Stability Peninsulas at the Neutron Drip Line Dmitry Gridnev 1, in collaboration with v. n. tarasov 3, s. schramm, k. A. gridnev, x. viñas 4 and walter greiner 1 Saint Petersburg State University, St. Petersburg, Russia; Frankfurt Institute for Advanced Studies, J.W.G. University, Frankfurt, Germany; 3 NSC Kharkov Institute of Physics and Technology, Kharkov, Ukraine; 4 University of Barcelona, Barcelona, Spain; Symposium on Exciting Physics, 13- November, South Africa
Nature 449, p. 1 (7)
σ r r r = (1 + σ ) δ ( ) + 1 1 σ δ δ ij V t x P r 3 3 6 t x P ρ α R δ r σ 1 (1 )[ ( ) ( ) ] t x P k r + r + r k r + r r r 1 r r r r r r r + t (1 + x P ) k δ ( ) k + (1 + ) ( ) ( ) + i W [ k δ ( ) k ]( σ + σ ), where r = r i r r 1 r r r i r r r i s s, ( ) j R = +, k = ( ) i j i j, k = ( i j ), t, x ( n =,1,,3), α, W - parameters. n n Skyrme Forces P σ 1 = (1 + r σ r iσ j ), i j Skyrme Forces used in the calculations Force t t 1 t MeV fm 3 MeV fm 5 t MeV fm 5 3 x x 1 x x 3 W MeV fm 3+3α MeV fm 5 SLy4-488,91 486,8-546,39 13777,,834 -,344-1, 1,354 13, 1/6 SkM* -645. 41. -135. 15595..9,,, 13. 1/6 Ska -16,78 57,88-67,7 8, -,,, -,86 15, 1/3 S3-118.75 395. -95. 14..45.. 1. 1. 1 SkI -1915.43 438.45 35.45 1548.9 -.11-1.738-1.534 -.178 1,6 1/4 SkI5-177.91 55.84-16.69 86.5 -.117-1.39-1.49.341 13.6 1/4 SkP -931.7 3.6-337.41 1879..9.653 -.537.181 1. 1/6 α
equations: r h r r r r r r [ r + U ( R) + W ( R) ( i)( σ ) Φ = e Φ m q ( R) Hartree Fock Equations with Skyrme Forces i i i Axially deformed oscillator basis for deformed nuclei (D) r r i r iλϕ Φ i ( R, σ, q) = χq ( q) C ( R, ) i αϕα σ Λ e ϕn (, ) ( ) ( ) ( ) r n R σ ψ zλσ = n r ψ r n z χ σ z Σ α π 1/ n ( z) N ( ) z nz z e ξ 1 ψ = β H n ξ ξ = zβ z z Nn = z n π z nz! 1 Λ η n! Λ Λ Λ r ψ n ( r) = N ( ) r n β η e L r n η η = r β N Λ n = r r ( nr + Λ)! Spherically symmetric nuclei are also solved through iterated differential equations (S) h lα ( lα + 1) d h 1 d h 3 R ( r) R ( r) R ( r) { U ( ) [ ( 1) ( 1) ] * α + α * α + q r + j j l l * + α α + α α + m q r dr m q r dr m q 4 1 1 Wq ( r)} Rα ( r) = eα Rα ( r) r BCS approximation with the paring constant G n,p =(19.5/)[1 ±.51(N - Z)/A] D: only with bound single-particle states. S: also quasistable continuum states under the centrifugal barrier are included.
Nuclear Chart (B vs Experiment) Z 1 SkM* 16 184 8 8 6 4 3 8 5 8 Exp. B Dobaczewski (SkM*) 4 8 1 16 4 N Green region shows the atomic nuclei which are experimentally known [Audi, Wapstra]. Solid line and black squares represent 1n drip line(λ n =) obtained in the B calculations with Skyrme forces SkM* [Stoitsov, Dobaczewski, Nazarewicz, Pittel, PR. C68 (3) 5431 ]. The orange arrow shows typical direction in the or B calculations.
