Nucleus-Nucleus Scattering Based on a Modified Glauber Theory

Commun. Theor. Phys. (Beijing, China) 36 (2001) pp. 313 320 c International Academic Publishers Vol. 36, No. 3, September 15, 2001 Nucleus-Nucleus Scattering Based on a Modified Glauber Theory ZHAO Yao-Lin, 1 MA Zhong-Yu 1,2,3 and CHEN Bao-Qiu 1,2 1 China Institute of Atomic Energy, P.O. Box 275-18, Beijing 102413, China 2 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China 3 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China (Received March 26, 2001) Abstract A modified microscopic Glauber theory has been extended to investigation of the reaction and elastic differential cross sections of various projectile-target collisions at low and intermediate energies. Through a systematic study, we find that the inclusion of the finite range interaction and Coulomb modifications plays very important role in the Glauber theory to reproduce the experimental data at these energy ranges. Usually the effect of the Coulomb modification is to decrease the reaction cross sections, on the contrary that of the finite range interaction modification increases them. The angular distributions calculated by the Glauber theory including these two corrections are in good agreement with the experimental data. PACS numbers: 25.70.-z, 11.80.Fv, 24.10.-i Key words: nucleus-nucleus scattering, Glauber theory, Coulomb modification 1 Introduction The recent experimental progress using radioactive nuclear beams has opened a new era in nuclear physics. The widespread use of radioactive beams has led to the establishment of a number of unusual properties of light exotic nuclei with characteristics greatly differing from those in stable nuclei predicted by the systematics. [1] One of the most important characteristics of the interaction of light exotic nuclei with stable target nuclei is the interaction cross section σ I or the total reaction cross section σ R. The interaction cross section σ I of light exotic nuclei scattering from stable nuclei has been measured in the pioneering work of Tanihata et al. [2] The analyses of the measured interaction cross sections for He, Li and Be isotopes at energy 790 MeV/nucleon are based on both empirical approach and Glauber approximation. It was concluded that the neutron density in neutron-rich isotopes among these nuclei is quite extended. This led to the hypothesis that a neutron halo, such as 11 Li, may exist in some of them. The most popular approach for analyzing the experimental data of the interaction cross section is the Glauber approximation, because in this approach the matter radii and density distributions of halo nuclei could be studied through the interaction cross section and reaction cross section. Of course a determination of the matter-density distribution of halo nuclei is not straightforward. At the present time, we have no modelindependent method for determining density distribution. Up to now, the Glauber model is one of the most widely adopted approaches among various models to connect the density distribution with cross sections. It is well known that the Glauber theory has had considerable success in describing high energy hadron-nucleus scattering, [3] and it was natural to extend the Glauber theory to nucleus-nucleus scattering. Karol [4] has derived an analytic expression for total reaction cross sections (σ R ) in nucleus-nucleus collisions at high energies. The only input data for the calculation are the parameters of the experimental density distribution of the two nuclei and the experimental nucleon-nucleon cross section. The validity of this model was tested by stable nuclei. A more realistic parametrization of the interaction cross sections was proposed by Kox et al. [5] The Coulomb effect and the energy dependence are taken into account in Kox parametrization. Therefore, Kox parametrization can be used to both high and low energies and over a wide mass range. A decrease of σ R with increasing energy above 30 MeV/nucleon is observed. Analysis of the experimental cross sections in terms of the Kox parametrization allows the general geometrical properties of light nuclei and the potentials for their interactions with stable nuclei to be established, but neither permits determination of the nuclear matter distribution nor explains the difference between the proton and neutron distributions. A modified Kox formula was suggested by Shen et al. [6] Meanwhile, Charagi et al. [7] proposed a Coulomb-modified Glauber model involving nuclear densities of two colliding nuclei and nucleon-nucleon cross section, a closed form of the analytic expression for nucleus-nucleus reaction cross section has been obtained from this model. They found that their expression can describe quite well the reaction cross sections over a wide energy range from a few MeV/nucleon to a few GeV/nucleon The project supported by National Natural Science Foundation of China under Grant Nos 10075080, 19847002, 19835010 and Major State Basic Research Development Program under Contract No. G200077407

