Structure factor amplitude reflections

Where Jo and S are the current density of the primary beam and the area of the irradiated sample, Z is the wave length, Ohki the structure factor amplitude, Q the volume cell, Z a factor that takes the microstructure of sample into account (Zm - for a mosaic single crystalline film, Zt - for a texture film), t is the sample thickness, dhu the interplanar spacing, a represents the mean angular distribution of the microcrystallites in the film, p is a multiplicity factor (accounts for the number of reflections of coincidence), R is a horizontal coordinate of a particular reflection in DP from textures and (p is the tilt angle of the sample. In the case of polycrystalline films, a local intensity is usually measured and the corresponding relation is ... 

Figure 13. Principle of direct methods using triplet relations. As shown in the lower right-hand image the trial structure eonsists of atoms which are located at the eomers of the unit eell. Aeeording to the Z2 formula (Sayre equation) a strict phase relation exists within a eertain set of three reflections (a triplet) with large normalized structure factor amplitudes Eu. Sueh a triplet or origin invariant sum is defined as hiEli + + h k h = 0 or hiEli +...

The direction and the periodicity of each cosine wave are given by its index u = hkl), the amplitude of the cosine wave is 2 F(u), proportional to the structure factor amplitude F(u). More importantly, the positions of the maxima and minima of the cosine wave (in relation to the unit cell origin) are determined by the structure factor phase ( )(u). If both the amplitudes F(u) and the phases ( )(u) of the structure factors for all reflections u are known, the potential cp(r) can be obtained by adding a series of such cosine waves. 

According to Ref. [11], the dynamical scattering effect was reduced by forcing the integral amplitudes of reflections to be equal to the corresponding structure factor amplitudes of the corresponding perfect... 

To obtain the Patterson function solely for the heavy atoms in derivative crystals, we construct a difference Pattersonfunction, in which the amplitudes are (AF)2 = (IFHpl — IFpl)2. The difference between the structure-factor amplitudes with and without the heavy atom reflects the contribution of the heavy atom alone. The difference Patterson function is... 

IF p + l with its center at the origin, representing the structure-factor amplitude of this same reflection in the nonanomalous scattering data set. The two points of intersection of these circles satisfy Eq. (6.13), establishing the phase of this reflection as either that of Fa or Fb. As with the SIR method, we cannot tell which of the two phases is correct. 

In words, for each reflection, we compute the difference between the observed structure-factor amplitude from the native data set IFobsl and the calculated amplitude from the model in its current trial location IFcalcl and take the absolute value, giving the magnitude of the difference. We add these magnitudes for all reflections. Then we divide by the sum of the observed structure-factor amplitudes (the reflection intensities). 

For each Bragg reflection, the raw data normally consist of the Miller indices (h,k,l), the integrated intensity I(hkl), and its standard deviation [ a[I) ]. In Equation 7.2 (earlier), the relationship between the measured intensity / [hkl] and the required structure factor amplitude F[hkl) is shown. This conversion of I hkl) to F hkl) involves the application of corrections for X-ray background intensity, Lorentz and polarization factors, absorption effects, and radiation damage. This process is known as data reduction.The corrections for photographic and diffractometer data are slightly different, but the principles behind the application of these corrections are the same for both. 

Direct methods, direct phase determination A method of deriving relative phases of diffracted beams by consideration of relationships among the Miller indices and among the structure factor amplitudes of the stronger Bragg reflections. These relationships come from the conditions that the structure is composed of atoms and that the electron-density map must be positive or zero everywhere. Only certain values for the phases are consistent with these conditions. 

The central problem in crystallography lies in obtaining the phase for every observed structure factor amplitude. We judge how correct a given set of phases is by the result does the electron density map make chemical sense For small molecules, very accurate data is usually available to high resolution (1 A or better), which allows the use of direct methods 9 to obtain the phases rapidly and correctly. The approach uses statistical relationships between the phases of certain reflections. Unfortunately, direct methods are not easily... 

The structure factor amplitude of the reflection (hkl) is equal to the structure factor amplitude of the centro-symmetrically related reflection (HliT), i.e.,... 

Figure 7. A vector diagram illustrating the effects of heavy atom anomalous scattering on the reflections hkl (denoted +) and filci (denoted —). Fpn is the average structure factor amplitude for the heavy atom derivative of the protein. FJ Ih imaginary part of the heavy atom structure factor amplitude which arises from anomalous scattering. Because FJl always advances the phase by fI/2, FpH (+) and Fpn (—) are no longer equal. The measured difference between these amplitudes can be used fbr phase determination.

In a crystallographic problem the parameters to be determined are the positional and thermal parameters for each atom (x,y,z,B). The observables are the structure factor amplitudes F bs(fj), where h represents the reflection hkl. Following the method of Lagrange, the best parameters are those for which the sum of the squares of the differences between the observed and calculated structure factors is a minimum, i.e.,... 

All methods of deduction of the relative phases for Bragg reflections from a protein crystal depend, at least to some extent, on a Patterson map, commonly designated P(uvw) (46, 47). This map can be used to determine the location of heavy atoms and to compare orientations of structural domains in proteins if there are more than one per asymmetric unit. The Patterson map indicates all the possible relationships (vectors) between atoms in a crystal structure. It is a Fourier synthesis that uses the indices, l, and the square of the structure factor amplitude f(hkl) of each diffracted beam. This map exists in vector space and is described with respect to axes u, v, and w, rather than x,y,z as for electron-density maps. 

The observed Bragg rod intensity / ( z) is actually a sum over those (h,k) reflections whose Bragg rods coincide at a particular horizontal angle or position. In the upper equation, the most important variation is due to the molecular structure factor amplitude The Debye-Waller factor = exp[-( jj[4 -i-... 

The relation (5.257) combined with the definition (5.231) will also provide the measured integrated reflectivity dependency by the structure factor amplitude of the non-deformed ideal crystal case ... 

The calculated structure factor, whether based on the Patterson synthesis or on the direct method, must be in agreement with the observed structure factor. The uncertainty involved in assigning the positional and thermal parameters in the calculation of the structure factor must be minimized so that the results will be close to the structure factor amplitudes (h) for each reflection h observed. The minimization procedure is called the refinement and is based on Legendre s (1806) classical suggestion. If we have a set of unknown parameters x, x2, , x that are related to a set of m observable values Oi, O2,. .Om by... 

The intensity lo(h) of the reflected wave is related experimentally to the square of the observed structure factor ) amplitude and expressed as follows ... 

---