Scattering of light nuclei
The exact treatment of nuclei starting from the constituent nucleons and the fundamental interactions among them has been a long-standing goal in nuclear physics. Above all nuclear scattering and reactions, which require the solution of the many-body quantum-mechanical problem in the continuum, represent an extraordinary theoretical as well as computational challenge for ab initio approaches. We present a new ab initio many-body approach which derives from the combination of the ab initio no-core shell model with the resonating-group method . By complementing a microscopic cluster technique with the use of realistic interactions, and a microscopic and consistent description of the nucleon clusters, this approach is capable of describing simultaneously both bound and scattering states in light nuclei. We will discuss applications to neutron and proton scattering on - and light -shell nuclei using realistic nucleon-nucleon potentials, and outline the progress toward the treatment of more complex reactions.
-title19 International IUPAP Conference on Few-Body Problems in Physics
To understand the evolution of the Universe, we need to understand nuclear reactions. Indeed, low-energy fusion reactions represent the primary energy-generation mechanism in stars, help determining the course of stellar evolution, and are crucial in the formation of the chemical elements. In addition, much of what we know about neutrino oscillations is established from neutrino emerging from the Sun following the -decay of reactions products, particularly B. The light-ion fusion reactions which encompass the Standard solar model need to be understood better, if solar neutrinos are to provide even more precise information on the neutrino oscillation properties. As an example, the BeB radiative capture is a rather poorly-known step in the nucleosynthetic chain leading to B, which in turn is the dominant source of the high-energy solar neutrinos (through -decay to Be) detected in terrestrial experiments.
Furthermore, nuclear reactions are one of the best tools for studying exotic nuclei, which have become the focus of the next generation experiments with rare-isotope beams. These are nuclei for which most low-lying states are unbound, so that a rigorous analysis requires scattering conditions. In addition, much of the information we have on the structure of such short-lived nuclei is inferred from reactions with other nuclei.
Unfortunately, the calculation of nuclear reactions represent also a formidable challenge for nuclear theory, the main obstacle being the treatment of the scattering states. In this paper we will present a brief overview of existing theoretical methods for nuclear reactions, highlighting in particular recent progress in the ab initio calculation of low-energy scattering of light nuclei.
1.1 Overview of reaction approaches
Because of their importance nuclear reactions attract much attention, and there have been many interesting new developments in the recent past. In this section we will give a brief overview of the theoretical efforts devoted to nuclear reactions, and in particular scattering of light nuclei. However, this is not intended to be completely exhaustive.
Nuclear reaction approaches may be classified according to two broad categories. The first category embraces the so-called microscopic approaches, in which all the nucleons involved in the scattering process are active degrees of freedom, and the antisymmetrization of the many-body wave functions is treated exactly.
In the three- and four-nucleon sectors there has been remarkable progress in the past ten years: the Faddeev (1), Faddeev-Yacubovski (2); (3), Alt-Grassberger and Sandhas
(AGS) (4); (5), hyperspherical harmonics (6), Lorentz integral transform methods (7); (8); (9), etc., are among the best known of several numerically exact techniques able to describe reactions observables starting from realistic nucleon-nucleon () and in some cases also three-nucleon () forces.
Going beyond four nucleons there are fewer ab initio or ab initio inspired methods able to describe reactions observables starting from realistic forces. Only very recently the Green’s function Monte Carlo (10), the no-core-shell model combined with the resonating group method (NCSM/RGM) (11); (12) and the fermionic molecular dynamics (13) have made steps in this direction.
Reactions among light nuclei are more widely described starting from semirealistic interactions with adjusted parameters within the traditional resonating-group method (14); (15); (16); (17); (18) or the generator coordinate method (19); (20); (21), which are microscopic cluster techniques.
