Reaction cross sections of proton scattering from carbon isotopes (A=8-22) by means of the relativistic impulse approximation

Reaction cross sections of carbon isotopes for proton scattering are calculated in large energy region. Density distributions of carbon isotopes are obtained from relativistic mean field results. Calculations are based on two procedures: the Glauber theory and relativistic impulse approximation, and results are compared with each other as well as with experimental data. A strong relationship between reaction cross section and root-mean-square radius is clearly shown for 12C using a model distribution.


I. INTRODUCTION
Unstable nuclei are fruitful objects for nuclear physics because they provide new information about halo structure, magic numbers, nuclear matter properties and many other things which are very different from stable nuclei. In order to obtain such information on the unstable nuclei, proton-elastic scattering is expected to be the most appropriate experiment besides electron scattering. The experimental observables are, however, restricted due to their unstableness, and reaction cross section or interaction cross section is considered at first. For unstable nuclei: 6,8 He, 11 Li, and 11,14 Be, the interaction cross sections have been found to be significantly large, and have provided their large root-mean-square radii [1,2]. The reaction cross section have been calculated for proton-rich isotopes, e.g. carbon [3,4], helium, lithium, beryllium, oxygen, and nitrogen in addition to carbon [5] in terms of the Glauber model based on the Glauber theory [6] and/or the eikonal model. Refs. [3,5] have considered unstable nuclei scattering from 12 C target, and Ref. [4] has calculated proton-nucleus scattering. Nuclear structures for the unstable nuclei have been provided by different ways: a Slater determinant generated from a phenomenological mean-field potential [3,4] , and relativistic [7] and nonrelativistic mean-filed theories in Ref. [5]. Reaction cross sections for 14,15,16 C isotopes scattering from 12 C target have been studied based on the g-matrix double-folding model and 14 C + n two-body model [8]. In addition to the neutron-rich isotopes, the angular distribution of proton-elastic scattering from the proton-rich isotopes like 9 C has been measured [9].
In the present study, densities of carbon isotopes are provided by the relativistic mean-field (RMF) results [10,11], and reaction cross sections for proton-elastic scattering from carbon isotopes are calculated with two prescriptions: the Glaubar model and the relativistic impulse approximation (RIA) [12,13]. As for target carbon nucleus, the mass numbers A=8 through A=22 are considered, e.i., proton-rich isotopes are also included * kaki.kaorii@shizuoka.ac.jp besides neutron-rich ones. The former isotopes are expected to have a large root-mean-square radius due to the Coulomb repulsive interaction between protons. It has been well known that the large reaction cross section or interaction cross section provides the large root-meansquare radius for the neutron-rich nuclei, however, for the proton-rich nuclei, the relationship between such radius and the reaction cross section is expected to be different from the relation with respect to the neutron-rich isotopes since the number of proton itself does not increase, and the NN amplitudes of proton-proton scattering are different frmo proton-neutron scattering. The present study, therefore, pays attention to the relationship between the root-mean-square radius and the reaction cross section for 8−22 C nuclei.
There is another interest in obtaining the information on the structure of unstable nuclei besides the root-meansquare radius form the restricted observables. The author has proposed a procedure for calcium [14,15], nickel [16] isotopes, and 208 Pb [17], in which the density distribution is assumed by the Woods-Saxon function, and two parameters of the function are determined with two observables. In the present study, showing a specific relationship between the reaction cross section and the root-mean-square radius is attempted for carbon isotopes rather than determining the density distributions.
In Sec. II, formulas on which the analysis is based are presented; Glauber model and RIA. Numerical results are given in Sec. III, In addition to the reaction cross sections, results calculated with RIA for proton-elastic scattering from 12,13,14 C, and comparison between numerical results and experimental data are also given in Sect. III. The summary and conclusion of present study appear in Sec. IV.

