The Hamiltonian of phonon system with the harmonic approximation is   .
Topics in Applied Physics
From this and the solution to dispersion relation, the phonon annihilation operator for the quantum treatment is defined as. The creation operator is the adjoint of the annihilation operator,. The phonon dispersion relation gives all possible phonon modes within the Brillouin zone zone within the primitive cell in reciprocal space , and the phonon density of states D p the number density of possible phonon modes.
This gives the Debye T 3 law at low temperature and Dulong-Petit law at high temperatures. From the kinetic theory of gases,  thermal conductivity of principal carrier i p , e , f and ph is. Thus, the larger the carrier density, heat capacity and speed, and the less significant the scattering, the higher is the conductivity. Phonons interact with other phonons, and with electrons, boundaries, impurities, etc.
The interaction rates are reviewed in  and includes quantum perturbation theory and MD. With the Debye model a single group velocity u p,g , and a specific heat capacity calculated above , this becomes. Slack phonon conductivity model mainly considering acoustic phonon scattering three-phonon interaction is given as  . This model is widely tested with pure nonmetallic crystals, and the overall agreement is good, even for complex crystals. Based on the kinetics and atomic structure consideration, a material with high crystalline and strong interactions, composed of light atoms such as diamond and graphene is expected to have large phonon conductivity.
Solids with more than one atom in the smallest unit cell representing the lattice have two types of phonons, i. Acoustic phonons are in-phase movements of atoms about their equilibrium positions, while optical phonons are out-of-phase movement of adjacent atoms in the lattice. Optical phonons have higher energies frequencies , but make smaller contribution to conduction heat transfer, because of their smaller group velocity and occupancy.
Phonon transport across hetero-structure boundaries represented with R p,b , phonon boundary resistance according to the boundary scattering approximations are modeled as acoustic and diffuse mismatch models. Thus, numerical techniques are used, and an electron configuration is approximated as product of simpler hydrogen-like atomic orbitals isolate electron orbitals. Molecules with multiple atoms nuclei and their electrons have molecular orbital MO, a mathematical function for the wave-like behavior of an electron in a molecule , and are obtained from simplified solution techniques such as linear combination of atomic orbitals LCAO.
The molecular orbital is used to predict chemical and physical properties, and the difference between highest occupied molecular orbital HOMO and the lowest unoccupied molecular orbital LUMO is a measure of excitability of the molecules. However, in a periodic lattice crystal , there is periodic crystal potential, so the electron Hamiltonian becomes . In practice, a lattice as many-body systems includes interactions between electrons and nuclei in potential, but this calculation can be too intricate. Thus, many approximate techniques have been suggested and one of them is density functional theory DFT , uses functionals of the spatially dependent electron density instead of full interactions.
The electron specific heat is based on the energy states and occupancy distribution the Fermi—Dirac statistics. In general, the heat capacity of electron is small except at very high temperature when they are in thermal equilibrium with phonons lattice.
Microscale and Nanoscale Heat Transfer (Topics in Applied Physics)
Electrons contribute to heat conduction in addition to charge carrying in solid, especially in metals. Various carriers electrons, magnons , phonons, and polarons and their interactions substantially affect the Seebeck coefficient. The change of the entropy-of-mixing upon adding an electron to a system is the so-called Heikes formula. Extending the Seebeck effect to spins, a ferromagnetic alloy can be a good example. Many vibrational effects with electrons also contribute to the Seebeck coefficient.
The softening of the vibrational frequencies produces a change of the vibrational entropy is one of examples. The vibrational entropy is the negative derivative of the free energy, i. The vibrational component in high-band gap materials such as B 13 C 2 is very important. Considering the microscopic transport transport is a results of nonequilibrium ,. Relating the thermoelectric coefficients to the microscopic transport equations for j e and q, the thermal, electric, and thermoelectric properties are calculated.
Electrons interact with other principal energy carriers. Electrons accelerated by an electric field are relaxed through the energy conversion to phonon in semiconductors, mostly optical phonon , which is called Joule heating. Energy conversion between electric potential and phonon energy is considered in thermoelectrics such as Peltier cooling and thermoelectric generator. Also, study of interaction with photons is central in optoelectronic applications i. Interaction rates or energy conversion rates can be evaluated using the Fermi golden rule from the perturbation theory with ab initio approach.
Fluid particle is the smallest unit atoms or molecules in the fluid phase gas, liquid or plasma without breaking any chemical bond. Energy of fluid particle is divided into potential, electronic, translational, vibrational, and rotational energies. The heat thermal energy storage in fluid particle is through the temperature-dependent particle motion translational, vibrational, and rotational energies.
The electronic energy is included only if temperature is high enough to ionize or dissociate the fluid particles or to include other electronic transitions. These quantum energy states of the fluid particles are found using their respective quantum Hamiltonian. From the Hamiltonian, the quantized fluid particle energy state E f and partition functions Z f [with the Maxwell—Boltzmann MB occupancy distribution ] are found as . With the energy states and the partition function, the fluid particle specific heat capacity c v,f is the summation of contribution from various kinetic energies for non-ideal gas the potential energy is also added.
Because the total degrees of freedom in molecules is determined by the atomic configuration, c v,f has different formulas depending on the configuration, . For the polyatomic ideal gas, N o is the number of atoms in a molecule. For dense fluids that the interactions between the particles the van der Waals interaction should be included, and c v,f and c p,f would change accordingly.
Conduction heat flux q k for ideal gas is derived with the gas kinetic theory or the Boltzmann transport equations, and the thermal conductivity is. For calculation of k f , the equilibrium MD with Green—Kubo relations , which express the transport coefficients in terms of integrals of time correlation functions considering fluctuation , or nonequilibrium MD prescribing heat flux or temperature difference in simulated system are generally employed.
Fluid particles can interact with other principal particles.
Microscale and Nanoscale Heat Transfer | Sebastian Volz | Springer
Vibrational or rotational modes, which have relatively high energy, are excited or decay through the interaction with photons. Gas lasers employ the interaction kinetics between fluid particles and photons, and laser cooling has been also considered in CO 2 gas laser. These interaction rates are also calculated through ab initio calculation on fluid particle and the Fermi golden rule. Photon is the quanta of electromagnetic EM radiation and energy carrier for radiation heat transfer.
The EM wave is governed by the classical Maxwell equations , and the quantization of EM wave is used for phenomena such as the blackbody radiation in particular to explain the ultraviolet catastrophe. The photon Hamiltonian for the quantized radiation field second quantization is  . Blackbody radiation among various types of photon emission employs the photon gas model with thermalized energy distribution without interphoton interaction.
From the linear dispersion relation i. For the case of optically thick medium of length L , i. Photons have the largest range of energy and central in a variety of energy conversions. Photons interact with electric and magnetic entities. For example, electric dipole which in turn are excited by optical phonons or fluid particle vibration, or transition dipole moments of electronic transitions. In heat transfer physics, the interaction kinetics of phonon is treated using the perturbation theory the Fermi golden rule and the interaction Hamiltonian.
The photon-electron interaction is .
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Photons also participate in ternary interactions, e. Examples are solid and molecular gas laser cooling. The spectral absorption coefficient for various electric entities are listed in the below table.
From Wikipedia, the free encyclopedia. See also: Thermoelectric effect. Microscale energy transport. Washington, D. New York: Oxford.
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New York: McGraw-Hill. Heat transfer physics 2nd ed. Cambridge: Cambridge University Press. Nanoscale and Microscale Thermophysical Engineering. Physical Review B. Bibcode : PhRvB.. Quantum-mechanical ab-initio calculation of the properties of crystalline materials. Berlin: Springer-Verlag.