No-Core Shell Model Analysis of Light Nuclei
The fundamental description of both structural properties and reactions of light nuclei in terms of constituent protons and neutrons interacting through nucleon-nucleon and three-nucleon forces is a long-sought goal of nuclear theory. I will briefly present a promising technique, built upon the ab initio no-core shell model, which emerged recently as a candidate to reach such a goal: the no-core shell model/resonating-group method. This approach, capable of describing simultaneously both bound and scattering states in light nuclei, complements a microscopic cluster technique with the use of two-nucleon realistic interactions, and a microscopic and consistent description of the nucleon clusters. I will discuss applications to light nuclei binary scattering processes and fusion reactions that power stars and Earth based fusion facilities, such as the deuterium-He fusion, and outline the progress toward the inclusion of the three-nucleon force into the formalism and the treatment of three-body clusters.
pacs:24.10.-i 24.10.Cn 21.60.De25.40.Lw 25.45.-z 21.45.Ff
Nuclei are aggregates of protons and neutrons interacting through forces arising from the underlying theory of quantum chromodynamics. Understanding how the strong force binds nucleons together in nuclei is fundamental to explain the very existence of the universe. Indeed, the mutual interactions between nucleons led to the formation of the lightest nuclei a few minutes after the Big Bang, and the following nuclear processes, producing heavier nuclei during stellar evolution and in violent events like supernovae, have been crucial in shaping the world we leave in. Therefore, one of the central goals of nuclear physics is to come to a basic understanding of the structure and dynamics of nuclei. The ab initio (i.e. from first principles) no-core shell model/resonating group method (NCSM/RGM) (1); (2) is a theoretical technique that attempts to achieve such a goal for light nuclei.
2 Ab initio Ncsm/rgm
In the ab initio NCSM/RGM approach the many-body wave function,
is expanded over a set of translational-invariant cluster basis states describing two nuclei (a target and a projectile composed of and nucleons, respectively) whose centers of mass are separated by the relative coordinate and that are traveling in a wave or relative motion (with the channel spin, the relative momentum, and the total angular momentum of the system):
Here, the antisymmetric wave functions and are eigenstates of the - and -nucleon intrinsic Hamiltonians, respectively, as obtained within the NCSM approach (3) and are characterized by the spin-parity, isospin and energy labels , and , respectively, where . Additional quantum numbers labeling these RGM-inspired continuous basis states are parity and total isospin . In our notation, all these quantum numbers are grouped into a cumulative index . The Pauli principle is enforced by introducing the appropriate inter-cluster antisymmetrizer, schematically
where the sum runs over all possible permutations of nucleons different from the identical one that can be carried out between the two different clusters, and is the number of interchanges characterizing them. Finally, the continuous linear variational amplitudes are determined by solving the orthogonalized RGM equations:
where and , commonly referred to as integration kernels, are respectively the overlap (or norm) and Hamiltonian matrix elements over the antisymmetrized basis (2), i.e.:
Here, is the microscopic nucleon Hamiltonian and is the total energy in the center of mass (c.m.) frame. For a detailed explanation of how norm and Hamiltonian kernels are obtained from the underlying nuclear interaction and the NCSM eigenvectors of target and projectile we refer the interested reader to Refs. (2) and (4).
Applications of the NCSM/RGM approach to the description of nucleon- and deuteron-nucleus type of collisions based on two-nucleon () realistic interactions have already led to very promising results (1); (2); (5); (4); (6); (7). In most instances, we employed similarity-renormalization-group (SRG) (8); (9) evolved potentials, and in particular, those obtained from the chiral NLO (10) interaction. Here we briefly review two of such applications, the calculation of the BeB radiative capture (6), and the study of the HHe and HeHe fusion reactions (7).
3.1 The BeB radiative capture
Recently, we have performed the first ab initio many-body calculation of the BeB radiative capture (6), the final step in the nucleosynthetic chain leading to B and one of the main inputs of the Standard Solar Model. This calculation was carried out in a model space spanned by -Be channel states including the five lowest eigenstates of Be (the ground and the ,, and first and second excited states) in an NCSM basis, and employed the SRG-NLO interaction with fm, where denotes the SRG evolution parameter (8). We first solved Eq. (4) with bound-state boundary conditions to find the bound state of B, and then with scattering boundary conditions to find the -Be scattering wave functions. Former and latter wave functions were later used to calculate the capture cross section, which, at solar energies, is dominated by non-resonant transitions from -Be - and -waves into the weakly-bound ground state of B. All stages of the calculation were based on the same HO frequency of MeV, which minimizes the g.s. energy of Be.