A. Glauber Model
Suppose that the projectile proton moves in the z direction and scatters from a target nucleus. The analysis of scattering amplitue based on the eikonal approximation provides the follwoing expression for the reaction cross section, where b is the impact parameter vector located in the xy plane, and χ(b) is the phase-shift function. In accordance with Ref. [6], the optical model is taken, then the average total phase is expressed in terms of the profile function as follows: where N corresponds to proton (p) or neutron (n), Γ pN (b) is the proton-nucleon profile function , and ρ N (r) is the target nuclear density. With Γ pN (b), the proton-nucleon scattering amplitude f pN (θ) is given in the eikonal approximation by where K is the wave number and q is the momentum transfer, q=2K sin θ 2 , and Γ pN (b) is conveniently parametrized as where α pN is the ratio of the real to the imaginary part of the pN scattering amplitude in the forward direction, β pN is the slope parameter of the pN elastic differential cross section, and σ tot pN is the pN total cross section due to the nuclear pN interaction. In the present study the parameter values tabulated in Ref. [4] are used. The difference between pp and pn interactions is taken into account as follows, In order to make calculations simple, the leading order of the cumulant expansion is taken, and yields The reaction cross section is caluclated by substituting this phase shift function into Eq. (1), and with both nuclear densities for proton and nuetron in the target nucleus. In the present calcuation, the densities are obtained from RMF results.

B. Relativistic Impulse Approximation
The Dirac equation containing the optical potential is described in momentum space as follows: where Ψ(p) is given by the Fourier transformation of the wave function in coordinate space: where natural unit (h = c = 1 ) is taken.
In accordance with the prescription of the RIA [12,13], the Dirac optical potential is given in momentum space bŷ whereρ p andρ n are density matrices for protons and neutrons, respectively. The trace is over the γ matrices with respect to the target nucleons and the subscript 2 in the trace corresponds to the target nucleons.
As discussed in Ref. [13] it is known that the nuclear density generally varies more rapidly with k than the NN amplitude and is largest at k = 0. Therefore taking the optimal factorization into account, the optical potentials are written in the well-known tρ forms: The relativistic density matrixρ depends only on the momentum transfer q, as follows: where each term is a Fourier transformation of a coordinate-space density; Nuclear densities, provided by the relativistic meanfield theory [11], are described in terms of upper and lower components as follows: where α represents the quantum numbers of the target nucleus.
In the generalized RIA [12,13] the Feynman amplitude for N N scattering is expanded in terms of covariant projection operators Λ ρ (p) to separate positive (ρ = +1) and negative (ρ = −1) energy sectors of the Dirac space. The invariant amplitudes, M , and kinetic covariants, κ n , are given bŷ where subscripts 1 and 2 correspond to the projectile and target nucleons, respectively. The covariant projection operator Λ ρ (p) is defined by Λ ρ (p) = 1 2m (ρ γ µ p µ + m), and kinetic covariants κ n are constructed from the Dirac matrices. The scalar Feynman amplitude, M ρ1ρ2ρ ′ 1 ρ ′ 2 n , consists of the direct and exchange parts, each of which represents a sum of four Yukawa terms characterized by coupling constants, masses and cutoff masses. In the present calculation, the IA2 parametrization of Ref. [12,13] is used.
By substituting Eq.(9) into Eq. (8) and replacing the momenta with appropriate operators, the coordinatespace Dirac equation is obtained as whereŨ (r) has five potential terms as in Ref. [13] and is described as follows, The local form of the optical potential is obtained by the prescriptions given in Ref. [13], namely the asymptotic value of the momentum operator and the angular averaged expression for nucleon exchange amplitudes, which have been expected to be rather good at high energy scattering. Equation (20) is written as two coupled equations for the upper (ψ U ) and lower (ψ L ) components, and solving forψ U and using the formψ(r) U = K(r)φ(r) in order to remove the first derivative terms yields the following Schrödinger equation for φ(r): where Coulomb potential V C is explicitly written. Although the IA2 potentials are used, it may be useful to display the form of the potentials for the simpler IA1 case. The Schrödinger equivalent potentials for IA1 parametrization are given as follows: This IA1 parametrization corresponds to well-known fiveterm expansion and is obtained by setting ρ i = ρ ′ i = +1 (i = 1, 2) and n max = 5 in Eq. (19), instead of n max = 13. In this case K(r) = √ A, and comes to 1 as r → ∞.

