China-Japan@Tsukuba: 26-28 June, 2017

Low-lying charge-exchange modes of
excitation relevant to beta decay
Kyoto Univ.

Kenichi Yoshida

Recent progress in microscopic description of beta-decay rate
proton-neutron (Q)RPA based on nuclear density-functional theory“no-core”“no adjustable parameters”
spherical
HFB

Engel et al., PRC60(1999)014302: A
Bai et al., PRC90(2014)054335

Niu, Colò et al, PRC94(2016)064328: A + phonon-coupling effect

HF-BCS
Fracasso, Colò, PRC72(2005)064310
RHB, RHFB
Paar, Nikšić et al., PRC69(2004)054303; Nikšić, Marketin et al., PRC71(2005)014308: A
Marketin et al., PRC93(2016)025805: A+FF
Niu, Niu et al., PLB723(2013)172; PRC95(2017)044301: A

(+), and [83] (⋆), measured half-lives [82] (circles), and the range ofhalf-lives reported in this paper (shaded region).

(×) superimposed upon the range of half-lives reported in this paper(shaded region).

Recent progress in microscopic description of beta-decay
half-lives of Ref. rate[17] lie, for the most part, right in the middleof our predictions and follow those of SV-min fairly closely
strongly affect r-process abundances near A = 160. (Neutron(the Q values are similar). Finally, Fang’s recent calculations
odd nuclei do not significantly affect the r process because[83] yield relatively short half-lives, shorter than even thosethey quickly capture neutrons to form even-N isotopes [1].)of UNEDF1-HFB most of the time. Still, the band of predicted
Figures 9 and 10 present new calculated half-lives in four
half-lives is relatively narrow among these three calculations
isotopic chains, with even-Z isotopes in Fig. 9 and odd-Z
even in the most neutron-rich nuclei. Reference [33] points out
isotopes in Fig. 10. The top panels of these figures include
that despite their differences, most global QRPA calculations
predictions with all five adjusted Skyrme EDFs, and the
produce comparable half-lives. Our results in both A ≃ 80 andbottom panels compare our calculations to measured values
A ≃ 160 nuclei support this observation.
[82] and the results of previous QRPA calculations [17,33,83]
HFB
Figures 9 and 10 (as well as Fig. 11, discussed momentarily)
where they are available. Our calculations span the (narrow)show Engelthat the etoverallpattern of β decays in the AA+FF≃ 160 region
Mustonen,
PRC87(2013)064302:
Aal., PRC94(2016)055802:
range Engel,of predictedhalf-lives in these isotopic
chains, with Shafer,
SLy5 and UNEDF1-HFB predicting the shortest half-lives forcontributionofFFtransitions
Yoshida,PTEP2013(2013)113D02:
A
the most neutron-rich isotopes, as one could expect from66
50%+
the analysis of Sec. III B 1. While the UNEDF1-HFB half-lives
Martiniareet uniformlyal., PRC89(2014)044306:
A of SLy5 actually are
64
short, however, those40%
62
the longest predictions (and the closest to measured values)
for nuclei nearer to stability. The variability of our half-life
60
30%
predictions in Figs. 9 and 10 is typical of all 70 nuclei in the58
calculation. The longest and shortest calculated half-lives for
56
20%
any nucleus in our set differ by a factor ranging from about 1.9
54
to 3.3, and this interval does not depend strongly on whether
52
10%
a nucleus has an even or odd number of protons.
50
The results of Refs. [17,33,83] are actually fairly similar to0%48
ours, spanning roughly the full range of our predicted values in
95
100
105
110
115
120
Fig. 9. (Neither Ref. [33] nor [83] report half-lives for odd-Z
Neutron number
nuclei and so cannot be a part of Fig. 10.) The half-lives ofRef. [33] are close to our own SkO′ half-lives, a result that
FIG. 11. Impact of first-forbidden β transitions in rare-earth
is unsurprising given that the EDF in that paper is a modified

proton-neutron (Q)RPA based on nuclear density-functional theory“no-core”“no adjustable parameters”

Proton number

deformed

Microscopic description of the nuclear β-decay
charge-exchange mode of excitation= superpositionof
2qp
excitations
ofaprotonandaneutron
begin
†
Ôi =

begin

end

αβ

1
T1/2

mother

=

Qβ

†
i
†
i
Xαβ âα,ν âβ,π − Yαβ âβ̄,π âᾱ,ν
begin

D

(gA /gV )2eﬀ ! !

=D2∗ T1/2 −Qβ − Ei )|⟨i|F̂K |0⟩|
! f (Z,q,i∗end
K Ei <Qβ M(gA /gV )2eﬀ !
begin
end

1

!

K Ei∗ <Qβ−∗f (Z, Qβ − Ei )|⟨i|F̂K |0⟩|αβE2qp <E ′"!
∗−Qβ − Ei ≃ ∆Mn−H − ωi + λν − λπ
†
end
F̂K
=
dr ψ̂π(rσ ′ )⟨σ ′ |σK |σ⟩ψ̂ν (rσ)*
2σσ ′= ∆Mn−H − ET,i
β
β
i
Q
+
0
+
!
B(E2
:
2
→
0
)
=
1
"
†
†
gs
i
†
i
J. Engel et al., PRC60(1999)014302
Ô
=
X
â
â
−
Y
âβ̄,π âᾱ,ν
!
16π
i
β,π
αβ
α,ν
αβ
2 !′ ! ′
+
†
1
(g
/g
)
A
V
αβ
∗(rσ),−
2F̂K = = dr ψ̂νeﬀ(rσ )⟨σ |σ
|σ⟩
ψ̂
K
π
f (Z, Qβ − Ei )|⟨i|F̂K |0⟩|
!
!
T
D
1/2
q,i
∗
†
†
′
*
†β
λ
K λEi <QσσMiαβ|Ψλ ⟩ = Γλ |Ψ0 "⟩=[Xαβ aα aβ − Yαβ aβ aα ]|Ψ0 ⟩E2qp <E ′
!
−
†"
′
′
αβ
!
F̂K =
dr
ψ̂
(rσ
)⟨σ
|σ
|σ⟩
ψ̂
(rσ).
2
K
ν
π
−
Q
†
′
′
+
0
+
F̂
=
dr
ψ̂
(rσ
)⟨σ
|σ
|σ⟩
ψ̂
(rσ)
K
ν
B(E2
:
2
→
0
)
=
K
π
1
gs
σσ ′
16π
† σσ′
†
[H, Γλ ]|Ψ0 ⟩ = ωλ Γλ |Ψ0 ⟩
!
†
λ
† †
λ
!
!
|Ψ
⟩
=
Γ
|Ψ
⟩
=
[X
a
a
−
Y
†
†
λ
0
i ††
i
λ
αβ α βαβ aβ aα ]|Ψ0 ⟩
†
i
†
i
Ô
= â Xαβâα,ν â−− Yαβâβ̄,πââᾱ,ν
β,πY
Ôi =
Xi αβ
â
â
αβ
β̄,π ᾱ,ν
β,π
α,ν
αβ
αβ
αβ
†
†
"
!
[H, Γλ ]|Ψ0 ⟩ = ωλ Γλ |Ψ0 ⟩
q,i
!
!
1
Mαβ
!
′
′ ′
T
′
q,i
P̂T =1,Tz =0,S=0 ≡drψ̂(rστ
)δ
⟨τ
|τ
|τ
⟩
ψ̂(r
σ̄
τ̄
)
σ,σ
0
E2qp
<E ′
M

