Effects of ground-state correlations on damping of giant dipole resonaces in LS closed shell nuclei
aa r X i v : . [ nu c l - t h ] F e b Effects of ground-state correlations on damping of giant dipole resonaces in LS closedshell nuclei Mitsuru Tohyama
Faculty of Medicine, Kyorin University, Mitaka, Tokyo 181-8611, Japan
The effects of ground-state correlations on the damping of isovector giant dipole resonances in LS closed shell nuclei O and Ca are studied using an extended random phase approximation (ERPA)derived from the time-dependent density-matrix theory. It is pointed out that unconventional two-body amplitudes of one particle–three hole and three particle–one hole types which are neglectedin most extended RPA theories play an important role in the fragmentation of isovector dipolestrength.
The random phase approximation (RPA) has beenused as the standard theory to study giant resonances.RPA describes giant resonances as highly collective statesconsisting of one particle (p) - one hole (h) excitations.Most observed giant resonances show strong fragmenta-tion of transition strength, however. For realistic de-scription of giant resonances, therefore, beyond RPA the-ories which include configurations higher than 1p–1h’sare needed. The second RPA (SRPA) [1, 2] includes thecoupling to 2p–2h configurations. The particle-vibrationcoupling or quasiparticle-phonon models [3] express p–h correlations included in the 2p–2h configurations byphonons. Our extended RPA (ERPA) derived from thesmall amplitude limit of the time-dependent density-matrix theory (TDDM) [4–6] consists of the coupledequations for one-body and two-body amplitudes andcontains the effects of ground-state correlations throughthe fractional occupation probability n α of a single-particle state α and the correlated part C of a two-bodydensity matrix. Special features of ERPA are that theone-body and two-body amplitudes are not restrictedto the usual 1p–1h and 2p–2h types. In this paper wedemonstrate that 1p–3h and 3p–1h configurations playan important role in the fragmentation of the isovectordipole strength in doubly LS closed shell nuclei O and Ca.The ground state used in ERPA is given as a sta-tionary solution of the TDDM equations. The TDDMequations consist of the coupled equations of motion forthe one-body density matrix n αα ′ (the occupation ma-trix) and the correlated part of the two-body densitymatrix C αβα ′ β ′ ( C ). The equations of motion for re-duced density matrices form a chain of coupled equa-tions known as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. In TDDM the BBGKY hier-archy is truncated by replacing a three-body density ma-trix with anti-symmetrized products of the one-body andtwo-body density matrices [4, 5]. The TDDM equationfor C αβα ′ β ′ contains all effects of two-body correlations;p–p, h–h and p–h correlations. A stationary solution ofthe TDDM equations can be obtained by using the gra-dient method [7].The ERPA equations are derived as the small ampli-tude limit of TDDM and are written in matrix form forthe one-body and two-body amplitudes x µαα ′ and X µαβα ′ β ′ [7] (cid:18) A BC D (cid:19) (cid:18) x µ X µ (cid:19) = ω µ (cid:18) S T T S (cid:19) (cid:18) x µ X µ (cid:19) , (1)where A , B , C and D are given as the expectation val-ues of the double commutators between Hamiltonian andeither one-body or two-body excitation operators while S , T (= T † ) and S are the expectation values of thecommutators between either one-body or two-body exci-tation operators. The effects of ground-state correlationsare included in Eq. (1) through n α and C . Each matrixelement of Eq. (1) is given explicitly in Ref. [8]. If theHF assumption is made for the ground state, Eq.(1) isreduced to the SRPA equation [1].The importance of each two-body configuration in thedamping of giant resonances may be estimated from thevalue of S . When C αβα ′ β ′ is neglected for simplicity, thediagonal element of S is given by [8, 9] S ( αβα ′ β ′ : αβα ′ β ′ ) = (1 − n α )(1 − n β ) n α ′ n β ′ − n α n β (1 − n α ′ )(1 − n β ′ ) , (2)where we assume that n αα ′ = δ αα ′ n α . In the case of theHF ground state where n α = 0 or 1, S is not vanishingonly for the 2p–2h and 2h–2p configurations: S is 1 ( − X µ pp ′ hh ′ ( X µ hh ′ pp ′ ).When the single-particle states are fractionally occupied,all two-body configurations can have non-vanishing val-ues of S . Let us assume that n α = ∆ for a particle stateand n α = 1 − ∆ for a hole state independently of α andthat ∆ is small. Then S ≈ −
