Soft mode concept

The atoms of a crystal vibrate around their equilibrium position at finite temperatures. There are lattice waves propagating with certain wavelengths and frequencies through the crystal [7], The characteristic wave vector q can be reduced to the first Brillouin zone of the reciprocal lattice, 0 q 7t/a, when a is the lattice constant. 

In general, each mode of the phonon dispersion spectra is collectively characterized by the relating energy, i.e. the frequency and wave vector k, and is associated with a specific distortion of the structure. 

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The soft mode concept can be extended to all distortive phase transitions (transitions with relatively small atomic displacements), even if they are only close to second order. In the case of a ferro-distortive transition, as for example in BaTiOs or KDP, the order parameter is proportional to the spontaneous electric polarization Fj. d F/ dp is not only proportional to co, but also to the dielectric susceptibility. This does not, however, mean that all components of the order parameter eigenvector must contribute to Ps. 

The position of Ti and Zr is again important in this context. While the b.c.c. phase in these elements has long been known to indicate mechanical instability at 0 K, detailed calculations for Ti (Petty 1991) and Zr (Ho and Harmon 1990) show tiiat it is stabilised at high temperatures by additional entropy contributions arising from low values of the elastic constants (soft modes) in specific crystal directions. This concept had already been raised in a qualitative way by Zener (1967), but the... 

The concept of quantum ferroelectricity was first proposed by Schneider and coworkers [1,2] and Opperman and Thomas [3]. Shortly thereafter, quantum paraelectricity was confirmed by researchers in Switzerland [4], The real part of the dielectric susceptibihty of KTO and STO, which are known as incipient ferroelectric compounds, increases when temperature decreases and becomes saturated at low temperature. Both of these materials are known to have ferroelectric soft modes. However, the ferroelectric phase transition is impeded due to the lattice s zero point vibration. These materials are therefore called quantum paraelectrics, or quantum ferroelectrics if quantum paraelectrics are turned into ferroelectrics by an external field or elemental substitution. It is well known that commercially available single crystal contains many defects, which can include a dipolar center in the crystal. These dipolar entities can play a certain role in STO. The polar nanoregion (PNR originally called the polar microregion) may originate from the coupling of the dipolar entities with the lattice [5-7]. When STO is uniaxially pressed, it turns into ferroelectrics [7]. 

In the presence of both order-disorder and displacive, as in the KDP family, the two dynamic concepts have somehow to be merged. It could well be that the damping constant Zs becomes somewhat critical too (at least in the over-damped regime of the soft mode), because of the bihnear coupling of r/ and p. It would, however, lead too far to discuss this here in more detail. The corresponding theory of NMR spin-lattice relaxation for the phase transitions in the KDP family has been worked out by Blinc et al. [19]. Calculation of the spectral density is here based on a collective coordinate representation of the hydrogen bond fluctuations connected with a soft lattice mode. Excellent and comprehensive reviews of the theoretical concepts, as well as of the experimental verifications can be found in [20,21]. 

Advanced experimental methods (e.g. inelastic neutron scattering and hyper-Raman scattering) have been applied effectively to studies of ferroelectrics, and several new concepts (e.g. soft modes of lattice vibrations and the dipole glass) have been introduced to understand the nature of ferroelectrics. Ferroelectric crystals have been widely used in capacitors and piezoelectric devices. Steady developments in crystal growth and in the preparation of ceramics and ceramic thin... 

Taking into account the fact that the solvation of ambident anions in the activated complex may differ considerably from that of the free anion, another explanation for the solvent effect on orientation, based on the concept of hard and soft acids and bases (HSAB) [275] (see also Section 3.3.2), seems preferable [366]. In ambident anions, the less electronegative and more polarizable donor atom is usually the softer base, whereas the more electronegative atom is a hard Lewis base. Thus, in enolate ions, the oxygen atom is hard and the carbon atom is soft, in the thiocyanate ion the nitrogen atom is hard and the sulfur atom is soft, etc. The mode of reaction can be predicted from the hardness or softness of the electrophile. In protic solvents, the two nucleophilic sites in the ambident anion must interact with two electrophiles, the protic solvent and the substrate RX, of which the protic solvent is a hard and RX a soft acid. Therefore, in protic solvents it is to be expected that the softer of the two nucleophilic atoms (C versus O, N versus O, S versus N) should react with the softer acid RX. 

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