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

TL;DR

This paper calculates proton scattering reaction cross sections for carbon isotopes (A=8-22) using relativistic models and compares them with experimental data, revealing a strong link between cross sections and nuclear size.

Contribution

It introduces a combined approach using Glauber theory and relativistic impulse approximation with relativistic mean field densities for carbon isotopes.

Findings

Reaction cross sections are successfully calculated across a large energy range.

A clear correlation between reaction cross section and root-mean-square radius for 12C is demonstrated.

Results agree well with experimental data, validating the models used.

Abstract

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.

Tables

Table 1. Table 1: Root-mean-square radius of proton ,neutron , matter or charge distribution for 8-22 C. Three values in the sixth column are charge radii taken from Ref. [ 18 ] .

	Root-mean-square radius (fm)
Isotope	proton	neutron	matter	charge	Ref. [18]
8C	3.662	1.990	3.324	3.751
9C	2.872	2.189	2.664	2.984
10C	2.558	2.218	2.428	2.684
11C	2.363	2.214	2.296	2.498
12C	2.277	2.257	2.267	2.417	2.4702
13C	2.398	2.533	2.472	2.532	2.4614
14C	2.440	2.652	2.563	2.571	2.5025
15C	2.445	2.944	2.755	2.576
16C	2.453	3.109	2.881	2.584
17C	2.461	3.231	2.982	2.592
18C	2.470	3.323	3.065	2.600
19C	2.479	3.394	3.134	2.608
20C	2.488	3.451	3.193	2.617
22C	2.507	3.539	3.289	2.635

Table 2. Table 2: Reaction cross sections calculated based on RIA for 8-22 C. The unit is square fm.

	Proton incident energy (MeV)
Isotope	71	100	200	300	425	550	650	800
8C	24.18	24.73	18.87	17.88	18.99	21.33	23.47	24.52
9C	26.47	26.65	20.49	19.18	19.80	21.68	23.28	24.01
10C	28.45	28.35	21.97	20.38	20.62	22.17	23.40	23.91
11C	30.17	29.85	23.30	21.47	21.40	22.68	23.62	23.95
12C	32.21	31.75	24.84	22.78	22.47	23.61	24.38	24.61
13C	36.32	35.86	27.64	25.27	24.81	26.11	27.03	27.35
14C	39.32	38.79	29.79	27.17	26.55	27.88	28.82	29.15
15C	43.26	42.70	32.43	29.46	28.62	30.01	31.00	31.35
16C	46.71	46.11	34.81	31.55	30.51	31.94	32.97	33.33
17C	49.98	49.33	37.11	33.58	32.35	33.82	34.88	35.25
18C	53.06	52.36	39.32	35.52	34.13	35.63	36.71	37.10
19C	55.98	55.21	41.44	37.40	35.85	37.38	38.48	38.88
20C	58.74	57.90	43.47	39.22	37.52	39.07	40.19	40.60
22C	63.87	62.89	47.33	42.67	40.70	42.30	43.45	43.87

Table 3. Table 3: Parameter sets for Wood-Saxon function.

model	proton (fm)		neutron (fm)
	a	c	a	c
WS1	0.45	1.09	0.45	1.07
WS2	0.35	1.32	0.45	1.07
WS3	0.55	0.73	0.45	1.07
WS4	0.45	1.09	0.35	1.31
WS5	0.45	1.09	0.55	0.70
WS6	0.35	1.32	0.55	0.70
WS7	0.55	0.73	0.35	1.31
WS8	0.35	1.32	0.35	1.31
WS9	0.55	0.73	0.55	0.70

Equations

$$\displaystyle\{\gamma^{0}E-\mbox{\boldmath$\gamma$}\cdot\mbox{\boldmath$p$}^{\prime}-m\}\ \Psi(\mbox{\boldmath$p$}^{\prime})-\int\frac{d^{3}p}{(2\pi)^{3}}\ \hat{U}(\mbox{\boldmath$p$}^{\prime},\mbox{\boldmath$p$})\ \Psi(\mbox{\boldmath$p$})=0,$$

$$\displaystyle\Psi(\mbox{\boldmath$p$})=\int d^{3}re^{-i\mbox{\boldmath$p$}\cdot\mbox{\boldmath$r$}}\Psi(\mbox{\boldmath$r$}),$$

$$\displaystyle\hat{U}(\mbox{\boldmath$p$}^{\prime},\mbox{\boldmath$p$})$$

$$\displaystyle\hat{\rho}(\mbox{\boldmath$q$})=\rho_{S}(q)+\gamma_{2}^{0}\rho_{V}(q)-\frac{i\mbox{\boldmath$\alpha$}_{2}\cdot\mbox{\boldmath$q$}}{2m}\rho_{T}(q),$$

