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2 Bulletin of the Polish Academy of Sciences. Technical Sciences

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Some new aspects of the characterization of Miller, Laue and directions indices for centred lattices

Michalski Edward

Bulletin of the Polish Academy of Sciences. Technical Sciences

2022

Vol. 70, nr 5

art. no. e143109

A review of the Miller, Laue and direction indices characterization was made. Excluding or allowing non-coprime indices, depending on whether the lattice is primitive or centred, were compared. The solution of the “spacing counting problem for centred lattices was proposed. It was shown that for centred lattices: (1) Laue indices nh nk nl can represent not only n-th order diffraction on (hkl) planes, but also the first order diffraction from a family of planes (nh nk nl); (2) “integral reflection conditions” are necessary, but not sufficient for the existence of given Miller indices. “Integral reflection conditions” for Laue indices hkl and other “conditions for Miller indices” (hkl) were distinguished. It was shown that in the case of centred lattices, the inference based on the value of n obtained from the equation of lattice planes, may not be correct. The homogeneity of the centred reciprocal lattices has been clarified. “Simple cubic cell with a base” as a choice of unit cell proposed by “general rule” was contrasted with: “unit cell, if not centred, must be the smallest one”. “Integral reflection conditions” for Laue indices and other, new “conditions for Miller indices”, resulting from transformation of centred lattices to unconventional primitive ones have been proposed. Examples of the not correct use of indices in the morphology and diffraction pattern descriptions were shown.

On the new characteristics of Miller indices for centered lattices

Michalski E.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2018

Vol. 66, nr 4

529--538

This work proposes and justifies a clarification of the description of the crystal structure with the use of centered lattices, and concerns also the following: (1) the graphical and analytical criterion for the existence of lattice planes, described by selected Miller indices, (2) the correct way to use the parametric equation of families of planes, (3) “geometric derivation of the integral reflection conditions” and “Laue indices of Bragg peaks versus Miller indices of families of lattice planes”, (4) the characteristics of Miller indices describing nodes of reciprocal lattices for centered direct lattices, (5) the characteristics of Miller indices describing the faces of single crystals and also (6) the characteristics of the information included in Miller indices. Reciprocal lattice nodes associated with families of lattice planes in direct lattices do not form the centered lattices in the reciprocal space themselves. The centered lattices in reciprocal space are created by points with coordinates equal to the Laue indices of Bragg reflections, which are allowed by the integral systematic absences. Parts of them are not associated with any of the direct lattice planes.