Nuclear Science Seminar (NSS) Nov.13, 2006 Weakly-bound and positive-energy neutrons in the structure of drip-line nuclei - from spherical to deformed nuclei
6. Weakly-bound and positive-energy neutrons harmonic oscillator potential finite-well potential The study of one-particle motion is the basis for understanding not only the single-particle mode but also the many-body correlation in nuclei.
The study of nuclei (far) away from the stability line becomes possible, due to The development of various kinds of Radioactive Nuclear Ion Beams (RNB) facilities - some are operational, while others being constructed. Europe ; GSI [ FAIR project ], Germany; GANIL [ SPIRAL2 project ], France; CERN [ Rex Isolde ], in Geneva ; Jyvaeskylae, Finland ; Catania, Italy USA ; [ RIA project ], MSU, ANL, Oak Ridge Japan ; RIKEN [ RIBF project ] soon operational! China ; Lanzhou The field has a close connection with the interests in Nuclear Astrophysics ; nucleosynthesis ( r-process, rp process, ) evolution of cosmos, neutron matter or neutron star,.,
drip-line : unstable against nucleon (proton or neutron) emission proton separation energy, S p = 0 proton drip line neutron separation energy, S n = 0 neutron drip line From the lecture notes by H.Sakurai at CISS04
Hartree-Fock (HF) potential is sometimes approximated by a Woods-Saxon potential, in order to understand the physics in a simple terminology and in a systematic way. Woods-Saxon potential V WS V(r) R 2a r V ( r) = V f ( r) WS where 1 f ( r) = r R 1+ exp a and 1/ 3 R = r 0 A a : diffuseness A : mass number
β-stable nuclei (S p S n 7-10 MeV) One-particle levels which contribute to many-body correlations V(r) r neutron drip line nuclei an important role of continuum and weakly-bound (bound many-body systems) one-particle levels V(r) V(r) Continuum levels r Γ r Importance of one-particle resonant levels with small width Γ in the many-body correlations. Obs. no one-particle resonant levels for s 1/2 orbits.
Hartree-Fock potential and one-particle energy levels V N (r) : neutron potential, V P (r) : proton nuclear potential, V P (r)+v C (r) : proton total potential A typical double-magic β-stable nucleus 208 Pb 82 126
Hartree-Fock potentials and one-particle energy levels V N (r) : neutron potential, V P (r) : proton nuclear potential ex. of neutron-drip-line nuclei ex. of proton-drip-line nuclei 3s1/2 50 50
6.1. Weakly-bound and positive-energy neutron levels in spherical potential Centrifugal potential + Woods-Saxon potential dependence on l + + + 2m x y z 2 2 2 2 V() r 2 2 2 2 2 2 2 1 1 1 cot ( ) 2 2 2 2 2 2 r = + + θ + + V r m r r r θ θ sin θ φ 2 2 2 1 1 ( ) = r + V() r 2 2 2 2m r r r centrifugal potential Woods-Saxon pot. centrifugal pot. W-S + centrifugal pot.
Height of centrifugal barrier + ( 1) 2 R h where R h > r 0 A 1/3 The height is higher for smaller nuclei and for orbits with larger l. ex. For the Woods-Saxon potential with R=5.80 fm, a=0.65 fm, r 0 =1.25 and V WS = 50 MeV ; l height of centrifugal barrier 0 0 MeV 1 0.4 2 1.3 3 2.8 4 5.1 5 8.2
Neutron radial wave-functions l = 4 1 Ψ ( ) ( ) ( ˆ njm r = Rnj r Xjm r) r l = 0 ε = 8MeV ε = 200keV halo
For a finite square-well potential V(r) The probability for one neutron to stay inside R 0 the potential, when the eigenvalue ε nl (< 0) 0 r l 0 1 2 3 R 0 0 2 Rn ( r) dr 0 1/3 3/5 5/7 Root-mean-square radius, r rms, of one neutron ; rrms r 2 In the limit of ε nl (<0) 0 1/2 r rms ( ) for l = 0 ε n ( ) ε n 1/4 for l = 1 finite value for l 2
Schrödinger equation for one-particle motion with spherical finite potentials H = + + + 2m x y z HΨ= εψ where 2 2 2 2 V() r 2 2 2 + V s () r 1 R () () ˆ nj rx jm j r r Ψ= 1 X () rˆ = C(,,; j m, m, m ) Y (, θφχ ) s 2 2 2 ( ) Y ( θ, φ) = ( + 1) Y ( θ, φ) jm j s j m 1/2, m m, m m m s (x, y, z) (r, θ, φ) The Shrödinger equation for radial wave-functions is written as d dr 2 ( + 1) 2m + ( ε nj V() r Vs() r ) Rnj () r = 0 r 2 2 2 ($) For example, for neutrons eq.($) should be solved with the boundary conditions; at r = 0 R () r = 0 at r large (where V(r) = 0) for for 2 2m ε < 0 R() r αrh( αr) where α = ε 2 and ε > 0 R () r cos( δ ) krj ( kr) sin( δ ) krn ( kr) δ : phase shift h j n : spherical Hankel function : spherical Bessel function : spherical Neumann function where h ( iz) j ( z) + in ( z) k 2m 2 = ε 2
