Energy-dependent optical model potentials for α and deuteron with ¹²C

It is well known that coupling effects due to breakup channels are important for reactions induced by weakly bound nuclei. A widely accepted model for evaluating breakup effects is the coupled discretized continuum channel (CDCC) method [1, 2]. ⁶He and ⁶Li, whose two-neutron and deuteron binding energies being only 0.972 and 1.47 MeV, respectively, are very good test cases for these studies. Within the cluster model, interactions between projectile and target nuclei are obtained with potentials between the target and the constituent clusters in the projectile, which are 2n and α for ⁶He [3], and deuteron and α for ⁶Li. One application of the CDCC method is to achieve an understanding, for weakly bound projectiles, of the dynamical polarization potentials (DPPs) that are due to the coupling of breakup channels to the elastic channel [4–7]. The DPPs are non-local and L-dependent, but their local and L-independent equivalents can be determined, and these depend on the incident energy. It is therefore necessary to study the properties of DPPs over a wide range of incident energies [8] and this in turn requires cluster–target interactions over a wide range of energies. Several measurements have been made for ⁶He and ⁶Li elastic scattering from ¹²C at energies between 10 and 100 MeV/nucleon [9–13], which are useful for systematic studies of DPPs due to the breakup couplings during their collisions with the target. However, unfortunately, most of those measurements were performed at incident energies where there are no corresponding α and deuteron elastic scattering data available, which are needed for the determination of the required α and deuteron optical model potentials (OMPs) at the appropriate energies. Because of this, the first step would be to determine the energy dependence of α and deuteron potentials with ¹²C so that reliable interpolations could be made at the energies needed.

Apart from their importance in studies of projectile breakup effects in ⁶He- and ⁶Li-induced reactions, the systematic behavior (energy and target-mass dependence) of α and deuteron OMPs themselves are interesting subjects. To a first approximation, the OMP of a projectile–target system at a certain incident energy can be evaluated with folding models that convolute nucleon–nucleon or nucleon–nucleus potential with nuclear density distributions of the projectile and/or target nuclei [4, 14]. Recently, global α– and deuteron–nucleus OMPs for heavy targets have been studied with a single-folding (SF) model approach by analyzing α and deuteron elastic scattering from ⁵⁸Ni, ⁹⁰Zr, ¹¹⁶Sn and ²⁰⁸Pb with incident energies between 10 and 100 MeV/nucleon [15]. The Bruyères Jeukenne–Lejeune–Mahaux (JLMB) model nucleon–nucleus potentials [16, 17] were used in that work. Energy-dependent SF model potential parameters were found giving a good account of both angular distributions of elastic scattering cross sections and total reaction cross sections for these two projectiles. It is well known that systematics of OMPs based on analysis at heavy-target regions (A_T ≳ 40) do not account well for elastic scattering at light-target regions (A_T ≲ 30) [18–20]. For this reason, α and deuteron OMPs with light targets, such as ¹²C, require separate study and that is the subject of this work.

There have been many phenomenological and semi-microscopical studies of α+¹²C potentials [21–25]. However, the energy dependence of these potentials within a sufficiently large energy range, namely within 10–100 MeV/nucleon, has not been established despite the fact that there exist abundant experimental data over a wide energy range. Unlike the case of the α particle, there are several global deuteron potentials available [26–28]. Our study of deuteron potential in this paper is nevertheless useful because it is interesting to compare the parameters and their energy dependence for deuteron and the α particle potentials obtained with the same approach, and the deuteron potential reported here was obtained with a different approach from that used in other studies.

In this paper, we derive the energy dependence of the α-¹²C and d-¹²C interactions by analyzing their corresponding elastic scattering data between 10 and 100 MeV/nucleon with the SF model as those made in [15]. The rest of this paper is organized as follows: the SF model approach is introduced in section 2, results of the energy-dependent potential parameters and their comparisons with other phenomenological OMPs [21, 27] against experimental data are given in section 3. Discussions of the nucleon-¹²C potentials used in the SF model and conclusions of this paper are given in section 4.

A detailed description of the SF model used in this work can be found in [15]. For simplicity, only necessary formulas are presented here. The SF model potentials for α-¹²C and d-¹²C systems at incident energy E_lab are as follows:

$$\begin{eqnarray} U_{\rm SF}^{\alpha }(R,E_{\rm lab}) &=& \sum _{i={\rm p,n}}\int \rho _{i}^{\alpha }({\bm r})U_{i2}(|{\bm s}|,E_{i}){\rm d}{\bm r}, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} U_{\rm SF}^{{\rm d}}(R,E_{\rm lab}) &=&\sum _{i={\rm p,n}}\int \phi _0^\ast ({\bm r})[U_{i2}(|{\bm s}|,E_i)] \phi _0({\bm r})\,{\rm d}{\bm r}, \end{eqnarray} \tag{ 2 }$$

where ${\bm R}$ is a vector from the center of mass (c.m.) of ¹²C to that of the projectile, ${\bm s}$ is ${\bm R}+{\bm r}$ for U^α_SF (with ${\bm r}$ being a vector from the c.m. of the α particle) and ${\bm R}\pm {\bm r}/2$ for U^d_SF (with ${\bm r}$ being a vector from the proton to the neutron in deuteron). $\rho _{i}^{\alpha }({\bm r})$ (i = p, n) are proton and neutron density distributions in ⁴He, which are Gaussian functions with parameter b = 1.1932 fm (version C in table I of [22]) in our calculations. This density distribution, which corresponds to the root-mean-square (rms) radius of ⁴He being 1.461 fm, was found to be most realistic for the α particle [4, 22]. $\phi _0({\bm r})$ is the ground state wavefunction of deuteron with Gaussian potential parameters V₀ = 72.25 MeV and r₀ = 1.484 fm for the n–p system. For simplicity, only the S component of deuteron wavefunction is used and ϕ₀ is normalized to unity. In equations (1) and (2), incident energies for the corresponding proton and neutron scattering from ¹²C are assumed to be

$$\begin{equation} E_{i={\rm p,n}}=E_{\rm lab}/A_{\rm P}, \end{equation} \tag{ 3 }$$

where A_P is the mass of the projectile.

