Elastic lattice distortion

The third contribution to the chemical potential is due to strain. If A and B atoms (ions) have different size, clustering results in elastic lattice distortions. By making a Fourier transformation, one can decompose the concentration profile into harmonic plane waves [D. DeFontaine (1975)]. The elastic energy contributions of these concentration waves are additive in the Unear elastic regime and yield Ea. Therefore, we may write... 

Developed by Englman and Halperin [4], the above steps represent the traditional way in the cooperative JT problem. It s conceptual advantage and important part is solving the respective one-cell JT problem (9) including the low-symmetry mean field (10) of all other distorted cells. In this way, both effects, the dynamic strengthening chemical bonds with low-symmetry lattice distortions and the intercell elastic coupling, are included. 

Because the energy state of a Jahn-Teller complex depends on the local lattice distortions, the macroscopic long-distant strain that produces an ultrasonic wave should influence it as well. The cross effect is initiated by the Jahn-Teller complexes (1) the dispersion (i.e., frequency-dependent variation of phase velocity) and (2) attenuation of the wave. In terms of the elastic moduli it sounds as appearance (or account) of the Jahn-Teller contribution to the real and imaginary parts of the elastic moduli. For a small-amplitude wave it is a summand Ac. Obviously, interaction between the Jahn-Teller system and the ultrasonic wave takes place only if the wave, while its propagation in a crystal, produces the lattice distortions corresponding to one of the vibronic modes. 

In the field of nanoscale materials, SIESTA has probably made its largest impact in the study of carbon nanotubes. This is a field which has captivated the attention of researchers for their unusual electronic and mechanical properties. Simulation and theory have played a major role, often providing predictions that have guided the way for experimental studies. Work done with SIESTA has spanned many aspects of nanotube science vibrational properties [239-241], electronic states [242-246] (including the effect of lattice distortions on the electronic states [247-250]), elastic and plastic properties [251-254], and interaction with other atomic and molecular species [255-259]. Boron nitride nanotubes have also received some attention [260, 261]. 

Elastic interaction occurs when the displacement fields from steps substantially superpose. Atoms located in the vicinity of steps tend to relax stronger compared to those farther away. The resulting displacements or lattice distortions decay with increasing distance perpendicular to the steps. Atoms situated in between two steps experience two opposite forces and cannot fully relax to an energetically more favorable position as would be the case with quasiisolated steps. The line dipoles at steps are due to Smoluchowski smoothing [160] and interact electronically. Only dipole components perpendicular to the vicinal surface lead to repulsion whereas parallel components would lead to attractive interaction. The dipole-dipole interaction seems to be weaker than the elastic one. For instance, steps on vicinal Ag(lll) have weak dipoles as was shown in a theoretical study [161]. Entropic interaction is due to the condition that steps may not cross and leads to an effective repulsive potential, the weakest interaction type. This contribution is always present and results from the assumption that cavities under the surface are unstable. Experiments and theory investigating steps on surfaces were recently reviewed [162]. 

The lattice distortion energy, W E ), generated by a solute atom in the bulk can be calculated by using elastic continuum theory as " ... 

For fields which are too small to cause a phase transition, there are two effects. The simplest is to favor a particular orientation of the crystallites without lattice distortion [82]. The second is electrostriction, namely a distortion of the cubic blue phase lattice due to applied fields. As in the previous discussion of elastic measurements, this distortion is fundamentally different from that which occurs in a conventional crystal where the lattice is composed of atoms at specific lattice points. In blue phases, it is the lattice of director orientations which does the distorting the molecules themselves continue to be free to diffuse through this lattice. 

Since the elastic constants are known for GdSb (Mullen et al., 1974) we can estimate the exchange-induced lattice distortion in this case. At the equilibrium distortions the gain in exchange energy is balanced by the deformation energy. 

The presence of spin slips in the magnetic structures leads through the magneto-elastic coupling to an induced modulation of the lattice. Magnetostriction is known to be significant in the lanthanides as expressed, for example, by their anomalous lattice distortions below the Neel point. 

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