First-order transitions defined

This implies that the exponents and y defined above are 0 = y = 2( = d) for a first-order transition. Since the symmetry around if = 0 is preserved for finite L, there is no shift of the transition. This feature is different, however, if we consider temperature-driven first-order transitions , since there is no symmetry between the disordered high-temperature phase and the ordered low-temperature phase. In order to understand the rounding of the delta-function singularity of the specific heat, which measures the latent heat for L- oo, it now is useful to consider the energy distribution, for which again a double Gaussian approximation applies ... 

We now distinguish solid state transformations as first-order transitions or lambda transitions. The latter class groups all high-order solid state transformations (second-, third-, and fourth-order transformations see Denbigh, 1971 for exhaustive treatment). We define first-order transitions as all solid state transformations that involve discontinuities in enthalpy, entropy, volume, heat capacity, compressibility, and thermal expansion at the transition point. These transitions require substantial modifications in atomic bonding. An example of first-order transition is the solid state transformation (see also figure 2.6)... 

In their statistical model for microphase separation of block copolymers, Leary and Williams (43) proposed the concept of a separation temperature Ts. It is defined as the temperature at which a first-order transition occurs when the domain structure is at equilibrium with a homogeneous melt, i.e.,... 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

If a first-order transition in a small system is driven by the temp>erature, for example, a transition taking place along an isobar, then it is typically spread out over a finite range of temperatures. The size of the transition region can be defined as the width AT of the finite p>eak in the curve of specific heat versus temperature. This peak replaces the -function singularity at the tran-... 

Fock (HF) one. This property is exploited (Sect. 4) for obtaining an equation in which both the correlation energy and the correlation matrices appear explicitly. These correlation matrices are defined here as the difference between a FCI-RDM and the corresponding UF-RDM. By applying the arguments given in [19] the exact structure of these correlation matrices can be expressed in terms of the 1 -RDM and of the first order transition RDM s. A calculation of the ground state of the beryllium atom illustrates the formalism. 

In the NMR study [335] a hysteresis of about 2.5% and a small discontinuity at 28.1 K in the temperature dependence of the order parameter was interpreted in terms of an evidence for a first-order herringbone transition. However, this assignment is based on NMR echo amplitudes which define only local order parameters. For the authors of the x-ray study [242] it seemed also probable that the herringbone orientational transition is a first-order transition. 

Within the compounds with equivalent sites for Ce, the specific heat of Cc4Bi3 also shows two well-defined peaks (Suzuki et al. 1987). This compound lies on the borderline between regions I and II, it is actually ferromagnetic for T < 3.5 K but after undergoing a first-order transition. The entropy gain including the transition at 3.8 K is 0.8I ln2 at 20 K. The field dependence of the specific heat in this compound was compared by the authors with that of CeBg, where a quadrupolar transition was observed by neutron diffraction. 

A plot of l/As vs. temperature T allows the extrapolated Curie-Weiss temperature and the corresponding C constant in the expression (Ae) = (T — Tq)/C (Table 25.1) to be measured. The Curie constant C is defined by the relation C = Np /k. The constant Tq is related to the exchange integral which takes into account the dipole-dipole interaction. In the case of a first order transition, Tq must be lower the T. ... 

A simple analysis of an irreversible first-order transition is the cold crystallization, defined in Sect 3.5.5. For polymers, crystallization on heating from the glassy state may be so far from equilibrium that the temperature modulation will have little effect on its rate, as seen in Fig. 4.122. The modeling of the measurement of heat capacity in the presence of large, irreversible heat flows in Fig. 4.102, and irreversible melting in Figs. 3.89 and 4.123, document this capability of TMDSC to separate irreversible and reversible effects. Little needs to be added to this important application. 

It should be noted that the system s response always lags behind the external effects or condition changes that are not necessarily related to phase transitions. These delays are defined with relaxation time. As a rule, the more gradually external conditions vary, the less is the relaxation time. However,. some first-order phase transitions show no decrease in the delays mentioned while slowing up the external condition change. It is in such cases that hysteresis and hysteresis phenomena are spoken of. Therefore, the manifestation of hysteresis is characteristic of first-order transitions (Brout, 1965 White and Geballe, 1979). 

A hypothesis has been put forward the asymptotic behaviour of the thermodynamic functions in the tricritical region is defined by competitive influence of the continuous transition region, on the one hand, and of the first-order transition region, on the other hand. This competition appears as the existence of the tricritic al region bounded by a crossover (Figure 1.39). 

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