Computing ATOMIC NUCLEI

Home , James Rainwater, Nuclear data, Nuclear matter

UNIVERSAL NUCLEAR ENERGY DENSITY FUNCTIONAL Computing ATOMIC NUCLEI

Petascale computing helps disentangle the nuclear puzzle. The goal of the Universal Nuclear Energy Density Functional (UNEDF) collaboration is to provide a comprehensive description of all nuclei and their reactions based on the most accurate knowledge of the nuclear interaction, the most reliable theoretical approaches, and the massive use of computer power.

Science of Nuclei the Hamiltonian matrix. Coupled cluster (CC) Nuclei comprise 99.9% of all baryonic matter in techniques, which were formulated by nuclear sci- the Universe and are the fuel that burns in stars. entists in the 1950s, are essential techniques in The rather complex nature of the nuclear forces chemistry today and have recently been resurgent among protons and neutrons generates a broad in nuclear structure. Quantum Monte Carlo tech- range and diversity in the nuclear phenomena that niques dominate studies of phase transitions in can be observed. As shown during the last decade, spin systems and nuclei. These methods are used developing a comprehensive description of all to understand both the nuclear and electronic nuclei and their reactions requires theoretical and equations of state in condensed systems, and they experimental investigations of rare isotopes with are used to investigate the excitation spectra in unusual neutron-to-proton ratios. These nuclei nuclei, atoms, and molecules. are labeled exotic, or rare, because they are not When applied to systems with many active par- typically found on Earth. They are difficult to pro- ticles, ab initio and configuration interaction duce experimentally because they usually have methods present computational challenges as the extremely short lifetimes. The goal of a compre- configuration space explodes rapidly. Thus other hensive description and reliable modeling of all models are needed in which the most important nuclei—light, heavy, and superheavy—represents degrees of freedom are identified and retained so one of the great intellectual opportunities for that a full treatment of all interactions among the physics in the twenty-first century. active particles can be avoided. This kind of The nuclear many-body problem is of broad approach to many-body quantum physics can be intrinsic interest. The phenomena that arise— found in many other fields, such as condensed shell structure, superfluidity, collective motion, matter physics, atomic and molecular physics, phase transitions—and the connections with and quantum chemistry. Density functional the- many-body symmetries, are also fundamental to ory (DFT), a tool of choice for complex nuclei, is fields such as atomic physics, condensed matter built on theorems showing the existence of uni- physics, and quantum chemistry. Although the versal energy functionals for many-body systems, interactions of nuclear physics differ from the elec- which include, in principle, all many-body corre- The rather complex tromagnetic interactions that dominate chemistry, lations. DFT has been spectacularly successful in nature of the nuclear materials, and biological molecules, the theoreti- condensed matter physics and chemistry, as was forces among protons cal methods and many of the computational tech- recognized by the 1998 Nobel Prize in chemistry, and neutrons generates niques to solve the quantum many-body awarded to Walter Kohn. In fact, it was the com- a broad range and problems are shared (figure 1). All basis expansion bined work of many dedicated researchers that diversity in the nuclear methods—configuration interaction in chemistry, culminated in finding remarkably accurate func- phenomena that can be interacting shell model in nuclear physics—use tionals for use in chemistry. A concerted effort observed. exactly the same technique, that of diagonalizing rooted in a fundamental understanding of inter-

42 S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG U N E D F

C O L L A

Nuclear Landscape B O R A T I O N Ab Initio Configuration Interaction Density Functional Theory

s 50

r P 28 20 82

8 50 2 28 20 2 8

Figure 1. The theoretical methods and computational techniques used to solve the nuclear many-body problem. On this chart of the nuclides in the (N,Z)-plane, the black squares represent stable nuclei and the yellow squares indicate unstable nuclei that have been produced and studied in the laboratory. The many thousands of these unstable nuclei yet to be explored are indicated in green (terra incognita). Except for the lightest nuclei, where it has been reached experimentally, the neutron drip line (the rightmost border of the nuclear landscape) has to be estimated on the basis of nuclear models—hence it is very uncertain due to the dramatic extrapolations involved. The red vertical and horizontal lines show the magic numbers, reflecting regions where nuclei are expected to be more tightly bound and have longer half-lives. The anticipated path of the astrophysical r-process responsible for nucleosynthesis of heavy elements is also shown (purple line). The thick dotted lines indicate domains of major theoretical approaches to the nuclear many-body problem. For the lightest nuclei, ab initio calculations (Green’s function Monte Carlo, no-core shell model, coupled cluster method), based on the bare nucleon–nucleon interaction, are possible (red). Medium-mass nuclei can be treated by configuration interaction techniques (interacting shell model, in green). For heavy nuclei, the density functional theory based on self-consistent/mean field theory (blue) is the tool of choice. By investigating the intersections between these theoretical strategies, one aims at nothing less than developing a unified description of the nucleus. nucleon interactions offers promise to achieve Practical Applications corresponding qualitative improvements in the Applications of nuclear physics in today’s global accuracy and applicability for nuclear physics. economy and national security are numerous. Recognizing that the nucleus is composed of They include the nuclear power industry and fermions, neutrons, and protons, DFT is the only nuclear medicine, as well as national defense. As tractable theory that can be applied across the has been illustrated many times in all fields of sci- Density functional theory entire table of nuclides. The new challenges faced ence, improved understanding of the microworld is built on theorems by the nuclear DFT are the presence of two kinds benefits society. Fusion and fission are excellent showing the existence of of fermions, the essential role of pairing, and the examples. The description of these fundamental universal energy need for symmetry restoration in finite, self- nuclear processes is still very schematic, yet nuclear functionals for many- bound systems. fission powers reactors that produce energy for the body systems.

