Dynamical matrix three-dimensional crystals

In Chap.3, the theory is generalized to three-dimensional crystals with a basis. Chapter 3 also contains a section dealing with the connection of lattice dynamics and the theory of elasticity. The force constants, dynamical matrix and dispersion relation are illustrated with the help of monoatomic crystals with fee structure. 

Behavior remarkably similar to that revealed by the one-dimensional model crystals is generally observed for lattice vibrations in three dimensions. Here the dynamical matrix is constructed fundamentally in the same way, based on the model used for the interatomic forces, or derivatives of the crystal s potential energy function, and the equivalent of Eq. (7) is solved for the eigenvalues and eigenvectors [2-4, 29]. Naturally, the phonon wavevector in three dimensions is a vector with three components, q = (qx, qy, qz)> and both the fiequency of the wave, co(q), and its polarization, e q), are functions... 

We start the discussion by formulating the Hamiltonian of the system and the equations of motion. The concept of force constants needs further examination before it can be applied in three dimensions. We shall discuss the restrictions on the atomic force constants which follow from infinitesimal translations of the whole crystal as well as from the translational symmetry of the crystal lattice. Next we introduce the dynamical matrix and the eigenvectors this will be a generalization of Sect.2.1.2. In Sect.3.3, we introduce the periodic boundary conditions and give examples of Brillouin zones for some important structures. In strict analogy to Sect.2.1.4, we then introduce normal coordinates which allow the transition to quantum mechanics. All the quantum mechanical results which have been discussed in Sect.2.2 also apply for the three-dimensional case and only a summary of the main results is therefore given. We then discuss the den-... 

Lanthanum molybdate, LaiMoiOg, has been reported to exhibit fast oxide ion conducting properties comparable with the conventional zirconia and ceria compositions. This compound presents a different crystal structure from all known oxide electrolytes, and consists of isolated [Mo04] units in a three-dimensional matrix of [LaiO]" " . La2Mo209 undergoes a reversible phase transition from the non-conductive monoclinic a-form to the highly conductive cubic -form at approximately 580 °C. Powder X-ray diffraction (XRD) of the phases a and fl are practically identical, because the structural phase transition a is actually a transition from a static to dynamic distribution of the oxygen defects... 

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