Electromagnetic Excitations and Responses in Nuclei from First Principles
R. B. Baker, K. D. Launey, N. Nevo Dinur, S. Bacca, J. P. Draayer, T. Dytrych
EElectromagnetic Excitations and Responses in Nuclei fromFirst Principles
R. B. Baker , K. D. Launey , N. Nevo Dinur , S. Bacca , J. P. Draayer andT. Dytrych Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada Institut f¨ur Kernphysik and PRISMA Cluster of Excellence, Johannes Gutenberg-Universit¨at Mainz, 55128 Mainz,Germany Department of Physics and Astronomy, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada Nuclear Physics Institute, Academy of Sciences of the Czech Republic, 250 68 ˘Re˘z, Czech Republic a) Corresponding author: [email protected]
Abstract.
We discuss the role of clustering on monopole, dipole, and quadrupole excitations in nuclei in the framework of the ab initio symmetry-adapted no-core shell model (SA-NCSM). The SA-NCSM starts from nucleon-nucleon potentials and, by ex-ploring symmetries known to dominate the nuclear dynamics, can reach nuclei up through the calcium region by accommodatingultra-large model spaces critical to descriptions of clustering and collectivity. The results are based on calculations of electromag-netic sum rules and discretized responses using the Lanczos algorithm, that can be used to determine response functions, and for He are benchmarked against exact solutions of the hyperspherical harmonics method. In particular, we focus on He, Be, and Oisotopes, including giant resonances and monopole sum rules.
INTRODUCTION
Nuclear response functions can provide valuable information about clustering, particularly by elucidating giant andpygmy resonances. For many nuclei, excitation strengths (e.g. monopole, dipole, etc.) can become fragmented, re-sulting in multiple peaks in the response function. These have been the focus of experiments for some time, e.g. see[1, 2, 3, 4], and they have provided ample opportunity for various theoretical approaches to successfully study theunderlying cluster structure, e.g. see [5, 6, 7, 8, 9, 10, 11, 12]. Here, we seek to determine if these features also emergefrom first principles. To achieve this, we utilize the ab initio symmetry-adapted no-core shell model [13, 14], whichprovides solutions in terms of a manageable number of collective and physically relevant basis states. Furthermore, inthis framework, we are able to determine the individual contribution of each intrinsic deformation in these resonances.
THEORETICAL FRAMEWORKSymmetry-Adapted No-Core Shell Model
The symmetry-adapted no-core shell model (SA-NCSM) utilizes emergent symmetries in nuclear physics to accountfor important collective correlations in nuclei, which allows one to reduce the computational complexity of nuclearstructure calculations and, hence, reach model spaces that are currently unfeasible. It uses a collective basis describ-ing deformation, plus rotations and vibrations thereof. Each basis state in this scheme is labeled schematically as | (cid:126)γ N ( λ µ ) κ L ; ( S p S n ) S ; JM (cid:105) . The SU(3) quantum numbers ( λ µ ) are associated with deformation, where, e.g. (0 0),( λ µ ) describe spherical, prolate, and oblate shapes, respectively, N is the total number of harmonic os-cillator (HO) excitation quanta, and L is the orbital angular momentum ( κ is multiplicity). The additional quantumnumbers (cid:126)γ are needed to distinguish among configurations carrying the same N ( λ µ ) and ( S p S n ) S labels. In this way, a a r X i v : . [ nu c l - t h ] D ec omplete shell-model basis is classified [13]. In the SA-NCSM, one can down-select the model space to only the phys-ically relevant basis states, and typical SA-NCSM model spaces include the complete model space (all basis states)up to N max = N max (the SA-NCSM model spaces are hencedenoted as (cid:104) (cid:105) N max or (cid:104) (cid:105) N max ). Here, we employ the SA-NCSM in an SU(3)-coupled basis to examine light nucleiup to oxygen isotopes. Using a realistic nucleon-nucleon NN potential, we first calculate the many-body Hamiltonianand then we find the eigenenergies and many-body wave functions via the Lanczos algorithm. An important feature ofthe SA-NCSM is that the center-of-mass (CM) motion can be factored out exactly [14]. This ensures the translationalinvariance of the SA-NCSM wave functions. Lorentz Integral Transform Method