Z 1 SkM* 16 184 8 Rn 8 8 76-78 Pb 6 5 3 4 8 8 4-4 O 76 Ar 15-15 Zr 16-13 Kr 11 Ni Exp (Ska, SkM*) B Dobaczewski (SkM*) 4 8 1 16 4 Red squares show isotopes stable with respect to one-neutron emission. Z is changed in the direction shown by the orange arrow. Stability enhancement should be checked on neutron magic numbers. N
a S n, (MeV) S n (Ska) 5 Exp λ n (Ska) 15 1n, n drip line 1 A=8 Z 14 Si (Ska) 1 Mg 1 Ne 8 O 6 4 O C 4 8 3 36 5-5 1 16 4 8 3 36 4 44 N One neutron separation energies for Oxygen and the drip line fragment The stability enhancement through adding neutrons. The last neutron is also confined by the centrifugal barrier (dotted line shows its density distribution).
Comparison of single particle spectra in S, D and RMF methods. Ska forces for and G1,,3 and NL3 for the RMF method Spherical with Ska Axially deformed with Ska
S n, MeV 5 а Oxygen S n Z 14 1 ÕÔ (SkM*) Si Mg 15 λ n λ n B 1 8 6 4 O Ne O C 1 16 4 8 3 36 N 5 1n, n drip-line β n,p -5 1 16 4 8 3 36 4 44,3,,1 b β n β p B A, -,1 -, -,3 1 16 4 8 3 36 4 44 The properties of oxygen isotopes in D and B methods with SkM* forces. The insertion represents the D neutron drip line. (a) - One neutron separation energies S n and neutron chemical potential. (b) - Neutron and proton parameters of deformation β n,p. A
ε i, MeV - -4 1f 5/ 1f 7/ p 1/ p 3/ 1d 3/,5,5,333 1f 5/ p 1/ 1f 7/ p 3/ -6-8 -1 s 1/ 1h 11/ 1g 7/ 36 Ο,167 38 Ο,333,5 4 Ο,667 4 Ο,833 1, 44 Ο,5 1d 3/ s 1/ p 3/ 1h 11/ The fragments of neutron single-particle spectra for extremely neutron rich isotopes of Oxygen and Nickel. - d 3/ 1g 7/ ε i (MeV) -4-6 3s 1/ d 5/ d 3/ 3s 1/ d 5/ -8 1g 9/ -1 98 Ni 1 Ni 1 Ni 14 Ni 16 Ni 18 Ni 11 Ni 1g 9/ 11 Ni
Z 1 SkM* 16 184 8 8 6 5 3 4 8 8 166 Zr(1i ) 4 13/ 4 56 Ba(1k 15/ ) 18 6 Fe(1h 11/ ) 74 S(d ) N=58 16 5/ 4 O(1f ) 8 7/ Exp. (Ska, SkM*) B Dobaczewski (SkM*) 4 8 1 16 4 N Red squares form the peninsulas of stability. In brackets: the quantum numbers of the stabilizing level.
8 Z 1 SkM* 16 184 6 16 8 6 3 5 8 5 184 4 56 Ba 166 4 Zr 4 1 14 16 18 4 8 8 166 4 Zr(1i 13/ ) 4 56 Ba(1k 15/ ) 18 6 Fe(1h 11/ ) 74 S(d ) N=58 16 5/ 4 O(1f ) 8 7/ Exp. (Ska, SkM*) B Dobaczewski (SkM*) 4 8 1 16 4 N Detailed picture of stable isotopes with respect to one neutron emission between N=16 and N=184.
