Correlation function interatomic

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

Fig. 6.86. Oxygen-oxygen pair correlation function obtained from molecular dynamic simulations on the adsorbed layer of a Pt(100) surface. Ax and Ay are the projections of the interatomic distances in the x- and indirections, respectively. They reflect the positions of the oxygen atoms on the top site of the platinum lattice, and the pronounced form of the peaks refers to their relatively small displacement. (Reprinted from E. Spohr, G. Toth, and K. Heinzinger, Electrochim. Acta 41 2131, copyright 1996, Fig. 10a, with peimission from Elsevier Science.)...

The crux of the method is that the relative positions of the heavy atoms in the two different crystals must be known. When nothing detailed is known of the molecular structure, it is not easy to obtain this information. Perutz (1956) devised methods based on Fourier syntheses of the Patterson type referred to in a later section, which give interatomic vector maps the combined data for the two heavy-atom derivatives, in special correlation functions, give the relative positions... 

Concerning the interatomic correlation function B k, t) we may transform Eqn. (3.1.18) into an integral from zero to infinity, provided T j t <... 

Concerning the interatomic correlation function we shall limit... 

The strength of the water-metal interaction together with the surface corrugation gives rise to much more drastic changes in water structure than the ones observed in computer simulations of water near smooth nonmetallic surfaces. Structure in the liquid state is usually characterized by pair correlation functions (PCFs). Because of the homogeneity and isotropy of the bulk liquid phase, they become simple radial distribution functions (RDFs), which do only depend on the distance between two atoms. Near an interface, the PCF depends not only on the interatomic distance but also on the position of, say the first, atom relative to the interface and the direction of the interatomic distance vector. Hence, considerable changes in the atom-atom PCFs can be expected close to the surface. 

Fig. 15. Anisotropic oxygen-oxygen pair correlation functions gooip, for the adsorbate molecules (left), the molecules in the second layer (middle), and the molecules in the bulk-like center of the water lamina (right) between Pt(lOO) surfaces, p is the transversal and z the normal part of the interatomic distance.

We will describe integral equation approximations for the two-particle correlation functions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27. 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the h q)emetted chain (HNC) approximation for charged systems, are readily obtained by fast Fourier transform methods... 

It is reasonable (see McTague and Birnbaum, 1971) to regard each pair of atoms (ij) in the fluid as a quasilinear diatomic molecule with an axis ry defined by the interatomic vector ry. Thus to each of the N(N-l) pairs (ij) of atoms we ascribe the polarizability tensor given by Eq. (14.2.1a) and a center-of-mass distance Ry = (r,- + iy)/2. It then follows from Section 7.B that the depolarized spectrum is the Fourier transform of the time-correlation function... 

At the present time, of all EXAFS-like methods of analysis of local atomic structure, the SEES method is the least used. The reason is that the theory of the SEES process is not sufficiently developed. However the standard EXAES procedure of the Fourier transformation has been applied also to SEES spectra. The Fourier transforms of MW SEES spectra of a number of pure 3d metals have been compared with the corresponding Fourier transforms of EELFS and EX-AFS spectra. Besides the EXAFS-like nature of SEES oscillations shown by this comparison, parameters of the local atomic structure of studied surfaces (the interatomic distances and the mean squared atomic deviations from the equilibrium positions [12, 13, 15-17, 21, 23, 24]) have been obtained from an analysis of Fourier transforms of SEES spectra. The results obtained have, at best, a semi-quantitative character, since the Fourier transforms of SEES spectra differ qualitatively from both the bulk crystallographic atomic pair correlation functions and the relevant Fourier transforms of EXAFS and EELFS spectra. 

The total pair correlation functions for the neutron diffraction, MD, and RMC of the all samples are compared in Fig. 4.7. While there is broad similarity between the MD and the neutron data, it is apparent that there are significant interatomic interactions that are not accounted for with the simple Buckingham potentials described in Sect. 4.2. The RMC total structme factors are in very good agreement with the corresponding neutron diffraction structure factors, as shown in Fig. 4.7a. 

It also becomes difficult conceptually to describe excitations in the liquid when the wavelength becomes comparable with the interatomic spacing. These are no longer waves in the usual sense, that is, oscillations in the density. One way to confront this issue is to choose as dynamic variables the longitudinal current fluctuations described by a correlation function //(o>, Q) = co S(Q, The current-current... 

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