To study the nuclear response of an external probe, we use the Lorentz integral transform (LIT) method [15, 16]. Anelectromagnetic response function is defined as R ( ω ) = (cid:88)(cid:90) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:68) ψ f (cid:12)(cid:12)(cid:12) ˆ O (cid:12)(cid:12)(cid:12) ψ (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) δ ( E f − E − ω ) , (1)where the sum runs over discrete and continuum final states of energy E f , ˆ O is the excitation operator associatedwith the probe, ω is the energy transferred by the external probe, | ψ (cid:105) and (cid:12)(cid:12)(cid:12) ψ f (cid:69) are the ground and excited states,respectively, and E is the energy of the ground state [16]. Given that the response function requires information aboutthe ground state and all the excited states the excitation operator can connect to, direct calculations of R ( ω ) can bechallenging. The Lorentz Integral Transform (LIT) method instead considers the quantity L ( σ, Γ ) = Γ π (cid:90) d ω R ( ω )( ω − σ ) + Γ , (2)which smears the response function with a Lorentzian peaked at σ that has width Γ . Inserting the response functionas defined above and invoking the closure relation, we can write L ( σ, Γ ) in terms of the solution to a Schr¨odinger-likeequation that is much simpler to solve. Typically, in the LIT method, one calculates the transform in Eq. (2) at large Γ values until convergence is reached in the model space expansion and then one inverts it to obtain the responsefunction. However, in the limit Γ →
0, the Lorentzian becomes a delta function L ( σ, Γ → = (cid:90) d ω R ( ω ) δ ( ω − σ ) = R ( σ ) , (3)and thus for small Γ , we obtain the discretized response function [16, 17]. While the discretized response function iscalculated with bound-state boundary conditions, it is useful to look at it if one wants to study where the discretizedexcited states are located in a given model space. When we calculate the discretized response function for small Γ andwe do not perform an inversion, we call this approach the “Lanczos response method” [16].In this study, we focus on several moments of the response function, also called sum rules, I n = (cid:90) d ω ω n R ( ω ) , (4)which, again with the help of the closure relation, one can rewrite as I n = (cid:68) ψ (cid:12)(cid:12)(cid:12) ˆ O † ( H − E ) n ˆ O (cid:12)(cid:12)(cid:12) ψ (cid:69) . (5)This means that such calculations depend on the given many-body Hamiltonian H , along with its ground state energyand wave function, and the operator ˆ O . Namely, it is a ground state expectation value of a new operator. Thus, the useof a bound-state basis is justified for its computation. Merging SA-NCSM and Lanczos response method
Our procedure for calculating the response function with ab initio
SA-NCSM wave functions involves the followingsteps. First, we employ the ab initio
SA-NCSM, with a given realistic interaction, to find the ground state waveunction for the nucleus of interest. The SA-NCSM results, in particular the binding energy and rms radius of theground state, are ensured to converge with respect to the maximum number of HO excitation quanta, N max . For achosen excitation operator ˆ O , we then construct a normalized pivot vector | v (cid:105) , given by | v (cid:105) = ˆ O | ψ (cid:105) (cid:113)(cid:68) ψ (cid:12)(cid:12)(cid:12) ˆ O † ˆ O (cid:12)(cid:12)(cid:12) ψ (cid:69) , (6)where for ˆ O we use the isoscalar monopole, isovector dipole, and isoscalar quadrupole operators defined in the usualways [5, 6, 18]. The square of the quantity in the denominator is referred