Upper critical dimensionality

Fig. 5. Lower and upper critical tielines in a quaternary system at different temperatures and a plot of the critical end point salinities vs temperature, illustrating lower critical endline, upper critical endline, optimal line, and tricritical poiat for four-dimensional amphiphile—oil—water—electrolyte-temperature...

Hc2- k-(ET)2Cu(NCS)2 gave higher upper critical magnetic field H 2 values in the two-dimensional plane than the Pauli limited magnetic field //pauii [226, 227]. 

These powers a, (3, 7, p and i/ are called the critical exponents. These exponents are observed to be universal in the sense that although Pc de-pends on the details of the models or lattice considered, these exponents depend the only on the lattice dimensionality (see Table 1.2). It is also observed that these exponent values converge to the mean field values (obtained for the loopless Bethe lattice) for lattice dimensions at and above six. This suggests the upper critical dimension for percolation to be six. 

Fig. 17. Schematic variation of the critical exponents of the order parameter fi (a), the order parameter response function y (b), and the correlation length v (c) with the spatial dimensionality, for Lhe m-veclor model. Upper (du) and lower (rf ) critical dimensionalities are indicated. Here m = 1 corresponds to the Ising model, m = 2 to the XY model, m = 3 to the Heisenberg model of magneLism, while the limit of infinitely many order parameter components (m —> oo) reduces to the exactly solved spherical model (Berlin and Kac 1952, Stanley, 1968).

At d = 1 one has a completely stretched chain with ly = 1. At d = 2 the exact result v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882 0.0011 [11]) and Monte Carlo simulation gives (i/ = 0.592 0.003 [12]), both values being in a good agreement. 

Figure 1. Three-dimensional phase model for polyethylene + ethylene mixtures with constant temperature cuts at 120, 160 and 200 °C (showing upper critical solution pressures) and a constant pressure cut (showing lower critical solution temperature). [ Adapted from ref 6].

The formation of IL/O microemulsions in mixtures of [bmim][BFJ (IL) and cyclohexane, stabilized by the nonionic surfactant, TX-lOO has been proved [30]. Three-component mixtures could form IL/O microemulsions of well-defined droplet size determined by fixing the water content (mole ratio of IL to TX-lOO) [30,48,49]. An upper critical point (T) was observed in the mixture [([bmim][BFJ/ TX-lOO)-I-cyclohexane] with fixed water content (mole ratio of [bmim][BFJ to TX-lOO) [50]. The mixture separated into two microemulsion phases of different composition but with the same composition below as occurred in other systems [48]. The microemulsion system, [bmim][BF ]/TX-100 +cyclohexane, could be regarded as a pseudobinary mixture of [bmim][BF ]/TX-100 IL droplets dispersed in the cyclohexane continuous phase. Therefore, the phase behavior could be depicted in a two-dimensional diagram with concentration of droplets along the abscissa and temperature along the ordinate. A coexistence curve of temperature (T) against a concentration variable, such as volume fraction ( ), could then be drawn in the same way as it was done for pseudobinary mixtures in AOT/water/decane micro-emulsions [48]. 

Exponent relations involving the dimensionality d explicitly are called hyperscaling relations. They only hold below the upper critical dimension Above they are destroyed by dangerously irrelevant variables. 

Experiments [19] and simulations [19,21-24] for d > 2 systems in which both reactants diffuse, support the above predicted values for a and (3. Indeed, Cornell et al [23] argue that the upper critical dimension is d = 2 and the MF approach should therefore be valid for d > 2. However, numerical simulations of ID systems show that the width exponent appears to be a 0.3 and the height exponent p 0.8 [23,24]. The origin of the difference between the exponents of ID systems and those of higher dimensional systems is due to fluctuations in the location of the front which are important in low dimensions and are neglected in the MF approach. 

Hubbard (13) elucidated a mathematical description of the change from one situation to another for the simplest case of a half-filled s band of a solid. His result is shown in Figure 11. For ratios of W/U greater than the critical value of 2/ /3 then a Fermi surface should be found and the system can be a metal. This critical point is associated with the Mott transition from metal to insulator. At smaller values than this parameter, then, a correlation, or Hubbard, gap exists and the system is an antiferromagnetic insulator. Both the undoped 2-1 -4 compound and the nickel analog of the one dimensional platinum chain are systems of this type. At the far left-hand side of Figure 11 we show pictorially the orbital occupancy of the upper and lower Hubbard bands. 

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