III. RESULTS
A. Root-mean-square radii and density distributions Figure 1 provides proton and neutron distributions for 8−22 C, which are vector densities calculated with Eq. (17) and correspond to hadron densities in the relativistic expression. Results are provided by relativistic mean field calculations [11]. In Fig.1, the panels of (a) and (b) are for 8−14 C and 15−22 C, respectively, and solid lines show neutron distributions and dashed lines proton ones. It is seen that neutron density is expanding with increasing mass number while proton density almost stays for 14−22 C, and becomes expanding for 8−11 C. Table 1 shows root-mean-square radius of proton, neutron, matter, or charge distribution for C isotopes, respectively. Values are also provided by relativistic mean field calculations [11]. As compared to the average values given in Ref. [18], which are shown in the sixth column, the charge radius in Tab.1 is about 2 % smaller for 12 C, and about 3 % larger for 13,14 C. The root-mean-square radius of proton is almost flat for neutron-rich carbon isotopes, however slightly increasing with increasing mass number. In this mass number region, the radius of neutron increases reasonably with increasing mass number. On the other hand, for the proton-rich carbon isotopes, the root-mean-square radius of proton is increasing drastically with decreasing mass number, and the radius of neutron is almost flat for 9−11 C, and is rather small for 8 C. This is reasonable behavior concerning the Coulomb interaction among protons, the number of which is much larger than that of neutrons. As a result, the rootmean-square radius of matter density becomes quite large though the mass number is small. In Fig.2 the root-mean-square radii of Tab.1 are shown with respect to mass number of carbon isotopes. Solid circles, squares, and triangles are results for proton, neutron, and matter densities, respectively. Open ones are corresponding to values appeared in Ref. [4] for 12−22 C. It is found that shell effects of neutrons are significantly small in results for relativistic mean-field calculations, especially in larger mass number.

B. Reaction corss sections
Calculated values of reaction cross section are given in Table II (RIA) and Table III (Glauber), respectively. Because the RIA calculations are available for the proton incident energies more than 50 MeV, the lowest energy in Tab.II is different from that in Tab.III. Figure 3 shows reaction cross sections as a function of the mass number at energies: 100, 425, and 800 MeV, respectively. Solid circles are results for RIA, ,and solid triangles for the Glauber calculation. Open triangles are corresponding to the results for Ref. [4], in which nuclear density distributions are merely different from the present Graluber calculation. As expected, mass number dependence of the reaction cross section between solid and open triangles is similar to that is seen in the rootmean-square radius of matter or neutron distribution in Fig.2. In Fig.3, the RIA calculation always gives larger values than Glauber calculation, and such difference between them seems to come from the difference of the NN interactions based on the calculations. In both calculations, the reaction cross sections for neutron rich isotopes reasonably increase with increasing mass number or the root-mean-square radius of the matter density distributions to which neutron densities mainly contribute. And the reaction cross sections for proton rich isotopes do not show large value corresponding to the large matter  radius. One of the reasons why such thing occur is that the cross section of pp-scattering is smaller than that of pn-scattering in low energy region, and comes to almost similar in high energy region. Therefore expanding proton distribution which gives large root-mean-square radius, dose not contribute to the reaction cross section as much as expanding neutron distribution dose for the neutron rich isotopes. In the figure for 800 MeV, the contribution of proton comes to appear comparing to the figures for lower energies. Another reason is that the expanding proton distribution gives low density because proton number is fixed with 6, and the contribution of such proton is also expected to become small.  [19] .
Comparison between results for calculations and the experimental data is shown in Fig.5 in the case of 12 C target. The horizontal axis of energy is logarithmic scale. The solid line is the result for RIA calculations with tensor density, the dashed line for RIA without tensor density, and dot-dashed line for Glauber calculations. The solid circles are experimental data taken from Ref. [20]. The Glauber calculations predict the energy dependence of the reaction cross section overall except for two values at 61 and 77 MeV. These experimental values seem to be inconsistent with the other data. The RIA results show good agreement with high energy data, accidentally with low energy ones. In general RIA calculations give significantly good predictions for proton-elastic scattering in the energy region higher than 300 MeV. These results are shown in the following section.