to as the non-energy weighted sum rule andis an important probe of the response function, as it represents the total transition strength from the ground state to allpossible excited states.From here, we use the same many-body Hamiltonian as the one used in the SA-NCSM calculations describedabove, and initiate the Lanczos algorithm with the pivot vector in Eq. (6) as our starting vector. The resulting Lanczoscoe ffi cients, i.e. the matrix elements of the tridiagonal matrix the Lanczos algorithm constructs, can then be used tocalculate the transform L ( σ, Γ ) and sum rules, as described in Ref. [16, 19].This combined approach has two notable advantages: 1) By utilizing the SA-NCSM’s ability to select only phys-ically relevant basis states, we can perform larger calculations than the traditional NCSM and thus include importantcontributions from higher harmonic oscillator shells and reach heavier nuclei. 2) Within the SA-NCSM framework,we can decompose the wave function and examine the contribution of each individual basis state and its associatedintrinsic deformation. This ability carries over to the response, where we can examine the contribution of each in-dividual deformation to peaks in the response and identify emergent patterns. While other NCSM approaches havetruncation techniques to accommodate large model spaces, this latter ability is novel. We also note that the SA-NCSMis not only a truncation scheme but a unique ab initio approach that reproduces nuclear collectivity up through the Caregion without e ff ective charges [20]. RESULTSBenchmark results for He As a benchmark study, we calculate the ground state wave function for He from SA-NCSM compared to hyperspher-ical harmonics (HH), which is an exact method. The HH truncates its model space in terms of K max , the hypersphericalmomentum, and is generally limited to smaller systems due to the use of Jacobi coordinates [16, 21]. Figure 1 showscomparisons between the ground state properties of He, specifically its energy and rms radius, as calculated by HH,conventional NCSM or equivalently, the complete-space SA-NCSM, and SA-NCSM with SU(3) selection as describedin Ref. [13], using the JISP16 realistic interaction. As can be seen there, all three methods are in good agreement witheach other. Further, Fig. 2 shows calculations for the non-energy weighted sum rule, i.e. (cid:68) ψ (cid:12)(cid:12)(cid:12) ˆ O † ˆ O (cid:12)(cid:12)(cid:12) ψ (cid:69) , for both themonopole and quadrupole operator. Again, all three methods are in good agreement. -29 -28 -27 -26 -25 -24 0 2 4 6 8 10 12 14 16 18 20 22 24 E ne r g y ( M e V ) K max / N max Bare HH, JISP16 NCSM, JISP16, ħΩ =25 SA-NCSM, JISP16, ħΩ =25 r m s r ad i u s ( f m ) K max / N max Bare HH, JISP16 NCSM, JISP16, ħΩ =25 SA-NCSM, JISP16, ħΩ =25 FIGURE 1. (Left) Ground state energy and (right) rms radius of He as a function of K max or N max for bare HH, conventionalNCSM, and SA-NCSM using the JISP16 interaction. IGURE 2.
Elastic and total monopole ( L =
0) and quadrupole ( L =
2) non-energy weighted sum rule (NEWSR) for He as afunction of K max or N max for bare HH, conventional NCSM, and SA-NCSM using the JISP16 interaction. Response functions for O Using the SA-NCSM, we can apply the Lanczos response method to light and medium-mass nuclei. Figure 3 showsthe response functions with SA-NCSM wave functions for O with isoscalar monopole and quadrupole excitationsusing the JISP16 interaction and
Γ = λ µ ) = (2 0) and (4 2), and the main contributions ( ∼ ∼ Ois dominated by a spherical configuration in the SA-NCSM, (0 0), an excited 0 + (2 + ) state that is strongly connectedto the ground state by a monopole (quadrupole) transition is expected to be dominated by (2 0) deformation [13].Indeed, the operator known to generate 1p-1h giant monopole (quadrupole) excitations is exactly of (2 0) SU(3) rank[18]. Similarly, the (4 2), spin-0 configuration is a deformed 2p-2h configuration known to dominate the third 0 + in O [22].
