Total structure factors

The data taken is normally presented as the total structure factor, F(Q). This is related to the neutron scattering lengths hi, the concentrations C , and the partial structure factor Sy(Q) for each pair of atoms i and j in the sample, by Equation 4.1-1 ... 

Bowron et al. [11] have performed neutron diffraction experiments on 1,3-dimethylimidazolium chloride ([MMIM]C1) in order to model the imidazolium room-temperature ionic liquids. The total structure factors, E(Q), for five 1,3-dimethylimidazolium chloride melts - fully probated, fully deuterated, a 1 1 fully deuterated/fully probated mixture, ring deuterated only, and side chain deuterated only - were measured. Figure 4.1-4 shows the probability distribution of chloride around a central imidazolium cation as determined by modeling of the neutron data. 

The atomic PDF is related to the probability to find a spherical shell around a generic atom (scattering center) in the material - it is defined as G(r) = Anp[p r)-p(, where p r) and po are, respectively, the local and average atomic number densities and r the radial distance. G(r) is the Fourier transform of the total structure factor Sid). ... 

The total structure factor, finally, is given by a product of SRoLe(Q>f) ... 

In analogy to the definition of the total charge density, we define the total structure factor Ftotal(H), which includes both the nuclei and the electrons, and is, excluding thermal effects, given by... 

To take into account the disparity in electrons, and hence scatting power, for the various atoms in the unit cell, the atomic number Zj has been introduced as a means of defining amplitudes for the component waves. The total diffracted wave for the entire crystal will be the product of the equation above with the total number of unit cells in the crystal.1 For a single unit cell, like that in Figures 5.12a and 5.12b, an atom s contribution to the total structure factor of one unit cell can also be illustrated in vector terms as in Figure 5.13. To put the expression in the correct units, it is necessary to multiply the summation by a constant, the volume of the unit cell. Here, that is simply V = a x b x c. Thus... 

There is an equivalent relationship between the partial structure functions, SapiQ) and the total structure function, S Q). In the Faber-Ziman scheme, the weighting is chosen such that each partial structure factor, Sa iQ), has the same property of the total structure factor that as Q go, So,p(Q)= 1- This means that we can define the Fourier couple between S iQ) and Ga (r) in the same way as for the total structure function, i.e. ... 

Fig. 5.7. Total structure factor S(K) of quench-condensed Au-Sb alloys [5.50]. The parameters indicate the Sb-content and Z, respectively. 2kF is given by solid vertical lines...

The above derivation can also be applied to colloidal or polymer-based liquids and is then used to calculate the so-called form factors of soft matter samples. The major difference between a monoatomic liquid and a polymer chain in the melt or in solution is that the total structure factor consists of two parts. The first is the inter-particle structure factor and the second the intra-particle structure factor. This second part is also often called the particle form factor P(q). Using Eq. (2.38) it is straightforward to calculate P(q) for a given soft matter sample. A good example is the form factor of a single polymer coil in a melt [88, 92]. The pair correlation function of such a coil is given by... 

If a structural reaction is rapidly and uniformly initiated in a crystal, what will the likely effects be on the Laue x-ray intensities The very weak intermolecular interactions characteristic of protein crystals make it implausible that existence of a certain tertiary structure in one molecule would favor a particular tertiary structure in its neighbors. That is, all molecules in the crystal are likely to behave independently of each other, as they do in solution. Their populations will evolve smoothly in time in a manner governed by the reactant concentrations and the rate constants. To preserve the x-ray diffraction pattern, spatial coherence must be maintained between molecules, but temporal coherence need not be. Thus, at any instant the crystal will contain many different conformations, each representing a different structural intermediate. The total structure factor for a particular reflection will be a weighted vector sum of the time-independent structure factors of each conformation, where the weights are the time-dependent fractional occupancies. In favorable cases [26), it may be possible to extract the individual time-independent structure factors from this sum, and hence to obtain directly the structure of each intermediate. 

RMC is a variation of the standard Metropolis Monte Carlo (MMC) method (Metropolis et al., 1953 see also Chapters l and 5). The principle is that we wish to generate an ensemble of atoms, i.e. a structural model, which corresponds to a total structure factor (set of experimental data) within its errors. These are assumed to be purely statistical and to have a normal distribution. Usually the level and distribution of statistical errors in the data is not a problem, but systematic errors can be. We shall initially consider materials that are macro-scopically isotropic and that have no long range order, i.e. glasses, liquids and gases. The basic algorithm, as applied to a monatomic system with a single set of experimental data, is as follows ... 

Calculate the difference between the measured total structure factor, Ae(Q), and that determined from the configuration, A (Q),... 

Move one atom at random. Calculate the new (n) radial distribution function, (r), and total structure factor, A (Q), and... 

