Low-lying Gamow-Teller excitations and betadecay properties of neutron-rich Zr isotopes

Niigata Univ.!
Kenichi Yoshida

ARIS2014@Tokyoβ-decay half-lives in neutron-rich Zr isotopes
S. NishimuraPRL
et al.,PRL106(2011)052502
106,052502 (2011)
103(a)

Sr(b)(Z=38)(c)
Zr(Z=40)

102

FRDM+QRPA [16]FRDM+QRPA [1]
KTUY+GT2
Previous
This work10
310(e)

Rb half-lives
Y
shortwere(Z=41)
Nbmeasured(Z=37)(Z=39)(Z=43)

(f)

(g)

(h)
Tc

102

1060 6264626466 68
66 68Number of Neutrons

70687072

74

FIG. 3 (color online). Neutron number dependence of !-decay

105

1TFR
1/10.158

100

0.1
10Ratio

10Half-life (ms)(d)
Mo(Z=42)

Ratio

Half-life (ms)Kr(Z=36)

PHYSICAL REVIEW LETTER

TFR
1/100

105

FIG. 4 (color onlinetopes. Another remarkable behavior at N ¼ 64 has been
alf-lifeof 620 % 150 ns was derived from the
数４０と中性子数５０が魔法数であり、その形状は球形となります（図１）。
Zr に中Hua et al. observed a bandobserved for Mo isotopes.
me spectra for these five " rays. Some low性子を１０個加えて中性子数を６０にすると、大変形領域に入って球形は急激に変化し、
crossing due to the rotation alignment of an h11=2 neutron
108-ray peaks from theZr isomer might notpair [23]. The shift of the band crossing to higher rotational
さらに中性子を加えると中性子数６４まで緩やかに変形度が大きくなることが知られidentified,and no information on "-" coinci106 Mo is interpreted as a consequence of thefrequency
inobtained due to the low statistics. Nevertheless,deformed subshell closure at N ¼ 64. The maximum of
stimated that the energy of the isomeric stateþ
1=Eð21 Þ at N ¼ 64 can also be interpreted as being due
more than 1 MeV. The ground-state band isto the deformed subshell closure at N ¼ 64 with !2 &sphericalup to 4þ ; thus, the spin
is likely more than or
0:47ð7Þ [5]r-process
for 104 Zr. path
The r-process path between A ¼ 110 and A ¼ 125 may
be affected by the weakening of the spin-orbit force, which
Zr isotopes
8+ 1594.9
is associatedwith the neutron skin [24]. The harmonic8+ 1550.2
neutron drip line
oscillator-like doubly magic nucleus of 110 Zr [24] or thedeformed630.1624.4
systematics of the 2+ energy
90

proton number

Deformed Zr isotopes on the r-process path473.7

neutron number(4+) 476.5
478.3 spherical
452.1number
4+ magic4+

図１326.5

(4+)

521.6

64347.9
原子核図表。
縦軸が陽子数。
青線で囲まれた領域は大変形領域であり、
324.4
2312.8 横軸が中性子数、(2+) 173.7
2+ 152.1
151.8
2+ 境界の点線部は推測である。赤２重線は、本研究によりジルコニウム（Zr）同位体に発見され2+ 139.3
173.7
151.8
152.1
139.3
00黒２重線は魔法数0+0+ た中性子の変形魔法数６４である。
0+0+000 （球形）を、緑２重線はZr 同位体を表す。
5060102

Kr (Z = 36)
Sr (Z = 38)
Zr (Z = 40)
Mo (Z = 42)
Ru (Z = 44)
Pd (Z = 46)
Cd (Z = 48)
Sn (Z = 50)

spherical magic number–1925.8+486.5

6+1/E(2 1) (MeV )

964.8

6+

8

deformedmagic number

104106

108

70

Neutron Number
ZrZr
Zr
Zr
紫の四角は元素合成の経路を示している。色付けされている領域は、β崩壊の半減期が測定さ80

þ Þ PRL106(2011)202501れている領域である。黒は安定な原子核で、青は半減期が長く、赤は短い。
T. Sumikama
et al.,
ound-state
bands of neutron-rich even-even Zr isoFIG. 3 (coloronline). 1=Eð2as a function of the neutron
1' 60. The energies of 100–104 Zr are taken from thenumber. The present results are for Zr isotopes with N ¼ 66 andbase [27].
68. Others are taken from the ENSDF database [27].

