MR-EDF calculations of odd-even nuclei Benjamin Bally 1 , B. Avez 1 , M. Bender 1 and P.-H. Heenen 2 1 Universit´ e Bordeaux 1; CNRS/IN2P3; Centre d’Etudes Nucl´ eaires de Bordeaux Gradignan, UMR5797, Chemin du Solarium, BP120, F-33175 Gradignan, France 2 Service de Physique Nucl´ eaire Th´ eorique, Universit´ e Libre de Bruxelles, C.P. 229, B-1050 Bruxelles, Belgium B. Bally MR-EDF calculations of odd-even nuclei
Introduction ◮ Treatment of even-even and odd-even nuclei on the same footing. ◮ Particle number and angular momentum restored GCM of cranked triaxial quasiparticle states. ◮ Motivations : • Odd-A nuclei represent half of the chart of nuclides. • Coupling of single particle and shape degrees of freedom. • Analysis of signatures for shell structure (separation energies, g factors, spectroscopic quadrupole moments, ...). • Analysis of the interplay of pairing correlations and fluctuations in shape degrees of freedom for the odd-even mass staggering. • Study of coexistence phenomena (shape, single-particle levels). B. Bally MR-EDF calculations of odd-even nuclei
Theoritical model B. Bally MR-EDF calculations of odd-even nuclei
Philosophy of the approach ◮ Energy Density Functional : E [ ρ, κ, κ ∗ ] � Ψ L | c † � Ψ L | c † i c † j c i | Ψ R � RL = j | Ψ R � � Ψ L | c j c i | Ψ R � κ ∗ ρ LR ij = κ LR ij = � Ψ L | Ψ R � � Ψ L | Ψ R � ij � Ψ L | Ψ R � ◮ First step : Single-Reference EDF (”Self-consistent mean field”, ”HFB”) B. Bally MR-EDF calculations of odd-even nuclei
SR-EDF : ”HFB” realization ◮ Minimization with constraint : δ ( E − λ � ˆ N � − λ q � ˆ Q � ) = 0 λ, λ q : lagrange multipliers ˆ N : particle number operator ˆ Q : multipole operator ◮ ”HFB” equations : � ˆ � � U � U � � ˆ h − λ ∆ = E h ∗ + λ − ˆ − ˆ ∆ ∗ V V ˆ h = ∂ E ˆ ∂ E ∂ρ : particle-hole field ∆ = ∂κ ∗ : particle-particle field ρ = V ∗ V T κ = V ∗ U T ◮ Self-iterative blocking : ( U ki , V ki ) ↔ ( V ki ∗ , U ki ∗ ) B. Bally MR-EDF calculations of odd-even nuclei
SR-EDF : ”HFB+LN” realization N 2 � − λ q � ˆ ◮ Minimization with constraint : δ ( E − λ � ˆ N � − λ 2 � ˆ Q � ) = 0 λ 2 : not a lagrange multiplier λ, λ q : lagrange multipliers ˆ N : particle number operator ˆ Q : multipole operator ◮ ”HFB+LN” equations : � ˆ � � U � U � � ˆ h − λ ∆ = E h ∗ + λ − ˆ ∆ ∗ − ˆ V V ˆ ˆ h = ∂ E ∂ E ∂ρ : particle-hole field ∆ = ∂κ ∗ : particle-particle field ρ = V ∗ V T κ = V ∗ U T ∗ , U ki ∗ ) ◮ Self-iterative blocking : ( U ki , V ki ) ↔ ( V ki B. Bally MR-EDF calculations of odd-even nuclei
SR-EDF : symmetries of the code ◮ Triaxial cranking code ” CR8 ”. P. Bonche, H. Flocard, P.-H. Heenen, S.J. Krieger and M.S. Weiss, Nucl. Phys. A 443 (1985) 39-63 R x , ˆ 2 h subgroup { ˆ y ,ˆ ◮ D TD S T P } : R x = e − i π ˆ J x . : ˆ X signature ˆ Pe − i π ˆ J y . y = ˆ T ˆ Y time-simplex : S T : ˆ Parity P . B. Bally MR-EDF calculations of odd-even nuclei
