Multicritical point

Figure A2.5.30. Left-hand side Eight hypothetical phase diagrams (A through H) for ternary mixtures of d-and /-enantiomers with an optically inactive third component. Note the syimnetry about a line corresponding to a racemic mixture. Right-hand side Four T, x diagrams ((a) tlirough (d)) for pseudobinary mixtures of a racemic mixture of enantiomers with an optically inactive third component. Reproduced from [37] 1984 Phase Transitions and Critical Phenomena ed C Domb and J Lebowitz, vol 9, eh 2, Knobler C M and Scott R L Multicritical points in fluid mixtures. Experimental studies pp 213-14, (Copyright 1984) by pennission of the publisher Academic Press.

Domb C and Lebowitz J (eds) 1984 Phase Transitions and Critical Phenomena vol 9 (London, New York Academic) oh 1. Lawrie I D and Sarbach S Theory of tricritical points oh 2. Knobler C M and Scott R L Multicritical points in fluid mixtures. Experimental studies. 

A point at which nematic, SmA and SmC phases meet was demonstrated experimentally in the 1970s [95, 96]. The NAC point is an interesting example of a multicritical point because lines of continuous transition between N and... 

S.M. Allen and J.W. Cahn. Phase diagram features associated with multicritical points in alloy systems. Bull. Alloy Phase Diagrams, 3(3) 287-295, 1982. 

To conclude this section on the dense random copolymer model, wc briefly discuss the spinodal criterion and ask whether critical or multicritical points can exist for a general parent distribution pW(a) (with p = 1). The criterion (53), applied to our one-moment free energy becomes... 

In the dense limit p0 —> 1, the earlier result (93) is recovered. To study critical and multicritical points, we use the moment free energy corresponding to (94),... 

In this section we present three different phenomena, resonances, crossover and multicritical points. We will discuss in general the applicability of FSS to resonances, the existence of multicritical points and the definition of the size of the critical region. Application of FSS to a specific Hamiltonian that presents the three phenomena shows the relation between them [156]. 

Note that it is not a true change in the phase transition (second order to first order). If such a change occurs, a new scaling relation appears and the curves with different N should cross at approximately the same point. This point is a particular case of critical point, called multicritical point in theory of phase transition [25], Multicritical points in few-body systems is the subject of the next subsection. 

We studied in the previous section several types of phase transition, namely, bound-virtual, bound-resonance, and so on. A characteristic of a phase transition is that two different solutions merge (a / 1), or coexist at the critical point (a = 1). Many-body and multiparameter Hamiltonians could present more complicated transitions, and we will call them multicritical points. 

One kind of a multicritical point is a point over a critical line where more than two different states coalesce. The common multicritical points in statistical mechanics theory of phase transition are tricritical points (the point that separates a first order and a continuous line) or bicritical points (two continuous lines merge in a first order line) (see, for example, Ref. 166). These multicritical points were observed in quantum few-body systems only in the large dimension limit approximation for small molecules [10,32]. For three-dimensional systems, this kind of multicritical points was not reported yet. 

As we show in Fig. 25, the multicritical point is related to the crossing of the bound state n = 2 line with the resonance n = 4 at the critical Re( ) = 0 energy. In this figure we show also the results for n = 3. The dashed lines in Fig. 25 describe virtual states. The cusp behavior is a reflection of a transition through a branch point with exponent of one-half from a virtual state associated with a real eigenvalue to a virtual state that is associated with a complex eigenvalue. 

Figure 25. Energies for the states n = 2, n = 3, and n = 4 (continuous line), and virtual state energies (dashed lines) and the real part of the complex energy for n = 3 and n = 4 (dot-dashed line) of Hamiltonian equation (101), calculated using Eqs. (110) and (111) with a = ac = 1.027, xo = 6. The multicritical point ( ) is located at (a, J) (1.027,4.932).

We have studied one-fluid model of binary fluids with polyamorphic components and found that multicritical point scenario gives opportunity to consider the continuous critical lines as the pathways linking isolated critical points of components on the global equilibria surface of binary mixture. It enhances considerably the landscape of mixture phase behavior in a stable region at the account of hidden allocation of other critical points in metastable region. 

Fig. 22. Schematic phase diagram of a system exhibiting crossover between ordinary critical phenomena along the line Tc(p < pm) and the multicritical point p — pm, rm = TC(P m)-Considering the approach to the critical line along an axis parallel to the T-axis one will observe multicritical behavior as long as one stays above the dash-dotted curve /CIOS5 = describing the center of the crossover region. Only in between the dash-dotted curve and the critical line Tc(p) (full curve) the correct asymptotic behavior for p < pln can be seen.

Now another multicritical point arises for the special case where K = 0 (cf. fig. 23), and then eq. (117) yields a Lifshitz point (Hornreich et al., 1975)... 

Fig. 46. Schematic order parameter (magnetization) profiles m(z) near a free surface, according to mean field theory. Various cases arc shown (a) Extrapolation length X positive. The transition of the surface from the disordered state to the ordered state is driven by the transition in the bulk ( ordinary transition ). The shaded area indicates the definition of the surface magnetization ms. (b) Extrapolation length X = oo. The transition of the surface is called "special transition ( surfacc-bulk-multicritical point ), (c), (d) Extrapolation length X < 0, temperature above the bulk critical temperature (c) or below it (d). The transition between states (c) and (d) is called the extraordinary transition , (c) Surface magnetic field Hi competes with bulk order (mi, > 0, 0 < H such that mi < -mb). In this case a domain of oppositely oriented magnetization with macroscopic thickness ( welting layer ) separated by an interface from the bulk would form at the surface, ir the system is at the coexistence curve (T < Tv, H = 0). From Binder (1983).

Of particular interest is the behavior near the multicritical point, where again a crossover scaling description applies (Binder and Landau, 1984, 1990)... 

Fig. 56. (a) Schematic phase diagrams of a semi-infinite lsing magnet in the vicinity of the bulk critical point Tc as a function of temperature T, bulk field H, and surface field Hi. In the shaded part of the plane H 0 the system (for T < rc) is non-wet, while outside of it (For T < Tc) it is wet. The wetting transition is shown by a thin line where it is second order and by a thick line where it is first order. First-order prewetting surfaces terminate in the plane H = 0 at the first-order wetting line. Critical and multicritical points are indicated... 

A 1984 volume reviews in detail theories and experiments [37] on multicritical points some important papers have appeared since that time. 

A critical point or possibly multicritical point is observed in volumetric adsorption measurements [348]. Based on the qualitative change in the behavior of the quasivertical risers as a function of temperature, it is suggested that this point is located at about 89 K for CO on graphite an identical experiment for N2 on graphite [189] yielded 82 K (see Section III.C). 

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