Harker construction

Figure 20 (a) The Harker construction for the SIR method. The circles intersect at two places, A and B, leading to phase ambiguity, (b) An alternative way of drawing the Harker construction with FP centered at the origin is often more convenient for more complicated phasing schemes, such as SIRAS (Figure 24). 

As an aside, let us return for a moment to SIR. How can anomalous scattering be used to break the phase ambiguity of SIR if both methods have similar Harker constructions Fortunately, the information from the isomorphous differences and the anomalous differences is not the same, but complementary. If, as is usually the case, the heavy atom is the only anomalous scatterer, the substructure is the same, that is, we can use the same AH for a reflection h, k, l. When the anomalous difference/" and its inverse are added to Fh, we can draw two circles of radii FPh + and FPh centered at the ends of each vector (Figure 24). With the FP -circle centered at the origin, these three circles only intersect at one point, which defines the phase of FP. This method is Single Isomorphous Replacement with Anomalous Scattering - or SIRAS. 

Figure 23 The Harker construction for the SAD method. The anomalous differences f" are drawn from the end of FH, giving rise to two circles. These two circles intersect at two points (just like in SIR), leaving the phase of FP (not shown for clarity) ambiguous.

Figure 24 The Harker construction for the SIRAS method. The phase ambiguity in SIR is broken by drawing two circles F+ph and F PH centered at the ends of the vectors F+ and F respectively. These circles have only one intersection with the circle FP centered at the origin, just like in MIR (Figure 21), leaving only one possible value for the phase of FP.

Figure 6.7 Harker Construction Solution of the X-ray crystallographic phase problem for each hkt-reflection by Harker construction. Heavy atom structure factor F (hkl) is completely solved by Patterson Function and plotted on complex plane (white) along with known modulus FBH(hkl). The second known modulus Fb(AW) is then included on a second complex plane yellow. Intersection points characterise the two possible solutions for Fi hkl) and two possible solutions for cni hkl) or ccp ), one of which is usually eliminated by inspection or with the aid of a second heavy atom derivative.

Based on their observations Ho and Harker constructed a theory attributing the low and intermediate temperature loss mechanism to localised doping of the primary and secondary crystalline phases to create isolated microscopic regions of finite electrical conductivity localised within the grains and along individual grain Junctions and grain boundaries. A distribution of electrical conductivity values was shown to explain the observed frequency dependence between sapphire and alumina. 

In practice, Harker diagrams, like those described in Figure 8.5 are not actually constructed. Instead, the probability of any phase on the circle being correct, given the structure amplitudes of two or more derivatives and the calculated structure factors of the heavy atoms, is computed at increments around the circle, say at every 5 or 10 degrees. Blow and Crick (1958) showed that the probablility formulation, given certain assumptions regarding the distribution of errors, for any reflection is... 

As with the isomorphous replacement method, the locations x, y, z in the unit cell of the anomalous scatterers must first be determined by Patterson techniques or by direct methods. Patterson maps are computed in this case using the anomalous differences Fi,u — F-h-k-i-Constructions similar to the Harker diagram can again be utilized, though probability-based mathematical equivalents are generally used in their stead. 

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