202501-3

drrL YL ψ † (rστ )⟨σ|σ|σ ′ ⟩⟨τ |τ |τ ′ ⟩ψ(rσ ′ τ ′ )KY,! PTEP2013,113D022λ †δ E†
λXαβ âα,ν âβ,π − Yαβâβ̄2δ[HEMF , T− ] ̸= 0v(r,r)=
τ 1 · τ 2responses+ αβdrrL Y ŝ inσres12
Self-consistent pnQRPA
forspin-isospin
L 1t)δsδρ1t3 (r 1 )δρ1t3 (r 2 )δs1t3 (r11t3 (r 2 )deformed nucleiÔλ† =!

Ôλ† =!

λ †λ′′′ ′
Xαβâα,ν â†β,π − Yŝαβâβ̄,πâᾱ,νψ † (rστ
2 )⟨σ|σ|σ ⟩⟨τ |τ t |τ ⟩ψ(rσ τ )1t =δ Eαβvres(r 1 , r 2 E[ρ(r),)=τ1 · τ2 +ρ̃(r)]
starting point: Skyrme
EDF
δρ1t3 (r 1 )δρ1t3 (r 2 )δs1t3 (r 1×zρ

= 14.7 fm × 14.4 fm

maxmaxvariation w.r.t densities2δ2E∆ρδ= E∆z = 0.6 fmE[ρ(r), ρ̃(r)]vres (r 1 , r 2 ) =Hartree-Fock-Bogoliubov
τ 1eq.
· τfor
σ1 · σ2 τ 1 · τ 2
The coordinate-space
2 +ground statesδρ1t3 (r 1 )δρ1t3 (r 2 )≤ 60
δs1t3J.E(r(ret2 )al., NPA422(1984)1032qp1 )δs1t3MeV
Dobaczewski

δE,
S− h− S=10.06
q+ =
δρq

E[ρ(r), ρ̃(r)]“s.p.” hamiltonian and pair potential:

δE
h = q,
δρqquasiparticle basisq = ν, π∆ν = 0.00 MeVδEh̃ = qδ ρ̃q

T =1vpp(r, r ′ ) =The proton-neutron quasiparticle RPA eq. for excited states∆π = 0.41 MeV'Ôλ =αβXαβ âα,ν âβ,π − Yαβ âβ̄,π âᾱ,ν

residual interactions derived self-consistently :*1 ρ00 (r)1 − Pσ 1 + Pτ
V0 1 −
δ(r222 ρ0[HMF , T− ] ̸= 0!
Collective excitation = coherent† superposition
excitations:† of 2qp
λ †λδEh̃ = qδ ρ̃qÔλ† =[HMF , T− ] ̸= 0"αβλ †λXαβâα,ν â†β,π − Yαβâβ̄,π âᾱ,νδ2E
δ2Evres (r 1 , r 2 ) =τ1 · τ2 +σ1 · σ2 τ 1 · τ 2
δρ1t3 (r 1 )δρ1t3 (r 2 )δs1t3 (r 1 )δs1t3 (r 2 )2E[ρ(r), ρ̃(r)]GT !
giantresonance"
δEδE!
q,t3 h =qF̂K =δρ′σ,σ τ,τ ′8h̃q† = q
dr ψ̂ (rστδ ρ̃)⟨σ|σ K |σ ′ ⟩⟨τ |τ t3 |τ ′ ⟩ψ̂(rσ ′ τ ′ )q = ν, πSkM*
mixed-type pairingP̂T† =1,T +=0,S=0R- (/MeV)z6!"=β2,ν = 0.38dr ψ̂ † (rστ )⟨τ |τ 0 |τ ′ ⟩ψ̂ † (rσ̃τ̃ ′ )σ = 0.41
β2,π✓sudden onset of deformation at N=60
SLy4: A. Blazkiewicz et al., PRC71(2005)054321