FIGURE 3. (Left) Monopole and (right) quadrupole response for O for a (cid:104) (cid:105) Γ = (cid:126)
Ω =
15 MeV. Note that both plots qualitatively agree with the results from the softer interaction usedin Ref. [23].
Moreover, as shown by Suzuki [24], the (0 0), (2 0), and (4 2) components in O have a very high overlap with acluster wave function of C and an alpha particle in their ground state with an associated relative motion. For furtherdiscussion, see Ref. [20]. pen-shell nucleus: Be As an example of an open-shell nucleus within the reach of this approach, Fig. 4a shows the isovector dipole responsefor Be. The ground state of Be is dominated by the configurations (2 2) and (3 0), and in the dipole response wefind that the two largest peaks are largely composed of (3 2), with the first peak being more than 35% comprised of(3 2), spin-0 and the second peak being more than 30% a combination of (3 2), spin-0 and spin-1.
FIGURE 4. (Left) Dipole response and (right) energy-weighted dipole response for Be for a (cid:104) (cid:105)
10 model space in the SA-NCSMusing the JISP16 interaction,
Γ = (cid:126)
Ω =
20 MeV.
Additionally, the results in Fig. 4b compare favorably, qualitatively, with the results for isovector dipole exci-tations in Be shown in Ref. [5]. Similar to these earlier results, the energy-weighted E − state to this response function. Asystematic study of the underlying physics as emerging within the SA-NCSM framework and detailed comparisonto results of Ref. [5] can provide further insight into the role of clustering and collectivity, and is the focus of ournear-future work. CONCLUSIONS
We have produced a new first-principle approach to study the underlying shapes and dynamics inherent in nuclearresponses. This technique combines the ab initio
SA-NCSM with the Lanczos response method, which together allowsus to examine the peaks in the response function in terms of individual deformation contributions. This new approachhas been benchmarked against the hyperspherical harmonics method for He and preliminary results for O and Beshow qualitative agreement with existing literature. Additionally, analysis of the O results in this framework showedcharacteristic signs of giant monopole and quadrupole resonances. Future studies will work toward identifying thesegiant resonance characteristics from first principles in nuclei up through the Ca region.
ACKNOWLEDGMENTS
This work was supported in part by the U.S. NSF (OIA-1738287, ACI-1713690), SURA, and the Czech SF (16-16772S), and benefitted from high performance computational resources provided by LSU ( ) andBlue Waters; the Blue Waters sustained-petascale computing project is supported by the National Science Foundation(awards OCI-0725070 and ACI-1238993) and the state of Illinois, and is a joint e ff ort of the University of Illinoisat Urbana-Champaign and its National Center for Supercomputing Applications. A portion of the computationalresources were provided by the National Energy Research Scientific Computing Center and by an INCITE awardfrom the DOE O ffi ce of Advanced Scientific Computing. Additional support was provided in part by the NaturalSciences and Engineering Research Council (NSERC), the National Research Council of Canada, by the DeutscheForschungsgemeinschaft DFG through the Collaborative Research Center [The Low-Energy Frontier of the StandardModel (SFB 1044)], and through the Cluster of Excellence [Precision Physics, Fundamental Interactions and Structureof Matter (PRISMA)]. T. D. acknowledges support from Michal Pajr and CQK Holding. EFERENCES [1] D. H. Youngblood, C. M. Rozsa, J. M. Moss, D. R. Brown, and J. D. Bronson, Phys. Rev. Lett. , 1188–1191 (1977).