S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG 43 UNIVERSAL NUCLEAR ENERGY DENSITY FUNCTIONAL A .

T O

V Experimental Evidence that Challenges Theory E Y While only about 300 combinations of protons and neutrons in nuclei are stable enough to exist in nature, several thousand nuclei can be synthe- sized in the laboratory, and even more can be cre- ated in stars. The chart of nuclei (figure 1, p43) shows all possible nuclei in the plane of the neutron number N and the proton (or atomic) number Z. In this landscape, the stable nuclei are bunched along the valley of beta-stability. The ensemble of the heaviest isotopes of each element forms a broken line in the nuclear chart, called the neutron drip line, to the right of beta stability. Universities Atomic nuclei beyond that line are unbound with Laboratories respect to neutron radioactivity. Coulomb repul- sion limits the existence on the proton-rich side Figure 2. The UNEDF collaboration includes researchers from six national laboratories of the nuclear landscape. and eight U.S. universities. These include Ames Laboratory, ANL, LBNL, LLNL, LANL, In 1963 Maria Goeppert-Mayer and J. Hans D. ORNL, Central Michigan University, Iowa State University, Michigan State University, Jensen received the Nobel Prize in physics for Ohio State University, San Diego State Univesity, the University of North Carolina, the explaining why nuclei containing certain num- University of Tennessee–Knoxville, and the University of Washington. bers of protons and neutrons (2, 8, 20, 28, 50, 82, and 126, for example) are extremely stable. These nation, and fusion, which is responsible for energy numbers are called “magic numbers.” This discov- production in stars, has the promise of providing ery had led to an extremely successful tool used a clean alternative source of energy. in describing nuclei called the “shell model.” In There is little question that the nuclear many- this model, the protons and neutrons—collec- body problem has high societal relevance. In the tively called the nucleons—move and mutually area of national defense, for instance, developing interact within a given subset of the shells that are a comprehensive description of nuclei aligns well well separated in energy from other shells in an with the goals of the National Nuclear Security average mean field generated by the remaining Administration (NNSA) Stockpile Stewardship particles. This separation means a nucleus can be Program, which entails an accurate and complete modeled as a core, representing the magic modeling of the behavior and performance of nucleus, and valence nucleons. This simple pic- devices in the nation’s aging nuclear weapons ture works well in many nuclei near the valley of stockpile. Improving the accuracy of that under- stability. standing is central to the continuing process of However, a significant new theme concerns certifying both the safety and the reliability of the shell structure near the particle drip lines and in stockpile without a resumption of nuclear test- the superheavy nuclei. Theoretical predictions ing. In short, understanding of nuclei and their and experimental discoveries in the last decade reactions is critical to providing a more secure indicate that nucleonic shell structure is being rec- homeland. ognized now as a more local concept. The experimental data indicate that the magic numbers in Important Questions Remain neutron-rich nuclei are not the immutable bench- In the last few years, perhaps more than ever, marks they were once thought to be. The magic there has been an especially productive interplay numbers at N=20 and N=28 fade away with neu- between theory and experiment in forging a tron number, and the new magic numbers at deeper understanding of the nuclear quantum N=14, N=16, and N=32 seem to appear. Nuclei far many-body problem. Yet, there are significant from stability have unusual properties as com- components missing from the current under- pared to their stable cousins. Excellent examples standing, and these must be addressed in order to are 6He and 11Li, where a two-neutron halo forms develop a comprehensive predictive theory of the around the 4He and 9Li cores. This radial exten- nucleus and nucleonic matter that will answer a sion of neutron matter in these nuclei is indeed number of fundamental scientific questions. As extreme: the halo radius of 11Li is the same as the discussed in the next section, these questions can radius of 208Pb. New experimental facilities such There is little question only be studied with access to new realms of as the proposed Facility for Rare Isotope Beams that the nuclear many- nuclei—those with proton and neutron numbers (FRIB) in the U.S., as well as present radioactive- body problem has high far different than those of the familiar nuclei beam facilities, will undoubtedly find more of societal relevance. found in nature. these surprising systems.