A second category is that of few-body methods describing scattering among structureless clusters. Here one starts form nucleon-Nucleus (usually optical) potentials fitted on some reaction observable, and the nucleus core is usually inert. There are exact techniques which can employ either local or non-local optical potentials like the Faddeev or AGS methods adopted by Deltuva (22); (23), and various approximated ones, like the continuum-discretized coupled channel equations (24); (25), distorted wave born approximations, or various adiabatic approximations (26), etc., which usually adopt local optical potentials.
Finally, there are also some recent attempts to describe reactions among light nuclei in an effective-field theory approach for halo nuclei (27); (28). Starting from experimental resonance parameters for the system under investigation, phase shifts and cross section are predicted at low energy.
2.1 Ab initio no-core shell model
The NCSM is a technique for the solution of the -nucleon bound-state problem. All (point-like) nucleons are active degrees of freedom, hence the difference with respect to standard shell model calculations with inert core. Starting from a microscopic Hamiltonian ( being the momentum of the ith nucleon and the nucleon mass)
containing realistic ( or plus () forces (both coordinate- and momentum-space interactions can be equally handled), the non-relativistic Schrödinger equation is solved by expanding the wave functions in terms of a complete set of -nucleon harmonic oscillator (HO) basis states up to a maximum excitation above the minimum energy configuration, with the HO frequency.
The choice of the HO basis in a complete space is motivated by its versatility. Indeed, this is the only basis which allows to work within either Jacobi relative coordinates or Cartesian single-particle coordinates (as well as easily switch between the two), while preserving the translational invariance of the system. Consequently, powerful techniques based on the second quantization and developed for standard shell model calculations can be utilized. As a downside, one has to face the consequences of the incorrect asymptotic behavior of the HO basis.
Standard, accurate potentials, such as the Argonne V18 (AV18) (29), CD-Bonn (30), INOY (inside non-local outside Yukawa) (31) and, to some extent, also the chiral NLO (32), generate strong short-range correlations that cannot be accommodated even in a reasonably large HO basis. In order to account for these short-range correlations and to accelerate convergence with respect to the increasing model space, the NCSM makes use of an effective interaction obtained from the original, realistic or potentials by means of a unitary transformation in a body cluster approximation, where is typically or (33). The effective interaction depends on the basis truncation and by construction converges to the original realistic or interaction as the size of the basis approaches infinity.
On the other hand, a new class of soft potentials has been recently developed, mostly by means of unitary transformations of the standard accurate potentials mentioned above. These include the (34), the Similarity Renormalization Group (SRG) (35) and the UCOM (36) potentials. A different class of soft phenomenological potential used in some NCSM calculations are the so-called JISP potentials (37). These so-called soft potentials are to some extent already renormalized for the purpose of simplifying many-body calculations. Therefore, one can perform convergent NCSM calculations with these potentials unmodified, or “bare.” In fact, the chiral NLO potential (32) can also be used bare with some success. NCSM calculations with bare potentials are variational with respect to the HO frequency and the basis truncation parameter .
2.2 Resonating-group method
The resonating-group method (RGM) (14); (15); (16); (17); (18); (38) is a microscopic cluster technique in which the many-body Hilbert space is spanned by wave functions describing a system of two or more clusters in relative motion. Here, we will limit our discussion to the two-cluster RGM, which is based on binary-cluster channel states of total angular momentum , parity , and isospin ,
In the above expression, and are the internal (antisymmetric) wave functions of the first and second clusters, containing and nucleons (), respectively. They are characterized by angular momentum quantum numbers and coupled together to form channel spin . For their parity, isospin and additional quantum numbers we use, respectively, the notations , and , with . The cluster centers of mass are separated by the relative coordinate
where are the single-particle coordinates. The channel states (2) have relative angular momentum . It is convenient to group all relevant quantum numbers into a cumulative index .
