Real space averaging

Real-space averaging A computational method for improvement of phases, when there are two or more identical chemical units in the crystallographic asymmetric unit. The electron densities of these identical units are averaged. Then a new set of phases is computed by Fourier transformation of the averaged structure, and with these a new map is synthesized with the observed F values. By iteration of this procedure, the electron density is improved. 

We now consider the first order correction to the average potential, i.e. Vn>. In real space representation, substituting Equations (11) and (13) into (10) gives [36]... 

Abstract Chemists may find it difficult to admit that their concepts and opportunities have always been strongly influenced by the available methods for characterization and analysis. Physics, has, of course, the lead when it comes to the visualization of single molecules in real space and to the detection of their specific, not ensemble-averaged properties. The challenge for chemistry is to provide molecules as objects of study which really disclose new concepts of structure and function. This chapter presents a chemical approach toward nanosciences which comprises (i) design and synthesis, (ii) immobilization, often using principles of self-assembly, (iii) visualization, e.g. by scanning probe... 

Subject to the caveat that there can be a significant shift in peak position when the total layer thickness is sub-micrometre we can determine the average composition of the MQW using the zeroth order, or average mismatch, peak. Asymmetric reflections are often used, both to determine any relaxation and to enhance the diffraction from thin layers. Let the period of the superlattice in real space be A, and the thickness of layers of A B i of composition x i and x 2 be D 1 and D 2 respectively. Then... 

Applying a mask function in real space is equivalent to combining many structure factors through a convolution in reciprocal space. This results in an improvement because the random error component of the structure factors will average out, whereas the true values of the structure factors will add up systematically. Fig. 10.4 gives a graphical example showing this phenomenon. 

In many ways, NCS averaging is the easiest density modification technique to intuitively understand, especially if considered in real space. In the process of NCS averaging, one simply takes all the different NCS related molecules in the as)un-metric imit, superimposes them, and then replaces their density with the average density. Because these... 

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