INFLUENCE OF COMPLEX CONFIGURATIONS ON THE PROPERTIES OF THE PYGMY - PowerPoint PPT Presentation

INFLUENCE OF COMPLEX CONFIGURATIONS ON THE PROPERTIES OF THE PYGMY DIPOLE RESONANCE Arsenyev Nikolay Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia XV th International Seminar on Electromagnetic Interactions of Nuclei «EMIN-2018» Moscow 8–11 October 2018

Outline Introduction Part I: Main ingredients of the model − Realization of QRPA − Phonon-phonon coupling Part II: Results and discussion − Details of calculations − Giant dipole resonance − Pygmy dipole resonance Conclusions N.Arsenyev

Introduction E 1 strength in (spherical) atomic nuclei N.Arsenyev Courtesy: N. Pietralla

Relevance of the PDR 1. The PDR might play an important role in nuclear astrophysics. For example, the occurrence of the PDR could have a pronounced effect on neutron-capture rates in the r-process nucleosynthesis, and consequently on the calculated elemental abundance distribution. S. Goriely , Phys. Lett. B436 , 10 (1998). 2. The study of the pygmy E 1 strength is expected to provide information on the symmetry energy term of the nuclear equation of state. This information is very relevant for the modeling of neutron stars. C. J. Horowitz and J. Piekarewicz , Phys. Rev. Lett. 86 , 5647 (2001). 3. New type of nuclear excitation: these resonances are the low-energy tail of the GDR, or if they represent a different type of excitation, or if they are generated by single-particle excitations related to the specific shell structure of nuclei with neutron excess. N.Arsenyev N. Paar, D. Vretenar, E. Khan, G. Col` o , Rep. Prog. Phys. 70 , 691 (2007).

MAIN INGREDIENTS OF THE MODEL N.Arsenyev

Realization of QRPA We employ the effective Skyrme interaction with the tensor terms in the particle-hole channel r 2 ) + t 1 k 2 + � � � � � � � 1 + x 0 ˆ 1 + x 1 ˆ r 2 ) � r 2 ) C = t 0 k ′ 2 δ ( � V ( � r 1 ,� δ ( � r 1 − � δ ( � r 1 − � r 1 − � r 2 ) P σ P σ 2 � � � k + t 3 r 1 + � r 2 � � � � 1 + x 2 ˆ � r 2 ) � 1 + x 3 ˆ k ′ · δ ( � r 2 ) ρ α + t 2 r 1 − � δ ( � r 1 − � P σ P σ 6 2 k ′ × δ ( � � � � + iW 0 ( � σ 1 + � σ 2 ) · r 1 − � r 2 ) and r 2 ) T = T k ′ ) − 1 � [( σ 1 · � k ′ )( σ 2 · � 3 ( σ 1 · σ 2 ) � k ′ 2 ] δ ( � V ( � r 1 ,� r 1 − � r 2 ) 2 k ) − 1 � r 2 )[( σ 1 · � k )( σ 2 · � 3 ( σ 1 · σ 2 ) � k 2 ] + δ ( � r 1 − � k ) − 1 � � ( σ 1 · � r 2 )( σ 1 · � 3 ( σ 1 · σ 2 )[ � r 2 ) � k ′ ) δ ( � k ′ δ ( � + U r 1 − � r 1 − � k ] T. H. R. Skyrme , Nucl. Phys. 9 , 615 (1959). D. Vautherin and D. M. Brink , Phys. Rev. C5 , 626 (1972). N.Arsenyev

Realization of QRPA The Hamiltonian includes the surface peaked density-dependent zero-range force in the particle-particle channel. � 1 − ρ ( r 1 ) � V pair ( � r 1 ,� r 2 ) = V 0 δ ( � r 1 − � r 2 ) , ρ c where ρ ( r 1 ) is the particle density in coordinate space, ρ c is equal to the nuclear saturation density. The strength V 0 is a parameter fixed to reproduce the odd-even mass difference of nuclei in the studied region. A. P. Severyukhin, V. V. Voronov, N. V. Giai , Phys. Rev. C77 , 024322 (2008). The starting point of the method is the HF-BCS calculations of the ground state, where spherical symmetry is assumed for the ground states. The continuous part of the single-particle spectrum is discretized by diagonalizing the HF Hamiltonian on a harmonic oscillator basis. J. P. Blaizot and D. Gogny , Nucl. Phys. A284 , 429 (1977). N.Arsenyev