4∆ for the 2p–2h configu-rations and the 3p–1h (1h–3p) and 1p–3h (3h–1p) config-urations, X µ pp ′ p ′′ h ( X µ hpp ′ p ′′ ) and X µ phh ′ h ′′ ( X µ hh ′ h ′′ p ), have S ≈ ∆( − ∆). S for other configurations are of higherorder of ∆. This suggests that the 3p–1h (1h–3p) and1p–3h (3h–1p) are the next order configurations to be in-cluded when the effects of ground-state correlations areconsidered. That these configurations play a particularlyimportant role for ground state correlations is also bornout by the fact that their inclusion to the RPA operatorleads to a destructor which annihilates exactly the Cou-pled Cluster Doubles (CCD) ground state [10]. In thefollowing we demonstrate the importance of such con-figurations in the damping of the isovector giant dipoleresonances (GDR’s) in O and Ca.Using a minimal single-particle space needed to cal-culate n α and C and to define the 3p–1h and 1p–3hconfigurations, we study the fragmentation of the isovec-tor dipole strength in O and Ca. In the case of Othe result in ERPA is compared with that in exact diag-onalization approach (EDA). The occupation probability n α and C in O are calculated within TDDM using the1 p / , 1 p / and 1 d / states for both protons and neu-trons. For Ca the 2 s / , 1 d / , 1 d / and 1 f / statesare used. The single-particle energies and wavefunctionsare calculated using the Skyrme III force [11]. A simpli-fied interaction which contains only the t and t terms ofthe Skyrme III force is used as the residual interaction [7].To reduce the dimension size, we only consider the 2p–2hand 2h–2p elements of C and use the iterative gradientmethod [7] to solve the TDDM equations. In the case of O we reduce the strength of the residual interaction by10 % to achieve convergence in the gradient method. The3p–1h and 1p–3h (and 1h–3p and 3h–1p) amplitudes aredefined by using the same single-particle states as thoseused in the ground-state calculations. In these restrictedsingle-particle space X µαβα ′ β ′ for GDR has no 2p–2h and2h–2p components.In the case of GDR in Ca we also perform a real-istic calculation following the procedures used in Refs.[7, 9]. The one-body amplitudes x µαα ′ are defined with alarge number of single-particle states including those inthe continuum: We discretize the continuum states byconfining the wavefunctions in a sphere with radius 15fm and take all the single-particle states with ǫ α ≤ j α ≤ / ~ . The same simple interaction asthat used in the ground-state calculation is used as theresidual interaction [7]. Since the residual interaction isnot consistent with the effective interaction used in thecalculation of the single-particle states, it is necessaryto reduce the strength of the residual interaction in thislarge space calculation so that the spurious mode cor-responding to the center-of-mass motion comes at zeroexcitation energy in RPA. We found that the reductionfactor f is 0.66 [9]. This factor is used in the A , B and C parts of Eq. (1). We define the 3p–1h, 1h–3p, 1p–3h and3h–1p amplitudes using the same single-particle statesas those used in the ground-state calculation. In thissingle-particle space there are no isovector dipole statesconsisting of 2p–2h and 2h–2p configurations.The occupation probabilities calculated in TDDM for O are shown in Table I. The results in EDA which areobtained using the same single-particle states and inter-action as those used in TDDM are given in the parenthe-ses. The results in TDDM agree well with the EDA re-sults. The occupation probabilities calculated in TDDMfor Ca are given in Table II. The deviation of n α fromthe HF values ( n α = 1 or 0) exceeds 10 %, indicating thatthe ground states of O and Ca are highly correlatedas other calculations [12, 13] have already suggested.In Fig. 1 the isovector dipole strength distributionscalculated in ERPA (solid lines) for O are comparedwith those in RPA (dashed line) and EDA (dot-dashed
