Real-space R-factor

DeGennes" also studied the poly c/(A-T)-poly d(A-T) copolymer, deriving equations for the melting curve and radius of gyration. For a isingle strand of j unbonded bases starting at r and ending at r (in real space), the factor... 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

In using cross-validation it is essential to avoid, or at least minimize, bias to the free R factor itself. In the era of emerging automated procedures for modelling and refinement a frequent mistake is to set aside a fraction of reflections for minimization of the residual in reciprocal space and, at the same time, to use all data for computation of electron density and model rebuilding. Since local adjustment of the model in real space is equivalent to global phase adjustment in reciprocal space, the free reflection set becomes biased towards the current model and loses its validation credibility. 

The Debye-Waller factor, Eq. 4, describes how the uncertainty in real space (u) determines the range of S(Q) in Q space. Now the exact converse happens with respect to the resolution of the measurement in Q space. If the Q resolution of the instrument is AQ, the PDF will have an envelope exp(- r ( AQ) ), and the oscillations in the PDF decay. Therefore in order to determine the PDF up to large distances it is important to use an instrument with high Q resolution. Since the PDF method was initially applied to glasses and liquids in which atomic correlation decays quickly with distance, this point was not... 

The progress of iterative real- and reciprocal-space refinement is monitored by comparing the measured structure-factor amplitudes IFobsl (which are proportional to (/obs ) /2) with amplitudes IFca(c I from the current model. In calculating the new phases at each stage, we learn what intensities our current model, if correct, would yield. As we converge to the correct structure, the measured Fs and the calculated Fs should also converge. The most widely used measure of convergence is the residual index, or R-factor (Chapter 6, Section V.E). 

Where A is a constant. The singularity q2 = - -2 implies screening in real space, i.e. an attenuation factor e r/ . 

The static structure of three-dimensional colloidal suspensions is usually determined experimentally, not by measuring directly g(r) in real space, but by measuring the static structure factor S(k) in the reciprocal space, which is the Fourier transform of the local particle-concentration correlation function. The radial distribution function is directly related to the Fourier transform of S(k), as it is explained below. Let us consider a system of N particles in a volume V. The local particle concentration p(r) at the position r is given by... 

If, on the other hand, the electron density map calculated at some resolution r is somehow improved in quality in real space, then if it is transformed to produce structure factors, phases at somewhat higher resolution, r + Ar, can be computed that do have some measure of validity. Improvement of an electron density map thus allows gradual extension of phases in reciprocal space to higher resolution, and ultimately to an electron density map of sufficient quality and detail that a model can be constructed. This is another example of those bootstrap, incremental procedures so common to X-ray crystallography. 

As noted already, most X-ray crystal structure refinement incorporates both real and reciprocal space refinement approaches, generally alternating one with the other. At the end of the refinement process, one should have both a low R factor and a featureless difference Fourier map. 

In evaluating a structure determination for accuracy and precision, it is usually prudent to inquire as to the quality of the electron density map according to which the final model was built, and to the properties and quality of the final model. The former question can be addressed in real space, that is, how good is the electron density map, and/or in reciprocal space, that is, how well determined were the phases, and how well measured the structure amplitudes that contributed to the map. The model, of course, can be judged by how well it predicts the diffraction intensities (the R factor), by how it explains chemical and biochemical questions, and how well it agrees with canonical stereochemical properties, such as bond lengths and angles. 

If the system is sufficiently disordered (i.e. a high Debye-Waller factor), the diffractometer resolution is sufficiently low and the Q range is sufficiently large, then the experimental structure factor can be transformed directly to gE(r) and the comparison made withgc(r) (Keen et ah, 1990b). However, this procedure is not applicable in many cases, i.e. there are still oscillations in AE(Q) at the maximum Q measured. In addition the low Q resolution leads to a loss of real space resolution in the RMC model, i.e. a broader distribution of atoms around their crystal sites. 

The classical rubber elasticity model considers, however, that the crosslink points are particular, such that the cut-off occurs by these points in real space. The corresponding calculations for a chain obliged to pass by several crosslinks are recalled in Ref The calculation for the junction affine model was accomplished by Ull-mann for R and by Bastide for the entire form factor for the case of the phantom network model, this was achieved by Edwards and Warner using the replica method. 

The static structure factor can be Fourier transformed into real space to give the pair correlation function, g(r), which is proportional to the probability of finding an atom at a position r relative to a reference atom at the origin. Due to the isotropic nature of liquids and glasses it is only necessary to consider the distance from the reference atom hereafter the vector dependence becomes a magnitude dependence. Therefore the pair correlation function, g(r), is expressed in terms of S(q) ... 

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