REACTION CROSS SECTIONS FOR EXOTIC NUCLEI IN THE GLAUBER APPROACH
G.D. Alkhazov, A.A. Lobodenko
Reaction cross sections serve as one of the main sources of information on sizes of exotic halo nuclei. The
nuclei sizes are determined from the measured reaction cross sections σr by comparing the calculated cross
sections for nucleus-nucleus scattering with their experimental counterparts. Therefore, it is important to
measure and calculate reaction cross sections with rather high accuracy. It is believed that at intermediate
energy the scattering cross sections can be calculated with high accuracy by using the Glauber multiple
scattering theory. However, a calculation of cross sections by the exact Glauber theory formula in the case of
nucleus-nucleus scattering is not a simple task, and previously such calculations for scattering involving
exotic nuclei were performed using some approximate approaches, most often an optical limit (OL)
approximation to the Glauber theory.
In the present study [1, 2], reaction cross sections for scattering involving halo nuclei have been
calculated for the first time by using directly the Glauber formula without any simplifying approximations.
The reaction cross sections were calculated for scattering of 6Li and the exotic 6He and 11Li nuclei with the
target 12C nuclei at energies of about 0.8 GeV per nucleon. According to the Glauber theory, the reaction
cross section for scattering of nuclei A and B may be calculated by the formulas


σr = 2 π { 1 – |SAB(b)|2 } b db,
(1)
0
AB
SAB(b) = < ΨBΨA |

k,l
[ 1 – γ(b – sk + sl)] | ΨAΨB >,
(2)
where SAB(b) is the S matrix for the interaction of nuclei A and B, γ(b – sk + sl) is the profile function for
free nucleon–nucleon (NN) interaction, b is the impact vector parameter, sk and sl are the nucleon transverse
coordinates, ΨA and ΨB are the nuclear wave functions, and A and B are the mass numbers of nuclei A and B.
In our calculations, the profile function γ(b) was parametrized as
γ(b) = σ(1 – iε)/8πβ for b ≤ 2β1/2, and γ(b) = 0 for b > 2β1/2,
(3)
where σ is the total cross section for NN interaction, ε is the ratio of the real to imaginary part of the free NN
scattering amplitude, and β is the relevant slope parameter. We used the following isospin-averaged values of
σ, ε, and β: σ = 4.25 fm2, ε = –0.18, and β = 0.2 fm2. The nuclei 6Li, 6He and 11Li were supposed to consist of
two valence (halo) nucleons and a nuclear core having 4 and 9 nucleons, correspondingly in 6He (6Li) and
11
Li. The core and halo nucleon density distributions were described by Gaussian functions. The root mean
square (rms) radius of the cores Rc in their centre-of-mass systems were taken to be 1.49 fm for 6He (and 6Li)
and 2.30 fm for 11Li. The calculations were performed for different assumed values of the rms halo radii Rh.
More realistic density distributions, corresponding to theoretical few-body calculations were also used. The
density distribution of the target nucleus 12C was described by a Gaussian, the rms radius of 12C being set to
2.32 fm. Multidimensional integration in (2) was performed by the Monte Carlo technique.
In calculations of reaction cross sections, the OL approximation was routinely used in the past. More
recently, a more accurate “few body” (FB) approximation was applied to deduce nuclear sizes of light halo
nuclei (the formulas for calculating σr in the OL and FB approximations are presented in [1]). In our study,
the reaction cross sections were calculated by the exact Glauber formula, using the OL and FB
approximations, and also with the aid of the “rigid target” (RT) approximation [3]. In the last case, one first
calculates the S matrix for the interaction of one of the nucleons of the exotic nucleus involved with the
target nucleus and then uses this matrix to calculate the total SAB matrix.
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Fig. 1. Calculated reaction cross sections σr for the interaction of (a) 6He (6Li) and (b) 11Li with the 12C
nuclei at the energy of 0.72 GeV per nucleon versus the assumed total rms matter radius RA of the
considered halo nucleus
Figure 1 presents the reaction cross sections for different assumed halo radii (and, therefore, for different
total matter radii RA) calculated using the exact Glauber formula (solid curves), and approximate formulas of
the OL approximation (dash-dotted curves), RT approximation (dotted curves), and the optical limit for the
case of a rigid target (dashed curves). One can see that an approximate calculation in the RT approximation
allows us to calculate the reaction cross sections significantly more accurately than that in the OL
approximation, which significantly overestimates the reaction cross sections, the difference between the OL
and the exact Glauber calculations being especially big for halo nuclei with a large halo radius.
Fig. 2. Calculated reaction cross sections σr and their experimental counterparts at the energy of 0.79 GeV
per nucleon for the interaction of the (a) 6He (and 6Li) and (b) 11Li nuclei with the 12C target nuclei for
different assumed total matter rms radii RA. Calculations were performed using the OL approximation
(curves 1) and the exact Glauber formula (all other results) with different assumed structures of the
considered exotic nuclei (see the text). The horizontal lines and the shaded corridors correspond to the
experimental cross sections with their errors
2
Figure 2 displays reaction cross sections σr calculated in the OL approximation (curves 1) and using the
exact Glauber formula (other results) for different matter radii RA and different assumed spatial structures of
6
He, 6Li and 11Li. Curves 2, 3, 4, and 5 correspond respectively to calculations for the cases when the
considered nuclei have no halo structure (Rh = Rc = RA), when these nuclei have a halo structure (Rh > Rc),
the halo nucleon motion being not correlated, when the halo nucleons form a di-neutron cluster with the rms
matter radius Rd = 3 fm, and when the di-neutron cluster matter radius is Rd = 2 fm. The stars in Fig. 2
correspond to calculations with the 6He and 11Li theoretical nuclear matter distributions [4, 5].
The performed calculations have shown that the OL approximation overestimates considerably the
reaction cross sections for halo nuclei. At the same time, a rather simple RT approximation allows one to
calculate the cross sections σr significantly more accurately, the calculated cross sections being very close to
those calculated by the exact Glauber formula. The calculations have also shown that the reaction cross
sections depend not only on the total rms matter radius of the investigated nuclei, but also on their spatial
structure – on nucleon correlations and on the radial shape of the nuclear matter distributions. Therefore, the
rms nuclear matter radii RA deduced from the experimental reaction cross sections σr are subject to influence
of the spatial structure of the investigated nuclei. In more detail, the obtained results are discussed in [1].
References
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5.
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G.D. Alkhazov et al., Nucl. Phys. A 280, 365 (1977).
J.S. Al-Khalili, J.A. Tostevin, I.J. Thompson, Phys. Rev. C 54, 1843 (1996).
I.J. Thompson, M.V. Zhukov, Phys. Rev. C 49, 1904 (1994).
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