Cluster sum rules for three-body systems with angular-momentum dependent interactions

Abstract

We derive general expressions for non-energy-weighted and energy-weighted cluster sum rules for systems of three charged particles. The interferences between pairs of particles are found to play a substantial role. The energy-weighted sum rule is usually determined by the kinetic energy operator, but we demonstrate that it has similar additional contributions from the angular momentum and parity dependence of two-and three-body potentials frequently used in three-body calculations. I. MOTIVATIONThe use of sum rules in quantum mechanics is well established and abundantly applied for many different systems [1,2]. The prominent examples are the transitions from a given quantum state induced by an electromagnetic multipole operator. For any multipole operator acting on an initial state, the sum of all the related transition probabilities multiplied by powers of the excitation energy are completely determined by the properties of the initial state [3,4].The sum rules exist in general for any many-body quantum system. Of specific interest are those systems where the constituents clusterize, such that the degrees of freedom can be divided into the internal ones corresponding to each cluster and those associated with the relative motion of the clusters [5]. Then the different multipole operators can be decomposed into terms depending on the intrinsic coordinates of each cluster and an additional term depending only on the relative coordinates of the centers of mass of the clusters. This operator structure then leaves two sum rules showing the same decomposition: the sum rules associated with each individual cluster (depending only on the properties of the initial cluster state) plus the cluster sum rule (depending on the properties of the few-body initial wave function). Examples are found in Refs. [6,7], where the dipole non-energy-weighted and dipole energy-weighted sum rules are obtained for many-body systems clusterizing into a two-body system. When a clusterized system can be properly described as a few-body system where the internal cluster degrees of freedom are frozen, only the cluster sum rules remain, corresponding to the much smaller Hilbert space of ground and excited states of the relative cluster motion. This kind of few-body descriptions have been extensively used in nuclear physics during the past 10-15 years in connection with halos and weakly bound states in general [5]. The most interesting and frequently investigated of these systems are approximated by a three-body structure. Extensions to excited three-body continuum states are now being pursued and attracting a lot of attention [8][9][10][11][12]. To get accurate three-body wave functions the Faddeev decomposition with different Jacobi coordinates is employed in coordinate space computations [13]. The unavoidable transformation from one set of Jacobi coordinates to another complicates the structure of the cluster sum rules, especially when more than one of the three particles is charged.The purpose of this work is to generalize the dipole twobody...

Tests of the discretized-continuum method in three-body dipole strengths

“…As expected, the energies of the pseudostates, and thus the E1 distributions, are sensitive to the choice of the basis. However all distributions present a maximum near E ≈ 1 MeV, where a broad 1 − three-body resonance is predicted by various models [2,19,12,23,4].…”

“…[24] Lag [24] where a maximum is hardly visible. The consistency between the pseudostate and continuum methods can be tested by the sum rules [25,23]. The non-energy-weighted (S 0 ) and energy-weighted (S 1 ) sum rules are defined as…”

Photodisintegration ofBe9through the1/2+state and cluster dipole resonance

“…Thus, the resonance strength is 11.3-14.2 mb MeV. Let us compare the resonance strength with the cluster dipole sum rule [50][51][52] for the two-body (n-8 Be) and threebody (n-α-α) systems:…”

Astrophysical reaction rate forBe9formation within a three-body approach

“…(9) can be easily expressed, in general, using different Jacobi systems. The relation between harmonic polynomials in different Jacobi sets is given by the expression [32] …”