C. Elastic scattering calculations for RIA
According to Eq.(20), the ovservables for protonelastic scattering from carbon isotopes are calculated with optical potentials based on the RIA, and are compared with experimental data. Figure 6 shows results for proton-12 C scattering at 150, 250, and 300 MeV; differential cross section and analyzing power . The solid line is the result for RIA calculations with tensor density, the dashed line for RIA without tensor density. Solid circles are experimental data from Ref. [21] (150 MeV), Ref. [22] (250 MeV) and Ref. [23] (300 MeV), respectively. Differential cross sections in the forward angle region: θ c.m. ≤ 40 degrees are well predicted for all energies shown here. For such low energy region, it is known that the RIA predictions for analyzing powers are not so good as those for differential cross sections, and calculations come to show good agreement with experimental data in the energies larger than 300 MeV, though such a comparison is not given in the figure due to absence of the analyzing powre data. Contributions of tensor densities are small for both differential cross sections and analyzing powers in these energies. Figures 7 and 8 show the results for proton-elastic scattering from 12 C and 13 C targets. In Fig.7 ,(a) Figure 8 shows differential cross section (a), analyzing power (b) and spin rotation (c) for 12 C and 13 C target at 500 MeV. Experimental data given by the solid circles are from Ref. [26], and the spin rotation shown by R is given D ss in the reference. The calculated results for differential cross sections predict well in the very forward region: θ c.m. ≤ 20 degrees, and results for spin obervables show good agreement with the data overall, especially the analyzing power for 13 C target, which is also seen in Fig.7 (a). In this case, the contributions of tensor density is significant around the first dip of analyzing powers, and make predictions fit to the experimental data. As seen in Fig. 1 (a), the density distribution of 13 C spreads much more than that of 12 C though only one neutron exceeds. The relativistic mean field results show, as given in Tab.I, that the charge radius of 12 C is smaller than the value of Ref. [18], while the charge radii for 13 C and 14 C are slightly lager than those of the reference. In the RIA calculations, the different results between 12 C and 13 C originate in the density distributions. Provided the spreading density distribution of 13 C shows good prediction for analyzing power, the elastic scattering data for 12 C are given by slightly spreading density of the target nucleus. In other words, the relativistic    mean field result for 12 C in the present calculations gives rather compact density distribution and may be modified to provide slightly spreading distribution in order to fit the experimental data.
In Fig.9 appears the differential cross section for 14 C target at 50 MeV. The solid circles are experimental data from Ref. [27], while the data have been taken at 40 MeV. The proton incident energy 50 MeV is the lowest one for RIA calculation here, therefore the prediction gives always small values comparing to the experimental data, even in the forward angle region. It is however seen that the angular distribution is overall predicted for 14 C tar-get.
In comparison with the experimental data of Ref. [9], though they are not shown here, the numerical results for 9 C target at 300 MeV also give small values as seen in Fig.7. The root-mean-square radius for nuclear matter in the reference, which has been determined from the data, has been 2.43±0.55 fm, and this is smaller than the value for the relativistic mean field results given in Tab.I: 2.664 fm while the value itself exists within a margin of error. As for the differential cross section calculated with RIA, the first dip position seems to exist in smaller angle than the experimental data. This phenomenon is consistent with the spreading density distribution of the target nucleus, e.i., the large root-mean-square radius of the nuclear matter. In other words, the experimental data seem to prefer the density distribution for 9 C with smaller matter radius than the relativistic mean field results. In the case of small number of neutron, nuclear densities provided by the relativistic mean field results show a tendency to expand, as seen in helium isotopes [28].

