Off‐shell scattering by Coulomb‐like potentials

We derive closed expressions for and interrelationships between off‐shell and on‐shell scattering quantities for Coulomb plus short‐range potentials. In particular we introduce off‐shell Jost states and show how the transition matrices are obtained from these states. We discuss some formulas connecting the coordinate and momentum representatives of certain quantities. For the pure Coulomb case we derive analytic expressions for the Jost state and the off‐shell Jost state for l=0 in the momentum representation

The Coulomb unitarity relation and some series of products of three Legendre functions

We obtain from the off‐shell Coulomb unitarity relation a closed expression for J∞l = 0(2l+1)Pl(x) ×Qliγ ( y) Ql−iγ (z), and we consider some related series of products of Legendre functions

Polarized fluorescence measurements on ordered photosynthetic antenna complexes Chlorosomes of Chloroflexus aurantiacus and B800-B850 antenna complexes of Rhodobacter sphaeroides

We have used a new and relatively easy approach to study the pigment-organization in chlorosomes from the photosynthetic bacterium Chloroflexus aurantiacus and in B800–850 antenna complexes of the photosynthetic purple bacterium Rhodobacter sphaeroides. These particles were embedded in compressed and uncompressed gels and the polarized fluorescence was determined in a 90° setup. Assuming both a rotational symmetric distribution of the particles in the gel and of the transition dipole moments in the particles, the order parameters <P2> and <P4>, describing the orientation of the symmetry axis of the particles with respect to the direction of gel expansion can be determined. Moreover, the direction parameters, describing the orientation of the absorption and emission dipole moments with respect to the symmetry axis of the particles can be obtained.The value of <P2> is essential for quantitative interpretation of linear dichroism measurements and usually it is estimated from theoretical approaches, which may lead to incorrect results. For the rod-like chlorosomes the value of <P2> appears to be the same as predicted by the theoretical approach of Ganago, A. O., M. V. Fok, I. A. Abdourakhmanov, A. A. Solov'ev, and Yu. E. Erokhin (1980. Mol. Biol. [Mosc.]. 14:381–389). The agreement with linear dichroism results, analyzed with this theoretical approach shows that the transition dipole moments are indeed in good approximation distributed in a rotationally symmetric way around the long axis of the chlorosomes. Moreover, it appears those BChl c molecules, which fluoresce, are oriented in the same way with respect to the symmetry axis as the rest of these pigments, with the dipole moments close to parallel to the long axis.The B800–850 complexes appear to orient like discs, whereas the transition dipoles of the BChl a 800- and 850-nm bands are oriented almost perpendicular to the symmetry axis. These findings are in agreement with the minimal model for these complexes proposed by Kramer, H. J. M., R. van Grondelle, C. N. Hunter, W. H. J. Westerhuis, and J. Amesz (1984. Biochim. Biophys. Acta. 156–165).The amount of orientation of the particles appears to vary for different gels and it is lower than predicted by the theory of Ganago et al., showing that application of their approach for these particles leads to incorrect interpretations.The approach that is used in this study allows determination of orientations of those dipole moments, which transfer their excitation energy to the fluorescing species, in contrast to linear dichroism measurements, where the orientations of all absorbing dipole moments are studied. For the polarized fluorescence measurements, the amount of orientation of the particles is determined experimentally, whereas for linear dichroism this amount has to be estimated from theoretical models. The value of <P2> that can be determined from the fluorescence measurements can, however, also be used for a quantitative interpretation of the linear dichroism results

Effective field theory of 3He

3He and the triton are studied as three-body bound states in the effective field theory without pions. We study 3He using the set of integral equations developed by Kok et al. which includes the full off-shell T-matrix for the Coulomb interaction between the protons. To leading order, the theory contains: two-body contact interactions whose renormalized strengths are set by the NN scattering lengths, the Coulomb potential, and a three-body contact interaction. We solve the three coupled integral equations with a sharp momentum cutoff, Lambda, and find that a three-body interaction is required in 3He at leading order, as in the triton. It also exhibits the same limit-cycle behavior as a function of Lambda, showing that the Efimov effect remains in the presence of the Coulomb interaction. We also obtain the difference between the strengths of the three-body forces in 3He and the triton.Comment: 18 pages, 6 figures; further discussion and references adde