TABLE I. Single-particle energies ǫ α and occupation proba-bilities n αα calculated in TDDM for O. The results in EDAare given in the parentheses. ǫ α [MeV] n α orbit proton neutron proton neutron1 p / -18.3 -21.9 0.920(0.914) 0.920(0.914)1 p / -12.3 -15.8 0.898(0.890) 0.897(0.890)1 d / -3.8 -7.2 0.087(0.094) 0.087(0.094)TABLE II. Single-particle energies ǫ α and occupation proba-bilities n αα calculated in TDDM for Ca. ǫ α [MeV] n α orbit proton neutron proton neutron1 d / -15.6 -22.9 0.923 0.9241 d / -9.4 -16.5 0.884 0.8842 s / -8.5 -15.9 0.846 0.8461 f / -3.4 -10.4 0.154 0.154 lines). The peak at 21.2 MeV in RPA consists of the1 p / → d / transitions and corresponds to GDR. Thedipole strength in ERPA is largely fragmented due tothe coupling to the 3p–1h and 1p–3h configurations: Thecontributions of the 3h–1p and 1h–3p configurations aresmall because they have negative energies. As mentionedabove, there are no dipole states consisting of the 2p–2h and 2h–2p configurations in the small single-particlespace used here. Since the unperturbed energies of the3p–1h and 1p–3h configurations are smaller than theGDR energy, the main GDR strength is shifted upwardin ERPA due to the coupling to these two-body con-figurations. The strength distribution in ERPA aboveGDR qualitatively agrees with the EDA result which alsoshows the fragmentation of GDR. ERPA shows the smallstrength distribution below GDR ( E < . Ca using the trun-cated single-particle space are shown in Fig. 2. The re-sult in RPA is depicted with the dashed line. The resid-ual interaction used in the A , B and C parts of Eq. (1)is multiplied by f = 0 .
66 to be compared with a morerealistic calculation shown below. The EDA results arenot given because it is hard to perform EDA calculationseven in the small single-particle space used for Ca. Thepeak at 15.9 MeV in RPA consists of the 1 d / → f / transitions and corresponds to GDR. As in the case of
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EDARPA S [ e f m ] E [MeV] ERPA
FIG. 1. Isovector dipole strength distributions calculated inERPA (solid lines), RPA (dashed line) and EDA (dot-dashedlines) for O. O the dipole strengths are fragmented in ERPA due tothe coupling to the 3p–1h and 1p–3h configurations. Inthis truncated single-particle space used for Ca thereare no dipole states consisting of the 2p–2h and 2h–2pconfigurations. Since the unperturbed energies of the 3p–1h and 1p–3h configurations are around the GDR energy,the dipole strength in ERPA is distributed both aboveand below GDR.In the following we show for Ca how the 3p–1h and1p–3h configurations affect the damping of GDR in amore realistic calculation which includes a large numberof single-particle states for x µαα ′ . The result of such anERPA calculation is shown in Fig. 3 with the solid line.The dotted line depicts the result in RPA. The distri-butions are smoothed with an artificial width Γ = 0 . LS closed shell nuclei O and Ca were studied us-ing an extended random phase approximation derivedfrom the time-dependent density-matrix theory. It waspointed out that the unconventional two-body ampli- tudes of one particle–three hole and three particle–onehole types which are neglected in most extended RPAtheories play an important role in the fragmentation ofthe dipole strength because the ground states of these nu-clei are highly correlated. Our results suggest that theseunconventional configurations should also be included in S [ e f m ] E [MeV] RPAERPA
FIG. 2. Isovector dipole strength distributions calculated inERPA (solid lines) and RPA (dashed line) for Ca.
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10 15 20 25 30050100150200 () [ m b ] E [MeV]
ERPA S ( E ) [ e f m / M e V ] E [MeV] RPA
FIG. 3. Isovector dipole strength functions calculated inERPA (solid line) and RPA (dotted line) for Ca. The distri-butions are smoothed with an artificial width Γ = 0 . extended RPA studies of GDR’s in these nuclei. [1] S. Dro˙zd˙z, S. Nishizaki, J. Speth, J. Wambach, Phys.Rep. 197 (1990) 1. [2] O. Vasseur, D. Gambacurta, M. Grasso, Phys. Rev. C 98(2018) 044313; D. Gambacurta, M. Grasso, O. Vasseur,[1] S. Dro˙zd˙z, S. Nishizaki, J. Speth, J. Wambach, Phys.Rep. 197 (1990) 1. [2] O. Vasseur, D. Gambacurta, M. Grasso, Phys. Rev. C 98(2018) 044313; D. Gambacurta, M. Grasso, O. Vasseur,