The former basis states can be used to expand the many-body wave function according to
However, to preserve the Pauli principle one has to introduce the appropriate inter-cluster antisymmetrizer, schematically
where the sum runs over all possible permutations that can be carried out among nucleons pertaining to different clusters, and is the number of interchanges characterizing them. Indeed, the basis states (2) are not anti-symmetric under exchange of nucleons belonging to different clusters.
The coefficients of the expansion (4) are the relative-motion wave functions . These are the only unknowns of the problem, to be determined solving the non-local integral-differential coupled-channel equations
where denotes the total energy in the center-of-mass frame. Here, the two integration kernels, specifically the Hamiltonian kernel,
and the norm kernel,
contain all the nuclear structure and antisymmetrization properties of the problem. The somewhat unusual presence of a norm kernel is the result of the non-orthogonality of the basis states (2), caused by the presence of the inter-cluster antisymmetrizer. The exchange terms of this antisymmetrization operator are also responsible for the non-locality of the two kernels.
The main inputs of the RGM method are the internucleon interaction; and the wave functions of the - and -nucleon clusters. Staring from the latters, in the traditional RGM the clusters internal wave functions are often (but not exclusively) translationally invariant HO shell-model functions of the lowest configuration or a linear superposition of such functions. The value of the HO size parameters are chosen ad hoc to reproduce properties of the nucleon clusters (such as size and/or binding energy, etc.) within the adopted interaction. This somewhat simplified description of the clusters internal wave functions is usually compensated by the use of semirealistic interactions, such as the Volkov (39) or Minnesota (40) potentials, with parameters that can be adjusted to reproduces important properties of the compound nucleus or reaction under study. The spin-orbit force, not present in the mentioned semi-realistic interactions, is sometimes added to the microscopic Hamiltonian. Exception to this general description of the RGM approach exist, particularly in the few-nucleon sector, where the method has been utilized in combination with realistic and forces (38). Finally, the treatment of the Coulomb interaction between charged clusters does not represent an issue in the RGM approach.
The advantage in expressing the RGM basis states (2) as antisymmetrized products of single-particle functions, and in particular Slater determinants, lies in the ability to carry out analytical derivations of the required matrix elements (7) and (8). Once the integration kernels are calculated, by solving the integral-differential coupled channel equations (6) subject to appropriate boundary conditions, one obtains bound-state wave functions and binding energies or scattering wave functions and scattering matrix, from which any other scattering and reaction observable can be calculated.
2.3 Ab initio NCSM/RGM approach
A new first-principles, many-body approach capable of simultaneously describing both bound and scattering states in light nuclei has been developed by combining the RGM with the ab initio NCSM (11); (12). This new approach complements the microscopic-cluster technique of the RGM with the utilization of realistic interactions and a consistent microscopic description of the nucleonic clusters, while preserving important symmetries such as Pauli exclusion principle, translational invariance, and angular momentum. More in detail, the formalism presented in Section 2.2 can be combined with the ab initio NCSM as follows.
First, we note that the Hamiltonian can be written as
where and are the ()- and -nucleon intrinsic Hamiltonians, respectively, is the relative kinetic energy and is the sum of all interactions between nucleons belonging to different clusters after subtraction of the average Coulomb interaction between them, explicitly singled out in the term ( and being the charge numbers of the clusters in channel )
Nuclear, , and point-Coulomb components of the two-body potential have been listed explicitly ( , denoting spin and isospin coordinates, respectively, of the ith nucleon). If the -nucleon Hamiltonian contains a force, the inter-cluster interaction will present also a contribution from the latter, denoted here with .
The cluster’s Hamiltonians and inter-cluster interaction are consistent, as they contain the same realistic potentials. The clusters internal wave functions are also treated consistently: and are obtained by diagonalizing and , respectively, in the model spaces spanned by the NCSM basis. Both the - and -nucleon model spaces are characterized by the same HO frequency and maximum number of excitations above the minimum configuration. At the same time, in calculating the Hamiltonian and norm kernels of Eqs. (7), and (8), all “direct” terms arising from the identical permutations in both and are treated exactly (with respect to the separation ) with the exception of . The latter and all remaining terms are localized and can be obtained by expanding the Dirac of Eq. (2) on a set of HO radial wave functions with identical frequency , and model-space size consistent with those used for the two clusters. In this respect, we note that, thanks to the subtraction of the average potential , is localized also in the presence of the Coulomb force.