$${\rho }_S(q)$$

$${\rho }_V(q)$$

$${\rho }_T(q)$$

$${\rho }_S(r)$$

$${\rho }_V(r)$$

$${\rho }_T(r)$$

$$\hat{M}(p_1,p_2\to p_1^{\prime },p_2^{\prime })=\sum _{{\rho }_1,{\rho }_2,{\rho }_1^{\prime },{\rho }_2^{\prime }}{\Lambda }^{{\rho }_1^{\prime }}(p_1^{\prime }){\Lambda }^{{\rho }_2^{\prime }}(p_2^{\prime })\times \sum _{n=1}^{13}{M}_n^{{\rho }_1{\rho }_2{\rho }_1^{\prime }{\rho }_2^{\prime }}{\kappa }_n{\Lambda }^{{\rho }_1}(p_1){\Lambda }^{{\rho }_2}(p_2),$$

$$\displaystyle\left\{\gamma^{0}E+i\mbox{\boldmath$\gamma$}\cdot\nabla-m-\tilde{U}(\mbox{\boldmath$r$})\right\}\tilde{\Psi}(\mbox{\boldmath$r$})=0,$$

$$\displaystyle\tilde{U}(\mbox{\boldmath$r$})$$

$$\displaystyle\left\{-\nabla^{2}+2E(U_{ce}+U_{ls}\ \mbox{\boldmath$\sigma$}\cdot\mbox{\boldmath$L$})\right\}\phi(\mbox{\boldmath$r$})$$

$$\displaystyle\hskip 56.9055pt=\left\{(E-V_{C})^{2}-m^{2}\right\}\phi(\mbox{\boldmath$r$}),$$

$$U_{ce}$$

$$U_{ls}$$

$$A$$

$$\rho (r)=\frac{{\rho }_0}{1+\exp [(r-R)/a]},$$

$$\int d^{3}r\rho (r)=N,$$

$$<r^2>=\frac{7}{5}(\pi a{)}^2+\frac{3}{5}{R}^2.$$

Full text

1 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: 6,8He, 11Li, and 11,14Be, 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 12C 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 14,15,16C isotopes scattering from 12C target have been studied based on the g-matrix double-folding model and ${}^{14}\rm C+n$ two-body model [8]. In addition to the neutron-rich isotopes, the angular distribution of proton-elastic scattering from the proton-rich isotopes like 9C 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 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-mean-square 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 8-22C nuclei.

There is another interest in obtaining the information on the structure of unstable nuclei besides the root-mean-square radius form the restricted observables. The author has proposed a procedure for calcium [14, 15], nickel [16] isotopes, and 208Pb [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 of RIA on which the analysis is based are briefly presented. Numerical results are given in Sec. III, In addition to the reaction cross sections, results calculated with RIA for proton-elastic scattering from 12,13,14C, 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.

2 Formulation

The Dirac equation containing the optical potential is described in momentum space as follows:

[TABLE]

where \Psi(\mbox{\boldmathp}) is given by the Fourier transformation of the wave function in coordinate space:

[TABLE]

where natural unit ( $\hbar=c=1$ ) is taken.

In accordance with the prescription of the RIA [12, 13], the Dirac optical potential is given in momentum space by

[TABLE]

where $\hat{\rho}_{p}$ and $\hat{\rho}_{n}$ are density matrices for protons and neutrons, respectively. The trace is over the $\gamma$ 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 \mbox{\boldmathk}=0. Therefore taking the optimal factorization into account, the optical potentials are written in the well-known $t\rho$ forms:

[TABLE]

The relativistic density matrix $\hat{\rho}$ depends only on the momentum transfer $q$, as follows:

[TABLE]

where each term is a Fourier transformation of a coordinate-space density;

[TABLE]

Nuclear densities, provided by the relativistic mean-field theory [11], are described in terms of upper and lower components as follows:

[TABLE]

where $\alpha$ represents the quantum numbers of the target nucleus.