Unique behavior of low-l orbits, as E nlj (<0) 0 Energies of neutron orbits in Woods-Saxon potentials as a function of potential radius Fermi level of neutron drip line nuclei Fermi level of stable nuclei R = radius, r 0 = 1.27 fm, V WS = 51 MeV Strength of the potential For stable nuclei (R/r 0 ) 3 A : mass number
Neutron one-particle resonant and bound levels in spherical Woods-Saxon potentials Unique behavior of l=0 orbits, both for ε nlj <0 and ε nlj >0 }One-particle resonant levels with width 1d 5/2 (6) 2s 1/2 (2) 1d 3/2 (4) π Rj ( r) sin kr + δj 2 2mε for r and kr r 2 width 2 Γ dδ dε ε = ε res δ (phase shift) Strength of the potential (π/2) res ε ε
Some summary of weakly-bound and positive-energy neutrons in spherical potentials (β=0) Unique role played by neutrons with small l ; s, (p) orbits (a) Weakly-bound small-l neutrons have appreciable probability to be outside the potential; ex. For a finite square-well potential and ε nlj (<0) 0, the probability inside is 0 for s neutrons 1/3 for p neutrons Thus, those neutrons are insensitive to the strength of the potential. Change of shell-structure (b) No one-particle resonant levels for s neutrons. Only higher-l neutron orbits have one-particle resonance with small width. Change of many-body correlation, such as pair correlation and deformation in loosely bound nuclei
6.2. weakly-bound neutron levels in Y 20 deformed potential One-particle levels in (Y 20 ) deformed harmonic oscillator potentials A.Bohr and B.R.Mottelson, vol.2, Figure 5-1. [321 3/2] [N n z Λ Ω ] asymptotic quantum numbers Parity π = (-1) N Each levels are doubly-degenerate with ±Ω 6 doubly-degenerate levels in sd-shell 3 Ω π =1/2 + (l min =0) 2 Ω π =3/2 + (l min =2) 1 Ω π =5/2 + (l min =2)}12 particles ex. components of Ω π =½ + are s 1/2, d 3/2, d 5/2, g 7/2,..
In Y 20 -deformed Woods-Saxon potentials I.H., Phys. Rev. C69, 041306R (2004) Structure of the [220 ½] wave function depending on ε Ω Probabilities of s 1/2, d 3/2, d 5/2 and g 9/2 s 1/2 radial wave functions halo one-particle energy eigenvalues obtained by adjusting the radius of Woods-Saxon potentials
For ε 0, the s-dominance will appear in all Ω π =1/2 + bound orbits. However, the energy, at which the dominance shows up, depends on both deformation and respective orbits. ex. three Ω π =1/2 + Nilsson orbits in the sd-shell ;
PRC69, 041306R (2004) Structure of the [330 ½] wave function depending on ε Ω
Radial wave functions of the [200 ½] level in Woods-Saxon potentials s 1/2 d 3/2 d 5/2 (The radius of potentials is adjusted to obtain respective eigenvalues ε Ω.) Bound state with ε Ω = -8.0 MeV. Bound state with ε Ω = -0.0001 MeV. Similar behavior to wave functions in harmonic osc. potentials. Wave functions unique in finite-well potentials.
I.H., Phys. Rev. C69, 041306R (2004) Deformed halo nuclei l min =0 Ω π =1/2 + neutron orbit s 1/2, as ε Ω 0. deformed core, irrespective of the size of deformation and the kind of one-particle orbits. The rotational spectra of deformed halo nuclei must come from the deformed core.
6.3. Positive-energy neutron levels in Y 20 -deformed potentials Ω π = 1/2 + s 1/2, d 3/2, d 5/2, g 7/2,g 9/2,.., components l min = 0 Ω π = 3/2 + d 3/2, d 5/2, g 7/2, g 9/2,.., components l min = 2 Ω π = 1/2 p 1/2, p 3/2, f 5/2, f 7/2, h 9/2,.., components l min =1 etc. The component with l = l min plays a crucial role in the properties of possible one-particle resonant levels. (Among an infinite number of positive-energy one-particle levels, one-particle resonant levels are most important in the construction of many-body correlations of nuclear bound states.)
W-S potential parameters are fixed except radius R. I.H., Phys. Rev. C72, 024301 (2005) Woods-Saxon l min = 0 l min = 2 l min = 1 [200 ½] [220 ½] [211 ½] potential strength (r 0 = 1.27 fm is used.)
