Function correlation

1 Correlation Function We now find how the structure factor is related to the local segment density p(r) defined by 

The pair distribution function is the statistical average of the product of the densities at rj and Y2. 

2 Relationship Between the Correlation Function and Structure Factor The statistical average in the definition of the structure factor 5 (k) in Eq. 2.60 is taken with respect to the pair distribution. With Eq. 2.62, Eq. 2.60 is rewritten to 

Note that p(0)/p is the segment density at r = 0 normalized by the average. We can therefore interpret (p(r)p(0) /p as measuring the average number of monomers per volume at r when there is already a monomer at r = 0. 

Equation 2.64 illustrates that the static structure factor, and hence the scattering pattern obtained in the light-scattering experiments, is the Fourier transform (see Appendix A2) of the autocorrelation function of the local segment density. 5(k) indicates which wave vector components are present in the correlation function. 

In a very dilute solution c with a good solvent, the polymer molecule swells and its size can be expressed (in terms of mean-square end-to-end distance) by 

In between we have a semidilute solution c with the threshold density p. A given chain now has two different sizes  

We introduce another dummy variable a such that 

Although both p and a are expressed in terms off — t, they are not equal they are two different variables. Squaring the function, we obtain 

If we impose a condition that/(f) is random, then the function (/(f — p)f(f — cj)) depends only on the time interval p a. We now have 

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Auto-correlation and Inter-correlation Functions allow a good discrimination between these two types of defects by quantifying the resemblance between the different echoes and their derivatives. 

Signal processing in mechanical impedance analysis (MIA) pulse flaw detectors by means of cross correlation function (CCF) is described. Calculations are carried out for two types of signals, used in operation with single contact and twin contact probes. It is shown that thi.s processing can increase the sensitivity and signal to noise ratio. 

In this report problem of information processing in MIA pulse flaw detectors by means of cross correlation function is considered. Such processing promises to increase the sensitivity and to reduce the noises, mainly the frictional ones. 

In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules detemiiued through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation fiinctions (g, g. . ) in liquid water. An important feature of these correlation functions is that... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

It follows that the exact expression for the pair correlation function is... 

By analogy with the correlation function for the ftilly coupled system, the pair correlation ftmction g(r A) for an intennediate values of A is given by... 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

Fisher M 1964 Correlation functions and the critical region of simple fluids J. Math. Phys. 5 944... 

Hemmer P C 1964 On van der Waals theory of vapor-liquid equilibrium IV. The pair correlation function and equation of state for long-range forces J. Math. Phys. 5 75... 

There are two approaches connnonly used to derive an analytical connection between g(i-) and u(r) the Percus-Yevick (PY) equation and the hypemetted chain (FfNC) equation. Both are derived from attempts to fomi fimctional Taylor expansions of different correlation fimctions. These auxiliary correlation functions include ... 

The singlet direct correlation function C r) is defined through the relationship C Ur) = + --/t]... 

Figure A2.4.11. Water pair correlation functions near the Pt(lOO) surface. In each panel, the frill curve is for water molecules in the first layer, and the broken curve is for water molecules in the second layer. From [30].

The second application is to temperature fluctuations in an equilibrium fluid [18]. Using (A3.2.321 and (A3.2.331 the correlation function for temperature deviations is found to be... 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

Forster D 1975 Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (New York Benjamin)... 

Zhu S-B, Lee J, Robinson G Wand Lin S H 1989 Theoretical study of memory kernel and velocity correlation function for condensed phase isomerization. I. Memory kernel J. Chem. Phys. 90 6335-9... 

Cao J and Voth G A 1995 A theory for time correlation functions in liquids J. Chem. Phys. 103 4211... 

Voth G A, Chandler D and Miller W H 1989 Time correlation function and path integral analysis of quantum rate constants J. Phys. Chem. 93 7009... 

Wahnstrom G and Metiu H 1988 Numerical study of the correlation function expressions for the thermal rate coefficients in quantum systems J. Phys. Chem. JPhCh 92 3240-52... 

Thachuk M and Schatz G C 1992 Time dependent methods for calculating thermal rate coefficients using flux correlation functions J. Chem. Phys. 97 7297-313... 

Lamellar morphology variables in semicrystalline polymers can be estimated from the correlation and interface distribution fiinctions using a two-phase model. The analysis of a correlation function by the two-phase model has been demonstrated in detail before [30,11] The thicknesses of the two constituent phases (crystal and amorphous) can be extracted by several approaches described by Strobl and Schneider [32]. For example, one approach is based on the following relationship ... 

Hwang L-P and Freed J H 1975 Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids J. Chem. Rhys. 63 4017-25... 

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