Order parameter definition

Figure 2.34 Definition of up/down order parameter R from X-ray powder diffraction patterns of iPP samples170,175 (a) unannealed sample (b) sample annealed at 167°C. (Reprinted with permission from Ref. 170. Copyright 1992 by Springer-Verlag.)... 

The form of the order parameter is given implicitly since by definition T=1 at 0 K and hence... 

Following the concepts on page 53 and 57 we find out a third concept for experimental strategies to change order parameters in the A/S sense regarding structure and reactivity (for definition of A/S see )). This concept fits for degenerated subsystems on the level... 

This definition is analogous to that of the more common spin relaxation generalized order parameter, although the sensitivity to motional timescales is very different. As will be discussed later, a further distinguishing characteristic is that RDC analysis can in principle allow the explicit separation and determination of the direction and extent of the motional asymmetry. 

A more complicated but solvable problem is a definition of the order parameter for antiferromagnetics, binary alloys, superconductors etc. The dimensionless units T/Tc and 77/770 (Fig. 1.4) allow us to present the behaviour of the order parameter rj = r) T) in a form universal for many quite different systems. Moreover, in some cases even quantitative similarities hold which concerns in particular the value of the exponent (3. (The value of 77 = 0 characterizes always disordered phase.)... 

This implies that, in general, temporal ordering parameters cannot be identified directly with physical time—they merely share one essential characteristic. This situation is identical to that encountered in the Lagrangian formulation of general relativity there, the situation is resolved by defining the concept of particle proper time. In the present case, this is not an option because the notion of particle proper time involves the prior definition of a system of observer s clocks—so that some notion of clock time is factored into the prior assumptions on which general relativity is built. 

Order parameters are usually derived from the measured spectral splittings through relationships such as Equation (2.269-b) thus the availability of good estimates of the magnetic tensors is an essential requirement to obtain accurate order parameters. The theoretical results suggest that the magnetic tensors obtained from calculations in a solvent, introduced by the polarizable continuum model, should definitely be a better choice than the tensors derived, as customary, from solid-state data or from calculations for molecules in a vacuum. 

In conclusion, field dependent single-crystal magnetization, specific-heat and neutron diffraction results are presented. They are compared with theoretical calculations based on the use of symmetry analysis and a phenomenological thermodynamic potential. For the description of the incommensurate magnetic structure of copper metaborate we introduced the modified Lifshits invariant for the case of two two-component order parameters. This invariant is the antisymmetric product of the different order parameters and their spatial derivatives. Our theory describes satisfactorily the main features of the behavior of the copper metaborate spin system under applied external magnetic field for the temperature range 2+20 K. The definition of the nature of the low-temperature magnetic state anomalies observed at temperatures near 1.8 K and 1 K requires further consideration. 

In all the other cases, with either a linear or a nonlinear recycling process or with both, the coefficients A and B are no longer zero at the same time, and a definite value of the order parameter

0, B becomes nonzero since /xqi,oo > 0. If the linear recycling exists as X > 0, not all the achiral substrate transform to chiral products but a finite amount remains asymptotically as a(t = oo) > 0. Therefore, nonzero values of ko, k or k2 k 2 give contributions to the coefficients A or B. 

When A is positive, as in the case of Fig. 6c, the coefficient of the cubic termB is also positive, and the velocity 4>i vanishes at three values of 0i in the range of - 1 < 0i < 1. This is possible if a strong quadratic autocatalysis > k 2 exists together with a linear recycling X > 0, or if a linear autocatalysis k and a nonlinear recycling x > 0 coexist. By following the direction indicated by the arrows for positive order parameter ends up at a definite value ... 

Since the total concentration a + r + s follows the time evolution d(a + r + s)/ dt = F - k(a + r + s), it approaches the steady state value F/k with a relaxation time 1 /k. This is a consequence of unbiased outflow (Eq. 47) of all reactants with the same rate k. Consequently, even though we are dealing with an open system under a flow, the analysis is similar to the closed system by replacing the total concentration c with the steady state value F/k. Instead of recycling, therefore, constant supply of the substrate allows the system to reach a certain fixed point with a definite value of the order parameter 0i, independent of the initial condition. 

As assumed, the relaxation time corresponds to the time required for the z -type channel migration of hydrogen to change the short-range order parameter value in e times. The temperature dependence of Vh has its usual definition given by a Boltzmann distribution ... 

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