The Log-Lin Metric for Generic Responses in Logarithmic Structures

The Log-Lin metric is keystone on the arch that joins experimental quasicrystal data with ideal structure: how does a periodic probe, $e.g. $an X-ray or electron beam, interact with an ``aperiodic'' solid to produce sharp diffraction in geometric space? Based on the known structure [1-2], quasi-structure factors are expanded in geometric series, where the metric serves to overlap the periodic wave onto a logarithmic grid [3]. The metric, now systematically analyzed and simulated, enables measurement from the atomic scale to high order superclusters. The metric is analytically derived from a mathematical constant (pi/tau) that converts the geometric series base tau to the same series base pi. The factor applies to physical clusters of extremely dense, binary, hard-sphere, icosahedral, unit cells.\\[4pt] [1] Bourdillon, A.J., \textit{Micron, }\textbf{51} 21-25, (2013): doi: 10.1016/j.micron.2013.06.004.\\[0pt] [2] Bourdillon, A.J.,\textit{ J. Mod. Phys. }\textbf{5} 488-496 (2014): doi.org/10.4236/jmp.2014.56060.\\[0pt] [3] Bourdillon,A.J.,\textit{ J. Mod. Phys. }\textbf{5} 1079-1084 (2014): doi.org/10.4236/jmp.2014.512109