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Charge-exchange scattering to IAS andimplications for the nuclear symmetry energy
Dao T. Khoa1, B.M. Loc1, and R.G.T. Zegers2
1Institute for Nuclear Science & Technology, VINATOM.
2NSCL and Department of Physics and Astronomy, MSU.
ARIS2014, Tokyo, June 6, 2014

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(a)

(b)

Figure : Isobaric analog states in the mirror nuclei 7 Li and 7 Be, and in the isobars A = 6JJ
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The Fermi transition (∆L = ∆S = 0 and ∆T = 1) between theisobaric analog states (IAS), induced by the charge-exchange (p , n) or(3He,t ) reaction, can be considered as “elastic” scattering of protonor 3He by the isovector part of the OP that flips the projectile isospin
I A.M. Lane, Phys. Rev. Lett. 8, 171 (1962)
I G.R. Satchler, Isospin in Nuclear Physics (Ed. D.H. Wilkinson,
North-Holland Pub., Amsterdam, 1969) p390
In the folding model, the isovector part of the OP is determined
exclusively by the neutron-proton difference in the nuclear densities
and the isospin dependence of the effective NN interaction.
I DTK, W. von Oertzen, A.A. Ogloblin, Nucl. Phys. A 602, 98(1996)
I DTK, E. Khan, G. Colò, N.V. Giai, Nucl. Phys. A 706, 61 (2002)
I DTK, B.M. Loc, D.N. Thang, Eur. Phys. J. A 50: 34 (2014)

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JJ
II
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charge-exchange scattering to the IAS

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Given the same isospin t = 1/2, the central nucleon-nucleus or
3
He-nucleus OP for the elastic scattering on a nonzero-isospin target
can be written as
U (R ) = U0 (R ) + 4U1 (R )t .T ,aA(1)where t is the isospin of the projectile and T is that of the target withmass number A, and a = 1 and 3 for nucleon and 3He, respectively.
The second term is the symmetry term of the OP, and U1 is known
as the Lane potential that contributes to both the elastic andcharge-exchange scattering. The knowledge of U1 is of fundamental
interest in the study of the isovector mode of direct nuclear
reaction.

JJIIJI
BackCloseGiven the isospins t and T for the projectile and target⇒ Tz = (N − Z )/2 and T˜z = Tz − 1 for the target A and its
isobaric analog Ã, respectively. Denoting the state formed by adding
proton or 3He to A is |aA >, and that formed by adding a neutron ortriton to Ã is |ãÃ > ⇒ Form factor of the charge-exchange
scattering to the IASt.T2√
F (R ) =< ãÃ|4U (R )|aA >=
2T U (R ).
cx

1aAaAz

1

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(2)

In the two-channel approximation
Ψ = |aA > χaA(R ) + |ãÃ > χãÃ(R ).

(3)

The elastic and charge-exchange scattering cross sections are obtained
from the solutions of the coupled-channel Lane equations[Ka + Ua (R ) − Ea ] χaA(R ) = −Fcx(R )χãÃ(R ),[Kã + Uã (R ) − Eã ] χãÃ(R ) = −Fcx(R )χaA(R ).
G.R. Satchler, Direct Nuclear Reactions (Oxford Univ. Press, 1983)

(4)
(5)

JJ
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Central nucleon-nucleus or nucleus-nucleus optical potential U isevaluated as a Hartree-Fock-type potential
U=X[< ij |vD|ij > + < ij |vEX|ji >],

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(6)

Using explicit proton (ρp ) and neutron (ρn ) densities, nucleon-nucleus
or nucleus-nucleus OP is obtained in terms of the isoscalar (UIS) andisovector (UIV ) parts as
U (E , R ) = UIS (E , R ) ± UIV (E , R ),(7)

where - sign pertains to the proton or 3He OP, and + sign to theneutron or triton OP ⇒ FF for both the (p , n) and (3He,t ) scatteringto the IASs2√
2Fcx (R ) =2Tz U1(R ) =UIV (R ).
(8)
aATzJJIIJIBackClose210(p,n) IAS, E =35 MeVp04810Ca

7/17-210-390x10Zr(mb/sr)d /d-41010120-6-6Snx10-810208-9Pbx10-1010Hybrid OP
Folded OP-12

100204060

80c.m.

100

120140160180(deg.)

Figure : CC description of the (p , n) scattering to the IAS of different targets at Ep = 35 MeV.
MSU data by R.R. Doering, D.M. Patterson, A. Galonsky, Phys. Rev. C 12, 378 (1975).

JJIIJI
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The folding + CC analysis of the charge-exchange scattering is helpful
for the study of the NM symmetry energy, when the same density- andisospin dependent effective NN interaction is used in the HF
calculation of the total NM energy density E

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1X
[hkk 0|vD|kk 0i + hkk 0|vEX|k 0k i],
E = Ekin +2 kk 0
where |k i are plane waves. The total NM energy per particle EE
ρn − ρp= E (ρ, δ) = E (ρ, δ = 0) + S (ρ)δ 2 + O (δ 4) + ..., δ =. (9)ρρThe folding model analysis of the (p , n) scattering to the IAS has put
some constraint on the nuclear symmetry energy S (ρ) at ρ . ρ0,within the empirical boundaries deduced from experiments.

