Crossover classical

Moreover, well away from the critical point, the range of correlations is much smaller, and when this range is of the order of the range of the intenuolecular forces, analytic treatments should be appropriate, and the exponents should be classical . The need to reconcile the nonanalytic region with tlie classical region has led to attempts to solve the crossover problem, to be discussed in section A2.5.7.2. 

Povodyrev et aJ [30] have applied crossover theory to the Flory equation ( section A2.5.4.1) for polymer solutions for various values of N, the number of monomer units in the polymer chain, obtaining the coexistence curve and values of the coefficient p jj-from the slope of that curve. Figure A2.5.27 shows their comparison between classical and crossover values of p j-j for A = 1, which is of course just the simple mixture. As seen in this figure, the crossover to classical behaviour is not complete until far below the critical temperature. 

Gillam M J 1987 Quantum-classical crossover of the transition rate in the damped double well J. Phys. C Solid State Phys. 20 3621... 

The occurrence of nonradiative losses is classically illustrated in Figure 3. At sufficiently high temperature the emitting state relaxes to the ground state by the crossover at B of the two curves. In fact, for many broad-band emitting phosphors the temperature dependence of the nonradiative decay rate P is given bv equation 1 ... 

Kramers formula for classical escape out of a metastable well in the case of moderate and strong damping [Kramers 1940]. In accord with the multidimensional theory predictions, the crossover temperature should be equal to... 

On the other hand, it is clear that in the classical regime, T> (T i is the crossover temperature for stepwise transfer), the transition should be step-wise and occur through one of the saddle points. Therefore, there should exist another characteristic temperature. r 2> above which there exist two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points atlT = T i. The existence of the second crossover temperature, 7, 2, for two-proton transfer has been noted by Dakhnovskii and Semenov [1989]. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

Vitamin K is a fat-soluble vitamin cofactor for the activation of factors II, VII, IX, and X in the liver. Almost all neonates are vitamin K-deficient at as a result of (1) insignificant transplacental vitamin K crossover, (2) lack of colonization of the colon by vitamin K-producing bacteria, and (3) inadequate dietary vitamin K intake (especially in breast-fed infants because human milk contains less vitamin K than infant formula or cow s milk). Vitamin K-deficiency bleeding (VKDB) refers to bleeding attributable to vitamin K deficiency within first 6 months of life. It occurs in three general time frames early (0-24 hours), classic (1-7 days), and late (2-12 weeks). Early onset occurs rarely and usually is associated with maternal ingestion of anticonvulsants, rifampin, isoniazid, and warfarin. Classic vitamin K-dependent bleeding usually results from the lack of prophylactic vitamin K administration in... 

Substitution of one ligand by another can generate, or alter, spin crossover characteristics. The systems studied early provide the classic illustration of this effect. Thus [Fe(py)4(NCS)2] is high spin at room temperature and does not undergo a thermal spin transition. Substitution of two of the pyridine molecules by a phenanthroline molecule gives [Fe (phen)(py)2 (NCS)2] which does undergo a thermal transition [99, 141], as does the species in which the remaining two pyridines are substituted [Fe(phen)2 (NCS)2]. As would be expected, T1/2 for the former complex (106 K) is lower... 

As was mentioned above, the [Fe(HB(3,5-(CH3)2pz)3)2] complex represents a classic example [27, 28] of an iron(II) spin-state crossover that may be induced in a high-spin complex upon cooling. The room temperature crystal structure of this complex [26] reveals a structure rather similar to that of [Fe(HB(pz)3)2], but with a substantially longer average iron-nitrogen bond... 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

Extension of the classical Landau-Ginzburg expansion to incorporate nonclassical critical fluctuations and to yield detailed crossover functions were first presented by Nicoll and coworkers [313, 314] and later extended by Chen et al. [315, 316]. These extensions match Ginzburg theory to RG theory, and thus interpolate between the lower-order terms of the Wegner expansion at T -C Afa and mean-field behavior at f Nci-... 

Fig. 5. The low temperature crossover diagram of a one-dimensional CDW. t and K are proportional to the temperature and the strength of quantum fluctuations, respectively. The amount of disorder corresponds to a reduced temperature tu 0.1. In the classical and quantum disordered region, respectively, essentially the t = 0 behavior is seen. The straight dashed line separating them corresponds to At 1, i.e., K 1, where At is the de Broglie wave length. In the quantum critical region, the correlation length is given by At- Pinning (localization) occurs only for t = 0, K

This means that there is a crossover temperature defined by (1.7) (or hpc = 2it/o> ), above which tunneling switches off, because the quasi-classical trajectories that give the extremum to the integrand in (2.1) cease to exist. This change in the character of semiclassical motion is universal for barriers of sufficiently general shape. 

Of special interest is the case where the barrier is parabolic, as in Eq. (1.5). Here, it is possible to examine the crossover between the classical and quantum regimes in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the transmission coefficient of the parabolic barrier [Landau and Lifshitz, 1981]... 

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

Abnormal hemoglobulins can be detected by electrophoresis, as shown in Figure 7.4, which includes a pattern observed in /3+-thalassemia and one in a newborn with a-thalassemia (possibly HbH disease). It should also be mentioned that, unless there is a coexisting hemoglobin abnormality resulting from a point mutation or crossover problem, the globin chains of classic a- and /3-thalassemia are perfectly normal. It is usually the quantities of either the a or the /3 chains that are decreased. Some frameshifts have been found near the terminus of the /3 chain that lead to frameshift mutations in certain areas. 

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