Order parameter reduction

With increasing values of P the molar volume is in progressively better agreement with the experimental values. Upon heating a phase transition takes place from the a phase to an orientationally disordered fee phase at the transition temperature where we find a jump in the molar volume (Fig. 6), the molecular energy, and in the order parameter. The transition temperature of our previous classical Monte Carlo study [290,291] is T = 42.5( 0.3) K, with increasing P, T is shifted to smaller values, and in the quantum limit we obtain = 38( 0.5) K, which represents a reduction of about 11% with respect to the classical value. 

Although transition across a critical point may proceed without any first-order discontinuity, the fact that there is a change of symmetry implies that the two phases must be described by different functions of the thermodynamic variables, which cannot be continued analytically across the critical point. The order parameter serves to compensate for the reduction in symmetry. Although it is a regular function of temperature it can develop a discontinuous derivative at the critical temperature. Likewise, several measurable... 

The (generally complex) quantity p plays a role analogous to the order parameter familiar from phase transitions. The fact that only one such parameter survives in the final equations illustrates the enormous reduction of degrees of freedom associated with the first bifurcation. Note also the similarity between equations (8) and (9) and the normal forms at which one arrives in the qualitative theory of differential equations in the vicinity of resonance points.3... 

In the present cases, in contrast, when the order parameter increases, the ESR lineshapes are narrowed, because the relaxation processes are less effective. Thus, narrowing in turn involves a reduction in the number of ex-dt states necessary to obtain convergence and thoeby a faster attainntont of the result. 

FIG. 2.12 Addition of a hydrotrope (that of Figure 2.9) to a lamellar hquid crystal gives a reduction of the order parameter of the surfactant hydrocarbon chain (o) addition of a surfactant gives no change in order ( ). 

Comparing the evaluated D values for the crystalline and LC phases, a reduction of D by a factor of 0.40 + 0.04 is observed, corresponding to an order parameter [122] of 0.80 + 0.08. This could be explained by postulating the presence of out-of-plane motion in addition to the axial rotation of the molecule about an axis perpendicular to the ring. The good agreement with the value of 0.84 obtained from an analysis of ID NMR lineshapes is to be noted [123]. It is to be emphasised, however, that the DQ MAS method is applicable to as-synthesised samples, i. e., there is no reliance upon isotopic labelling. 

Fig. 28. Reduction of the squared AF moment in UPt3 with hydrostatic pressure p (above), it vanishes at a critical pressure of — 3.8 kbar. Simultaneously the SC Tq splitting is reduced and also vanishes at pc staying zero for larger pressures (Hayden et al., 1992). This supports the idea that a two-fold degenerate SC order parameter (e.g., E2m) is split by an AF symmetry breaking field according to eqs. (69), (70).

Formally, a is the electron-phonon coupling parameter defined by eqn (2.38), but it is often convenient to regard it as a semiempirical parameter. Notice that a positive value of A corresponds to a reduction in the bond length, and vice versa. It is this term in that couples the electrons to the lattice, and corresponds to eqn (2.40) with / = 0. A plays the role of an order parameter, whereby a nonzero value indicates a broken symmetry. 

Muthukumar projmsed the most general version of the theory of systems P- -F,MWL in which the Gibbs mixing potential is given with due account of fluctuations in the order parameter (the concentration of polymer segments) (Eiquation 5.6-36). This approach is based on the step-by-step application and the development of the Kdwards formalism. A peculiar feature of this approach is the reduction of the three-body interactions to the effective two-body ones. 

Another issue to be investigated is the excitation spectrum inside the vortex core, where the penetration of the magnetic field leads to local breakage of the time-reversal symmetry, resulting in local reduction of the order parameter. STM is a suitable tool to detect the spatial variation of the excitation spectrum inside the core. 

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