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Fundamentals of lattice dynamics in the harmonic approximation
International Tables for Crystallography (2013). Vol. D, Section 2.1.2, pp. 286-294 [ doi:10.1107/97809553602060000911 ]
Fundamentals of lattice dynamics in the harmonic approximation 2.1.2. Fundamentals of lattice dynamics in the harmonic approximation 2.1.2.1. Hamiltonian and equations of motion | | In order to reduce the complexity of lattice dynamical considerations, we describe the crystal's periodicity by the smallest unit needed to generate the whole (infinite) lattice by ...
Density of states and the lattice heat capacity
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.7, pp. 291-292 [ doi:10.1107/97809553602060000911 ]
Density of states and the lattice heat capacity 2.1.2.7. Density of states and the lattice heat capacity The total energy stored in the harmonic phonon system is given by the sum over all phonon states (): Related thermodynamic quantities like the internal energy or the heat capacity are determined by the frequency ...
Amplitudes of lattice vibrations
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.6, pp. 290-291 [ doi:10.1107/97809553602060000911 ]
Amplitudes of lattice vibrations 2.1.2.6. Amplitudes of lattice vibrations Lattice vibrations that are characterized by both the frequencies and the normal coordinates are elementary excitations of the harmonic lattice. As long as anharmonic effects are neglected, there are no interactions between the individual phonons. The respective amplitudes depend on the excitation ...
Eigenvectors and normal coordinates
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.5, p. 290 [ doi:10.1107/97809553602060000911 ]
Eigenvectors and normal coordinates 2.1.2.5. Eigenvectors and normal coordinates The plane-wave solutions (2.1.2.10) of the equations of motion form a complete set of orthogonal functions if q is restricted to the first Brillouin zone. Hence, the actual displacement of an atom ([kappa]l) can be represented by a linear combination ...
Eigenvalues and phonon dispersion, acoustic modes
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.4, pp. 288-290 [ doi:10.1107/97809553602060000911 ]
Eigenvalues and phonon dispersion, acoustic modes 2.1.2.4. Eigenvalues and phonon dispersion, acoustic modes The wavevector dependence of the vibrational frequencies is called phonon dispersion. For each wavevector q there are 3N fundamental frequencies yielding 3N phonon branches when is plotted versus q. In most cases, the phonon dispersion is displayed for ...
The dynamical matrix
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.3, pp. 287-288 [ doi:10.1107/97809553602060000911 ]
The dynamical matrix 2.1.2.3. The dynamical matrix If the ansatz (2.1.2.10a) is inserted into the equation of motion (2.1.2.9), the following eigenvalue equation is obtained: The summation over all primitive cells on the right-hand side of equation (2.1.2.15) yields the Fourier-transformed force-constant matrix which is independent of l ...
Stability conditions
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.2, p. 287 [ doi:10.1107/97809553602060000911 ]
Stability conditions 2.1.2.2. Stability conditions Not all of the elements of the force-constant matrix are independent. From its definition, equation (2.1.2.5), it is clear that the force-constant matrix is symmetric: Moreover, there are general stability conditions arising from the fact that a crystal as a whole is in mechanical ...
Hamiltonian and equations of motion
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.1, pp. 286-287 [ doi:10.1107/97809553602060000911 ]
Hamiltonian and equations of motion 2.1.2.1. Hamiltonian and equations of motion In order to reduce the complexity of lattice dynamical considerations, we describe the crystal's periodicity by the smallest unit needed to generate the whole (infinite) lattice by translation, i.e. the primitive cell. Each individual primitive cell may be characterized ...
Thermal expansion, compressibility and Grüneisen parameters
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.8, pp. 292-294 [ doi:10.1107/97809553602060000911 ]
Thermal expansion, compressibility and Grüneisen parameters 2.1.2.8. Thermal expansion, compressibility and Grüneisen parameters So far, we have always assumed that the crystal volume is constant. As long as we are dealing with harmonic solids, the thermal excitation of phonons does not result in a mean displacement of any atom. ...