At , the largest model space achievable for the present calculation within the full NCSM basis, the Be g.s. energy is very close to convergence as indicated by a fairly flat frequency dependence in the range MeV, and the vicinity to the result obtained within the importance-truncated NCSM (11); (12). With the chosen value fm for the SRG evolution of the NLO interaction, we obtain a single bound state for B with a separation energy of 136 keV, which is quite close to the observed one (137 keV). This is very important for the description of the low-energy behavior of the BeB astrophysical S-factor, known as . While for a complete ab initio calculation one should include also the three-nucleon () interaction induced by the SRG evolution of the potential as well as the SRG-evolved attractive initial chiral force, we note that in the -range - fm, and, in very light nuclei, the former interaction is repulsive and the two contributions cancel each other to a good extent (13); (14).
Figure 1 compares the resulting astrophysical factor with several experimental data sets. Energy dependence and absolute magnitude follow closely the trend of the indirect Coulomb breakup measurements of Shümann et al. (15), while somewhat underestimating the direct data of Junghans et al. (16). The resonance due to the capture, particularly evident in these and Filippone’s data and missing in our results, does not contribute to a theoretical calculation outside of the narrow B resonance and is negligible at astrophysical energies (17). The operator, for which any dependence upon two-body currents needs to be included explicitly, poses more uncertainties than the Siegert’s operator. In addition, the treatment of this operator within the NCSM/RGM approach is slightly complicated by the contributions coming from the core (Be) part of the wave function. Nevertheless, we plan to calculate its contribution in the future.
Our calculated MeV b is on the lower side, but consistent with the latest evaluation (expt)(theory) (17). The 0.7 eV b uncertainty was estimated by studying the dependence of the S-factor on the harmonic oscillator (HO) basis size as well as the influence of higher-energy excited states of the Be target. More precisely, we performed calculations up to within the importance-truncation NCSM scheme (11); (12) including (due to computational limitations) only the first three eigenstates of Be. The and S-factors are very close. In addition, the convergence in the number of Be states was explored by means of calculations including up to 8 Be eigenstates in a basis (larger values are currently out of reach with more then five Be states). This last set of calculations is presented in Fig. 2, from which it appears that, apart from the two states, the only other state to have a significant impact on the is the second , the inclusion of which affects the separation energy and contributes somewhat to the flattening of the -factor around MeV. For these last set of calculations we used SRG-NLO interactions obtained with different values with the intent to match closely the experimental separation energy in each of the largest model spaces. Based on this analysis, we conclude that the use of an HO model space is justified and the restriction to five Be eigenstates is quite reasonable.
3.2 The HHe and HeHe fusion reactions
In the following we present the first ab initio many-body calculations of HHe and HeHe fusion reactions (7) starting from the SRG-NLO interaction with fm, for which we reproduce the experimental -value of both reactions within . These reactions have important implications first and foremost for fusion energy generation, but also for nuclear astrophysics, and atomic physics. Indeed, the deuterium-tritium fusion is the easiest reaction to achieve on earth and is pursued by research facilities directed at reaching fusion power. Both HHe and HeHe affect the predictions of Big Bang nucleosynthesis for light-nucleus abundances. In addition, the deuterium-He fusion is also an object of interest for atomic physics, due to the substantial electron-screening effects presented by this reaction.
The model spaces adopted are characterized by HO model basis sizes up to with a frequency of MeV and channel bases including -He (-He), -H (-He), -H (-He) and -H (-He) binary cluster states. Here, and denote - and deuterium excited pseudostates, respectively, and the H (He) and He nuclei are in their ground state.
The results obtained for the HeHe S-factor are presented in Figure 4. The deuteron deformation and its virtual breakup, approximated by means of pseudostates, play a crucial role in reproducing the observed magnitude of the S-factor. Convergence is reached for . The typical dependence upon the HO basis sizes adopted is illustrated by the HHe results of Fig. 4. The convergence is satisfactory and we expect that an calculation, which is currently out of reach, would not yield significantly different results. While the experimental position of the HeHe S-factor is reproduced within few tens of keV and we find an overall fair agreement with experiment (if we exclude the region at very low energy, where the accelerator data are enhanced by laboratory electron screening), the HHe S-factor is not described as well with fm. Due to its very low activation energy, the HHe S-factor, particularly the position and height of its peak, is extremely sensitive to higher-order effects in the nuclear interaction, such as the force (not yet included in the calculation) and missing isospin-breaking effects in the integration kernels (which are obtained in the isospin formalism). With a very small change in the value of the SRG evolution parameter we can compensate for these missing higher-order effects in the interaction and reproduce the position of the HHe S-factor. This led to the theoretical S-factor of Fig. 4 (obtained for fm), that is in overall better agreement with data, although it presents a slightly narrower and somewhat overestimated peak. This calculation would suggest that some electron-screening enhancement could also be present in the HHe measured S-factor below 10 keV c.m. energy. However, these results cannot be considered conclusive until more accurate calculations using a complete nuclear interaction (that includes the force) are performed. Work in this direction is under way.
4 Recent developments
Here we outline some of our more recent efforts in the development of the NCSM/RGM approach, namely the progress toward the inclusion of the three-nucleon force into the formalism, and the treatment of three-body clusters and their dynamics.
