Soft-mode transition

Whereas the first microscopic theory of BaTiOs [1,2] was based on order-disorder behavior, later on BaTiOs was considered as a classical example of displacive soft-mode transitions [3,4] which can be described by anharmonic lattice dynamics [5] (Fig. 1). BaTiOs shows three transitions at around 408 K it undergoes a paraelectric to ferroelectric transition from the cubic Pm3m to the tetragonal P4mm structure at 278 K it becomes orthorhombic, C2mm and at 183 K a transition into the rhombohedral low-temperature Rm3 phase occurs. 

Sodium iodide, values of properties. See Alkali halides Sodium sulphide, selenide, and telluride. See Fluorites Soft-mode transition, 210... 

This also pertains to the phase transformation in a solid that is the result of a gradual displacement of atoms from their original position in a soft mode transition. By way of illustration, we adopt a one-dimensional model and suppose that the gradual shift in atomic location may be correlated with anharmonic terms in the lattice vibration of the participating atoms. This is quantified via the relation... 

V = V2X — V4X = IV2 — kgTV4/V2]x, from which one deduces that the vibration frequency varies as [V2 — feBTV4/V2], which is of the form v i o(l that agrees with the formulation (7.5.5). Surprisingly, soft-mode transitions do indeed conform to this very simple model. 

Some materials undergo transitions from one crystal structure to another as a function of temperature and pressure. Sets of Raman spectra, collected at various temperatures or pressures through the transition often provide useftil information on the mechanism of the phase change first or second order, order/disorder, soft mode, etc. 

Condition (2) is also quite common. For instance, in crystals it results in a reduced sound velocity, v q) when q approaches a boundary of the Brillouin zone [93,96], a direct result of the periodicity of a crystal lattice. In addition, interaction between modes can lead to creation of soft mode with qi O and corresponding structural transitions [97,98]. The importance of nonlocality at fluid interfaces and the corresponding softening of surface modes has been demonstrated recently, both theoretically [99] and experimentally [100]. 

Also known for some time is a phase transition at low temperature (111K), observed in studies with various methods (NQR, elasticity measurement by ultrasound, Raman spectrometry) 112 temperature-dependent neutron diffraction showed the phase transition to be caused by an antiphase rotation of adjacent anions around the threefold axis ([111] in the cubic cell) and to lower the symmetry from cubic to rhombohedral (Ric). As shown by inelastic neutron scattering, this phase transition is driven by a low-frequency rotatory soft mode (0.288 THz 9.61 cm / 298 K) 113 a more recent NQR study revealed a small hysteresis and hence first-order character of this transition.114 This rhombohedral structure is adopted by Rb2Hg(CN)4 already at room temperature (rav(Hg—C) 218.6, rav(C—N) 114.0 pm for two independent cyano groups), and the analogous phase transition to the cubic structure occurs at 398 K.115... 

Although this classic picture evolved from "soft, mononuclear transition metal complexes suffices to explain a great deal of carbon monoxide chemistry, it is not clear that it is complete or accurate for understanding processes whereby CO is reduced, deoxygenated, and/or polymerized to form methane, long-chain hydrocarbons, alcohols, and other oxocarbons, especially in cases where heterogeneous catalysts or "hard" metals are involved (6, 7, ,9,J 0). This deficiency of information has led to the search for new modes of carbon monoxide reactivity and to attempts to understand carbon monoxide chemistry in nontraditional environments ... 

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]. 

Figure 6 shows the influence of the pressure of e T) on both ST018-92(a) and SCT(0.007) [21]. We first note the large shift in the transition to lower temperature for STO 18. The initial slope is dTcdP = - 20 K/kbar, a large effect. Second, there is a large decrease in the ampHtude of the peak with pressure. At 0.70 kbar, the transition is completely suppressed, and the e (T) response closely resembles that of STO 16 at 1 bar. These pressure effects are characteristic of displacive ferroelectrics in the quantum regime and can be understood in terms of the soft-mode theory. The situation is similar for SCT(0.007), as shown in Fig. 6b. In the case of SCT(0.007), ferroelectricity completely disappears at 0.5 kbar.

