Low-symmetry structures

The problem of symmetry breaking (SB) is well known and multiply discussed in literature. Briefly, we can formulate it as follows. The Hamiltonian of any system of particles forming the Universe is totally symmetric with respect to rotations and reflections in the isotropic space-time, as well as transmutations of identical and equivalent particles, whereas the real objects of the material world composed by these particles do not possess such symmetry. This is seen already from the examples that we live in a world of particles, not antiparticles, and in condensed matter, we have mostly low-symmetry structures. This circumstance can be expressed by the statement that the world is in a state of broken symmetry. An obvious explanation of the contradiction between the totally symmetric Hamiltonian and the broken symmetry of the real world is that the latter is not a solution of its Schrodinger equation. 

The above derivation of the effective Hamiltonian is only complete when, for some reasons, the uniform strains of the crystal are not relevant. This is clearly the case for crystals with low concentration of Jahn-Teller impurities. Contrary to that, bulk deformations often arise in its low-symmetry structural phases of Jahn-Teller crystals [2,11]. The uniform strains describing the bulk deformations of the crystal cannot be reduced to a combination of phonon modes, as it was first pointed out by... 

The potential energy surfaces of the reactions in equations 10 and 11 are more complex than that of equation 14 and involve reaction path bifurcation and low symmetry structures in each channel. The salient features of structures of these complexes and TSs are shown... 

Substances with these low-symmetry structures yield powder patterns which are almost impossible to index by graphical methods, although the patterns of some orthorhombic crystals have been indexed by a combination of graphical and analytical methods. The essential difficulty is the large number of variable parameters involved. In the orthorhombic system there are three such parameters (flf, b, c), in the monoclinic four (a, b, c, P), and in the triclinic six a, b, c, oc, p, y). If the structure is known, patterns of substances in these crystal systems can be indexed by comparison of the observed sin 0 values with those calculated for all possible values of hkl. 

The above observations and discussions indicate that HEU frameworks behave differently compared to most synthetic zeolites with disordered or partly disordered Si, A1 distribution. Even for a given Si/Al ratio the exact exchange behavior of a HEU framework can not be predicted based on the existing knowledge. One of the reasons is the low topological symmetry (C2/m) of the HEU framework compared to the cubic frameworks of e.g. LTA or FAU. In low symmetry structures the distribution of Si and Al, or the existence of numerous polymorphs, plays a much more important role than in a high symmetry framework. In summary, we have to accept the conclusion [14] that for any sophisticated practical application of natural clinoptilolites specific studies on representative samples from the deposit that is being examined for its exploitation potential have to be carried out. 

Now, the qualitative symmetry principle of crystal chemistry [48] first states that there is a tendency for highly symmetric atomic arrangements. Second, symmetry reductions result from specific atomic properties but they tend to be small. Third, whenever low-symmetry structures are obtained due to phase transformations and solid-state chemical reactions, the initial symmetry is indirectly (macroscopically) preserved by the formation of crystallographic domains. 

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