Structure factor, description

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

The obtained static structure factors agree well with the experimental ones [4], all trends of the peak positions are reproduced correctly. There are only small deviations from the experiments (i) due to the pseudopotential (slighly too small bond lengths which correspond to slightly too large peak positions in the reciprocal lattice) and (ii) correct positions but a wrong trend in the heights of the prepeaks. For a detailed description see Ref. [7]. 

Under the simplifying assumption that the reflexions are independent of each other, K, can be written as a product over reflexions for which experimental structure factor amplitudes are available. For each of the reflexions, the likelihood gain takes different functional forms, depending on the centric or acentric character, and on the assumptions made for the phase probability distribution used in integrating over the phase circle for a discussion of the crystallographic likelihood functions we refer the reader to the description recently appeared in [51]. 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

The crossover from 0- to good solvent conditions in the internal relaxation of dilute solutions was investigated by NSE on PS/d-cyclohexane (0 = 311 K) [115] and on PDMS/d-bromobenzene(0 = 357K) [110]. In Fig. 45 the characteristic frequencies Qred(Q,x) (113) are shown as a function of t = (T — 0)/0. The QZ(Q, t) were determined by fitting the theoretical dynamic structure factor S(Q, t)/S(Q,0) of the Zimm model (see Table 1) to the experimental data. This procedure is justified since the line shape of the calculated coherent dynamic structure factor provides a good description of the measured NSE-spectra under 0- as well as under good solvent conditions. 

WAXS the scattering is completely described by the interactions of neighboring atoms along a single chain, the so-called single-chain structure factor. Cf. descriptions of the Ruland method [14] in textbooks [7,22],... 

In the same paper (Yamamoto 1996) an authoritative description is given of several interrelated topics such as super-space group determination, structure determination, indexing of diffraction patterns of quasicrystals, polygonal tiling, icosahedral tiling, structure factor calculation, description of quasicrystal structures, cluster models of quasicrystals. 

Later chapters will deal with a more complete description of the diffraction process, but we now have enough to discuss the selection of radiations and techniques. If the structure factor and scattering strength of the radiation are high, the penetration is low and the rocking curve is broad. This is the case with electron radiation. For X-rays and even more for neutrons, the structure and absorption factors are small, penetration is high and rocking curves are narrow. These factors have three main consequences for X-rays and also for neutrons ... 

Figure 3.10 compares the same experimental data, with a best fit to the Rouse model (Eq. 3.19). Here a good description is observed for small Q-values (Q<0.14 A 0, while at higher Q important deviations appear. Similarly, the simulations cannot be fitted in detail with a Rouse structure factor. Recently this result was confirmed by an atomistic computer simulation on PE molecules of different lengths. Again, at high Q the Rouse model predicts a too-fast decay forSpair(Q,0 [53]. 

Inserting the Rouse rate W(, 3Q9 K)=(7 0.7)xl0 AVns (Table 3.2) obtained from single chain structure factor measurement into Eq. 3.18 the solid line is obtained. It quantitatively corroborates the correctness of the Rouse description at short times. The data also reveal clearly a transition to a law, though Eq. 3.36 would predict the dotted line. The discrepancy explains itself in considering the non-Gaussian character of the curve-linear Rouse motion (Eq. 3.38). Fixing and d to the values obtained from the single chain struc-... 

Fig. 4.3 Scaling representation of the spin-echo data at the first static structure factor peak Qmax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve for 1,4-polybutadiene at Qmax=l-48 A L The scale r(T) is taken from a macroscopic viscosity measurement [130]. Inset Temperature dependence of the non-ergodicity parameter/(Q) near the lines through the points correspond to the MCT predictions (Eq. 4.37) (Reprinted with permission from [124]. Copyright 1988 The American Physical Society)...

A phenomenological description of the dynamic structure factor at this Q-value by KWW functions ... 

Figure 5.3 presents NSE results obtained on PIB at 470 K together with a fit with the Rouse dynamic structure factor Eq. 3.19. The Rouse model provides a good description of the spectra for Q<0.15 A In this range, the elementary... 

