Transition higher order

First of all the term stress-induced crystallization includes crystallization occuring at any extensions or deformations both large and small (in the latter case, ECC are not formed and an ordinary oriented sample is obtained). In contrast, orientational crystallization is a crystallization that occurs at melt extensions corresponding to fi > when chains are considerably extended prior to crystallization and the formation of an intermediate oriented phase is followed by crystallization from the preoriented state. Hence, orientational crystallization proceeds in two steps the first step is the transition of the isotropic melt into the nematic phase (first-order transition of the order-disorder type) and the second involves crystallization with the formation of ECC from the nematic phase (second- or higher-order transition not related to the change in the symmetry elements of the system). 

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

Thermodynamic arguments [23] indicate that the transition from the normal to the superconducting state at zero field does not involve a latent heat and therefore must be a higher-order transition. Experimental evidence indicates that it is second-order transition. 

Fig. 5 Magnetic phase diagram of [Mn(Cp )2][Pt(tds)2] M(T) (filled diamonds) M(H) (//] (filled triangles), H (filled inverted triangles), x (T) (open circles) x (H) (open squares) Tt is the tricritical temperature I denotes the first-order MM transition II denotes a second-order transition (AF-PM phase houndary) and III denotes a higher order transitions (from a PM to a FM like state). From [45]...

By contrast the approach here is based on the theory of quantum transitions and is similar in approach to Bardeen s theory of tunneling (34). Further, in the present development, terms corresponding to higher-order transitions contain products of FC factors for different virtual transitions which results in additional orders of smallness in a perturbative sense. (This is additional justification for limiting consideration here to eq. 46.) This is in contrast to the theory of Schatz and Ross (36) and C. Villa et al. (37,38), which leads to a single FC factor. 

Note that the terms corresponding to higher-order transitions contain products of FC factors for different virtual transitions this results in additional orders of smallness in a perturbative sense. We note that the theory of Schatz, Ross, and co-workers (36-38) contains only a single FC factor. 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

The compositional dependence of the structural O -O transition at 7jt and the 0 -R transition at Tor can be clearly followed by monitoring the temperature dependence of the resistance (Mandal et al., 2001) monitoring the variation with x of the higher-order transition at T from the resistance curve R(T) is more subtle and has been accomplished with further aid from the thermoelectric power a(T) measured on single crystals (Zhou and Goodenough, 2000). The transition from polaronic to itinerant electronic behavior in the paramagnetic R-rhombohedral phase has not been studied. 

Although they differ in detail, it may be accepted that the basic unit of the cluster is a tetrahedron with one interstitial iron (most likely Fe3+ [52, 53] surrounded by four vacancies on the nearest octahedral site, which is found locally in the magnetite structure. The wiistite structure is then understood to have these unit tetrahedra arranged in some ordered manner. From this point of view, the measurements suggesting three phases separated by second- or higher-order transitions within the wiistite phase [22, 22a, 78] can be interpreted as successive loss of different types of order as the temperature is raised or the number of the unit tetrahedra decreases (the reduction proceeds). However, no definite conclusions have yet been drawn and indeed, the existence of these three subphases is still disputed [19, 20, 23, 24, 28]. 

The classical Clapeyron equation adequately predicts the features of first-order phase transitions, and this has been established for a number of examples of first-order transitions effected by the deliberate variation of temperature or pressure. Second- or higher-order transitions are not readily explained by classical thermodynamics. Unlike the case of first-order transitions, where the free-energy surfaces of the two phases... 

It has been pointed out that the central ( 2, transition does not experience any first-order quadrupole interaction. The absence of first-order broadening effects is a general property of symmetric (m, - m) transitions. There are cases where this can be a distinct advantage, the most direct instance being for integer spin nuclei (e.g. D and both 1=1) where there is no ( /2, — /2) transition. The main problem is to excite and detect such higher-order transitions, for which there are two separate approaches. The sample may either be irradiated and detected at the multiple quantum frequency (called overtone spectroscopy) or the MQ transition can be excited and a 2D sequence used to detect the effect on the observable magnetisation. 

The low-temperature properties of H2Se were studied by Kruis and Clusius [37KRU/CLU] who found that the solid phase was characterised by two phase transitions at 80 to 87 K and at 172.5 K, respectively. The temperature span of the lower transition is indicative of a higher order transition, and the higher transition takes place between two isotropic phases. The heats of transformation for the two transitions are (1.574 + 0.002) and (1.116 + 0.002) kJ-mol, respectively. H2Se(cr) melts at (207.430 + 0.018) K with an enthalpy of fusion of ... 

In systems where inter-hydrogen forces are important, usually because of the close proximity of hydrogen neighbours, dispersion is clearly seen in the one quantum spectrum, as in Fig. 6.22. The higher order transitions from such systems would not then be expected to appear as sharp transitions, since the two phonon spectrum should resemble the self convolution of the one phonon spectrum ( 2.6.3). Occasionally, however, sharp transitions are observed, at E(2 o just below the energy of 2E(i o) If -d is the full width at half height of the (1<—0) transition. Then, provided that... 

This notion is easily generalized to higher-order transitions. Thus a second-order transition is described as one in which the second partial derivatives of... 

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