RIA RELATED THEORY PROGRAMS

The theory group has several programs that are directly related to experimental research at RIA facilities or to questions of astrophysical interest.

A characteristic feature of nuclei far from stability is the close vicinity of the continuum. This presents new opportunities and new challenges, both to structure and reaction theory. The conventional methods for stable nuclei have been to use the Shell Model to describe the structure and coupled-channels calculations to describe reactions. Both methods were in fact pioneered and developed here at Argonne, by Maria Goeppert-Mayer, Dieter Kurath, Stan Cohen, and Bob Lawson (Shell Model), and by Malcolm MacFarlane and Steve Pieper (PTOLEMY), respectively. Both methods consider a finite number of basis of states. The close vicinity and strong influence of the continuum in unstable nuclei, however, makes it difficult or impracticable to use these methods.

Our theoretical efforts are currently directed towards RIA-type experiments that are or can be performed at existing low-energy, heavy-ion facilities, such as ATLAS at Argonne, Holifield at Oak Ridge, and at fragmentation facilities, such as Michigan State University. The aim has been not only to provide theoretical support for the analyses and interpretation of such experiments but also to test the structure models we have developed. Our activities can roughly be divided into the following categories:

Ab Initio calculations of light nuclei
Structure and reactions of halo nuclei
Structure of heavy drip-line nuclei
Reactions of interest to astrophysics
Reaction theory for nuclei far from stability
Hadronic models of nuclei
Nuclear structure in heavy and superheavy elements
Superdeformation
N-P pairing

AB INITIO CALCULATIONS OF LIGHT NUCLEI

A major goal of nuclear physics is to understand the stability, structure, and reactions of nuclei as a consequence of the interactions among individual nucleons. Realistic nucleon-nucleon (NN) potentials that accurately describe NN scattering data are very complicated, including spin, isospin, tensor, spin-orbit, quadratic momentum-dependent, and charge-independence-breaking terms. In addition, multi-nucleon forces are found to be non-negligible and are empirically required to reproduce few-nucleon bound state energies. This makes the the solution of the nuclear many-body problem extremely challenging, even for relatively small systems [1].

We have been developing realistic two- and three-nucleon potentials and the quantum many-body methods necessary to evaluate them in light nuclei. The many-body methods we use are two quantum Monte Carlo (QMC) methods: variational (VMC) and Green's Function (GFMC). To date they have been successfully applied to all nuclei up to mass A=10 [2]. Some results for nuclear spectra are shown in the figure above, where we compare calculated energies of ground and excited states with experiment for a number of nuclei. The left-most bars show the results for our realistic Argonne v₁₈ (AV18) NN potential [3] alone, while the middle bars show the results with the Illinois-2 (IL2) three-nucleon potential [4] added. This AV18/IL2 model currently reproduces 53 levels in A=3--10 nuclei with an rms deviation of only 740 keV.

One of the interesting aspects for RIA-related physics is elucidating the nature of the three-nucleon force in neutron-rich systems, such as ^6,8He, or in neutron drops - collections of neutrons artificially confined by an external potential well [5]. These studies can help to formulate more schematic models, like Skyrme energy-density functionals, that can be used for much larger nuclei. The 3N force also plays an important role in the equation of state for dense neutron matter and neutron stars [6].

Aside from calculating energies of nuclear states, the VMC and GFMC methods are being used to study a wide variety of other properties, such as elastic and transition form factors [7], spectroscopic factors [8], and weak decay rates [9]; additional applications specific to halo nuclei are discussed below.

[1] S. C. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci. 51, 53 (2001)
[2] S. C. Pieper, K. Varga, and R. B. Wiringa, Phys. Rev. C 66, 044310 (2002)
[3] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995)
[4] S. C. Pieper, V. R. Pandharipande, R. B. Wiringa, and J. Carlson, Phys. Rev. C 64, 014001 (2001)
[5] B. S. Pudliner, et al., Phys. Rev. Lett. 76, 2416 (1996)
[6] R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C 38, 1010 (1988)
[7] R. B. Wiringa and R. Schiavilla, Phys. Rev. Lett. 81, 4317 (1998)
[8] L. Lapikás, J. Wesseling, and R. B. Wiringa, Phys. Rev. Lett. 82, 4404 (1999)
[9] R. Schiavilla and R. B. Wiringa, Phys. Rev. C 65, 054302 (2002)

STRUCTURE AND REACTIONS OF HALO NUCLEI

The ab initio QMC calculations described above are being used to study the structure of some of the lightest halo nuclei like ^6,8He [1]. The figure above shows predicted densities compared to those deduced from experiment. In addition, a novel application of Monte Carlo integration techniques to the Glauber model analysis of reactions of halo nuclei [2], makes it possible to use the sophisticated QMC wave functions for some high-energy reaction studies.

