Search for states with enlarged radii in
excited states of
12
B
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1. Introduction
Recently the evidence of the excited states of light nuclei with nonstandard sizes and enlarged radii, located closely and above the particleemission threshold, was convincingly demonstrated (see, e.g., Ref. [1] and references therein). The existence of neutron halos in the shortlived excited states of some stable and radioactive nuclei was revealed, in particular, by the ANC analysis of the neutrontransfer reactions [2,3,4]. Thus Liu et al. [2] reported the observation of halos in the first (1/2
+
, E
x
= 3.089 MeV) excited state of
13
C, and the second (2¯, E
x
= 1.674 MeV) and third (1¯, E
x
= 2.621 MeV) excited states of
12
B using the (d,p) reactions on
12
C and
11
B, respectively, by analyzing the data with the ANC method.
In this article, we present the results of the measurement and analysis of the
11
B(d,p)
12
B reaction populated the 1
+
g.s. and five excited states up to the neutronemission threshold: 0.953 MeV 2
+
, 1.674 MeV 2¯, 2.621 MeV 1¯, 2.723 MeV 0
+
, and 3.389 MeV 3¯ states, at incident deuteron energy E
lab
= 21.5 MeV. We determine preliminary values of the spectroscopic factors and the ANC coefficients by comparing the results with the measured differential cross sections. Afterwards we calculate the preliminary rms radii of the last neutron in the final state of
12
B.
One of the main our aims is to estimate the radius of the 3.389 MeV 3¯ state located in the continuum spectrum and to answer the question whether there exists a state, in which a neutron halo is formed by the valence neutron occupying the l = 2 orbital.
2. Experiment
The differential cross sections of deutron elastic scattering from
11
B and the
11
B(d,p)
12
B reaction leading to formation of the 1
+
g.s. and the excited 0.95 MeV 2
+
, 1.67 MeV 2¯, 2.62 MeV 1¯, 2.72 MeV 0
+
, and 3.39 MeV 3¯ states were measured at incident deutron energy E
lab
= 21.5 MeV in the angular range (5
∘
 85
∘
). The measurements were performed at Jyvaskyla University cyclotron using the Large Scattering Chamber (LSC) having a diameter 1500 mm. The scattered particles were detected with four ΔE  E telescopes, each performing measurements at two angles. Selfsupporting
11
B foil of thickness 0.275 mg/cm
2
served as a target. The beam current was about 20 particle nA. The experiment employed a beam monochromatization system, which made it possible to obtain a total energy resolution of about 70 keV. A typical proton spectrum for the excitation of
12
B states up to E
x
∼
3.5 MeV obtained at Θ
c.m.
= 18
∘
is shown in Fig. 1. The
12
B excitation peaks are clearly separated in this spectrum, and contamination from other ions (carbon, oxygen) is well separated. We used the standard expansion procedure to describe peaks with Gaussian functions, whereas the peak positions and widths were fixed in accordance with table values, and the area under the peak was the only free parameter. A separation of the neighbouring 2.62 MeV 1¯ and 2.72 MeV 0
+
states of
12
B was made possible owing to a good energy resolution assured by the monochromatization of the beam (see Fig. 1, low part). The experimental differential cross sections of d +
11
B elastic scattering and the
11
B(d,p)
12
B (g.s., 0.95, 1.67, 2.62, 2.72, 3.39 MeV) reaction are shown in Fig. 2.
Figure 1
A typical proton spectrum at Θ
c.m.
= 18
∘
.
3. Analysis of the results
Compoundnucleus (CN) analysis of the differential cross sections is carried out within the HauserFeshbach formalism of the statistical CN model by using the computer code CNCOR [5]. In Fig. 2, the results of CN calculations (dotted lines) are shown in comparison with the experimental differential cross sections at beam energy 21.5 MeV. The analysis showed that the CN mechanism provides less than 0.1% of the cross section values at forward angles and about 13% at medium angles 60
∘
80
∘
. So the transfer mechanism predominates for this reaction at given energy.
Figure 2
Differential cross sections of the
11
B(d,p)
12
B reaction at E
lab
= 21.5 MeV leading to the 1
+
ground state and excited 0.954 MeV 2
+
, 1.674 MeV 2¯, 2.621 MeV 1¯, 2.723 MeV 0
+
, and 3.389 MeV 3¯ states of
12
B obtained in the present work (points) in comparison with the CRC (dashed line) and CN (dotted line) calculations, and their sum (solid line).
The Coupledreactionchannels (CRC) method for direct reactions includes the onestep DWBA as a special case, so the finiterange neutron transfer mechanism and the elastic scattering in the entrance and exit channels are calculated within the same formalism. CRC calculations are carried out with the code FRESCO [6] to fit the data.
Four first excited states studied belong to the discreet spectrum, whereas the last 3.39 MeV 3¯ state lying 0.019 MeV above the neutronemission threshold is located in the continuum. Nevertheless, in view of the small positive energy and the structure of the proton angular distribution similar to those of the preceding bound states, we used the effective positive small (0.01 MeV) binding energy of valence neutron.
The results of the transfer calculations in the CRC model (dashed lines) are shown in Fig. 2. The solid curves represent the incoherent sum of the CN model and direct transfer calculations. The calculations reproduce the positions and shapes of the first extremes very well.
Neutrontransfer reactions with light nuclei are mainly peripheral, that is they actually sensitive only to the surface part of the neutroncore potential.
Calculations of the ANCs with different sp wave functions allow us to answer the question whether there exist states in
12
B that are formed due to the peripheral neutron transfer reactions.
As in the sp approximation, the exact overlap function is approximated by the twobody sp wave function, the rms neutron radius is approximately determined by the rms radius of the sp wave function of neutron. To estimate the weight of the asymptotic part of the wave function a coefficient is introduced called D
1
. It characterises the spatial extension of the neutron wave functions to the asymptotic potential region. It is accepted in Refs. [2] that if D
1
(R
N
)
>
50%, than the state is assumed to be a “real” halo.
Our preliminary calculations showed that the rms radii of the last neutron in the second 1.674 MeV 2¯ and third 2.621 MeV 1¯ excited states of
12
B far exceed those for the g.s. and the first 0.953 MeV 2
+
excited state. Exactly, for the 2¯ state, the excess is a factor of 1.6, and for the 1¯ state, it is a factor of 2.1, with respect to the rms radius of the ground state. Summary on preliminary values of R
h
and D
1
for current work in comparison with results from [2] can be seen in Table 1.
The large values of D
1
coefficient, more than 50%, indicate that the last neutron spends more than 50% of its time outside the range of the core potential. In this case we can conclude that the rms radius of the last neutron wave function can be associated with the halo radius. From preliminary results of current work condition D
1
(R
N
)
>
50% is strictly fulfilled only for the 2.621 MeV 1¯ state and R
h
(1¯) = 7.4
±
0.2 fm. For 1.674 MeV 2¯ state D
1
(R
N
) is a bit less than 50% but R
h
is rather large so this state can be a candidate for halo. This result partly confirms a conclusion made by Liu and collaborators in Ref. [2] that the second (1.674 MeV 2¯) and third (2.621 MeV 1¯) excited states in
12
B are the neutron halo states. Nevertheless, our analysis indicated that the halo radii in these states are considerably large.
Table 1
Preliminary results of ANC calculations (last 2 columns) in comparison with results of [2].