4.1 Scattering and three-nucleon force
In past applications for light-ion reactions, we omitted the interaction induced by the SRG renormalization of the potential as well as the initial chiral force. By neglecting induced forces, we introduced a dependence on the SRG parameter , which was then chosen so that the particle separation energies were well reproduced. For low-energy thermonuclear reactions, this is a dominant effect, and overall such a procedure led to (never obtained before) realistic results. However, a truly accurate ab initio description demands the inclusion of both induced and chiral interactions.
While the inclusion of the force into the NCSM/RGM formalism is conceptually straightforward, in practice it poses major challenges having to deal with: 1) the large number of matrix elements, which makes it essential to work within the JT-coupled scheme; and 2) the appearance of kernels depending on the three-body densities of the target already for nucleon-nucleus processes, which demands new efficient computing strategies for applications with -shell targets to be possible. Figures 6 and 6 present initial results for HeHe scattering phase shifts with inclusion of the force. The use of SRG-evolved interactions facilitates the convergence of the calculation within . At the same time, Fig. 6 highlights the influence of induced and initial components of the force on the resonant phase shifts. The largest splitting between and is found when both induced and chiral forces are included (NN+NNN curve). It should be noticed that these results are still preliminary, as not all relevant excitations of the He nucleus have been taken into account yet. In particular, the resonances are sensitive to the inclusion of the first six excited states of the He (2) (here only the g.s. is included). We will complete this study in the coming months.
We have also obtained first results for the He scattering phase shifts and the ground state of Li including both SRG-induced and chiral forces. The model spaces adopted so far contain only the g.s. of the He and nuclei within a HO basis size. While calculations for larger values and including pseudo-excited states of the deuteron (fundamental to model the deformation and virtual breakup of this nucleus) are under way, these preliminary results are already very promising. In particular, the inclusion of the force produces a change in position and splitting of the and scattering phase shifts, which were not well described in our former calculation with only the part of the SRG interaction. The force has also the effect of increasing the binding energy of the Li nucleus.
4.2 Three-cluster dynamics
A proper description of Borromean halo nuclei and three-body breakup reactions (but also virtual breakup effects) within the NCSM/RGM approach requires the inclusion of three-cluster channel states and the treatment of the three-body dynamics.
At present we have completed the development of the formalism for the treatment of three-cluster systems formed by two separate nucleons in relative motion with respect to a nucleus of mass number . Preliminary results for the g.s. energy of He within a He(g.s.) cluster basis and an , MeV HO model space, are compared to NCSM calculations in Table 1. The interaction adopted is the SRG-NLO potential with fm. With such a low value of , at the binding energy calculations are close to convergence in both NCSM/RGM and NCSM approaches. The MeV difference observed is due to the excitations of the He core, included only in the NCSM at present. Contrary to the NCSM, in the NCSM/RGM the He(g.s.) wave functions present the appropriate asymptotic behavior. This will be essential in describing He excited states in the continuum, such as, e.g. the soft dipole resonance. Work towards the solution of the three-cluster NCSM/RGM equations with continuum boundary conditions is under way.
We presented an outline of the NCSM/RGM, an ab initio many-body approach capable of describing simultaneously both bound and scattering states in light nuclei, by complementing the RGM with the use of realistic interactions, and a microscopic and consistent description of the nucleon clusters, obtained via the ab initio NCSM. We discussed applications to fusion reactions that power stars and Earth based fusion facilities, such as the BeB radiative capture, and the HHe and HeHe fusion reactions. Finally, we outlined the progress toward the inclusion of the force into the formalism and the treatment of three-cluster dynamics, and presented an initial assessment of -force effects on HeHe scattering, as well as preliminary results for the g.s. energy of He computed within a He(g.s.) NCSM/RGM three-cluster basis. Since the publication of the first results (1); (2); (5), obtained for nucleon-nucleus collisions, the NSCM/RGM has grown into a powerful approach for the description of binary reactions starting from realistic forces. A truly accurate ab initio description of light-ion fusion reactions and light exotic nuclei that encompasses the full force and the three-cluster dynamics is now within reach.
- thanks: Computing support for this work came from the LLNL institutional Computing Grand Challenge program and the Jülich supercomputer Centre. Prepared in part by LLNL under Contract DE-AC52-07NA27344. Support from the U. S. DOE/SC/NP (Work Proposal No. SCW1158), the NSERC Grant No. 401945-2011, and from the U. S. Department of Energy Grant DE-FC02-07ER41457 is acknowledged. TRIUMF receives funding via a contribution through the Canadian National Research Council. This work is supported in part by the Deutsche Forschungsgemeinschaft through contract SFB 634 and by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse.
- thanks: Presented at the 20th International IUPAP Conference on Few-Body Problems in Physics, 20 - 25 August, 2012, Fukuoka, Japan
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- journal: Few-Body Systems (FB20)
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