If the adopted potential generates strong short-range correlations, we employ consistent NCSM effective interactions derived from it. More specifically, the cluster eigenstates are obtained by employing the usual NCSM effective interaction (33). However, in place of the bare potential entering we adopt a modified effective interaction, which avoids renormalizations related to the kinetic energy. Following the notation of Ref. (33), at the two-body cluster level this is given by , where is the effective Hamiltonian derived from , with . Note that in the limit and, for each model space, the renormalizations related to the kinetic energy and the HO potential introduced in are compensated by the subtraction of . The kinetic-energy renormalizations are appropriate within the standard NCSM, but they would compromise scattering results obtained within the NCSM/RGM approach, in which the relative kinetic energy and average Coulomb potential between the clusters are treated exactly (that is, are not truncated within a finite HO model space).
To give a somewhat more in depth description of the formalism involved in the calculation of the matrix elements (7) and (8), here we will present examples of algebraic expressions derived within the single-nucleon projectile basis, i.e., for binary-cluster channel states (2) with (with channel index ). In this model space, the norm kernel is rather simple and is given by
where it is easy to recognize a direct term, in which initial and final state are identical (corresponding to diagram of Fig. 1), and a many-body correction due to the exchange part of the inter-cluster anti-symmetrizer (corresponding to diagram of Fig. 1). As the exchange is a short-range operator, in calculating its matrix elements we replaced the delta function of Eq. (2) with its representation in the HO model space. Such HO expansion is appropriate whenever the operator is short-to-medium range.
The presence of the inter-cluster anti-symmetrizer affects also the Hamiltonian kernel, and in particular the matrix elements of the interaction:
where are the eigenenergies of the -nucleon cluster. If no forces are present in the Hamiltonian one obtains a “direct” term involving interaction and exchange of one of the nucleons in the first cluster with the nucleon () of the second cluster (see diagrams () and () of Fig. 1), and an “exchange” term involving the interaction of the th nucleon with one of the nucleons, accompanied by the exchange with a second of such nucleons. Diagram () of Fig. 1 describes this latter term. These two potential kernels, which together constitute the matrix element , have the following expressions:
The inclusion of a interaction in the Hamiltonian is straightforward, and amounts to extra “direct” and “exchange” potential kernels, which can be obtained in a similar way.
Being translationally-invariant quantities, the Hamiltonian and norm kernels (7, 8) can be “naturally” derived working within the NCSM Jacobi-coordinate basis. However, particularly for the purpose of calculating reactions involving -shell nuclei, it is computationally advantageous to introduce Slater-determinant (SD) channel states of the type
in which the eigenstates of the -nucleon fragment are obtained in the SD basis (while the second cluster is still a NCSM Jacobi-coordinate eigenstate), and is the vector proportional to the center of mass coordinate of the -nucleon cluster. Indeed, it easy to demonstrate that translationally invariant matrix elements can be extracted from those calculated in the SD basis of Eq. (17) by inverting the following expression:
Here represents any scalar and parity-conserving and translationally-invariant operator (, , etc.), and , are general HO brackets for two particles with mass ratio . We exploited both Jacobi-coordinate and SD channel states to verify our results.
As an example, the single-nucleon projectile “exchange” part of the norm kernel within the Jacobi-coordinate basis for a system of nucleons is given by:
Here and are the coefficients of the expansion of initial and final two-nucleon target wave functions, respectively, with respect to the HO basis states depending on the Jacobi, spin, and isospin coordinates , , and , respectively,
where are the HO quantum numbers corresponding to the harmonic oscillator associated with , while , and are the spin, total angular momentum, and isospin of the two-nucleon channel formed by nucleons 1 and 2, respectively. Note that the basis (21) is anti-symmetric with respect to the exchange of the two nucleons, . Finally, are the general HO brackets for two particles with mass ratio .
