ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 8, с. 1044-1053
= ЯДРА
CLUSTER MODEL WITH CORE EXCITATIONS. THE nBe EXAMPLE
© 2014 S. N. Ershov1)*, J. S. Vaagen2), M. V. Zhukov3)
Received July 1,2013
Bound states and low-lying resonances of the 11 Be one-neutron halo nucleus have been calculated within a two-body cluster model with core excitation. The lowest 10Be core excited 2+ state was considered as a quadrupole vibration. Shallow potentials were applied for neutron—core interaction, preventing motion in Pauli forbidden orbits. A good description of available experimental data including dipole excitations of 11Be was obtained. For the bound 1/2" excited state the [p3/2 ® 2+1\1/2- wave-function component gives the dominant contribution to the structure, which differs from results obtained in cluster models with deep potentials.
DOI: 10.7868/S0044002714070071
We cordially dedicate this work to Spartak T. Belyaev on the occasion of his 90th birthday
1. INTRODUCTION
Theory of nuclear structure has made a long way on the road to understanding how nuclei are constructed, from the basic principles of neutron— proton arrangement of an atomic nucleus to the modern calculations that apply only realistic two- and many-body nucleon—nucleon interactions. Nowadays, driven by spectacular progress in experimental technique, the focus of the interest is moving from explanations of stable nuclear properties to understanding the nuclear structure at the limit of stability and beyond. Many remarkable phenomena were discovered along this way, including new types of nuclear structure — halos, the evolution of magic numbers when nuclei are approaching the driplines, et cetera. Still many new discoveries, like in exploration of envisioned islands of stability in the sea of instability, lay ahead of us. Many scientists brought tremendous contributions to the development of nuclear theory, and Spartak T. Belyaev is one of the pioneers who put in place the first rocks in the foundation of a microscopic theory of the nuclear structure and modern approaches to analysis of the many-body
1)1 Joint Institute for Nuclear Research, Dubna, Russia.
2)Institute of Physics and Technology, University of Bergen, Norway.
3) Fundamental Physics, Chalmers University of Technology, Goteborg, Sweden.
E-mail: ershov@theor.jinr.ru
systems. His seminal article [1] led to understanding of many nuclear phenomena as effects of nucleon pairing correlations, which explain the energy gap in one-particle spectra of the non-magic nuclei, the difference of the momenta of inertia in deformed nuclei from rigid-body values, etc. This breakthrough achievement gave a strong impetus to the development of microscopic theories of nuclear structure.
Structure of light nuclei has always demanded a special treatment due to sizeable changes in nuclear properties with changing number of nucleons. The 11 Be nucleus is a good example of the nuclear structure peculiarity. The neutron separation energy of11 Be is only 0.5 MeV, significantly smaller than in neighboring isotopes. There is a parity inversion near the ground state of11 Be, which is not a 1/2", as one would expect from the spherical shell model, but a 1/2+ state. That such an inversion should occur in 11 Be was already suggested a long time ago [2]. But, the notion of a mean field is not so well established in nuclei of this very light mass region and models starting directly from an effective two-body interaction have failed to reproduce the inversion without an ad hoc renormalization. Considerable theoretical efforts have been put in to reproduce this level inversion in a systematic manner: the inclusion of correlations beyond a mean-field method by the variational shell model [3], a variation of single-particle energies via vibration coupling [4], and many others. Note, that the most ambitious ab initio no-core shell-model calculations [5] still have not been successful in parity inversion description.
Few-body cluster approaches often suggest conceptually simple models that successfully explain
many features of light nuclei [6]. They use phe-nomenological neutron—core interactions fitted to measured resonances and virtual states. The first applications of few-body models to description of the 11 Be structure and other two-cluster systems have been done in works [7—10]. In this work we develop a variant of a two-body cluster model with shallow potentials and apply it to calculations of bound states and low-lying resonances in the 11 Be nucleus.
2. MODEL DESCRIPTION 2.1. General Formulation
Few-body cluster models are often used for a description of the nuclear structure of weakly bound nuclei. They have attractive features supplying in the most transparent way the asymptotic behavior and continuum properties of weakly bound systems. Such models assume a separation in the internal cluster (core) degrees of freedom and the relative motion of few-body constituents. Such separation is only an approximation. To extend the applicability of cluster models, core excitations have to be taken into account. For fixed total angular momentum a coupling with excited core states having different spins involves additional partial waves into the consideration. This allows to account for some emergent core degrees of freedom and get a more realistic description of nuclear properties. It is an analog to increasing the number of shells within the framework of shell-model approaches.