314 ZHAO Yao-Lin, MA Zhong-Yu and CHEN Bao-Qiu Vol. 36 for several systems of colliding nuclei. In this article, we shall describe the nucleus-nucleus scattering cross sections based on the generalized Glauber theory and make a systematic investigation to the nucleusnucleus scattering taking account of the Coulomb and finite range interaction modifications. Especially we shall concentrate on the nucleus-nucleus scattering at low and intermediate energy domains to test the validity of the Glauber theory in these energy regions. The arrangement of the article is as follows. The formalism of the extended Glauber theory for nucleusnucleus reaction cross section is given in Sec. 2. The details of the calculations are described in Sec. 3, and the results calculated in the Glauber theory and discussions are given in Sec. 4. 2 Glauber Theory for Nucleus-Nucleus Scattering 2.1 Scattering Amplitude One of the widely used models for analyzing the interaction and the reaction cross sections of nucleus-nucleus scattering is the Glauber model. The process of nucleusnucleus reaction can be described in the projectile rest frame as P( ψ 0 )+T( K,θ 0 ) P( q,ψ α )+T( K q,θ β ),(1) where P and T indicate projectile and target, respectively. At the initial stage of the reaction the projectile with the intrinsic wavefunction ψ 0 is at rest in the projectile rest frame, while the target nucleus with the intrinsic wavefunction θ 0 approaches the interaction region with momentum h K. At final stage of reaction the projectile goes to the state specified by ψ α with momentum transfer h q. The target nucleus receives the momentum transfer h q and goes to the state specified by the wavefunction θ β. It is understood that α = 0 and β = 0 stand for the corresponding ground states. The scattering amplitude for nucleus-nucleus scattering of Eq. (1) in the Glauber theory is written as F αβ ( q ) = ik 2π i P d bexp( i q b) ψ α θ β 1 [1 Γ ij ( b + s i t j )] ψ 0 θ 0, (2) here Γ ij ( b+ s i t j ) is the profile function for N-N scattering, and i, j indicate nucleons at the projectile and target nuclei respectively. We choose the z-axis to be perpendicular to the wave vector transfer q = k i k f, where k i and k f are the initial and final relative momenta. The initial relative coordinate of the two nuclei r is decomposed as r = b + z z, where b is the impact parameter vector, perpendicular to the z-direction. r i = s i +z i z is the coordinate of the ith nucleon in the projectile with respect to the center of mass of the projectile and η j = t j + z j z the coordinate of the jth nucleon in the target with respect to its center of mass. b, s i and t j lie in a plane perpendicular to K. Therefore, the elastic scattering amplitude can be written as F 00 ( q ) = ik d b exp( i q b ) 2π [ 1 ψ 0 θ 0 [1 Γ ij ( ] b+ s i t j )] ψ 0 θ 0.(3) Thus the optical phase-shift function χ PT ( b ) for nucleusnucleus scattering is defined by exp[iχ PT ( b )] = ψ 0 θ 0 [1 Γ ij ( b+ s i t j )] ψ 0 θ 0.(4) 2.2 Cross Sections The cross section for nucleus-nucleus scattering of process described in Eq. (1) is given by 1 σ αβ = k 2 d q F αβ( q ) 2 = d b ψ 0 θ 0 1 [1 Γ ij ( b + s i t j )] ψ α θ β ψ α θ β 1 [1 Γ ij ( b + s i t j )] ψ 0 θ 0.(5) The reaction cross section is obtained by summing σ αβ over the possible final states α, β, except for αβ = 00, σ R = αβ 00 = d b σ αβ { 1 ψ 0 θ 0 i P [1 Γ ij ( b+ s i t j )] ψ 0 θ 0 2 } = d b {1 exp[iχ PT ( b )] 2 }. (6) The interaction cross section represents the probability that the projectile loses at least one nucleon after a collision with the target nucleus, and can thus be obtained by summing σ αβ over all possible states α and β except for α = 0, σ I = = α 0,β d b σ αβ ψ 0 i P { 1 ψ 0 θ 0 i P [1 Γ ij ( b + s i t j )] ψ 0 [1 Γ ij ( } b + s i t j )] ψ 0 θ 0. (7) In order to calculate those cross sections discussed above one has to deal with two types of matrix elements through phase-shift functions, which appear in Eqs (6) and (7).