State 
R
h
(fm) [2] 
D
1
(%)[2] 
R
h
(fm) current work 
D
1
(%) current work 
g.s. 1
+

3.16
±
0.32 
19.9 
3.6
±
0.2 
11 
0.953 2
+


3.95
±
0.20 
16 
1.674 2


4.01
±
0.61 
53.6 
5.57
±
0.26 
38 
2.621 1


5.64
±
0.90 
66.8 
7.4
±
0.4 
53 
2.723 0
+


5.6
±
0.3 
30 
3.389 3



6.5
±
0.3 
28 
Let us pay particular attention to the fifth 3.389 MeV 3¯ excited state that is localized 0.019 MeV above the neutronemission threshold. As this state belongs to the continuum spectrum, it makes impossible a correct estimation of the ANC, even though the calculation was carried out with a very small positive neutron binding energy (ε = 0.1 MeV) and the asymptotic behaviour of the sp and Hankel functions become the same at very long distance, about 50 fm. Nevertheless, we could approximately estimate the rms radius of the last neutron wave function by determining the rms radius of the sp wave function. It was found to be R
rms
= 6.5 fm, which is a factor of 1.8 larger than that of the g.s. and is comparable with the rms radii of the last neutron wave function in the 1.674 MeV 2¯ state and the 2.621 MeV 1¯ state of
12
B. But D
1
is less than 50%. Thus the
12
B the 3.389 MeV 3¯ excited state can only be considered as probable candidate to neutron halo. If this state is truly neutron halo, this observation would indicate to the first halo excited state with a nonzero orbital momentum l = 2.