C. S. Shastry

Resonant states and transmission coefficient oscillations for potential wells and barriers

The oscillatory behavior of the transmission coefficient T as a function of energy is examined for an attractive square well and a rectangular barrier. We calculate T using resonant state boundary conditions and demonstrate that the maxima in T are correlated with the broad resonances generated by these potentials. For barrier potentials the maxima signify resonances occurring at energies above the barrier height. It is shown that the resonance position and width can also be generated from the complex poles of the amplitude of the transmitted plane wave. We also explain the relation between the positions of the resonances generated by the square well and the rectangular barrier to the energy eigenvalues of the corresponding rigid box with the same range. We show for a potential with an attractive well and a repulsive barrier that T exhibits oscillations when the particle energy is below the barrier, implying that in many cases the simple WKB type barrier penetration expression for T is not adequate. These...

Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

AbstractThis paper proves that for N attractive delta function potentials the number of bound states (Nb) satisfies 1 ≤ Nb ≤ N in one dimension (1D), and is 0 ≤ Nb ≤ N in three dimensions (3D). Algebraic equations are obtained to evaluate the bound states generated by N attractive delta potentials. In particular, in the case of N attractive delta function potentials having same separation a between adjacent wells and having the same strength λV, the parameter g=λVa governs the number of bound states. For a given N in the range 1–7, both in 1D and 3D cases the numerical values of gn, where n=1,2,..N are obtained. When g=gn, Nb ≤ n where Nb includes one threshold energy bound state. Furthermore, gn are the roots of the Nth order polynomial equations with integer coefficients. Based on our numerical calculations up to N=40, even when N becomes large, 0 ≤ gn ≤ 4 and

STUDY OF ALPHA DECAY OF SUPER HEAVY ELEMENTS USING S-MATRIX AND WKB METHODS

\frac{{\Sigma g_n }} {N} \simeq 2

Threshold conditions and bound states for locally periodic delta potentials

and this result is expected to be generally valid. Thus, for g > 4 there will be no threshold or zero energy bound state, and if g≈ 2 for a given large N, the number of bound states will be approximately N/2. The empirical formula gn = 4/[1+exp((N0 − n)/β)] gives a good description of the variation of gn as a function of n. This formula is useful in estimating the number of bound states for any N and g both in 1D and 3D cases.

Analysis of 16O + 28Si elastic scattering in the laboratory energy range 50.0 MeV to 142.5 MeV

A comparative study of the S-matrix and the WKB methods for the calculation of the half widths of alpha decay of super heavy elements is presented. The extent of the reliability of the WKB methods is demonstrated through simple illustrative examples. Detailed calculations have been carried out using the microscopic alpha-daughter potentials generated in the framework of the double-folding model using densities obtained in the relativistic mean field theories. We consider alpha-nucleus systems appearing in the decay chains of super heavy parent elements having A = 277, Z = 112 and A = 269, Z = 110. For negative and small positive log T-1/2 values the results from both methods are similar even though the S-matrix results should be considered to be more accurate. However, when log inverted perpendicular(1/2) values are large and positive, the width associated with such state is infinitesimally small and hence calculation of such width by the S-matrix pole search method becomes a numerically difficult problem. We find that overall, the WKB method is reliable for the calculation of half lives of alpha decay from heavy nuclei.

Fusion, resonances and scattering in12C+12C reaction

We present a systematic study of the conditions for the generation of threshold energy eigen states and also the energy spectrum generated by two types of locally periodic delta potentials each having the same strength λV and separation distance parameter a: (a) sum of N attractive potentials and (b) sum of pairs of attractive and repulsive potentials. Using the dimensionless parameter g = λV a in case (a) the values of g = gn, n = 1, 2, …, N at which threshold energy bound state gets generated are shown to be the roots of Nth order polynomial D1(N, g) in g. We present an algebraic recursive procedure to evaluate the polynomial D1(N, g) for any given N. This method obviates the need for the tedious mathematical analysis described in our earlier work to generate D1(N, g). A similar study is presented for case (b). Using the properties of D1(N, g) we establish that in case (a) the critical minimum value of g which guarantees the generation of the maximum possible number of bound states is g = 4. The corresponding result for case (b) is g = 2. A typical set of numerical results showing the pattern of variation of gn as a function of n and several interesting features of the energy spectrum for different values of g and N are also described.