Z 1 SkI 16 184 8 8 6 5 3 5 4 8 36 5 Te(1k 15/ ) 8 18 6 Fe(1h 11/ ) 7 Si(d ) N=58 14 5/ 4 O(1f ) 8 7/ 164 38 Sr(1i 13/ ) Exp. (SkI) B Dobaczewski (SkM*) 4 8 1 16 4 N The same as in the previous figures but for SkI forces. The location of the peninsulas is the same, but for N=58, 16, 184 the peninsulas edges appear at larger Z then for SkM* forces. For N=184 the peninsula edge corresponds to Tellurium with A=36.
Z 1 Sly4 16 184 8 8 6 3 5 48 64 Gd(1k 15/ ) 4 8 8 4 1 Ne(1f 7/ ) 11 Zn(1h ) 3 11/ 8 Ti(d ) 5/ 17 Ru(1i ) 44 13/ Exp. (Sly4) B Dobaczewski (Sly4) 4 8 1 16 4 The same as in the previous figures but for Sly4 forces. The location of the peninsulas is the same but the edges locate at smaller Z compared to SkM* and SkI forces. N
S n МeV 3 а Ska 66 Pb S n МeV 3 b SkM* 66 Pb 1 4 Ba 1 N=184 N=184 4 Ba -1-38 Xe S n (Ska) 5 56 6 64 68 7 76 8 84 88 Dependences of one-neutron separation energies S n and two-neutrons separation energies S n for the isotones with N=184 for the Ska, SkM* forces on Z. The red line shows the S n obtained by the B method [PR. C68 (3) 5431 ]. Z -1 - S n МeV 5 4 3 1-1 - 38 Xe S n (SkM*) 5 56 6 64 68 7 76 8 84 88 c 4 Ba SkM* 66 Pb N=184 5 56 6 64 68 7 76 8 84 88 Z S n (SkM*) S n BDobaczewski (SkM*) Z
Comparison with grid calculations Empty squares: spherical equations with resonances incuded in BCS Black filled squares: Deformed with only bound states in BCS
V(r),МeV 15 1 φ i (r)*1 4 Ba V(r),МeV 15 1 4 Ba 5-5 -1-15 1k 15/ h=8.48 МeV 4 6 8 1 1 14 16 r,фм g 7/ Ska 4 6 8 g 1 1 14 16 7/ r, Fm V(r),МeV 15 15 V(r),МeV 36 Te 48 Gd 5-5 -1-15 1k 15/ SkM* h=8.57 МeV potentials of the last filled level of the isotones with N=184. Red lines show the last filled level. 1 1 5-5 -1 1k 15/ h=7.6 МeV 4 6 8 1 1 14 16 r,fm 3d 3/ -5 SkI 5-1 3d 3/ 4 6 8 1 1 14 16 r,fm 1k 15/ Sly4 h=8.7 МeV -15-15 The mechanism of stability enhancement at N=184 (the edge of stability peninsula). The appearance of peninsulas beyond the drip line is due to the complete filling of the corresponding neutron subshells with large values of angular momentum. Being partially filled these shells locate in the continuum but descend into the discrete spectrum getting further filled.
S n, (MeV) S n, (MeV) 14 1 1 8 Gd S n (SkM*) Exp λ n (SkM*) 5 15 Gd (SkM*) Exp B (SkM*) 6 1 4 5 48 Gd - -4 14 16 18 4 6-5 A 14 16 18 4 6 A Left: One-neutron separation energies Gd with SkM* forces compared to B calculations [PRC 68 (3) 5431 ]. Right: the same for two neutrons separation energy.
Chemical Potential for Gd isotopes λ n, (MeV) 14 1 1 Gd λ n (SkM*) λ n B (SkM*) 8 6 4 14 16 18 4 6 A
Deformation parameters and Root Mean Square Radii for Gadolinium,5,4,3, β β m (SkM*) β B Dobaczewski (SkM*) Gd <r n,p >1/, (fm) 6,6 6,4 6, 6, r n (SkM*) r p (SkM*) r n B (SkM*) r p B (SkM*) Gd,1 48 Gd 5,8 5,6 48 Gd, 5,4 -,1 5, -, 5, -,3 14 16 18 4 6 A 4,8 14 16 18 4 6 A Empty circles on the left are unstable isotopes. Isotope 48 Gd belongs to the peninsula of stability.