100Zr✓fragmentation of strength distribution
due to deformation

separable pnRPA:
P. Urkedal et al., PRC64(2001)054304

4P̂T† =0,S=1,Sz =0!"2=dr ψ̂ † (rστ )⟨σ|σ 0 |σ ′ ⟩ψ̂ † (rσ̃τ̃ )✓SkM* and SLy4 give almost the sameσ,σ ′excitation energies of GTGRg0’=0.94 (SkM*), 0.90 (SLy4)010266E*GR (MeV)

Total
K=0K=18R- (/MeV)

98Zr

42422SkM*
SLy4202180-505

1 MeV smearing width

1015ET (MeV)

202530

98

100

102

104

106

108

110

112

Aexcitation energy w.r.t. the g.s of daughter

E[ρ(r), ρ̃(r)]q = ν, π

GTGR: the need of self-consistency
10SLy4 + mixed-type pairing
R- (/MeV)

8β2,ν = 0.38β2,π = 0.41
104Zrβ2,ν = 0.39β2,π = 0.4364δEh̃ = qδ ρ̃q✓a repulsive character of the residual interaction
raises the peak energyq = ν, πβ2,ν = 0.38
thelow-lying
strengths
are absorbed to the✓β2,π = 0.41
high-energy peakstrong collectivity of the GTGR
β2,ν = 0.39
β2,π = 0.43

2✓the collectivity generated by the Landau-Migdal

010

approximation is weak
98Zr

QRPA
LM2qp

8R- (/MeV)δE
h = q,
δρq6vph (r 1 r 2 ) = N0−1 [f0′ τ1 · τ2 + g0′ σ1 · σ2 τ1 · τ2 ] δ(r 1 − r 2 )LM parameter:
M. Bender et al., PRC65(2002)054322

420051015E* (MeV)

202530self-consistency is needed for aquantitative description of theGTGR

β2,ν = 0.38β2,π = 0.41

T=0 (S=1) pairing✓affects the GT response

if we have (a) T=1
pairing
condensate(s)β = 0.392,ν

due to the coupling between theβp-h =and0.43p-p excitations
2,πwe may see the effect in the low-lying states that are generated by 2qp
excitations around the Fermi levels−1′′v(rr)=N[fτ·τ+g
σ1 · σ2 τ1 · τ2 ] δ(r 1 − r 2 )doesnotaffect
thegsproperties
inN>Z
ph1212✓00nuclei
0a form of the interaction or an np-pairing EDF is seldom known
t3
F̂1K=!"

drrY1K ψ̂ † (rστ )⟨τ |τ t3 |τ ′ ⟩ψ̂(rστ ′ )Take the simplest one;
σ1 + Pσ 1 − Pτ
T =0′vpp (r, r ) =V0 δ(r − r ′ )22

the pairing strength determined to reproduce the β-decay half-life of 100Zr (7.1 s)

Low-lying GT statest3F̂1K=0.25

110Zr

22

|⟨π[N n3 Λ]Ω = Λ +−11/2|t′ − σ+1 |ν[N′ n3 Λ]Ω = Λ − 1/2vph (r 1 r 2 ) = N0 [f0 τ1 · τ2 + g0 σ1 · σ2 τ1 · τ2 ] δ(r 1constructed dominantly by

π[413]7/2 ⊗ ν[413]5/2

0Strength (/MeV)

2106Zr

0.05
102Zr! " particle-like particle-liket3F̂1K=drrY1K ψ̂ † (rστ )⟨τ |τ t3 |τ ′ ⟩ψ̂(rστ108Zr✓T=0 pairing 3effective0.2σ0.15

