ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 4, с. 533-537
= ЯДРА
CONTINUUM RESONANCES WITH SHIELDED COULOMB-LIKE POTENTIAL AND EFIMOV EFFECT
©2014 D. S. Tusnski1)*, M. T. Yamashita1)**, T. Frederico2), L. Tomio1)>3)
Received 29 January, 2013
Motivated by the possibility of the second energy level (0+) of 12C (in a three-alpha model) to turn into an Efimov state, we study a simple non-realistic toy model formed by three bosons interacting by the phenomenological s-wave Ali—Bodmer potential plus a Coulomb interaction. An artificial three-body potential was used to create a resonance with energy close to the energy of the 0+ of12C, 0.38 MeV. The strength of the Coulomb potential is decreased until the energies of the two alpha pairs are zero. The system was placed inside a harmonic trap and a stabilization method has been used to calculate the energies of the resonances. We found that the shielded-Coulomb potential, which keeps the long tail, is not able to produce the Efimov effect. The energy of the three alphas decreases only to 0.19 MeV when the two-body energy crosses the threshold to become bound.
DOI: 10.7868/S0044002714040163
We cordially dedicate this work to Vladimir B. Belyaev on the occasion of his 80th birthday
1. INTRODUCTION
The theoretical prediction of Efimov states in 1970 [1] was made in the context of nuclear physics. However, the experimental achievement of this counterintuitive phenomenon has been made in an ul-tracold gas of cesium atoms [2]. Since then, once realized that such effect is not a simple mathematical artifact emerging in the limit of zero two-body binding, lot of papers have been published in this topic, starting a new branch of quantum few-body physics, called "Efimov Physics" [3].
Originally, the Efimov effect was derived for a system of three spinless neutral particles of equal mass and their possible implications have been considered for the nuclei 12C and 3H [1]. This effect is manifested by the appearance of an infinite number of three-body bound states as the two-body energy tends to zero, or equivalently the two-body scattering length tends to infinity, which can be explained by the appearance of an effective potential proportional to
''Instituto de Física Teórica, UNESP — Universidade Estad-
ual Paulista, Sa o Paulo, Brazil.
2)Instituto Tecnologico de Aeronautica — DCTA, S. J. dos
Campos, Brazil.
3)Universidade Federal do ABC, Santo Andre;, Brazil.
E-mail: saracol@ift.unesp.br
E-mail: yamashita@ift.unesp.br
1/R2 (R2 — r1_2 + + , where rj is the distance between particles ij) [4]. The energy ratio between two consecutive states is given by e2n/s0, where the constant s0 depends on the mass ratio of the particles (for equal masses so ^ 1.006 24). Their root-mean-square radii, a/(R2), also present a constant ratio between two consecutive states given by en/s0. Nowadays, we know that this effect is possible for several systems with different masses and also for more than three particles [5].
Efimov states are very weakly bound systems, their wave functions are extremely extended in space and located outside the range of the potential, such that their properties are independent of the details of the short-range potential [6]. Obviously, these characteristics cannot be expected to appear in a deeply bound and compact nuclear system such as the 12C ground state. The 12C, approximated by three alpha particles, also presents a Coulomb interaction, such that the collapse of the system, known as Thomas collapse [7], is avoided (note that the appearance of Efimov effect is closely related to the Thomas collapse [8]). Moreover, 8Be does not have zero energy and we do not have the freedom, as in atomic systems, to manipulate the two-body energy. These reasons might be enough to exclude the possibility to observe the Efimov effect in such non-exotic nuclei4).
4)For a recent report on alpha cluster description of nuclei, see [9]. In particular, the treatment of three-body charged particles is discussed by Belyaev and collaborators in [10].
As exposed in the last paragraph, the freedom we have in atomic condensates to manipulate the two-body scattering length by using the Feshbach resonance [11] is not present in nuclei. However, the Coulomb potential could eventually be shielded in a plasma with an electronic density enough to decrease the energies of12C and 8Be [12], such that Higa et al. conjectured that "the 12C Hoyle state is a remnant of a Efimov state that appears in the unitary limit" [13]. In this paper, we use a very simple non-realistic toy model to make a first investigation of the former conjecture. We study a system of three bosons (3B) in an s-wave state, where the two bosons (2B) interact by the phenomenological s-wave Ali— Bodmer potential [14]. This is a very preliminary calculation which may give a first insight about the conjecture. The paper is organized as follows. In Section 2, we detail the method used to calculate the resonances. In Section 3 we present our results. The conclusions are presented in Section 4.
2. FORMALISM
We calculated a three-boson system with total angular momentum zero. For the two-body interaction we used the phenomenological s-wave Ali—Bodmer potential given by
V(Tjj) = 500e-(°'7rij)2 - 130e-(0A75rij)2, (1)
where rij is the distance between the bosons i and j. All energies and distances are given in MeV and fm.