44 S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG U N E Why is nuclear structure changing in the exotic D F

C O L environment? There are several good reasons for L A B O R this. First, the nuclear mean field is expected to A T I O strongly depend on the orbits being filled. Sec- N ond, many-body correlations, such as supercon- ductivity, involving weakly bound and unbound nucleons become crucial when the nucleonic binding gets small. Third, the nucleus is an open quantum system. The presence of states that are unbound to particle emission may have a significant impact on nuclear properties. All this could have a profound impact on the understanding of element production in the Universe as a number of important nucleosynthesis processes—especially those producing nuclei heavier than iron— occur in very neutron-rich or neutron-deficient nuclei. Nature does not have the luxury of deal- ing only with stable nuclei. A robust nuclear-theoretical capability is required in order to understand stable and exotic nuclei that are the core of matter and the fuel of stars.

The UNEDF Project On the theoretical side, recent developments of powerful conceptual, analytic, algorithmic, and computational tools enable scientists to peer into the inner workings of nuclei with far greater pre- Figure 3. The Universal Nuclear Energy Density Functional collaboration, showing the cision than previously possible. These new tools main participants and the main physics (red font) and computational (blue font) make researchers optimistic that the goal of themes. developing a comprehensive, quantitative, and predictive theory of the nucleus and nucleonic matter is indeed achievable. universities (figure 2). The collaboration also The purpose of SciDAC’s Universal Nuclear involves a number of scientists based in Europe Energy Density Functional (UNEDF) project is to and Japan. Figure 3 shows the main research areas formulate the next generation of nuclear struc- and methods used to achieve the end goal. ture and reaction theory. The mission of the project is threefold: Nuclear Forces and the Energy Density Functional The past few years have witnessed the re-emergence •Find an optimal energy density functional of the more basic approach to understanding using all knowledge of nucleonic, Hamiltonian, nuclei. In ab initio strategy, the basic interactions and basic nuclear properties among protons and neutrons are treated explicitly and the many-body problem is solved with as •Apply DFT and its extensions to validate the few approximations as possible. DFT provides a functional using all the available relevant way to systematically map the many-body prob- nuclear structure data lem onto a one-body problem without explicitly involving inter-nucleon interactions; here the fun- •Apply the validated theory to properties of damental entity is the energy functional that interest that cannot be measured, in particu- depends on one-body densities and currents. The lar the properties needed for reaction theory, following includes descriptions of both the ab ini- such as cross sections relevant to NNSA pro- tio and DFT approaches, with an emphasis on the grams latter because of its broad applicability across the entire table of nuclides. The activities to be supported fall into different The types of computations needed to describe areas of nuclear theory and computer science, but physical phenomena always depend on the Understanding nuclei the goal can only be achieved by working at the energy scale and the number of degrees of free- and their reactions is interfaces among these areas. The collaboration dom of the problem, and the nuclear many-body critical to nuclear power, involves theoretical physicists and computer sci- problem is no exception (figure 4, p46). Quantum nuclear medicine, and entists from six national laboratories and eight chromodynamics (QCD), the theory of strong national security.

S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG 45 UNIVERSAL NUCLEAR ENERGY DENSITY FUNCTIONAL U N E D

F interactions, governs the dynamics and proper-

Degrees of Freedom

C Energy (MeV) O L

L ties of quarks and gluons that form baryons and A B O R

A mesons (“Probing Matter at Subnuclear Scales,” T I O N s SciDAC Review, Summer 2007, p36). Hence, it is n o

r Quarks, Gluons also responsible for the forces that bind nuclei. In d

a this area, significant progress is being made by H

f computing properties of inter-nucleon forces o

s using the effective field theory (EFT), which starts c i

s 940

y from an effective Lagrangian that retains the basic

h Neutron Mass

P symmetries of QCD and is constructed in terms Constituent Quarks of nucleon and pion fields and their derivatives. A power-counting scheme enables one to write down various terms of the nuclear interaction in 140 a systematic way. The unknown low-energy cou- Pion Mass pling strengths that appear in the expansion must Baryons, Mesons be determined from experiment, or eventually from lattice QCD. An important breakthrough in nuclear theory came more than 30 years ago with the realization that three-body forces are an integral part of the 8 Proton Separation Energy in Lead nuclear problem because two-nucleon forces i e

