Elastic external fields

Dipolar ions like CN and OH can be incorporated into solids like NaCl and KCl. Several small dopant ions like Cu and Li ions get stabilized in off-centre positions (slightly away from the lattice positions) in host lattices like KCl, giving rise to dipoles. These dipoles, which are present in the field of the crystal potential, are both polarizable and orientable in an external field, hence the name paraelectric impurities. Molecular ions like SJ, SeJ, Nf and O J can also be incorporated into alkali halides. Their optical spectra and relaxation behaviour are of diagnostic value in studying the host lattices. These impurities are characterized by an electric dipole vector and an elastic dipole tensor. The dipole moments and the orientation direction of a variety of paraelectric impurities have been studied in recent years. The reorientation movements may be classical or involve quantum-mechanical tunnelling. 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

In many cases of transport in solids, the atoms (ions) of one sublattice of the crystal are (almost) immobile. Here, we can identify the crystal lattice with the external (laboratory) frame and define the fluxes relative, to this immobile sublattice (to = 0). v° is bk-Xk (Eqn. (4.51)) where Xk is the sum of all local forces which can be applied externally (eg., an electric field), or which may stem from fields induced by the, (Fickian) diffusion process itself (eg., self-stresses). An example of such a diffusion process that leads to internal forces is the chemical interdiffusion of A-B. If the lattice parameter of the solid solution changes noticeably with concentration, an elastic stress field builds up and acts upon the diffusing particles, it depends not only on the concentration distribution, but on the geometry of the bounding crystal surfaces as well. 

According to the formula (7) absorption spectrum for conductivity electrons in bulk metal should be a smooth curve down to co->0. Ag film in the near UV range demonstrates spectrum of such type. Appearance of a near UV absorption peak in a spectrum of M nanocrystal is caused by the surface charges that resulted from displacement of conductivity electrons under action of an external field. These charges create in a nanocrystal the internal field directed against external one [16]. For conductivity electrons this internal field plays a role of quasi-elastic bonds between valent electrons and cations in a crystal lattice. 

Let ub take a cylinder filled with gas and a piston holding the gas in the cylinder, and let us consider the action of some continuous field of force on this Bystem. As long ob the piston is not moved and the external field of force is unchanged, the above condition is fulfilled. However, if we move the piston, for example (considering the action of the piston as a field of elastic forces), the condition is violated in this case we act on the gas, inasmuch as we change in time the existing external field of force. Cf. Section 23b. 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

As shown in Sect. 3.2, elastic materials with tailor-made anisotropy can be prepared under an external field. Jolly et al. have found 0.6 MPa maximum increases in the shear modulus for iron-loaded elastomer [17]. In this work, the shear elastic behavior perpendicular to the columnar structure has been investigated. To the best of our knowledge, no information is available concerning other experimental situations. Depending on the direction of the... 

On the basis of Eqs. 12 and 13, the elastic modulus measured under uniform external field can be expressed as the sum of two contributions ... 

The elasticity theory has been used in dealing for example with the response of liquid crystals to external fields (electric, magnetic, mechanical force), defects, etc. 

Sanfeld et al. (1990) have theoretically analysed the competition between chemical reactions at the surface and in the volume. They define a surface elasticity which is determined by the kinetics of all these processes and conclude that the effect of capillary forces may be considered as an external field acting on the reactive system. Their conception started from the principle of De Donder, as did our general description of relaxation in chemical reactions. To generalise relaxation phenomena at interfaces can be described as ordinary chemical reactions. In principle there is no distinction between the application of the laws to chemical kinetics in bulk phases and at interfaces. 

For an external field pardlel to tlK helicoidal axis, the cholesteric pitch varies depending on the relative magnitudes of 22 and 33 (the bending elastic constant ) 53). 

The thermodynamical equilibrium of nematics would correspond to a spatially uniform (constant n(r)) director orientation. External influences, like boundaries or external fields, often lead to spatial distortions of the director field. This results in an elastic increment, fd, of the volume/ree energy density which is quadratic in the director gradients [2, 3] ... 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

Since the flexoelectric effect is associated with curvature distortions of the director field it seems natural to expect that the splay and bend elastic constants themselves may have contributions from flexoelectricity. The shape polarity of the molecules invoked by Meyer will have a direct mechanical influence independently of flexoelectricity and can be expected to lower the relevant elastic constants.The flexoelectric polarization will generate an electrostatic self-energy and hence make an independent contribution to the elastic constants. In the absence of any external field, the electric displacement D = 0 and the flexoelectric polarization generates an internal field E = —P/eo, where eq is the vacuum dielectric constant. Considering only a director deformation confined to a plane, and described by a polar angle 9 z), and in the absence of ionic screening, the energy density due to a splay-bend deformation reads as ... 

All above means that the ferroics can be regarded as a general notation for the materials, where at T < (so-called low-temperature phase) some reorientable physical quantities (order parameters) spontaneously appear. Latter order parameters can be of vector (spontaneous electric polarization, spontaneous magnetization) or tensor (second order tensor like spontaneous deformation or higher order tensors like elastic moduli and piezoelectric coefficients) nature. In low-temperature phase ferroics can usually split into domains, their switching being possible by the external fields. 

It has already been indicated in the Sect. 1.1 that the primary, secondary and higher-order ferroics are defined by the number of external fields necessary to switch the ferroic from one orientational state to another. In particular, the primary ferroics can be switched under the application of one kind of physical fields (magnetic, electric, elastic). Besides three mentioned types of primary ferroics, there are six more types of secondary ones, where the difference between orientational (domain) states thermodynamic potential (free energy) Ag is proportional either to square or to the product of external fields as it is shown in the Table 1.1. 

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