Pseudoscalar Potential

A theory of Lin-Liu et al. [62], [63] and Zakhlevnykh [64], which requires at least two independent pseudoscalar potential parameters for one and the same molecule, predicts a helix inversion for a rigid molecule when each parameter introduces a contribution of different sign and with a different temperature dependence. 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

The Nijmegen potential [13], published in 1978, is a nonrelativistic r-space OBEP. As a late representative of this model, it is one of the most sophisticated examples of its kind. It includes all nonstrange mesons of the pseudoscalar, vector, and scalar nonet. Thus, besides the six mesons... 

In the optical activity arising from higher-order cross-terms, the effects are in most cases expected to be orientation-dependent. Pseudoscalar terms are the only ones which survive in random orientation (molecules in solution or liquid phase). At the same order of perturbation as El-Ml there is a product of the electric dipole and electric quadrupole transition operators (E1-E2). Since the latter product involves tensors of unequal rank, the result cannot be a pseudoscalar and this term would not, therefore, contribute in random orientation but can be significant for oriented systems with quadrupole-allowed transitions. The E1-E2 mechanism was developed by Buckingham and Dunn and recognized by Barron" as a potential contribution to the visible CD in oriented crystals containing the [Co(en)3] " ion. 

According to experiments, the n mesons exist in three charge states Jt, Jt , n. From a pseudoscalar theory, taking into account all three charge states in a symmetric way, the following expression was obtained (see, e.g., Bohr and Mottelson 1969) for the asymptotic behavior of the so-called one-pion exchange nucleon-nucleon potential Vopep)-... 

The potential shows similarity to the interaction potential of two remote, pointlike magnetic dipoles, but the electromagnetic field has vectorial character, while the character of the meson field is pseudoscalar (the spin of tu meson is zero). 

The parity nonconserving aspect of the potential is evident from the pseudoscalar combinations of the vectors in Eq. (11). This interaction will mix electronic states of opposite parity. As pointed out earlier, only mixing between Si/2 and pi/2 states need be considered usually, because other states make a negligible contribution at the origin. 

Fig. 28. Real optical potential strengths in infinite nuclear matter of the scalar and vector potentials for medium energy protons based on the RIA model with pseudoscalar invariant (solid curves) and the one-meson exchange model using the pseudovector invariant with explicit direct and exchange terms (dashed curves). Results of Dirac phenomenology [Ha 90] for p + °Ca elastic scattering (potentials evaluated at r = 0) are shown by the dotted curves.

For incident beam energies of a few hundred MeV the scalar and vector potentials of the IA2 are typically one-half those of the RIA. TTiis is shown in fig. 30 (from ref. [Ot88]), where the real and imaginary parts of the IA2 and RIA (labelled lAl) scalar and vector potentials for p -I- " Ca at 200 MeV are compared. We have already discussed the use of the pseudoscalar invariant in the RIA and how this choice caused the strengths of the scalar and vector opticd potentials to diverge at low energy. The bulk... 

The invariants with l + m + k odd are pseudoscalars and therefore the corresponding coupling constants / " (ry) are pseudoscalars as well. These terms can appear only in the interaction potential between chiral molecules. The first nonpolar chiral term of the general expansion (Eq. 30) reads ... 

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