Comparison ofS-matrix and WKB methods for half-width calculations

Abstract

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...

Quasi-bound states, resonance tunnelling, and tunnelling times generated by twin symmetric barriers

“…The state |φ n has complex energy eigenvalue E n = E nr − iΓ nr /2 and obeys the boundary conditions of eqs (13) and (14). Substituting eq.…”

“…Evaluations of such states are very important in areas like nuclear scattering and decay of unstable nuclei via alpha decay, proton emission etc. as evident from refs [14][15][16][17][18][19]. A detailed procedure exists for the study of such states in 3D scattering theory.…”

“…The non-relativistic quantal theory of 3D potential scattering provides a unified analysis of bound states below the threshold energy E = 0 and continuum above the threshold [9][10][11][12][13][14][15]. The bound states are fully normalizable negative energy states; however, above the threshold the regular radial wave functions for positive energy behave like sin(kr − lπ/2 + δ l ) for large r, where δ l is the partial wave phase shift.…”

“…Hence it is appropriate to use the term quasi-bound (QB) states for these sharp resonances. Analytic S-matrix (SM) theory of potential scattering [9][10][11][12][13][14][15] provides a unified picture of bound states and resonances in terms of the pole structure of partial wave SM. Evaluations of such states are very important in areas like nuclear scattering and decay of unstable nuclei via alpha decay, proton emission etc.…”

From nuclear reactions to compact stars: A unified approach

“…The half life of a parent nucleus decaying via proton emission is calculated using the WKB barrier penetration probability. The WKB method is found quite satisfactory and even better than the Smatrix method for calculating half widths of the α decay of superheavy elements [64]. For the present calculations, the zero point vibration energies used here are given by Eq.…”