Field correlation function

A and B are instrumental factors and g(T) the electric field correlation function which is related to the translational diffusion coefficient D by... 

One of the interesting consequences of eqs. (11-25) and (11-26) is the dependence of the probability of the molecule being in a given nonstationary state on the time correlations in the coupled radiation field. In most experimental studies the radiation field employed consists of a superposition of many frequencies with random phases. It is convenient to represent that form of field in terms of a correlation function d>(t, t"), which is defined in eq. (6-16). Introducing, because of the polychromaticity of the radiation field, the averages of eqs. (11-25) and (11-26), choosing the same representation for the field correlation function as did Bixon and Jortner, and using the conservation of probability, we find for the probability of dissociation of the molecule the relation ... 

In dynamic or quasi-elastic light scattering, a time dependent correlation function (i (0) i (t)) = G2 (t) is measured, where i (0) is the scattering intensity at the beginning of the experiment, and i (t) that at a certain time later. Under the conditions of dilute solution (independent fluctuation of different small volume elements), the intensity correlation function can be expressed in terms of the electric field correlation function gi (t)... 

In Chap. B II.4 we have shown that the angular dependence of the first cumulant of the electric field correlation function can be obtained by integration over the particle-scattering factor. This rule remains valid also for copolymers but is restricted to Gaussian behavior of the subchains. Although the whole q-region can be covered by this integration, which in most cases has to be carried out numerically, it is useful to discuss the... 

In a dynamic light scattering experiment, the measured intensity-intensity time-correlation function g<2)(tc), where tc is the delay time, is related to the normalized electric field correlation function g(1)frc), representative of the motion of the particles, by the Siegert relation [18] ... 

The electric field correlation function can also be expressed in terms of a cumulant expansion ... 

In the next section we first introduce the main ingredients of a FT by mimicking as far as possible what is done in QFT. To be illustrative, in Sec. 3 we show how it is possible to derive the virial expression for the pressure in a FT. In Sec. 4 we show that we have at our disposal some relations between field-field correlation functions that are obtained from symmetry arguments or from the fact that fields are dummy variables. In Sec. 5 on two examples, we derive some results starting from Dyson-like equations. Finally, in Sec. 6, leaving the purely microscopic level we introduce a model showing the existence of a demixion transition at a metal-solution interface using a mean field approximation. Brief conclusions are presented in Sec. 7. 

Dynamic light scattering. Dynamic light scattering measurements provide the electric field correlation function g(q, t), which can be expressed for a polydisperse system of particles as (see [90,91])... 

Electric field correlation function Intensity correlation function Enthalpy... 

The characteristic sample position dependency for the poly(acrylic acid) sample MBAAm-2 is illustrated in Fig. 10. Several measurements of g f) at different sample positions are shown (scattering angle 6 = 90°). After each measurement the sample was rotated in the measuring cell to adjust another position. Each position yields a different intensity correlation function g f) coruiected with a different value for Xp. The resulting field correlation functions represent the fully fluctuating component. They all are described by one curve g " f). 

Maeda and Saito (1973) have calculated the spectrum of light scattered from optically anisotropic rigid rods whose length is >q J. This calculation is even more complex than that for the integrated intensity (zero-time scattered-field correlation function) for the same model given in Appendix 8.B, and will, therefore, not be given here. The interested reader should consult the article by Maeda and Saito. 

Figure 6. The field correlation function for the linear polarization configuration of the instrument,...

What is the bcisis for using spectroscopic methods such as QELSS, FRAP, FRS, or FCS to measure Dg or Dp In each case physical theory links the spectroscopic correlation function to fluctuations in microscopic variables that describe the liquid. For QELSS, the spectrum is determined [16] by the field correlation function t) of the scattered light, which mses from the scattering molecules via... 

Provencher s CONTTN [29], except that the former directly minimizes the sum of the squared differences between experimental and calculated intensity-intensity g2(t) correlation functions using non-linear programming. For a system exhibiting a distribution of relaxation times, the field correlation function g t) (g2(0=gi (t)+l) is... 

The field correlation function g directly reflects any changes in the microstructure of the suspension. In the case of purely diffusive processes, i.e. when the displacement of scattering objects follows Pick s second law, an exponential time dependency is found. 

To quantitate the spectral lineshape, we applied ( ) non-linear least squares methods based on the simplex algorithm. The intensity-intensity correlation function g (field-correlation function g (q,t) via... 

Linear response theory [152] is perfectly suited to the study of fluid structures when weak fields are involved, which turns out to be the case of the elastic scattering experiments alluded to earlier. A mechanism for the relaxation of the field effect on the fluid is just the spontaneous fluctuations in the fluid, which are characterized by the equilibrium (zero field) correlation functions. Apart from the standard technique used to derive the instantaneous response, based on Fermi s golden rule (or on the first Bom approximation) [148], the functional differentiation of the partition function [153, 154] with respect to a continuous (or thermalized) external field is also utilized within this quantum context. In this regard, note that a proper ensemble to carry out functional derivatives is the grand ensemble. All of this allows one to gain deep insight into the equilibrium structures of quantum fluids, as shown in the works by Chandler and Wolynes [25], by Ceperley [28], and by the present author [35, 36]. In doing so, one can bypass the dynamics of the quantum fluid to obtain the static responses in k-space and also make unexpected and powerful connections with classical statistical mechanics [36]. 

U(k) is no longer a non-negative function and yields instability in the mean field structure factor. The mean field correlation function, furthermore, shows a peak at r = 0,... 

The time dependence of S(q,t) is thus entirely determined by a two-particle, two-time correlation function, the field correlation function (or dynamic structure factor) g q, t). Up to a normalization (which is of no consequence in this calculation because normalizations cannot affect the physical time dependence)... 

The time dependence of the field correlation function is therefore determined by the time correlation function of a spatial Fourier component of the concentration, namely... 

X being the particle displacement along q. Berne and Pecora further show for the same very restrictive conditions that the field correlation function is a single exponential... 

Figure 4.1 Demonstration that QELSS does not in general measure the mean-square particle displacement, a) Field correlation function g q,t) expected for bidisperse Brownian particles, b) The solid line is the alleged displacement - log(g q, t)) as computed from part a of this figure and the erroneous Eq. 4.21. The true mean-square particle displacement is shown by the dashed line. The true displacement differs radically at large t from the nominal displacement predicted by the Gaussian approximation, Eq. 4.21.

Dynamic light scattering examines the time dependence of the field correlation function. There is an enormous literature, much contradictory, on direct calculation of g q, t) from the forces between the diffusing particles. This section treats the direct calculation, but only for the simplest of model systems, namely a suspension of colloidal spheres. There are corresponding calculations for nondilute polymer molecules, but these calculations are even more complicated than what follows, in part because neighboring beads on the same chain are required to stay attached to each other. The presentation here shows the tone of the approach, based on papers by Carter and Phillies(25) and Phillies(26,27). Several excellent alternative treatments are available, e.g., Beenakker and Mazur(28,29) and Tokuyama and Oppenheim(30,31). 

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