SR-EDF : symmetries of the code ◮ Triaxial cranking code ” CR8 ”. P. Bonche, H. Flocard, P.-H. Heenen, S.J. Krieger and M.S. Weiss, Nucl. Phys. A 443 (1985) 39-63 R x , ˆ 2 h subgroup { ˆ y ,ˆ ◮ D TD S T P } : R x = e − i π ˆ : ˆ J x . X signature ˆ Pe − i π ˆ y = ˆ T ˆ J y . Y time-simplex : S T : ˆ Parity P . even-even vacua ˆ R x | Φ � = | Φ � ˆ P | Φ � = | Φ � ˆ S T y | Φ � = | Φ � B. Bally MR-EDF calculations of odd-even nuclei
SR-EDF : symmetries of the code ◮ Triaxial cranking code ” CR8 ”. P. Bonche, H. Flocard, P.-H. Heenen, S.J. Krieger and M.S. Weiss, Nucl. Phys. A 443 (1985) 39-63 R x , ˆ 2 h subgroup { ˆ y ,ˆ ◮ D TD S T P } : R x = e − i π ˆ J x . : ˆ X signature ˆ Pe − i π ˆ J y . y = ˆ T ˆ Y time-simplex : S T : ˆ Parity P . even-even vacua odd-even nuclei ˆ R x | Φ � = | Φ � ˆ R x | Φ � = ± i | Φ � ˆ P | Φ � = | Φ � ˆ P | Φ � = ± | Φ � ˆ S T y | Φ � = | Φ � ˆ S T y | Φ � = | Φ � B. Bally MR-EDF calculations of odd-even nuclei
Philosophy of the approach ◮ Energy Density Functional : E [ ρ, κ, κ ∗ ] � Ψ L | c † � Ψ L | c † i c † j c i | Ψ R � RL = j | Ψ R � � Ψ L | c j c i | Ψ R � κ ∗ ρ LR κ LR ij = ij = ij � Ψ L | Ψ R � � Ψ L | Ψ R � � Ψ L | Ψ R � ◮ First step : Single-Reference EDF (”Self-consistent mean field”, ”HFB”) ✔ takes into account static collective correlations. ✘ loss of quantum numbers and selection rules for transitions. ◮ Second step : Multi-Reference EDF (”Beyond mean field”) B. Bally MR-EDF calculations of odd-even nuclei
MR-EDF : symmetry restoration Angular-momentum restoration operator : rotation in real space � 2 π � 2 π � π � �� � MK = 2 J + 1 ˆ d γ D ∗ J ˆ P J d β sin( β ) MK ( α, β, γ ) R ( α, β, γ ) d α 8 π 2 � �� � 0 0 0 Wigner function K is the z component of angular momentum in the body-fixed frame. Projected states are given by + J + J � � P Z ˆ P N | q � = f J ,κ ( K ) ˆ P J MK ˆ | JMq κ � = f J ,κ ( K ) | JMKq � K = − J K = − J f J ,κ ( K ) are the weights of the components K and determined variationally B. Bally MR-EDF calculations of odd-even nuclei
MR-EDF : configuration mixing via the ”Generator Coordinate Method” Superposition of angular-momentum restored SR-EDF states � | JMKq � + J � � projected mean-field state | JM ν � = f J ν ( q , K ) | JMKq � f J ν ( q , K ) weight function q K = − J � JM ν | ˆ δ H | JM ν � = 0 ⇒ Hill-Wheeler-Griffin equation δ f ∗ J ν ( q , K ) � JM ν | JM ν � + J � � � � H J ( qK , q ′ K ′ ) − E J ,ν I J ( qK , q ′ K ′ ) f J ,ν ( q ′ , K ′ ) = 0 q ′ K ′ = − J with H J ( qK , q ′ K ′ ) = � JMKq | ˆ H | JMK ′ q ′ � energy kernel I J ( qK , q ′ K ′ ) = � JMKq | JMK ′ q ′ � norm kernel Angular-momentum projected GCM gives the ◮ correlated ground state for each value of J ◮ spectrum of excited states for each J B. Bally MR-EDF calculations of odd-even nuclei
