Correlation function orientational

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... 

Other orientational correlation coefficients can be calculated in the same way as tf correlation coefficients that we have discussed already. Thus, the reorientational coiTelatio coefficient of a single rigid molecule indicates the degree to which the orientation of molecule at a time t is related to its orientation at time 0. The angular velocity autocorrelatio function is the rotational equivalent of the velocity correlation function ... 

The above results show that the structure of the system with the molecules self-assembled into the internal films is determined by their correlation functions. In contrast to simple fluids, the four-point correlation functions are as important as the two-point correlation functions for the description of the structure in this case. The oil or water domain size is related to the period of oscillations A of the two-point functions. The connectivity of the oil and water domains, related to the sign of K, is determined by the way four moleeules at distanees eomparable to their sizes are eorrelated. For > 0 surfactant molecules are correlated in such a way that preferred orientations... 

Viovy,J.L. and Monnerie, L. Fluorescence Anisotropy Technique Using Synchroton Radiation as a Powerful Means for Studying the Orientation Correlation Functions of Polymer Chains. Vol. 67, pp. 99—122. 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

Of course, knowledge of the entire spectrum does provide more information. If the shape of the wings of G (co) is established correctly, then not only the value of tj but also angular momentum correlation function Kj(t) may be determined. Thus, in order to obtain full information from the optical spectra of liquids, it is necessary to use their periphery as well as the central Lorentzian part of the spectrum. In terms of correlation functions this means that the initial non-exponential relaxation, which characterizes the system s behaviour during free rotation, is of no less importance than its long-time exponential behaviour. Therefore, we pay special attention to how dynamic effects may be taken into account in the theory of orientational relaxation. 

For one-dimensional rotation (r = 1), orientational correlation functions were rigorously calculated in the impact theory for both strong and weak collisions [98, 99]. It turns out in the case of weak collisions that the exact solution, which holds for any happens to coincide with what is obtained in Eq. (2.50). Consequently, the accuracy of the perturbation theory is characterized by the difference between Eq. (2.49) and Eq. (2.50), at least in this particular case. The degree of agreement between approximate and exact solutions is readily determined by representing them as a time expansion... 

The behaviour of orientational correlation functions near t = 0 carries information on both free rotation and interparticle interaction during collisions. In the impact approximation this information is lost. As far as collisions are considered as instantaneous, impact Eq. (2.48) holds, and all derivatives of exponential Kj(t) have a break at t = 0. However,... 

Fig. 2.5. Time evolution of the second derivative of the orientational correlation function. The long-time asymptotics described by impact theory is shadowed.

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

According to Eq. (2.13), the spectra we are interested in are given by a Fourier transform of the orientational correlation functions of the corresponding order Similarly to what was done in Chapter 3, the correlation functions for linear and spherical molecules may be represented as a superposition of the partial (marginal) components... 

Euler angles Q define the e orientation, and an orientational correlation function of 1th order is introduced in the usual way ... 

It must be stressed that every Un(t,J) brings dq(0) to a different coordinate system. Consequently, the averaged operator (A7.13) is actually a weighted sum of the quantities in differently oriented reference systems. It can nevertheless be used to find the scalar product (A7.7), that is the orientational correlation function. 

Chandler D. Translational and rotational diffusion in liquids. I. Translational single-particle correlation functions. J. Chem. Phys. 60, 3500-507, (1974). Translational and rotational diffusion in liquids. II. Orientational single-particle correlation functions. J. Chem. Phys. 60, 3508-12 (1974). 

Here the summation is over molecules k in the same smectic layer which are neighbours of i and 0 is the angle between the intermolecular vector (q—r ) projected onto the plane normal to the director and a reference axis. The weighting function w(rjk) is introduced to aid in the selection of the nearest neighbours used in the calculation of PsCq). For example w(rjk) might be unity for separations less than say 1.4 times the molecular width and zero for separations greater than 1.8 times the width with some interpolation between these two. The phase structure is then characterised via the bond orientational correlation function... 

Fig. 8. The distance dependence of the bond orientational correlation function gs (r ) found for the mesogen GB(4.4, 20.0, 1, 1) in the smectic A (.) and the smectic B (—) phases...

Reorientational relaxation times, tJ can be estimated from the assumed exponential decay of the orientational correlation functions cf(/), defined as the average of the / I.egendre polynomial of cos 0, ... 

For the analysis of the dynamical properties of the water and ions, the simulation cell is divided into eight subshells of thickness 3.0A and of height equal to the height of one turn of DNA. The dynamical properties, such as diffusion coefficients and velocity autocorrelation functions, of the water molecules and the ions are computed in various shells. From the study of the dipole orientational correlation function... 

All these features were observed experimentally for solutions of 3-amino-/V-methylphthalimide, 4-amino-/V-methylphthalimide, and for nonsubstituted rhoda-mine. The results were observed for cooled, polar solutions of phthalimides, in which the orientational relaxation is delayed. Exactly the same spectral behavior was observed [50] by picosecond spectroscopy for low viscosity liquid solutions at room temperature, in which the orientational relaxation rate is much higher. All experimental data indicate that correlation functions of spectral shifts Av-l(t), which are used frequently for describing the Time Dependent Stokes Shift, are essentially the functions of excitation frequency. 

As pointed out by Stribeck [139,171] g (x) is, as well, suitable for the study of oriented microfibrillar structures and, generally, for the study of ID slices in deliberately chosen directions of the correlation function. This follows from the Fourier-slice theorem and its impact on structure determination in anisotropic materials, as discussed in a fundamental paper by Bonart [16]. 

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