Triangle Diagram with Off-Shell Coulomb T-Matrix for (In-)Elastic Atomic and Nuclear Three-Body Processes

The driving terms in three-body theories of elastic and inelastic scattering of a charged particle off a bound state of two other charged particles contain the fully off-shell two-body Coulomb T-matrix describing the intermediate-state Coulomb scattering of the projectile with each of the charged target particles. Up to now the latter is usually replaced by the Coulomb potential, either when using the multiple-scattering approach or when solving three-body integral equations. General properties of the exact and the approximate on-shell driving terms are discussed, and the accuracy of this approximation is investigated numerically, both for atomic and nuclear processes including bound-state excitation, for energies below and above the corresponding three-body dissociation threshold, over the whole range of scattering angles.Comment: 22 pages, 11 figures, figures can be obtained upon request from the Authors, revte

Generalized Faddeev equations in the AGS form for deuteron stripping with explicit inclusion of target excitations and Coulomb interaction

Theoretical description of reactions in general, and the theory for $(d,p)$ reactions, in particular, needs to advance into the new century. Here deuteron stripping processes off a target nucleus consisting of ${A}$ nucleons are treated within the framework of the few-body integral equations theory. The generalized Faddeev equations in the AGS form, which take into account the target excitations, with realistic optical potentials provide the most advanced and complete description of the deuteron stripping. The main problem in practical application of such equations is the screening of the Coulomb potential, which works only for light nuclei. In this paper we present a new formulation of the Faddeev equations in the AGS form taking into account the target excitations with explicit inclusion of the Coulomb interaction. By projecting the $(A+2)$-body operators onto target states, matrix three-body integral equations are derived which allow for the incorporation of the excited states of the target nucleons. Using the explicit equations for the partial Coulomb scattering wave functions in the momentum space we present the AGS equations in the Coulomb distorted wave representation without screening procedure. We also use the explicit expression for the off-shell two-body Coulomb scattering $T$-matrix which is needed to calculate the effective potentials in the AGS equations. The integrals containing the off-shell Coulomb T-matrix are regularized to make the obtained equations suitable for calculations. For $NN$ and nucleon-target nuclear interactions we assume the separable potentials what significantly simplifies solution of the AGS equations.Comment: 34 pages, 13 figure

Scattering theory with the Coulomb potential

Basic features of a new surface-integral formulation of scattering theory are outlined. This formulation is valid for both short-range and Coulombic long-range interactions. New general definitions for the potential scattering amplitude are given. For the Coulombic potentials the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the amplitudes of breakup, direct and rearrangement scattering in a Coulomb three-body system are presented

Scattering theory for arbitrary potentials

The fundamental quantities of potential scattering theory are generalized to accommodate long-range interactions. New definitions for the scattering amplitude and wave operators valid for arbitrary interactions including potentials with a Coulomb tail are presented. It is shown that for the Coulomb potential the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure.Comment: To be published in Phys Rev

A renormalisation group approach to two-body scattering in the presence of long-range forces

We apply renormalisation-group methods to two-body scattering by a combination of known long-range and unknown short-range potentials. We impose a cut-off in the basis of distorted waves of the long-range potential and identify possible fixed points of the short-range potential as this cut-off is lowered to zero. The expansions around these fixed points define the power countings for the corresponding effective field theories. Expansions around nontrivial fixed points are shown to correspond to distorted-wave versions of the effective-range expansion. These methods are applied to scattering in the presence of Coulomb, Yukawa and repulsive inverse-square potentials.Comment: 22 pages (RevTeX), 4 figure

N-d scattering above the deuteron breakup threshold

The complex Kohn variational principle and the (correlated) Hyperspherical Harmonics technique are applied to study the N--d scattering above the deuteron breakup threshold. The configuration with three outgoing nucleons is explicitly taken into account by solving a set of differential equations with outgoing boundary conditions. A convenient procedure is used to obtain the correct boundary conditions at values of the hyperradius $\approx 100$ fm. The inclusion of the Coulomb potential is straightforward and does not give additional difficulties. Numerical results have been obtained for a simple s-wave central potential. They are in nice agreement with the benchmarks produced by different groups using the Faddeev technique. Comparisons are also done with experimental elastic N--d cross section at several energies.Comment: LaTeX, 13 pages, 3 figure