S-MATRIX-BASED UNIFIED CALCULATION OF Q-VALUES AND HALF-LIVES OF α-DECAY OF SUPER HEAVY ELEMENTS

We have successfully carried out an optical model analysis of 16O + 28Si differential scattering cross section σ(θ) in the laboratory energy range EL from 50.0 MeV to 142.5 MeV and angular range θ from 0° to 100° using the LC potential suggested by Lee and Chan. This potential is very successful in explaining the differential cross section σ(θ) at EL = 50.0 and 55.0 MeV in the entire centre of mass angle range reproducing the enhanced back angle oscillations correctly. The LC potential has only seven parameters, including the Coulomb radius parameter, which is much smaller than the number of parameters used in recent calculations on the system. In the light of an excellent fit to the data, we make a detailed analysis of different features of the LC potential and compare it with the typical heavy-ion potential having a Woods–Saxon (WS) form factor used in the analysis of 18O + 58Ni at EL = 60.0 MeV. These features include the behaviour of the real effective potential around grazing partial wave (lg), angular momentum (l) dependence of classical deflection function (Θ), reflection function |Sl| and the term |1 − Sl|2 governing the total partial wave elastic cross section. We also calculate the barrier region resonance positions generated by the partial waves around lg. We find that the orbiting phenomenon generated by the flatter, surface transparent effective potential and the resulting coherent behaviour of only a small number of partial waves in the surface region, resulting in the interference of absorptive region and Coulombic region amplitudes, are primarily responsible for back angle oscillations. As a quantum-mechanical consequence of orbiting behaviour, we find close clustering of resonances corresponding to a set of partial waves around lg which is the special feature of the LC potential. On the basis of this, we envisage that 16O + 28Si is likely to show large enhanced back angle oscillations at EL = 66.0 and 72.0 MeV. It will be interesting if this is experimentally investigated.

Interesting features of transmission across locally periodic delta potentials

The variation of fusion cross-section (σJfus) with energy in the12C+12C collision is linked to the underlying resonance phenomenon through the behavior of reaction cross-section (σ) of which σfus is taken as a part. The calculation of σfus is done through an energy-dependent imaginary potential in the optical model potential (OMP). Through dispersion relation, such an imaginary potential gives rise to energy-dependent real potential which is incorporated in the OMP. In our calculation, a form of potential for the nuclear part which has a soft repulsive in-built core is introduced based on similar works done earlier. The calculated results of σfus are used to explain the oscillatory structure, astrophysical S-factor and the decreasing trend at higher energies of the experimental σfus data in the case of12C+12C system with remarkable success. The potential used for fusion calculation is tested for fitting elastic scattering data at some energies and is found good in forward angles. Further improvement of the fitting of these data is obtained by incorporating a coupling potential in the surface region. About twenty resonances are observed in our calculation in the specific partial waves and some of them are found close to the experimentally identified resonances in12C+12C reaction. Thus, we provide an integrated and comprehensive analysis of fusion, resonance and scattering data in the best studied case of12C+12C reaction within the framework of optical potential model.

Threshold conditions and bands in attractive Dirac comb

The Q-values and half-lives of several heavy α decaying systems are calculated using the relativistic mean field (RMF) theory-based microscopic α-daughter nucleus potential. A unified procedure is adopted, using analytic S-matrix method and treating α-decay as the decay of the resonance state of the α-daughter nucleus system. The resonance parameters are obtained from the pole positions of the S-matrix in the complex k-plane and using these Q-values and half widths are evaluated. The calculation reproduces the experimental results well. We find that the present unified approach gives a good description of the data and compare well with those obtained by empirical formulae.

Transmission and scattering by an absorptive potential

We study the theory of transmission of electrons through N delta potential barriers as well as wells. Some of the interesting features like the correlation between resonance peak positions and box states, number of peaks in transmission band and bound states are analyzed for locally periodic attractive, repulsive and pair of attractive and repulsive potentials.