l alone could not account for the binding energy of c u Protons, Neutrons the triton, or alpha particle. Of course, it is a nat- N

f ural idea since nucleons are not point particles. o

s At the time several models of three-nucleon c i

s forces were developed, but the capability to cal- y

h 1.32

P culate the effect of three-body forces in all but the Vibrational State in Tin smallest nuclei did not occur until the last 10 years. In fact, the beauty of EFT has been to sys- Nucleonic Densities tematically define what those forces should look and Currents like. The main ingredient of DFT is the energy den- 0.043 sity functional that depends on densities and cur- Rotational State in Uranium rents representing distributions of nucleonic matter, spins, momentum, and kinetic energy and Collective Coordinates their derivatives (gradient terms). Standard functionals used in nuclear DFT calculations have Figure 4. The basic elements (degrees of freedom) of strongly-interacting matter been parameterized by means of about 10 cou- depend on the energy of the experimental probe and the distance scale. The building pling constants that are adjusted to basic proper- blocks of the theory of strong interactions, quantum chromodynamics (QCD), are quarks ties of nuclear matter (such as saturation density, and gluons. Hadrons (baryons and mesons) can often be described by the dynamics of binding energy per nucleon) and to selected data the effective (or constituent) quarks, with the gluon degrees of freedom being integrated on magic nuclei. The functionals are augmented out. The classical nuclear physics problem is an effective approximation to QCD. It by the pairing term which describes nuclear involves a strongly interacting quantum mechanical system of two fermionic species, superfluidity. When not corrected by additional protons and neutrons. A common starting point for nuclear physics is an inter-nucleon phenomenological terms, standard functionals interaction, represented by a potential or by a set of meson-exchange forces. For complex reproduce total binding energies with a root nuclei, calculations involving all protons and neutrons become prohibitively difficult. mean square error of about 2 MeV; however, they Therefore, a critical challenge is to develop new approaches that identify the important have been successfully tested over the whole degrees of freedom of the nuclear system and are practical in use. Such a strategy is nuclear chart on a broad range of phenomena and similar to what is being used in other fields of science, in particular in condensed matter usually perform quite well when applied to energy physics, atomic and molecular physics, and quantum chemistry. Of particular importance differences, radii, and nuclear moments and defor- is the development of the energy density functional, which may lead to a comprehensive mations. Historically, the first nuclear energy den- description of the properties of both finite nuclei and extended asymmetric nucleonic sity functionals appeared in the context of matter. Here, the main building blocks are the effective fields represented by local proton Hartree–Fock or Hartree–Fock–Bogoliubov and neutron densities and currents. Finally, for certain classes of nuclear models, in methods and zero-range interactions such as the particular those representing emergent many-body phenomena that happen on a much Skyrme force. However, it was realized after- lower energy scale, the effective degrees of freedom are collective coordinates describing wards that—in the spirit of DFT—an effective various vibrations and rotations and the large-amplitude motion as seen in fission. interaction could be secondary to the functional,

46 S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG G R A P that is, it is the density functional that defines the H :

U N force. This is the strategy that the UNEDF collab- E D F

C oration will follow. O L L A B O

The density and gradient dependences of the R A T I O functional are poorly known. To make progress, N P H O T

a concerted effort will be required to study the O :

O R

nuclear DFT when applied to finite nuclei and N bulk nuclear matter. A promising approach to a L systematic density expansion, rooted in fundamental inter-nucleon forces, is given by EFT supplemented by renormalization group methods. By evolving EFT interactions to lower momentum, they are softened to the extent that con- structing a microscopic energy density functional based on many-body perturbation theory becomes feasible. By varying the cutoff, this approach allows theoretical error estimates, which may be coupled to the optimization of parameters from experiment through efficient global fits, with systematic error and covariance analysis.