However, complete many-body calculations of larger nuclei are difficult and often impracticable, so it is useful to resort to much simpler descriptions that grasp the essential physics of a given problem. One example is the three-body model of two-neutron halo nuclei, which we developed early on for ¹¹Li [3,4]. The model contained the basic Borromean features, namely, that the ⁹Li-neutron and the neutron-neutron subsystems are both unbound but the scattering lengths are fairly large so that the combined ⁹Li-n-n three-body system becomes bound.

Our early work was based on a density-dependent, contact interaction between the two neutrons. The strength of this interaction was calibrated to produce an infinite neutron-neutron scattering length at low density, far from the ⁹Li core. Moreover, core recoil effects were ignored. These simplifications made it possible to calculate the ground state [3] and the dipole response [4] of ¹¹Li, effectively as a two-body problem, using a two-body Greens function technique.

We have tested the model against numerous measurements, as, for example, the production of ¹¹Li in pion-double-charge exchange reactions on ¹¹B [5], fragmentation cross sections at high energy [6], and fragment momentum distributions in Coulomb dissociation experiments [7]. Although the model is simple and schematic, it reproduces many essential features of the experiments. Of particular interest is the effect of correlations between the two valence neutrons, which is probed in the pion-double-charge exchange reaction [5], and in the relative momentum distributions of the fragments [7].

One uncertainty in the model is the ⁹Li-neutron interaction which was poorly know in the early 90's. A better empirical knowledge was obtained in the mid-90's. We used this opportunity to improve our calculations [8] by using more realistic ⁹Li-neutron and neutron-neutron interactions, and also to treat recoil effects in the three-body system correctly. The new model was used to analyze high-energy nuclear induced breakup reactions [9]. The analysis showed that equal amounts of s and p waves in the ¹¹Li halo were favored by the data.

[1] S. C. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci. 51, 53 (2001)
[2] K. Varga, S. C. Pieper, Y. Suzuki, and R. B. Wiringa, Phys. Rev. C 66, 034611 (2002)
[3] G. F. Bertsch and H. Esbensen, Annals of Physics 209, 327 (1991)
[4] H. Esbensen and G. F. Bertsch, Nucl. Phys. A542, 310 (1992)
[5] H. Esbensen, D. Kurath and T.-S. H. Lee, Phys. Lett. B287, 289 (1992).
[6] H. Esbensen and G. F. Bertsch, Phys. Rev. C 46, 1552 (1992)
[7] H. Esbensen, G. F. Bertsch, and K. Ieki, Phys. Rev. C 48, 326 (1993)
[8] H. Esbensen, G. F. Bertsch, and K. Hencken, Phys. Rev. C 56, 3054 (1997)
[9] G. F. Bertsch, K. Hencken, and H. Esbensen, Phys. Rev. C 57, 1366 (1998)

STRUCTURE OF HEAVY DRIP-LINE NUCLEI

The structure of heavy, proton-rich nuclei has been studied experimentally, both at Argonne and at Oak Ridge, by measuring the proton decay of nuclei that are just outside the proton drip-line. >From the measured life-time and the energy of the emitted proton one can put constraints of the single-particle properties of the decaying state, such as the angular momentum and spectroscopic factor. In some cases it has been possible to observe the decay, not only to the ground state but also to excited states of the daughter nucleus. The energies and branching ratios associated with decays to different final states put further constraints on the structure of the decaying nucleus.

We have developed a coupled-channels description of a proton interacting with a deformed core, a so-called particle-rotor model [1]. The model has been applied to analyze the data for the proton decay of nuclei that are known or expected to be deformed, and a fairly consistent picture has emerged. Thus it appears that the spectroscopic factors we extract are in good agreement with estimates based on BCS pairing.

A numerical problem we faced was to calculated extremely narrow widths, of the order of 10^-20 MeV associated with life-times in the range of milliseconds to seconds. This would be very difficult in ordinary scattering theory, or by treating the decaying state as a Gamow state with a complex energy. We adopted and developed a different technique, which is based on the Gell-Mann Goldberger transformation [2].