E =Q -E

ET,i

!

E

daughterlow-lying GT states aredecisive for the half-lifeE"Spin-isospin begin
response
s1,µσ ,µτ (rt) =!!
σ,σ ′ τ,τ ′

begin

†

′ ′

′

⟨ψ (rστ t)ψ(rσ τ t)⟩⟨σ|σ µσ |σ ⟩⟨τ |τ µτ |τ ⟩end
λ−iωλ tρ(r) + [δs1,µσ ,µτ (r)e+ c.c]

end′

in linear-response of TDDFT

1 !!
′
′
physical observables
spin-isospin
responseρ̃0,µσin(rt)=⟨ψ(rστ t)ψ(r σ̄ τ̄ t)⟩⟨σ|σ µσ |σ ⟩2!" ′ τ
!2
σ,σ
!
!
λ
!
!
!
!!
1! drf (r)δs1,µσ ,µτ (r)!!¯′
′
†
′
′
′
′
ρ̃1,µ⟨ψ(rt)= t)ψ(rσ τ⟨ψ(rστt)ψ(rσ̄
τ
t)⟩⟨τ
|τ
|τ
⟩
µ
τ
τ
s1,µσ ,µτ (rt) =(rστ
t)⟩⟨σ|σ
|σ
⟩⟨τ
|τ
|τ
⟩
µσ
µτ
2 σ
′
τ,τ
begin
σ,σ ′ τ,τ ′end
beginstrength distribution: w/ isospin-change µτ = ±1
Gamow-Teller
"#
"2
!!""
"
!
2
µτ end # !"
λ
"
!
S
(ω)
=
drδs
(r)
δ(ω
−
ω
)
!
!
λ
1,µ
,µ
λ
σ ψ(rσ̄τ̄ ) =σ
τ""
1 S µτµ(ω)!
!
(−2σ)(−2τ
)ψ(r
−σ
−τ
)
′
=
drδs
(r)
δ(ω
−
ω
)
′
λ
µσ
1,µ
,µ
ρ̃0,µσ (rt) =⟨ψ(rστt)ψ(rσ̄τ̄t)⟩⟨σ|σ
|σ
⟩
σ
τ
!
!
λ
µσ
"#
"2
2 ′ τ
λ
GT matrix!
element
states:" to low-lying
"
σ,σ
end
λ
"
"
S(ω)
=
drδs
(r)
key quantity to" beta-decay!!
1,µσrate,µτ" δ(ω − ωλ )

begin

1
ρ̃1,µτ (rt) =
2

σ

τ,τ ′

′
′
¯
⟨ψ(rστ t)ψ(rσ̄ τ t)⟩⟨τ |τ µτ λ|τ ⟩µτ = ±1ρ̃ν (r) = ⟨ψν (r ↓)ψν (r ↑)⟩µ = ±1

Skyrme energy-density functional
approach!
!
!

!

E =E = drH(r)drH(r)= drH(r)Energy functional: E = EdrH(r)!

E = drH(r)H = Hkin +HSkyrme +Hem
H = Hkin +HSkyrme +Hem
Hkin=+H
Hkin
+H
H +H
= Hkin+HSkyrme+Hem
Energy density:
Skyrmeem
H=H+Ht
Skyrme" "
em#
$
even
odd
HSkyrme =
H
+
H
t
tt3
tt3
#
$
"
"
t
#
$
"
t3 =−tt=0,1"
even odd odd
even=H
HSkyrme= t
HH+ H+
Skyrme energy density: HSkyrme
tt
tt
tt
tt3$
#
3
3
t
3
"
"
#
$
" " t=0,1
t=0,1 t3 =−t even
odd
t
=−t
even
odd
3
←
→2
ρ
∆ρ
even
2
τ
∇J
J
H
=
H
+
H
HSkyrme== Ct ρtt3 + Ct ρtt3 ∆ρ
Htttt33+ C+
H
HttSkyrme
τtttt
tt33 ∇ · Jtt3 + Ct J tt3
t ρtt33
tt33+ Ct ρtt
3
←
→
ρ
∆ρ
t
=−t
t=0,1
even
2
τ
∇J
J
3
Htt3 = Ctt=0,1
ρtt3 + tC3t=−tρtt3 ∆ρtt3 + Ct ρtt3 τtt3 + Ct ρtt3 ∇ · Jtt3 + Ct J 2tt3
→2
ρ 2
∆ρ
even
τ
∇J
J←
Htt3 = Ct odd
ρtt3 +s C2 t ρ∆stt3 ∆ρtt3 + C
ρ
τ
+
C
ρ
∇
·
J
+
C
J
tt
tt
tt
tt
t
t
t
tt3
3
3
3
3
j
∇j
2
T
∇s
2
Htt3 = Ct stt3 +Ct stt3 ·∆stt3 +Ct stt3 ·Ttt3 +Ct (∇·stt3 ) +Ct jtt3 +Ct stt3 ·∇×jtt3

