Sayre equation

This formula is also known as Sayre equation, since David Sayre discovered around 1952 a similar expression that was later confirmed by the theoretical derived Z2 formula. The practical meaning of the Z2 formula is that the sign S of the normalised structure factor amplitude Eh depends on the signs of structure factors Ek and Eh-k- For example, if Ek is positive and Eh-k is negative, the product of these two is negative ( + => - ) and hence the... 

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 Sayre equation [12] is algebraic rather than probabilistic in origin and is derived from the expression relating the electron density and its square ... 

Deficiencies in intensities, which occur in x-ray powder dififiaction as well as in single crystal electron diffiaction, may cause problems even in early stages of ab initio structure analysis. Nevertheless, examples for successful use of the tangent formula or Sayre equation for structure determination from ED data have been worked out [14]. Other direct methods, like maximum entropy can provide us with an envelope of the molecules in the cell, which delivers an idea of its orientation [15]. An alternative approach to ab initio structure determination is the calculation of the gas phase conformation of an initial model for subsequent refinement by energy minimization [16]. 

The conunonly used methods for solving the phase problem required for stmcture solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets (0 = 4>h + 4>k + -h-k The probabihty of 0 >= 0 increases with the magnitude of the product of the normalized stmcture factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

In the centrosymmetric structures, the relationships between the signs of the reflections forming a triplet (Eq. 2.140) are described by the Sayre equation ... 

For each permutation, the phases of all other reflections are generated using Sayre equations. Thus, direct methods always result in more than one array of phases, and the problem is reduced to selecting the correct solution, if one exists. Several different figures merit and/or their combinations have been developed and are used to evaluate the probability and the relationships between phases. Thus, the solutions are sorted according to their probability - from the highest to the lowest. Then each solution is analyzed and evaluated starting from the one that is most probable. 

The form in which the Sayre equation is commonly used today is due to Hughes [79] ... 

Using Equation (37) for the probability of relation (32) takes care of the fact that not all products required for the Sayre equation are available. A strict equation has become a probability relation. One of the corresponding equations for the probability that relation (33) is correct in the noncentrosymmetric case reads as follows ... 

The Hughes form of the Sayre equation (34) is for the centrosymmetric case in the noncenlro-symmetric case, at least as far as physical information is concerned, it corresponds to the tangent formula. In contrast to Equation (33), here the. sum is weighted with the normalized structure factors, and all pairs of structure factors possible in a special case are taken care of ... 

The stream function satisfying the fourth-order differential equation, used by Haberman and Sayre (H2) is... 

Whitmore (F4) for A < 0.6 lie between the expressions of Haberman and Sayre and of Francis. The Haberman and Sayre result shows about 1% less deviation, but the equation due to Francis has the virtue of simplicity. Experimental results due to Sutterby (S7) for a < 0.13 with Re- 0 agree with the Faxen, Haberman, and Francis curves which are virtually indistinguishable in this range. The Ladenburg result is only accurate for A < 0.05. 

The surface velocities of Haberman and Sayre (HI), when used in the thin concentration boundary layer equation for circulating spheres, Eq. (3-51), yield the mass transfer factors and X d shown in Fig. 9.7 for k <2. For a fluid sphere in creeping flow the relationship between the mass transfer factors is... 

Modeling particulate transport, or various process phases, has been attempted only relatively recently. Sayre (20) gave a very sound basis for further work by using a momentum solution of the two-dimensional convection-diffusion equation characterizing particle transport when additional terms for sedimentation, bed adsorbance, and re-entrainment (erosion) are included. He showed, with extensive hypothetical calculations, which hydrodynamic parameters are important and how they could be quantified. He was also able to show that his concept of bed adsorbance and re-entrainment requires further elucidation and indicated that there might be a turbulence effect on the sedimentation step. Hahn et al. 

Again, the left-hand side could be rewritten as in Eq. (6). Equation (11) contains the assumption that molecular and turbulent diffusion are independent processes and thus additive, as indicated by terms of the form (Dm + e. ). Mickelsen (63) made and justified this assumption. As noted by Sayre and Chang (82), however, it is really only of academic interest in open-channel flows, as the turbulent diffusion coefficients are typically several orders of magnitude greater than Dm- For this reason, Dm will be neglected in subsequent presentations of the equation. 

In this section, minimal attention has been paid to the methods for solving the stream function equations since the basic purpose has been to illustrate the method for obtaining the correction It should be noted that the solutions in the form of stream functions for many regular geometric shapes are available and can be used to calculate the first correction In cylindrical and spherical coordinates, the general solutions have been provided by Haberman and Sayre (1958) and their method can be applied to many of the separable coordinate systems (Happel and Brenner, 1983). 

The best-known equation for direct phase determination is that of David Sayre 78] ... 

One attempt to replace the probability relations with true equations for phase determination was that of R. Rothbauer [133], based on earlier work by Sayre [78], and Wooli-son [134]. The future must show whether the equations derived can be used for practical structure determination [135], Their use for the determination of the absolute scaling factor and the overall temperature factor has already been tested in practice [136]. A publication by R. Rothbauer shows, that solving the phase problem generally is still up to date [0]... 

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 ... 

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