Case 1: Zonal data sets – view down the chain axis. Such data are from the least optimal projection of the polymer packing, because of extensive atomic overlap along the chain axis. Initially, symbolic addition was used to find phase values for hk0 data sets from six representative polymers, including three complicated saccharide structures (Dorset, 1992). Most of the determinations were strikingly successful. Later, an unknown data set from the polysaccharide chitosan was obtained from Grenoble (Mazeau et al., 1994) and direct phase determination was able to find a correct model (Dorset, 1995b). More recently, other polymers have been tested [including one case where an electron micrograph provided many of the starting phase terms (Dorset, 1995b)] also comparing favourably with the solution found by energy minimization of a linkage model. For all examples considered so far, the projected symmetry was centrosymmetric.
Later, it was found that a partial phase set provided by symbolic addition could be expanded to the complete zone by the Sayre equation (Dorset et al., 1995). In all of these tests (Dorset, 1995b), there were only one or two examples where there were serious deviations from the phase terms found by other methods. Even in these instances, the potential maps could still be used as envelopes for the actual projection of the chain structure (Dorset, 1992).
Case 2: Zonal data set – view onto the chain axes. Electron diffraction data from a projection onto the polymer chain axes would be more useful if individual atomic positions were to be resolved. An interesting example where such a view can be obtained is an data set from the polypeptide polyγmethylLglutamate. Electron diffraction data were collected from stretched films by Vainshtein & Tatarinova (1967). In projection, the cell constants are , c = 6.83 Å with planegroup symmetry pg. As shown in Table 4.5.3.1, there were 19 unique intensity data used for the analysis. After initial phase assignment by symbolic addition, a correct solution could be visualized which, after Fourier refinement (Dorset, 1995b), differed from the original one by a mean phase difference of only 6°.
    (previous)  (this study) 
002 
0.48 
0.72 
0.57 
−63 
−51 
004 
0.43 
0.38 
0.31 
49 
73 
006 
3.01 
1.47 
0.88 
1 
−3 
100 
1.48 
2.12 
2.37 
0 
0 
200 
1.03 
1.04 
1.06 
0 
0 
300 
0.30 
0.65 
0.89 
0 
0 
400 
0.35 
0.15 
0.46 
0 
0 
500 
0.23 
0.07 
0.04 
180 
180 
101 
0.75 
1.02 
0.67 
−169 
−178 
201 
0.32 
0.31 
0.42 
90 
108 
102 
0.42 
0.48 
0.56 
17 
14 
202 
0.40 
0.33 
0.64 
41 
43 
103 
0.95 
0.85 
0.77 
88 
90 
203 
0.51 
0.36 
0.42 
91 
88 
303 
0.12 
0.06 
0.31 
92 
87 
403 
0.13 
0.04 
0.54 
90 
90 
104 
0.66 
0.45 
0.27 
−22 
−13 
105 
0.55 
0.28 
0.29 
−26 
−7 
106 
1.75 
0.69 
0.58 
5 
−5 
 This study  Vainshtein & Tatarinova (1967) 
x  z  x  z 

0.048 
0.000 
0.042 
0.000 
C′ 
0.067 
0.331 
0.092 
0.330 
O 
0.281 
0.335 
0.300 
0.330 
N 
0.000 
0.161 
−0.025 
0.175 

The progress of this structure analysis can be reviewed to give a representative example. Since the h00 reflections have centrosymmetric phases, the value was chosen as a single origindefining point. From highprobability threephase invariants (assessed after calculation of normalized structure factors ), one could assign . Symbolic values were then given to three other phases, viz. ; ; . From this entire basis, other values could be found from highly probable threephase invariants, as follows: (These invariant relationships include phase interactions among symmetryrelated Miller indices characteristic of the plane group.) Additionally could be specified to complete origin definition for the zone. It was then possible to permute values of a and b to arrive at test phase values for this subset, i.e. to generate a multiple set of solutions. When , , the map in Fig. 4.5.3.1 was observed. After finding trial atomic positions for Fourier refinement (assuming that two carbonatom positions were eclipsed in this projection), the final phase set was found as shown in Table 4.5.3.1. Although the crystallographic residual to the observed data, calculated with the model coordinates, was rather large (0.32), there was a close agreement with the earlier determination.

Figure 4.5.3.1
 top  pdf  Initial potential map for polyγmethylLglutamate (plane group pg) found with phases generated by the Sayre equation.