←
→
←
→
ρ
∆ρ
even
2
τ
∇J
J
2
ρ
∆ρ
j 2
∇j
evenH
2
τ
∇J
J
2
odd
s 2
∆s
T
∇s
2
H
=
C
s
+C
s
·∆s
+C
s
·T
+C
(∇·s
)
+C
j
+C
s
·∇×j
=
C
ρ
+
C
ρ
∆ρ
+
C
ρ
τ
+
C
ρ
∇
·
J
+
C
J
tt
tt
tt
tt
tt
tt
=
C
ρ
+
C
ρ
∆ρ
+
C
ρ
τ
+
C
ρ
∇
·
J
+
C
J
tt
t
t
tt
t
tt
t
t
t
tt
3
3
3
3
3
3
tt
tt
tt
tt
tt
tt
3
3
3
3
tt
tt
tt
tt
tt
tt
t
t
tt
tt
t
t
t
tt3
3
3
3
3
3
3
t
tt3
tt3
t
t
t
tt3
3
3
3
3
3
3
3 t
3
Poorly known (poorly constrained):

j 2
∇j
odd
sT-odd2∆sT∇s2
Skyrmeenergydensitycoupling
constantsfor
Htt3 = Ct stt3 +C
s
·∆s
+C
s
·T
+C
(∇·s
)
+C
j
+C
s
·∇×j
tt
tt
tt
tt
tt
tt
tt3
t tt3
t
t
t
t
3
3
3
3
3
3

vanishes for ground-state of even-even nuclei

vector-isovector density

j 2
j ∇j
∇j
2
T
∇s
odds 2 s 2∆s
∆s
T
∇s 2
s
+C
s
·∆s
+C
s
·T
+C
(∇·s
)
+C
j
+C
s
·∇×j
H=ttC
=
C
s
+C
s
·∆s
+C
s
·T
+C
(∇·s
)
+C
j
+C
s
·∇×j
tt
tt
tt
tt
tt
tt
tt
tt
tt
tt
tt
tt
tt
t
t
t
tt
t
t
t
tt
3
3
3
3
3
3
tt3
t
t
t
tt
t
t
t
tt
3
3
3
3
3
3
Isovector(t=1)
coupling
constants
3
3
3
3
3
3

d

less information on nuclei with neutron (proton) excessH

odd

Gamow-Teller MEfor SAMi (R ≈ 42) and SGII (R ≈ 10) while it is equal toabout 4 for the other four interactions. Within RPA þ PVC,
half-life
obtained with the sets SkM* and SIII falls
Many-body correlations essential inthe
lowenergywithin the experimental error. It has to be stressed that the
208Pb, SGII

132Sn
spherical and normal-fluid nucleus

200

Strength (/MeV)

150

RPA

100

single peak

50
0
50

predicts
is highoverestvery sm

Phonon coupling
unp.

0
0

5

spreadinglowering
10 15 20
ET (MeV)

25

30

Most of the strengths are gathered in
the high-energy giant resonanceT1/2: ∞ →finite

Tiny low-lying strengths

Y. F. Niu et al., PRL114(2015)142501
FIG. 2 (color online). Experimental data related to β decay from nuclei 132 Sn

β-decay study at RIBF for the rare-earth elements production of the r-process
week ending
17 FEBRUARY 2017

PHYSICAL REVIEW LETTERS
PRL
118, S.
072701(2017) et al., PRL118(2017)072701
J. Wu,
Nishimura
104

57 La

Half-life (ms)

55 Cs

103

104

105

59 Pr

104

103

61Pm

63 Eu

105

65 Tb

4

10

103

67 Ho

105

104

102

104

103

102

102

103
103
102

102
100 102 104 106 108 110

107
104
3

10

58 Ce

56 Ba

104

103

60 Nd

105

5
62 Sm 10

104
104

2

10

102

103

64Gd

106

105

104

103

10

102

102

role of MB correlations?

66 Dy

103

10

far from the magic numbers
strongly deformed in space/gauge space

103

10

Half-life (ms)106

Rare-earth nuclei

103This Work
Previous
FRDM+QRPAKTUY+GT2
RHB+pn-RQRPA

86 88 90 92 94 96 98 100 88 90 92 94 96 98 100102 92 94 96 98 100 102 104 94 96 98 100 102 104 106 96 98 100 102 104 106 108 100 102 104 106 108 110

Number of neutrons N

FIG. 3. Systematic trends of β-decay half-lives from this work (solid circles) and previous measurements (open triangles) [25] withneutron number for thirteen elements. The measurements are compared to predictions of three theoretical models: FRDM þ QRPA [26](green), KTUY þ GT2 [27,28] (red), and RHB þ pn-RQRPA [30] (blue). The shaded areas are the currently known-masses region foreach element [31], except for the unknown mass of 165
65 Tb (red arrow).

distributions below Qβ, canceling the effect of about 2 MeV
increase in Qβ [see Figs. 4(c), 4(d)]. Since the ground-state(GS) deformation changes very little along this sequence of160
61 Pm99 , which is somewhat intriguing (see Fig. 3), butsimilar features were not found in other elements.
To evaluate the impact of the newly measured half-lives on

Nuclear DFT cal.

assuming the spherical shapereasonably produces the obs.
Marketin et al., PRC93(2016)025805

Roles of pairing and def. on beta-decay: naive picturephase-space factor

sensitive to shell structure
and pairing/deformation(both T=0 and T=1)
=f ·V

T =1

Eβ:

∗
Qβ − Ei ≃ ∆Mn−H − ωi + λν − λπ= ∆Mn−H − ET,i

2 !
/g
)
A
V eﬀ

D

−
K =

K Ei∗ <Qβ

"
!
σσ ′

†
Ôi =

!

!
αβ

energy of emitted beta particle

E*0

Vpn-pair−∗2f (Z, Qβ − Ei )|⟨i|F̂K |0⟩|0

distortion of Fermi surfaces†
dr ψ̂π(rσ ′ )⟨σ ′ |σK |σ⟩ψ̂ν (rσ)†
i†iXαβ âα,ν âβ,π − Yαβ âβ̄,π âᾱ,ν

0w/ pn-pairing
w/ like-particle pairing, deformation

GT strength

V

T =0

0

w/o pn-pairing
w/ like-particle pairing, deformationE*
w/o like-particle pairing, deformation

E*: excitation energy wrt daughter nucleus

Pairing and deformation for low-lying GT states
3

162Nd

T=1
superfluidity

2

0
3

μσ=0
2 μσ=±1120spherical

Low-lying strengths are reduced
distortion of Fermi surfaces

Quadrupole deformation ≃2+ phonon condensation
non-perturbed phonon couplingfragmentation and appearance of low-lying states

T1/2 = 0.53 s

10
0
-8

Energies are shifted higher
pairing gaps

T1/2 = 0.31(20) s

T1/2 = 0.14 s
deformation

Strength (/MeV)

1

0
30

T1/2 = 0.27 s

Exp.

