Soft order parameters

Soft order parameters characterize flexible societies open to an exchange of ideas and offering a large range of intellectual and material options to their members. However, a society with order parameters that are too soft may degenerate into an orientationless or even chaotic state. 

A basic concept in the reconstruction theory of solid surfaces is the soft phonon approach of displacive structural transitions. An essential property of these structural phase transitions is the existence of an order parameter which... 

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. 

The coupling to the soft mode introduces here oscillatory terms that can become important close to Tc. The fact that the amplitude, as well as the coherence length of the order parameter fluctuations, increase on approaching Tc from either side also probably brings higher order coupling terms into... 

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... 

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. 

The fact that the order parameter vanishes above does not mean that Nature does not have an inkling of things to come well below (or above) T. Such indicators are indeed found in many instances in terms of the behaviour of certain vibrational modes. As early as 1940, Raman and Nedungadi discovered that the a-) transition of quartz was accompanied by a decrease in the frequency of a totally symmetric optic mode as the temperature approached the phase transition temperature from below. Historically, this is the first observation of a soft mode. Operationally, a soft mode is a collective excitation whose frequency decreases anomalously as the transition point is reached. In Fig. 4.4, we show the temperature dependence of the soft-mode frequency. While in a second-order transition the soft-mode frequency goes to zero at T, in a first-order transition the change of phase occurs before the mode frequency is able to go to zero. 

We can understand the relation between soft modes and order parameters in phase... 

Schmahl WW, Swainson IP, Dove MT, Graeme-Barber A (1992) Landau free energy and order parameter behavior of the a-p phase transition in ciistobalite. Z Kristallogr 201 125-145 Sollich P, Heine V, Dove MT (1994) The Ginzburg interval in soft mode phase transitions Consequences of the Rigid Unit Mode picture. JPhys Condensed Matter6 3171-3196 Strauch D, Domer B (1993) Lattice dynamics of a-quartz. 1. Experiment. J Phys Condensed Matter 5 6149-6154... 

Other structural effect, such as a soft optic mode, which provides the driving mechanism. Lattice distortions then occur by coupling of the spontaneous strain to the order parameter. Under these conditions, the general form of Equation (4) may be given as... 

Some insight into the mechanism by which Pc is changed is provided by the frequency of the soft mode responsible for the phase transition. In the stability field of the high symmetry (tetragonal) phase, the inverse order parameter susceptibility, x of the order parameter varies as... 

More recently, Lacks and Weinhoff [265] have examined the mechanical stability of soft spheres as a function of size polydispersity. In their study they minimize the potential energy of an fee array of polydisperse spheres, and examine how the structure of the minimum-energy configuration changes with increasing polydispersity. An appropriate structural order parameter for the fee solid is found to drop precipitously for polydispersities in the range of 10% to 15%. The absence of energy minima in the ordered phase indicates that such a system is not mechanically stable, and therefore cannot be thermodynamically stable. 

Phase transitions in which the square of the soft-mode frequency or its related microscopic order parameter goes to zero continuously with temperature can be defined as second order within the framework of the Ginzburg-Landau model [110]. The behavior is obviously classical and consistent with mean field... 

Fig. 8.5. Profile of the lowest order parameter mode (thick line) and of one of the highly excited modes for a LC heterophase system in contact with (a) ordering and (b) disordering substrates in the proximity of the phase transition. Dashed line denotes the spatial variation of the mean-field scsdar order parameter. In all cases T — Tjvj- Inset The corresponding spectrum of the relaxation rates chciracterized by a soft lowest order parameter mode.

The bead dynamics is realized by the integration of the equations of motion for the beads. A trajectory is generated through the system s phase space. All thermodynamic observables (e.g. density fields, order parameters, correlation functions, stress tensor, etc.) can be constructed from suitable averages. An immense advantage over conventional molecular dynamics and Brownian dynamics is that all forces are soft , thus allowing... 

Ha)nvard et al. (2005) performed a comprehensive investigation of LaAlOs in the temperature range of 10-750 K and determined the crystal structure, dielectric relaxation, specific heat, birefringence, and the frequencies of the two soft modes via Raman spectroscopy. While all these experiments show that the behaviour at the critical point around Tc = 813 K is consistent with a second-order transition, some evidence for an additional anomalous behaviour below 730 K have been shown. This anomaly was explained by a biquadratic coupling between the primary order parameter of the transition and the hopping of intrinsic oxygen vacancies. 

The dynamics of the electroclinic effect is, in fact, the dynamics of the elastic soft mode. From Eqs. (13.18) and (13.19) follows that the switching time of the effect is defined only by viscosity and the term a(T — T ) and is independent of any characteristic size of the cell or material. It means that the relaxation of the order parameter amplitude is not of the hydrodynamic type controlled by term Kq (K is elastic coefficient). For the same reason Xg is independent of the electric field in agreement with the experimental data, shown in Fig. 13.9b. At present, the electroclinic effect is the fastest one among the other electro-optical effects in liquid crystals. 

Coexistence of FM order and superconductivity under pressure The experimental phase diagram of FM collapse under pressure and simultaneous appearance of superconductivity is shown in fig. 43. The critical pressure for disappearance of FM order is pc2 = 16-17 kbar. The SC phase appears between pc = 10 kbar and pc2 = 16 kbar which is also the critical pressure for the FM-PM transition. The critical temperature Tx p) of the jc-phase hits the maximum of Tdp) at the optimum pressure pm = 12.5 kbar. As mentioned before the nature of the order parameter in the jc-phase remains elusive. The coincidence of maximum Tc with vanishing jc-phase order parameter suggests that the collective bosonic excitations of the X-phase which supposedly become soft at pm mediate superconductivity and not quantum critical FM spin fluctuations which are absent due to the persisting large FM... 

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