Computing Nuclei in Ab Initio and Configuration Interaction Approaches Given a realistic nuclear interaction, the next step is to solve the quantum many-body problem for Figure 5. ORNL’s Jaguar, the world’s second fastest computer, enables certain a given nucleus with that interaction. Several ab nuclear calculations only dreamt of a few years ago. As an example, Jaguar was used initio techniques are being used today to calculate for the first ever ab initio computation of neutron-rich helium nuclei using coupled nuclear properties directly, each of which has cluster theory (shown in the figure on the side of the computer). The figure shows the advantages and disadvantages. The Green’s func- binding energy of these nuclei, while the inset indicates the width, related to lifetime. tion Monte Carlo (GFMC) technique has been vig- 12 Experimental data are marked in red. The calculated masses show a systematic orously pursued in light nuclei (up to C) since deviation from experiment; this can be attributed to a three-body force, missing in the the mid-1990s and demonstrates that one can calculation. build nuclei from scratch. The GFMC is naturally parallel and requires specialized load-balancing algorithms to efficiently scale to thousands of lating the Nuclear Mass Table on Jaguar,” p49). The main ingredient of processors. CC theory requires the solution of coupled non- DFT is the energy density Another ab initio approach, the no-core shell linear algebraic equations and fast matrix-matrix functional that depends model (NCSM), involves diagonalization of the multiplies. on densities and currents representing distributions nuclear Hamiltonian in a basis of independent- Given the dramatic rise in computing power, of nucleonic matter, particle states. Parallel Krylov techniques are used predicted to reach petaflop-scale within three spins, momentum, and to find the lowest energy levels and wave func- years and moving toward exascale computing, kinetic energy and their tions in these computations. The methods used researchers are likely to soon have the computa- derivatives (gradient rely on sparse algorithms as well. Both the GFMC tional power to pursue ab initio calculations using terms). and NCSM approaches scale exponentially with CC techniques in very massive nuclei. A petas- the number of nucleons; a recent variation of cale coupled-cluster calculation will probably GFMC, auxiliary field diffusion Monte Carlo involve 100 nucleons in 1,000 orbitals. This is (AFDMC), has only polynomial scaling. The CC within the realm of the possible if algorithms can method represents yet another approach to the be generated that will scale to enormous num- nuclear problem. This technique is particularly bers of compute cores. Efforts in this direction appropriate for closed-shell or sub-shell nuclei. It are underway. This same computational power has the advantage that it scales very gently (poly- should enable ab initio calculations using either nomially) with increasing numbers of nucleons. GFMC (or its derivatives) or NCSM into the mass A complex-energy version of the CC theory, par- 20–40 regions. Because of the complementary ticularly useful for description of open exotic nature of the methods, it will be important that nuclei, was recently developed to calculate widths all three methods advance to take advantage of of states in the helium isotopic chain and is being petascale computing and beyond (sidebar “Com- run on ORNL’s Jaguar (figure 5; sidebar “Calcu- putational Scaling of Ab Initio Techniques,” p50).

S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG 47 UNIVERSAL NUCLEAR ENERGY DENSITY FUNCTIONAL Nuclear DFT: 2007 20 SLy4 exp U U N N E th E Nuclear DFT: 2007 D D F F

C 124 20 C O 18 O L D1S exp L L SLy4 L A 126 A B 72 10 88 B O 128Monte Carlo Kr th O R Ru R A 26 A T 10 T I I O O N 16 122 18 124 N D1S 118 126 10 116 120 fp-gds 128 11422 14 n 10 1 122 o 16 i 112 s 118 n 120 e 110 fp-g 116 y m

18 r ) 12 i 10 9/2 114

Hs o V 1 D 14 e e Sg e h 112 M T c 60 0.1 ( Rf Zn

a 68 α No p Se 110

10 y

S 14 Q r ) Fm 10 12

Hs o n V

Cf e o e

i Sg t 56 h M T

a Ni 0.1 ( r Rf 8 2007 u 10 0.01 α No g 10 i 10 f Q Fm n 48 + + 2 2 o Cr 1994 B(E2, 0 2 ) Cf(e b ) 6 C 6 8 10 fp 0.01 0.1 110 0.01 4 44 Cm Configuration Interaction Experiment + + 2 2 Ti Pu (Interacting Shell Model) B(E2, 0 2 ) (e b ) 102 6 144 164 184 20 22 24 26 28 30 32 34 36 38 40 42 44 46 0.01 0.1 110 Neutron Number 4 N = Z Cm Experiment Pu Figure 6. Configuration space dimension of the interacting shell model for fp-shell nuclei. 144 164 184 U

N Neutron Number E D F

C O L

L Figure 8. An example of large-scale systematic DFT

A SkP

B 100 O R

A calculations for complex nuclei produced by the UNEDF T I O N collaboration. Alpha-decay energies (Q values) for even- α 80 even heavy and superheavy nuclei with 96 ≤ Z ≤ 118

r calculated with the energySkP density functional SLy4. They e 100 b

m 60 are compared to experimental data (closed symbols).