At the same time, as mentioned above, the matrix elements of the operators , , and can be more intuitively derived working
within the SD basis of Eq. (17). Using the second-quantization formalism, they can be related to linear combinations of matrix elements of creation and annihilation operators between -nucleons SD states. As an example, the case of the exchange operator yields:
Here, are one-body density matrix elements of the target nucleus and . Next we extract the corresponding translationally-invariant matrix elements, , by inverting Eq. (18) for and . The final step follows easily from Eq. (12).
Due to the exchange terms of the intercluster antisymmetrizers, norm and potential kernels are non-local and appear as surfaces in three dimensions such as, e.g., those shown in Figs. 2 and 3. The latter figures present results of single-channel calculations carried out using - cluster channels with the particle in its g.s. (note that the index is simply replaced by the quantum number ). The interaction models adopted are the NLO potential (32), and the potential (34) derived from AV18 (29) with cutoff fm.
Orthogonalization of the RGM equations
An important point to notice, is that Eq. (6) does not represent a system of multichannel Schrödinger equations, and do not represent Schrödinger wave functions. This feature, which is indicated by the presence of the norm kernel and is caused by the short-range non-orthogonality induced by the non-identical permutations in the inter-cluster anti-symmetrizers, can be removed by introducing normalized Schrödinger wave functions
where is the square root of the norm kernel, and applying the inverse-square root of the norm kernel, , to both left and right-hand side of the square brackets in Eq. (6). By means of this procedure, known as orthogonalization and explained in more detail in Ref. (12), one obtains a system of multichannel Schrödinger equations:
where are the energy eigenvalues of the -th cluster (), and are the overall non-local potentials between the two clusters, which depend upon the channel of relative motion, while do not depend upon the energy of the system.
The two-cluster NCSM/RGM formalism within the single-nucleon projectile outlined in the previous section, can be used to calculate nucleon-nucleus phase shifts below three-body break threshold, by solving the
system of multi-
channel Schrödinger equations (24) with scattering boundary conditions. In the next sections we will review part of the results for neutrons scattering on H, He and Be and protons scattering on He, using realistic potentials, which were first presented in Refs. (11) and (12), and present some new calculations.
3.1 Convergence with respect to the HO model space
To study the behavior of our approach with respect to the HO model space, we have performed NCSM/RGM scattering calculations for the systems, using the potential (34), which is “soft” and we treated as “bare”, and the NLO interaction (32), which generates strong short-range correlations, thus requiring the use of effective interactions. In particular, for this convergence tests, we restricted our binary-cluster basis to target-nucleon channel states with the target in its g.s. (corresponding to channel indexes of the type ).
Results obtained for are presented in Fig. 4. The overall convergence is quite satisfactory, with a weak dependence on .
Figure 5 presents the convergence rate (achieved by using two-body effective interactions tailored to the HO model-space truncation) obtained for the same - scattering phase shifts with the NLO potential. Clearly, the NLO results converge at a much slower rate than the ones. However, a gradual suppression of the difference between adjacent values with increasing model-space size is visible, although the pattern is somewhat irregular for the phase shifts.
Although not shown, the - phase shifts present analogous convergence properties.
The next figure, Fig. 6 compares the results for the previously discussed and NLO interactions, and those obtained with the CD-Bonn potential (30). The NCSM/RGM calculations for the latter potential were carried out using two-body effective interactions, and present a convergence pattern similar to the one observed for NLO. Clearly, the and phase shifts are sensitive to the interaction models, and, in particular, to the strength of the spin-orbit force. This observation is in agreement with what was found in the earlier study of Ref. (10). Following a behavior already observed in the structure of -shell nuclei, CD-Bonn and NLO interactions yield about the same spin-orbit splitting. On the contrary, the larger separation between the and resonant phase shifts is direct evidence for a stronger spin-orbit interaction.