Philosophy of the approach ◮ Energy Density Functional : E [ ρ, κ, κ ∗ ] � Ψ L | c † � Ψ L | c † i c † j c i | Ψ R � RL = j | Ψ R � � Ψ L | c j c i | Ψ R � κ ∗ ρ LR κ LR ij = ij = ij � Ψ L | Ψ R � � Ψ L | Ψ R � � Ψ L | Ψ R � ◮ First step : Single-Reference EDF (”Self-consistent mean field”, ”HFB”) ✔ takes into account static collective correlations. ✘ loss of quantum numbers and selection rules for transitions. ◮ Second step : Multi-Reference EDF (”Beyond mean field”) ✔ takes into account fluctuations in collective degrees of freedom. ✔ restoration of quantum numbers and selection rules for transitions. B. Bally MR-EDF calculations of odd-even nuclei
Preliminary results about 49 Cr B. Bally MR-EDF calculations of odd-even nuclei
Choice of the functional ◮ Skyrme parametrization SIII. ◮ Delta force pairing (strength : 300 MeV for neutrons and protons). ◮ No coulomb exchange. ◮ Regularization of the functional to avoid the ”pole problem”. M. Bender, T. Duguet, P.-H. Heenen, D. Lacroix, Int. J. Mod. Phys. E 20 (2011) 259-269 B. Bally MR-EDF calculations of odd-even nuclei
Non-projected energy surface From the lowest quasiparticle with � J z � π ≈ | 2 . 5 | − B. Bally MR-EDF calculations of odd-even nuclei
Non-projected → PNR energy surface → B. Bally MR-EDF calculations of odd-even nuclei
− energy surface PNR → PNR+AMR : J π = 5 2 5 2 → B. Bally MR-EDF calculations of odd-even nuclei
− energy surface PNR+AMR : J π = 5 2 5 2 B. Bally MR-EDF calculations of odd-even nuclei
49 Cr versus 48 Cr 5 2 VS M. Bender, B. Avez, P.-H. Heenen, unpublished B. Bally MR-EDF calculations of odd-even nuclei
− energy surface PNR+AMR : J π = 5 2 5 2 B. Bally MR-EDF calculations of odd-even nuclei
J and K decompositions β = 0 . 27 γ = 00 . 0 ◦ β = 0 . 27 γ = 00 . 0 ◦ B. Bally MR-EDF calculations of odd-even nuclei
− energy surface PNR+AMR : J π = 5 2 5 2 B. Bally MR-EDF calculations of odd-even nuclei
J and K decompositions β = 0 . 33 γ = 15 . 3 ◦ β = 0 . 33 γ = 15 . 3 ◦ B. Bally MR-EDF calculations of odd-even nuclei
− energy surface PNR+AMR : J π = 5 2 5 2 B. Bally MR-EDF calculations of odd-even nuclei
J and K decompositions β = 0 . 00 γ = 00 . 0 ◦ β = 0 . 00 γ = 00 . 0 ◦ B. Bally MR-EDF calculations of odd-even nuclei
− energy surface PNR+AMR : J π = 7 2 7 2 B. Bally MR-EDF calculations of odd-even nuclei
Incomplete GCM (3 points) 5 2 → Experiment : T.W. Burrows, Nuclear Data Sheets 109 (2008) 1879-2032 B. Bally MR-EDF calculations of odd-even nuclei
Cranking ◮ δ ( E − λ � ˆ N � − λ 2 � ˆ N 2 � − λ q � ˆ Q � − ω � ˆ J x � ) = 0 ω : lagrange parameter √ ◮ � ˆ J 2 − K 2 J x � = K = 5 ( → see K decompositions) 2 ◮ Projection of cranked states should improve the moments of inertia. B. Bally MR-EDF calculations of odd-even nuclei
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