Normalized scaled structure factor

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. 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

In certain applications, e.g. when the normalized structure factors should be calculated (see section 2.14.2), the knowledge of the approximate scale factor is required before the model of the crystal structure is known. This can be done using various statistical approaches [e.g. see A.J.C Wilson, Determination of absolute from relative x-ray intensity data,... 

In their basic form DM exploit two types of prior information the positivity of the electron density map (this condition may be relaxed, e.g., for neutron diffraction, see Section 8.4.7), and the atomicity (the electrons are non-dispersed into the unit cell but concentrated around the nuclei). This information, apparently trivial, is very useful to succeed in all the steps of a modern DM procedure (1) scaling of the observed intensities and normalization of the structure factors (2) estimate of the structure invariants (3) application of the tangent formula (4) crystal structure completion and refinement. 

Scaling of the Observed Intensities and Normalization of the Structure Factors... 

Once scale and average thermal factors have been determined, normalized structure factors IFhl can be calculated as follows ... 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

Fig. 48a. Normalized inverse scattering intensity NS l(q, e) observed in the Monte Carlo simulation of a block copolymer model on the simple cubic lattice (see Fig. 44) plotted vs the normalized inverse temperature eN. b Reciprocal structure factor S (q ) -l(cxrcfes, left scale) and q ( squares, right scale) plotted vs temperature for a nearly symmetric diblock copolymer of polystyrene/poly (cis— 1,4) isoprene (Mw = 15 700). Filled symbols refer to cooling, open symbols to heating runs. The straight tine indicates the extrapolation to a spinodal temperature (T,) that occurs above the actual transition temperature (Tmst). where the data show a jump. From Stuhn et al. [323],...

X-ray reflectivity results are sometimes reported in terms of structure factor. In this case, the data are presented as the square root of the measured integrated intensities. Suitable normalization enables the conversion of this scale to a plot of scattering amplitude in units of electron scattering. 

R-value (normally <0.07) due to good-quality crystals with no disorder and a high (usually >10) ratio of the number of observed reflections to the number of refined parameters. The R-value is a measure of the level of disagreement between the properly scaled observed structure factors (Fobs) and calculated structure factors (Fcaic)- It is usually given as %, i.e., an R-value of 0.07 is reported as 7%. 

Experience in the chemical cleaning industry has shown that no two iron oxides are alike. The heat history, density, and impurities may have a very large effect on the actual rate of deposit removal. Azuma and Kametani made systematic studies of the effect of sample preparation on oxide dissolution. The formation temperature of the oxide markedly influenced the dissolution rate. The physical properties apparently controlling the reaction were the surface area (calculated by a permeability method) and a normalizing factor that is dependent on the aggregate crystal structure. Baud and Perrier and Fields also demonstrated the importance of the scale structure (layers and microcracks). McPherson showed that the presence of chromium-containing spinels has a marked influence on deposits found in the superheat or reheat sections of power boilers. 

Finally, is a scaling factor, depending inter alia on crystal size, mosaic spread, quality of the crystal beam intensity, etc. Most commonly this factor is omitted, so that during data reduction relative structure factors are calculated rather than observed structure factors (F ). The - rel values are normally converted to during structure solution, by comparing the values of with the calculated structure factors F, thus calculating k in the equation below. K and k are the same for all reflections. 

Obviously, these structural changes make the transfer of force-constants from the neutral B3 molecule to the B3+ radical inadequate. Instead, we tentatively transferred the scale factors optimized for the neutral molecule to the quantum-mechanical force-field of B3+ and calculated the corresponding scaled normal frequencies. We obtained a clear correspondence between many of the frequencies experimentally observed in Cl doped B3 crystals (Fig. 3(a)) and the calculated scaled frequencies (Fig. 3(b)). We also observed that some of the calculated scaled frequencies in the neutral B3 molecule are present in the spectra of the Cl doped crystals (Fig. 3(c)). This fact tells us that there is some portion of unoxidized B3 molecules in the sample and gives additional proof for the validity of the SQMF calculations performed on the neutral B3 molecule. 

The results presented above once again demonstrate the transferability of the scale factors between molecules containing similar structural elements (benzene and B3) as well as between different states of oxidation of a given molecule (neutral B3 and the B3+ radical). The detailed results of the SQMF analysis (normal mode assignment and the scaled factors) will be published elsewhere. 

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