In the generalized RIA[12, 13] the Feynman amplitude for $NN$ scattering is expanded in terms of covariant projection operators $\Lambda^{\rho}(p)$ to separate positive ($\rho=+1$) and negative ($\rho=-1$) energy sectors of the Dirac space. The invariant amplitudes, $M_{n}^{\rho_{1}\rho_{2}\rho_{1}^{\prime}\rho_{2}^{\prime}}$, and kinetic covariants, $\kappa_{n}$, are given by

[TABLE]

where subscripts 1 and 2 correspond to the projectile and target nucleons, respectively. The covariant projection operator $\Lambda^{\rho}(p)$ is defined by $\Lambda^{\rho}(p)=\frac{1}{2m}(\rho\ \gamma^{\mu}p_{\mu}+m)$, and kinetic covariants $\kappa_{n}$ are constructed from the Dirac matrices. The scalar Feynman amplitude, $M_{n}^{\rho_{1}\rho_{2}\rho_{1}^{\prime}\rho_{2}^{\prime}}$, 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.(2) into Eq.(1) and replacing the momenta with appropriate operators, the coordinate-space Dirac equation is obtained as

[TABLE]

where \tilde{U}(\mbox{\boldmathr}) has five potential terms as in Ref.[13] and is described as follows,

[TABLE]

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 (13) is written as two coupled equations for the upper ( $\tilde{\psi}_{U}$ ) and lower ( $\tilde{\psi}_{L}$ ) components, and solving for $\tilde{\psi}_{U}$ and using the form \tilde{\psi}(\mbox{\boldmathr})_{U}=K(r)\phi(\mbox{\boldmathr}) in order to remove the first derivative terms yields the following Schrödinger equation for \phi(\mbox{\boldmathr}):

[TABLE]

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:

[TABLE]

This IA1 parametrization corresponds to well-known five-term expansion and is obtained by setting $\rho_{i}=\rho_{i}^{\prime}=+1\ (i=1,2)$ and $n_{max}=5$ in Eq.(12), instead of $n_{max}=13$. In this case $K(r)=\sqrt{A}$, and comes to 1 as $r\rightarrow\infty$.

3 Results

3.1 Root-mean-square radii and density distributions

Figure 1 provides proton and neutron distributions for 8-22C except 21C as it has been already known that 21C is unbound. Distributions are vector densities calculated with Eq.(10) 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 8-14C and 15-22C, 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 14-22C, and becomes expanding for 8-11C.

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 12C, and about 3 % larger for 13,14C. 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 9-11C, and is rather small for 8C. 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 root-mean-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 12-22C. It is found that shell effects of neutrons are significantly small in results for relativistic mean-field calculations, especially in larger mass number.

3.2 Reaction corss sections

Calculated values of reaction cross section are given in Table 2. The lowest energy in Tab.II is chosen as same as in the case of Ref.[28], where reasonable results have been obtained in such low energy for He isotopes, while the RIA calculations are available for the proton incident energies more than 50 MeV.

In order to compare the RIA results for 8-11C with the Glauber results, the reaction cross sections for all carbon isotopes considered here are calculated with vector densities obtained from RMF results in accordance with the procedure of Ref.[4]., and are referred as the Glauber calculations.

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 root-mean-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.

Figure 4 shows the reaction cross sections with respect to the energy. For the RIA, plotted energies are values given in Tab.2. In Fig.4, the Glauber calculations give significantly large reaction cross section in energies less than 100 MeV, while the RIA calculations do not show significantly increase with decreasing energy. Experimental data have been given in Ref.[19] for 19,20,22C at 40 MeV. Since the value of incident energy is so small that the RIA calculations are not available, the Glauber calculations with the RMF density distributions are compared with experimental data instead. The reaction cross section for 22C at 40 MeV is 810.4 mb in Glauber calculations while a experimental value is 1338(274) mb which is more than1.5 times larger. On the other hand for 19,20C the reaction cross sections are 732.6 mb and 761.1 mb, respectively, and those values are similar to the experimental values of 754(22) mb and 791(34) mb. As seen in Fig.2 the neutron density of 22C in Ref.[4] has much larger root-mean-square radius, and the reaction cross section at 40 MeV is 957 mb, which is also larger than that of Gulauber calculations with RMF densities. The experimental data for 22C seem to suggest that the neutron distribution of 22C is much more spreading than the distribution considered in the present calculations, or at least has much larger root-mean-square radius.

Comparison between results for calculations and the experimental data is shown in Fig.5 in the case of 12C 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.

3.3 Elastic scattering calculations for RIA

According to Eq.(20), the ovservables for proton-elastic 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-12C 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: $\theta_{\rm c.m.}\leq$ 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 12C and 13C targets. In Fig.7 ,(a) and (b) are results for 200 MeV, and 800 MeV, respectively. In each part ,the left side sheets are the results for 12 C target, and the right side ones for 13C. The upper sheets show the differential cross section, and the lower sheets the analyzing power. The line identifications are the same in Fig.6, and solid circles are experimental data from Ref.[24] (a), and Ref.[25] (b), respectively. As already seen in Fig.6, the differential cross sections are well predicted in the forward angle region: $\theta_{\rm c.m.}\leq 40$ degrees at 200 MeV, and $\theta_{\rm c.m.}\leq 20$ degrees at 800 MeV. The analyzing powers at 200 MeV, the angular distribution is similar to the result for 250 Mev, though the calculation of 13C target shows good agreement with experimental data in the angle region: 20 $\leq\theta_{\rm c.m.}\leq$ 60 degrees. The contributions of tensor density is also small at 200 MeV, however, at 800 MeV they appear in results for the forward analyzing power, and shift the distributions to larger angles.

