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Physical Factors in Phase Formation
Published in Daniel D. Pollock, PHYSICAL PROPERTIES of MATERIALS for ENGINEERS 2ND EDITION, 2020
where ri, is the distance between the ion being considered and the ith ion. The Madelung constant is about 1.75 for the NaCl structure, but varies between 1.63 and 25.0, depending on the type of ionic lattice. Multiplying both sides of Equation 10-13 by NAZe () NAZeϕ=−NA(Ze)2αmr
An Introduction to Crystal Structures
Published in Elaine A. Moore, Lesley E. Smart, Solid State Chemistry, 2020
Elaine A. Moore, Lesley E. Smart
where NA is the Avogadro number, 6.022 × 1023 mol−1 (note that the expression is multiplied by NA and not by 2NA, even though there are NA cations and NA anions present; this avoids counting every interaction twice!). The value of the Madelung constant is dependent only on the geometry of the lattice and not on its dimensions; values for various structures are given in Table 1.15.
Symbols, Terminology, and Nomenclature
Published in W. M. Haynes, David R. Lide, Thomas J. Bruno, CRC Handbook of Chemistry and Physics, 2016
W. M. Haynes, David R. Lide, Thomas J. Bruno
the complex has a hydrophilic outer surface consisting of all the protein and the polar ends of any phospholipids. [5] Liter (L)* - A synonym for cubic decimeter. [1] Lithosphere* - The outer layer of the solid earth, extending from the base of the mantle to the surface of the crust. Lorentz contraction - The reduction in length of a moving body in the direction of motion, given by the factor (1-v2/c2)1/2 , where v is the velocity of the body and c the velocity of light. Also known as the FitzGerald-Lorentz contraction. Lorentz force - The force exerted on a point charge Q moving at velocity v in the presence of external fields E and B. It is given (in SI units) by F = Q(E + v × B). Loss angle () - For a dielectric material in an alternating electromagnetic field, is the phase difference between the current and the potential difference. The function tan is a measure of the ratio of the power dissipated in the dielectric to the power stored. Low energy electron diffraction (LEED) - See Techniques for Materials Characterization, page 12-1. Lumen (lm)* - The SI unit of luminous flux, equal to cd sr. [1] Luminous flux () - The intensity of light from a source multiplied by the solid angle. The SI unit is lumen. [1] Lux (lx)* - The SI unit of illuminance, equal to cd sr m-2. [1] Lyddane-Sachs-Teller relation - A relation between the phonon frequencies and dielectric constants of an ionic crystal which states that (T/L)2 = ()/(0), where T is the angular frequency of transverse optical phonons, L that of longitudinal optical phonons, (0) is the static dielectric constant, and () the dielectric constant at optical frequencies. Lyman series - The series of lines in the spectrum of the hydrogen atom which corresponds to transitions between the ground state (principal quantum number n = 1) and successive excited states. The wavelengths are given by 1/ = RH(1-1/n2), where n = 2,3,4,... and RH is the Rydberg constant for hydrogen. The first member of the series (n = 12), which is often called the Lyman- line, falls at a wavelength of 1216 Å, and the series converges at 912 Å, the ionization limit of hydrogen. Mach number (Ma) - A dimensionless quantity used in fluid mechanics, defined by Ma = v/c, where v is velocity and c is the speed of sound. [2] Macromolecule - A molecule of high relative molecular mass (molecular weight), the structure of which essentially comprises the multiple repetition of units derived, actually or conceptually, from molecules of low relative molecular mass. [8] Madelung constant* - A constant characteristic of a particular crystalline material which gives a measure of the electrostatic energy binding the ions in the crystal. Magnetic field strength (H) - An axial vector quantity, the curl of which is equal to the current density, including the displacement current. [1] Magnetic induction (B) - An axial vector quantity such that the force exerted on an element of current is equal to the vector product of this element and the magnetic induction. [1] Magnetic moment - See Dipole moment, magnetic. Magnetic susceptibility (m, )* - Defined by m = (µ-µ0)/µ0, where µ is the permeability of the medium and µ0 the permeability of a vacuum. [1] Magnetization (M) - Defined by M = (B/µ0)-H, where B is magnetic induction, H magnetic field strength, and µ0 the permeability of a vacuum. [1]
Topological and real charge of Bjerrum defects in ices Ih and Ic
Published in Molecular Physics, 2020
For the calculation of the ‘real’ Coulomb charge of a Bjerrum defect, we use the convergence factor technique known from calculations of the Madelung constant or deriving the Ewald sums [26]. To calculate the charge at position , a sum over all charges weighted by a Gaussian function of certain width σ is made. Since the position of the Bjerrum charge is not accurately known, it has to be obtained by minimisation or maximisation. Finally, the limit is performed. The formulas for the respective Bjerrum charges are where the sum is over positions of all charges in the system.
First-principles study of point defects with different valences on the carrier activity, lifetime, absorption spectrum, and redox reaction of AlN (Li/Na/K) system
Published in Philosophical Magazine, 2022
Zhichao Wang, Qingyu Hou, Yuqin Guan, Zhenchao Xu, Chunwang Zhao
Formation energy (Ef) is an important physical quantity to judge the stability of the doped system and the ease of doping the difficulty of atom doping of the doped system. The calculation formula of Ef [52,53] is as follows: where , and represents total energy of the Al34MHiN36 (M = Li/Na/K), Al35MHiN35(M = Li/Na/K) and Al36N36 system. The team’s previous research [14] has already explained the chemical potential. represents Fermi level, and denote the total energy at the VBM of the undoped AlN system. The last term, , is a correction term set to remove the errors coming from the supercell approach of first-principles calculations under the periodic boundary condition. Regarding the Ef calculations of charged defects, the errors could be mainly introduced by long-range Coulomb interactions between the image defect charges [54,55]. Several schemes have been proposed for this correction, such as the Makov and Payne (MP) scheme [56], the alignment-only scheme [57], the Lany and Zunger scheme [54], and the Freysoldt, Neugebauer, and Van de Walle scheme [58]. Komsa et al. [55] evaluated the performance between different corrections. Given the situation of this work, we take the MP scheme, by which the term is given as: where is the monopole Madelung term, while is the third-order quadrupole electrostatic correction. Specifically, is the state of the monopole aperiodic charge, α is the Madelung constant that could be easily calculated with the Ewald method, and ϵ is the relative dielectric constant of the crystal. Q is the quadrupole moment of the defect charge, and L is the cubic root of the supercell volume. The Ef of all doped systems under different q are shown in Appendix Fig. A1(a–f).