We have also developed a description of the proton decay from near-spherical nuclei which includes couplings to vibrational excitations of the core, i.e., a particle-vibration model [3].

[1] H. Esbensen and C. N. Davids, Phys. Rev. C 63, 014315 (2001)
[2] C. N. Davids and H. Esbensen, Phys. Rev. C 61, 054302 (2000)
[3] C. N. Davids and H. Esbensen, Phys. Rev. C 64, 034317 (2001)

REACTIONS OF INTEREST TO ASTROPHYSICS

The abundance of elements in the universe has primarily developed from proton and neutron capture reactions, sometimes on nuclei far from stability. One uncertainty in calculating this abundance is that the capture rates on unstable nuclei are often poorly known, because measurements with unstable targets can be very difficult or simply impossible. Even stable nuclei can have very small capture rates that make experimental determinations difficult. It is therefore extremely important to have other methods of determining the reaction rates.

One possibility is to calculate reaction rates from the most accurate and reliable nuclear structure models on the market. This can be done for lighter nuclei where the QMC calculations provide an accurate description of nuclei. First calculations of radiative captures to ⁶Li [1], ⁷Li, and ⁷Be [2] have been made; these reactions with stable nuclei are important for big bang nucleosynthesis. In the case of ⁶Li the cross section is so small at the relevant energies that experiments are extremely difficult, while the experimental situation for capture to ⁷Be is very uncertain, as shown in the above figure. The calculation's accurate reproduction of the ⁷Li S-factor suggests that lower data for the ⁷Be S-factor may well be the most reliable. We plan to extend these studies in future to other nuclei amenable to QMC calculations, such as ⁸B.

Another possibility is to measure the Coulomb dissociation of unstable nuclei, and use detailed balance to infer the rate of the inverse radiative capture reactions. This avenue is possible at a RIA facility where beams of unstable nuclei can be produced with sufficient intensity. This is already possible for lighter nuclei at the existing fragmentation facilities. Thus we have taken part in the analysis of the ⁸B Coulomb dissociation experiments, which have been performed at Michigan State University [3,4]. The analysis showed that the Coulomb dissociation technique can be made into a reliable tool for inferring radiative capture rates.

[1] K. M. Nollett, R. B. Wiringa, and R. Schiavilla, Phys. Rev. C 63, 024003 (2001)
[2] K. M. Nollett, Phys. Rev. C 63, 054002 (2001)
[3] B. Davids et al., Phys. Rev. Lett. 86, 2750 (2001)
[4] B. Davids et al., Phys. Rev. C 63, 065806 (2001)

REACTION THEORY FOR NUCLEI FAR FROM STABILITY

We have taken part in the development of the reaction theory and the models that have been used in analyses of radioactive beam experiments over the past decade. Early on we pointed out the significance of using the eikonal model in analyses of the nuclear induced breakup of halo nuclei at high energies [1]. We later applied this model to lower energies and discussed the significance of stripping and diffraction dissociation [2]. We have also tested the validity of the eikonal approximation at low beam energies and have shown that it deviates from dynamical calculations with a characteristic 1/E beam energy dependence [3].

We have also investigated the Coulomb dissociation of halo nuclei. Early on we applied first-order perturbation theory, which is a fairly reasonable approximation for neutron halo nuclei [4]. For proton halo nuclei [5], on the other hand, the first-order approximation becomes unreliable at low beam energies. The reason is a dynamic polarization effect, which is of order Z³ in the target charge Z. It is caused by an interplay of higher-order E1 and E2 transitions [6,7]. This mechanism does not play any role in the Coulomb dissociation of a neutron halo nucleus because the E2 strength is strongly suppressed compared to what it is in a proton halo nucleus.

Another complication of the dissociation of proton halo nuclei is that the Coulomb form factors are sensitive to close collisions, where the proton-core distance is larger than the projectile target distance [6,7]. The combined effect of dynamic polarization and close collisions explains nicely the strong reduction that has been observed in the dissociation cross section of ⁸B at low energy [7].

