CHIN. PHYS. LETT. Vol. 26, No. 2 (2009) 022503
Optical Potential Parameters for Halo Nucleus System 6 He+12 C from Transfer
Reaction 11 B(7 Li, 6 He)12 C *
WU Zhen-Dong(吴振东)** , LIN Cheng-Jian(林承键), ZHANG Huan-Qiao(张焕乔), LIU Zu-Hua(刘祖华),
YANG Feng(杨峰), AN Guang-Peng(安广朋), ZHANG Chun-Lei(张春雷), ZHANG Gao-Long(张高龙),
JIA Hui-Ming(贾会明), XU Xin-Xing(徐新星), BAI Chun-Lin(白春林), YU Ning(喻宁), JIA Fei(贾飞)
China Institute of Atomic Energy, Beijing 102413
(Received 9 September 2008)
The optical potential parameters for the halo nucleus system 6 He+12 C are extracted from fits to the measured
angular distributions of 11 B(7 Li, 6 He)12 C reaction at energies of 18.3 and 28.3 MeV with distorted-wave Born
approximation analysis. The characters of the obtained optical potential parameters are basically consistent with
the results extracted from the fits to the elastic-scattering angular distributions in the literature.
PACS: 25. 70. Hi, 24. 10. Eq, 24. 10. Ht
Optical model is one of the most successful models to describe the nuclear reaction mechanism for
its simplicity, figurativeness and usability. The optical potential parameter is one of the most important
quantities, which describes the interaction in nucleus–
nucleus collision. It is closely related to the internal
structure of the colliding nuclei. There are many differences between the halo nuclei[1−4] and the tightly
bound nuclei. For example, the halo nuclei have large
nuclear radius, large nuclear surface diffuseness, small
separation energy and so on. These properties cause
the difference of the optical potential parameters between the halo nucleus systems and the tightly bound
nuclei systems.
6
He is a well-known neutron-halo nucleus which
can be easily described as a tightly bound 𝛼 core plus
two valence neutrons (𝑆2𝑛 = 972.4 ± 0.8 keV). The
weak binding of the valence nucleons leads the wavefunctions to tunnel out of the core potential and causes
the valence nucleons to form the ‘halo’ with a large
spatial extension. Thus the study of the interaction
potentials of 6 He with other nuclei is very interesting.
In principle, the optical potential parameters of the
halo nucleus systems can be extracted from the fits to
the measured elastic-scattering angular distribution,
the inelastic excitation function, transfer angular distribution and fusion excitation function, etc. Usually,
extracting the optical potential parameters from fits to
the elastic scattering angular distribution is the most
direct method. For halo nucleus systems, the elasticscattering angular distribution is necessary with the
radioactive nucleus beam.[5] At present, the intensity
and quality of radioactive nucleus beam limit the experimental precise. Therefore, the optical potentials
of the halo nucleus systems have not been sufficiently
studied due to the lack of elastic-scattering angular
distribution data.
In view of this situation, the transfer reaction with
stable projectile and target combination can be used
as an alternative method[6] to extract the optical potential parameters of the halo nucleus systems, which
may become an useful complement method. In this
Letter, we extract the 6 He+12 C optical potential parameters from the measuring angular distributions
of 11 B(7 Li, 6 He)12 C reaction at energies of 18.3 and
28.3 MeV.
x
x
b
A
+
a/bx
b
A
+
B/Ax
Fig. 1. Schematic representation of transfer reaction.
A schematic representation of transfer reaction is
shown in Fig. 1. Here 𝑥 is a transfer particle. According to the formal theory for transfer reaction, the differential cross-section from entrance channel 𝛼 = 𝑎+𝐴
to exit channel 𝛽 = 𝑏 + 𝐵 in 𝑎 + 𝐴 → 𝑏 + 𝐵 reaction
can be exactly written as
(︁ 𝑑𝜎 )︁
𝑑Ω
(︁ 4𝜋 2 )︁2 𝜇 𝜇 𝑘
𝛼 𝛽 𝛽
|⟨𝜙𝛽 |𝑇 |𝜙𝛼 ⟩|2
𝑘𝛼
ℎ̄2
(︁ 4𝜋 2 )︁2 𝜇 𝜇 𝑘
𝛼 𝛽 𝛽
=
|𝑇𝑓 𝑖 |2 ,
𝑘𝛼
ℎ̄2
=
𝛽𝛼
(1)
where 𝜇𝛼 , 𝜇𝛽 , 𝑘𝛼 𝑘𝛽 , 𝜙𝛼 and 𝜙𝛽 are the reduced
masses, the wave vectors, and the wavefunctions of
the entrance and exit channels, respectively. 𝑇 is the
transition matrix and 𝑇𝑓 𝑖 is the element of transition
matrix from the entrance channel to the exit channel.