Here, we use the two-body (core + neutron) cluster model for calculation of wave functions for one-neutron halo nucleus 11 Be. Halo nuclei are examples of extreme clusterization into a veil of valence neutrons and a core (C). Cluster models assume that the nuclear wave function of a nucleus with A nucleons is factorized into a sum of products from two parts, ^(r-1, ...,ta ) = E i■ ■ -,rAC )A(r). The first 0i(r1,. . . , rAc) is a core wave function describing the motion of the AC nucleons within core, being in the state i. The second, ^(r) describes the corresponding relative motion of the core center of mass (c.m.) and the valence nucleon. Here, the radius r is a relative distance between the c.m. of a core and the valence nucleon.
The wave function ^(r) is a solution of the coupled Schrodinger equations in a general form,
(T + ei - E)Ur) + E Vij(r) = 0, (1)
where T is the kinetic energy operator for relative motion of the valence nucleon and the core c.m., the
energy ei is the excitation energy of the core being in a state i, and the matrix elements
Vn (r) =
Ac \
EV(r, rfc) k=1
= ( ^i(r1,...,TAo )
(r1,...,rAc )
describe the core—neutron binary interactions.
A nuclear bound-state wave function is characterized by the total angular momentum J and its projection M on a quantization axis. Assuming that the core states have spins I and the other quantum numbers necessary for a complete core definition denoted by n, the bound-state wave function ^ JM can be decomposed in the following way:
^ jm (r 1,..., rA )= (2)
JM '
where 7 = {l,s,j,n,I} is an abbreviation for a set of quantum numbers, which indicates the coupling of angular momenta (the relative orbital angular momentum l and the spin s of the valence (halo) nucleon moving relative to the core c.m. are coupled to the total angular momentum j, and thereafter j is coupled with the core spin I to the total angular momentum J), \sv is the spin function of the valence neutron with spin s and its projection v, and xJ(r) is a radial part of the bound-state wave function. Inserting the decomposition (2) into the Schrodinger equation (1) and projecting out the angular parts, the set of coupled equations for radial wave functions xJ(r) can be obtained:
2/j,
d2 _ ¿(¿ + 1) dr2 r2
= - £ *
+ fy - E)xJ(r)= (3)
J
(r)Xy(r).
Here, ¡i is the reduced mass for a system consisting of the valence nucleon and the core, and corresponding matrix elements VYJY/, (r) are generated from binary interactions,
VyJy(r) = ( [Yi(r) ® Xsj ® tni
JM
(4)
Ac
E*(r, r*) k=1
Y (r) ® Xs' ® ^ n' I '
JM
These matrix elements define the dynamics of the system. Thus, the model for the core wave functions and the interaction between the valence nucleon and the core has to be specified for particular applications.
Y
Y
X
X
j
In this approach the continuum wave function ^<vniMI(k; rirA), which describes the nucleon having spin projection v and moving with the energy E = h2k2/(2^) relative to the core being in the state (nIMI), can be written as
(±)
v,nIMj
(k;ri, ■■■,rA) =
(5)
^ (lmisv\jm3)(jmjIMi\JMj)Yl*mi(k) x
IjJmimj M j 1
Un'I'
Y'
JMj
Here, the radial wave functions xJ> Y(k, r) constitute a matrix whose columns are characterized by the incoming channels 7 and elements in the column have index 7' which corresponds to the outgoing channel. Asymptotically at r ^^ radial wave functions, normalized to the 5 function on the linear momentum k, can be presented as
x
J',Y r) =
1
y/2it ^/k
Y' kY
(6)
x (H^kr)5Y'Y - H<+\kY' r)Sy,Y
nucleus can be calculated: n
^el(E) =
k2 (2s + 1)(2Ij + 1)
x£(2Jf + 1)\1 - Sl>f\2, f
n 1
^m (E) =
(7)
(8)
The linear momentum = ^/(2/j,/h2)\E — e7| corresponds to a relative motion in the channel 7. When energy E is larger than the core excitation energy eY, the channel 7 is open, if not, then the channel is closed. Functions H<±)(x) are the Coulomb functions describing in- and outgoing waves. The quantity SY', Y is the S-matrix element for the outgoing wave in channel 7' from an incoming wave in channel 7. Representation (6) is justified if channels Y and Y are both open. When a channel Y is closed, the respective component of wave functions asymptotically goes to zero just as for a bound state case. The radial wave functions xJ' Y(k,r) for any incoming open channel 7 satisfy the same set of the coupled radial equations (3) as the bound states. Recently, we have developed [11 — 13] a new method for solution of the coupled system of the radial Schrodinger equations whic
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