[1] G. F. Bertsch, H. Esbensen, and A. Sustich, Phys. Rev. C 42, 758 (1990)
[2] K. Hencken, G. Bertsch, and H. Esbensen, Phys. Rev. C 54, 3043 (1996)
[3] H. Esbensen and G. F. Bertsch, Phys. Rev. C 64, 014608 (2001)
[4] H. Esbensen, G. F. Bertsch, and C. A. Bertulani, Nucl. Phys. A581, 107 (1995)
[5] H. Esbensen and G. F. Bertsch, Nucl. Phys. A600, 37 (1996)
[6] H. Esbensen and G. F. Bertsch, Nucl. Phys. A706, 383 (2002)
[7] H. Esbensen and G. F. Bertsch, Phys. Rev. C 66, 044609 (2002)

HADRONIC MODELS OF NUCLEI

We have developed a general hadronic model based on a relativistic field description [1,2]. The model interactions are derived adopting SU(3) chiral symmetry constraints and contain the lowest-mass multiplets of baryons and scalar, pseudoscalar, vector and axialvector mesons. Within this approach it is possible to generate the masses of the hadrons via spontaneous symmetry breaking as well as to describe the properties of infinite nuclear matter [1,2] and to study the structure of finite nuclei and hypernuclei [3,4,5] at a competitive level of accuracy using the same set of model parameters.

A major goal of pursuing a broad approach of this kind is to investigate the influence of hadronic interactions on a range of observables including the properties of nuclei and neutron stars [5]. In the RIA facility, by approaching the neutron dripline, especially the isospin effects in hadronic physics will become important. For theory this entails the need to extrapolate the nuclear forces to large isospins, which can only be controlled by studying the impact of varying isospin-dependent interactions for a wide range of observables. In this spirit we varied several isospin-dependent chirally invariant interaction terms in our model [8] and studied the influence of those terms on the neutron skin of heavy nuclei and on the radii of neutron stars as shown in the Figure. The graph shows the correlation of the neutron skin Δ r_np of ²⁰⁸Pb and the radius of a neutron star with a typical mass of 1.4 solar masses. The strengths of isospin interactions are varied. The shaded band denotes the expected accuracy of an upcoming experiment measuring the neutron distribution in Pb.

As extension of our studies we also investigate isospin-dependent flow and fragment production patterns in heavy-ion collisions at RIA-relevant energies. In the same framework we have already successfully reproduced experimental particle production yields at very high beam energies [7].

[1] P. Papazoglou, J. Schaffner, S. Schramm, D. Zschiesche, H. Stöcker and W. Geiner, Phys. Rev C 55, 1499 (1997)
[2] P. Papazoglou, S. Schramm, J. Schaffner, H. Stöcker and W. Greiner, Phys. Rev. C 57, 2576 (1998)
[3] P. Papazoglou, D. Zschiesche, S. Schramm, J. Schaffner-Bielich, H. Stöcker and W. Greiner, Phys. Rev. C 59, 411 (1999)
[4] Ch. Beckmann, P. Papazoglou, S. Schramm, D. Zschiesche, H. Stöcker, W. Greiner, Phys. Rev. C 65, 024301 (2002)
[5] S. Schramm, Phys. Rev. C 66, 064310 (2002)
[6] M. Hanauske, D. Zschiesche, S. Pal, S. Schramm, H. Stöcker, W. Greiner, Ap. J. 537, 958 (2000)
[7] D. Zschiesche, S. Schramm, J. Schaffner-Bielich, H. Stöcker, W. Greiner, Phys. Lett. B 547, 7 (2002)
[8] S. Schramm, preprint nucl-th/0210053, submitted to Phys. Lett. B

NUCLEAR STRUCTURE IN HEAVY AND SUPERHEAVY ELEMENTS

What is the heaviest element that can be synthesized? This is a question that has intrigued nuclear physicists for more than 60 years. A RIA facility should provide an answer to this question. Theory can provide guidance as to which isotopes are most likely to be observable by tying model parameters to known structural features in known elements. The best way to extrapolate to superheavy elements is to utilize data on the heaviest known elements.

We have been engaged in a long term collaboration with experimentalists in the study of heavy elements and have used these data to make predictions about the stability of superheavy elements. An early result of such extrapolations was the realization that there is a strong [1] spherical neutron shell stabilization starting at roughly 178 neutrons rather than at the 184 neutron closed shell. From our studies of the heaviest elements, we found that there is a large gap [2] in the level spacings of deformed actinides at N=162 and N=164, suggesting exta stabilization for nuclei with such neutron numbers.

