ЯДЕРНАЯ ФИЗИКА, 2012, том 75, № 12, с. 1624-1630
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
SENSITIVITY OF CROSS SECTIONS FOR ELASTIC NUCLEUS-NUCLEUS SCATTERING TO HALO NUCLEUS DENSITY DISTRIBUTIONS
©2012 G. D. Alkhazov, V. V. Sarantsev*
Petersburg Nuclear Physics Institute NRC KI, Gatchina Received November24, 2011; in final form, February 14, 2012
In order to clear up the sensitivity of the nucleus—nucleus scattering to the nuclear matter distributions in exotic halo nuclei, we have calculated differential cross sections for elastic scattering of the 6He and 11 Li nuclei on several nuclear targets at the energy of 0.8 GeV/nucleon with different assumed nuclear density distributions in 6He and 11 Li.
1. INTRODUCTION As has been shown in [1, 2], the elastic proton-nucleus scattering at intermediate energy in inverse kinematics is an efficient means of studying nuclear matter density distributions in exotic halo nuclei. An analysis of the measured cross sections for small-angular elastic proton-nucleus scattering allows one to determine not only the total nuclear size, but also the sizes of the nuclear core and neutron halo. At the same time, as has been shown in [1], the sensitivity of the cross sections for elastic proton-nucleus scattering is not sufficient for obtaining information on the matter distribution in the nuclear far periphery, which contains small amount of the nuclear matter. The authors of paper [3] discuss the sensitivity of the reaction cross sections (that is, of the integrated cross sections for all inelastic processes) to the nuclear density distribution at large distances from the nuclear center. They come to the conclusion that the reaction cross sections for nucleus-nucleus scattering are significantly more sensitive to the matter density at the nuclear periphery than the reaction cross sections for proton-nucleus scattering. The results of paper [3] allow us to suppose that the cross sections for elastic nucleus-nucleus scattering are also more sensitive to the density at the nuclear periphery than the cross sections for elastic proton-nucleus scattering. If it is really so, then it would be reasonable to perform the relevant experiments, for example, at the future nuclear facility FAIR at Darmstadt.
2. CROSS SECTION CALCULATIONS In the present work, in order to find out the sensitivity of the cross sections for nucleus-nucleus
elastic scattering to the matter density at the nuclear periphery, we have performed calculations of the cross sections for elastic scattering of exotic nuclei 6He and 11 Li on protons and nuclear targets 4He, 9Be, 12C, 58 Ni, 90Zr, and 208 Pb at the 800 MeV/nucleon energy of the incident nuclei, different matter distributions being assumed in the 6Heand 11 Li nuclei. Theoretical and experimental investigations have shown that the cross sections for elastic proton-nucleus and nucleus-nucleus scattering at the intermediate energy (~0.5—1 GeV/nucleon) can be calculated fairly accurately with the help of the Glauber theory [4]. It should be noted, however, that calculations of the cross sections for nucleus-nucleus scattering with the exact formula of the Glauber theory is a rather complicated task. In the present work, the cross sections for nucleus—nucleus scattering were calculated using the Glauber theory within the "rigid-target" approximation [5], that is, at first the amplitude of scattering of one nucleon on the nuclear target was calculated, and then this amplitude was used in the calculations of the cross sections for scattering of an exotic nucleus consisting of several nucleons. As was shown in [6], the reaction cross sections for scattering of exotic nuclei on nuclear targets calculated within the rigid-target approximation are very close to those calculated with the exact Glauber formula. The main formulas used in the present paper for calculation of the cross sections da/dt for elastic nucleus—nucleus scattering are given below:
E-mail: saran@pnpi.spb.ru
da П 2
(1)
(2)
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x J d2bexp(fqb)d3rid3r2 ... d3rA x
AT
A
Ac
Ac+Ah
p(ri, ..., rA) = !! Pc(rj) II Ph(v3), (5)
PT (rl,...,rAT ) = n PT (rj ).
(6)
j=1
X P(ri ,...,rA^1 -[]> - r(b - Sj)]
r(b) = J d3rid3r2 ... d3tat x (3)
x pt(ri,..., rATH1 - II [1 - Y(b - Sm)] > .
I m=i J
Here, t is the four-momentum transfer squared (for elastic scattering t = -q2, where q is the three-momentum transfer), k is the value of the wave vector of the incident exotic nucleus, F(q) is the amplitude of nucleus—nucleus scattering, b is the impact parameter vector, p(ri,..., rA) and pT(ri,..., rAT) are the many-body densities correspondingly of the studied exotic nucleus and the target nucleus, ri,..., rA, ri,..., rAT and si,..., sa, si,..., sat stand for the radius vectors of the nucleons in these nuclei and their transverse coordinates, A and AT are the total numbers of nucleons in the exotic and target nuclei, r(b) is the profile function of scattering of one nucleon of the studied exotic nucleus on the target nucleus, and 7(b) is the profile function of the nucleon—nucleon interaction. Spin-independent isospin-averaged amplitude of the free nucleon— nucleon (NN) scattering was employed, the traditional high-energy parametrization of this amplitude and the corresponding profile-function
7(b) = ajvjv(1 ~i€NN) 1 x (4) 2
x exp (-¿b)
being taken with the following parameters: the total cross section aNN = 42.5 mb, the ratio of the real to imaginary part eNN = -0.18, and the amplitude slope /3NN = 0.2 fm2. The Coulomb scattering contribution was taken into account using the well-known formulas [7].