As the channel is dominated by the repulsion between the neutron and the particle induced by the Pauli exclusion principle (see also Figs. 2, 3), the short-range details of the nuclear interaction play a minor role on the phase shifts. As a consequence, we find very similar results for all of the three adopted potential models.
3.2 Test of the NCSM/RGM approach in the four-nucleon sector
A stringent test-ground to study the performances of the NCSM/RGM approach within the single-nucleon projectile basis is provided by the four-nucleon system. Numerically exact calculations for the sector have been already successfully performed within accurate few-body techniques, such as Faddeev-Yacubovsky, AGS, and hyperspherical harmonics methods (2); (3); (4); (5); (6); (41).
Figures 7 and 8 show -H and -He phase shifts, respectively, calculated with the NLO potential (32) together with results obtained by Deltuva and Fonseca (4); (42) from the solution of the AGS equations ( symbols) using the same interaction. The convergence behavior of The NCSM/RGM calculations was achieved using two-body effective interactions tailored to the model-space truncation, as outlined in Sec. 2.3. For the , and partial waves, the increase in model-space size produces gradually smaller deviations with a clear convergence towards the results. The rest of the phase shifts, particularly the , show a more irregular pattern. Nevertheless, in the whole energy-range we find less than deg absolute difference between the phases obtained in the largest and next-to-largest model spaces.
Concerning the comparison to the highly accurate AGS results, in general the agreement between the two calculations worsens as the relative kinetic energy in the c.m. frame, , increases. This discrepancy is a manifestation of the the influence played by closed channels not included in our basis states, that is, target-nucleon channel states with the target above the g.s., and 2+2 configurations, both of which are taken into account by the AGS results. Because these states correspond to the breakup of the system, it is not feasible to include them in the current version of the NCSM/RGM approach, which so far has been derived only in the single-nucleon projectile basis. However, we are planning on extending our approach to be able to account for the target breakup, and these development will be discussed in future publications. Nevertheless, this test of the NCSM/RGM approach in the sector clearly indicates that one has to pay attention not only to the convergence with respect to the HO model-space size , but also to the convergence in the RGM model space, which is enlarged by including excited states of the nucleon clusters in the binary-channel basis states.
3.3 HeHe scattering
A better scenario for the application of the NCSM/RGM approach within the single-nucleon projectile basis is the scattering of nucleons on He. This process is characterized by a single open channel up to the He breakup threshold, which is fairly high in energy. In addition the low-lying resonances of He are narrow enough that they can be reasonably reproduced by diagonalizing the four-body Hamiltonian in the NCSM model space, and consistently included as closed channels in the NCSM/RGM model
In Fig. 9 we explore the effect of the inclusion of the first six excited states of the He on the - scattering phase shifts obtained with the NLO interaction. More specifically, in addition to the single-channel results (dotted line) discussed in Sec. 3.1, we show coupled-channel calculations for five different combinations of He states, i.e., g.s., (dash-dotted line), g.s., (dash-dot-dotted line), g.s., (dash-dash-dotted line), g.s., (dashed line), and g.s., (solid line).
The use of these five different combinations of ground and excited states (also shown in the legends of Fig. 9) indicates that the phase shifts are well described already by coupled channel calculations with g.s. and first (the phase shifts obtained in the four larger Hilbert spaces are omitted for clarity of the figure). On the contrary, the negative parity excited states have relatively large effects on the phase shifts, and in particular the and mostly on the , whereas the and on the . These negative parity states influence the phase shifts because they introduce couplings to the -wave of relative motion. Though also couples to in the channel, the coupling of the states is dominant for the phase shifts.