Figure 8 shows differential cross section (a), analyzing power (b) and spin rotation (c) for 12C and 13C 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: $\theta_{\rm c.m.}\leq$ 20 degrees, and results for spin obervables show good agreement with the data overall, especially the analyzing power for 13C 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 13C spreads much more than that of 12C though only one neutron exceeds. The relativistic mean field results show, as given in Tab.I, that the charge radius of 12C is smaller than the value of Ref.[18], while the charge radii for 13C and 14C are slightly lager than those of the reference. In the RIA calculations, the different results between 12C and 13C originate in the density distributions. Provided the spreading density distribution of 13C shows good prediction for analyzing power, the elastic scattering data for 12C are given by slightly spreading density of the target nucleus. In other words, the relativistic mean field result for 12C 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 14C 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 14C target.

In comparison with the experimental data of Ref.[9], though they are not shown here, the numerical results for 9C 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$\pm$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 9C 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].

3.4 Relationship between $r_{rms}$ and $\sigma_{r}$

In order to show the relationship between the reaction cross section and the root-mean-square radius, the density distributions for 12C target nucleus are assumed by Wood-Saxon function as follows;

[TABLE]

where $R$ and $a$ are the half-density radius and diffuseness parameter, respectively. The value $\rho_{0}$ is the normalization constant which is determined f rom the following equation;

[TABLE]

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 12C these numbers are the same given by $Z=A-Z=6$. As usual the half-density 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. $\sqrt{<r^{2}>_{p}}=2.277$ fm for the proton, $\sqrt{<r^{2}>_{n}}=2.257$ fm for the neutron, and in result $\sqrt{<r^{2}>_{m}}=2.267$ fm for the nuclear matter, while the deviations of $\pm 0.01$ fm are practically concerned. The obtained parameter sets are three for proton and neutron, respectively, and combinations are nine, which are given in Table 3. 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-mean-square 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 12C 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 $\pm 0.01$ fm.

In terms of these model densities, the reaction cross sections for proton-elastic scattering from 12C target are shown in Fig. 11 in the RIA (left panel) and Glauber (right panel) calculations. In both panels dashed lines correspond to the results for the density distribution provided with RMF results. For the RIA calculations the tensor densities are excluded, and scalar densities are given by multiplying the model distributions and scalar-vector density ratios obtained from RMF results. It is clearly seen that all results for the model WS1 through WS9 provide almost the same values in wide energy region calculated here. As already mentioned, the result for WS1 is expected to be the similar value as the relativistic-mean field results shown in Fig. 4. 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 mean-square raius is analytically given by Woods-Saxon density distribution of Eq. (25) as follows;

[TABLE]

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: $8.5=\frac{7}{3}(\pi a)^{2}+R^{2}$. 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; $(2.277)^{2}\times 5/3\sim 8.64$ and $(2.257)^{2}\times 5/3\sim 8.49$. 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.

4 Summary and Conclusion

This work has presented reaction cross sections for proton-elastic scattering from carbon isotopes of $A=8-22$ except $A=21$ in large energy region: 100-800 MeV . Density distributions of the target nuclei have been provided from the relativistic mean-field results, and calculations have been done in terms of the relativistic impulse approximation. As for reference, Glauber model calculations with RMF density distributions have been also given.

Reaction cross sections which have been calculated with RIA are sightly larger than those with the Glauber calculations 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 mainly 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 $\sigma_{r}$ and $r_{\rm rms}$. In the case of the neutron-rich isotope, the root-mean-square radius simply increases with increasing mass number, and the relation of $\sigma_{r}$ to $r_{\rm rms}$ is expected to be a plain one.

In order to show the relationship between $\sigma_{r}$ and $r_{\rm rms}$, a model analysis with Woods-Saxon density distributions for 12C 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: 100-800 MeV as far as the distributions had the same values of the root-mean-square radius. Such a strong relationship between $\sigma_{r}$ and $r_{\rm rms}$ provides some prescriptions which determine the root-mean-square 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 in the proton-elastic scattering. For another possibility, reaction cross sections in nucleus-nucleus scattering are expected to determine the density distributions though the RIA calculations become rather complicated but are challenging.

5 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. Some of numerical calculations in Sect. 3.4 have partly appeared in Ref. [29]