Die Symmetriegruppen der amorphen und mesomorphen Phasen

One-dimensional disorder in Zr9M11(M = Ni, Pd, Pt) and low-temperature atomic mobility in Zr9Ni11

“…In diffraction experiments, deviations of atoms from an ideal crystal structure (positional disorder) may cause the replacement of intensity localized in Bragg positions by diffuse intensity. A classification scheme for different types of disorder, ranging from an amorphous glass to a perfect crystal, was derived by Hermann in 1931 using group theory [1]. Mabis [2] has provided illustrations of these eighteen different structural types in real and reciprocal space.…”

A systematic approach to reduce the independent tensor components by symmetry transformations: a commented translation of “Tensors and Crystal Symmetry” by Carl Hermann

“…We now focus on the complex axes (16) and the basic elastic equations D i k , = C im kn S n m , whereby we first pretend that there are 81 different components C im kn . Equation 17becomes for this special case: For the horizontal r e f l e c t i o n p l a n e, 17 the equation is fulfilled when the number of indices 3 is even; on the other hand, in the presence of this refection plane, the following components must disappear; C 11 13 , C 11 23 , C 21 13 , C 12 13 , C 33 31 , the same applies for all index permutations of these and when interchanging the indices 1 and 2.…”

“…For n-fold r o t a t i o n a x e s with n > 4, only the following components are allowed 18 : C 11 11 , C 12 12 , C 12 21 , C 13 13 , C 31 31 , C 13 31 , C 31 13 , C 12 33 , C 33 12 , C 33 33 , the same holds when 1 and 2 are interchanged. For the fourfold axis C 11…”

“…For the threefold, fourfold and sixfold axes, the following component vanishes C 11 12 = 1 4 (c 11 11 − c 22 22 − c 12 12 + c 11 22 + ic 11 21 + ic 22 12 + c 21 22 − c 22 11 − ic 11 12 − ic 22…”

What has the Penrose tiling to do with the icosahedral phases? Geometrical aspects of the icosahedral quasicrystal problem

“…Our preliminary but immediate answer must be that it is a texture. Textures of icosahedral symmetry were mentioned as a possibility by Hermann (1931) since there is no restriction on the symmetry elements which may occur in point groups.…”