No. 3 Nucleus-Nucleus Scattering Based on a Modified Glauber Theory 315 They are ψ 0 θ 0 [1 Γ ij ( b + s i t j )] ψ 0 θ 0, (8) ψ 0 θ 0 [1 Γ ij ( b + s i t j )] ψ 0 ψ 0 [1 Γ ij ( b + s i t j )] ψ 0 θ 0. (9) Usually these matrix elements of the phase-shift functions can be calculated with the approximations of an independent particle model and the optical limit. As an illustrative example we evaluate the quantity of Eq. (8). Using the independent particle model the nuclear ground state wavefunction is factorized and we have ψ 0 ( r 1, r 2,...) 2 = i P θ 0 ( η 1, η 2,...) 2 = ρ i ( r i ), (10) ρ j ( η j ), (11) where ρ i ( r i ), ρ j ( η j ) are the density distributions of the ith nucleon in the projectile, jth nucleon in the target, respectively. Then we define G( b,λ) = d r 1 d r 2 d η 1 d η 2 ψ 0 ( r 1, r 2,...) 2 θ 0 ( η 1, η 2,...) 2 i P [1 λγ ij ( b + s i t j )]. (12) One can find that G( b,λ) at λ = 1 is equal to the matrix element of Eq. (8). Noting G( b,λ = 0) = 1, we expand ln G( b,λ) as a power series in λ, lng( b,λ)=ln G( b,0)+ λ ( lng( b,λ) ) ( + λ2 2 lng( b,λ) 1! λ λ=0 2! λ )λ=0 2 + =λg ( b,0)+ 1 2 λ2 (G ( b,0)+g ( b,0) 2 )+.(13) The terms higher than the lowest order in the cumulant expansion of Eq. (13) represent multiple scattering of the nucleons. If all the higher-order terms are neglected, the sought matrix element is approximated by G( b,λ = 1) = exp(g ( b,0)). (14) The approximation introduced in Eq. (14) is called the optical limit. Using an independent particle model and the optical-limit approximation, we can get the expression for the matrix element of Eq. (8), ψ 0 θ 0 i P [1 Γ ij ( b+ s i t j )] ψ 0 θ 0 = exp[iχ PT ( b )]= exp(g ( { b,0))= exp d rρ P ( r ) d ηρ T ( η )Γ( } b+ s t ),(15) where ρ P ( r ) and ρ T ( η ) are density distributions of the projectile and target, which are normalized to the mass numbers of the projectile and the target, respectively. In a similar fashion, one can also get the expression for the matrix element of Eq. (9), ψ 0 θ 0 [1 Γ ij ( b + s i t j )] ψ 0 θ 0 i P where Γ [1 ij ( b + s i t j )] ψ 0 ψ 0 { = exp d rρ P ( r ) d ηρ T ( η )Γ( b + s t ) + 1 A P d r ρ P ( r ) d η ρ T ( η )Γ ( b + s t ) } d rd r d ηρ P ( r )ρ P ( r )ρ T ( η )Γ ( b + s t )Γ( b + s t ) = exp{ 2 Im χ PT ( b ) + χ PPT ( b, b )}, (16) χ PPT ( b, b ) = 1 A P and A P is the mass number of the projectile. d rd r d ηρ P ( r )ρ P ( r )ρ T ( η )Γ ( b + s t )Γ( b + s t ), (17) In the partial wave decomposition, the elastic scattering amplitude f(θ) is given by f(θ) = f c (θ) + 1 2ik (2l + 1) l exp(2iσ l )(S l 1)P l (cos θ), (18) where f c (θ) and σ l are the Coulomb scattering amplitude and the Coulomb phase shift of the lth partial wave, respectively. P l (cos θ) is the Legendre polynomial. The nuclear elastic scattering matrix element S l is given in the Glauber model with the approximation for the impact parameter b l = (l + 1/2)/k by S l = ψ 0 θ 0 [1 Γ ij ( b l + s i t j )] ψ 0 θ 0. (19)

316 ZHAO Yao-Lin, MA Zhong-Yu and CHEN Bao-Qiu Vol. 36 Using an independent particle model and the optical-limit approximation, we can also obtain the expression for S l, { S l = exp d rρ P ( r ) d ηρ T ( η )Γ( } b l + s t ). (20) Finally, the elastic differential cross section dσ/dω of the nucleus-nucleus scattering is given by 3 Calculations dσ el dω = f(θ) 2. (21) 3.1 Density Distribution Parameters In order to simplify the calculations, the density distributions for the projectile and target are expanded in terms of multiple Gaussian functions with appropriate coefficients ρ i (0) and ranges a i, ρ( r ) = ) ρ i (0)exp ( r2 a 2. (22) i i In our calculations, the parameters ρ i (0) and a i for the nucleus 12 C are taken from Ref. [8] which are obtained by fitting the experimental electron scattering data. And those for other nuclei are taken from Ref. [7], where the parameters are adjusted to the experimentally determined characteristic of the nuclear surface. 3.2 Profile Function for N-N Scattering The N-N scattering profile function Γ( b + s t ) appearing in Eqs (15) (17) and (20) can be calculated as follows: [9] Γ( 1 b + s t ) = d 2πik NN exp( i b i s+i t )f NN ( ),(23) where f NN ( ) is the N-N scattering amplitude at the momentum transfer and k NN is the relative wave number. Taking account of the finite range interaction the N-N scattering amplitude f NN ( ) is given by f NN ( ) = f NN (0)exp ( r2 0 2 ), (24) 4 where f NN (0) is the forward N-N scattering amplitude while the second factor describes the dependence in terms of finite range parameter r 0. The forward N-N scattering amplitude f NN (0) can be written using the optical theorem as f NN (0) = k NN 4π σ NN(i + α NN ), (25) where σ NN is the total cross section for the N-N scattering while α NN is the ratio of the real to the imaginary part of the forward N-N scattering amplitude. The σ NN averaged over neutron and proton numbers is discussed in the following subsection. 