Survey of groundstate octupole deformations of atomic nuclei

Survey of ground-state octupole deformations of atomic nuclei Yuchen Cao and Witold Nazarewicz, Michigan State University/FRIB-TA Topical Program: Hadronic Electric Dipole Moments in the FRIB Era: From the Proton to Protactinium MSU, August 12 -23, 2019 Menu • Nuclear shapes • Ground-state octupole deformations: a survey • Conclusions FRIB-TA Topical Program 2019 1

Shapes FRIB-TA Topical Program 2019 2

Nuclear shapes The first evidence for a non-spherical nuclear shape came from the observation of a quadrupole component in the hyperfine structure of optical spectra. The analysis showed that the electric quadrupole moments of the nuclei concerned were more than an order of magnitude greater than the maximum value that could be attributed to a single proton and suggested a deformation of the nucleus as a whole. • Schüler, H. , and Schmidt, Th. , Z. Physik 94, 457 (1935) • Casimir, H. B. G. , On the Interaction Between Atomic Nuclei and Electrons, Prize Essay, Taylor’s Tweede Genootschap, Haarlem (1936) The question of whether nuclei can rotate became an issue already in the very early days of nuclear spectroscopy • Thibaud, J. , Comptes rendus 191, 656 ( 1930) • Teller, E. , and Wheeler, J. A. , Phys. Rev. 53, 778 (1938) • Bohr, N. , Nature 137, 344 ( 1936) • Bohr, N. , and Kalckar, F. , Mat. Fys. Medd. Dan. Vid. Selsk. 14, no, 10 (1937) FRIB-TA Topical Program 2019 3

Reflection-asymmetric nuclear shapes Remarks on pear-shaped nuclei, Vilen Strutinsky, At. Energ. 4, 150 (1956) Physica, 22, 1166 (1956) FRIB-TA Topical Program 2019 4

Stability of Pear-Shaped Nuclear Deformations, K. Lee and D. R. Inglis Phys. Rev. 108, 774 (1957) FRIB-TA Topical Program 2019 5

P. Butler and WN, Rev. Mod. Phys. 68, 349 (1996) FRIB-TA Topical Program 2019 6

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Intrinsic dipole moment P. Butler, J. Phys. G 43, 073002 (2016) P. Butler and WN, Nucl. Phys. A 533, 249 (1991) FRIB-TA Topical Program 2019 8

Mean-Field Theory ⇒ Density Functional Theory Degrees of freedom: nucleonic densities • • • mean-field ⇒ one-body densities Nuclear DFT zero-range ⇒ local densities • two fermi liquids finite-range ⇒ gradient terms • self-bound particle-hole and pairing channels • superfluid Does not have to be related to a force Has been extremely successful. A broken-symmetry generalized product state does surprisingly good job for nuclei. Constrained by microscopic theory: ab-initio functionals provide quasi-data! Not all terms are equally important. Usually ~12 terms considered Some terms probe specific experimental data Pairing functional poorly determined. Usually 1 -2 terms active Most popular: Skyrme, Gogny, Covariant, Fayans, DME-based Can be extended into multi-reference DFT (GCM) and projected DFT FRIB-TA Topical Program 2019 9

Shell correction approach FRLDM (2002) Axial and reflection. FRIB-TA asymmetry of the nuclear ground state Topical Program 2019 P. Möller et al, Atomic Data and Nuclear Data Tables 94, 758 (2008) 10

DFT: Gogny vs. Barcelona-Catania-Paris Ba Axial octupole collective Hamiltonian Ra FRIB-TA Topical Program 2019 L. M. Robledo et al. , Phys. Rev. C 81, 034315 (2010) 11

RDFT: DD-PC 1 FRIB-TA Topical Program 2019 K. Nomura et al. , Phys. Rev. C 89, 024312 (2014) 12

S. E. Agbemava, A. V. Afanasjev, P. Ring, Octupole deformation in the ground states of even-even nuclei: A global analysis within the covariant density functional theory, Phys. Rev. C 93, 044304 (2016) FRIB-TA Topical Program 2019 13

L. M. Robledo, J. Phys. G 42, 055109 (2015) Gogny D 1 M L. M. Robledo and R. R. Rodrıguez-Guzman, J. Phys. G 39, 105103 (2012) FRIB-TA Topical Program 2019 14

Robledo & Bertsch, PRC 84, 054302 (2011) FRIB-TA Topical Program 2019 15

Global calculations of octupole deformations Y. (Maxwell) Cao, WN, E. Olsen, S. Agbemava FRIB-TA Topical Program 2019 16

Equilibrium deformation 224 Th 154 Dy 114 Ba 146 Ba FRIB-TA Topical Program 2019 17

Equilibrium deformation FRIB-TA Topical Program 2019 18

Octupole energy FRIB-TA Topical Program 2019 19

Number of cases (based on 7 models) 114 Ba 146 Ba 200 Gd (170 Gd is the heaviest known) FRIB-TA Topical Program 2019 20

Number of cases, heavy nuclei (9 models) 224, 226 Ra, 226, 228 Th, 228 Pu FRIB-TA Topical Program 2019 21

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Correlating Schiff Moments in the Light Actinides with Octupole Moments J. Dobaczewski et al. , Phys. Rev. Lett. 121, 232501 (2018) The result is a dramatically reduced uncertainty in intrinsic Schiff moments. Direct measurements of octupole moments in odd nuclei will reduce the uncertainty even more. The only significant source of nuclear-physics error in the laboratory Schiff moments will then be the intrinsic matrix elements of the time-reversal noninvariant interaction produced by CP-violating fundamental physics. FRIB-TA Topical Program 2019 26

Conclusions • Very rich data; need to be fully analyzed. o Octupole softness o Systematic uncertainties • The usual suspects remain the best cases • To be done: o Search for best parity doublet candidates in odd-A nuclei o Schiff moment calculations FRIB-TA Topical Program 2019 27