More recently a similar data set, collected from oriented crystal `whiskers' of poly(poxybenzoate) in plane group pg was analysed. Again the Sayre equation, via a multisolution approach, was used to produce a map that contained 13 of 18 possible atomic positions for the two subunits in the asymmetric unit. The complete structure was observed after the remaining five atom sites were identified in two subsequent cycles of Fourier refinement (Liu et al., 1997) and the average atomic positions were found to be within 0.2 Å of the model derived from an energy minimization.
Case 3: Threedimensional data – single crystal orientation. The first data set from a chainfolded lamella for a direct structure analysis was a centrosymmetric set (space group ) from poly(1,4transcyclohexanediyl dimethylene succinate), composed of 87 reflections (Brisse et al., 1984). The phase determination was quite successful and atomic positions could be found as somewhat blurred density maxima in the threedimensional maps (Dorset, 1991a). A model was constructed from these positions and the bonding parameters optimized to give the best fit to the data (R = 0.29).
Noncentrosymmetric threedimensional intensity sets (orthorhombic space group ) from the polysaccharides mannan (form I) (Chanzy et al., 1987) and chitosan (Mazeau et al., 1994) were also collected from tilted crystals. In both cases, direct phase determination by symbolic addition via an algebraic unknown was successful, even though the data were not sampled along the chain repeat. For the former polymer, a monomer model could be fitted to the blurred density profile, much as one would fit a polypeptide sequence to a continuous electrondensity map (Dorset & McCourt, 1993; Dorset, 1995c). If the Sayre equation were used to predict phases and amplitudes within the `missing cone' of unsampled data, then the fit of the monomer could be much more highly constrained.
Case 4: Threedimensional data – two crystal orientations. The optimal case for collection of diffraction data is when two orthogonal projections of the same polymer polymorph can be obtained, respectively, by selfseeding and epitaxic orientation. While tilting these specimens, all of reciprocal space can be sampled for intensity data collection.
Polyethylene crystals were used to collect 50 unique maxima (Hu & Dorset, 1989) and, via symbolic addition, the centrosymmetric phases of 40 reflections (space group Pnma) could be readily determined (Dorset, 1991b). The structural features were readily observed in the threedimensional potential maps (Fig. 4.5.3.2a), and atomic coordinates (with estimated values for hydrogenatom positions) could be refined by least squares (Dorset, 1995b) to give a final R value of 0.19.

Figure 4.5.3.2
 top  pdf  Crystal structures of linear polymers determined from threedimensional data. (a) Polyethylene; (b) poly(caprolactone); (c) poly(1butene), form (III).

Poly(caprolactone) was epitaxically crystallized on benzoic acid and, with hk0 data from solutioncrystallized samples, a unique set of 47 intensities was collected for the noncentrosymmetric orthorhombic unit cell (space group ) (Hu & Dorset, 1990). Direct phase determination was achieved via symbolic addition, using one algebraic unknown to assign values to 30 reflections (Dorset, 1991c). Atomic positions along the chain repeat, including the carbonyl position, were clearly discerned in the [100] projection (Fig. 4.5.3.2b) and the threedimensional model was constructed to fit to the map calculated from all phased data, yielding a final crystallographic residual R = 0.21. This independent determination was able to distinguish between two rival fibre Xray structures, in favour of the one that predicted a nonplanar chain conformation. Because of the methylene repeat, this is actually a difficult structure to solve by automated techniques. For example, the tangent formula and SnB (Miller et al., 1993) could only find chain zigzag positions and not the position of the carbonyl oxygen atom (Dorset, 1995b).
The most complicated complete polymer crystal structure solved so far by direct methods using electron diffraction data (Dorset et al., 1994) was based on 125 unique data (space group ) from isotactic poly(1butene), form (III), using orthogonal molecular orientations crystallized in Strasbourg (Kopp et al., 1994). Initially, the standard NQEST figure of merit (FOM) (De Titta et al., 1975) was not suitable for identifying the correct solution among the multiple sets generated with the tangent formula. A solution could only be found when a separate phase determination was carried out with the hk0 data to compare with the multiple solutions generated. More recently, the minimal principle (Hauptman, 1993), used as a FOM with the tangent formula or with a multiple random structure generator, SnB, correctly identified the structure on the first try (Dorset, 1995b). The maps clearly show individual carbonatom positions in a 4_{1} helix that parallels 2_{1} helices of the space group (Fig. 4.5.3.2c). After Fourier refinement, the crystallographic residual was R = 0.26. The previous powder Xray diffraction determination was based on only 21 diffraction maxima, some of which had as many as 15 individual contributors.