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Symmetry energy
100N
N80

N

9/17=1.5

V1=1.3

V1=1.0V1S (MeV)

CDM3Y6s

60

40

200
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8-3(fm )Figure : HF prediction of S (ρ) compared to empirical data by M.B. Tsang et al. Prog. Part.
Nucl. Phys. 66, 400 (2011); L. Trippa et al. Phys. Rev. C 77, 061304(R) (2008); DTK, H.S.
Than, Phys. Rev. C 71, 044601 (2005).

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11048Ca(p,n)

48

Sc

IAS

10/17010-1(mb/sr)

10E

p= 45 MeV-2d /d10

N=1.5

N=1.3

N=1.0

V1V10

V110

CDM3Y6s

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p= 35 MeV

-2

10020

40

60

80

c.m.

100(deg.)

120140160

180

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2.5

2.0

STIFF

1.5

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CDM3Y3
1.0

CDM3Y40.50.0/M M Solar

CDM3Y62.0

SOFT
1.5

1.0
CDM3Y3s
CDM3Y4s
0.5

CDM3Y6s

0.0510R15(km)Figure : NS gravitational mass vs. its radius obtained with the EOS’s given by the stiff- and softtype NN interactions. HF results from D.T. Loan, N.H. Tan, DTK, J. Margueron, Phys. Rev. C
83, 065809 (2011), empirical data (x-ray burster 4U 1608-52) deduced by A.W. Steiner, J.M.
Lattimer, E.F. Brown, Astrophys. J. 722, 33 (2010).

JJIIJI
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The experimental resolution of (3He,t ) experiment is now much higher
than that of (p , n) reaction [Y. Fujita, B. Rubio, W. Gelletly, Prog.
Part. Nucl. Phys. 66, 549 (2011).]The difference between the neutron and proton densities or theneutron skin ∆Rnp of the target can be well probed in the folding
model analysis of the charge-exchange (3He,t ) scattering to IAS atmedium energies (when the two-step processes can be neglected, andthe t −matrix interaction can be used in the folding calculation).
∆Rnp = hrn2i1/2 − hrp2i1/2,12/17(10)was found by the structure studies to be strongly correlated withthe slope of the symmetry energy of nuclear matter ⇒Information on the density dependence of the symmetry energy.
M.B. Tsang et al., Phys. Rev. C 86, 015803 (2012)

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10

290

10t

90

Nb,IAS

E =420 MeV
lab

1

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0d /d(mb/sr)

103Zr( He, )1010-1

Rnp= 0.09 ± 0.03 fm
HFB density, R = 0.07 fm
npBest-fit,

-20.0

0.5

1.0

1.5

2.0

2.5

3.0(deg)c.m.

Figure : Folding model analysis of 90 Zr(3 He,t )90 Nb∗IAS data [R.G.T. Zegers et al., Phys. Rev.
Lett. 99, 202501 (2007)] given by the charge-exchange FF based on the empirical density by Rayet al., Phys. Rev. C 18, 1756 (1978), and HFB density by M. Grasso et al., Phys. Rev. C 64,064321 (2001).

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10

2208

3tPb( He, )10Bi

IAS,E =420 MeVlab

114/170d /d(mb/sr)

10208

10-1

Rnp= 0.16 ± 0.04 fm
HFB density, R = 0.17 fm
npBest-fit,

10-20.0

0.5

1.0

1.5

2.0

2.5

3.0(deg)c.m.

Figure : Folding model analysis of 208 Pb(3 He,t )208 Bi∗IAS data [R.G.T. Zegers et al., Phys. Rev.
Lett. 99, 202501 (2007)]. See details in B.M. Loc, DTK, R.G.T. Zegers, Phys. Rev. C 89,024317 (2014).

JJ
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3Pb( He, )

10Bi

IAS,

E =420 MeVlab

115/170d /d(mb/sr)

10208

10-1IV density ~ PREX data

10-20.0

0.5

1.0

1.5

2.0

2.5

3.0(deg)c.m.

Figure : Folding model analysis of 208 Pb(3 He,t )208 Bi∗IAS data given by the charge-exchange FF
based on the empirical density adjusted to reproduce the PREX data ∆Rnp ≈ 0.33+0.16−0.18 fm
[S. Abrahamyan et al. (PREX Collaboration), Phys. Rev. Lett. 108, 112502 (2012)].

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Conclusion
The consistent folding model study of the charge-exchange (p , n)scattering to the IAS has been done using the isospin dependent
OP and charge-exchange FF given by the density dependent
CDM3Y6 interaction
I Some preference for the stiff behavior of the nuclear symmetryenergy with the increasing NM density has been established.
390I The folding model analysis of the ( He,t ) scattering the IAS ofZrand 208Pb at Elab = 420 MeV has been done to deduce the neutron
skin for these nuclei. The best-fit neutron skin values found by ouranalysis are in a good agreement with the existing database.

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