Mean-field calculation based on proton-neutron mixed energy density functionals Koichi Sato (RIKEN Nishina Center) Collaborators: Jacek Dobaczewski (Univ. of Warsaw /Univ. of Jyvaskyla ) Takashi Nakatsukasa (RIKEN Nishina Center) Wojciech Satuła (Univ. of Warsaw ) PRC 88(2013) (R).

Pairing between protons and neutrons (isoscalar T=0 and isovector T=1) Goodman, Adv. Nucl. Phys.11, (1979) 293. Energy-density-functional calculation with proton-neutron mixing ？ p n Proton-neutron mixing: Single-particles are mixtures of protons and neutrons EDF with an arbitrary mixing between protons and neutrons superposition of protons and neutrons Isospin symmetry Protons and neutrons can be regarded as identical particles (nucleons) with different quantum numbers In general, a nucleon state is written as Perlinska et al, PRC 69, (2004) A first step toward nuclear DFT for proton-neutron pairing and its application Here, we consider p-n mixing at the Hartree-Fock level (w/o pairing)

Basic idea of p-n mixing Let’s consider two p-n mixed s.p. wave functions standard unmixed neutron and proton w. f. standard n and p densities They contribute to the local density matrices as p-n mixed densities (spin indices omitted for simplicity)

Hartree-Fock calculation including proton-neutron mixing (pnHF) Extension of the density functional Invariant under rotation in isospin space i=1,…,A Extension of the single-particle states isovector isoscalar Perlinska et al, PRC 69, (2004) can be written in terms of not invariant under rotation in isospin space Energy density functionals are extended such that they are invariant under rotation in isospin space Standard HF pnHF isovector isoscalar

HFODD(1997-) Skyrme energy density functional Hartree-Fock or Hartree-Fock-Bogoliubov No spatial & time-reversal symmetry restriction Harmonic-oscillator basis Multi-function (constrained HFB, cranking, angular mom. projection, isospin projection, finite temperature….) We have developed a code for pnHF by extending an HF(B) solver J. Dobaczewski, J. Dudek, Comp. Phys. Comm 102 (1997) 166. J. Dobaczewski, J. Dudek, Comp. Phys. Comm. 102 (1997) 183. J. Dobaczewski, J. Dudek, Comp. Phys. Comm. 131 (2000) 164. J. Dobaczewski, P. Olbratowski, Comp. Phys. Comm. 158 (2004) 158. J. Dobaczewski, P. Olbratowski, Comp. Phys. Comm. 167 (2005) 214. J. Dobaczewski, et al., Comp. Phys. Comm. 180 (2009) J. Dobaczewski, et al., Comp. Phys. Comm. 183 (2012) 166.

w/o Coulomb force Total isospin of the system Total energy should be independent of the orientation of T. (and w/ equal proton and neutron masses) Check of the code How to control the isospin direction ? Test calculations for p-n mixing invariant under rotation in isospace EDF with p-n mixing is correctly implemented? All the isobaric analog states should give exactly the same energy “Isobaric analog states”

w/ p-n mixing and no Coulomb Initial state: HF solution w/o p-n mixing (e.g. 48 Ca (Tz=4,T=4) ) Isocranking term Final state p-n mixed state HF state w/o p-n mixing iteration Isocranking calculation : Input to control the isospin of the system HF eq. solved by iterative diagonalization of MF Hamiltonian. Analog with the tilted-axis cranking for high- spin states isospin

Isocranking calculation for A=48 w/o Coulomb No p-n mixing at |Tz|=T Energies are independent of The highest and lowest weight states are standard HF states We have confirmed that the results do not depend on φ. 48 Ca 48 Ni 48 Ti 48 Cr 48 Fe

Result for A=48 isobars with Coulomb For T=4 48 Ca 48 Ni gives No p-n mixing at |Tz|=T Energies are dependent on Tz (almost linear dependence) Shifted semicircle method PRC 88(2013) (R).

Our framework nicely works also for IASs in even-even A=40-56 isobars T=1 triplets in A=14 isobars 14 O(g.s) 14 C(g.s) 14 N Excited 0 +, T=1 state in odd-odd 14 N Time-reversal symmetry conserved (The origin of calc. BE is shifted by 3.2 MeV to correct the deficiency of SkM* functional) (excited 0 + ) 14 N: p-n mixed, 14 C,O: p-n unmixed HF

Summary We have solved the Hartree-Fock equations based on the EDF including p-n mixing Isospin of the system is controlled by isocranking model The p-n mixed single-reference EDF is capable of quantitatively describing the isobaric analog states For odd(even) A/2, odd(even)-T states can be obtained by isocranking e-e nuclei in their ground states with time-reversal symmetry. Augmented Lagrange method for constraining the isospin. Remarks: See PRC 88(2013) (R). Benchmark calculation with axially symmetric HFB solver: Sheikh et al., PRC, in press; arXiv:

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Assume we want to obtain the T=4 & Tz=0 IAS in 48Cr (w/o Coulomb) 48 Ca (Tz=4) (b) Starting with the highest weight state. 48 Cr (Tz=0) (a) Starting with the T=0 state 48 Cr (Tz=0) isospin

Illustration by a simple model W. Satuła & R. Wyss, PRL 86, 4488 (2001). To get T=1,3, ・・ states, we make a 1p1h excitation At each crossing freq., Four-fold degeneracy at ω=0 In this study, we use the Hamiltonian based on the EDF with p-n mixed densities. ( isospin & time-reversal) (a) Starting with the T=0 state

48Ca (Tz=4) Standard HF The size of the isocranking frequency is determined from the difference of the proton and neutron Fermi energies in the |Tz|=T states. ~11MeV ⑳ p-n mixed Isocranking calc. p-n unmixed (b) Starting with the highest weight state.

48Ca (Tz=|T|=4) ~11MeV 48Ni (Tz=-|T|=-4) ~11MeV ⑳ ⑳ Standard HF Isobaric analog states with T=4 in A=48 nuclei We take the size of the isocranking frequency equal to the difference of the proton and neutron Fermi energies in the |Tz|=T states. Fermi energy

With Coulomb interaction Initial : final : :violates isospin symmetry p-n mixed stateHF state w/o p-n mixing The total energy is now dependent on Tz larger is favored

Shifted semicircle Coulomb gives additional isocranking freq. effectively w/ Coulombw/o Coulomb semicircle w/o Coulomb w/ Coulomb Difference of p and n Fermi energies 0 (MeV)