Realization of QRPA The residual interaction in the particle-hole channel V ph res and in the particle- particle channel V pp res can be obtained as the second derivative of the energy density functional H with respect to the particle density ρ and the pair density ˜ ρ , respectively. δ 2 H δ 2 H V ph V pp res ∼ res ∼ ρ 2 . δρ 1 δρ 2 δ ˜ ρ 1 δ ˜ G. T. Bertsch and S. F. Tsai , Phys. Rep. 18 , 125 (1975). We simplify V res by approximating it by its Landau-Migdal form � k 1 , � � � � � k 2 V res ( � k 1 , � � k 2 ) = N − 1 F l + G l σ 1 · σ 2 + ( F ′ l + G ′ l σ 1 · σ 2 ) τ 1 · τ 2 P l , 0 k 2 F l = 0 where τ i is the isospin operator, and N 0 = 2 k F m ∗ /π 2 � 2 with k F and m ∗ standing for the Fermi momentum and nucleon effective mass. A. B. Migdal , Theory of Finite Fermi Systems and Applications to Atomic Nuclei (Wiley, New York, 1967). N.Arsenyev

Realization of QRPA Moreover we keep only Landau parameters F 0 and F ′ 0 . Thus, we can write the residual interaction in the following form: � � V ( a ) r 2 ) = N − 1 F ( a ) 0 ( r 1 ) + F ′ ( a ) res ( � r 1 ,� ( r 1 )( τ 1 · τ 2 ) δ ( � r 1 − � r 2 ) , 0 0 where a = { ph , pp } is the channel index. The expressions for F 0 and F ′ 0 in terms of the Skyrme force parameters can write in the following form: � 3 � 4 t 0 + 1 16 t 3 ρ α ( α + 1 )( α + 2 ) + 1 F ph 2 [ 3 t 1 + ( 5 + 4 x 2 ) t 2 ] 0 = N 0 8 k F , � 1 � 4 t 0 ( 1 + 2 x 0 )+ 1 24 t 3 ρ α ( 1 + 2 x 3 )+ 1 F ′ ph 2 [ t 1 ( 1 + 2 x 1 ) − t 2 ( 1 + 2 x 2 )] = − N 0 8 k F , 0 � � 1 − ρ ( r ) 0 ( r )= 1 F pp 4 N 0 V 0 , ρ c F ′ pp ( r )= F pp 0 ( r ) . 0 N. V. Giai and H. Sagawa , Phys. Lett. B106 , 379 (1981). N.Arsenyev A. P. Severyukhin, V. V. Voronov, N. V. Giai , Phys. Rev. C77 , 024322 (2008).

Realization of QRPA We introduce the phonon creation operators λµ i = 1 � � Q + � jj ′ A + ( jj ′ ; λµ ) − ( − 1 ) λ − µ Y λ i X λ i jj ′ A ( jj ′ ; λ − µ ) , 2 jj ′ A + ( jj ′ ; λµ ) = � C λµ jm j ′ m ′ α + jm α + j ′ m ′ . mm ′ The index λ denotes total angular momentum and µ is its z -projection in the laboratory system. One assumes that the ground state is the QRPA phonon vacuum | 0 � and one-phonon excited states are Q + λµ i | 0 � with the normalization condition � 0 | [ Q λµ i , Q + λµ i ′ ] | 0 � = δ ii ′ . Making use of the linearized equation-of-motion approach one can get the QRPA equations � A � � X � X � � B = ω . −B −A Y Y Solutions of this set of linear equations yield the one-phonon energies ω and the amplitudes X , Y of the excited states. N.Arsenyev P. Ring and P. Schuck , The Nuclear Many Body Problem (Springer, Berlin 1980).

