Continuum Coupling and Pair Correlation in Weakly Bound Deformed Nuclei
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International Journal of Modern Physics Ec (cid:13)
World Scientific Publishing Company
CONTINUUM COUPLING AND PAIR CORRELATION INWEAKLY BOUND DEFORMED NUCLEI
HIROSHI OBA
Graduate School of Science and Technology, Niigata University, Niigata 950-2181, [email protected]
MASAYUKI MATSUO
Department of Physics, Faculty of Science, Niigata University, Niigata 950-2181, [email protected]
We formulate a new Hartree-Fock-Bogoliubov method applicable to weakly bound de-formed nuclei using the coordinate-space Green’s function technique. An emphasis is puton treatment of quasiparticle states in the continuum, on which we impose the correctboundary condition of the asymptotic out-going wave. We illustrate this method withnumerical examples.
1. Introduction
The RI-beam facilities in the new generation will increase significantly the numberof experimentally accessible nuclei, especially in medium and heavy mass regions.We may reach nuclei close to the neutron drip-line in the 10 ≤ Z ≤
20 and N ≥ , , , , ovember 5, 2018 11:25 WSPC/INSTRUCTION FILE JFS H. Oba and M. Matsuo show that the coordinate-space Green’s function technique enables us to formulatethe deformed continuum HFB method in which the nucleon waves satisfy a properboundary condition of the asymptotic out-going wave.
2. Deformed continuum HFB method using the Green’s function
We first describe the quasi-particle motion in the HFB mean-fields consisting of theparticle-hole field and the pair field which are both deformed. The axial symmetryis assumed. The quasiparticles of the Bogoliubov type have two-component wavefunctions ψ (1 , ( r σ ), for which we use the radial coordinate system and the partialwave expansion ψ ( i ) ( r σ ) = L max X L φ ( i ) L ( r ) y L (ˆ r σ ) (1)with L ≡ ( jlm ) and y L (ˆ r σ ) being the spin spherical harmonics. The HFB equationis then written as a coupled-channel Schr¨odinger equations9 for the radial wavefunctions { φ ( i ) L ( r ) } where the quantum number L represents the “channel”. Notethat the energy spectrum of the quasiparticle consists of discrete and continuumparts, which are separated by the energy condition E < | λ | and E > | λ | ( λ is theFermi energy) as is in the spherical case.4We can construct the exact Green’s function for the quasiparticle motion inour deformed HFB problem. It is an extension of the spherical theory of Ref.3 todeformed cases, and we accomplished this by employing a general prescription10of constructing the exact Green’s function for a deformed potential scatterer. Herethe HFB Green’s function (a 2 × G ( r σ, r ′ σ ′ , E ) = N c X L,L ′ y L (ˆ r σ ) g LL ′ ( r, r ′ , E ) y † L ′ (ˆ r ′ σ ′ ) . (2)The coupled-channel radial Green’s function g LL ′ ( r, r ′ , E ) is constructed as a linearcombination of products of “regular solutions” { φ I( i ) LL ′ ( r ) } (i.e. those satisfying theboundary conditions φ I( i ) LL ′ ( r ) → r l δ LL ′ δ ij at the origin r →
0) and “out-going wavesolutions” { φ O( i ) LL ′ ( r ) } (those connected to the proper asymptotic form φ O( i ) LL ′ ( r ) → r − H + l ′ ( k i r ) δ LL ′ δ ij for r → ∞ where H + l ′ ( kr ) is the out-going Hankel function).We calculate the density ρ ( r ) and the pair density ˜ ρ ( r ) using the HFB Green’sfunction thus constructed. The generalized density matrix R ( r σ, r ′ σ ′ ) = (cid:18) ρ ( r σ, r ′ σ ′ ) ˜ ρ ( r σ, r ′ σ ′ )˜ ρ ∗ ( r ˜ σ, r ′ ˜ σ ′ ) δ rr ′ δ σσ ′ − ρ ( r ˜ σ, r ′ ˜ σ ′ ) (cid:19) = 12 πi Z C G ( r σ, r ′ σ ′ , E ) dE, (3)which is a sum of the wave functions of all the quasiparticle states including the con-tinuum states, is calculated using a contour integral of the HFB Green’s function3.Incorporating this way of calculating densities into the standard iterative algorithm,we obtain the HFB ground state after convergence.ovember 5, 2018 11:25 WSPC/INSTRUCTION FILE JFS Continuum Coupling and Pair Correlation in Weakly Bound Deformed Nuclei ρ ( r ) and ρ ( r ), of the neutron density (left panel).The monopole and quadrupole parts, ˜ ρ ( r ) and ˜ ρ ( r ), of the neutron pair density (right). Thedashed curves are the results obtained with the box boundary condition.
3. Numerical analysis
We shall demonstrate with numerical examples how the deformed continuum HFBworks. We adopt for simplicity a deformed Woods-Saxon potential as the particle-hole field, but we perform the HFB iteration to obtain the selfconsistent pair field.We use the density-dependent delta interaction (DDDI) acting in the singlet pair, v pair = v (cid:0) − η ( ρ n ( r ) / . . (cid:1) δ ( r − r ′ ) , (for neutrons) where v is fixed toreproduce the nn -scattering length a = −
18 fm. We consider Mg and assume adeformation β = 0 .
