Structure factors for

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Figure Al.3.29. Pair correlation and structure factor for liquid silicon from experiment [41],...

Figure 10 Elastic incoherent structure factors for lipid H atoms obtained from an MD simulation of a fully hydrated DPPC bilayer, and quasielastic neutron scattering experiments on DPPC bilayers at two hydration levels for (a) motion in the plane of the bilayer and (b) motion m the direction of the bilayer normal.

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

For this range of values of u the structure factor for (154) is much greater than that for (037), if aluminum atoms are at (a) and (b) the observation that the latter plane reflects much more strongly than the former despite its smaller interplanar distance accordingly eliminates this arrangement. 

The reflecting powers of Mn and Fe are nearly the same, and may be taken equal without serious error. This reduces the number of distinct structures to three namely, 1 ab, %abc, and 3, of which 1 ab depends on two parameters and the others on one. It is possible to decide among them in the following way. Let us assume that the contribution of oxygen atoms to the intensity of reflection in various orders from (100) is small compared with the maximum possible contribution of the metal atoms that is, with 32M. The metal atom structure factor for structure 1 for (/a 00) is... 

There accordingly remains only structure 3. We may take 8(Mn,Fe) in 8e rather than 8i, which leads to the same arrangements. The structure factor for various orders from (100) is then... 

Now there are two physically distinct arrangements of the metal atoms corresponding to w = 0.030, the first with u = 0.030, and the second with u = — 0.030 and it is not possible to distinguish between them with the aid of the intensities of reflection of X-rays which they give. Let us consider the positions 24e. The structure factor for 24e is ... 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

Let us consider a collection H = (fr, h2,. . . , hA/) of symmetry-unique reflexions. We denote by Fj[ the target phased structure factor amplitude for reflexion h/, and with F rag the contribution from the known substructure to the structure factor for the same reflexion. We are interested in a distribution of electrons q( ) that reproduces these phased amplitudes, in the sense that, for each structure factor in the set of observations H,... 

For a number of 1907 acentric reflexions up to 0.463 A resolution, the mean and rms phase angle differences between the noise-free structure factors for the full multipolar model density and the structure factors for the spherical-atom structure (in parentheses we give the figures for 509 acentric reflexions up to 0.700A resolution only) were (Acp) = 1.012(2.152)°, rms(A( >) = 2.986(5.432)° while... 

Saunders, M., Bird, D M., Zaluzec, N.J., Burgess, W.G., Preston, A.R. and Humphreys, C.J. (1995) Measurement of low-order structure factors for silicon from zone-axis CBED patterns, Ultramicroscopy, 60, 311-323. 

Table 1. Dynamic structure factors for Rouse and Zimm dynamics... 

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). 

Fig. 12a, b. Dynamic structure factor for two polyethylene melts of different molecular mass a Mw = 2 x 103 g/mol b Mw = 4.8 x 103 g/mol. The momentum transfers Q are 0.037, 0.055, 0.077, 0.115 and 0.155 A-1 from top to bottom. The solid lines show the result of mode analysis (see text). (Reprinted with permission from [36]. Copyright 1994 American Chemical Society, Washington)... 

Following the mode analysis approach described in Section 3.2.1, the spectra at different molecular masses were fitted with Eqs. (32) and (33). Figure 13 demonstrates the contribution of different modes to the dynamic structure factor for the specimen with molecular mass 3600. Based on the parameters obtained in a common fit using Eq. (32), S(Q,t) was calculated according to an increasing number of mode contributions. 

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

Regime of creep In the time range t > tr, the chain creeps out of the tube. Doi and Edwards give a simple argument for the shape of the dynamic structure factor for the range QRG 1 [5]. For the parts of chain still in the original tube S(Q, t) = 12/Q V2 and for those parts of the chain which have already crept out of the tube all correlations have subsided and S(Q, t) = 0. It follows... 

Fig. 24 NSE spectra from the PEP homopolymer (above) and the triblock (below) at 492 K. The solid lines are the result of a fit with the Ronca model [49] the dashed line presents the expected dynamic structure factor for Rouse relaxation corresponding to the highest Q-value. ( Q = 0.058 A"1 V Q = 0.068 A"1 Q = 0.078 A"1 A Q = 0.097 A"1 o Q = 0.116 A"1 Q = 0.116A 1). The arrows mark the crossover time xe. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

The long-time behavior (Q(Q)t) > 1 of the coherent dynamic structure factors for both relaxations shows the same time dependence as the corresponding incoherent ones... 

Fig. 49a-c. Spectral contributions of the two modes (0, (2)), characterized by T1 and r2, respectively, to the dynamic structure factor for different contrast conditions. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)... 

Pauling, L. and Corey, R. B. (1951), Atomic coordinates and structure factors for two helical configurations of polypeptide chains , Proceedings of the National Academy of Sciences (USA),... 

Refinement takes place by adjusting the model to find closer agreement between the calculated and observed structure factors. For proteins the refinements can yield R-factors in the range of 10-20%. An example taken from reference 10 is instructive. In a refinement of a papain crystal at 1.65-A resolution, 25,000 independent X-ray reflections were measured. Parameters to be refined were the positional parameters (x, y, and z) and one isotropic temperature factor parameter... 

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