[2] D. H. Youngblood, Y.-W. Lui, and H. L. Clark, Phys. Rev. C , 2748–2751 (1998).[3] Y.-W. Lui, H. L. Clark, and D. H. Youngblood, Phys. Rev. C , p. 064308 (2001).[4] Y. Gupta, U. Garg, J. Matta, D. Patel, T. Peach, J. Ho ff man, K. Yoshida, M. Itoh, M. Fujiwara, K. Hara,H. Hashimoto, K. Nakanishi, M. Yosoi, H. Sakaguchi, S. Terashima, S. Kishi, T. Murakami, M. Uchida,Y. Yasuda, H. Akimune, T. Kawabata, and M. Harakeh, Phys. Lett. B , 343–346 (2015).[5] Y. Kanada-En’yo, Phys. Rev. C , p. 024322 (2016).[6] Y. Kanada-En’yo, Phys. Rev. C , p. 054307 (2016).[7] T. Yamada, Y. Funaki, T. Myo, H. Horiuchi, K. Ikeda, G. R¨opke, P. Schuck, and A. Tohsaki, Phys. Rev. C , p. 034315 (2012).[8] T. Kawabata, H. Akimune, H. Fujita, Y. Fujita, M. Fujiwara, K. Hara, K. Hatanaka, M. Itoh, Y. Kanada-En’yo, S. Kishi, K. Nakanishi, H. Sakaguchi, Y. Shimbara, A. Tamii, S. Terashima, M. Uchida, T. Wakasa,Y. Yasuda, H. Yoshida, and M. Yosoi, Phys. Lett. B , 6 – 11 (2007).[9] Y. Chiba, M. Kimura, and Y. Taniguchi, Phys. Rev. C , p. 034319 (2016).[10] M. Kimura, “Cluster states in stable and unstable nuclei,” in Progress of time-dependent nuclear reactiontheory (Bentham Science Publishers, 2016) to be published; arXiv:1612.02086 [nucl-th].[11] M. Kimura, Phys. Rev. C , p. 044319 (2004).[12] Y. Suzuki and W. Horiuchi, “Clustering in light nuclei with the correlated gaussian approach,” in EmergentPhenomena in Atomic Nuclei from Large-Scale Modeling: A Symmetry-Guided Perspective , edited by K. D.Launey (World Scientific Publishing Co., 2017) Ch. 7.[13] K. D. Launey, T. Dytrych, and J. P. Draayer, Prog. in Part. and Nucl. Phys. , 101–136 (2016).[14] T. Dytrych, K. D. Launey, J. P. Draayer, P. Maris, J. P. Vary, E. Saule, U. Catalyurek, M. Sosonkina, D. Langr,and M. A. Caprio, Phys. Rev. Lett. , p. 252501 (2013).[15] V. D. Efros, W. Leidemann, and G. Orlandini, Phys. Lett. B , 130–133 (1994).[16] V. D. Efros, W. Leidemann, G. Orlandini, and N. Barnea, J. of Phys. G: Nucl. and Part. Phys. , p. R459(2007).[17] M. Miorelli, S. Bacca, N. Barnea, G. Hagen, G. R. Jansen, G. Orlandini, and T. Papenbrock, Phys. Rev. C , p. 034317 (2016).[18] C. Bahri, J. Draayer, O. Castanos, and G. Rosensteel, Phys. Lett. B , 430–436 (1990).[19] N. Nevo Dinur, N. Barnea, C. Ji, and S. Bacca, Phys. Rev. C , p. 064317 (2014).[20] K. D. Launey, A. Mercenne, G. H. Sargsyan, H. Shows, R. B. Baker, M. E. Miora, T. Dytrych, and J. P.Draayer, “Emergent clustering phenomena in the framework of the ab initio symmetry-adapted no-core shellmodel,” in this proceedings collection, (2018).[21] N. Barnea, W. Leidemann, and G. Orlandini, Phys. Rev. C , p. 034003 (2006).[22] D. J. Rowe, G. Thiamova, and J. L. Wood, Phys. Rev. Lett. , p. 202501 (2006).[23] C. Stumpf, T. Wolfgruber, and R. Roth, arXiv:1709.06840 [nucl-th] .[24] Y. Suzuki, Nucl. Phys. A448