Techniques to Treat the Continuum Applied to Electromagnetic Transitions in 8Be

Abstract

Bremsstrahlung emission in collisions between charged nuclei is equivalent to nuclear gamma decay between continuum states. The way the continuum spectrum can be treated is not unique, and efficiency and accuracy of cross section calculations depend on the chosen method. In this work we describe, relate, and compare three different methods in practical calculations of inelastic cross sections, that is, by (i) treating the initial and final states as pure continuum states on the real energy axis, (ii) discretizing the continuum states on the real energy axis with a box boundary condition, and (iii) complex rotation of the hamiltonian (complex scaling method). The electric quadrupole transitions, 2 + → 0 + and 4 + → 2 + , in α + α scattering are taken as an illustration.Keywords First keyword · Second keyword · More IntroductionThe emission of bremsstrahlung in a collision between two charged particles constitutes an important background effect in Coulomb deexcitation processes. In a classical picture, this phenomenon is understood as the energy radiated due to the deceleration of a charged particle when deflected by another charged particle. The radiated energy is just the kinetic energy lost in the deceleration process. In a quantum mechanical picture, the process can be seen as the γ-emission due to the decay of a two-body system from some two-body continuum state into another continuum state of lower energy. A detailed derivation of the cross section for this kind of processes can be found in [1].The fact that these transitions involve continuum structures, both in the initial and final states, introduce technical as well as conceptual difficulties in the calculations. First, the treatment of the continuum spectrum itself, which can be handled in different ways, but in numerical studies almost necessarily by some kind of discretization. Second, the unavoidable matrix elements between the continuum states involved in the calculation are diverging and thus not well defined without some regularization prescription. These two problems, how to treat the continuum and how to obtain converged results, are possible sources of uncertainty that deserves detailed investigations.Different methods were previously used to compute the bremsstrahlung cross sections for various combinations of nuclei. In [2] α − α collisions were investigated in connection with the resonant electric

Resonant and scattering states in the α+α system from the nonlocalized cluster model

“…Various aspects of the α + α system have been studied theoretically by many authors using the RGM [44,45], the GCM [46][47][48][49], the quantum Monte Carlo method [50][51][52], the THSR wave function [11,12], the cluster effective field theory [53,54], the complex-scaled cluster model [55][56][57][58], the lattice effective field theory [59], the configuration interaction technique [60,61], the δ-shell potential method [62], etc. Also, the experience on studying the α + α system would help extend our method further to the α + α + α system, which could contain more exotic structures such as gaslike α-condensates [6], linear-chain structures [18], etc.…”

Three-body bremsstrahlung and the rotational character of theC12spectrum

“…This procedure follows very closely the one described and tested in Ref. [21] for the two-body system of 8 Be. In this way we can test the validity of the generalization to three-body systems.…”

“…Therefore, the question arises about a possible rotational (or in general collective) character for a sequence of continuum states of considerable width. This problem was recently investigated in previous works [19][20][21]. This was done by computation of the electric quadrupole cross sections for bremsstrahlung emission after transitions between the 8 Be states, which were described as two-body systems made of two alpha particles.…”

Rotational character of the12Cspectrum investigated through inelastic cross sections via photon emission

“…[3], in the case of continuum to continuum transitions, a direct calculation of the transition strength from Eq. (3) is not possible.…”

“…All the states in the 8 Be spectrum are unbound (resonances), with the ground state only about 0.1 MeV above the threshold for emission of two alpha particles. This spectrum was recently investigated in [1][2][3] by computation of the electric quadrupole cross sections for bremsstrahlung emission after transitions between the 8 Be-states, which were described as two-body systems made of two alpha particles. These calculations required a careful treatment of the continuum wave functions and clarification of the definition of the cross section for transitions between states with a non well-defined energy.…”