Phonons and Elasticity in Disordered binary Crystals


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RAS, Tadeus Markus, 2017. Phonons and Elasticity in Disordered binary Crystals

@phdthesis{Ras2017Phono-39350, title={Phonons and Elasticity in Disordered binary Crystals}, year={2017}, author={Ras, Tadeus Markus}, address={Konstanz}, school={Universität Konstanz} }

2017-06-22T09:44:28Z Ras, Tadeus Markus eng Phonons and Elasticity in Disordered binary Crystals 2017 The lack of a unique, classical microscopic reference state impedes the study of elasticity in non-ideal crystals at finite temperature. Not only do thermal lattice vibrations contribute to that lack for their part but they also promote the creation of point defects. Moreover, phason flips cause an ambiguity of the reference configuration intrinsic to quasicrystals. These processes crucially go along with an unclear microscopic definition of the displacement field.<br /><br />This thesis generalizes previous approaches to isothermal crystal elasticity, focusing on the periodic case discussed in Part II. The two equivalent approaches considered can be distinguished by their nature as static/thermodynamic respectively hydrodynamic. Primarily, the linear, dissipationless and isothermal Zwanzig–Mori equations of motion from [WF10] are generalized to binary periodic crystals. This includes the derivation of a dynamical block matrix whose diagonalization yields both acoustic and optical phonon dispersion relations. The elastic constants follow from the acoustic branches by the method of long waves. The central input parameter to the theory is the direct correlation function which can be obtained from classical Density Functional Theory (DFT). As a proof of principle, dispersion relations are computed for several binary hard sphere model systems, based on a simple DFT approach to freezing. Dispersion relations are further computed from the equilibrium statistics of a binary hard sphere crystal Molecular Dynamics simulation. To that end, the definition of the linear response displacement field from [WF10] is generalized in a species-wise manner.<br /><br />A discussion of the properties of the dynamical matrix in the long-wavelength limit is performed with the result of potential couplings of the total momentum density to non-hydrodynamic variables. While these couplings can be ruled out in systems with inversion symmetry, there are strong indications for a lowering of the elastic constants in the opposite case. This seems in conflict with phenomenological hydrodynamics of crystals and raises fundamental questions about the correct application of the Zwanzig–Mori formalism. More specifically, the presence of reversible contributions in the memory matrix comes into question. In reaction to that, two working-hypotheses are formulated that allow to apply the formalism in the conventional way, yet appear incompatible in the light of the findings.<br /><br />Therefore, a “second witness is called to court” in the form of the thermodynamic DFT approach to crystal elasticity. It is formulated for binary periodic crystals of arbitrary symmetry. By dropping the requirement of inversion symmetry and by the inclusion of point defect fluctuations, this thesis adds to the scope of previous binary approaches. Additionally, the equilibrium condition at given external strain and defect fluctuations is discussed on the level of arbitrary internal strain. This is expected to provide a straightforward access to the treatment of more complicated internal strains. The subsequent example includes only a single sublattice displacement as “optical” internal strain parameter. Its coupling to external strain is derived and related to results known from the classical low-temperature potential expansion. Whether this potential expansion approach has been correctly recovered by [Wal09] in the case of lattices without inversion symmetry is studied on the example of the honeycomb lattice.<br /><br />In a separate part, approaches to the description of acoustic phonons in quasicrystals within the same Zwanzig–Mori formalism are explored. The characteristic “dense” structure of the reciprocal lattice for such systems poses challenges to which no conclusive answer could be given. The author hopes that the approaches made may serve as a starting point for future research. Nonetheless, encouraged by earlier works in that direction, the thermodynamic approach to elasticity is modified from periodic crystals and applied to a specific two-dimensional quasicrystal example. 2017-06-22T09:44:28Z Ras, Tadeus Markus

Dateiabrufe seit 22.06.2017 (Informationen über die Zugriffsstatistik)

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