3. Using the Runge-Kutta-Nystrom method we solve numericallythe coupled-channel equation within an interval r = [0 , r max ] ( r max = 15 fm) witha step size ∆ r = 0 . η = 0 .
76 is chosen to produce the neutronpairing gap around ∆ ∼ . j z ) quantum number is Ω max = 21 /
2. For comparison, weperformed also the HFB calculation using the same model but with a box boundarycondition assuming an infinite wall at r = r max . In the following we show resultsfor neutrons.Figure 1 shows the radial profile of the monopole and quadrupole parts ρ ( r ) and ρ ( r ) of the neutron density ρ ( r ) = P λ ρ λ ( r ) Y λ (ˆ r ), and the corresponding ˜ ρ ( r )and ˜ ρ ( r ) of the neutron pair density ˜ ρ ( r ). We obtain exponential asymptotics herethanks to the proper boundary condition, and it is in contrast to the results obtainedwith the box boundary condition (the dashed curves in Fig.1). ρ ( r ) and ρ ( r ) havethe same exponential slope, indicating that we can define the deformation of theequi-density surfaces in the asymptotic region. This kind of deformed exponentialtail is also seen in the neutron pair density. But the ratio of ˜ ρ ( r ) against ˜ ρ ( r ) issignificantly smaller than that of the normal density. This points to that the pairdensity in the tail has smaller deformation than that of the normal density.The quasiparticle spectrum above the threshold energy E th = | λ | should be con-ovember 5, 2018 11:25 WSPC/INSTRUCTION FILE JFS H. Oba and M. Matsuo
Fig. 2. The occupation number density n ( E ) (left panel) and the pair number density ˜ n ( E ) (rightpanel) for neutrons, plotted with the solid curves. The results obtained with the box boundarycondition are also plotted with the dashed curves. The inset is a magnification of ˜ n ( E ), andwe compare it with the result (the dotted curve) obtained with a Woods-Saxon potential whosebottom is shifted up by +2 MeV. tinuous, and it is indeed the case in our formulation. Figure 2 shows the occupationnumber density n ( E ) and the pair number density ˜ n ( E ) which are defined by n ( E ) = 1 π Im X σ Z d r G ( r σ, r σ, − E − iǫ ) , ˜ n ( E ) = 1 π Im X σ Z d r G ( r σ, r σ, − E − iǫ ) . (4)They quantify contributions of the quasiparticle state at energy E to the neutronnumber R ρ ( r ) d r = N and to R ˜ ρ ( r ) d r . Here is shown only for the neutron Ω = 1 / ǫ = 25 keV (a discrete state wouldhave artificial FWHM=50 keV).In this numerical example there is no discrete quasiparticle states below | λ | ( λ = −
889 keV), and all the quasiparticle states are embedded in the continuum
E > | λ | .The quasiparticle states corresponding to deep hole Woods-Saxon orbit appear bothin n ( E ) and ˜ n ( E ) as narrow resonances. It is observed also that the non-resonantcontinuum states have a significant contribution to the pair number density ˜ n ( E ).This figure shows also that the box-discretized calculation has difficulty to describethe non-resonant continuum states. The lowest energy resonance is not describedwell by a single state in the box-discretized calculation, and it is because thisresonance has a rather large width. This resonance corresponds to the [310] orbitin the deformed Woods-Saxon potential, which is, in the absence of the pairing, abound state with the single-particle energy e = −
798 keV.Naturally we expect most dramatic effect of the weak binding on this state.The inset shows how the spectrum ˜ n ( E ) changes when the depth of the Woods-Saxon potential is artificially shifted (made shallower) by 2 MeV. The Woods-Saxonsingle-particle energy e and the Fermi energy λ changes from e, λ = − , − e, λ = − , −
88 keV. We see in the inset of Fig.2 a dramatic increase inthe width of the lowest-energy resonance, which apparently originates from theovember 5, 2018 11:25 WSPC/INSTRUCTION FILE JFS
Continuum Coupling and Pair Correlation in Weakly Bound Deformed Nuclei weak binding. Note however that the peak energy of the resonance stays almostconstant. This implies that the effective pairing gap of this resonant quasiparticlestate is unchanged, if we estimate the effective pairing gap through the relation E qp = p ( e − λ ) + ∆ . This observation is different from that in Ref.9 claiming areduction of the effective pairing gap due to the weak binding effect. The differenceoriginates from the fact that we here use the selfconsistent pairing field generatedfrom the DDDI, whose force strength becomes large at low densities. The pair fieldextending to far outside the nucleus plays important roles.
4. Conclusions
We have formulated the deformed continuum HFB method which is designed forweakly bound deformed nuclei. We utilized here the exact construction of the quasi-particle Green’s function for deformed HFB mean-fields on the basis of the coupled-channel representation. The proper boundary condition of the out-going wave isimposed on the continuum quasiparticles. We have analyzed numerically effects ofthe continuum coupling and the weak binding on the pair correlation. It is foundthat the quasiparticle states in the non-resonant continuum play significant role togenerate the pair correlation. It is also suggested that the effective pairing gap ofweakly bound orbits is not reduced very much, provided that the pairing interactionhas the surface enhancement.
Acknowledgments
The work is supported by the Grant-in-Aid for Scientific Research(No.20540259)from the Japan Society for the Promotion of Science, and also by the JSPS Core-to-Core Program, International Research Network for Exotic Femto Systems(EFES).
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