Strengths are concentrated on a single state w/ high energy
-6

-4
-2
ET (MeV)0ν1h9/2 → π1h11/2

SLy4 + pairing functional of M. Yamagami et al., PRC80(2009)064301

Pairing and deformation for low-lying GT states
106Zr

0.3
0.2

w/ T=0 pairing

SLy4
T1/2 = 0.21 s(a)GT strength

0.1

S. Nishimura et al., PRL106(2011)052502
Exp.

T1/2 = 0.186(11) s

T=0 pairing

0

w/o T=0 pairing
0.2
K=0
μσ=0
μσ=±1
K=1
0.1

(b)

T1/2 = 0.41 s
deformation
T=1 superfluidity0

w/o T=0 pairing
1.5
spherical1(c)T1/2 = 0.07 sphase space

0.5
0

(d)

4

10
2
10
100
10-2
-10

-8

-6

-4

ET (MeV)KY, JPS Conf. Proc. 6(2015)020017

-2

0

2

Effect of MB correlations on β-decay ratedepends very much on the nuclide (shellstructure)Summary (I)TDDFT gives an intuitive picture of nuclear dynamics
by looking at the density distributions
Linear response is a powerful method to investigate vibration of densities
allowing the breaking of symmetries: rotational symmetry in space/gauge space,
we can include the many-body correlations in a simple way(Q)RPA on top of the deformed state includes (partly) the non-perturbed phonon
coupling effectβ-decay rates are sensitive to the details of the strengths in low-energy
nuclear deformation and superfluidity

half-lives reported in this paper (shaded region).

strongly affect r-process abundances near A = 160. (Neutronodd nuclei do not significantly affect the r process because
they quickly capture neutrons to form even-N isotopes [1].)Figures 9 and 10 present new calculated half-lives in four
isotopic chains, with even-Z isotopes in Fig. 9 and odd-Z
isotopes in Fig. 10. The top panels of these figures include
predictions with all five adjusted Skyrme EDFs, and thebottom panels compare our calculations to measured values[82] and the results of previous QRPA calculations [17,33,83]
where they are available. Our calculations span the (narrow)range of predicted half-lives in these isotopic chains, with
SLy5 and UNEDF1-HFB predicting the shortest half-lives forthe most neutron-rich isotopes, as one could expect fromthe analysis of Sec. III B 1. While the UNEDF1-HFB half-lives
are uniformly short, however, those of SLy5 actually are
the longest predictions (and the closest to measured values)
for nuclei nearer to stability. The variability of our half-life
predictions in Figs. 9 and 10 is typical of all 70 nuclei in thecalculation. The longest and shortest calculated half-lives forany nucleus in our set differ by a factor ranging from about 1.9
to 3.3, and this interval does not depend strongly on whether
a nucleus has an even or odd number of protons.
The results of Refs. [17,33,83] are actually fairly similar toours, spanning roughly the full range of our predicted values inFig. 9. (Neither Ref. [33] nor [83] report half-lives for odd-Z
nuclei and so cannot be a part of Fig. 10.) The half-lives ofRef. [33] are close to our own SkO′ half-lives, a result thatis unsurprising given that the EDF in that paper is a modified
version of SkO′ (and that we use the same pnFAM code). The(shaded region).

half-lives of Ref. [17] lie, for the most part, right in the middleof our predictions and follow those of SV-min fairly closely(the Q values are similar). Finally, Fang’s recent calculations[83] yield relatively short half-lives, shorter than even thoseof UNEDF1-HFB most of the time. Still, the band of predictedhalf-lives is relatively narrow among these three calculations
even in the most neutron-rich nuclei. Reference [33] points outthat despite their differences, most global QRPA calculations
produce comparable half-lives. Our results in both A ≃ 80 andA ≃ 160 nuclei support this observation.
Figures 9 and 10 (as well as Fig. 11, discussed momentarily)
show that the overall pattern of β decays in the A ≃ 160 region

Low-lying charge-exchange dipole modes in very-neutron rich nuclei(spherical systems for simplicity)

66

50%+

Proton number

64
40%

62
60

30%

58
56

20%

54
52

10%

50
48

95

100

105

110

115

120

0%

Neutron number

Shafer,Engel et βal.,transitions
PRC94(2016)055802
FIG. 11. Impactof first-forbidden
in rare-earth
nuclei that are important for the r-process nuclei, with the Skyrme
EDF SV-min.