3.3 Parametrization of Nucleon-Nucleon Cross Sections The free nucleon-nucleon cross sections with an energy and isospin dependence are taken from Ref. [7], where Charagi formula can be expressed by σ np = 70.67 18.18/β + 25.26/β 2 + 113.85β, (26) σ pp = σ nn = 13.73 15.04/β + 8.76/β 2 + 68.67/β 4. (27) The coefficients in Eqs (26) and (27) were obtained by a least-square fit to the experimental nucleon-nucleon cross section data over a wide energy range from 10 MeV to 1 GeV. σ np and σ nn are in unit of mb and β = v/c. The averaged nucleon-nucleon cross section σ NN is calculated by the following expression σ NN (E) = N PN T σ nn + Z P Z T σ pp + N P Z T σ np + N T Z P σ np A P A T, (28) where A P, A T, Z P, Z T and N P, N T are the mass numbers, charge and neutron numbers of the projectile and target, respectively. 3.4 Coulomb Field Correction Due to the presence of the strong Coulomb field in the heavy nucleus collisions, the straight trajectory approximation of nucleons is not adequate for heavily charged nuclei at relatively low bombarding energies. The deviations of the Rutherford orbits from the straight line are also marked for grazing partial wave. The neglect of the Coulomb effect leads to an overestimation of the reaction cross section and an underestimation of the grazing angle in the angular distribution. A simple modification to mend these shortcomings is such that the trajectory can be related to the value of the distance of closest approach b in a Coulomb field rather than to the impact parameter b, which are related as kb = µ + (µ 2 + k 2 b 2 ) 1/2, (29) where k is the wave number and µ = Z P Z T e 2 / hv is the Sommerfeld parameter. It retains the same form that the standard Glauber optical limit can be obtained as a particular case. 4 Results and Discussions In the framework of the modified Glauber theory, we calculated the reaction cross sections for some light nuclei 6 Li, 12 C, 16 O, 20 Ne and angular distributions for 16 O scattering from various targets at low and intermediate energies. Through a systematic analysis and comparison with the experimental data, the effects of the finite range interaction and Coulomb field modifications are discussed and clarified. In our calculation the range parameter r 0 is fixed and chosen to be 1.0.

No. 3 Nucleus-Nucleus Scattering Based on a Modified Glauber Theory 317 4.1 Nucleus-Nucleus Reaction Cross Section We have studied the variation of reaction cross section over an energy range from 10 MeV/nucleon to 300 MeV/nucleon. Figures 1a 1d show the reaction cross sections of 12 C + 12 C, 12 C + 27 Al, 16 O + 12 C and 20 Ne + 12 C, where the experimental data (solid squares) are taken from various experiments. [5,10 13] The theoretical results are calculated in the modified Glauber theory with the corrections of the finite range interaction and Coulomb field, which are plotted by solid circles. For comparison we also show the results calculated only with Coulomb (open circles) or finite range corrections (open diamonds), and the open up triangles represent the results without any modifications. We also show the plot of reaction cross sections σ R as a function of (A 1/3 P +A1/3 T )2 in Fig. 2. The definition of symbols are the same as those in Fig. 1. The experimental data are taken from Refs [5], [10], [14] and [15]. Fig. 1 The variations of reaction cross section for (a) 12 C + 12 C, (b) 12 C + 27 Al; (c) 16 O + 12 C and (d) 20 Ne + 12 C over an incident energy. The solid circles and the open up triangles correspond to the results in Glauber theory with and without modifications, respectively. The open circles and diamonds are those with only Coulomb or finite range modifications, respectively. The solid squares are the experimental data, which are taken from Refs [5] and [10] [13]. From these figures one could find that in general the cross sections calculated in Glauber theory without modifications are smaller than the experimental data, the deviations are larger for lighter nuclei at low and intermediate energies. Therefore the modifications of Glauber theory are necessary in order to describe the low and medium energy nucleus-nucleus collisions. The repulsive Coulomb field makes the effective impact constant larger, therefore the inclusion of the Coulomb field modification makes reaction cross sections smaller. On the contrary, the finite range modifications take account the spreading of the nucleon-nucleon scattering at low momentum transfers. As a result, the calculated