Bessel function expansions of Coulomb wave functions

From the convergence properties of the expansion of the function Φ_l∝r^(−l−1)F_l in powers of the energy, we successively obtain the expansions of F_l and G_l as single series of modified Bessel functions I_(2l+1+n) and K_(2l+1+n), respectively, as well as corresponding asymptotic approximations of G_l for ‖η‖→∞. Both repulsive and attractive fields are considered for real and complex energies as well. The expansion of F_l is not new, but its convergence is given a simpler and corrected proof. The simplest form of the asymptotic approximations obtained for G_l, in the case of a repulsive field and for low positive energies, is compared to an expansion obtained by Abramowitz

R-matrix and K-matrix analysis of elastic alpha-alpha scattering

The R- and K-matrix parametrizations are analyzed and compared for the elastic alpha-alpha scattering at center-of-mass energies below 40 MeV. The two parametrizations differ in their definitions of the resonance energy which can lead to quite different results. The physical values of the best-fit parameters are compared with those computed for a potential model. The existence of a broad resonance near 9 MeV is not supported by the data or by the potential model. We discuss the positive and negative aspects for both parametrizations.Comment: 14 pages with 4 figure

Nontrivial eigenvalues of the Liouvillian of an open quantum system

We present methods of finding complex eigenvalues of the Liouvillian of an open quantum system. The goal is to find eigenvalues that cannot be predicted from the eigenvalues of the corresponding Hamiltonian. Our model is a T-type quantum dot with an infinitely long lead. We suggest the existence of the non-trivial eigenvalues of the Liouvillian in two ways: one way is to show that the original problem reduces to the problem of a two-particle Hamiltonian with a two-body interaction and the other way is to show that diagram expansion of the Green's function has correlation between the bra state and the ket state. We also introduce the integral equations equivalent to the original eigenvalue problem.Comment: 5 pages, 2 figures, proceeding

Analysis of the total 12C(α,γ)16O cross section based on available angular distributions and other primary data

Because a knowledge of the 12C/16O ratio is crucial to the understanding of the later evolution of massive stars, new R- and K-matrix fits have been completed using the available angular distribution data from radiative α capture and elastic α scattering on 12C. Estimates of the total 12C(α,γ)16O rate at stellar energies are reported. In contrast with previous work, the analyses generally involve R- and K-matrix fits directly to the primary data, i.e., the energy- and angle-dependent differential yields, with all relevant partial waves fitted simultaneously (referred to here as surface fits). It is shown that, while the E1 part of the reaction is well constrained by a recent experiment on the β-delayed α-particle decay of 16N, only upper limits can be placed on the E2 ground state cross section factor which we take conservatively as SE2(300)<140 keV b. Simulations were then carried out to explore what kind of new data could lead to better restrictions on SE2(300). We find that improved elastic scattering data may be the best short-term candidate for such restrictions while significantly improving S(300) with new radiative capture data may require a longer-term effort. Theoretical models and estimates from α-transfer reactions for the E2 part of 12C(α,γ)16O are then discussed for comparison with the R- and K-matrix fits of the present work

K-matrix analysis of the 12C(α,γ) reaction at low energy

The most recent data for the 12C(α,γ)16O reaction are parametrized in terms of a K matrix in order to derive the astrophysical S(E) factor at stellar energies. This straightforward parametrization introduces neither boundary condition constants nor channel radii. To constrain the free parameters, all the available data for the phase shifts δl(l=1,2) of 12C(α,α)12C were simultaneously fitted with those for the E1 and E2 radiative captures. For each of the three sets of capture data we have analyzed, χ2 tests have been performed with different types of energy-dependent background terms, namely, polynomials and nonresonant echo poles (in the sense of McVoy). The introduction of such poles is motivated by the falling of the δl phase shift at higher energies. On the basis of the present analysis, it is concluded that, from the data available, one can derive an allowed range for S(0.3), from 0.00 to 0.17 MeV b. No confidence can be given to a so-called best value of S(0.3) within this range because such a value is dependent on both the data set analyzed and the type of background terms introduced into the parametrized K matrix

Level matrix, 16N β decay, and the 12C(α,γ)16O reaction

The level matrix corresponding to the scrK-matrix parametrization of a resonant nuclear reaction is derived and applied to the spectrum of α particles emitted following 16N β decay. The parametrized spectrum is fitted to data simultaneously with the E1 capture cross section of the 12C(α,γ)16O reaction and the p-wave phase shift of 12C(α,α)12C. Our analysis shows that new measurements of the α spectrum from 16N β decay could be used to significantly reduce the uncertainty of the 12C(α,γ)16O astrophysical S factor at 0.3 MeV. Various constraints on the parameters are analyzed and suggestions are made for further reducing the uncertainty in this crucial reaction rate

Completeness of the Coulomb scattering wave functions

Completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is the basic ingredient of quantum mechanics, plays an important role in nuclear reaction and nuclear structure theory. However, until now, there was no a formal proof of the completeness of the eigenfunctions of the two-body Hamiltonian with the Coulomb interaction. Here we present the first formal proof of the completeness of the two-body Coulomb scattering wave functions for repulsive unscreened Coulomb potential. To prove the completeness we use the Newton's method [R. Newton, J. Math Phys., 1, 319 (1960)]. The proof allows us to claim that the eigenfunctions of the two-body Hamiltonian with the potential given by the sum of the repulsive Coulomb plus short-range (nuclear) potentials also form a complete set. It also allows one to extend the Berggren's approach of modification of the complete set of the eigenfunctions by including the resonances for charged particles. We also demonstrate that the resonant Gamow functions with the Coulomb tail can be regularized using Zel'dovich's regularization method.Comment: 12 pages and 1 figur

WHO Clinical Staging of HIV Infection and Disease, Tuberculosis and Eligibility for Antiretroviral Treatment: Relationship to CD4 Lymphocyte Counts.

SETTING: Thyolo district, Malawi. OBJECTIVES: To determine in HIV-positive individuals aged over 13 years CD4 lymphocyte counts in patients classified as WHO Clinical Stage III and IV and patients with active and previous tuberculosis (TB). DESIGN: Cross-sectional study. METHODS: CD4 lymphocyte counts were determined in all consecutive HIV-positive individuals presenting to the antiretroviral clinic in WHO Stage III and IV. RESULTS: A CD4 lymphocyte count of < or = 350 cells/microl was found in 413 (90%) of 457 individuals in WHO Stage III and IV, 96% of 77 individuals with active TB, 92% of 65 individuals with a history of pulmonary TB (PTB) in the last year, 91% of 89 individuals with a previous history of PTB beyond 1 year, 81% of 32 individuals with a previous history of extra-pulmonary TB, 93% of 107 individuals with active or past TB with another HIV-related disease and 89% of 158 individuals with active or past TB without another HIV-related disease. CONCLUSIONS: In our setting, nine of 10 HIV-positive individuals presenting in WHO Stage III and IV and with active or previous TB have CD4 counts of < or = 350 cells/microl. It would thus be reasonable, in this or similar settings where CD4 counts are unavailable for clinical management, for all such patients to be considered eligible for antiretroviral therapy