ture, another cross-section enhancement,

locatedtor, two broad peaks are clearly visible in all
four spectra. The smooth curves are fits throughabout 10 MeV higher in excitation energy, wasobservedPHYSICAL
with the reaction
8 DECEMBER 1980
VOLUME 4$, +UMBER 23
LETTERS t) at 130 MeV
REVIEW 90Zr('He,and confirmed in 45-MeV (p, n) spectra' and in
4
data
at
Q-Valueforhigherenergies.
Isovector
Giant
Resonances
Excited
Systematics
It was suggestedn)(p,—1Pbcomponent
andIsotopesn) Reactions
Nb, Mo,Sn, Tbyby(p,Marty'
that on
thisZr,peakis theand Aaron A
Galonsky
H. P. DeVito,
M. Austin, dipole
W. A.of
Sterrenburg,
the giant Sam
resonance.
charge(electric)
Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824
the (p, n) reaction, issuchexchange reaction,
7 July as
1980)(Q, eceived
T —peaks
excitesuch
1 strength.
onlyat way
found previously withtwo broadis usedto study
reaction
45 Mevto
The (P, n)thenow been observed with all but one of seventeen targets fromthe target Zr.
They have
Thus
far
these
have
been
observedphenomena
Zr to pb. Energy systematics favor the conclusion that these peaks are antfanalogs
whetherZr. Toindetermine
occurdeterThe first theyexperimental
Ml andinEl resonances
the target nucleus.
of the giant only
are reported. Their low values
minations of T, T —1 splittings of the giant El resonanceobtained'have
specmore
wegenerally
n)
(p,
+
be
as due to ainterpreted
canin comparison to T, T 1 splittings observed previously
~
~'
512~
~Zr, ~Nb,of thefor
effective
potential.
isospintensor parttra
seventeen
nucleiF. PETROVITCH AND W.G. LOVE94 98 98 100Mo 112 118 120 I 122 124Sn and 208Pb)
I+, r
9
9
PACS numbers: 25. 40.Ep, 24.30.Cz
IAS
p+
=45
MeV. With one exception we find the two
E~\
The (p, n) studies were performed with the p+, r-I
Many studies with (p, n) reactions have focused
nuclei1n addition,
the sys- system atneutron time-of-flightfeature forin theall
neutronspec-studied.
beam-swinger
on the most prominent peaks' All targets
MlStateofUniversity.
of ground states
tematicof the Michigan
locationthese peaks
is were isoanalogs variation
tra, i.e, on the isobaric& 9570, and had thickenriched, mostasto theon '0Zr at a bombardtopically(IAS). In the (p, g) reaction
evidencefor
their
rz= r-Iinterpretationstrongnesses of 1-10 mg/cm2, The neutron detector
ing energy of 45 MeV two other features were ob—Twithwidethe7 cmofM1
antianalogs,
isospingiant& 7.6 cm thick) of DAUGHTER
1,
NUC
diam
about
4
a
served. ' The first was
a
MeV
was
cylinder
(12.
peak
;;\
scintillatorThese
placed 7 m from thewhich E1
interpreted as the
not far above the IAS andwas excitations
liquidnuclei.
in theNE-213
parent& —1 component (&\ =target isospin) of the pretarget.
data (GT) or spin-flipthe first observationof GT E1
dicted' giant Gamow-Teller represent
Figure 2 shows neutron time-of-Qight spectra
TARGET NUCLEUS
andoutsidefor Kthefirstto the IAS a~strength
addition
fourprovide
Sn isotopes.
this peaktransition. Alternatively,
may be of '0Zr for= T)ofnucleusstatesthe1.
Sometarget
FIG.
(T,
and the sharp y-ray
31 MeV
thought of as the antianalog
energygiant magnetic body about
theoof data
forneutron
time ofa asubstantial
testing=andtheirT) and antianalogs (isospin
analogs(isospin
discriminastate'
thefeapulse-shapethrough
The
in
the
other
leakingZr.
dipolepeaktarget,ries ofthe locationthesevisible in= allisospinsplittingtwo broadlocatedand tor,peaks areofclearlyenhancement,T —1) in the T, = T —1 nucleus resulting from a (P, n)ture, another cross-section
throughand are fits reaction.
excitation energy,The
Theasmooth
about 10 MeV higher infour spectra.
states of
was relevant
resonances.
targetcurves
The target states are the ground state and90Zr('He, t) at 130
observed with the reaction
E MeV
nucleus are illustrated inisobaricits
daughterthe M1 and F. l giant resonant states. Isospin geometry
45-MeV
spectra'and confirmed in
and in(p, n)P.
Petrovichandfavorsthe three transitions indicated.
strongly
4 W. G. Love, NPA354(1981)499c1oFig.
A,,=(-IIJ+'
data
athigherenergies.
wasItsuggestedn)(p,Analog E1

Anti-analog E1

'

("

~

5oc

I

~

I

~

I

I

~

GT=Anti-analog M1

~

I

I

I

400

2

3oc

&
I
x
r”

2oc

100

(Mb’)

103 _

I

I

1

:

45,

23
P HY
VOLUME2345, NUMBERSICAL REVIP HY SICAL REVI EXV LETTERS

8 DECEMBER 1980

fQf get20

10 20

fQf get
1

I

I

~

IA5

22—

IA5

IAS

I

~

IAS

Sn(p, nj
E&='t S Me V

E&='t S Me V

El

9„=7.59„=7.5

20—

20—

20—

20—

18

18

16

16

[p, n)
Ep='t S Me V

[p, n)
Ep='t S Me V

30
90 V]
E„(Me

90

El

18

16

16

)

1'f

1'f

)

1'i

1'i
E

E
C

1

I

22—
Sn(p, nj

18

10

12

C

12

LLI

12

LLI

Ltj

b 10

b 10

b 10

20

30
E„[MeVj

20

Ltj

b 10

20

20

FIG 2 Neutron spectre, for 112, 120 t22, 124Sn(P sl
120 t22, 124Sn(PNeutron spectre,
sl for "Sn, '2'Sn, andmeasuredforat112,
7.5'.
The baselines
~

12

90

30
90
E„[MeVj
~

FIG 2

E

E

30
E„(MeV]

FIG. 3. Neutron spectra for ~2Zr, 93Nb, '4Mo,"Sn,Physics Letters B 720 (2013) 428–432

Contents lists available at SciVerse ScienceDirect

Physics Letters B
www.elsevier.com/locate/physletb

A. Krasznahorkay et al. / Physics Letters B 720 (2013) 428–432

pnRQRPAJ=30,32,34,36,38★:DD-ME2

4

Anti-analog giant dipole resonances and the neutron skin of nuclei
A. Krasznahorkay a,∗ , N. Paar b , D. Vretenar b , M.N. Harakeh caInstitute for Nuclear Research, Hungarian Academy of Sciences (MTA Atomki), H-4001 Debrecen, P.O. Box 51, HungarybPhysics Department, Faculty of Science, University of Zagreb, CroatiacKernfysisch Versneller Instituut, University of Groningen, Groningen, The Netherlands

a r t i c l e

i n f oArticle history:
Received 27 August 2012
Received in revised form 15 January 2013
Accepted 24 February 2013
Available online 27 February 2013
Editor: J.-P. Blaizot
Keywords:
Neutron-skin thickness
Giant dipole resonances
Relativistic random-phase approximation
Rare-isotope beams

1. Introductiona b s t r a c t
We examine a method to determine the neutron-skin thickness of nuclei using data on the chargeexchange anti-analog giant dipole resonance (AGDR). Calculations performed using the relativistic proton–
neutron quasiparticle random-phase approximation (pn-RQRPA) reproduce the isotopic trend of theexcitation energies of the AGDR, as well as that of the spin-flip giant dipole resonances (IVSGDR), incomparison to available data for the even–even isotopes 112–124 Sn. It is shown that the excitation energiesof the AGDR, obtained using a set of density-dependent effective interactions which span a range of thesymmetry energy at saturation density, supplemented with the experimental values, provide a stringent
constraint on value of the neutron-skin thickness. For 124 Sn, in particular, we determine the value! R pn = 0.21 ± 0.05 fm. The result of the present study shows that a measurement of the excitationenergy of the AGDR in ( p , n) reactions using rare-isotope beams in inverse kinematics, provides a valuable
method for the determination of neutron-skin thickness in exotic nuclei.
© 2013 Elsevier B.V. Open access under CC BY license.