As experimental spectroscopic studies go to higher mass actinides, we continue [3-7] to utilize such data to make more accurate extapolations of the properties of superheavy elements. The figure shows our estimate of shell stabilization energies based on studies of the heaviest elements. We find large stabilization effects for all elements from Z=112 to Z=120.

[1] R. R. Chasman, in Proceedings of the International Symposium on Superheavy Elements, Lubbock, Texas, 1978, edited by M. A. K. Lodhi
[2] R. R. Chasman, I. Ahmad, A. M. Friedman, and J. R. Erskine, Rev. Mod. Phys. 49, 833 (1977)
[3] R. R. Chasman and I. Ahmad, Phys. Lett. B 392, 255 (1997)
[4] I. Ahmad, A. M. Friedman, R. R. Chasman, and S. W. Yates, Phys. Rev. Lett. 39, 12 (1977)
[5] I. Ahmad, R. R. Chasman, A. M. Friedman, and S. W. Yates, Phys. Lett. B 251, 338 (1990)
[6] I. Ahmad, et al., Nucl. Phys. A646, 175 (1999)
[7] I. Ahmad, R. R. Chasman and P. R. Fields, Phys. Rev. C 61, 044301 (2000)

SUPERDEFORMATION

With increasing nuclear charge, the line of stability goes through nuclei with larger and larger neutron excesses. Because of the curving of the line of stability, high-spin superdeformation studies are typically restricted to nuclei on the proton rich side of stability, because both beams and targets must be stable or long-lived nuclei. One of the attractive possibilities of a RIA facility is the opportunity to use secondary beams with neutron excesses and produce high-spin states in nuclei that approach the line of stability. This will provide an opportunity to extend the search for superdeformed and hyperdeformed states to new regions of the periodic table.

Experimental searches for superdeformed shapes are difficult and theoretical guidance [1-3] is necessary. For a theoretical calculation to be useful, it must describe many shape possibilities. In addition to describing the shapes in superdeformed minima, a useful theoretical treatment must also give an accurate description of the fission barrier and the barriers between normal and superdeformed shapes. If either barrier is too low, the superdeformed minimum will not accommodate any states. We have developed a four-dimensional [4-8] description of nuclear shapes that includes an explicit description of necking in. This approach gives an accurate treatment of fission barriers in addition to an accurate treatment of deformed, superdeformed and hyperdeformed nuclear shapes. Our calculations [9] include nuclei that have too large a neutron excess to be accessible with today's accelerators. By accessible we mean that these nuclides can be produced at sufficiently high spins to populate very deformed shapes. In the figure, we display the calculated energy surface of ¹⁰⁸Cd at an anglular momentum of I=60 hbar. At this angular momentum, the minimum in the energy surface corresponds to a very extended shape with major to minor axis ratios of 2.3:1. With present day accelerators, such angular momenta are not accessible.

[1] R. R. Chasman, Phys. Lett. B 187, 219 (1987)
[2] R. R. Chasman, Phys. Lett. B 219, 227 (1989)
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[4] R. R. Chasman, Phys. Lett. B 302, 134 (1993)
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[8] R. R. Chasman, Phys. Rev. Lett. 80, 4610 (1998)
[9] R. R. Chasman, Phys. Rev. C 64, 024311, (2001)

N-P PAIRING

In most nuclei, the valence neutrons and protons occupy different shells, and pairing interactions involving n-p pairs are suppressed. For nuclides near the N=Z line, one expects that the scattering of T=1 n-p pairs should play a role that is as important as n-n and p-p pair scatterings. An additional aspect of nuclear structure that is unique to the region of the N=Z line is the existence of T=0 n-p pairing. In lighter nuclei, level densities are low and accordingly pairing effects are suppressed. The RIA accelerator will afford an opportunity to make heavier nuclides near the N=Z line and to learn something about pairing in these nuclei.

We are developing a treatment of n-p pairing that goes far beyond the quasi-particle approach. We have developed [1,2] a triple projection method that gives a more accurate treatment of the pairing Hamiltonian and corresponds roughly to isospin projection. In the figure, we display low-lying states calculated for several odd-odd nuclei using constant pairing matrix elements. In the upper part of the figure, we show calculated spectra for equally spaced single-particle levels and in the lower part of the figure, we show results for a system in which some single particle levels are bunched in the vicinity of the Fermi level.

[1] R. R. Chasman, Phys. Lett. B 524, 81 (2002)
[2] R. R. Chasman, Phys. Lett. B 553, 204 (2003)