u N = Z N

n 0 o 80 t o

r 40 P

N=Z=20, initiallyr calculated in the 1980s), and fp- e

10 shell nuclei (rangingb from N=Z=20 to N=Z=40,

m 60 20 Two-Neutron fully investigatedu in the 1990s–2000s).N = Z The next N Separation Energy N = 2Z frontier will n be nuclei in the gds-shell region. 0 30 o (MeV) t o 0 While the computationalr 40 capability of the shell 0 20 40 60 80 100 120 140 160 180 P model has been the reason for this changing 10 Neutron Number emphasis across20 shells, it is clear that either significant approximations or technology break- Two-Neutron Figure 7. An example of large-scale systematic density functional theory (DFT) N = 2Z Separation Energy 30 calculations for complex nuclei produced by the UNEDF collaboration. Results of the throughs must occur in order to tackle the next (MeV) shell because when0 going to heavier systems with deformed DFT calculations of two-neutron separation energies for 1,553 particle-bound 0 20 40 60 80 100 120 140 160 180 even–even nuclei with Z ≤ 108 and N ≤ 188. many active nucleons, the configuration space explodes rapidly, resulting in combinatorialNeutron Number growth in the complexity of calculations (figure 6). It is expected that efforts with NCSM will DFT equations present a nonlinear eigenvalue deliver benefits to standard shell model compu- problem that must be solved iteratively. New tations as both methods rely on Krylov tech- wavelet expansion techniques are being developed niques to diagonalize the Hamiltonian matrix. to achieve better accuracy for the nuclear problem. Auxiliary field Monte Carlo methods which have The ongoing work with configuration interac- been developed and run on Jaguar demonstrate tion techniques is also of note. The interacting an alternative to diagonalization that holds sig- shell model, in which the configuration space is nificant promise for certain aspects of the physics truncated by involving valence nucleons only, can of medium mass and heavier nuclei, where the be used to make detailed studies of nuclear struc- dimensions of the model space reach 1,030. ture in small regions of the nuclear chart. The method was applied to p-shell nuclei (ranging Nuclear Energy Density Functional Theory from N=Z=2 to N=Z=8, initially calculated in the Because nuclei are self-bound objects, they pro- 1960s), sd-shell nuclei (ranging from N=Z=8 to duce their own confining potential, or mean field.

48 S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG Calculating the Nuclear Mass Table on Jaguar

To more precisely determine the nuclear nucleus in the case of a spherical shape, on and among the tagged nuclei are some with energy density functional, one has to solve another node the same properties in the case unphysical neutron or proton numbers. nuclear density functional theory (DFT) of a football-like shape, and on a third node Next, the process moves on to the equations for all nuclei in the chart of nuclei. deal with the case of a disc-like shape. This computation of nuclear properties, going from DFT models estimate the number of particle- can be done for all bound nuclei at the same the highest to the lowest priority. The codes bound nuclei to be about 6,000. time, thus making it possible to calculate are designed in such a way that several Sophisticated computer codes are needed to various nuclear properties across the chart of hundred nuclei can be processed solve the equations (a huge nonlinear the nuclides. A quick turnaround is essential simultaneously, taking advantage of the eigenvalue problem in three dimensions) and for benchmarking and optimizing the functional parallel architecture of Jaguar. When one obtain even very basic quantities like the by confronting calculated nuclear properties nucleus is done, the researchers change its atomic mass or nuclear radius. On a modern with experimental data. tag. When the first unbound nucleus in an workstation, the determination of the This is not as easy as it sounds because isotopic chain is encountered, that signals that properties of one nucleus using standard DFT researchers do not know the exact position of the neutron drip line has been reached: thus codes can easily take up to a few hours. the border between bound and unbound nuclei all isotopes beyond that point can be tagged Obviously, it is not realistic to study the whole (a drip line) for a given functional. To use with the completion tag and ignored. For nuclear chart in this way. On top of that, not Jaguar efficiently and not waste its resources even–even nuclei, the entire mass table can only are the ground-state properties of nuclei computing nonexistent unbound nuclei having be completed in less than a day, and of interest, but also their excited states, often the positive chemical potential, the UNEDF scientists obtain at the end the position of the characterized by different intrinsic shapes. collaboration designed a technique that allows neutron and proton drip lines, all atomic How can scientists get all this information computing only those nuclei that are stable, masses, one- and two-particle separation for several thousand nuclei in a reasonable without actually knowing which ones are energies, charge radii, deformation properties, timescale? This is where the large-scale stable. For the current algorithm, it begins with and so on. For odd-mass (and odd–odd) computing facilities provided by the an estimate for the valley of stability. To each nuclei, the calculation scheme is more Department of Energy Office of Science’s nucleus in this valley a certain priority flag is complicated because a number of competing Advanced Scientific Computing Research assigned. The closer the nucleus is to the low-energy configurations must be considered program come into play. For example, with valley of stability, the higher the priority. All and the size of the problem increases due to

ORNL’s Jaguar supercomputer, researchers can nuclei are tagged with Z < Zmax and N < Nmax. the absence of self-consistent symmetries, compute on one node the properties of one At this point, no calculation has been made, such as time reversal. Nuclear DFT: 2007 20 exp