The Pb radius experiment (PREX) at JLAB [5] has initiateda new line of research based on the parity-violating elastic electron scattering to measure the neutron density radius R n , which in
An interesting phenomenon in nuclear structure is the formaturn allows to determine the neutron-skin thickness from ! R pn =tion of a skin of neutrons on the surface of a nucleus, and its
Fig. 4. The difference in the excitation energy of the AGDR and the IAS for
Fig. 3. Calculated values of the neutron-skinthicknessfortheeven–even
Sn
isoR
−
R
,
where
R
is
the
radiusoftheprotondensity
distribunppevolution with mass number in an isotopic chain [1]. A precise
124targetnucleusSn, calculated with the pn-RQRPA using five relativistic effect
topes
as
a
function
of
the
mass
number(filledcirclesconnected
by
the
solid
tion. Although parity-violating elastic electron scattering provides
measurement of the thickness of neutron skin is important notinteractions characterized by the symmetry energy at saturation a4 = 30, 32,
a model obtained
independentmeasurementof ! R pn ,absorption
its current precisionline), compared
to experimental
resultswiththe antiproton
only because this quantity represents
a basic nuclear
property,is far from satisfactory and the method cannot be applied to un36 and 38 MeV (squares), and the interaction DD-ME2 (a4 = 32.3 MeV) (star). T
but also because its value constrains the symmetry energy termJumpei Yasuda1,∗ , Tomotsugu Wakasa1 , Midori Okamoto1 , Masanori Dozono1 ,Kichiji Hatanaka3 , Munetake Ichimura2 , Sho Kuroita1 , Yukie Maeda4 , Tetsuo Noro1 ,Yasuhiro Sakemi5 , Masaki Sasano2 , and Kentaro Yako6

Prog. Theor. Exp. Phys. 2013, 063D02 (8 pages)
1
DOI: 10.1093/ptep/ptt038
Department of Physics,
Kyushu University, Higashi, Fukuoka 812-8581, Japan
2
RIKEN Nishina Center for Accelerator-Based Science, The Institute of Physical and Chemical Research,

Wako, Saitama 351-0198, Japan
3
ResearchCenter for
Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan
Extraction of anti-analog giant
dipoleresonance
4
208 Pbof Applied Physics, University of Miyazaki, Miyazaki, Miyazaki 889-2192, Japanand neutron skin thickness5 Department
for
Cyclotron and Radioisotope Center, Tohoku University, Sendai, Miyagi 980-8578, Japan
6
1,∗
1
1 , Masanori
1,
forNuclear Study,of Tokyo, Bunkyo, Tokyo 113-0033, Japan
Jumpei Yasuda , Tomotsugu Wakasa , MidoriCenter
Okamoto
DozonoUniversity∗
1 , Yukie Maeda4 , Tetsuo Noro1 ,Kichiji Hatanaka3 , Munetake Ichimura2 , Sho Kuroita
E-mail:yasuda@phys.kyushu-u.ac.jp
Yasuhiro Sakemi5 , Masaki Sasano2 , and Kentaro Yako6
Received January 23, 2013; Revised March 26, 2013; Accepted April 11, 2013; Published June 1, 2013
1
Department of Physics, Kyushu University, Higashi, Fukuoka 812-8581, Japan
2

RIKEN Nishina Center for Accelerator-Based Science, The Institute of Physical and Chemical Research,
Wako, Saitama 351-0198, Japan...............................................................................
3
Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047,
Japan
The
anti-analog
giant dipole resonance (AGDR) was separated from other excitations such as
4
Department of Applied Physics, University of Miyazaki, Miyazaki, Miyazaki 889-2192, Japan
208
5
Pb( p⃗, n⃗) reaction at athespin-dipole
resonancebymultipoledecomposition
analysisofthe
Cyclotron and Radioisotope Center, Tohoku University, Sendai, Miyagi 980-8578, Japan
6
Center for Nuclear Study, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
bombarding energy of T p = 296 MeV. The polarization transfer observables were found to be∗
E-mail: yasuda@phys.kyushu-u.ac.jp

useful for carrying out this separation. The energy difference between the AGDR and the isobaric
Received January 23, 2013; Revised March 26, 2013; Accepted April 11, 2013; PublishedJune
1, 2013analogstate(IAS) was determined to be !E = 8.69 ± 0.36 MeV, where the uncertainty includes
contributions. Theoretical calculations using the proton-neutron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .both. . . . . .statistical. . . . . . . . . . .and
. . . . .systematic......
The anti-analog giant dipole resonance (AGDR) was separated
from other excitations
such as random phase approximation predicted a strong correlation between
relativistic
quasi-particle
the spin-dipole resonance by multipole decomposition analysis of the 208 Pb( p⃗, n⃗) reaction at a
!E and the neutron skin thickness !R pn . Under the assumption that the correlation predicted inbombarding energy of T p = 296 MeV. The polarization transfer observables were found to be
this model
is correct,
the present !E value corresponds to a neutron skin thickness of !R pn =useful for carrying out this separation. The energy difference between
the AGDR
and the isobaric
analog state (IAS) was determined to be !E = 8.69 ± 0.36 MeV,
where
uncertainty
includes
0.216
±the
0.046
± 0.015
fm, where the first and second uncertainties are the experimental andboth statistical and systematic contributions. Theoretical calculations using the proton-neutron
theoreticaluncertainties,
relativistic quasi-particle random phase approximation predicted
a strong correlation
between respectively.

