Relativistic kinematics

Fig. 3. A comparison of the eigenvalues of the outermost valence electrons for Pu using relativistic, semi-relativistic and non-relativistic kinematics and the local density approximation (LSD). Dirac-Fock eigenvalues after Desclaux are also shown. The total energies of the atoms (minus sign omitted), calculated with relativistic and non-relativistic kinematics are also shown. HF means Hartree Fock...

Kinematical relativistic effects are caused by the fact that in the vicinity of the nucleus the electrons acquire high velocities, at a substantial fraction of the velocity of light. The direct influence of the relativistic kinematics (the so-called direct relativistic effect) is thus largest in the vicinity of the nucleus. However, as far as their impact on chemistry is concerned, relativistic effects are most important in the valence shells, which despite the small velocities of outer electrons are still strongly affected by relativistic kinematics (Schwarz et al. 1989). In particular, valence s and p orbitals possess inner tails they are core-penetrating orbitals, which means that there is a nonvanishing probability of finding their electrons close to the nucleus and thus... 

Since the operators f and f2 occur only at the level of the calculation of the spatial spin-orbit integrals over atomic orbitals, Breit-Pauli spin-orbit coupling operators and DKH spin-orbit coupling operators can be discussed on the same footing as far as their matrix elements between multi-electron wave functions are concerned. These terms constitute, by definition, the spin-orbit interaction part of the operator H+ (Hess etal. 1995). The spin-independent terms characteristic of relativistic kinematics define the scalar relativistic part of the operator, and terms with more than one cr matrix (not considered here) contribute to spin-spin coupling phenomena. 

In view of its field-theoretical basis, this energy functional not only accounts for the relativistic kinematics of both electrons and photons, but, in principle, also for all radiative corrections. With the Ritz principle,1 avoiding the question of interacting u-representability (Dreizler and Gross 1990), we may then formulate the basic... 

The time-dependence of wave packets moving according to the Dirac equation usually cannot be determined explicitly. In order to get a qualitative description of the relativistic kinematics of a free particle, we investigate the temporal behavior of the standard position operator. With Ho being the free Dirac operator, we consider (assuming, for simplicity, h = l from now on)... 

The result about the velocity obtained above is not what we expect from classical relativistic kinematics. In classical mechanics, the connection between the kinietic energy E, the momentum p, and the velocity v can be described by the formula... 

With —eE = V(/>ei this expression is twice as large as the spin-orbit term in (99). But there is still another contribution coming from an effect in relativistic kinematics the Thomas precession. As the composition of two boosts with nonparallel velocities contains a rotation, one finds that an accelerated frame of reference performs an additional precession with the frequency... 

The result (146) overestimates the correct spin-orbit interaction (see section 4.6) by a factor 2. This can be explained by noting [7], that in the just-given derivation one ignores the Thomas precession, which has to do with relativistic kinematics - and is ignored in the nrl of quantum mechanics, and which compensates half of (146). In addition there are also spin-independent effects of relativistic kinematics (see section 4.6). 

The induced field arises with a prefactor c , and vanishes in the nrl of electrodynamics (see section 2.9). It must therefore be ignored in a theory that is consistently at the nrl. The expression (146) is a relativistic effect, even if it is due to a correction to the nrl of electrodynamics. To consider only the relativistic corrections to electrodynamics, and not those - of the same order in c - due to relativistic kinematics, is highly inconsistent. 

To summarize, we have the somewhat puzzling situation that external magnetic fields, the origin of which we don t know, give rise to a non-relativistic interaction with moving electrons and with electron spin, while magnetic fields that are obviously created by moving electrons, enter only as relativistic effects, which are comparable in size to effects of relativistic kinematics, and should not be separated from the latter. 

Our DKH values and the ZORA results for g values are rather close in many cases although for such large Ag shifts as for the heavy-atom radical PdH the absolute difference can become significant. In general, the accuracy of calculated g values for c -metal species, at all levels of approximations presently described in the literature, is notably lower than that of p-element compounds [35] see also Chapter 9 of this volume. Detailed reasons for this misrepresentation still have to be found. Obviously, for g tensors of (heavy) d-metal open-shell systems, one can profit from two-component methods that directly treat spin-orbit interaction which may become too large to be amenable to a perturbation treatment. One may also benefit from specially adapted xc functionals that reflect the relativistic kinematics of the electrons and the Breit contribution to the electron-electron interaction [57]. Despite of the very minor effect of these relativistic corrections on many molecular observables [66], notable changes of the very sensitive g tensor are anticipated. 