T =0vpp(r, r ′ ) =0.1
0.050.25√

1 2,π+ Pσ 1 − Pτ
T =0′vpp (r, r ) =V0 δ(r − r ′ )0.1

0V δ(r − r )(r, r ) =|⟨π[N n3 Λ]Ω = Λ + 1/2|tn3 Λ]Ω = Λ − 1/2⟩| =β − σ+1=|ν[N0.43

0.15

0.25

drrY1K ψ̂ † (rστ )⟨τ |τ t3 |τ ′ ⟩ψ̂(rστ ′ )0σ pprule for GT22
selection
β2,ν = 0.39

w/ T=0 pairing
w/o T=0 pairing

0.2

β2,π = 0.41!" σt3†′F̂1K=drrY
ψ̂(rστ)⟨τ|τ|τ⟩ψ̂(rστ1Kt3′
1 + Pσ 1 − Pτ
T =0′v

104Zr!"

1 + Pσ 1 − Pτ
V0 δ(r − r ′ )22neutrons added|⟨π[N n3 Λ]Ω = Λ + 1/2|t− σ+1 |ν[N n3 Λ]Ω = Λ − 1/2

106Zr

100Zr

108, 110Zr

0.2

π[413]7/2 ⊗ ν[413]5/2
3 hole-likeparticle-like

0.15
0.1

✓T=0 pairing ineffective

0.05
0-10-8-6-4-20-12 -10

ET (MeV)-8-6-4-20β2,ν = 0.38β2,π = 0.41

Beta-decay half-lives
100

SkM* (w/o T=0 pairing)
SLy4 (w/o T=0 pairing)Exp.

✓β2,ν = 0.39
Fermi’s golden rule
β2,π = 0.43

N. B. Gove, M. J. Martin,
At. Data −1Nucl.
10(1971)205′ Data Tables′vph (r 1 r 2 ) = N0 [f0 τ1 · τ2 + g0 σ1 · σ2 τ1 · τ2 ] δ(r 1 − r 2 )Gamow-Teller strengths✓Fermi and!"tincludedF̂1K=

10

3T1/2 (s)σdrrY1K ψ̂ † (rστ )⟨τ |τ t3 |τ ′ ⟩ψ̂(rστ ′ )T =0vpp(r, r ′ ) =1 + Pσ 1 − Pτ
V0 δ(r − r ′ )22✓SkM* produces longer half-lives|⟨π[N n3 Λ]Ω = Λ + 1/2|t− σ+1 |ν[N n3 Λ]Ω = Λ − 1/2⟩| =primarily due to a small Q-value

1π[413]7/2 ⊗ ν[413]5/2

Q-value calculated approximately
Qβ − = ∆Mn−H + B(A, Z + 1) − B(A, Z)≃ ∆Mn−H + λν − λπ − E0
E0 = min[Eν + Eπ ]0.1

cf. J. Engel et al., PRC60(1999)014302
360

626466

N6870

72

√

Beta-decay half-lives with T=0 pairing
100

SkM* (w/ T=0 pairing)
SkM* (w/o T=0 pairing)
SLy4 (w/ T=0 pairing)
SLy4 (w/o T=0 pairing)
Exp.

✓Strength of T=0 pairing determined at
N=60

SLy410T1/2 (s)✓reproduces well the observed isotopic
dependence with T=0 pairing!
✓Effect of the T=0 pairing is small
beyond N=68
1SkM*

✓gives a strong deformed gap at N=640.160

626466

N68

7072

Summary
Fully-selfconsistent deformed pnQRPA is developed in a Skyrme EDF
KY, PTEP2013,113D02
framework
Microscopic and quantitative description of spin-isospin excitations in nuclei
with arbitrary mass number whichever they are spherical or deformed,
located around the stability line or close to the drip line

Deformation effects on spin-isospin responses
Tiny deformation splitting in Gamow-Teller excitation
Fragmentation of GTGR

Effects of T=0 pairing
Low-lying GT states: Sensitive to the location of the Fermi levels
Beta-decay half-lives are shortened due to the attractive nature