Single particle spectra for 48 Gd Two different codes were used: allowing spherical shape and allowing possible deformation for SkM* forces. Red color shows the last filled level.
Proton and neutron density distributions for Oxygen Interesting relations between proton R p and neutron R n root mean square radii (SkM* forces) For nuclei in the middle of stability valley R n R p.1. fm 4 O R n R p 1.9 fm R n / R p 1.44 48 Gd R n R p.77 fm R n / R p 1.14 4 Ba R n R p.94 fm R n / R p 1.17
+BCS calculations of proton and neutron density distributions for 16,4 O and 146,48 Gd with SkM* forces.
The same for isotones with N = 184 (from 4 Ba up to 66 Pb excluding 58 W).
The same for isotones with N = 58 from Z=86 (Radon) up to Z=14.
Angular distribution of a Halo D θ x y θ θ 5 (, ˆ, ˆ) = cos sin ρ Ψ( ρ, θ, xˆ, yˆ) dρ ρ tan = y / x = x + y θ D( θ, xˆ, yˆ) 1 4 π 4 sin π θ
-E, МeV а 38 36 34 3 -E -E B S SkM* 1n drip line S n, -λ n, МeV 1 8 6 4 c S n -λ n -λ n B 74 S (N=58) S n, МeV V 3 15 1 5-5 b 4 5 6 7 8 S n S n B 4 5 6 7 8 β n,p -,4,3,,1 d 4 5 6 7 8 -,1 β -, n 74 S (N=58) β -,3 p β B -,4 A 4 5 6 7 8A Properties of neutron rich Sulphur calculated by D and B methods using SkM* forces. Dot line is 1n drip line. Fig. (a): Total energy Fig.(b): Two-neutron separation energies Fig. (c): One-neutron separation energies and chemical potentials Fig. (d): Neutron and proton deformation parameters,
Total energies of uranium isotopes One-neutron separation energies and neutron chemical potentials of uranium isotopes
Two-neutron separation energies of uranium isotopes Parameters of deformation β of uranium isotopes
Which is the heaviest Uranium isotope? Relativistic mean field and +BCS calculation. Left: assuming spherical symmetry. Right: axially deformed case. Model N (1d) N (d) Etot (d) NL-Z 58 58 1 MeV SkM* 46 1987 MeV SkI4 58 18 1975 MeV SLy6 184 6 199 MeV
Peninsula on the shell N=58
-178-171 -171 SkM* N =(1771) Pb 8-1714 E, MeV -1716-1718 -17 Pb 3-17 -174 4 6 8 1 Q m,barn Constrained calculations of 8,3 Pb energy surface using Ska forces.
