Neutron pair distribution function

C.E. White, J.L. Provis, A. Llobet, T. Proifen, J.S.J. van Deventer, Evolution of local structure in geopolymer gels an in situ neutron pair distribution function analysis, J. Am. Ceram. Soc. 94 (2011) 3532-3539. 

Fig. 19 Neutron pair distribution function analysis of (a) bulk BaTiOs and (b) 5 nm BaTiOs nanoparticles. Clear differences are noted in the profile of both samples. The bulk sample is well described by the tetragonal P mm model above 4 nm, while the split peak for the Ti-O distance at 2 nm is better fit to a rhombohedral Kim model (see inset). For the 5 nm sample, the sample is well described by the P mm tetragonal model. From (b) it is also possible to see the contributions made by the benzyl alcohol capping group. 2010 American Chemical Society. 

The structure of a liquid is conventionally described by the set of distributions of relative separations of atom pairs, atom triplets, etc. The fundamental basis for X-ray and neutron diffraction studies of liquids is the observation that in the absence of multiple scattering the diffraction pattern is completely determined by the pair distribution function. 

The diffraction experiments to collect pair-distribution functions (PDF) are typically done at synchrotrons or neutron spallation sources since high quahty data at large momentum transfers Q = AnsmBIl. > 20 A- are required to reduce termination errors at low real-space distances. The atomic PDF G(r) is defined as... 

The importance of the pore size in Prussian blue analogues is supported by differential pair distribution function analysis of X-ray and neutron scattering data of hydrogen- and deuterium-loaded Mn3[Co(CN)6]2 [119]. This shows that no evidence for adsorption interactions with unsaturated metal sites exists and that the hydrogen molecules are disordered about the center of the pores defined by the cubic framework. In conclusion, experimental results indicate that optimum pore dimensions in Prussian blue analogues are predominantly responsible for the heat of adsorption at low loadings rather than the polarizing effect of open metal sites. 

The modified potential function method is the implicit way to include the polarizability in the calculation by modifying the potential function based on the physical properties in solid state so that it reproduces the pair distribution function obtained by the X-ray and neutron diffraction measurements in the liquid state. 

The isotropic nature of a liquid implies that any structure factor, S(k), obtained from a scattering experiment (typically X-ray or neutron) on that liquid will contain no angular dependence (of the molecules). Thus, the Fourier transform of any S(k) will yield a radial distribution function. Recently developed techniques of isotopic substitution [5-7] have been utilized in neutron diffraction experiments in order to extract site-site partial structure factors, and hence site-site radial distribution functions, gap(r). Unfortunately, because g p(r) represents integrals (convolutions) over the full pair distribution function, even a complete set of site-site radial distribution functions can not be used to reconstruct unambiguously the full molecular pair distribution function [2]. However, it should be mentioned at... 

Probably the most notable work on the structure in liquid water based upon experimental data has been that of Soper and co-workers [6,8,10,30,46,55]. He has considered water under both ambient and high temperature and pressure conditions. He has employed both the spherical harmonic reconstruction technique [8,46] and empirical potential structure refinement [6,10] to extract estimates for the pair distribution function for water from site-site radial distribution functions. Both approaches must deal with the fact that the three g p(r) available from neutron scattering experiments provide an incomplete set of information for determining the six-dimensional pair distribution function. Noise in the experimental data introduces further complications, particularly in the former technique. Nonetheless, Soper has been able to extract the principal features in the pair (spatial) distribution function. Of most significance here is the fact that his findings are in qualitative agreement with those discussed above. 

The use of difference methods offers a means whereby a detailed picture of ionic hydration can be obtained 22). For neutron diffraction, the first-order isotopic difference method (see Section III,A) provides information on ionic hydration in terms of a linear combination of weighted ion-water and ion-ion pair distribution functions. Since the ion-water terms dominate this combination, the first-order difference method offers a direct way of establishing the structure of the aquaion. In cases for which counterion effects are known to occur, as, for example, in aqueous solutions of Cu + or Zn +, it is necessary to proceed to a second difference to obtain, for example, gMX and thereby possess a detailed knowledge of both the aquaion-water and the aquaion-coun-terion structure. 

Ci and Cj are the concentrations of atomic species i and j, and bi and bj are the corresponding neutron scattering lengths. gy r) is the pair distribution function, defined such that the probability of finding an atom of species j within the range of distances r to r + dr from an atom of atomic species i is equal to Cj4w gij(r)dr. It is common to define the quantity T(r) as... 

The pair distribution function leads to the pair correlation function, which illustrates how the local order found near a given molecule is lost as distance from the molecule increases. This quantity is of fundamental theoretical interest and may be determined in X-ray and neutron scattering experiments, as discussed below. The definition of the pair correlation function g r j) is... 

In the case of neutron diffraction, the radiation is scattered by the atomic nuclei, not by the electrons. It turns out that nucleons such as H and have very different scattering amplitudes. This means that isotope effects are very important in developing experimental strategies. Soper and Phillips [8] used data for the structure function obtained in mixtures of normal and heavy water to extract values of the partial structure factors for water. In this way they were able to determine all of the pair distribution functions for water from their diffraction data. These are gHnW. g oHW) and gooW- More details of their experimental results are given in section 2.10. 

Figure 4.1 Pair distribution function of liquid argon at 84 K obtained by neutron scattering measurement.

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. 

Narten AH, Thiessen WE, Blum L (1982) Atom pair distribution functions of liquid water at 25°C from neutron diffraction. Science 217 1033-1034... 

Further information was obtained from the radial distribution function p(r), which is the probability of finding another atom at a distance r from the reference atom. Normally, the Si-O, Si-Si and 0-0 distances cannot be separated, and the interpretation of p(r) is therefore difficult. With three different types of radiation (X-ray, neutrons, electrons), a separation of the three partial radial distribution fimctions is possible. In Figure 18.1, the pair distribution functions Gsj-si( ), Gsi-o( ) and Go-o( ) of SiO are shown,where G is the probability of finding an element j in a distance r from atom i. 

Among the techniques that probe the average or long-range structure, powder X-ray diffraction (PXRD) and neutron diffraction (ND) will be briefly discussed. Techniques that provide local, atomistic information that will be mentioned are nuclear magnetic resonance (NMR) spectroscopy. X-ray photoelectron spectroscopy, X-ray absorption spectroscopy, and pair distribution function (PDF) analysis. A brief introduction to the underlying theory of each technique will be provided along with relevant examples to illustrate the type of information that can be... 

Breger J, Dupre N, Chupas P, Lee P, Proffen T, Parise J, Grey C (2005) Short- and long-range order in the positive electrode material, Li(NiMn)(0.5)O-2 a joint X-ray and neutron diffraction, pair distribution function analysis and NMR study. J Am Chem Soc... 

The original and most extensively employed method of evaluating g R) experimentally is the study of the X-ray diffraction pattern by liquids. Recently, diffraction of neutrons has been found increasingly useful for this purpose. The principal idea of converting diffraction patterns into pair distribution functions is common to both methods, though they differ both in experimental detail and scope of information that they provide. 

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