. . .that. .the. . correlation. . . . . . . predicted. . . . . .in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
!E and the neutron skin thickness !R pn . Under the assumption
this model is correct, the present !E value corresponds to a neutron skin thickness of !R pn =Subject Index
D13, D22
0.216 ± 0.046 ± 0.015 fm, where the first and second uncertainties are the experimental andtheoretical uncertainties, respectively.

...............................................................................
Subject Index

D13, D22

1.

Introduction

PHYSICAL REVIEW C 92, 034308 (2015)

Constraints on the neutron skin and symmetry energy from the anti-analog giant
dipole resonance in 208 Pb
Li-Gang Cao,1,2,3,4 X. Roca-Maza,5,6 G. Colò,5,6,3 and H. Sagawa7,8,3

OLÒ, AND H. SAGAWA)

22

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
2State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
3
Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China
4Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China
5
Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy
6
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Milano, via Celoria 16, 20133 Milano, Italy
7
Exp. Data
from Ref.
[57] Japan
Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu,
Fukushima965-8580,
8Exp.
Data from Ref. [54]RIKEN, Nishina Center, Wako, 351-0198,
Japan
SAMi-J(Received 23 April 2015; published 10 September
2015)

PHYSICAL REVIEW C 92, 034308 (2015)

9.5

EAGDR-EIAS(MeV)

SJ27
SJ29
SJ31
SJ33
SJ35

20

1

24

9.0

We investigate the impact of the neutron skin thickness, !Rnp , on the energy difference between the anti-analog
giant dipole resonance (AGDR), EAGDR , and the isobaric analog state (IAS), EIAS , in a heavy nucleus such as
208
Pb. For guidance, we first develop a simple and analytic, yet physical, approach based on the droplet model thatlinearly connects the energy difference EAGDR − EIAS with !Rnp . To test this correlation on more fundamental
grounds, we employ a family of systematically varied Skyrme energy density functionals where variations on
the value of the symmetry energy at saturation density J are explored. The calculations have been performed
within the fully self-consistent Hartree-Fock (HF) plus charge-exchange random phase approximation (RPA)
framework. We confirm the linear correlation within our microscopic approach and we can compare our resultswith available experimental data in 208 Pb in order to extract a preferred value for !Rnp and, in turn, for thesymmetry energy parameters. Averaging the results from two available experimental data, our analysis gives!Rnp = 0.236 ± 0.018 fm, J = 33.2 ± 1.0 MeV, and a slope parameter of the symmetry energy at saturation
L = 97.3 ± 11.2 MeV. The errors include the experimental uncertainties and a lower-limit estimate of model
uncertainties. These results are consistent with those extracted from different experimental data albeit L and!Rnp are somewhat large when compared to previous estimations based on giant resonance studies. Possible
hints whether model dependence can explain this difference are provided.

8.5

8.0
0.12

DOI: 10.1103/PhysRevC.92.034308

SJ27

DD-ME

0.16

0.20
0.24
∆Rnp(fm)

0.28

PACS number(s): 21.60.Jz, 21.65.Ef, 21.10.Sf, 24.30.Cz

I. INTRODUCTION
alreadydifference
approved with bothof improving
FIG.
4. (Color online) The energy
E the aim−Eofthe reachedFirst-forbidden vector (FFV) modes= charge-exchange dipole ~ anti-analog dipole
ÔKµ =!

dr""
σσ ′ τ τ ′effect of isospin mixing(?) not discussed

rY1K δσ,σ′ ⟨τ |τµ |τ ′ ⟩ψ̂ † (rστ )ψ̂(rσ ′ τ ′ )F (r, r ′ ) = δ(r, r ′ )r L YL

focus on μ=-1 excitations in neutron-rich nuclei
F (r, r ′ ) = 1⟨τ |τ± |τ ′ ⟩(anti-)analog of pygmy
dipole(Low-energy
dipole)mode??
!
m1 =dωωS(ω) =1⟨Ψ0 |[Ô, [Ĥ, Ô]]|Ψ0 ⟩2
understanding of PDR
(LED)
in
terms
of
iso-triplet
states!
general mechanism
emergenceof the PDR
m = for
dωω
S(ω)
p

p

S(ω) ="⟨Ψi |Ô|Ψ0 ⟩δ(ω − ωi )appearance of the low-lying mode and its effect on FF transitions?
iδτ τ ′δσσ′δτ τ ′

100
30 (a)

Ca

(c)

Ni

30

25

Strength (fm2/MeV)

35 (b)

SkM*, Γ=2.0 MeV
Sn

80

25

20

60
20

15
15
N=
10 56
54
52

5

40
N=
110

10 N=
66

50

5
40

20 100

62
58

90

54
30

0
-30 -20 -10 0 10 20 30 40
ET (MeV)050-20 -10 0 10 20 30 40
ET (MeV)0-20 -100

10 20
ET (MeV)30

40

Mechanism for the occurrence of the low-lying FFV modecross-shell (N→N-1) excitation in low-energy−1!ω excitationNneutrons!are weakly bound†δρ+ (r) (quasi= ⟨λ| )neutronsψ̂ν (rσ)ψ̂πin(rσ)|0⟩arethe continuum when |λ|≃0N-1σδρ− (r) = ⟨λ|!

†ψ̂π (rσ)ψ̂ν (rσ)|0⟩σprotons are deeply bound!!

σσ ′ τ τ ′

′

′

†

′

′

rY1K ⟨σ|σµσ |σ ⟩⟨τ |τµτ |τ ⟩ψ̂ (rστ )ψ̂(rσ τ )⎧1⎪⎪− √ (τ + τ ) (µ = +1)πνOnset of low-lying FFV mode: sensitive to the shell structuresummed strength in low-energy (ω < 15 MeV)SkM*
Fraction (%)

8 SkM*

Ca

6

Ni

Sn

4
2
0

34 40

60
80
Neutron number

100

strong shell effectOnset of low-lying FFV mode: sensitive to the shell structuresummed strength in low-energy

30 (a)

Ca

Ca

6

Ni

25

Sn

4
2
0

34 40

60
80
Neutron number

100

Is this because the occupation of f5/2 begins?
No, occupation of g9/2 begins due to pairing
νg9/2 → πf7/2

Strength (fm2/MeV)

Fraction (%)

8 SkM*

20

15

N=
10 56
54
52

5

50
40
30

0
-30 -20 -10 0 10 20 30 40
ET (MeV)Onset of low-lying FFV mode: sensitive to the shell structuresummed strength in low-energy

