Atomic distribution function

Figure A2.3.8 Atom-atom distribution functions aiid for liquid water at 25 °C detemrined...

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

Filming of atomic motions in liquids was thus accomplished. More specifically, the above experiment provides atom-atom distribution functions gpv(F, t) as they change during a chemical reaction. It also permits one to monitor temporal variations in the mean density of laser-heated solutions. Finally, it shows that motions of reactive and solvent molecules are strongly correlated the solvent is not an inert medium hosting the reaction [58]. 

The purpose of this section is to show how the above theory can be applied in practical calculations. For the time being, only quasistatic processes have been studied in detail the subsequent discussion will thus focus on this limit. Two sorts of quantities enter into the theory the atom-atom distribution functions r) in a given electronic state j and the corresponding populations nj t). The total atom-atom distribution function gnv(r, t) is then... 

Here all functions fn(P) are atomic distribution functions. We will discuss subsequently the Born approximation of (3.67). 

Now we are ready to write a kinetic equation for the atom distribution function fa(Pt). For this reason we introduce the deviations of the mean values (fluctuations) by... 

Interference between X-rays scattered at different atomic centres occurs in exactly the same way as for an atom. The scattered amplitude becomes a function of an atomic distribution function. In an amorphous fluid, a gas or non-crystalline solid the function is spherically symmetrical and the scattering independent of sample orientation. It only depends on the radial distribution of scattering centres (atoms). 

Partial Interference and Atomic Distribution Function of Liquid Ag—Sn... 

Harasima, A. Atomic distribution functions of liquids. T. Phys. Soc. Tapan 8, 590 (1953). 

Even within the 6-A limit, variations between the predicted atom-atom distribution functions can be quite small. All of the distribution functions predicted by rigid molecule potentials rise from zero more steeply than the experimental curve. There are two reasons for this. First, the rather slow initial increase in the experimental first-neighbor peak is most likely an artifact. Second, the very sharp increase shown by essentially all of the calculated functions is a consequence of the repulsive potential used in these models. This form of repulsion is much too strong, and the softer exponential repulsion gives a slower increase in goo( )-Models that allow for polarization or internal relaxation give a better description of this increase although the value of the maximum is usually overestimated. 

Of interest also are the results concerning deviations of the atomic fluctuations from simple isotropic and harmonic motion. As discussed in Chapt. XI, most X-ray refinements of proteins assume (out of necessity, because of the limited data set) that the motions are isotropic and harmonic. Simulations have shown that the fluctuations of protein atoms are highly anisotropic and for some atoms, strongly anharmonic. The anisotropy and anharmonicity of the atomic distribution functions in molecular dynamics simulations of proteins have been studied in considerable detail.193"197 To illustrate these aspects of the motions, we present some results for lysozyme196 and myoglobin.197 If Ux, Uy, and Uz are the fluctuations from the mean positions along the principal X, Y, and Z axes for the motion of a given atom and the mean-square fluctuations are... 

After 100 ps, the anisotropy introduced by the pump laser has disappeared due to the interaction with the solvent. Thus, with the isotropy of the liquid system before and after laser excitation, the contribution to the signal from the solute can be described by an equation similar to (44), with the quantum distribution of intemuclear positions replaced by paverage(R tp), i.e., the time-dependent I-I atom-atom distribution function. 

From the analysis of the experimental scattering data it was, in particular, possible to extract the time-dependent I-I atom-atom distribution function. 

Diverse investigations of the miscellaneous ternary systems Sn-Pb-Cd, Sn-Ga-In, Sn-Sb-Bi, " Pb-In-Sb, and Pb-Bi-Hg have been undertaken by a number of Russian authors. It was concluded from the results of an ultrasonic study of Sn-Pb-Cd liquid solutions that intermetallic compounds are not formed in this system. Thermodynamic analysis of the Sn-Sb-Bi system shows both positive and negative deviations from ideality in the liquid state.Finally, interpretation of the atomic distribution functions of Pb-In-Sb solutions has led to the conclusion that the melts have a microheterogeneous structure in the fusion... 

A pair correlation function is sometimes used, which is related to the atomic distribution function by ... 

A comparison has been made of the radial atomic distribution functions of three forms of selenium the liquid at 250 and 350 °C, the amorphous solid (quenched from 550 to 0 °C) at 20 °C, and as hexagonal crystals at ambient temperature. In the liquid, the number of parallel atomic chain... 

An alternative approach to the problem is the isotopic substitution method. Here one uses the same alloy prepared with different isotopes having different neutron scattering factors (Mizoguchi et al., 1978 Kudo et al., 1978). In the amorphous substitution method several alloys A, are used, where x is fixed and B or A is replaced by a component of similar size and chemical affinity but different scattering factor (Chipman et al., 1978 Williams, 1982). In these methods it is tacitly assumed that the atomic distribution functions in the alloy series are the same or, at least, do not differ much. 

Laser-Etching Process. In certain photo-etching processes, a laser pointed at a target area rg causes an emission of surface atoms with a gaussian distribution in velocities, i.e., we have the initial state for the single atom distribution function as (see Fig. 2.4)... 

For this system, obtain the solution to the Liouville equation for the time dependent, single atom distribution function under a constant external force. Show that the initial distribution function satisfies the normalization condition... 

In liquids and solutions a chemical shift model should ideally account for the dynamical disordering of the solvent structures. This calls for models that are based on a decomposition of the intermolecular contributions to the shift and a parameterization of these contributions in terms of solvent stmcture, for example, atom-atom distribution functions. Such models should ideally account for the dependence of shift on temperature and pressure. From the distribution functions the shifts can be derived as well as the full photoelectron spectral function, including shift, width, and asymmetry, upon condensation. A basic assumption is that photoionization is vertical, meaning that both initial and final states can be associated with the same nuclear conformation. This approximation is well grounded considering the time scales between the photoelectron process and the rearrangement of the solvent molecules, which means that the solvent is not in equilibrium with respect to the final state. A common assumption behind such models is also that the internal solute nuclear motion is decoupled from external forces. This means that the spectral function/ can be written as a convolution of internal and external parts,/ and/ / respectively,... 

Petkov V, BiUinge SJL, Shastii SD, Himmel B. High-resolntion atomic distribution functions of disordered materials by high-energy X-ray diffraction. J Non-Cryst Solids 2001 293-295 726-730. 

One way which has been used extensively in the past in order to gain information on the local order in amorphous polymers consists of analyzing the structure in terms of an atomic distribution function. It is governed both by the atoms belonging to the same chain (intramolecular contributions) as well as by atoms belonging to different chains (intermolecular contributions). This is shown schematically in Figure 6. 

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