CONTINUUM SHELL MODEL

The consistent treatment of continuum states, both in nuclear structure and reactions, is an old problem which has been a playground of the continuum shell model (CSM) [24,25]. In the CSM, including the recently developed Shell Model Embedded in the Continuum (SMEC) [26], the scattering states and bound states are treated on an equal footing. So far, most applications of the CSM, including SMEC, have been used to describe limiting situations in which there is coupling to one-nucleon decay channels only. There have been only a few attempts to treat the multi-particle case and, unfortunately, the proposed numerical schemes, due to their complexity, have never been adopted in shell-model calculations. Recently, we formulated and tested the multiconfigurational shell model in the complete Berggren basis [27], the so-called Gamow Shell Model (GSM). (For application to two-particle resonant states, see also Ref. [28].) By going into the complex momentum (or energy) plane, GSM overcomes a number of difficulties pertaining to the traditional CSM; in particular, it can easily be applied to systems containing several valence neutrons.

The main idea behind GSM is the use of Gamow (or resonant) states [29] - generalized eigenstates of the time-independent Schrödinger equation with complex energy eigenvalues. These states correspond to the poles of the S-matrix in the complex energy plane lying on or below the positive real axis; they are regular at the origin and satisfy purely outgoing asymptotics.

In GSM, the single-particle (s.p.) basis corresponds to eigenstates of a spherical single-particle finite potential (such as a Woods-Saxon potential). The generalized completeness relation involving Gamow states [30,31] can be written as:

$$\sum_{n} \vert\phi_{nj}\rangle \langle \tilde{\phi}_{nj}\vert... ...L_+} \vert\phi_j(k)\rangle \langle \phi_j(k^*)\vert \, dk = 1,$$ (3)

where $\phi_{nj}$ are the Gamow states carrying the s.p. angular momentum, j, n stands for all the remaining quantum numbers, $\phi_j(k)$ are the scattering complex-momentum states, and the contour L₊ in the complex k-plane has to be chosen in such a way that all the poles in the discrete sum in Eq. (3) are contained in the domain between L₊ and the real energy axis. (In practical calculations, the integral in Eq. (3) is discretized.) If the contour L₊ is chosen reasonably close to the real energy axis, the first term in (3) represents the contribution from bound states and narrow resonances, while the integral part accounts for the non-resonant continuum. Gamow resonances and the Berggren basis (3) have been employed in a number of calculations involving one-body continuum [32]. Examples are s.p. level density calculations [33] and studies of deformed proton emitters [34,35].

The crucial problem pertaining to the interpretation of the CSM results is the selection of states associated with resonant excitations of the system. Bound states can be clearly identified, because the imaginary part of their energy must be zero. No equally simple criterion exists for resonance states. Fortunately, the coupling between scattering states and resonant states is usually weak; hence, one can determine the physical resonances by considering first the subspace of Gamow states (the so-called pole expansion) and then by adding the non-resonant continuum. In the following example of GSM calculations, we shall consider the case of ^6-9He with the inert ⁴He core and 2-5 active neutrons in the p shell. (For details and more examples, including the chain of neutron-rich oxygen isotopes, see Refs. [27,36].) Our aim is not to give the precise description of these light nuclei (for this, one would need a realistic Hamiltonian and a large configuration space), but rather to illustrate the method and underlying features.

A description of the neutron-rich helium isotopes, including Borromean nuclei ^6,8He, is a challenge for the GSM. ⁴He is a well-bound system with the one-neutron emission threshold at 20.58MeV. On the contrary, the nucleus ⁵He, with one neutron in the p shell, is unstable with respect to the neutron emission. Indeed, the $J^{\pi}=3/2_1^{-}$ground state of ⁵He lies 890keV above the neutron emission threshold and its neutron width is large, $\Gamma$=600keV. The first excited state, 1/2₁^-, is a very broad resonance ($\Gamma$=4MeV) that lies 4.89MeV above the threshold. In our GSM calculations, the states in ⁵He are viewed as one-neutron resonances outside of the ⁴He core. For the s.p. field, we took a Woods-Saxon potential and for the residual interaction we assumed the surface-delta interaction.

  

Figure 5: Experimental (EXP) and predicted(GSM) binding energies of ^6-9He as well as energies of $J^\pi$=2⁺ states in ^6,8He. The resonance widths are indicated by shading.

As seen in Fig. 5, GSM calculations reproduce the most important feature of ⁶He and ⁸He: the ground state is particle-bound, despite the fact that all the basis states lie in the continuum. In spite of a very crude Hamiltonian, rather limited configuration space, etc., the calculated ground state energies reproduce surprisingly well the experimental data. The neutron separation energy anomaly (i.e., the increase of the neutron separation energy when going from ⁶He to ⁸He) is reproduced. Also, the energies of excited 2₁⁺ states are in fair agreement with the data. As discussed in Refs. [27,36], the contribution from the non-resonant continuum to the ground state wave functions of Borromean systems ⁶He and ⁸He is very large.