Spin-orbit potential

Figure 6.9 Radial dependence of the strength of the spin-orbit potential.

A magnetic potential may not be real. An example is the spin—orbit potential. This is due to the coupling of the magnetic moment of the electron to the solenoid magnetic field caused by the orbital motion of its... 

The spin—orbit matrix element The spin—orbit potential (3.174) is... 

This form for the spin—orbit potential acts only in diagonal direct potential matrix elements. It may be considered as an additional term in the potential (7.61)... 

In this case, the analytical fit parameters A, n, a refer to Vy — VL or Vf — VL, rather than Vy, Vf. Thirdly, one has to be careful about the spin-orbit potentials Vf° which often contain numerical factors (such as 2/(2/ + 1), cf. 

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

Below we use the RMF model which previously has been successfully applied for describing ground states of nuclei at and away from the )3-stability line. For nucleons, the scalar and vector potentials contribute with opposite signs in the central potential, while their sum enters in the spin-orbit potential. Due to G-parity, for antiprotons the vector potential changes sign and therefore both the scalar and the vector mesons generate attractive potentials. 

Realistic nuclear structure theory has traditionally been formulated in a framework which is tailored to the features of the simple mean-field shell model including its strong single-particle (sp) spin-orbit potential [1,2]. The introduction of a small model space which supposedly includes the essential configurations to describe the low-energy excitations, plays a central role in this traditional approach. An important early example of this method is provided by the seminal work of... 

With the core potentials and spin-orbit operators given in the forms of Eqs. (2.1) and (2.2), existing programs for nonrelativistic calculations can be adapted to include relativistic effects. In the literature, the coefficients of the spin-orbit potentials are not always defined in the same manner. In our implementation of the spin-orbit interaction in NWChem, the spin-orbit potential is defined as the A in (2.3) scaled by... 

Stevens and coworkers have published shape-consistent one-component quasirelativistic pseudopotentials, i.e. sLi-igAr (Stevens et al. 1984), igK-syLa and 72Hf-86Rn (Stevens et al. 1992). Cundari and Stevens (1993) presented the corresponding parameters for the lanthanides sgCe-jiLu. Spin-orbit potentials have not been published but may be derived (Stevens and Krauss 1982) since the reference data are taken from all-electron DHF calculations. 

M. Casarrubios, L. Seijo. The ab initio model potential method. Relativistic Wood-Boring valence spin-orbit potentials and spin-orbit-corrected basis sets from B(Z= 5) to Ba(Z=56). /. Mol. Struc. (THEOCHEM), 426 (1998) 59-74. 

Doing so yields the so-called Schrddinger equivalent potential (SEP). The central, spin independent part of this potential is determined by the sum of the scalar and vector Dirac potentials plus smaller, but important, quadratic terms. The spin-orbit potential depends mainly on the vector-scalar difference. Thus the relatively small spin independent nuclear potential arises from the near cancellation between the strong scalar and vector potentials, while the relatively strong nuclear spin-orbit force comes about from the constructive addition of the two parts of the Dirac potential. 

The Schrddinger equivalent of the RIA optical potential and the nonrelativistic impulse approximation optical potential used in the calculations leading to fig. 2 are compared in hg. 4. The real, central SEP (solid curve) is repulsive in the interior, being about 30 MeV more positive than the NRIA value (dashed curve). The real, central SEP is attractive in the tail region, where it is comparable to the NRIA potential. The imaginary central potential and both parts of the spin-orbit term of the SEP are stronger than the NRIA predictions. The stronger spin-orbit potentials and the so-called wine-bottle-bottom shape of the real, central SEP account for the improvement in the description of the p + Ca 500 MeV data provided by the RIA. 

From these figures we observe the following effects due to medium modifications (1) Re(fg) is increased overall, yielding a less attractive real, central optical potential. (2) The magnitude of Im(fo) is suppressed at low momentum transfer but is slightly enhanced for >3fm (3) Re(fo ) (which determines the imaginary part of the pA spin-orbit potential) is enhanced in magnitude at low q. (4)... 

Im(fo ) (which determines the real spin-orbit potential) is enhanced in overall magnitude. (5) The size of the effects decreases rather slowly with increasing incident beam energy. (6) The corrections depend nonlinearly on /cp. The continued importance of medium corrections at 800 MeV is somewhat unexpected. 

The correlation potential in eq. (3.97) corresponds to that in eqs. (42), (43) and (44) in ref. [Ra79] for the Pauli, SRD, and Pauli-SRD cross term, respectively. I ng range correlations due to center-of-mass constraints and intrinsic, permanent deformations have also been included as in eq. (50) in ref. [Ra79] and eqs. (7) and (8) in ref. [Ra83c], respectively. Following a similar derivation, the local form of the Pauli correlation contribution (largest part) to the spin-orbit potential can be obtained the explidt result is in eq. (56) in ref. [Ra79]. The correlation potentials in these references... 

Efforts to use relativistic dynamics to describe nuclear phenomena began in the 1950s with application to infinite nuclear matter. Johnson and Teller [Jo 55] developed a nonrelativistic field theory for interacting nucleons and neutral, scalar mesons which served as a catalyst for Duerr, who, in a landmark paper [Du 56], developed a relativistic invariant version of the Johnson and Teller model which included both scalar and vector meson fields. He showed that nuclear saturation and the strong spin-orbit potential of the shell model could be readily understood. He also predicted a single particle potential which qualitatively reproduced the real part of the central optical potential well depth and its energy dependence for incident kinetic energies up to 200 MeV. 

We now define spin-free and spin-orbit potentials,... 

Here we note that Uf appears as a weighted average of what may be regarded as the potentials for two spin-orbit split levels, and f/ " is related to the difference between these two potentials. The factor of 2 in the numerator of the spin-orbit potential is included for representation of the final expression in terms of the spin operator. Rearranging these expressions, we can write... 

To obtain a similar simplification for the spin-orbit potential, we note that... 

This enables us to write the various terms involving the spin-orbit potential as... 

By far the majority of relativistic pseudopotential calculations include only the spin-free pseudopotential at the SCF stage, and add the spin-orbit potential afterwards if they add it at all. There are a few where both the spin-free and spin-orbit effects are included at the SCF stage and the calculations performed with double-group symmetry, just like four-component calculations. 

There is a variety of approaches to the development of pseudopotential spin-orbit operators. The spin-orbit potential is mostly expressed in the semi-local form... 

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