Phase angle determination

In any of the reciprocal space methods, which are based exclusively on the use of the observed structure factors, the powder diffraction pattern must be deconvoluted and the integrated intensities of all, or as many as possible, individual Bragg reflections determined with a maximum precision. Only then, Patterson or direct phase angle determination techniques may be employed to create a partial or compete structural model. Theoretical background supporting these two methods was reviewed in section 2.14. 

Figure 6.12. The cross-sections of the three-dimensional Fourier map of LaNi4 ssSno.is at Z = 0 (left) and at Z= 1/2 (right) calculated using structure amplitudes listed in Table 6.4 and phase angles determined by the La atom placed in the 1(a) site. The triplets of numbers indicate the coordinates of the strongest peaks in the unit cell. The following groups of peaks are symmetrically equivalent to one another 0,0,0 1,0,0 0,1,0 and 1,1,0 (all four correspond to the La atom in 1(a) and peak No. 1 mTable6.6) l/3,2/3,0 and 2/3,l/3,0 (peakNo. 2 in Table 6.6), and l/2,0,l/2 0,l/2,l/2 l/2,l/2,l/2 l,l/2,l/2 and 1/2,1,1/2 (peakNo. 3 in Table 6.6).

Table 6.6. The three-dimensional electron density distribution in the symmetrically independent part of the unit cell of LaNi4 85Sno,i5 calculated using the observed structure factors determined from Le Bail s extraction Table 6.4) and phase angles determined by the...

Proceeding slowly, we add the third peak as Ge and change the z-coordinate of Rh from 0.662 to 0.665 as had been determined from the latest electron density map Table 6.18). All distances remain normal and the residual lowers to Rp = 32.1 % the Fourier map calculated using phase angles determined by all three independent atoms is shown in Table 6.19. 

Table 6.27. The three-dimensional nuclear density distribution in the symmetrically independent part of the unit cell of CeRhGea calculated using the observed structure factors determined from Le Bail s extraction Table 6.23) and phase angles determined by Ge in 4(b) with z = 0.000, Ge in 2(a) with z = 0.355, Ce in 2(a) with z = 0.754, and Rh in 2(a) with z = 0.113 in the space group I4mm (Rp = 16.3 %). ...

If the protein has anomalous scatters in its molecule, the difference in intensity between the Bijvoet pairs, Fm/( + ) and Fa /( —), can be used for the phase angle determination. In the MAD method the wavelength... 

Only one of the crystallized proteins has yet matured into a crystal structure. This is the L7/L12 CTF (Leijonmarck et al., 1980 Leijonmarck and Liljas, 1982). The structure was initially solved with phase angles determined from isomorphous heavy atom derivatives at 2.6 % resolution. Subsequently the structure has been refined and the phase angles extended stepwise to 1.7 % resolution. 

Permanent deformation The rutting resistance of the binder is represented by the stiffness of the binder at high temperatures that one would expect in use. This is represented by G /sin(5), where G is the complex shear modulus and 5 is the phase angle determined by the dynamic shear rheometry, DSR, measured at 10 rad s (1.59 Hz). The complex modulus can be considered as the total resistance of the binder to deformation under repeated shear, and consists of elastic modulus, G and loss modulus, G" (recoverable and non-recoverable components). The relative amounts of recoverable and non-recoverable deformation are indicated by the phase angle, 5. The asphalt binder will not recover or rebound from deformation if d = 90°. 

The presence of a heavy atom greatly simplifies the structure determination since the prominent Patterson peaks are those corresponding to vectors between heavy atoms. Once the heavy atoms are located, the other atoms can be positioned from Fourier electron density maps using the phase angles determined by the heavy-atom coordinates. The ease of solving such structures is partly responsible for the rapid growth of metal organic chemistry. 

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