Lattice Dynamics

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As presented, crystal structures, crystal lattices, and lattice dynamics is presented in detail. In the presentations, the hamiltonian of the system is expressend in terms of Born Oppenheimer approximations. In the Born Oppenheimer approximation, the Hamiltonian of the system is treated as Nuclear Hamiltonian and quantum Harmonic Hamiltonian. This approximatiation constists two parts, electron - atom interaction and electron nuclear interactions. Electronic and energy band structures is presented for different nanomaterials . This structures are illustarted by using Born Oppenheimer approximations, in conjuction with Kohn-sham equations, and density fuctional theory. In the Born approximations, energy surfaces was illustrated at equilibrium conditions Fourier transform is employed to describe harmonic approximations and force constant. Phonon problem is treated in terms of quantum mechanical picture as Quantum harmonic oscillator and Coherent quantum states. Phonon dispersion relation is also presented for model materials such as Diamond. In this, circumstance, important phenomenon such as energy quantization and second quantization s were presented.Flux quantization, and conductance quantization are also an integral part of this topics. Phonon problem is presented for magnetic, superconducting, and multiferroic materials. Examples of such materials, includes, Paramagnetic iron (Fe), BaSrO3, BaTiO3, and etc. Moreover, topological systems, PbO2, topological superconductors, Semiconductor monolayers, semiconductor bi layers, and semiconductor multilayers are the focus area of research. As electron phonon coupling and many of the important features related with these system, is presented, It is expected that photon, nucleus interactions and electron nuclear nteractions governed by Brahemstrhalung are responsible for many important radiations such as B-radiation, X-ray radiations, and gamma radiations which have an important applications in radiation therapy and tracer.