It has been widely recognized that the Ught scattering technique yields essential information on a dynamic mechanism of ferroelectric phase transition because it clearly resolves the dynamics of the ferroelectric soft mode that drives the phase transition. Quantum paraelectricity is caused by the non-freezing of the soft mode. Therefore, the isotope-exchange effect on the soft mode is the key to elucidating the scenario of isotopically induced ferroelectricity. 

Measurements of NMR for Ti, Ti [33], and Sr [34,35] were carried out for STO 16 and STO 18-96. Ti and Sr nuclear magnetic resonance spectra provide direct evidence for Ti disorder even in the cubic phase and show that the ferroelectric transition at Tc = 25 K occurs in two steps. Below 70 K, rhomb ohedral polar clusters are formed in the tetragonal matrix. These clusters subsequently grow in concentration, freeze out, and percolate, leading to an inhomogeneous ferroelectric state below Tc. This shows that the elusive ferroelectric transition in STO 18 is indeed connected with local symmetry lowering and impHes the existence of an order-disorder component in addition to the displacive soft mode [33-35]. Rhombohedral clusters, Ti disorder, and a two-component state are found in the so-called quantum paraelectric... 

Keywords Order-disorder transitions Soft mode Phase segregation Nucleation Takagi defects... 

It should be noted that the theory described above is strictly vahd only close to Tc for an ideal crystal of infinite size, with translational invariance over the whole volume. Real crystals can only approach this behaviour to a certain extent. Flere the crystal quality plays an essential role. Furthermore, the coupling of the order parameter to the macroscopic strain often leads to a positive feedback, which makes the transition discontinuous. In fact, from NMR investigations there is not a single example of a second order phase transition known where the soft mode really has reached zero frequency at Tc. The reason for this might also be technical It is extremely difficult to achieve a zero temperature gradient throughout the sample, especially close to a phase transition where the transition enthalpy requires a heat flow that can only occur when the temperature gradient is different from zero. 

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. 

If, however, the transition is of a pure displacive nature, the fluctuation amplitude of the order parameter is critical and is by no means temperature-independent. Since the soft mode is an under-damped lattice vibration (at least outside the close vicinity of Tc), defined by its frequency a>s and damping constant Tj, the spectral density is a Lorentzian centred at s and the... 

Except for coi (transition frequencies of the nuclear spin Hamiltonian) all values are temperature-dependent. From the previous subsection the behaviour of a>s is known. From the anomalous contribution to the birefringence which is proportional to (Sp ) we get the information concerning Ai. If we assume that the damping of the soft mode is non-critical (which is generally accepted), Eq. 10 describes a transition from an under-damped mode to an over-damped one as Tc is approached from either side. 

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]. 

They can serve therefore as a test for Ti dispersion. In Fig. 12 the relaxation results are shown for D-RADP-15. The solid lines are a fit of the theory [19] to the data. Above Tc the lit is excellent, whereas below Tc it probably suffers from the fact that the phase transition is already diffuse and only nearly of second order. This proves that a soft mode component is needed to explain the data. Furthermore, the fact that the ratio ti/t2 remains unchanged above and below Tc proves that the order parameter fluctuations are in the fast motion regime on both sides of the transition. 

Another feature that is still puzzling scientists is the role of the soft mode in structural order-disorder transitions. As already mentioned in Sect. 3 the energy U in theoretical treatments is based predominantly on short-range interactions and often the local fields are replaced by a mean field to mimic some long-range properties. In the solid solution D-RADP-x the FE soft mode suffers from the lack of translational invariance, so that only for x < 0.32 sufficient coherence can be achieved to provide a successful contribution to... 

We have demonstrated in this contribution that the structural order-disorder transitions in the KDP family are by far more complex than a pseudo-spin model, for example, could describe. Geometrical constraints and couphng to the soft mode, as well as cluster formation of nanometric size, indicate that here the conventional picture of cooperative phenomena has to be revised. 

---