SLy4 U N th E D DFT provides the rigorous theoretical foundation F

124 O

18 L D1S

L for a self-consistent description of the nucleus in

126 A B 10 O

128 R

A terms of one-body densities and currents that T I O

122 N 16 118 build the mean field. As discussed previously, the 116 120 challenge consists in relating the nuclear DFT to 114 14 1 the approaches based on the realistic inter- 112 110 nucleon interactions and/or directly to low- y r ) 12 energy QCD derivations. The hope for the longer

Hs o V e e Sg h

M term is to find a universal functional that would T 0.1 ( Rf

α 10 No cover the entire chart of nuclei. Q Fm Cf The nuclear DFT efforts have been quite suc- 8 cessful in describing a wide variety of nuclear data 0.01 with very good precision across the nuclear chart B(E2, 0+ 2+) (e2b2) 6 (figures 7, 8, and 9). The various parameteriza- tions usually work quite well in regions where 0.01 0.1 110 4 nuclear masses and other properties are experi- Cm Experiment Pu mentally determined, but extrapolations into very Comparison between experimental and 144 164 184 Figure 9. neutron-rich nuclei have been problematic. The theoretical excitation energies of the lowest 2+ states in Neutron Number next-generation facilities should enable theorists 519 even–even nuclei. Calculations are based on a to obtain a functional parameterization that will The hope for the longer microscopic collective Hamiltonian in five dimensions in describe bulk properties of all nuclei. Various term is to find a universal which the potential energy and the tensor of inertia are nuclear data along long isotopic and isotonic functional that would obtained from constrained triaxial DFT using the Gogny chains are needed to constrain the isovector part cover the entire chart of D1S functional. of the energy functional. More specifically, one nuclei. 100 SkP

S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG 80 49 r e b

m 60 u N = Z N

n 0 o t o r 40 P 10 20 Two-Neutron N = 2Z Separation Energy (MeV) 30 0 0 20 40 60 80 100 120 140 160 180 Neutron Number UNIVERSAL NUCLEAR ENERGY DENSITY FUNCTIONAL

Computational Scaling of Ab Initio Techniques

The Green’s function Monte Carlo (GFMC) ab initio approach. It uses a basis expansion used in nuclear physics. It is able to precisely approach projects out the many-body wave technique. The underlying single-particle states calculate ground-state properties of closed- function in coordinate, isospin, and spin space represent the mean field of the nucleus and are shell (or sub-shell) nuclei and their neighbors. for a given interaction. This method was the typically taken as harmonic oscillator states. The Recent calculations in 16O and 40Ca indicate an first to use modern two- and three-nucleon method begins by calculating an effective error of 1% for the ground-state energies in interactions to solve nuclear problems. The interaction in the given basis states using a these nuclei. The calculations were performed technique uses Monte Carlo sampling of high- renormalization group approach. The many-body on Jaguar using up to 1,000 single-particle dimensional integrals to obtain observables. Hamiltonian can then be constructed within the basis states and incorporating triples Memory requirements scale like the number of basis of Slater determinants and diagonalized to corrections to the standard CCSD (coupled pairs of particles, A(A–1)/2, times the number obtain ground-state and excitation energies and cluster with single and double excitations) of spin-isospin states in the nucleus. These other observables. The method uses Krylov calculations. Nuclear CC methods have numbers range from 32 in 4He to 540,000 in space techniques to obtain spectral information. recently been extended to work with three- 12C, and 93 million in 16O. Computation time The calculation of 10 converged energy levels body Hamiltonians, and first calculations were also scales exponentially. Calculations for 12C within a given space requires approximately 100 performed in 4He. The CC methods solve are 530 times more expensive than those for iterations. The effective dimension of the coupled nonlinear sets of equations. The 8Be. The method uses local potentials, with the Hamiltonian matrices that this method can largest calculation to date involved roughly 10 best results being obtained with a particular currently reach is O(108). Recent runs on Jaguar million unknowns. These calculations scale as two-body force called AV18 and a three-body using 4,000 processors and with an effective a polynomial in the number of particles and in potential called Illinois-2. Together, these dimension of 156 million produced results for the number of single-particle states. Overall 6 2 4 3 5 potentials yield a root mean square error of He with a 2% estimated error in the final result scaling is O(no nu ) for CCSD and O(no nu ) for 750 keV for all levels calculated up to the due to model space restrictions for the ground CCSDT (coupled cluster with single, double,

A=10 systems. Calculations of excited states state. The many-body space in NCSM grows and triple excitations) where no is the number in 12C will require several million hours of Blue exponentially with the number of nucleons. of occupied orbitals (the number of particles)

Gene/P-equivalent time in the coming year. The coupled cluster (CC) method is a recent and nu is the number of unoccupied orbitals in The no-core shell model (NCSM) is another addition to the family of ab initio techniques the basis space.