Energy gaps of 3 Pb using SkM* forces. 1,,8 SkM* N =(1771) Pb 3 x n p n,p, MeV,6,4,, 4 6 8 1 Q m,barn
Mechanism of stability enhancement on the example of Pb isotopes
Density distribution in 3 Pb Left: normally deformed Right: superdeformed
β n,p β n,p,8,6 а Pb Ska,8,6 d Pb SkI,4,4 β n, β n, β p, β p,,8,6,4 b β n β p Pb SkM*,8,6,4 e β n β p Pb S3,,,,,8,6,4 c β n β p Pb SLy4,8,6,4 f β n β p Pb SkP,,,, 16 4 8 3 36 A 16 4 8 3 36 A Proton and neutron deformation parameters using Ska, SkM*, SkI, S3 and SkP forces
,8,6 a Hg SkM*,8,6 b Pb SkM* β n β p,4,4 β n,p,,,, 1 16 4 8 3 36 4,8,6 c 1 16 4 8 3 36 4,8,6 d,4,4 β n,p, Po SkM*, Rn SkM*,, 1 16 4 8 3 36 4 A 1 16 4 8 3 36 4 A Proton and neutron deformation parameters of Hg, Pb, Po, Rn using SkM* forces
<r n,p >1/, (fm) 6,4 6, 6, 5,8 5,6 a Hg SkM* 6,4 6, 6, 5,8 5,6 b Pb SkM* r n r p 5,4 1 16 4 8 3 5,4 1 16 4 8 3 36 4 44 6,4 c 6,4 d <r m) n,p >1/, (fm 6, 6, 5,8 5,6 5,4 Po SkM* 6, 6, 5,8 5,6 16 4 8 3 36 16 4 8 3 36 4 A A Rn SkM* Proton and neutron root mean square radii for Hg, Pb, Po, Rn using SkM* forces
V(r), МeV 6 7 Rn а V(r), МeV 6 8 Rn b 4 4 (1k 13/ ) 3f 7/ (h 11/ ) (3f 7/ ) (3d 3/ ) 4 6 8 1 1 14 16 18 r,fm 1k 13/ 3f 7/ 4 6 8 1 1 14 16 18 (h 11/ ) (3f 7/ ) r,fm - h 11/ 3d 3/ - h 11/ V(r), МeV 6 4 314 Rn 3f 7/ 4 6 8 1 1 14 16 18 c (1l 17/ ) (1k 13/ ) (h 11/ ) (3f 7/ ) r,фм Ska - 1k 13/ h 11/ 1l 17/ potentials for 7,8,314 Rn obtained with Ska forces in S approximation. Quantum numbers are shown for each one particle level. Red lines show the last filled level. Blue line show empty levels with a large angular momentum.
-E, МeV 44 а 4 4 Ar SkM* -E -E B 1n drip line S n, -λ n, МeV 1 8 6 4 c S n -λ n -λ n B 76 Ar (N=58) 38 15 1 b 5 6 7 8 S n S n B -,,1 d 5 6 7 8 S n, МeV 5-5 5 6 7 8 β n,p n, -,1 β n 76 Ar (N=58) β p -, β B A 5 6 7 8 A The properties of neutron rich Argon calculated within D and B with SkM* forces. Dotted line is 1n drip line. a) Total energy b) Two-neutron separation energy c) S n and neutron chemical potentials λ n d) Neutron and proton deformation parameters β n,p
-E, МeV 188 186 184 а -E -E B Rn SkM* 1n drip line S n, -λ n, МeV 5 4 3 1-1 c S n -λ n -λ n B 8 b 6 8 3 3 S n -,8,6 d 6 8 3 3 β n S n, МeV 4-4 S n B β n,p,4,, -, β p β B -,4-8 6 8 3 3 A -,6 6 8 3 3 A The properties of neutron rich Radon calculated within D and B with SkM* forces. Dotted line is 1n drip line. a) Total energy b) Two-neutron separation energy c) S n and neutron chemical potentials λ n d) Neutron and proton deformation parameters β n,p
Conclusions The peninsulas of nuclei stable with respect to emission of one or two neutrons can exist beyond the drip line that was previously known theoretically. The numbers N which correspond to the peninsulas of stability are close or coincide with the known magic numbers. The stability restoration through adding more neutrons and formation of stability peninsulas appear for various Skyrme forces. The isotones with N=3, 58, 8, 16, 184 which belong to the peninsulas of stability have spherical shape. The mechanism of stability restoration is due to the complete filling of neutron subshells lying in the discrete spectrum and having large values of angular momentum. These shells being partially filled are located in the continuum. Our results are in a good agreement with the results obtained by the B method with Skyrme forces up to neutron drip line. The obtained results show that the structure of the drip-line can be rather complex and reflects the shell structure. The drip line may extend further beyond what it was generally assumed previously.