The physical quantities 𝑘𝛼 , 𝑘𝛽 , 𝜙𝛼 , 𝜙𝛽 and 𝑇
can be deduced from the Schrödinger equation for
* Supported by the National Natural Science Foundation of China under Grant No 10575135, and the National Basic Research
Programme under Grant No 2007CB815003-2.
** Email: [email protected]
c 2009 Chinese Physical Society and IOP Publishing Ltd
○
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CHIN. PHYS. LETT. Vol. 26, No. 2 (2009) 022503
the transfer reaction system when the optical potential 𝑈𝛼 , the bound potential 𝑈𝐴 (for 𝑎 = 𝑏 + 𝑥),
the optical potential 𝑈𝛽 , the bound potential 𝑈𝐵 (for
𝐵 = 𝐴 + 𝑥) of entrance channel and exit channel are
known. So, the transfer reaction differential crosssection (𝑑𝜎/𝑑Ω)𝛽𝛼 can be taken as a function of 𝑈𝐴 ,
𝑈𝐵 , 𝑈𝛼 and 𝑈𝛽 ,
(𝑑𝜎/𝑑Ω)𝛽𝛼 = 𝑓 (𝑈𝐴 , 𝑈𝐵 , 𝑈𝛼 , 𝑈𝛽 ).
(2)
than 0.01∘ which can be neglected. Thus only the
counting statistic errors are considered.
The measured transfer reaction angular distributions were analysed within the framework of
Distorted-Wave Born Approximation (DWBA) with
the computer code PTOLEMY.[8] The nucleusnucleus interaction optical potential consists of the
Woods–Saxon form and Coulomb potential of a uniform charged sphere and it can be written as
Usually, the bound potentials 𝑈𝐴 and 𝑈𝐵 can be obtained from reproducing the 𝑥 particle separation energies. The optical potentials 𝑈𝛼 can be extracted
from fits to the experimental elastic-scattering angular distribution data. When the potentials 𝑈𝐴 , 𝑈𝐵 ,
𝑈𝛼 and the transfer angular distribution (𝑑𝜎/𝑑Ω)𝛽𝛼
are known, then the exit channel optical potential 𝑈𝛽
can be deduced. The spectroscopic factors should be
considered in the calculation of deducing 𝑈𝛽 .
𝑈 (𝑟) = −
𝑉
𝑖𝑊
−
𝑟−𝑅𝑖
1 + exp( 𝑟−𝑅
)
1
+
exp(
𝑎
𝑎𝑖 )
⎧
𝑧𝑍𝑒2
⎪
⎪
,
⎨
𝑟
+
(︁
𝑧𝑍𝑒2
𝑟2 )︁
⎪
⎪
⎩
3− 2 ,
2𝑅𝐶
𝑅𝐶
𝑅= 𝑟0 𝐴1/3 ,
𝑅𝑖 = 𝑟𝑖0 𝐴1/3 ,
when 𝑟 ≥ 𝑅𝐶 ,
(3)
when 𝑟 < 𝑅𝐶 ,
𝑅𝐶 = 𝑟𝑐 𝐴1/3 ,
(4)
where 𝑉 , 𝑊 , 𝑟0 , 𝑟𝑖0 , 𝑎, 𝑎𝑖 are the potential depth, radius and diffuseness parameters for real and imaginary
part, respectively; 𝑟𝑐 is Coulomb radius parameter.
They are the adjustable parameters.
Fig. 2. Δ𝐸-ER spectrum of 7 Li +11 B reaction at 𝐸Lab =
28.3 MeV. ER: energy residual.
The experiment was performed at the HI-13 tandem accelerator of China Institute of Atomic Energy. A natural boron target in thickness about
100 µg/cm2 evaporated onto a 20 µg/cm2 carbon foil
backing was bombarded by collimated 7 Li beams of
18.3 and 28.3 MeV. The 7 Li beam current is in the
range of 200–400 enA. The Q3D magnetic spectrometer with a multi-layer position-sensitive focal plane
gas detector was employed to detect the exit products.
The entry slit of the Q3D magnetic spectrometer was
fixed in the experiment. Figures 2 and 3 show the ∆𝐸ER and ∆𝐸-POS spectra of the products recorded by
a focal plane detector at 13.5∘ for the 28.3 MeV 7 Li
+11 B reaction.
The relative angular distributions of the 11 B(7 Li,
6
He)12 C reaction were obtained by analysing the experimental data with the software tool PAW (Physics
Analysis Workstation).[7] The uncertainties of the
transfer angular distributions are mainly from the entry angle error of the Q3D magnetic spectrometer and
the statistic errors. In our experiment, the entry angle error of the Q3D magnetic spectrometer is smaller
Fig. 3.