needs (difference of) masses and measures of col- Reliable extrapolation is possible only with the lectivity and of the shell evolution in unknown establishment of theoretical uncertainties. Con- regions, where predictions of currently used func- sequently, construction of new energy density tionals disagree. Data on large deformations (at functionals should be supplemented by sensitiv- low and high angular momentum) and multipole ity analysis. It is not sufficient to predict proper- strength distributions in neutron-rich nuclei will ties of exotic nuclei by extrapolating properties also be extremely valuable. All these data will be of those measured in experiment. The UNEDF used to determine the coupling constants charac- collaboration must also quantitatively determine terizing the functional (a many-dimensional opti- errors related to such extrapolations. Moreover, mization problem). for experimental work it is essential that an At the heart of DFT lies the correlation energy improvement gained by measuring one or two that is rooted in quantum effects and symmetry more isotopes be quantitatively known. From a breaking. The 1975 Nobel Prize in physics was theoretical perspective, scientists must know the awarded to Aage Niels Bohr, Ben Roy Mottelson, confidence level with which the parameters of the and Leo James Rainwater for showing that a large functional are determined. part of those correlations can be included by con- Defining a universal energy density functional sidering symmetry-breaking, independent-par- requires computations of ground-state energies ticle states. However, for finite systems, and other observable properties across the chart quantitative description often requires symmetry of nuclei. This effort also requires the extensive Defining a universal restoration. For this purpose, one can apply a vari- use of computation in order to calculate the prop- energy density functional ety of techniques, such as projection methods and erties of several thousand nuclei numerous times. requires computations of the generator coordinate method. Ideally, approx- Once the next-generation functional has been ground-state energies imations would be worked out that would allow obtained, further refinements will be necessary. and other observable avoiding full-scale collective calculations, but DFT breaks laboratory-system symmetries such properties across the would be based on calculations performed on the as particle number, parity, and angular momen- chart of nuclei. top of self-consistent mean fields. tum. These symmetries should be restored in

50 S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG U N

E The advent of D F

O terascale—and very L L A B

O shortly petascale— R A T I O computing platforms N is paving the way for today’s progress in theoretical studies of the physics of nuclei.

Figure 10. The figure shows the energy surface of the transuranic element 258Fm calculated within the self- consistent nuclear density functional theory with the SkM* functional as a function of two collective variables: the

total quadrupole moment Q20 representing the elongation of nuclear shape, and the total octupole moment Q30 representing the left–right shape asymmetry. Indicated are the two static fission valleys: asymmetric path aEF leading to asymmetric mass split of fission fragments, and symmetric-compact path sCF corresponding to a division into nearly spherical fragments. Experimentally, a transition is observed from an asymmetric distribution of mass splits in neutron-deficient fermium isotopes to a more symmetric distribution when getting closer to 264Fm. The density functional theory calculations explain this phenomenon in terms of shell effects in the emerging fission fragments approaching the doubly-magic 132Sn nuclei. In calculations, all possible nuclear shapes, including triaxial and reflection-asymmetric (pear-like) shapes, are allowed.

order to compare theoretical results to observed power available at this moment in time. Until nuclear excitation spectra. Projection techniques, recently, such an undertaking was hard to imag- performed self-consistently, will require about ine, and even at the present time such an ambi- 1,000 computational cycles for each nucleus. The tious endeavor would be far beyond what a single modern energy density functional should also researcher or a traditional research group could lead to a microscopic theory of nuclear reactions carry out. But the prospects look good: the and fission (figure 10). Here, the challenges UNEDF collaboration is witnessing breakthrough include the proper treatment of the resonant and calculations of nuclear properties that the previ- non-resonant continuum, development of the ous two generations of scientists had only begun microscopic optical potential and theory for indi- to dream about. ● rect reactions, and description of the large-amplitude nuclear collective motion. Contributors: For the UNEDF collaboration: Dr. George F. Bertsch, University of Washington; Dr. David J. Dean, Outlook ORNL; and Dr. Witold Nazarewicz, University of Tennessee The advent of terascale—and very shortly petas- and ORNL cale—computing platforms is paving the way for today’s progress in theoretical studies of the Further Reading RIA Theory Bluebook: A Road Map physics of nuclei. The UNEDF collaboration has www.orau.org/ria/RIATG/Blue_Book_FINAL.pdf planned a comprehensive study of all nuclei based on the most accurate knowledge of the strong Scientific Opportunities with a Rare-Isotope Facility in the nuclear interaction, the most reliable theoretical United States approaches, and the massive use of the computer www.nap.edu/catalog.php?record_id=11796

S CIDAC REVIEW W INTER 2007 WWW. SCIDACREVIEW. ORG 51