30 (a)

Ca

Ca

6

Ni

25

Sn

4
2
0

34 40

60
80
Neutron number

100

Neutrons start occupying the loosely-bound d5/2
λ=-0.4~-0.6 MeVStrength (fm2/MeV)

Fraction (%)

8 SkM*

20

15

N=
10 56
54
52

5

50
40
30

0
-30 -20 -10 0 10 20 30 40
ET (MeV)Onset of low-lying FFV mode: sensitive to the shell structuresummed strength in low-energy

Fraction (%)

8 SkM*

Ca

6

Ni

Sn

4
2
0

34 40

52

60
8084
Neutron number

occupation of d5/2

100

occupation of f7/2

Microscopic structure of the low-lying FFV mode in Ni isotopes

88Ni

70

N=52~56

QRPA

60

ν2d5/2 → π2p3/2

Γ=0.1 MeV

50

N=58~
ν2d5/2 → π2p3/2
ν3s1/2 → π2p1/2
s-wave q.p.resonanceν7/2+ → π2p1/2
discretised continuum besides the g7/2 hole-like resonance

Strength (fm2/MeV)

40
30
20
10
0
70

νd5/2 → πp3/2

unp.

60
50
40
30
20
10
0
-20

νd3/2 → πp1/2, νs1/2 → πp1/2
ν7/2+ → πf5/2
-15

-10

ET (MeV)-50Appearance of the FFV modes

N+1

cross-shell (N→N±1) excitation for negative-parity excitationNneutrons are weakly boundN-1

protons are deeply bound: low-lying mode
protons are in the continuum: giant resonance and pygmy?

π

ν

ħω0

30 (a)

100

Ca

80

25

20

60

anti-analog
15 PDR

20

-ħω0

15

N=
10 56
54
52

5

(c)

Ni

30

25

Strength (fm2/MeV)

35 (b)

SkM*, Γ=2.0 MeV
Sn

40
N=
110

10 N=
66

50

5
40

20 100

62
58

90

54
30

0
-30 -20 -10 0 10 20 30 40
ET (MeV)050-20 -10 0 10 20 30 40
ET (MeV)0-20 -100

10 20
ET (MeV)30

40those of the Hartree-Fock plus random-phase approximation(HF+RPA) [57] presented by the dashed lines with open

calculation
becomesatsmaller
the58–60
result andfrom around
HF+RPA.
cusps
also appear
aroundthan
N=
N=
This is while
due tothe
the concave
pairing effect.
In thethe
72–74,
ones canbe HF
seencalculation,
at N = 60–62.
54
groundstatein
Ca corresponds
to the fullof
These cusps are mostprominent aroundthe occupation
proton subshell
S.(Z
Ebata
PRC90(2014)024303
the
pal.,
orbit.
However,the HF+BCS
calculation
6
1/2 they
6C≈neutron40),etwhile
becomeweakerapproaching
the magic
8O
numbers, Z → 28 and Z → 50. This suggests that these may
Ne105
be associatedwith the ground-state deformation.
6
12Mg
14Si
Let
us
brieflycomment on the deformation effect. In Ne
RPA - CS
16
4
RPA - Oand Mgneutron-richisotopes, the present results suggest that
18Ar
5
RPA - Ne
20Ca
the
onset
of
deformation in the ground state increases fPDR .
RPA - Mg
3
RPA - Si
However,thein the present mass region is more com4
RPA -behaviorS- Ar
plex. fPDR RPA
decreaseat the onset of deformation around; then,2
RPA - Cathey 3decrease again near N = 74, which corresponds to thedisappearance of the deformation back to the spherical shape.
1
2 IV, we discuss the effect of deformation in more details.
In Sec.
0
The next jump in fPDR at N = 82 → 84 is clearly identified.
4
8
12
16
20
24
28
This1 suggests that the definition of the “low-ℓ orbits” is
Neutron Number
different between light and heavy systems. The nuclei around0N=
82 are16all calculated
to be spherical,
it cannot
FIG. 7. (Color online) Fraction of the low-lying E1 strength fPDR
20
24
28
32 thus 36
40 be theeffect of deformation. The
single-particle
in Eq. (13) as functions of the neutron number for C, O, Ne and Mg
NeutronNumberorbit just above theN = 82 shell gap is f7/2 . In light nuclei, when the Fermi levelisotopes. The solid lines with filled symbols show the present results

Appearance of anti-analog-PDR

12

8

Ca

Sn

6
4

of Cb-TDHFB, while the dashed lines with open symbols show thoseof HF+RPA.

2
0

Ni

FIG. 8. (Color online) Same as Fig. 7, but for Si, S, Ar and Ca
isotopes.
6

28Ni

30Zn

024303-540

60
80
Neutron number

occupation of weakly-bound p3/2 (N>28), d5/2(N>50),
and f7/2 and f5/2 (N>82)

32Ge

5

100
fPDR [%]

Fraction (%)

10

fPDR [%]

fPDR [%]

summed strength below GDR (except the low-lying states)

34Se

36Kr

38Sr

4

40Zr

42Mo

44Ru

3

46Pd

48Cd
50Sn

2
1
0

28

36

44

52

60

68

Neutron Number

76

84

92

Summary (II)✓ LLFFV state appears uniquely in very neutron-rich nuclei−1!ω excitation✓ strong shell effectsteadyselection
rule
due
to
deeply-boundprotonorbitals!
†
ρ+ (r) = ⟨λ| weakψ̂νcollectivity(rσ)ψ̂π (rσ)|0⟩σ! # of neutron hole states satisfying the selection rule is limited†ρ− (r) = ⟨λ|ψ̂π (rσ)ψ̂ν (rσ)|0⟩✓ effect on the beta-decay rate and beta-delayed neutron emission in futureσ!!

σ′ τ τ ′axial-vector (spin-dipole) modes also should be considered
′

′

†

′

′

rY1K ⟨σ|σµσ |σ ⟩⟨τ |τµτ |τ ⟩ψ̂ (rστ )ψ̂(rσ τ )✓ emergence of (anti)analog PDR below giant resonance

⎧
1
⎪
⎪
−
(τ
+
τ
)
(µ
=
+1)
√
x
y
⎪
⎪