The R-matrix theory

The different facets of the $R$-matrix method are presented pedagogically in a general framework. Two variants have been developed over the years: $(i)$ The "calculable" $R$-matrix method is a calculational tool to derive scattering properties from the Schr\"odinger equation in a large variety of physical problems. It was developed rather independently in atomic and nuclear physics with too little mutual influence. $(ii)$ The "phenomenological" $R$-matrix method is a technique to parametrize various types of cross sections. It was mainly (or uniquely) used in nuclear physics. Both directions are explained by starting from the simple problem of scattering by a potential. They are illustrated by simple examples in nuclear and atomic physics. In addition to elastic scattering, the $R$-matrix formalism is applied to transfer and radiative-capture reactions. We also present more recent and more ambitious applications of the theory in nuclear physics.Comment: 93 pages, 26 figures. Rep. Prog. Phys., in pres

Equivalence of the Siegert-pseudostate and Lagrange-mesh R-matrix methods

Siegert pseudostates are purely outgoing states at some fixed point expanded over a finite basis. With discretized variables, they provide an accurate description of scattering in the s wave for short-range potentials with few basis states. The R-matrix method combined with a Lagrange basis, i.e. functions which vanish at all points of a mesh but one, leads to simple mesh-like equations which also allow an accurate description of scattering. These methods are shown to be exactly equivalent for any basis size, with or without discretization. The comparison of their assumptions shows how to accurately derive poles of the scattering matrix in the R-matrix formalism and suggests how to extend the Siegert-pseudostate method to higher partial waves. The different concepts are illustrated with the Bargmann potential and with the centrifugal potential. A simplification of the R-matrix treatment can usefully be extended to the Siegert-pseudostate method.Comment: 19 pages, 1 figur

Comparison of potential models of nucleus-nucleus bremsstrahlung

At low photon energies, the potential models of nucleus-nucleus bremsstrahlung are based on electric transition multipole operators, which are derived either only from the nuclear current or only from the charge density by making the long-wavelength approximation and using the Siegert theorem. In the latter case, the bremsstrahlung matrix elements are divergent and some regularization techniques are used to obtain finite values for the bremsstrahlung cross sections. From an extension of the Siegert theorem, which is not based on the long-wavelength approximation, a new potential model of nucleus-nucleus bremsstrahlung is developed. Only convergent integrals are included in this approach. Formal links between bremsstrahlung cross sections obtained in these different models are made. Furthermore, three different ways to calculate the regularized matrix elements are discussed and criticized. Some prescriptions for a proper implementation of the regularization are deduced. A numerical comparison between the different models is done by applying them to the $\alpha+\alpha$ bremsstrahlung.Comment: submitted to Phys. Rev.

Time-dependent analysis of the nuclear and Coulomb dissociation of 11Be

The breakup of 11Be on carbon and lead targets around 70 MeV/nucleon is investigated within a semiclassical framework. The role of the 5/2+ resonance is analyzed in both cases. It induces a narrow peak in the nuclear-induced breakup cross section, while its effect on Coulomb breakup is small. The nuclear interactions between the projectile and the target is responsible for the transition toward this resonant state. The influence of the parametrization of the 10Be-n potential that simulates 11Be is also addressed. The breakup calculation is found to be dependent on the potential choice. This leads us to question the reliability of this technique to extract spectroscopic factors.Comment: 9 pages, 6 figures, to be published in the Proceedings of the Second Argonne/MSU/JINA/INT RIA Workshop on Reaction Mechanisms for rare Isotope Beams (2005

Are Prices ‘Sticky’ Online? Market Structure Effects and Asymmetric Responses to Cost Shocks in Online Mortgage Markets

We analyze daily mortgage rates posted by online lenders at the price comparison site, Microsurf. While cost shocks occurred almost daily in our sample, quoted mortgage rates are surprisingly rigid: Only 16 percent of the posted rates represent changes. However, firms that adjusted rates in response to cost shocks did so quite rapidly; about 98 percent of a cost shock was passed through within two days of the cost shock. Duration analysis reveals that the observed rigidity in rates systematically depends on market structure: Online mortgage rates are 30 to 40 percent more durable in concentrated markets than in markets where there are many competitors. We also find that rates posted online tend to exhibit downward stickiness; rate adjustments in response to cost increases are about twice the corresponding adjustments for cost decreases.mortgage rate, price adjustment, price rigidity, price dispersion

He23+_2^{3+} and HeH2+^{2+} molecular ions in a strong magnetic field: the Lagrange mesh approach

Accurate calculations for the ground state of the molecular ions He$_2^{3+}$ and HeH$^{2+}$ placed in a strong magnetic field $B\gtrsim 10^{2}$ a.u. ($\approx 2.35 \times 10^{11}$G) using the Lagrange-mesh method are presented. The Born-Oppenheimer approximation of zero order (infinitely massive centers) and the parallel configuration (molecular axis parallel to the magnetic field) are considered. Total energies are found with 9-10 s.d. The obtained results show that the molecular ions He$_2^{3+}$ and HeH$^{2+}$ exist at $B > 100$\,a.u. and $B > 1000$\,a.u., respectively, as predicted in \cite{Tu:2007} while a saddle point in the potential curve appears for the first time at $B \sim 80$ a.u. and $B \sim 740$ a.u., respectively.Comment: 8 pages, 1 figure, 2 tables. arXiv admin note: text overlap with arXiv:0912.104

Analysis of the 6^6He β\beta decay into the α+d\alpha+d continuum within a three-body model

The beta-decay process of the $^6$He halo nucleus into the alpha+d continuum is studied in a three-body model. The $^6$He nucleus is described as an alpha+n+n system in hyperspherical coordinates on a Lagrange mesh. The convergence of the Gamow-Teller matrix element requires the knowledge of wave functions up to about 30 fm and of hypermomentum components up to K=24. The shape and absolute values of the transition probability per time and energy units of a recent experiment can be reproduced very well with an appropriate alpha+d potential. A total transition probability of 1.6E-6 s$^{-1}$ is obtained in agreement with that experiment. Halo effects are shown to be very important because of a strong cancellation between the internal and halo components of the matrix element, as observed in previous studies. The forbidden bound state in the alpha+d potential is found essential to reproduce the order of magnitude of the data. Comments are made on R-matrix fits.Comment: 18 pages, 9 figures. Accepted for publication in Phys.Rev.