Partial pair distribution functions

Figure 4. Partial pair-distribution functions gij(r) as computed from an RMC procedure in liquid acetonitrile for N-CH3 pairs (stars) and contributions firom the nearest n neighbors of a central molecule (full lines).

Expression (3.23) allows us to determine the partial pair distribution functions at a point by measurement of the intensity of the energy losses in a simple two-component system, particularly across the gas-liquid critical lines. Also for a statistically determined system, (3.23) allows us to determine the charge density distribution in the neighborhood of a localized excitation in a binary complex. 

In organic polymers, there is always more than one kind of atoms present. Even a hydrocarbon polymer, the simplest among polymers in chemical structure, is made up of two types of atoms, carbons and hydrogens. To specify the short-range order in a hydrocarbon polymer, three partial pair distribution functions, g cc(r), gHH(r)> and ch(t)> are needed. The definition of gch(/0, for example, is as follows. We pick an arbitrary C atom and look at the volume element dr at a position displaced from it by r. We note the number of H atoms present in this volume element to be equal, on the average, to ncw(r) dr. The function gcn(r) is obtained on normalization as... 

With m atomic species, there are m(m + l)/2 partial pair distribution functions gap(r) that are distinct from each other. When only a single intensity function I(q) is available from experiment, no method of ingenious analysis can lead to determination of all these separate partial pair distribution functions from it. Different and independent intensity functions I(q) may be obtained experimentally when measurements are made, for example, with samples prepared with some of their atoms replaced by isotopes. When a sufficient number of such independent intensity functions is available, it is then possible to have all the partial pair distribution functions gap(r) individually determined, as will be elaborated on shortly. When only a single intensity function is available from x-ray or neutron scattering, however, what can be obtained from a Fourier inversion of the interference function is some type of weighted average of all gap (r) functions. The exact relationship between such an averaged function and gap(r)s is as follows. 

The x-ray atomic scattering factor f(q) of a C atom is about six times that of an H atom, and therefore when a hydrocarbon polymer is studied by means of x-ray scattering, gcc(r) makes the predominant contribution to g(r). In such a case, g(r) may be taken as a reasonable approximation to gcc(r), which is usually the object of primary interest. However, when the weighting factors wa for different atomic species are comparable, each of the partial pair distribution functions gap(r) makes a substantial contribution to g(r), and this makes meaningful interpretation of the latter difficult. 

For each of the three interference functions /x( )> h( )> and b(q) determined as mentioned above, Equation (4.19) is applicable, where the iap (q)s [i.e., cc( ) ch( )> and i hh( )] are common among the three cases and are unknown and yet to be determined. The weighting factors wa, on the other hand, depend, as seen from (4.16), on the scattering lengths ba and assume different values in the three measurements. For a given q value, Equation (4.19) therefore constitutes a set of simultaneous linear equations with three unknowns iap(q), whose values can be determined by solving the simultaneous equations. The partial pair distribution functions gap(r) are then obtained from them by Fourier inversion as implied by (4.18). 

The structure of binary alloys can be handled similarly the partial structure factor, nd the partial pair distribution function, related by ... 

Fig. 1.1 The Faber-Ziman Sap k) a, /3 = M, X) and Bhatia-Thornton Su k) (/, / = N, C) partial structure factors for liquid and glassy ZnCl2. The points with vertical (black) error bars are the measured functions in (a) and (c) for the liquid at 332(5) °C [ 16] and in (b) and (d) for the glass at 25(1) °C [15, 16]. The solid (red) curves are the Fourier backtransforms of the corresponding partial pair-distribution functions after the unphysical oscillations at r-values smaller than the distance of closest approach between the centres of two atoms are set to the calculated Unlit at r = 0. The broken (green) curves in (a) are from the polarisable ion model of Sharma and Wilson [63] for the Uquid at 327 °C...

Fig. 1.2 The Faber-Ziman partial structure factors Sap(k) and partial pair-distribution functions gafi (r) (a, /S = M, X) as calculated for models using two different values for the anion polarisability ax [61]. The curves in (a) and (b) correspond to a rigid ion model (RIM) with ax = 0, while the curves in (c) and (d) correspond to a polarisable ion model (PIM) with ax = 20 au. The introduction of anion polarisability leads to the appearance of an FSDP in S mm (k) at kpsop 1.2 A and to an alignment of the principal peaks in aU three Safi(k) functions atkpp 2 A. The alignment of the principal peaks in (c) arises from in-phase large-r oscillations in the ga r) functions shown in (d)...

Comparing structural information from X-ray and neutron diffraction provides a very valuable way to validate MD simulation results of glasses. In some simple systems, the partial pair distribution function or partial structure factors of all atom pairs can be determined experimentally and they provide excellent validations for simulated structures. However, as the composition becomes more complicated and more elements included, larger number of pair contributions will complicate the comparison and the validation becomes more and more difficult in multicomponent glass systems. For example, for binary oxides, e.g. sodium silicate, there are six partial pair distribution functions, but for a four component systems, for example the bioactive glass composition, there are a total of fifteen partials contributions. The overlap between partial contributions makes it very challenging to assign the peaks and to determine the quality of comparison and hence the validation of the simulated structure models. 

It is also possible to compare the reciprocal space structure factors, either from neutron or X-ray diffraction measiu ements. The partial structure factors are first calculated through Fourier transformation the partial pair distribution functions using... 

The local structure of the modified network can be characterized further by computing partial pair distribution functions gap r). We now can count all P-atoms as nearest neighbors of an o-atom which are closer in distance from a selected a-atom than the location of the first minimum of gop(r). In this way, one can characterize the corresponding coordination number z for... 

PPDF partial pair distribution function SHS self-propagating high-temperature... 

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