From the volume of the unit cell V = 131 103 Å, the molecular mass M = 1703, and the experimental density of 1.025 g/cm3, the number of molecules per unit cell was calculated using equation (34) as 47.6 and then rounded as 48. As the structure developed from the hexagonal columnar phase we assumed that this basic structure was preserved, implying that there were two columns per unit cell of the P212121 pseudohexagonal phase. Therefore it was concluded that there were twenty four PBI 12-2 molecules per each column and hence twelve PBI 1-2 molecules in an asymmetric unit. 3D Reconstructed Electron Density Maps According to the symmetry of the space group P212121, the structure factor must be F200 real (phase angle 0 or ) while and must be imaginary (phase angle /2). The F110 F201 contribution of and to the electron density map is essentially to produce columns F200 F110 without helicity. Their phases are chosen so that the cores of the columns, formed by the aromatic parts of the molecules, have higher electron density than the alkyl-comprising surroundings. The addition of 201 and 111 reflections makes the columns helical, but the columns are all at the same height (same ( c and d). The sense of the helix (left or right) determines the phase z) angle of 201. In this case the phase angle is chosen arbitrarily to be +/2 corresponding to columns. After the phases of the first three peaks had been determined, was added and its phase F111 angle was varied between 0 and 2 in /6 increments. The resulting 12 electron density maps were examined using isoelectron at different levels. The best phase angle was chosen so as to produce the smoothest helical columns. This procedure is qualitatively equivalent to the maximum entropy method, where the map is the one containing least structural “correct” detail. The next step was the addition of with its phase angle chosen according to the same F211, criterion. As shown in Figures b and b d, the addition of introduces a vertical shift of F211 alternate columns up and down the The difference to the maps caused by the inclusion z-axis. of is further highlighted in . All intensities and phase angles used in the calculation of F211 the maps in b, d are listed in .
From the volume of the unit cell V = 131 103 Å, the molecular mass M = 1703, and the experimental density of 1.025 g/cm3, the number of molecules per unit cell was calculated using equation (34) as 47.6 and then rounded as 48. As the structure developed from the hexagonal columnar phase we assumed that this basic structure was preserved, implying that there were two columns per unit cell of the P212121 pseudohexagonal phase. Therefore it was concluded that there were twenty four PBI 12-2 molecules per each column and hence twelve PBI 1-2 molecules in an asymmetric unit. 3D Reconstructed Electron Density Maps According to the symmetry of the space group P212121, the structure factor F200 must be real (phase angle 0 or ) while F110 and F201 must be imaginary (phase angle /2). The contribution of F200 and F110 to the electron density map is essentially to produce columns without helicity. Their phases are chosen so that the cores of the columns, formed by the aromatic parts of the molecules, have higher electron density than the alkyl-comprising surroundings. The addition of 201 and 111 reflections makes the columns helical, but the columns are all at the same height (same z) ( c and d). The sense of the helix (left or right) determines the phase angle of 201. In this case the phase angle is chosen arbitrarily to be +/2 corresponding to columns. After the phases of the first three peaks had been determined, F111 was added and its phase angle was varied between 0 and 2 in /6 increments. The resulting 12 electron density maps were examined using isoelectron at different levels. The best phase angle was chosen so as to produce the smoothest helical columns. This procedure is qualitatively equivalent to the maximum entropy method, where the “correct” map is the one containing least structural detail. The next step was the addition of F211, with its phase angle chosen according to the same criterion. As shown in Figures b and b d, the addition of F211 introduces a vertical shift of alternate columns up and down the z-axis. The difference to the maps caused by the inclusion of F211 is further highlighted in . All intensities and phase angles used in the calculation of the maps in b, d are listed in .
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