Lower 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...

Russian researchers [70] have found that in glass tubes the same Ku value is obtained at Bd > 6. This discrepancy caused Eichhom [71] to carry out a detailed dimensional-analytical examination. He first discovered that the lower critical vG corresponded to a critical film thickness and to a critical shear rate in the phase boundary G/L. Therefore, there are three parameters independent of each other, which could be regarded as target quantities. However, vG can be measured more accurately and more easily than the others, therefore, it is accepted as the target quantity. 

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).

We have analyzed the thermodynamic, magnetic and ultrasound attenuation data on oriented saaiples of the hlgh-T superconductors within the context of anisotropic Glnzburg-LSndau theory for coupled, even-parity superconducting states. We are able to present a consistent Interpretation of the data In terms of the coexistence of a quasi-two-dimensional d-wave state, with critical temperature T. - T and a more Isotropic mixed (s+d)-wave state with critical tempertaure T < T We predict the possibility of a "kink" in the temperature dependence of the lower critical field near 0 9T, which should be tested by experiments on single crystals. 

The shape of characteristic isobaric T[x) and isothermal p x) sections can be deduced from the three-dimensional phase diagrams in the upper row of Fig. 4 p x) sections are also given in Figs. 15c and d (for details see the Refs. [2-6,10,12]). Systems exhibiting lower critical solution temperatures (LCST) or closed loops in isobaric T x) diagrams have also been studied extensively at pressures even up to about 3 GPa [1,3,70,71,81,82,85]. 

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].

Working in very dilute solution the dimensional changes with temperature of the polystyrene coil in a poor solvent, cyclohexane, have been studied. Two domains are recognized one (the 0-region) where R is independent of the reduced temperature t [t = (T— 0)/O] in the second the coil collapses and R t -i/ in accordance with predicted behaviour. This system has been studied further in the region of the lower critical solution temperature (LCST), that is near the solvent critical point. The temperature range was 70— 220 °C and in this r on Edwards equation is modified to ... 

Peskin and Raco (P3) have given a theoretical analysis of both ultrasonic and electrostatic atomization from the point of view of liquid instability. They conclude that atomization with low frequency ac will require about twice the field strength as dc but that, by going to high frequency, lower fields are possible with conducting liquids. The value for the critical field for atomization given by these authors for a dc field is, however, smaller than that which would be calculated from Eq. (39) by a factor of (1/32)1/2. This presumably reflects the simplified one-dimensional model used in their derivation. 

Space remains for only a brief glance at detection in higher dimensions. The basic concept of hypothesis testing and the central significance of measurement errors and certain model assumptions, however, can be carried over directly from the lower dimensional discussions. In the following text we first examine the nature of dimensionality (and its reduction to a scalar for detection decisions), and then address the critical issue of detection limit validation in complex measurement situations. 

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