Long-range correlation/order

There has been much activity in the study of monolayer phases via the new optical, microscopic, and diffraction techniques described in the previous section. These experimental methods have elucidated the unit cell structure, bond orientational order and tilt in monolayer phases. Many of the condensed phases have been classified as mesophases having long-range correlational order and short-range translational order. A useful analogy between monolayer mesophases and die smectic mesophases in bulk liquid crystals aids in their characterization (see [182]). 

Besides being used as a tool for scientific research, siUca aerogels can be the cause for new scientific phenomena. For example, the long-range correlations of the disorder in siUca aerogels are beheved to be responsible for the intriguing observations of the superfluid transitions in He and He and on the ordering of He— He mixtures (75). 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

In the C °o limit, all the sites are bound the average correlation g(C is determined by the mth-order correlation function, which is 5 for the cyclic and 5 for the open linear system. This is true within the pairwise additive approximation for direct interaction, and neglecting long-range correlations. 

This long-range correlation effect shows up in both the first-order density matrix and the exchange-correlation hole for finite systems [19]. We concentrate here on the exchange-correlation hole. The general asymptotic form of the pair density is then... 

An important property of the Fermi surface is that the volume (in /c-space) that it encloses is not altered by the interaction between the electrons, unless long-range antiferromagnetic order is set up. This was first shown by Luttinger (1960). We shall make use of this theorem in Chapter 4, Section 3 in discussing metal-insulator transitions due to correlation. 

Therefore the model avoids two main difficulties the large amount of computer time which is normally needed for simulations and the loss of structural information which occurs in simple theoretical models (mean-field models) which do not take into account the structural aspects of the adsorbate layer. Mean-field-kind models fail in the prediction of phase transitions of the second order because at these points the long-range correlations appear. They also fail in describing the system s behaviour in the neighbourhood of the point of first-order kinetic phase transition. 

Evaluating the model (A 7.1) one finds that it describes a second order pha.se transition occurring for h = 0 and a critical value r = re. The transition is signaled by long range correlations of the spin field or power-type... 

Before investigating of long-range correlations of Jahn-Teller distortion using equation (37), we analyse in addition the effect of the exchange interaction between localized spins on the ordering of these distortions. 

For infinite systems there is another type of result concerning long-range magnetic order, which is implicated in making a material magnetic in the sense that there be a phase transition at the onset, as well as hysteresis. This Neel-state type of ordering concerns the spin correlation function... 

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