The derivation of the ZORA approach is valid only for the one-electron Dirac equation with an external potential. Thus, the theory must be extended in order to obtain the relativistic many-electron ZORA Hamiltonian with the electron-electron Coulomb or Breit interaction. The many-electron ZORA Hamiltonian may be defined in several ways. In the present study, we neglect the relativistic kinematics correction to the electron-electron interaction, which yields the simplest many-electron ZORA Hamiltonian, that is, the one-electron ZORA Hamiltonian with the electron-electron Coulomb operator in the non-relativistic form,... 

Obviously, the relativistic momentum pi is also a Lorentz 4-vector, where we have introduced the relativistic (kinematic) 3-momentum... 

If ipn is the approximate wave function, then in (12.2.1) the factor li/>n(0)P can be corrected into (0) 2(1 — yo s/7r). The correction factor, in this approach, is uncomfortably large. The relativistic kinematic terms, which are ignored in (12.2.1), are likely to be relevant only for decay into T+r pairs. 

In the traditional approaches to nuclear physics, nuclear structure and nuclear reactions at low energies are studied using nonrelativistic many-body theory. The state of the nucleons is described by a nonrelativistic Hamiltonian, and the interactions are taken to be static potentials which are assumed to arise from meson exchange. The average kinetic energy of a bound nucleon and the nuclear binding potential (in the two-component Pauli representation) are both much less than the nucleon mass, so that, in the traditional approach, neither relativistic kinematics nor dynamics should be needed in descriptions of low energy nuclear phenomena. 

At medium energies, where the projectile speed approaches that of light, relativistic kinematics are introduced into the Schrodinger equation in order to produce the correct de Broglie wave length and relativistic density of states. According to the traditional view, however, the use of relativistic... 

It is useful to clarify the various ways in which relativity enters the nucleon-nucleus scattering problem. With respect to kinematics, the nucleon-nucleus center-of-momentum (COM) system wave number and reduced mass, which appear in the Schrodinger equation, are replaced by corresponding relativistic quantities. Also the transformation of the NN scattering amplitude in the NN COM system (where it is known) to the nucleon-nucleus COM frame (where it is needed for the calculation) is done using a proper Lorentz boost [Me 83a]. Both of these procedures for accounting for relativistic kinematics are included in the nonrelativistic scattering calculations done here and shown in this work. 

For a local, spherically symmetric optical potential describing spin- projectiles (protons) scattering from even-even target nuclei, we therefore solve the radial Schrodinger equation with relativistic kinematics. 

The nucleon-nucleon t-matrices used to calculate the optical potential must also use consistent relativistic kinematics, to be discussed in detail in section 3.3. The whole procedure for introducing relativistic kinematics into the nonrelativistic multiple scattering theory will be illustrated and checked using Coulomb scattering as an example in section 3.4. 

A different relativistic kinematic substitution is used by Arellano et al. [Ar 90a, Ar 90b] and Picklesimer et al. [Pi 84]. They replace the quantity (E - c q- ) with the total relativistic kinetic energy in the projectile-nucleus COM system that is,... 

In fig. 10 the separate effects of Wigner rotation and the NN COM energy shift are shown. These results, from ref. [Me 83a], are from nonrelativistic impulse approximation on-shell tp model calculations. Three calculations are considered. The first uses only the M0ller factor the second additionally includes the Wigner rotation matrix the third uses the full, relativistic kinematic transformation by additionally including the shift in the NN COM system energy as a function of momentum... 

The proton-nucleus Coulomb scattering amplitude is obtained by solving the Schrodinger equation with relativistic kinematics (eq. 3.30) with the Coulomb potential from eq. (3.59). The p 4 Coulomb amplitude, to order a, is given by... 

To summarize, the relativistic kinematic prescription is to replace COM system momenta by their relativistic values, to replace the reduced mass by the reduced total energy (eq. 3.29), and to use the M0ller factor, eq. (3.37). This prescription yields the correct, relativistic proton-nucleus Coulomb scattering amplitude to order a. The usual proton-nucleus Coulomb potential must also be multiplied by the relativistic correction factor rj (eq. 3.56). 

We have provided a pedagogical derivation of the traditional, nonrelativistic form of multiple scattering theory based on the optical potential formalism. We have also discussed in detail each of the important advances made over the past ten years in the numerical application of the NR formalism. These include the full-folding calculation of the first-order optical potential, off-shell NN t-matrix contributions, relativistic kinematics and Lorentz boost of the NN t-matrix, electromagnetic effects, medium corrections arising from Pauli blocking and binding potentials in intermediate states, nucleon... 

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