Mean-field calculation based on proton-neutron mixed energy density functionals

1. Mean-field calculation based on proton-neutron mixed energy density functionals PRC 88(2013) 061301(R). Koichi Sato (RIKEN Nishina Center) Collaborators: JacekDobaczewski (Univ. of Warsaw /Univ. of Jyvaskyla ) Takashi Nakatsukasa (RIKEN Nishina Center) WojciechSatuła (Univ. of Warsaw )

2. Energy-density-functional calculation with proton-neutron mixing 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 Proton-neutron mixing: Single-particles are mixtures of protons and neutrons EDF with an arbitrary mixing between protons and neutrons n p ？ Here, we consider p-n mixing at the Hartree-Fock level (w/o pairing) A first step toward nuclear DFT for proton-neutron pairing and its application Pairing between protons and neutrons (isoscalar T=0 and isovector T=1) Goodman, Adv. Nucl. Phys.11, (1979) 293. Perlinska et al, PRC 69 , 014316(2004)

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

4. Hartree-Fock calculation including proton-neutron mixing (pnHF) Extension of the single-particle states i=1,…,A Extension of the density functional Invariant under rotation in isospin space isoscalar isovector Perlinska et al, PRC 69 , 014316(2004) can be written in terms of not invariant under rotation in isospin space pnHF isovector isoscalar Standard HF Energy density functionals are extended such that they are invariant under rotation in isospin space

5. HFODD(1997-) We have developed a code for pnHF by extending an HF(B) solver • 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….) http://www.fuw.edu.pl/~dobaczew/hfodd/hfodd.html 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) 2391. J. Dobaczewski, et al., Comp. Phys. Comm. 183 (2012) 166.

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

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

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

9. Result for A=48 isobars with Coulomb Shifted semicircle method PRC 88(2013) 061301(R). For T=4 Energies are dependent on Tz (almost linear dependence) No p-n mixing at |Tz|=T gives 48Ni 48Ca

10. T4states in A=40-56 isobars T=1 triplets in A=14 isobars 14O(g.s) (excited 0+) 14N 14C(g.s) Excited 0+, T=1 state in odd-odd 14N Our framework nicely works also for IASs in even-even A=40-56 isobars Time-reversal symmetryconserved 14N: p-n mixed , 14C,O: p-n unmixed HF (The origin ofcalc. BE is shifted by 3.2 MeV to correct the deficiency of SkM* functional)

11. 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 isocrankinge-e nuclei in their ground states with time-reversal symmetry. Remarks: Augmented Lagrange method for constraining the isospin. See PRC 88(2013) 061301(R). Benchmark calculation with axially symmetric HFB solver: Sheikh et al., PRC, in press; arXiv:1403.2427

12. Backups

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

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

15. (b)Starting with the highest weight state. Isocranking calc. 48Ca (Tz=4) ㉘ ㊽ ㊽ - ~11MeV ⑳ p-n unmixed p-n mixed 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.

16. How to determine the size of ? Isobaric analog states with T=4 in A=48 nuclei 48Ca (Tz=|T|=4) 48Ni (Tz=-|T|=-4) ㉘ ㉘ ~11MeV ~11MeV ⑳ ⑳ Fermi energy Standard HF Standard HF We take the size of the isocranking frequency equal to the difference of the proton and neutron Fermi energies in the |Tz|=T states. This choice of enable us to avoid the configuration change.

17. With Coulomb interaction :violates isospin symmetry but independent of azimuthal angle The total energy is now dependent on Tz Initial : final : HF state w/o p-n mixing p-n mixed state larger <Tz> is favored 　　・・・ With Coulomb, the s. p. Routhians depend on the titling angle

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