Phase of the structure factor

Figure 1 Fourier synthesis of the projected potential map of Xi2S along the c-axis. Amplitudes and phases of the structure factors are calculated from the refined atomic coordinates of Ti2S and listed in Table 1. The space group of Xi2S is Pnnm and unit cell parameters a= 11.35, fc=14.05 and c=3.32 A.

Images taken along the other 12 zone axes were processed in a similar way and amplitudes and phases of the structure factors were extracted. The results of image processing are summarised in Table 1 and Fig. 4. Three zone axes, [010], [120] and [5 18 0] have pmg symmetry while the other 9 zone axes have only p2 symmetry. The number of unique reflections from... 

The true phases of the structure factors will, in general, be different from the phases calculated with the independent-atom model. In centrosymmetric structures, with phases restricted to 0 or n, only very few weak reflections are affected. In acentric structures, only the reflections of centrosymmetric projections, such as the hkO, hOl, and Okl reflections in the space group P212,21, are invariant. 

When it is assumed that the phases of the structure factors are unknown, the analysis proceeds well, after fixing the origin of the cubic unit cell by choosing the sign of the strong (111) reflection. This corresponds to a direct structure determination without any prior knowledge of the structure, and supports the value of the maximum entropy method in the early stages of structure determination. 

In words, the difference Patterson function is a Fourier series of simple sine and cosine terms. (Remember that the exponential term is shorthand for these trigonometric functions.) Each term in the series is derived from one reflection hkl in both the native and derivative data sets, and the amplitude of each term is (IFHpI — IFpl)2, which is the amplitude contribution of the heavy atom to structure factor FHp. Each term has three frequencies h in the u-direction, k in the v-direction, and l in the w-direction. Phases of the structure factors are not included at this point, they are unknown. 

Figure 6.11 Real and imaginary anomalous-scattering contributions alter the magnitude and phase of the structure factor.

Barker, D. The determination of the phases of the structure factors of non-centrosymmetric crystals by the method of double isomorphous replacement. Acta Cryst. 9, 1-9 (1956). 

To save paper, we are here compressing notation to the vector forms for position x = x, y, z and indexes h = h,k t, and using Euler s form of the complex numbers. We will expand them back to their full forms later. Again, it is unnecessary to actually multiply by NP, which we do not even know, since it is a constant and doesn t change the relative intensities or phases of the structure factors. 

Here we have encountered the crucial, essential difficulty in X-ray diffraction analysis. It is not experimentally possible to directly measure the phase angles hki of the structure factors. The best that our sophisticated detectors can provide are the amplitudes of the structure factors Fhki but not their phases. Thus we cannot proceed directly from the measured diffraction pattern, the measured intensities, through the Fourier equation to the crystal structure. We must first find the phases of the structure factors. This central obstacle in structure analysis has the now infamous name, The Phase Problem. Virtually all of X-ray diffraction analysis, not only macromolecular but for all crystals, is focused on overcoming this problem and by some means recovering the missing phase information required to calculate the electron density. 

A Patterson map, different for each space group, is a unique puzzle that must be solved to gain a foothold on the phase problem. It is by finding the absolute atomic coordinates of a heavy atom, for both small molecule and macromolecular crystals, that initial estimates (later to be improved upon) can be obtained for the phases of the structure factors needed to calculate an electron density map. 

Worse still, the phases, not the structure factor amplitudes F, dominate the shape of the calculated electron density. If, as in Figure 15, we FT a picture of a duck and a cat, and then calculate the inverse transform using duck amplitudes and cat phases, only the shape of the cat is discernible. This is exactly what happens to electron density when the phases of the structure factors are incorrect. The features of the electron density, largely determined by the phases, will also be wrong. This is known as model bias (see Section 9.03.10.2). 

Lp is the Lorentz polarization factor, A and E are absorption and extinction factors, and IF /1 the absolute magnitude of the structure factor. The sign or phase of the structure factor, a /, is not directly determinable from the diffraction experiment. 

Even when the unit cell, the space group and Bragg intensities have been found, the solution of the crystal structure, i.e. to determine the positions of the atoms, is not straightforward. As mentioned above, this is an inherent problem with X-ray and neutron diffraction since the information about the phases of the structure factor is lost in the experiments (see Eq. (5.2)). Ab... 

