Electron density from measured reflections

Are all three of these parameters accessible in the data on our films We will see in Chapter 5 that the measurable intensity Ihkl of one reflection gives the amplitude of one Fourier term in the series that describes p(x,y,z), and that the position hkl specifies the frequency for that term. But the phase a of each reflection is not recorded on the film. In Chapter 6, we will see how to obtain the phase of each reflection, completing the information we need to calculate p (x,y,z). 

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 

2Access to Kevin Cowtan s Book cf Fourier is provided at the CMCC Home Page, www.usm.maine.edu/ rhodes/CMCC. 

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Figure 6 Artist s view of lipid/protein interaction derived from the electron density model from XR measurements and available structural data. The lipid monolayer is dipalmitoyl phosphatidyl ethanolamine (DPPE) and the protein is a membrane surface layer protein from B. sphaericus (from ref. [28]). The structural model sketched was based [28, 29] on the electron density profile (black line) inverted from measured reflectivity data as described in the text.

Now, the problem of determining the crystal structure can be reahzed. As the structure amphtude Fhki can be derived from the measurement of intensity of X-ray reflection, the phase angle hki cannot be directly determined and if these phases of the structure factor are known, then the crystal structure is known as one can compute the electron density from (8.4) and hence the positions of the atoms giving rise to the measured electron densities. Therefore, the lack of knowledge of the phases of the structure factors prevents from directly computing the electron density map and hence determines the positions of the atoms. Patterson suggested as an aid the use of the following equation instead of (8.4) [1,6,7] ... 

Two helices are packed antiparallel in the orthorhombic unit cell. Association of the helices occurs through a series of periodic carboxylate potassium water - carboxylate interactions. An axial projection of the unit-cell contents (Fig. 23b) shows that the helices and guest molecules are closely packed. This is the first crystal structure of a polysaccharide in which all the guest molecules in the unit cell, consistent with the measured fiber density, have been experimentally located from difference electron-density maps. The final / -value is 0.26 for 54 reflections, of which 43 are observed, and it is based on normal scattering factors.15... 

Using the refinement parameters, electron density maps were calculated. Figure 1 shows an example derived from CU96. According to all refinement results the intensities of the strong reflections were too low. Therefore, they were omitted for the electron density studies. The problem is currently under study. Additionally, low-temperature measurements are planned for the near future. 

The multipole model reduces the crystal electron density to a number of parameters, which can be fitted to experimental structure factors. For CU2O, structure factors for the (531) and higher-order reflections out to (14,4,2) were taken from X-ray measurements. Weak (ooe) (with o for odd and e for even) and very weak (eeo) reflections were also taken from X-ray work. Fig. 6 shows a three-dimensional plot of the difference between the static crystal charge density obtained from the multipole fitting to... 

For obvious reasons, Fourier transformations are widely used to solve problems in X-ray crystallography [129]. With innumerable replications of a molecule in a crystal, all being oriented the same way, approximate periodic boundary conditions are given. Periodic functions become discrete when Fourier transformed. In fact, the diffraction pattern of an X-ray shot on a crystal amounts to the Fourier transform of the square of the absolute values of the real space function [130]. The measurements of intensities and different reflection angles from the crystal relate to the Fourier transform of the electron densities in the crystal. 

The high-resolution C spectrum has also been measured, and the following values obtained chemical shift (in acetone) 62.4 ppm upheld from CSg (130.9 ppm downfield from tetramethylsilane) J (geminal) CH, 205 Hz J (vicinal) CH, 13.4 Hz. The chemical shift value is in accord with an empirical equation based on the number and position of the nitrogen atoms in several five-membered heterocyclics, and also reflects the 7T-electron density of the system as calculated by the extended Hiickel method - and by the simple MO method. spectra of v-triazole and of its 1- and 2-methyl derivative have also been obtained. ... 

It is well established [154] that MES probes the initial electronic structure while photoemission samples the final state, after ejection of the electron. Both MES I.S. and XPS b.e. shifts are sensitive to the local chemical environment of the atom [155, 156]. However, the I.S. reflects the electronic charge density, of predominantly s- character, within the nuclear volume, while XPS b.e. shifts are a measure of the overall electronic charge density from all the shells. 

While d is proportional to the s-electron density at the nucleus, valuable information about the d (and other) electrons can still be obtained from this parameter due to screening effects. That is, the addition of a d electron reduces, via screening, the effective nuclear charge felt by the s electrons thereby leading to an expansion of the s-electron cloud and a decrease in the electron density at the nucleus. A measurement of 8 thus reflects to some extent the entire electron distribution surrounding the nucleus, giving information about both the atom and its bonding characteristics. 

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