There is an important case which is intermediate between small bounded systems and macroscopic fully extended systems, namely the description of the surface region of a macroscopic metal. The correlation functions which describe density fluctuations in the surface region are extremely anisotropic and of long range, very unlike their counterparts in the bulk, and the thermodynamic limit must be taken with sufficient care. Consider the static structure factor for a large system of N particles contained within a volume Q,... 

In these terms, I will restate a central problem of crystallography In order to determine a structure, we need a full-color version of the diffraction pattern — that is, a full description of the structure factors. But diffraction experiments give us only the black-and-white version, the intensities of the... 

As I stated in Chapter 2, computation of the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to convert a Fourier-series description of the reflections to a Fourier-series description of the electron density. A reflection can be described by a structure-factor equation, containing one term for each atom (or each volume element) in the unit cell. In turn, the electron density is described by a Fourier series in which each term is a structure factor. The crystallographer uses the Fourier transform to convert the structure factors to p(.x,y,z), the desired electron density equation. 

Equation (5.18) tells us, at last, how to obtain p(pc,y,z). We need merely to construct a Fourier series from the structure factors. The structure factors describe diffracted rays that produce the measured reflections. A full description of a diffracted ray, like any description of a wave, must include three parameters amplitude, frequency, and phase. In discussing data collection, however, I mentioned only two measurements the indices of each reflection and its intensity. Looking again at Eq. (5.18), you see that the indices of a reflection play the role of the three frequencies in one Fourier term. The only measurable variable remaining in the equation is Fhkf Does the measured intensity of a reflection, the only measurement we can make in addition to the indices, completely define Fhkp Unfortunately, the answer is "no."... 

First consider Eq. (5.18) (p in terms of Fs). Each term in this Fourier-series description of p(x,y,z) is a structure factor representing a single X-ray reflection. 

Equation (5.18) tells us how to calculate p(jc,y,z) simply construct a Fourier series using the structure factors Fhkl. For each term in the series, h, k, and 1 are the indices of reflection hkl, and Fhkl is the structure factor that describes the reflection. Each structure factor Fhkl is a complete description of a diffracted ray recorded as reflection hkl. Being a wave equation, Fhkl must specify frequency, amplitude, and phase. Its frequency is that of the X-ray source. Its amplitude is proportional to (- j /)1/2, the square root of the measured intensity Ihkl of reflectionhkl. Its phase is unknown and is the only additional information the crystallographer needs in order to compute p(x,y,z) and thus... 

Although this equation is rather forbidding, it is actually a familiar equation (5.15) with the new parameters included. Equation (7.8) says that structure factor Fhk[ can be calculated (Fc) as a Fourier series containing one term for each atom j in the current model. G is an overall scale factor to put all Fcs on a convenient numerical scale. In the /th term, which describes the diffractive contribution of atom j to this particular structure factor, n- is the occupancy of atom j f- is its scattering factor, just as in Eq. (5.16) Xj,yjt and zf are its coordinates and Bj is its temperature factor. The first exponential term is the familiar Fourier description of a simple three-dimensional wave with frequencies h, k, and / in the directions x, y, and 7. The second exponential shows that the effect of Bj on the structure factor depends on the angle of the reflection [(sin 0)/X]. 

The description of a given crystal by a lattice and a basis is not unique. One might e.g. choose to double the size of the unit cell by including more atoms into the basis. This would also lead to a different reciprocal lattice. This seams to lead to a contradiction, since the diffraction pattern should only depend on the crystal and not how we choose our description. As we will see in example A.2, the choice of a different basis leads to a change in the structure factor so that the combination of reciprocal lattice and structure factor always leads to the same diffraction pattern. 

Furthermore, using Hamilton s test, all the differences were significant at the 5% level of probability. Thus it was concluded that for all combinations of other factors, the structure factor set of Yokouchi et al gave the best description of the final model. Further considerations were therefore restricted to this set. 

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

X-ray crystallography seeks to obtain the best model to describe the periodic electron density in a crystal by a least squares fit of the parameters of the model (used to calculate structure factors) against the observed structure factors derived from the diffraction experiment. All models used are atomic in nature, but vary in the complexity of the description of the atomic electron density. 

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