If Fhid is known for a large number of hkl reflections, (1.98) can be inverted to obtain pu(r) and hence the positions of all the atoms in the unit cell. Such an endeavor is called the crystal structure analysis and is explained in more detail in Section 3.3. The intensity of reflection, observed at s = r%kl is equal to F/ / 2. The absolute value of Fhki can therefore be obtained as the square root of the observed intensity of the hkl reflection, but the intensity data do not provide any direct information about the phase angle of the complex Fw A major task in crystal structure analysis is solving the phase problem to determine the phases of the structure factors. 

In the typical practice of the XSW technique, however, only a limited set of hkl measurements is taken, and the analysis resorts to comparing the measured / and values to those predicted by various competing structural models. The procedures of structural analysis using fH and will be described in more detail in a later section of this chapter. It should be stressed that the Bragg XSW positional information acquired is in the same absolute coordinate system as used for describing the substrate unit cell. This unit cell and its origin were previously chosen when the structure factors FH and Fs where calculated and used in Equations (9), (10), and (12). As previously derived and experimentally proven (Bedzyk and Materlik 1985), the phase of the XSW is directly linked to the phase of the structure factor. This is an essential feature of the XSW method that makes it unique namely, it does not suffer from the well known phase problem of X-ray diffraction. 

Estimation of phases of the structure factor. Both the phase and amplitude of the contribution of the heavy atom to the structure factor are computed to yield Fh. 

Direct determination of the phases of the structure factors, possibly followed by successive Fourier synthe.ses of the electron density (direct methods)... 

The differences relation of (5.32) t3 e indicates how, by exactly knowing the amplitude and randomly the phase of the structure factor, an error... 

In principle, it should now be possible to construct a map of electron density within the cell from a set of X-ray structure factors - but there is a major stumbling block. The structure factors are not just numbers but are complex quantities corresponding to sums of wave motions, and therefore have both amplitudes and phases (Eqs 10.8-10.10). All detectors measure intensities integrated over a period of time, so aU we can obtain are the moduli of the structure factors, Ffj i. The so-called crystallographic phase problem is therefore to deduce the phases of the structure factors, as well as the amplitudes. If we manage to do this, we have solved the structure. Note that the term structure solution is used specifically to describe the initial approximate identification of the atom positions within the unit cell, and is distinct from the subsequent refinement of those positions. 

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] ... 

Unfortunately the structure factors are complex numbers, not only amph-tudes F(h,k,l), but also the phases of the structure factors, T(h,k,l), are necessary to determine this complex (Figure 7.1). Until now, the frequency of X-rays is so high that no instrument has been able to directly measure the phase of this type of electromagnetic wave. 

The use of this equation obviously requires a knowledge of the phases of the structure factors. Once the phase problem is solved, the moduli of the observed stmcture factors can be used in a Fourier synthesis with calculated phases, to find the centroids of the electron density peaks, which indicate the positions of atomic nuclei. If the phases are nearly correct, the Fourier map will reveal the position of most if not all of the atoms. The Fourier synthesis is, in brief, a convenient way of transforming the phase information in terms of phase angles into the same information in terms of atomic positions (recall equations 5.28 to 5.30). If the first try does not reveal the position of all atoms, then a Fourier synthesis using as coefficients the differences between observed and calculated structure factors, plus the nearly correct phases, will reveal the position of the missing atoms. 

X-ray scattering experiments measures the structure factor F(h,k,l) or strictly the intensity I of diffraction spots proportional to the square of the structure factor, i.e., I F(h, k, /)p. In order to obtain /, and thereby the atomic positions using eq. (2.24) we also need the phase of the structure factor. Various techniques have therefore been developed for solving this so-called phase problem. 

Direct methods constitute the third important approach to the phase problem. Certain physical properties of crystals (such as the fact that the electron density is always positive) place restrictions on the magnitudes and phases of the structure factors. For centrosymmetric crystals, the structure factors are real, and the phase problem is therefore one of sign determination. Sayre s equation for intense reflections is an example of sign determination by direct methods ... 

---