Correlation radius

The variable is called the correlation radius. Near the critical point —> oo and h(r) decays according to the power law. 

In its turn Fig. 6.2 illustrates the effect of the initial concentration on the static tunnelling recombination kinetics. The latter is defined by a competition of three distinctive scales - the tunnelling recombination radius ro, mean distance between particles Iq = n(0) 1/d and lastly, the time-dependent correlation radius . At long time curves corresponding to different initial concentrations could be coincided by their displacements along ordinate axis, which confirms existence of the universal asymptotic decay law. 

According to the general definitions of the coil and the globule241, the macromolecule is in the coil state, if the fluctuations of the monomer concentration within the macromolecule are of order of the monomer concentration itself and the correlation radius of the fluctuations of concentration is of order of the macromolecular dimensions while in the globular state the concentration fluctuations are small compared with the concentration and the correlation radius is considerably smaller than the globular dimensions. 

The Stokes region of a dipole-forbidden resonance is formed by long-range interparticle interactions. According to our model, an essential change of EELS correlational structure takes place close to the critical point where the correlational radius tends to infinity. This effects corresponds qualitatively to those of section III. A. 

Magnetic columns FeNi(P) concern to a class of nanostructured materials. Therefore, for interpretation of the data on the approach magnetization to saturation, the special advanced techniques should be used [1], A field of local magnetic anisotropy Ha, correlation radius of casual magnetic anisotropy Rc, a field of anisotropy, the size of the basic unit of a nanostructured magiwtic, the stochastic magnetic domain can be determined with this technique. These parameters can be associated with the GMR effect. 

Figure 1. Correlation radius Rc against xi for the 1-propanol — water system. The broken line is for the alcohol-rich cluster and the sohd line is for the water-rich cluster experimental data from , ref 15 O, ref 17 , ref 19.

We may refer to Rc as the correlation radius and to Vc as the correlation volume. In general, Rc would be dependent on the species i and s. Therefore, in (8.8) we shall take the largest Rc which fulfills relation (8.7) for all i. It should be noted that Rc has the true meaning of the correlation distance in the sense that, beyond Rc, there exists no correlations due to intermolecular forces. 

It should be noted that our definition of Reis hi terms of the pair correlation function as in (8.7). Mansoori and Ely (1985) defined the correlation radius (or the radius of the sphere of influence ) as the distance R(, for which the integral J [gjj(-R) — l]47tR2 dR is zero for all pairs of species i and j. This is an unacceptable definition of a correlation radius. Because of the oscillatory nature of gij(R)> one can have more than one Rc for which this integral is zero. Therefore, such a definition of Rc does not confer the meaning of a correlation radius. 

Equation (8.9) is valid for any Ra > Rc i.e., when Ra is at least as large as the correlation radius of the system. Thus, having all the Gl , and a choice of a volume Va, we can calculate the local composition for any mixture. Clearly, if Ra is very large then xf (s, Ra) will approach the bulk composition and we shall miss the local character of xf. Therefore, we have to choose Ra large enough to take into account all the effects of s on its environment, but not too large that the local composition is washed out. 

Let us denote the first coordination-sphere radius, given by the position of the first maximum Ur), by r. The radius r, at which fat r) becomes (and remains at r > rlf) negligibly small, is a pair correlation radius. The correlation radius r s is equal to the mean distance between /i-type polyhedra and boundary... 

Further discussion is appropriate. Should the field variable lie parallel to t, an appropriate scaling equation for the correlation radius might be, = A, [ (k p -t J/(tcr+t ) ]... 

It is known from both experiments and theoretical work that the range of the pair correlation functions gap R) is only a few molecular diameters. This means that there exists a correlation radius Rcorr, such that for R > RcoRRy gap(R) is nearly unity. Alternatively, there is no correlation at distances beyond the correlation radius. 

Here ct = d z— )yl2> is effective dipole moment, y is Lorenz factor, which takes into account the difference between average and local field acting on the impurity ion, fc is correlation radius of the host lattice [27]. 

Above consideration shows that the dipoles or spins situated in the sphere of correlation radius have to be coherently oriented. This phenomenon occurs also for not very large correlation radii. As a result the short-range polar order has to appear in the clusters with the size of correlation radius. The orientations of the... 

This means that to consider the spontaneous flexoelectric effect influence on the substance physical properties one has to rewrite all earlier analytical expressions for long nanorods and nanowires without flexoelectric effect [8, 78] by the substitution g 2 and X for g and Xs in the expressions for the corresponding property. Note that for polydomain (if any) wires the predicted effect of R decrease with /44 increase should lead to the decrease of the intrinsic domain-wall width defined as 2 R. Below we demonstrate the spontaneous flexo-effect influence on the critical parameters (temperature and radius) of size-induced phase transition and correlation radius using the results [8,78] obtained without flexoeffect. 

The main effect from the flexoelectric coupling is the altering of transition temperature via the renormalization of the extrapoiation iength and the gradient term (see Eq. (4.22)). Due to the same reasons flexoelectric coupling leads to the renormalization of correlation radius as ... 

Fig. 4.28 (a, b) Correlation radius dependences vs. nanowire radius for different flexoelectric coefficients marked near curves (in V) and T = 20, 400 °C. (c, d) Correlation radius dependences vs. flexoelectiic coefficients /44 for different wire radius marked near curves (in nm) and T = 20, 400 °C. PbTi03 material parameters are the same as in Fig. 4.26 [88]... 

The renormalized correlation radius dependences on nanowire radius and flexoelectric coefficients /44 are reported in Fig. 4.28a-d for PbTiOs material parameters at room temperature. 

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