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Electrons in Semiconductors
Published in Hualin Zhan, Graphene-Electrolyte Interfaces, 2020
In general, it is still impossible to obtain the global analytical solution of Eq. 2.1 without any assumptions at this stage, though a recent study has proven the existence of a solution for long-range interactions [Gressman and Strain (2010)]. Therefore, the following assumptions are made for the BTE when investigating electron transport in solid materials using the Drude model. The distribution function is Maxwell-Boltzmann. In statistical mechanics, classical particles which are described by Newton's laws are distinguishable from each other, and their trajectories are traceable. The Pauli exclusion principle does not apply to these particles, and therefore their population is described by a Maxwell-Boltzmann distribution.An electron, as a classical particle, is only scattered by the atoms or ions in the material. The time taken by a collision process is negligible and the trajectory of an electron is unaffected between two collision events.Electron-electron interaction is ignored. This is because in statistical mechanics the distribution function of particles describes the probable state of the system. In other words, the population of electrons at each energy level (i.e., the distribution function) already accounts for electron-electron interaction.
Quantum Primer
Published in Thomas M. Nordlund, Peter M. Hoffmann, Quantitative Understanding of Biosystems, 2019
Thomas M. Nordlund, Peter M. Hoffmann
Average values naturally occur in statistical phenomena, where processes are repeated more than once or when several or many identical (or almost identical) objects perform the same process and the outcome of the process can produce various values. In formal statistical applications, a distribution function is generally used to describe the probability that each outcome results. We will develop some of these statistical distribution functions in Chapter 14. A quantum wave function is similar to a statistical distribution function. If |ψ(x)|2 is the probability per unit distance that the particle is located at position x, then for normalized wave functions () |ψ(x)|2dx = ψ*(x)ψ(x)dx = probability the particle is located between x and x + dx
Semiconductor Materials
Published in Jerry C. Whitaker, Microelectronics, 2018
The energy distribution function (under thermal equilibrium conditions) is given by the Fermi-Dirac distribution function F(E)=[1+exp(E−EfkBT)]−1
Iteration Methods with Multigrid in Energy for Eigenvalue Neutron Diffusion Problems
Published in Nuclear Science and Engineering, 2019
Luke R. Cornejo, Dmitriy Y. Anistratov, Kord Smith
Particle transport processes play an important role in various physical phenomena. For a wide class of problems, the basis for mathematical modeling of particle interactions with matter in a physical system is the linear Boltzmann equation.1 Its solution is the particle distribution function in the phase-space and time which enables the determination of various characteristics of particle population and prediction of behavior of the physical system. The physics of nuclear reactors is dominated by the neutron transport and neutron-nuclide interactions. To design and analyze a nuclear reactor it is necessary to model the dynamics of the neutron population. A particular question about performance of the nuclear system is to determine physical conditions under which there is balance between neutron production and loss. These conditions can be found by solving a certain type of eigenvalue problem for the steady-state Boltzmann equation.
Nanofluid double diffusive natural convection in a porous cavity under multiple force fields
Published in Numerical Heat Transfer, Part A: Applications, 2020
Here, is the velocity component of in the y direction. and the Hartmann number is the viscosity of the fluid. is the magnetic field, and represents the electrical conductivity. Macroscopic quantities such as density and velocity can be obtained from the distribution function.