Bicritical point

Fig. 29. Phase diagram of the model Eq. (22) for coadsorption of two kinds of atoms in the temperature-coverage space. Circles indicate a second-order phase transition, while crosses indicate first-order transitions. Point A is believed to be a tricritical point and point B a bicritical point. The dashed curve shows the boundary from the Blume-Capel model on a square lattice with a nearest-neighbor coupling equal to 7 in the present model (for - 0 Eq. (22) reduces to this model), only the ordered phase I then occurs. From Lee and Landau. )...

The values of the exponents for ordinary critical points or bicritical points (where two phases become identical) are called nondassical, because (unlike the exponents in van der Waals and other classical equations) they are not multiples of 1/2. 

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. 

At this point, we note that while the phase diagram of fig. 21(a) with a bicritical point where lines of m = 2-component ordering and m — 1-... 

Barois, Frost and Lubensky have used the phenomenological model, within the framework of the mean field theory, to construct phase diagrams involving the polymorphic forms of the A phase. They have predicted three kinds of critical points, the A -Aj critical point, the Aj-A tricritical point and the A, N-A bicritical point, the salient features of which are summarized below. 

The mean field theory predicts a bicritical point when the second order N-Ai and N-A<, phase boundaries meet a first order A -Aj boundary. However, when the effect of fluctuations is taken into account, the existence of such a point becomes doubtful. On the experimental side, phase diagrams with an Aj-Nj -A, point have been reported, (where Nr is the re-entrant nematic, see 5.6.2) the topology of these diagrams resembling that of the magnetic bicritical point. But high-resolution experiments carried out subsequently in the immediate vicinity of an... 

More recently, Richard, Guttmann and Jensen [92] have given a very persuasive conjecture as to the exact nature of the scaling function at the bicritical point (x, 1). In their work, it was natural to work with rooted SAP, and in that case the conjectured form of the scaling function was found to be... 

Figure 3. Chemical potential versus temperature phase diagram. Solid line and diamonds represent the Gas-LDL coexistence line, whereas solid line and triangles represent the LDL-HDL coexistence line. The temperature of maximum density (TMD) is represented by a solid line and stars. The dashed line is a critical line, named X-line, and the solid line is another critical line named r-line. X-line emerges from the Gas-LDL coexistence line at a tricritical point Toy and meets the r-line at the LDL-HDL coexistence line at a bicritical point...

Another interesting behavior under elevated pressures was observed [127] in binary mixtures of 4-(2 -methylbutyl) phenyl 4 - -octyl biphenyl-4-carboxylate (CE8) and 4- -dodecyloxy biphenyl-4 -(2 -methylbutyl) benzoate (C12) which show TGB phases close to a virtual N -SmA-SmC triple point. Krishna Prasad et al. observed, in a mixture which shows a direct SmA-N transition, that the appearance of the TGBa phase between SmA and N can be induced by pressure. From the topology close to the SmA-TGB-N point, it was concluded that this point is a critical end point rather than a bicritical point as predicted by Renn and Lubensky [5]. In a mixture with different concentrations which shows a SmC -TGBA transition, the... 

The point at which the two minima merge together is called a critical point, more precisely a bicritical point because it corresponds to the situation in which two phases merge into one. Other situations can occur, such as a monociitical point in which one of the minimum vanishes without merging with the other, or a tricritical point when three minima merge together [26,27]. 

When the incommensurability parameter is further increased, a new SmAj-SmAj line appears (Fig. 6 c) terminating at a mean field bicritical point B where the N, SmAj and SmAd phases meet, and a triple point T where the SmAj, SmA2, and SmAj phases coexist. 

A bicritical point is a point in the temperature-concentration or pressure-temperature plane at which two second-order phase boundaries and one first-order phase boundary meet. In order to find such a bicritical point in the pressure-temperature plane Rahr et al. [105] studied the pressure-temperature phase behavior of binary mixtures of4-(4- -butyloxybenzoyloxy)-4 -nitroazo-benzene and 4-(4- -nonyloxybenzoyloxy)-4 -cyanoazobenzene. From the p-T phase diagrams of the pure components which are also presented a bicritical behavior of their mixtures can be expected. Two mixtures with molar fractions of about 0.80 for the cyano component show an abrupt change of... 

Since the SmAj-N and SmA -N transitions are expected to be second order (low Ja n/Jn-i) values [97]) and the SmA,-SmAd transition first-order (symmetry arguments), the meeting point should be a bi-critical point. Moreover the topology of the p-T phase diagram obtained resembles that of a diagram exhibiting known bicritical points [106]. 

For n=2, H g / dr (x -y ) implies a higher transition temperature for the x (if g < 0) or the y component, i.e. uniaxial symmetry. This yields crossover to king model behavior. The XY behavior occurs only at the bicritical point, defined by The two Ising... 

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