Report

Mean-field calculation based on proton-neutron mixed energy density functionals PRC 88(2013) 061301(R). Koichi Sato (RIKEN Nishina Center) Collaborators: Jacek Dobaczewski (Univ. of Warsaw /Univ. of Jyvaskyla ) Takashi Nakatsukasa (RIKEN Nishina Center) Wojciech Satuła (Univ. of Warsaw ) Energy-density-functional calculation with proton-neutron mixing superposition of protons and neutrons Isospin symmetry = = +1 2 = 1 0 Protons and neutrons can be regarded as identical particles (nucleons) with different quantum numbers = = −1 2 = 0 1 In general, a nucleon state is written as = 1 2 Proton-neutron mixing: Single-particles are mixtures of protons and neutrons EDF with an arbitrary mixing between protons and neutrons ( , ) c , c , ' ( , ) c , ' c , , p , n p, n 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) p ？ n 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. 1 ()=1 (, ) 2 ()=2 (, ) They contribute to the local density matrices as standard n and p densities p-n mixed densities Hartree-Fock calculation including proton-neutron mixing (pnHF) Extension of the single-particle states i ,n a (n) i , ,n i j, p a ( p) j , , p a (n) i , , n ai, , p ( p) i=1,…,A Extension of the density functional E Skyrme [n , p ] E Skyrm e [0, ] isoscalar Invariant under rotation in isospin space isovector Perlinska et al, PRC 69 , 014316(2004) can be written in terms of 0 , 3 not invariant under rotation in isospin space isoscalar 0 n p Standard HF pnHF isovector 1 np pn 2 i np i pn 3 n p Energy density functionals are extended such that they are invariant under rotation in isospin space We have developed a code for pnHF by extending an HF(B) solver HFODD(1997-) 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. • • • • • 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….) 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 Tz T Total isospin of the system Total energy should be independent of the orientation of T. Ty T Tx 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 ? Isocranking calculation Analog with the tilted-axis cranking for high- spin states Isocranking term : 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 Tz isospin T Tz T Ty Tx HF state w/o p-n mixing p-n mixed state iteration Ty Tx By adjusting the size and titling angle of , we can obtain isobaric analog states Isocranking calculation for A=48 w/o Coulomb 48Ni Tz 48Ca 11(6.0) MeV for T 4 (2) 48Fe 48Ti Tx 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. 48Cr Result for A=48 isobars with Coulomb Shifted semicircle method ( sin , 0 , cos off ) PRC 88(2013) 061301(R). For T=4 90 12 . 0 MeV off 8 MeV Energies are dependent on Tz (almost linear dependence) No p-n mixing at |Tz|=T 90 gives Tˆz 0 48Ni 48Ca T≅4states in A=40-56 isobars T=1 triplets in A=14 isobars 14O(g.s) 14N (excited 0+) 14C(g.s) Our framework nicely works also for IASs in even-even A=40-56 isobars Excited 0+, T=1 state in odd-odd 14N Time-reversal symmetry conserved 14N: p-n mixed , 14C,O: p-n unmixed HF (The origin of calc. BE is shifted by 3.2 MeV to correct the deficiency of SkM* functional) 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. 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 Backups Assume we want to obtain the T=4 & Tz=0 IAS in 48Cr (w/o Coulomb) (a) Starting with the T=0 state Tz 48Cr Tz = (Tz=0) (Tz=0) − t Ty Tx 48Cr Tx = , y = = , = (b) Starting with the highest weight state. Tz 48Ca (Tz=4) = , = isospin Tz − t Ty Tx Ty 48Cr Ty Tx (Tz=0) (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. x (b) Starting with the highest weight state. 48Ca (Tz=4) Isocranking calc. ㉘ n =0 - ~11MeV p z ⑳ ㊽ ≠0 /2 z /2 x ㊽ Tz Tx Tz x Tx = = 4 = 4 ≠ 4 Standard HF p-n unmixed p-n mixed The size of the isocranking frequency is determined from the difference of the proton and neutron Fermi energies in the |Tz|=T states. 11 . 0 z How to determine the size of ? Isobaric analog states with T=4 in A=48 nuclei 48Ni (Tz=-|T|=-4) 48Ca (Tz=|T|=4) Tz ㉘ ㉘ Tx p Tz n x Tx ~11MeV z ~11MeV ⑳ 180 Standard HF n Fermi energy p ⑳ 0 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. 11 . 0 This choice of enable us to avoid the configuration change. With Coulomb interaction U Coulomb ( z ) :violates isospin symmetry The total energy is now dependent on Tz but independent of azimuthal angle Initial : Tz final : ˆ T Tz T Ty Ty Tx HF state w/o p-n mixing Tx p-n mixed state larger <Tz> is favored ・ ・ ・ With Coulomb, the s. p. Routhians depend on the titling angle w/ Coulomb w/o Coulomb Shifted semicircle semicircle x 0 x z n p (T z 4 ) p n (T z 4 ) n p (T z 4 ) z Coulomb gives additional isocranking freq. effectively off Tˆz w/o Coulomb ( x , y 0, z ) w/ Coulomb ( x , y 0 , z off ) (MeV) n p (T z 4 ) Difference of p and n Fermi energies