International Tables for Crystallography (2013). Vol. D, Section 2.1.2, pp. 286-294 [ doi:10.1107/97809553602060000911 ]
Fundamentals of lattice dynamics in the harmonic approximation 2.1.2. Fundamentals of lattice dynamics in the harmonic approximation 2.1.2.1. Hamiltonian and equations of motion | | In order to reduce the complexity of lattice dynamical considerations, we describe the crystal's periodicity by the smallest unit needed to generate the whole (infinite) lattice by ...
Density of states and the lattice heat capacity
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.7, pp. 291-292 [ doi:10.1107/97809553602060000911 ]
Density of states and the lattice heat capacity 2.1.2.7. Density of states and the lattice heat capacity The total energy stored in the harmonic phonon system is given by the sum over all phonon states (): Related thermodynamic quantities like the internal energy or the heat capacity are determined by the frequency ...
Amplitudes of lattice vibrations
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.6, pp. 290-291 [ doi:10.1107/97809553602060000911 ]
Amplitudes of lattice vibrations 2.1.2.6. Amplitudes of lattice vibrations Lattice vibrations that are characterized by both the frequencies and the normal coordinates are elementary excitations of the harmonic lattice. As long as anharmonic effects are neglected, there are no interactions between the individual phonons. The respective amplitudes depend on the excitation ...
Eigenvectors and normal coordinates
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.5, p. 290 [ doi:10.1107/97809553602060000911 ]
Eigenvectors and normal coordinates 2.1.2.5. Eigenvectors and normal coordinates The plane-wave solutions (2.1.2.10) of the equations of motion form a complete set of orthogonal functions if q is restricted to the first Brillouin zone. Hence, the actual displacement of an atom ([kappa]l) can be represented by a linear combination ...
Eigenvalues and phonon dispersion, acoustic modes
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.4, pp. 288-290 [ doi:10.1107/97809553602060000911 ]
Eigenvalues and phonon dispersion, acoustic modes 2.1.2.4. Eigenvalues and phonon dispersion, acoustic modes The wavevector dependence of the vibrational frequencies is called phonon dispersion. For each wavevector q there are 3N fundamental frequencies yielding 3N phonon branches when is plotted versus q. In most cases, the phonon dispersion is displayed for ...
The dynamical matrix
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.3, pp. 287-288 [ doi:10.1107/97809553602060000911 ]
The dynamical matrix 2.1.2.3. The dynamical matrix If the ansatz (2.1.2.10a) is inserted into the equation of motion (2.1.2.9), the following eigenvalue equation is obtained: The summation over all primitive cells on the right-hand side of equation (2.1.2.15) yields the Fourier-transformed force-constant matrix which is independent of l ...
Stability conditions
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.2, p. 287 [ doi:10.1107/97809553602060000911 ]
Stability conditions 2.1.2.2. Stability conditions Not all of the elements of the force-constant matrix are independent. From its definition, equation (2.1.2.5), it is clear that the force-constant matrix is symmetric: Moreover, there are general stability conditions arising from the fact that a crystal as a whole is in mechanical ...
Hamiltonian and equations of motion
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.1, pp. 286-287 [ doi:10.1107/97809553602060000911 ]
Hamiltonian and equations of motion 2.1.2.1. Hamiltonian and equations of motion In order to reduce the complexity of lattice dynamical considerations, we describe the crystal's periodicity by the smallest unit needed to generate the whole (infinite) lattice by translation, i.e. the primitive cell. Each individual primitive cell may be characterized ...
Thermal expansion, compressibility and Grüneisen parameters
International Tables for Crystallography (2013). Vol. D, Section 2.1.2.8, pp. 292-294 [ doi:10.1107/97809553602060000911 ]
Thermal expansion, compressibility and Grüneisen parameters 2.1.2.8. Thermal expansion, compressibility and Grüneisen parameters So far, we have always assumed that the crystal volume is constant. As long as we are dealing with harmonic solids, the thermal excitation of phonons does not result in a mean displacement of any atom. ...
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