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Proton Decay from Excited States in Spherical Nuclei

Şerban Mişicu, N. Cârjan, Patrick Talou

Published Jan 1, 1998• 5 citations
OpenAlex

Abstract

Based on a single particle model which describes the time evolution of the wave function during tunneling across a one dimensional potential barrier we study the proton decay of $^{208}$Pb from excited states with non-vanishing angular momentum $\\ell$. Several quantities of interest in this process like the decay rate $\\lambda$, the period of oscillation $T_{osc}$, the transient time $t_{tr}$, the tunneling time $t_{tun}$ and the average value of the proton packet position $ r_{av} $ are computed and compared with the WKB results.

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Based on a single particle model which describes the time evolution of the wave function during tunneling across a one dimensional potential barrier we study the proton decay of 208^{208}208Pb from excited states with non-vanishing angular momentum ell\\ellell. Several quantities of interest in this process like the decay rate lambda\\lambdalambda, the period of oscillation ToscT_{osc}Tosc​, the transient time ttrt_{tr}ttr​, the tunneling time ttunt_{tun}ttun​ and the average value of the proton packet position ravr_{av}rav​ are computed and compared with the WKB results.
The most intuitive and direct way to study proton emission from unbound nuclear states is to start with a proton wave packet localized inside the well of the p-nucleus interacting potential and follow its time evolution by solving numerically the time-dependent Schrödinger equation (TDSE). Then the physical properties of the decaying system can be inferred from its solution. The most important calculable quantities are: the tunneling probability and the decay rate. In the present paper we consider bi-dimensional axially symmetric nuclear shapes, allowing the treatment of both spherical and deformed nuclei. The solution of TDSE is obtained by the Crank-Nicolson method. We discuss also the treatment of numerical boundary conditions, the preparation of the initial wave function as a quasi-stationary state and the calculation of the decay rate using the flux. In application we reveal the influence of the deformation on the decay rate.
The time evolution of single proton quasi-stationary states emitted from axially deformed nuclei is studied by solving numerically the two-dimensional time dependent Schrödinger equation. Angular distributions of emission with respect to the nuclear axis of symmetry are obtained. It is shown that the main escape paths are determined by the quantum numbers of the metastable state, rather than by the deformation of the potential. Combinations of such numerical studies of multidimensional quantum tunneling and angular distribution measurements from oriented deformed nuclei could provide invaluable nuclear structure information and imaging of Nilsson orbitals.
The movement of a proton which moves in a one-dimensional effective double-minimum potential V(x) and can pass by means of the tunnel effect from one well into the other was investigated. The lingering times of the proton in the potential wells (a) and (b) were calculated for more than 150 potential models as functions of the height Vmax of the potential barrier separating the two wells, the energy difference ΔV, and separation d of the minima. In the determination of the lingering times, it was assumed that the proton at time t = 0 is to be found with certainty in one of the wells and can be represented by means of a statistical operator related to the eigenstates of a one-well Hamiltonian H(a)orH(b), respectively. The times determined on this basis were compared with those which result with a quasiclassical method from the classical frequency of oscillation of a particle in a well and from the penetration coefficient of the potential barrier. The deviation between the values determined with both methods is largest if an eigenvalue of H(a) becomes approximately equal to an eigenvalue of H(b). For the potential model V(x) a polynomial of fourth degree in the position coordinate x was chosen. The eigenfunctions of H(a)andH(b) as well as those of the effective Hamiltonian H corresponding to V(x) were represented as linear combinations of eigenfunctions of the harmonic oscillator with angular frequency ω. ω was chosen so that the mean square deviations of the approximated eigenvalues of H became as small as possible for the states of lowest energy.
We examine a one-dimensional model of a decaying quantum system, e.g., one that could simulate decay. A particle is initially confined in a region and leaks out, tunnelling through a potential barrier. The same model has been examined before, primarily to study decay properties which are determined by the wavefunction of the particle within the potential barrier. However, the wavefunction of this model (and of any similar models) outside the potential barrier has not been obtained explicitly as a function of position and time. We obtain the wavefunction outside as well as inside the potential barrier by solving the time-dependent Schrödinger equation, and explore various features of the space-time development of the system. In an early study by Winter it was pointed out that the probability current just outside the barrier, after a long time, may fluctuate and can be negative, i.e., inward. We find that under certain circumstances the amplitude of the fluctuations increases significantly as the distance from the barrier increases. We propose a simple approximate wavefunction that works well when the decay is very slow. The Gamow wavefunction, commonly used to describe the -decay process, is not appropriate at large distances. Our wavefunction can be used anywhere and is normalized in the usual manner.
This study investigates three-dimensional quantum tunneling through analytical solutions of the Schr¨odinger equation in Cartesian and spherical coordinates. By develop ing transmission coefficients for both square and spherical potential barriers, we establish a theoretical framework for α-decay via the Gamow model. The analysis reveals how angular momentum and energy dependence govern tunneling probabilities, with experimental valida tion showing agreement within 15% for high-energy decays. The results bridge fundamental quantum mechanics with nuclear physics applications.