Unobserved reflections

In contrast to single-crystal work, a fiber-diffraction pattern contains much fewer reflections going up to about 3 A resolution. This is a major drawback and it arises either as a result of accidental overlap of reflections that have the same / value and the same Bragg angle 0, or because of systematic superposition of hkl and its counterparts (-h-kl, h-kl, and -hkl, as in an orthorhombic system, for example). Sometimes, two or more adjacent reflections might be too close to separate analytically. Under such circumstances, these reflections have to be considered individually in structure-factor calculation and compounded properly for comparison with the observed composite reflection. Unobserved reflections that are too weak to see are assigned threshold values, based on the lowest measured intensities. Nevertheless, the number of available X-ray data is far fewer than the number of atomic coordinates in a repeat of the helix. Thus, X-ray data alone is inadequate to solve a fiber structure. 

After completion of the MEM enhancement, it becomes possible to evaluate the reflections missing from the summation. In a Fourier summation, the amplitudes of the unobserved reflections are assumed to be equal to zero, while the MEM technique provides the most probable values. 

Rpl =0.179, Rp2 = 0.202, Ral =0.207, and Ra2 = 0.249. Statistical tests show that model a2 can be rejected in favor of aj. The choice between the parallel models pj and p2 is more difficult but examination of model p2 shows that this model cannot be fully hydrogen bonded, and hence p2 is rejected in favor of pj, which also gives better x-ray agreement. Thus models pj and aj were taken as the most likely parallel and antiparallel models for further refinement. At this point the unobserved data were included, calculating weighted R and R" where w = l for observed and w=l/2 for unobserved reflections. F(hkl) for an unobserved reflection was set at two thirds an assigned threshold and was included only if the calculated structure amplitude exceeded the threshold. The final residuals for the two models were Rpi = 0.233, R = 0.299, and Rj = 0.215, R j = 0.270. Application of the Hamilton statistical test (13) to these data indicates that the a model can be rejected at the 99.5% level. 

Rp2 =0.188, Rai=0.195, and Ra2=0.171. These results suggest that pj can be eliminated but do not allow for selection between the other three. However examination of the intermolecular contacts shows only model a2 is fully acceptable, whereas serious short contacts occur for the other three. Non-bonded. constraints were incorporated in the refinements but these were not successful in removing the bad contacts nor was the situation improved by refinement using the full data including 41 unobserved reflections (at w = 1/2). ... 

Unobserved reflections(up to the smallest d-spacing at which a reflection was observed on a given layer line) were included in the refinement if their calculated structure factor was greater than the threshold value, otherwise they were omitted. Since all of the authors disregarded unobserved reflections in their original work, it was necessary to estimate threshold values. 

The photographs from which our original data set was obtained were used to do this, taking as a guide the intensities of weak reflections superimposed upon a background similar in intensity to that where the unobserved reflection would be expected. Having established values for our own data set, it was assumed that threshold intensities of unobserved reflections in other sets would be in a similar ratio to the intensities of nearby observed reflections. 

In adopting this procedure we have reduced the independence of the original data sets. However, it has been our recent experience that omission of unobserved reflections can lead to bad choices of final refined models. A good model (i.e. one for which the calculated intensity of most unobserved reflections is low) will only be slightly affected by the precise choice of threshold, since only those few reflections whose calculated intensity is greater than this will contribute to the refinement. 

A general practice is to classify those Bragg reflections that have measured intensities less than an arbitrary multiple (usually two or three) of their estimated standard deviations as unobserved reflections. The term unobserved " is an unfortunate one since it also includes some weak reflections whose intensities have been measured. Reflections that are classified as weak or unobserved cannot simply be discarded from the data set they are needed for statistical analysis purposes and may contain relevant information about the structure. ... 

Unobserved reflections Bragg reflections that are too weak to be measured by the apparatus in use. The term is also used for Bragg reflections for which the intensity I is less than n(r[I), where n is chosen (usually 2-3). 

Figure 5.17. Powder diffraction pattern of Fe7(P04)s collected on a Scintag XDS2000 diffractometer using Cu Ka radiation in a step scan mode with A20 = 0.02° and counting time 30 sec. The three sets of vertical bars illustrate the following top - positions of the observed Bragg peaks, middle - positions of Bragg peaks calculated using ITO solution No. 1 (correct), and bottom - the same calculated using ITO solution No. 2 (incorrect) both solutions are listed in Table 5.24. Filled circles indicate unobserved reflections and filled triangle indicates the only observed reflection below 20 = 20°, which was left unindexed in the solution No. 2.

Refinement requires an iterative approach owing to the nonlinearity of the problem. In principle, refinement with either R t or R2 should converge to the same result (if weights w(1) and w(2) are consistent), but the latter, besides being possibly more convenient for noncentrosym-metric structures, obviates a certain problem about unobserved reflections (the intensities of which are not significantly above background). The problem of what to do about unobserved reflections has nettled... 

Additionally the high value for (11) = 1.068 shows that there are a lot of weak or unobserved reflections. XPREP confirms the twinning with its obverse/reverse twin test ... 

The ramie cellulose I-l,3-diaminopropane complex structure is the most crystalline of the diaminopropane complexes so far examined, and has a (metrically) orthorhombic unit cell that can contain only one chain which dictates a parallel chain structure. Elemental analysis and density measurements point to one molecule of 1,3-diamlnopropane per glucose residue. Refinement proceeded in a manner similar to that described above for the ethylenedlamlne complex, based on Intensity data for 24 observed and 12 unobserved reflections. 

For unobservable reflections for which Fp(n) a lFp(n)l the agreement is taken to be exact and nothing is added to either the numerator or the denomi-... 

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