reaction cross sections increase at low energy regions with inclusion of the finite range corrections. Those two effects tend to cancelling each other for heavy nuclei, but make larger corrections to the cross sections for lighter nuclei. This is clearly shown in Fig. 2. At the same energy with the modifications the improvements on the reaction cross sections for light target nuclei are very notable, but for heavy target nuclei are not. In general, the modified Glauber theory could give very good description of experimental data. To make a qualitative estimate of those effects one may use the difference factor, [16] d = σ R(Exp) σ R (Gla) σ R (Gla) where σ R (Exp) and σ R (Gla) are the experimental data and the results obtained by Glauber model, respectively.,

318 ZHAO Yao-Lin, MA Zhong-Yu and CHEN Bao-Qiu Vol. 36 One finds that with the modifications the largest difference factor is reduced from 9 percent to 2 percent. It turns out that at low and intermediate energies the finite range interaction and Coulomb field are very important factors to the light nuclei scattering. Fig. 2 Reaction cross sections for (a) 38 MeV/u 6 Li; (b) 83 MeV/u 12 C; (c) 93.9 MeV/u 16 O and (d) 30 MeV/u 20 Ne as functions of (A 1/3 P + A1/3 T )2. The definitions of symbols are the same as those in Fig. 1, and the data are taken from Refs [5], [10], [14] and [15]. 4.2 Angular Distributions of Elastic Scatterings The importance of the finite range corrections at low and intermediate energies in the Glauber theory can be further illustrated from the comparisons of the differential cross sections. We also calculate the elastic angular distributions of 16 O scattering from various targets. Figure 3 shows the angular distributions for 16 O scattering from 12 C at four different energies (216 MeV, 311 MeV, 608 MeV and 1503 MeV). The solid squares are experimental data taken from Ref. [10], the solid and dashed-dotted curves represent the results calculated by the Coulomb-modified Glauber theory with or without inclusion of the finite range modification, respectively. And figure 4 shows the angular distributions of 16 O scattering by 12 C, 40 Ca, 90 Zr and 208 Pb at energy 1503 MeV, respectively. The experimental data are taken from Refs [10] and [14]. In the calculations the parameter α NN for the light target 12 C is taken as 0.9 and for heavy targets such as 40 Ca, 90 Zr, 208 Pb, α NN is taken as 1.07, which is the same as that in Ref. [9]. From these figures one can see that the phase of the diffraction patterns without the finite range modification is shifted. The results obtained in the modified Glauber theory with finite range modification are in better agreement with the experimental data. It not only gives correct absolute values of the differential cross sections but also well describes the diffraction patterns. These two figures clearly show the importance of the finite range interaction for both light and heavy nucleus scatterings at low and intermediate energy ranges. In summary, we have systematically analyzed the reaction cross sections and elastic differential cross sections of nucleus-nucleus scatterings by means of microscopic Glauber theory. The results show that it is possible to give a satisfactory description of the elastic scattering data between heavy ions at relative low bombarding energy within the Glauber model. It turns out that the finite interaction range and Coulomb modifications appear to play very important roles in nucleus-nucleus scattering at low and intermediate energies. The effect of Coulomb modification gives the reduction of the reaction cross section, while the finite range modification increases the reaction cross section. And the improvement on the reaction cross section of integrated effects of two modifications is very notable for light nucleus, but not for heavy nucleus because of its strong Coulomb field.

No. 3 Nucleus-Nucleus Scattering Based on a Modified Glauber Theory 319 Fig. 3 The elastic angular distributions of 16 O + 12 C at various energies: (a) 216 MeV; (b) 311 MeV; (c) 608 MeV; (d) 1503 MeV. The solid lines represent results in modified Glauber theory with finite range r 0 = 1.0 and Coulomb modifications, the dashed-dotted lines are those only with Coulomb modification (r 0 = 0). The solid squares represent the experimental data taken from Ref. [10]. Fig. 4 The elastic angular distributions of 16 O with various targets at the energy 1503 MeV. (a) The target is 12 C; (b) The target is 40 Ca; (c) The target is 90 Zr; (d) The target is 208 Pb. The definitions of symbols are the same as those in Fig. 3, and the data are taken from Refs [10] and [14].

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