We also included the following three-body potential to create artificially a resonance close to the experimental energy of 0.38 MeV for the 0+ state
V3(R) = -9Me-(R/6)2
(2)
where R = ^Jr22 + r|3 + r^. We should mention here that the two-body potential used in this toy model does not consider higher partial waves, in particular d and g waves, which give a significant contribution to the properties of 12C [15]. Much more realistic calculations can be found in [16, 17].
The energy of the resonances was calculated by using a stabilization method [18, 19]. For this purpose, the system was put inside a harmonic trap
Vt
h2 r2
trap
~ 2m '
(3)
where m is the mass of the alpha particle and Ti are the single-boson coordinates. The size of the trap, b, is varied and the resonance appears as a "plateau" revealed by the discretized-continuum states, which present the noncrossing behavior at the resonance energy.
The eigenvalues are approached by E = ^j^y-,
where H is the Hamiltonian of the system and the wave function was expanded by a sum given by
N
\i>) = £ CiS \fr),
(4)
i=1
where N is the number of terms of the basis, large enough to have a converged value for E, and S is a symmetrizer operator. The functions 0i have been chosen as correlated Gaussians [20, 21]
4>i = exp
NP
■ E
j>k=1
a
(i) jk (rj
(rj — rk)2
(5)
where NP is the number of bosons, {r1, r2, ...,rNp } are the single-boson coordinates and ajl is a parameter related to the extension of the wave function of each pair jk.
Equation (5) can be written in terms of the Jacobi coordinates as follows. The relative coordinates, given by {x^ x2,..., xNp(xNp is chosen as thecenter-of-mass coordinate), can be written in terms of the single boson coordinates by using a transformation matrix U:
NP
J^Uij ri, (6)
j=1
Xi
such that the functions 0i can now be written in a compact notation as
^i(x) = exp
2
(7)
where XX is a row matrix with the relative coordinates and A(i is a symmetric positive-definite (NP — 1) x x (NP — 1) matrix containing the a parameters to be defined conveniently. The success to obtain a good result for E is very closely related to the choice of the matrices
A(i)
which means the parameters ajk (j > k = 1NP ). This choice can be made, for example, using a stochastic search, where the matrices are sorted in a certain interval. Another way is by using a geometric progression in which the
parameters are written as ajk = 1/(bokPi-1)2 (i = = 1,..., N), with b0 and p chosen to give a good description of the wave function.
The energy is then determined by solving the generalized eigenvalue problem,
N
N
Hij Cj = bij Cj,
j=1
j=1
MEPHA^ OH3HKA tom 77 № 4 2014
CONTINUUM RESONANCES WITH SHIELDED COULOMB-LIKE POTENTIAL 535
Fig. 1. Eingenvalues (solid lines) for two (a) and three bosons (b) as a function of the trap size, b. Both spectra present the avoid crossings close to the resonance energies. The parameters (Er, K) of the fitted function E(b) = Er + Kb-4 are given by (0.080, 341.10) and (0.38, 856.72), respectively, for the two- and three-body systems (dashed lines).
Im E
B
B
N + 1
Two-body cut
Three-body cut
УУУУУУУУУУУУУУУЛК
ШУШУУУУУШ
* E2 Second Riemann sheet Three-body virtual state
Re E
3
Im E
B
B
N + 1
Three-body cut
У/УУУУУУУУУУУУУУУУЛ
ReE
Three-body resonant state
N
3
Fig. 2. Schematic representation of the analytical cut structure for a three-particle system. The upper diagram shows on the left side of the vertical axis the two-body cut defined by the two-body bound-state energy, E2 = —|E21, and the movement of the energy of an excited three-body bound state B^+1 when |E21 is increased. In this case, the bound state becomes virtual, with the corresponding pole, located in the positive imaginary axis of the momentum plane, moving to the negative imaginary axis. In the lower frame the two-body system is virtual (no two-body cut) and the three-body state evolves into a resonance.
where i — 1,... ,N and the Hamiltonian and overlap matrices are given by
Hj — (Sfa\H), bij — {Sfc). (9) The evaluation of the matrix elements is facilitated
because the mean value of the kinetic energy as well as the overlap and the potential can be calculated analytically since fa has been chosen as a Gaussian function.
The results for E3 and E2 are presented in Fig. 1.
E2, E3, MeV 0.4
0.3
0.2
0.1
0
0.19 MeV
1.00 0.99 0.98 0.97 0.96 0.95
Fig. 3. Two- and three-boson energies (real part) as a function of A. For A = 0.943 E2 is zero and E3 = = 0.19 MeV.
For large b we expect to fit a curve like E(b) = Er
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