One-particle neutrons in deformed finite-well potentials Coupled eqs. in coordinate space with correct asymptotic behavior are solved. Single-particle wave-function in Y 20 deformed Woods-Saxon potential, 1 Ψ ( ) ( ) ( ˆ Ω r = R jω r X jω r) r j which satisfies HΨ = ε Ψ Ω Ω Ω where 1 X () ˆ Ω r = C(,,; j m, m, Ω) Y (, θφχ ) j s m 1/2, m m, m s 2 s Taking the lowest-order term in deformation parameter β of the deformed Woods-Saxon potential, the coupled equations for the radial wave-functions are written as dr + r V() r V () r R () r X V X R () r 2 d ( 1) 2m 2m + 2 2 2 [ ε Ω s ] j Ω = 2 j Ω coupl ' j ' Ω ' j ' Ω ' j ' where V WS V ( r) = f ( r), V coupl ( r ) = βk( r) 20( rˆ) Y 1 f ( r) = r R 1+ exp a, k( r) = rv WS df ( r) dr
For ε Ω R < 0 jω ( r) rh ( α r) b In the calculation, do not confine the system to a finite box! for r where h ( iz) j ( z) + in ( z) and α 2 2 b mε 2 Ω For ε Ω R jω where > 0 ( r) cos( δω) rj( αcr) sin( δω) rn( αcr) π sin( α + ) c r δ Ω m ε Ω 2 2 αc 2 2 for r δω expresses eigenphase, common to all (l,j) components. A.U.Hazi, Phys.Rev.A19, 920 (1979) and references quoted therein. A given eigenchannel with a given eigenphase : radial wave-functions of all (l,j) components behave asymptotically in the same way.
A one-particle resonant level with ε Ω is defined so that one of eigenphases δ Ω increases through (1/2)π as ε Ω increases. δ δ Ω (1/2)π ε Ω ε res When one-particle resonant level in terms of one eigenphase is obtained, the width Γ of the resonance is calculated by 2 Γ dδ Ω d εω res ε Ω = ε Ω Phys. Rev. C72, 024301 (2005)
I.H., Phys. Rev. C73, 064308 (2006) Some comments on eigenphase ; 1) For a given potential and a given ε Ω there are several (in principle, an infinite number of) solutions of eigenphase δ Ω. 2) The number of eigenphases for a given potential and a given ε Ω is equal to that of wave function components with different (l,j) values. 3) The value of δ Ω determines the relative amplitudes of different (l,j) components. 4) In the region of small values of ε Ω ( > 0), only one of eigenphases varies strongly as a function of ε Ω, while other eigenphases remain close to the values of nπ.
Radial wave functions of the [200 ½] level s 1/2 d 3/2 d 5/2 The potential radius is adjusted to obtain respective eigenvalue (ε Ω < 0) and resonance (ε Ω > 0). Bound state with ε Ω = 0.1keV Resonant state with ε Ω = +100 kev Existence of resonance d component Width of resonance s component OBS. Relative amplitudes of various components inside the potential remain nearly the same for ε Ω = 0.1keV + 100 kev.
Bohr & Mottelson, vol.ii, Table 5-9 δ=0.4 with modified oscillator potential Namely, [220 ½]> = - 0.523 s 1/2 > - 0.285 d 3/2 > + 0.803 d 5/2 > [211 ½ ]> = 0.419 s 1/2 > + 0.735 d 3/2 > + 0.533 d 5/2 > [200 ½] > = 0.743 s 1/2 > - 0.615 d 3/2 > + 0.265 d 5/2 >
Relative probability of s 1/2 component inside the W-S potential P( s 1/ 2 ) = d 5/ 2 V ( r) d 5/ 2 + s 1/ 2 d 3/ 2 V ( r) s 1/ 2 V ( r) d 3/ 2 + s 1/ 2 V ( r) s 1/ 2 In order that one-particle resonance continues for ε Ω >0, P(s 1/2 ) at ε Ω =0 must be smaller than some critical value. The critical value depends on the diffuseness of the potential. One-particle shell-structure change for ε Ω (<0) 0 produces the large change of P(s 1/2 ) values of respective [N n z ΛΩ] orbits as ε Ω (<0) 0. One-particle resonance
Some summary of weakly-bound and positive-energy neutron levels in Y 20 deformed potential. Unique role played by neutrons with Ω π =1/2 +, where l min =0. OBS. The number of Ω π =1/2 + levels in deformed potentials is larger than that of s 1/2 level in spherical potentials. ex. In the sd-shell a half of Nilsson levels have Ω π =1/2 +! (a) Bound Ω π =1/2 + neutron orbit s 1/2, as ε Ω 0. (b) For ε Ω > 0 and usual size of deformation, neutron Ω π =1/2 + level can hardly survive as one-particle resonance. Possible resonance quickly decays via the admixed s 1/2 component. (c) Only one-particle resonant states with larger Ω-values having smaller Γ in the continuum may contribute to the many-body correlation in the bound many-body system. Possible consequence on ; deformed or spherical shape of neutron drip line nuclei? pair-correlation vs. deformation in the many-body correlation?