D. Relationship between rrms and σr
In order to show the relationship between the reaction cross section and the root-mean-square radius, the density distributions for 12 C target nucleus are assumed by Wood-Saxon function as follows; where R and a are the half-density radius and diffuseness parameter, respectively. The value ρ 0 is the normalization constant which is determined f rom the following where N is atomic number Z for the proton, and A − Z for the neutron. The number A is the mass number of the target nucleus, and for 12 C these numbers are the same given by Z = A − Z = 6. As usual the halfdensity radius is given by R = cA 1/3 , therefore a and c are determined freely. In the present calculations, these parameters are chosen so that the root-mean-square radius is the same as the result for the relativistic mean filed calculations given in Tab.I, i.e. < r 2 > p = 2.277 fm for the proton, √ < r 2 > n = 2.257 fm for the neutron, and in result √ < r 2 > m = 2.267 fm for the nuclear matter, while the deviations of ±0.01 fm are practically concerned. The obtained parameter sets are three for proton and neutron, respectively, and combinations are nine, which are given in Table IV. For the diffuseness parameter, a = 0.35, 0.45, 0.55 fm are first taken, and the half-density radii are determined so that the root-meansquare radius is obtained with the value of the relativistic mean-field results. The distributions of a = 0.45 fm are similar to the results for the relativistic-mean-field calculations, therefore the model WS1 corresponds to 12 C in Fig.1 (a). The results for a = 0.35 fm are compressed distributions and for a = 0.55 fm are spreading ones. Figure 10 shows density distributions for the models; WS1 through WS9 corresponding to Tab.IV. Solid, dashed, and dash-dotted lines are results for neutron, proton, and nuclear matter, respectively. There is much variety found between density distributions, while they have the same root-mean-square radii within the error of ±0.01 fm.
In terms of these model densities, the reaction cross sections for proton-elastic scattering from 12 C target are calculated based on the Glauber model, which are shown in Fig. 11. It is clearly seen that all results for the model WS1 through WS9 provide almost the same values in  wide energy region: 40-1000 MeV. As already mentioned, the result for WS1 is expected to be the similar value as the relativistic-mean field results shown in the Glauber part of Fig. 4, and that is found though they are not given in Fig.11. It is concluded from Fig.11 that there is a strong relationship between the reaction cross section and the root-mean-square radius, and is also that the reaction cross sections determine the parameter sets of Woods-Saxon density distributions, which provide a specific root-mean-square radius. For calcium and nickel isotopes, the reaction cross section and the mean-square radius have shown almost the same behaviors as the functions of parameters for Woods-Saxon density distributions based on the RIA calculations [15,16]. The meansquare raius is analytically given by Woods-Saxon density distribution of Eq. (25) as follows; Figure12 shows the relationship between the half-density radius and diffuseness, which are given in Tab. IV. Circles and triangles are results for neutron and proton, respectively. The solid line corresponds to a part of the ellipse: 8.5 = 7 3 (πa) 2 + R 2 . The number of the left hand side is determined in accordance with the values which are caluclated with the mean-square radii of proton and neutron as follows; (2.277) 2 × 5/3 ∼ 8.64 and (2.257) 2 × 5/3 ∼ 8.49. As already mentioned, the half-density radius is searched with respect to the given diffuseness parameter so that the root-mean-square radius is the same as the result for the relativistic mean filed calculations. Figure 12 confirms that these searched parameter sets completely satisfy Eq. (27). In order to determine the whole distribution of the target nucleus, another observable has been considered in the previous works [15,16], e.g. the first dip position of the differential cross section. Two observables: the reaction cross section and the first dip position of the differential cross section, in principle have been able to determine two parameters of Woods-Saxon function while the experimental errors have significantly affected the accuracy in the determination of the parameters.

IV. SUMMARY AND CONCLUSION
This work has presented reaction cross sections for proton-elastic scattering from carbon isotopes of A = 8 − 22 in large energy region: 100-1000 MeV . Density distributions of the target nuclei have been provided from the relativistic mean-field results, and calculations have been done in terms of two prescriptions: the Glauber theory and the relativistic impulse approximation.
Reaction cross sections which have been calculated with RIA are sightly larger than those with the Glauber theory in the whole energy region considered here. The behavior with respect to the energy is similar in both calculations, i.e. significantly decreasing with increasing energies smaller than 200 MeV, showing minimum values at around 300-400 MeV, and after that slightly increasing with increasing energies. These phenomena are attributed to the NN amplitudes on which both prescriptions have based in the calculations.
As expected, the reaction cross sections increase with increasing mass number of carbon isotopes, however, the root-mean-square radius shows much larger value for the isotopes whose mass numbers are less than 12 due to the expanding proton distributions. Such expansion is caused by both repulsive Coulomb interaction and small number of neutron which gives rise to attractive nuclear interaction. Contributions of expanding proton distributions have been slightly seen while those of neutron distributions have significantly appeared. For the proton-rich isotopes, effects of decreasing mass number and increasing root-mean-square radius contribute to the reaction cross section in the opposite direction each other. Therefore it is rather complicated to find the direct relationship between σ r and r rms . In the case of the neutron-rich isotope, the root-mean-square radius simply increases with increasing mass number, and the relation of σ r to r rms is expected to be a plain one.
In order to show the relationship between σ r and r rms , a model analysis with Woods-Saxon density distributions for 12 C nucleus has been done. It has been shown that various distributions with different parameters provided almost the same values of the reaction cross in the large energy region: 100-1000 MeV as far as the distributions had the same values of the root-mean-square radius. Such a strong relationship between σ r and r rms provides some prescriptions which determine the root-meansquare radius directly from the reaction cross section at least for the neutron rich nuclei. Besides the reaction cross section, however, another observable is necessary to obtain the whole profile of the density distribution for the target nucleus.

ACKNOWLEDGMENTS
The author acknowledges the use of the code of the relativistic mean field calculation provided from Y. Sugahara. Numerical calculations in this paper were performed using the facilities at the Information Processing Center of Shizuoka University, and partly using the computing service at Institute for Information Management and Communication, Kyoto University. Numerical calculations of part D in Sect. III have partly appeared in Ref. [29]