 | Acta Crystallographica Section B Acta Crystallographica Section B STRUCTURAL SCIENCE, CRYSTAL ENGINEERING AND MATERIALS |
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![[Figure 7]](bk5084fig7mag.jpg)
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Figure 7
The superspace approach for aperiodic structures. At the top a periodic one-atom structure is shown with four equivalent positions P, P′, P′′ and P′′′ (a). In the aperiodic structure the four atoms have been shifted out of their original positions (small black circles) by apparently arbitrary amounts (b). (1 + 1)-Dimensional sketch (c) of the superspace approach around the atomic positions of (b). A section defined by the superspace vectors as1 and as4 is shown, with as4 perpendicular to a1 and the angle between a1 and as1 defined by the α component of q (see text). Average positions (cf a) depicted as dashed vertical lines, `modulated positions' as solid curves (red). Please note that these red curves correspond to the red lines shown in the top row of Fig. 2 ![[link]](../../../../../../logos/arrows/b_arr.gif) (a). The positions of the atoms in three-dimensional space (P, P′, P′′, P′′′) can be derived from the intersections of the three-dimensional space line R (parallel to a1) and the atomic modulation functions. Shifting P, P′, P′′, P′′′ into the unit cell on the left by lattice translations (dotted lines) gives the shape of the AMF of this atom, or, in other words: the AMF representing atom P in superspace is the mathematical expression of all the positions of P in the three-dimensional crystal. R can be drawn for different values of t, the phase of the modulation (here: t = 0). A more detailed definition of t is given in §1.3. (d) Superspace section with two unit cells to illustrate the significance of t. To calculate all interatomic distances on R it is sufficient to calculate all distances for the first unit cell from t = 0 to t = 1 (see text). For example, the distance d2 at t = t2 in (d) is equivalent by translational symmetry to the distance between P′′ and P′′′ on the three-dimensional line R in (c) where t = 0. |
![[Figure 7]](bk5084fig7.jpg)
| STRUCTURAL SCIENCE CRYSTAL ENGINEERING MATERIALS |
ISSN: 2052-5206
