Structure factor calculated

Approximate atomic coordinates were obtained by assuming the effective metallic radius of magnesium to be about 1-60 A and the radii of aluminum and zinc to be about 1-40 A. The corresponding calculated structure factors were in fairly good agreement with those obtained from the observed intensities. The preliminary atomic coordinates are given in Table 1. 

Table 1. Observed and calculated structure factors far Ag3Co(CN)e.

In order to generate a set of calculated structure factors Fc(Q) from a set of coordinates, it is necessary to introduce a model for the time variation of the electron density. The usual assumptions in macromolecular crystallography include harmonic isotropic motion of the atoms and in addition, the molecular scattering factor is expressed as a superposition of atomic scattering factors. With these assumptions the calculated structure factor (equation III.2) is given by.27... 

We briefly discuss in this section the results of the valence MaxEnt calculation on the noisy data set for L-alanine at 23 K we will denote this calculation with the letter A. The distribution of residuals at the end of the calculation is shown in Figure 5. It is apparent that no gross outliers are present, the calculated structure factor amplitudes being within 5 esd s from the observed values at all resolution ranges. 

Figures 13c-e shows how the agreement between experimental data and the calculated structure factor improves if more and more relaxation modes are included. In Fig. 13e, finally, very good agreement between theory and experiment can be noted.

The results of the X-ray study are contained in Tables II-IV. Anisotropic B s and a table of observed and calculated structure factors are available (H)). 

A preliminary structural model of a protein is arrived at using one of the methods described above. Calculated structure factors based on the model generally are in poor agreement with the observed structure factors. The agreement is represented by an R-factor defined as found in equation 3.9 where k is a scale factor ... 

Calculate structure factors (by CCP4 sfall) for the transformed coordinates xnew in your own space group. 

The difference Ap(r) between the total electron density p(r) and a reference density pref(r), is a measure for the adequacy of the reference density in representing the system. Difference densities Ap(r) are obtained by Fourier summation in which the coefficients AF are equal to the difference between the observed and calculated structure factors. If k is the scale factor, as defined in chapter 4, the difference structure factor AF is given by... 

The crystal is placed in an X-ray diffraction apparatus (camera or diffractometer) where the X-ray pattern is recorded photographically or by measuring the intensity of the X-rays electronically. The resulting intensity values are used to obtain the observed structure factors which constitute the fundamental experimental data from which the crystal and molecular structures are derived. The structure derived is used for calculating structure factors that are compared with the experimental structure factors during the period when the derived structure is being modified to fit the experimental data. 

The residual index, or R factor, gives a measure of the difference between the observed and calculated structure factors and therefore of how well the structure has refined. It is defined as... 

Another useful equation due to Cruickshank (1950) is the relation of the standard deviation of electron density a(p) and the discrepancies between observed and calculated structure factors ... 

Normally, if the assumed model for a crystal structure has an R value of 0.5 and resists attempts to refine to a lower residual, then the model structure is rejected as false, and a new model is tried until a fit between the observed and calculated structure factors yields an acceptable residual (R < 0.25). (Other models were tried for this complex, but they either gave Fourier maps which were uninterpretable or they converged to the present model). However, the normal crystal structure is solved with data obtained from crystals which have dimensions of the order of 0.1 mm. In the crystals available for this experiment, two of the dimensions were of the order of 0.01 mm. Thus, long exposures were required to give a small number of relatively weak diffraction spots. (Each Weissenberg photograph was exposed for five days with Cuka radiation 50 kv., 20 ma. loading, in a helium atmosphere). 

The results of this crystal structure analysis should then be viewed as having a reliability intermediate between standard single crystal structures where R is often 0.1, and crystalline polymer structures where only qualitative agreement is expected between observed and calculated structure factors. 

Table II. Atomic Coordinates of Model Structure and the Observed and Calculated Structure Factors...

Structure refinements of the AH 200 and AH 300 samples were conducted in the same way. Unit cell constants, final atomic parameters, and R indexes are given in Table I. (Observed and calculated structure factors are available from the authors.) Interatomic distances and angles are given in Table II. Estimated errors on the population and position of the cations may in some cases be greatly underestimated especially for atoms with low occupancy factors. 

A listing of observed and calculated structure factors is available for all 4 crystals by interlibrary loan from Hamilton Library at this university see your librarian. 

I will discuss the iterative improvement of phases and electron-density maps in Chapter 7. For now just take note that obtaining the final structure entails both calculating p(x,y,z) from structure factors and calculating structure factors from some preliminary form of p(x,y,z). Note further that when we compute structure factors from a known or assumed model, the results include the phases. In other words, the computed results give all the information needed for a "full-color" diffraction pattern, like that shown in Plate 3d, whereas experimentally obtained diffraction patterns lack the phases and are merely black and white, like Plate 3e. 

---