Δ𝐸-POS spectrum of 7 Li +11 B reaction at
𝐸Lab = 28.3 MeV. POS: position of incident particles.
In DWBA calculations, the entrance channel optical potentials for the 7 Li+11 B system at energies
18.3 and 28.3 MeV are the same which is taken
from Neuschaefer[9] (row A in Table 1).
The
bound potentials of 7 Li = p + 6 He and 12 C = p + 11 B
were deduced from reproducing the last proton separation energies with the code PTOLEMY. The
ground state theoretical spectroscopic factors 0.888
for 7 Li = p + 6 He [10] and 2.85 for 12 C = p + 11 B [11]
were adopted. In the searching for optical potential
parameters, the experimental transfer reaction angular distributions were renormalized to the theoretical
values. The renormalization factor 𝑘𝑁 was calculated
by minimizing the goodness of fit 𝜒2 ,
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𝜒2 =
𝑛
∑︁
(𝜎𝑖𝑇 − 𝑘𝑁 𝜎𝑖 )2
.
(𝑘𝑁 ∆𝜎𝑖 )2
𝑖=1
(5)
CHIN. PHYS. LETT. Vol. 26, No. 2 (2009) 022503
Table 1. Optical potential parameters for
results of gradient descent searching.
System
A
B
C
7 Li +11 B
6 He +12 C
6 He +12 C
11 B(7 Li, 6 He) 12 C
reactions. Here row A taken from Ref. [9], rows B and C are the
𝐸Lab
(MeV)
𝐸c.m.
(MeV)
𝑉
(MeV)
𝑟0
(fm)
𝑎
(fm)
𝑊
(MeV)
𝑟𝑖0
(fm)
𝑎𝑖
(fm)
𝑟𝑐
(fm)
𝑘𝑁
𝜒2
34
25.75
34.91
20.78
17.16
23.27
184.2
160.129
320.00
0.72
1.00
1.00
0.75
0.802
0.703
11.1
27.321
25.007
1.33
1.271
1.133
0.61
0.50
0.50
0.96
1.200
1.2
34.839
10.283
13.498
494.774
To extract the reliable optical potential of the
He +12 C system, a grid searching on optical potential parameters was carried out. Then the best obtained 6 He +12 C optical potential parameters of the
grid searching were used as the initial potential parameters in another gradient descent searching for
6
He +12 C optical potential parameters again. Finally,
the optimal potential parameters of 6 He +12 C are obtained by the gradient descent searching in rows B and
C in Table 1. The theoretical transfer reaction angular
distributions calculated by using our 6 He+12 C optical
potential parameters for exit channel are compared
with the experimental data for the 11 B(7 Li, 6 He)12 C
reaction at 18.3 and 28.3 MeV in Fig. 4. The comparison shows a good agreement between them.
It should be mentioned that there are more than
one set of optical potential parameters with similar
goodness of fit to the transfer reaction angular distribution during our searching for the optical potential
parameters. It may be due to the Igo ambiguity of the
optical potentials.[12]
6
diffuseness should be caused by the large spatial extension of the 6 He halo. It is also consistent with the characteristic of the diffuseness parameters for real part
potential extracted from the fits to elastic-scattering
angular distributions of halo nucleus systems.
In summary, the angular distributions of 11 B(7 Li,
6
He)12 C reaction at energies of 18.3 and 28.3 MeV
have been measured. The optical potential parameters of the 6 He+12 C halo nucleus system are extracted
from the fits to the measured transfer reaction angular
distributions by means of DWBA approach. The theoretical transfer angular distribution calculated with
the 6 He+12 C optical potential parameters are in good
agreements with the experimental data. The obtained
6
He+12 C optical potentials show the characteristics of
halo nucleus systems, which are consistent with those
extracted from the fits to the elastic-scattering angular distributions. Thus the transfer reaction with stable projectile and target combination can be used as
an alternative method to extract the optical potential
parameters of the halo nucleus systems. It may be a
useful method.
References
Fig. 4. Angular distribution of
11 B(7 Li, 6 He)12 C.
The obtained 6 He+12 C optical potential parameters in Table 1 show two characteristics. First, the radius parameters for the real part potential are smaller
than those for the imaginary part potential. Second,
the diffuseness parameters for real part potentials are
larger than 0.65 fm which is taken from the systematic analysis elastic-scattering angular distributions of
tight bound nuclei systems. These characteristics are
in agreement with the theoretical prediction[13] of the
optical potential of halo nucleus systems. The larger
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