The calculated Q values and half widths of α-decay of superheavy elements using both the S-matrix and the WKB methods are analyzed. The calculations are carried out using the microscopically derived α-daughter potentials for the parents appearing in the α-decay chain of super heavy element (A = 277, Z = 112). Both the S-matrix and the WKB methods though yield comparable results for smaller, in fact negative log τ 1/2 values, the former is superior. However, for the case of positive log τ 1/2 it is found that the S-matrix method though exact, runs into some numerical difficulties. With the discovery of many superheavy elements beyond Z = 100 and their decay processes involving, among others, α-decay chains have revived interest in the careful analysis of Q value of α-decay and the corresponding decay constant. The α-decay of super heavy nuclei has been intensively investigated in recent years [1][2][3][4][5][6][7][8][9][10][11]. If α-decay is understood as a two body phenomena involving the daughter and parent nucleus, a proper approach requires a reliable input of α-daughter nucleus potential. These potentials are introduced either phenomenological [e.g., Woods-Saxon (WS) shape with adjustable parameters] or are generated microscopically in the tρρ approximation (double folding model) using explicitly the calculated nuclear (both proton and neutron) densities. Once such a potential is given the usual procedure of calculating Q value and decay constant is to use WKB type approximations to obtain the energies of long-lived states of the effective potential. In the α-decay problem, the effective potential is the sum of the nuclear potential, electrostatic potential and the centrifugal term. This potential generates a huge pocket in between the Coulomb barrier and the centrifugal barrier and in principle can generate bound states with E < 0 and resonant states with E > 0 with finite lifetime. For a typical α-decay system of our interest, the Coulomb barrier height is about 25 MeV, whereas Q values are in the range 5-10 MeV. Because of the long Coulomb tail, at energies near Q value, the barrier width is quite large of the order of 7-8 fm. As a result the resonance which pertains to α-decay will have in most cases, extremely narrow width. Hence resonance energies can be calculated using WKB method applicable for bound states. Thus in such cases, the positive energy resonant states and bound state energies can be expected to satisfy the equationwhere k 2 and V eff (r) are energy and effective potential respectively inh 2 = 2m = 1 units; they have dimension L −2 . The effective potential V eff includes the centrifugal term (l + 1/2 ) 2 /r 2 , as required in the WKB formula. Here r 1 and r 2 denote the two inner turning points.When this formula is used, in general, one gets a number of positive eigenvalues; however, for the study of α-decay, the eigenvalue which corresponds to the Q value of the α-decay is to be identified. Once this eigenvalue is identified, the decay constant can be calculated using the WKB formula in...

In analogy with the definition of resonant or quasi-bound states used in three-dimensional quantal scattering, we define the quasi-bound states that occur in onedimensional transmission generated by twin symmetric potential barriers and evaluate their energies and widths using two typical examples: (i) twin rectangular barrier and (ii) twin Gaussian-type barrier. The energies at which reflectionless transmission occurs correspond to these states and the widths of the transmission peaks are also the same as those of quasi-bound states. We compare the behaviour of the magnitude of wave functions of quasi-bound states with those for bound states and with the above-barrier state wave function. We deduce a Breit-Wigner-type resonance formula which neatly describes the variation of transmission coefficient as a function of energy at below-barrier energies. Similar formula with additional empirical term explains approximately the peaks of transmission coefficients at above-barrier energies as well. Further, we study the variation of tunnelling time as a function of energy and compare the same with transmission, reflection time and Breit-Wigner delay time around a quasi-bound state energy. We also find that tunnelling time is of the same order of magnitude as lifetime of the quasi-bound state, but somewhat larger.

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 τ 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 τ 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.

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 features of T are likely to hold for most attractive potentials and flat repulsive barriers. We also discuss the attractive modified Poschl–Teller type potential for which T does not show oscillations as a function of energy.

We examine the behavior of transmission coefficient T across the rectangular barrier when attractive potential well is present on one or both sides and also the same is studied for a smoother barrier with smooth adjacent wells having Woods-Saxon shape. We find that presence of well with suitable width and depth can substantially alter T at energies below the barrier height leading to resonant-like structures. In a sense, this work is complementary to the resonant tunneling of particles across two rectangular barriers, which is being studied in detail in recent years with possible applications in mind. We interpret our results as due to resonant-like positive energy states generated by the adjacent wells. We describe in detail the possible potential application of these results in electronic devices using n-type oxygen-doped gallium arsenide and silicon dioxide. It is envisaged that these results will have applications in the design of tunneling devices.