3. DENSITY DISTRIBUTIONS
In the present calculations it was assumed that
the 6He and iiLi nuclei consist of the nuclear core
of four and nine nucleons, correspondingly, and two
halo neutrons. The many-body densities in the projectile exotic nucleus and in the target nucleus were
presented as products of one-body densities:
Here, pc(rj) and ph(rj) are the one-body densities of the core and halo of the exotic nucleus, correspondingly; pT(rj) is the one-body density of the target nucleus; Ac and Ah are the numbers of the core and halo nucleons (Ac + Ah = A).
The matter density distributions in the core were described by Gaussian distributions
Pc(rj) = (3/2nR2 )3/2 exp(-3r2/2R2), (7)
whereas the density distributions in the halo were described by a 1p-shell harmonic oscillator-type function for ii Li
Ph(rj) = (5/3)(5/2nRh)3/2(rj/Rh)2 x (8) x exp(-5r2/2Rh),
and by a Gaussian for 6He. Here, Rc and Rh are the root-mean-square (rms) radii of the core and halo matter density distributions. Calculations with the halo density distributions containing long density "tails" (to be discussed later) were also performed. Note that the rms radius Rm of the total matter density distribution is connected with the core and halo radii Rc and Rh as
Rm = [(AR + AhRh )/A]
i/2
(9)
j=i
j=Ac+i
Figures 1 and 2 show the 6He matter density distributions applied in the calculations. The solid curve in Fig. 1 corresponds to the nuclear density distribution with a halo: Rc = 1.95 fm, Rh = 2.88 fm, and Rm = 2.30 fm. The dashed and dotted curves in Fig. 1 correspond to the density distributions without a halo (Rc = Rh), the dashed curve stands for Rm = = 1.95 fm, and the dotted curve stands for Rm = = 2.30 fm. The solid curve in Fig. 2 corresponds to the solid curve in Fig. 1. The halo density distribution in this version of the calculations, as it has been already said, is described with a Gaussian distribution. The density in this distribution decreases with increasing the distance r from the nuclear center faster than it is predicted by theory. The dashed curve in Fig. 2 shows the 6He density distribution where we have added a tail, which decreases exponentially with the radius r increasing. The resulting density distribution with this tail corresponds to the theoretical nuclear matter distribution FC of [8]. The halo rms radius in this case is equal to R'h = 3.34 fm. Figure 2 shows also by the dotted curve the 6He density distribution without a tail, but with the increased halo radius: Rh = R'h = = 3.34 fm. (The core radius Rc in these versions of the 6He density distribution is Rc = 1.95 fm.)
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ALKHAZOV, SARANTSEV
p(r), fm 3
0 2 4 6 8
r, fm
Fig. 1. The nuclear matter density distributions in 6He applied in the cross-section calculations with a halo structure (solid curve) and without a halo (dashed and dotted curves). For the parameters of the density distributions see the text.
Figure 3 shows the 11 Li density distributions used in the calculations. The solid curve in Fig. 3 corresponds to the density distribution with a halo (however, without a tail): Rc = 2.50 fm, Rh = = 5.86 fm, Rm = 3.37 fm. The dotted curve shows the density distribution without a halo (Rc = Rh, Rm = = 3.37 fm), and the dashed curve shows the density distribution with a halo and the tail corresponding to the theoretical density distribution P2 of [9]. (In this version of the 11 Li density distribution, Rc = 2.50 fm, Rh = 6.65 fm, Rm = 3.63 fm.)
The matter density distribution of the 4He target nucleus was described by a Gaussian distribution with the rms matter radius Rm = 1.49 fm [1]. For heavier nuclear targets, the Fermi distribution
p(r) - (10)
- [1 + w(r/Ro)2]/{1 + exp[(r - Ro)/a]} was used with the following parameters:
Ro = 1.95 fm, a = 0.60 fm, w = 0
(Rm = 2.68 fm) for 9Be [10]; R0 = 2.12 fm, a = 0.52 fm, w = 0 (Rm = 2.52 fm) for 12C [10];
R0 = 3.17 fm, a = 0.59 fm, w = -0.12 (Rm = 3.15 fm) for 28Si [11];
R0 = 4.23 fm, a = 0.55 fm, w = -0.13 (Rm = 3.76 fm) for 58Ni [12];
p(r), fm 3
10 0 4 ^ 8 12
r, fm
Fig. 2. The nuclear matter density distributions in 6He applied in the cross-section calculations with a density tail (dashed curve) and without a density tail (solid and dotted curves).
p(r), fm 3
0 4 8 12 16
r, fm
Fig. 3. The nuclear matter density distributions in 11 Li applied in the cross-section calculations without a h
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