As this procedure is iterated x will initially decrease until it reaches an equilibrium value about which it will fluctuate. The resulting configuration should be a three-dimensional structure that is consistent with the experimental total structure factor within experimental error. Statistically independent configurations may then be collected. In MMC, configurations are normally assumed to be independent if separated by N accepted moves, but in practice we normally use at least 5TV moves. 

The difference between RMC and MMC is simply that in RMC the difference between calculated and measured total structure factors (x2) is sampled (minimized), while in MMC the potential energy (UjkT) is sampled. Otherwise the two algorithms are identical. It is particularly important that RMC uses a proper Markov chain, so that the final structure should be independent of the initial configuration. This makes the method an ab initio structural determination akin to those discussed in Chapter 5 rather than a refinement. However, in some circumstances the method is deliberately used as a refinement. This involves only accepting moves that decrease x2 in step (6) and corresponds to setting T = 0 in MMC. The use of the RMC algorithm in practice has been discussed in detail elsewhere (McGreevy et al., 1990, 1994). 

Two of the most common systematic errors that occur in experimental total structure factors are small normalization errors in the form of additive and multiplicative constants, particularly for X-ray data because of the Q dependence of the form factor. Such errors can be accounted for within the RMC algorithm, since the RMC structure factors are correctly normalized (given the correct density, atomic composition, etc.). The required multiplicative factor which minimizes x2 is given by... 

The algorithm described above is specifically for modelling a single set of diffraction data which could be obtained using either X-rays, neutrons or electrons. The fit may be either to the structure factor or to the radial distribution function, though the former is recommended because the distribution of errors in the latter may be highly non-uniform. In practice a fit is normally made first to the radial distribution function, then to a subset of the total structure factor points, and finally to all the structure factor points. This considerably reduces the time required. 

The total structure factors that have been referred to above are obtained by measuring the total scattered intensity at a particular angle, i.e. including both... 

In normal diffraction studies of crystalline materials the structure factor that is measured is only elastic (Bragg) scattering, though often total scattering is actually measured and it is simply assumed that all sharp features are elastic scattering and all broad features are not broad features are then simply subtracted from the measured scattering pattern. Here the elastic structure factor will be referred to as S(Q) to distinguish it from the total structure factor... 

Figure 6.3 Experimental total structure factor for YBa2Cu306 95 (+), direct RMC fit (solid curve) and the difference (lowest solid curve). The solid horizontal line shows the incoherent scattering contribution. The sloping solid line is the diffuse scattering cafculated from the RMC modef and the sloping dashed line is the background fitted by...

The CTR shape is sensitive to the termination of the crystal surface. Calculations show (Fig. 5B) that CTR data are sensitive not only to the presence of the crystal termination but also to the detailed termination of the lattice. Here, we compare the scattering intensity for a semi-infinite lattice in which the outermost surface layer is ideally terminated with a bulk-like termination, or has been modified, either by its position, ds, or its scattering strength, fs. The total structure factor of the crystal with a modified surface is just the sum of individual structure factors for all atoms in the crystal (a table of commonly used structure factors is given in Appendix 3). Conceptually, this quantity can be broken into two parts, consisting of contributions from the modified surface layer, Fsurf, and from the semi-infinite substrate, Fsub -... 

The structure factor in Equation (14) can be rewritten in a form that allows for substantial conceptual simplification. Any mineral-fluid interface consists of three parts, the semi-infinite bulk crystal structure that is known in principle a priori, the distorted surface region (whose depth is not known a priori) and the fluid layer above the surface. The total structure factor, Ftot, can be written as follows ... 

The quantity measured in such an experiment is the differential neutron scattering cross section da/dQ, of which the interference part (or total structure factor) can be expressed as a linear sum of the partial structure factors, S(Q). The structure factor describes the spatial distribution of scattering centres (the atomic nuclei) of the sample in question. Thus, in the total structure factor, all distances between all scatterers are present, weighted according to the concentration of each particular type of atom, c, and their scattering length, b. The differential neutron scattering cross section can be written as ... 

The mean cross-section is a function of the total structure factor Fr(k)... 

The total structure factor F (k) contains the simultaneous information on aD correlation functions of adjacent atoms, intermolecular as well as intramolecular. As a consequence, reports in the literature have been limited to solvents with few atoms, e.g. HjO. Furthermore, the measured signal is very insensitive towards changes in solvent structure so that only highly concentrated solutions can be examined. The impottant results obtained from diffraction methods are the. .. OHj distances, the tilt angles between the. .. O axis and the plane of the water molecule and hydration numbers 241.242) configuration of CdIi"JT . S complexes (S == HjO or dimethylsulfoxide) were intestigated recently... 

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