Compromise spacing effect

The Compromise Spacing Effect is a physical effect that occurs in diffraction by quasicrystals .
Diffraction
Diffraction is the method used to reveal atomic structure. The method results from scattering of electrons, X-rays or neutrons by parallel planes of atoms. In crystals, the interplanar spacing is unique for each diffracted beam; but in quasicrystals, each diffracted beam is produced by multiple interplanar spacings in modified Bragg diffraction. The quasi-lattice parameter results from a compromise between such interplanar spacings. The parameter is larger than had been assumed by extrapolating Bragg’s law for crystals to quasicrystals.
Structure factors
Structure factors for superclusters of quasicrystalline Al<sub>6</sub>Mn alloy were at first close to zero when calculated following Bragg’s law. This was inconsistent with experimental data. However, when the diffraction angle was scanned, realistic factors were revealed that matched, without error, experimental beam intensity rankings. The factors peaked systematically at angles 5.6% smaller than Bragg angles for all indices. In consequence, the measured quasi-lattice parameter was re-evaluated larger, by a similar fraction. The explanation for the systematic angular shift has been described analytically.
Logarithmically periodic solids
The ideal of a logarithmically periodic solid is based on an icosahedral subcluster of 1 manganese atom surrounded by 12 aluminum atoms. This dense subcluster appears to be the driving force for the structure. The subclusters expand by edge sharing into icosahedral clusters and superclusters of order 1,2,3… infinity. The structure factors were calculated by the addition of individual scattering amplitudes from all the atoms in a supercluster order 3, containing 250,000 atom sites. The accuracy was within computer truncation errors. The method overcame problems of icosahedral symmetry; of real indices (neither integral as in crystals, nor six-dimensional as in overcomplicated theories); of logarithmically periodic order; and of multiple interplanar spacings. The method applied to both short-range and long-range logarithmic ordering of atomic planes.
The solid can be tiled in 2-dimensions on a closed, dodecahedral surface and in 3-dimensions on concentric, pseudo space filling icosahedra.
Dimensions in model building
The larger quasi-lattice spacing is especially significant for model building. The Compromise Spacing Effect is consistent with the subclusters. Dimensions in other models require correction since it should be assumed the Compromise Spacing Effect applies equally to them.
Sharp diffraction
The method also reveals why diffraction in quasicrystals is sharp even when glassy defects are evident. The structure factor simulations show that this is because the Compromise Spacing Effect acts also as an angular, narrow band-pass filter in the diffraction.

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