Correlations length

To And f as a function of the concentration and temperature, let us assume that it obeys the power law 

A similar argument for the theta region leads to f = a p the correlation length is independent of the temperature. 

For the low-temperature region, it is impossible to find the solution for which is independent of n. This indicates that it is impossible for the compact globules to 

The concept of screening survives in semidilute conditions. At the overlap concentration, by definition, the average distance between the chains is proportional to the Flory radius (Rp) in a good solvent. Therefore we can guess the following form for  

At p = p, the mean square end-to-end distance of a labeled chain should be proportional to R suggesting the general scaling form 

Thus a blob is an effective step along the contour of the chain containing many (g) segments. Furthermore, let us assume that (i) the segments inside the blob obey the excluded volume chain statistics so that g and (ii) the njg blobs obey the random walk statistics such that 

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For the case of a double-D coil we multiply each matrix element with an element shifted by a constant distance of the same line. This is done in x- and y-direction. The distance between the two elements is the correlation length X for filtering in x-direction and a second correlation length for the movement in y-direction. Thus one gets two new matrices Ax and Ax for the filtering from the left to the right (positiv x-direction) and vice versa (negativ x-direction). 

In case of a gradiometric excitation with a double-D coil, this algorithm enhances the response of the crack, while other signals like artificial peaks and plateaus are supressed. The calculation can be done using different correlation lengths X in order to obtain additional information about the depth in wliich the crack is located. 

We present two optical methods for characterizing wire surfaces. These methods allow us to measure the roughness and the correlation length of the surface. It is also possible to identify qualitatively, at a glance, the variations of the roughness along a wire or among its different zones. 

The divergence m the correlation length is characterized by the critical exponent v defined by... 

The correlation length C = T -T diverges with the exponent v. Assuming that when T>T the site... 

The correlation length follows from the above relation, since... 

A feature of a critical point, line, or surface is that it is located where divergences of various properties, in particular correlation lengths, occur. Moreover it is reasonable to assume that at such a point there is always an order parameter that is zero on one side of the transition and tliat becomes nonzero on the other side. Nothing of this sort occurs at a first-order transition, even the gradual liquid-gas transition shown in figure A2.5.3 and figure A2.5.4. 

Based on this equation, one can make a Debye-Bueche plot by plotting [i (q)] versus q and detemiine the slope and the intercept of the curve. The correlation length thus can be calculated as [21]... 

The quantitative analysis of the scattering profile in the high q range can be made by using the approach of Debye et aJ as in equation (B 1.9.52). As we assume tiiat the correlation fiinction y(r) has a simple exponential fomi y(r) = exp(-r/a ), where is the correlation length), the scattered intensity can be expressed as... 

Flere we discuss only briefly the simulation of continuous transitions (see [132. 135] and references therein). Suppose that tire transition is characterized by a non-vanishing order parameter X and a corresponding divergent correlation length We shall be interested in the block average value where the L... 

A fingerprint of a continuous phase transition is the divergence of the correlation length at the critical temperature 7", with for... 

FIG. 12 Inverse effective correlation length in units of the lattice constant a = 4.26 A as a function of temperature the extrapolated transition temperature Tq = 25.02 0.08 K, as obtained independently from energy cumulants, is marked by a dotted line. Full lines correspond to a fit assuming a simple hnear dependence + C l - r/FqI expected near Tq for a first-order transition, whereas dashed... 

Now, assume that we are getting closer to the critical point of our transition, i.e., to the point of the second-order transition. In the case of a uniform system the critical region can be described by the divergent correlation length of statistical fluctuations [138]... 

All the above scaling relations have one common origin in the behavior of the correlation length of statistical fluctuations, in a finite system [140,141]. Namely, the most specific feature of the second-order transition is the divergence of at the transition point, as is described by Eq. (22). In the finite system, the development of long-wavelength fluctuations is suppressed by the system size limitation can be, at the most, of the same order as L. Taking this into account, we find from Eqs. (22) and (26) that... 

One can easily extend the above analysis to dilute and semi-dilute solutions of EP [65,66] if one recalls [67] from ordinary polymers that the correlation length for a chain of length / in the dilute limit is given by the size R of the chain oc When chains become so long that they start to overlap at I I (X the correlation length of the chain decreases and reflects... 

A crossover scaling for and R of a polymer chain in a quenched random medium may be developed if one introduces a correlation length... 

In the real space the correlation function (6) exhibits exponentially damped oscillations, and the structure is characterized by two lengths the period of the oscillations A, related to the size of oil and water domains, and the correlation length In the microemulsion > A and the water-rich and oil-rich domains are correlated, hence the water-water structure factor assumes a maximum for k = k 7 0. When the concentration of surfac-... 

Finally, we assume that the fields 4>, p, and u vary slowly on the length scale of the lattice constant (the size of the molecules) and introduce continuous approximation for the thermodynamical-potential density. In the lattice model the only interactions between the amphiphiles are the steric repulsions provided by the lattice structure. The lattice structure does not allow for changes of the orientation of surfactant for distances smaller than the lattice constant. To assure similar property within the mesoscopic description, we add to the grand-thermodynamical potential a term propor-tional to (V u) - -(V x u) [15], so that the correlation length for the orientational order is equal to the size of the molecules. 

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