Coulombic fields

The first tenn is the Coulomb field of the ion, and the second is the potential due to the ion atmosphere at an effective distance equal to 1/k. For a univalent aqueous electrolyte at 298 K,... 

The function /(r) is a force-switching function that goes smoothly from 1 ar r = 0 to 0 at r = Tc. The long-range part of the field, i.e., what remains from the complete Coulomb field ... 

The hydrogen atom presented a unique opportunity in the development of quantum mechanics. The single electron moves in a coulombic field, free from fhe effecfs of infer-elecfron repulsions. This has fwo imporfanf consequences fhaf do nof apply fo any atom wifh fwo or more elecfrons ... 

In an electron-excited X-ray spectrum the discrete X-ray lines are superimposed on a continuous background this is the well-known bremsstrahlung continuum ranging from 0 to the primary energy Eq of the electrons. The reason for this continuum is that because of the fundamental laws of electrodynamics, electrons emit X-rays when they are decelerated in the Coulomb field of an atom. As a result the upper energy limit of X-ray quanta is identical with the primary electron energy. 

If the external field is the Coulomb field of a point charge located at the origin in particular, if... 

Hence, in the limit A->0, the differential cross-section diverges. This is due to the long range character of the Coulomb field, which entails that the expansion (10-259) of KA does not converge. Alternatively, the asymptotic condition is not valid in this case. 

For tiie case of a Coulomb field, special methods exist that reduce the equation to a form where the nonrelativistic solutions can be used.11 Thus, if we denote by tji the wave function of the electron moving in a Coulomb field, then t/i obeys the Dirac equation... 

For the case of the Coulomb field, particularly elegant methods of solution are due to ... 

It should be stressed at this point that, as we shall see, the in and out negaton-positon and electromagnetic fields given by Eqs. (11-56), (11-57), and (11-62) are ill-defined. For the matter field, the reason is that the Coulomb field has an infinite range, and, hence, charged particles, no matter how far apart, still interact with one... 

Co-representation matrices explicit forms, 733 multiplication of, 731 of the nonunitary group, 732 Corliss, L. M., 757 Corson, E. M., 498 Coulomb field Dirac equation in, 637 Coulomb gauge, 643,657,664 Counting functions, 165 Covariance matrix, 160 Covariant amplitude of one-particle system, 511 of one, two, etc. particle systems, 511... 

Mohr, P.J. (1992) Self-energy correction to one-electron energy levels in a strong Coulomb field. Physical Review A, 46, 4421-4424. 

The liquid organic salts are a new class of polar idiases for gas chromatography (56,57). They differ from the phases considered so far in that they contain ions that provide strong interactions associated with the Coulombic fields. Some representative examples of typical liquid org2utic salts and their chromatographic... 

Lekner J (1991) Summation of coulomb fields in computer-simulated disordered-systems. Physica A 176(3) 485-498... 

The methodology of stochastic treatment of e-ion recombination kinetics is basically the same as for neutrals, except that the appropriate electrostatic field term must be included (see Sect. 7.3.1). This means the coulombic field in the dielectric for an isolated pair and, in the multiple ion-pair case, the field due to all unrecombined charges on each electron and ion. All the three methods of stochastic analysis—random flight Monte Carlo (MC), independent reaction time (IRT), and the master equation (ME)—have been used (Pimblott and Green, 1995). 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

These results imply that the use of the representative single ion-pair distribution in the ionization produced by low-LET irradiation in liquid hydrocarbons can be approximately justified even though the track itself has considerable contribution from multiple-ion-pair spurs and short tracks. It also means that even in the case of an isolated ion-pair, the long-time limit of the existence probability is perturbed by the long-range coulombic field. 

In order to avoid overparametrization, we can use the point charge electrostatic model (PCE model) [19, 20], in which N ligands are represented by their point charges (Zt). The Hcf generated by a charge distribution can be written in its most primitive form as a sum of Coulomb fields created by the charges. The Aq CF parameters can then be calculated by the following expression ... 

It is not obvious how a stable nuclear atom can be rationalized in terms of a static charge distribution in a Coulomb field, where the force between two charges is given by... 

The Sehriidinger equation governing tile radial wave functions for positive energy slates in n Coulomb field is... 

---