Phonon-phonon coupling ( PPC ) To take into account the effects of the phonon-phonon coupling (PPC) in the simplest case one can write the wave functions of excited states as �� � R i ( J ν ) Q + � Q + λ 1 µ 1 i 1 Q + P λ 1 i 1 � � Ψ ν ( JM ) = JMi + λ 2 i 2 ( J ν ) | 0 � λ 2 µ 2 i 2 JM i λ 1 i 1 λ 2 i 2 with the normalization condition � 2 � � R 2 � P λ 1 i 1 i ( J ν ) + 2 λ 2 i 2 ( J ν ) = 1 . i λ 1 i 1 λ 2 i 2 V. G. Soloviev , Theory of Atomic Nuclei: Quasiparticles and Phonons (Inst. of Phys., Bristol 1992). N.Arsenyev

Phonon-phonon coupling ( PPC ) Using the variational principle in the form � � δ � Ψ ν ( JM ) |H| Ψ ν ( JM ) � − E ν [ � Ψ ν ( JM ) | Ψ ν ( JM ) � − 1 ] = 0 , one obtains a set of linear equations for the unknown amplitudes R i ( J ν ) and P λ 1 i 1 λ 2 i 2 ( J ν ) : � U λ 1 i 1 λ 2 i 2 ( Ji ) P λ 1 i 1 ( ω Ji − E ν ) R i ( J ν ) + λ 2 i 2 ( J ν ) = 0 ; λ 1 i 1 λ 2 i 2 � U λ 1 i 1 λ 2 i 2 ( Ji ) R i ( J ν ) + 2 ( ω λ 1 i 1 + ω λ 2 i 2 − E ν ) P λ 1 i 1 λ 2 i 2 ( J ν ) = 0 . i U λ 1 i 1 λ 2 i 2 ( Ji ) is the matrix element coupling one- and two-phonon configurations: Q + λ 1 i 1 Q + U λ 1 i 1 � � λ 2 i 2 ( Ji ) = � 0 | Q Ji H J | 0 � . λ 2 i 2 These equations have the same form as the QPM equations, but the single-particle spectrum and the parameters of the residual interaction are calculated with the Skyrme forces. N.Arsenyev A. P. Severyukhin, V. V. Voronov, N. V. Giai , Eur. Phys. J. A22 , 397 (2004). A. P. Severyukhin, N. N. Arsenyev, N. Pietralla, V. Werner , Phys. Rev. C90 , 011306(R) (2014).

Phonon-phonon coupling ( PPC ) Distribution of coupling matrix elements U λ 1 i 1 λ 2 i 2 ( Ji ) between the one- and two-phonon configurations in the PPC calculation of the GDR strength function for 208 Pb 100 ) -1 (Ji) ) (M eV 10 1 2 i i 1 2 P (U 1 0.1 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 i U ( Ji) ( Me V) 1 1 i 2 2 A. P. Severyukhin, S. ˚ Aberg, N. N. Arsenyev R. G. Nazmitdinov , Phys. Rev. C97 , 059802 (2018). N.Arsenyev More details: report Nazmitdinov (Thursday, October 11, at 16:00)

RESULTS AND DISCUSSION N.Arsenyev

Details of calculations SLy5 vs SLy5+T We use the Skyrme interactions SLy5 and SLy5+T. The SLy5+T involve the tensor terms added without refitting the parameters of the central interaction (the tensor interaction parameters are α T =-170 MeV · fm 5 and β T =100 MeV · fm 5 ). The pairing strength V 0 =-270 MeV · fm 3 is fitted to reproduce the experimental neutron pairing energies near 48 Ca. E. Chabanat et al. , Nucl. Phys. A635 , 231 (1998). G. Col` o et al. , Phys. Lett. B646 , 227 (2007). 15 (a) (a) 450 Energy (M eV ) (M eV ) 10 400 1n S 5 350 20 22 24 26 28 30 32 34 36 38 40 20 22 24 26 28 30 32 34 36 38 40 N N 0.5 25 (b) (b) 20 (M e V ) (fm ) 15 np 2n 10 R 0.0 S 5 0 20 22 24 26 28 30 32 34 36 38 40 20 22 24 26 28 30 32 34 36 38 40 N N N.Arsenyev N. N. Arsenyev, A. P. Severyukhin, V. V. Voronov, N. V. Giai , Phys. Rev. C95 , 054312 (2017). M. Wang et al. , Chin. Phys. C36 , 1603 (2012). J. Birkhan et al. , Phys. Rev. Lett. 118 , 252501 (2017).