyacetylene. They assume that the proton spin is relaxed by collisions with solitons moving along the polymer chain. From the observed dependence of the relaxation time they conclude that the soliton motion is one-dimensional with an anisotropy of at least /5 j. > 4 x 10. ... 

The essential element of the formalism is a mechanism of energy loss of solitons. The traveling ID soliton (single file in Figure 2.33c) transfers energy to the SG sublattice. Soliton motion creates a perturbation of the SG sublattice, which emanates along... 

Figure 5b). Each of them interrupts the phase of the lattice distortion. Each of these two particle-like excitations can move along the distorted lattice without changing the total energy. The conductivity of undoped or lightly doped polyacetylenes is predominantly due to soliton motions.

For charge transport along chains with conjugated double bonds the mechanism of soliton motion has been proposed (this would be a theory for R-j in the terminology of the previous section). If the conjugation defect of Fig. h is called soliton , this term is used to stress two features of this defect it does not disperse while it moves (just as solitary water waves do not disperse. This is where the name comes from) and it has certain symmetry properties. If most polymer chains in today s polyacetylene films are very short, solitons will not be able to move very far since they are confined to the polyene chains. Therefore the non-dispersivity will be hard to test. In addition solitons will not be important for electrical conductivity under these circumstances. 

In a non-exhaustive literature search, a brief account is given here on the study of phonon and its vibrations. Corso et al. did an extensive study of density functional perturbation theory for lattice dynamics calculations in a variety of materials including ferroelectrics [93]. They employed a nonlinear approach to mainly evaluate the exchange and correlation energy, which were related to the non-linear optical susceptibility of a material at low frequency [94], The phonon dispersion relation of ferroelectrics was also studied extensively by Ghosez et al. [95, 96] these data were, however, related more with the structure and metal-oxygen bonds rather than domain vibrations or soliton motion. In a very interesting work, a second peak in the Raman spectra was interpreted by Cohen and Ruvalds [97] as evidence for the existence of bound state of the two phonon system and the repulsive anharmonic phonon-phonon interaction which splits the bound state off the phonon continuum was estimated for diamond. 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

A soliton is a solitary wave that preserves its shape and speed in a collision with another solitary wave [12,13]. Soliton solutions to differential equations require complete integrability and integrable systems conserve geometric features related to symmetry. Unlike the equations of motion for conventional Maxwell theory, which are solutions of U(l) symmetry systems, solitons are solutions of SU(2) symmetry systems. These notions of group symmetry are more fundamental than differential equation descriptions. Therefore, although a complete exposition is beyond the scope of the present review, we develop some basic concepts in order to place differential equation descriptions within the context of group theory. 

With the connection of PDEs, and especially soliton forms, to group symmetries established, one can conclude that if the Maxwell equation of motion that includes electric and magnetic conductivity is in soliton (SGE) form, the group symmetry of the Maxwell field is SU(2). Furthermore, because solitons define Hamiltonian flows, their energy conservation is due to their symplectic structure. 

To create a soliton, a finite energy Es is needed, which is greater than the barrier height V0. Therefore, when /3V0>1, the appearance of a soliton is a rare event which can be neglected when analyzing the motion of the chain. 

Aside from solitons, there is a continuous spectrum of torsional modes, or rotons. These excitations are the eigenstates of the linearized Hamiltonian. To obtain their spectrum, one replaces the last term in (7.68) with a harmonic potential. This approximation implies that the vibrational amplitude of a rotor must be small enough compared with the large amplitude motion of a rotor participating in a soliton. The frequencies of rotons obey the dispersion equation ... 

Such an effect is not due to a trivial Joule heating of the samples. From a detailed experimental study based mainly on electrical [68], optical [70], and x-ray [71] measurements, this effect has been attributed to the motion of charged soliton-like defects existing in the TCNQ chains. A soliton in a dimerized chain is expected to have the following general form [47] ... 

After the authors of Ref. 47, the motion of charged solitons in dimerized chains requires a relatively low excitation energy, and it could take place according to the following scenario. At low electric field, this motion is hindered by the three-dimensional interactions between molecular chains. The current is then due to a few defects more or less free from these interactions. As the field is increased, more and more charged solitons are driven to motion and by a cooperative effect of these mobile solitons, a critical field th is finally attained at which value the three-dimensional order is lost. The solitons are then able to move in one chain almost independent of the others, and this new degree of freedom is believed to... 

Furthermore, quantitative characterizations of the spin motion in trans-(CH) have been performed by measurements of the proton NMR relaxation time 7 and analysis of the ESR line width [70,71]. The spin motion can be described in terms of highly one-dimensional diffusion. The diffusion rate along the chains is very fast D[ 1013 rad/s (i.e., a diffusion coefficient of ca. 5 x 10 3 cm2/s), and the anisotropy is extremely high Z>j /Z>x > 105. The very high anisotropy is also an argument for the soliton picture neutral... 

Earlier than all of this Holstein (10) described his optic polaron, in which the deformation of the ID chain is an optic deformation. His was the first polaron solution which was a Solitary Wave or Soliton. In such a deformation there is no change in lattice density in the polaron there is only a rearrangement of atoms without a change of density. In the case of the optic polaron there is no analytical solution for the moving polaron. However it is clear that on increase of the optic polaron energy due to motion there is no perturbation as the velocity goes through the sound velocity. In the pure optic polaron the sound velocity is not in the model at the outset. It is in the motion of the polaron at velocities up to the sound velocity that the profound difference between the acoustic and optic polarons occurs the difference in properties of the polarons at rest is leas Important. 

The stress we lay on the difference between optic and acoustic deformation is a reaction to the enormous literature on exclusively optic polarons in conjugated polymers. Yet as the Davydov Soliton and SWAP show, it is the acoustic deformation which dominates the motion of the polaron. In so far as it is motion which is the essential quality of a Soliton, it is the Acoustic deformation which it is essential to consider. In reality... 

In the continuum model, the motion of a kink occurs without energy barriers if Q is irrational. For rational fl the distribution of ground state must be discrete, which makes it impossible to transform the phases continuously without extra energy. The dynamic solutions of Eq. (33) exploit the isomorphism with nonlinear relativistic wave equations [107,108] and a moving kink (soliton) can be interpreted as an elementary excitation with energy k(u)... 

More than two decades ago Pople and Walmsley (1962) discussed the existence of an unpaired ir electron in the trans-(CH)x as shown in Fig. 12a, which accompanies a localized nonbonding MO level. This has been actually confirmed by ESR analyses (Shirakawa et al., 1978) and ENDOR (Kuroda and Shirakawa, 1982) observations. The lineshape of the ESR shows the motional narrowing from a highly mobile tt electron in the (CH), chain even down to 10 K (Goldberg et al., 1979 Weinberger et al., 1980). It has been concluded that there is one unpaired electron per approximately 3000 carbon atoms. This unpaired tt electron in the trans-(CH), chain is frequently referred to as the bond alternation domain wall, phase kink, or soliton. 

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