The nucleon–target potentials, U_i2, in equations (1) and (2) are results of the Lane-consistent JLMB model [16, 17], which was derived from the nucleon potential in nuclear matter (NM) with the improved local density approximation (iLDA) [16, 29]:

$$\begin{equation} \fl U_{\rm \textit {i}2}(s,E_i) = \frac{1}{(t\sqrt{\pi })^3}\int \frac{U_{\rm NM}(\rho (r^\prime ),\alpha (r^\prime ) , E_i)}{\rho (r^\prime )} \exp (-|{\bm s}-{\bm r}^\prime |/t^2)\rho (r^\prime )\,{\rm d}{\bm r}^\prime , \end{equation} \tag{ 4 }$$

where ρ(r') = ρ_p(r') + ρ_n(r') is the NM density at position ${\bm r}^\prime$ from the c.m. of the target with ρ_p and ρ_n being proton and neutron densities, respectively, α(r') = [ρ_n(r') − ρ_p(r')]/ρ(r') is the asymmetry term, and t is the range of Gaussian form factor, which can take different values for real and imaginary potentials, denoted as t_r and t_i, respectively. A two-parameter Fermi function with parameters ρ₀ = 0.194 fm⁻³, c = 2.214 fm and a = 0.425 fm (section 2 in table II of [22]) is used in this paper for the nucleon density distribution of ¹²C. The corresponding rms radius of ¹²C being 2.332 fm is very close to its experimental value, 2.33 ± 0.01 fm [30]. In the JLMB model, the NM potential for a given NM density ρ = ρ_p + ρ_n and asymmetry α = (ρ_n − ρ_p)/(ρ_n + ρ_p) is

$$\begin{eqnarray} &&\fl U_{\rm NM}(E_i)_{\rho ,\alpha }=\lambda _{V}(E_i)[V_0(\tilde{E_i})\pm \lambda _ { V1}(E_i)\alpha {}V_1(\tilde{E_i})] + {\rm i}\lambda _{W}(E_i)[W_0(\tilde{E_i})\pm \lambda _{W1}(E_i)\alpha {}W_1(\tilde{E_i})], \nonumber \\ \end{eqnarray} \tag{ 5 }$$

in which the '+' and '−' signs are for neutron and proton, respectively. The nucleon energies, $\tilde{E_i}$, used in calculations of the isoscalar (V₀ and W₀) and isovector (V₁ and W₁) potentials are E^c.m._p − V_C(s), and E^c.m._n, respectively, with

$$\begin{equation} E_{i}^{\rm c.m.}=\frac{E_{i}A_{\rm T}}{A_{\rm T}+1} \quad (i={\rm p,n}), \end{equation} \tag{ 6 }$$

where A_T is the mass of the target nucleus. For Coulomb corrections to the proton incident energies [31, 29], the Coulomb potential for proton is evaluated assuming a uniform charge distribution of the target nucleus with radius R_C = r_C × A^1/3_T where r_C = 1.123A^1/3_T + 2.35A^−1/3_T − 2.07A_T⁻¹ (in femtometers) [31]. Details of calculations for the isoscalar and isovector potentials and their corresponding normalization factors λ_V, λ_W, λ_V1 and λ_W1 can be found in [16, 17]. The particular form of equation (4), in which ρ(r') appears in both the numerator and the denominator, means that we have adopted the target position prescription of the iLDA in [16], which was found to be the best prescription for nucleon potentials [17].

The nucleon potentials used in SF model calculations are for free nucleons, while the nucleons in a projectile are not free. Due to the compositeness of projectile nuclei [32], SF potentials have to be renormalized for describing nucleus–nucleus scattering. Coulomb potentials also have to be added. Finally, with the SF model, a potential that accounts for nucleus–nucleus scattering takes the following form:

$$\begin{equation} U(R,E_{\rm lab})=N_{\rm r}{\rm Re}[U_{\rm SF}(R,E_{\rm lab})] +{\rm i}N_{\rm i}{\rm Im}[U_{\rm SF}(R,E_{\rm lab})] +V_{\rm C}(R), \end{equation} \tag{ 7 }$$

where N_r and N_i are renormalization factors for the real and imaginary parts, respectively. The Coulomb potential, V_C, is calculated in the usual way with radius R_C = r_C(A^1/3_T + A^1/3_P) for α and R_C = r_CA^1/3_T for deuteron with r_C = 1.3 fm. It is the task of this work to study the energy dependence of the parameters t_r, t_i, N_r and N_i by analyzing α+¹²C and d+¹²C elastic scattering data. Deuteron spin–orbit potentials are not included in this paper.

3.1. Data analysis

Throughout this work, as in [15], we require t_i to be equal to t_r because the former was found to be not as well determined as the latter by angular distribution data. We designate them commonly by t_ri in the following text. For each set of experimental data, the SF potential with t_ri values from 0.8 to 2.0 fm with steps of 0.05 fm was calculated and corresponding N_r and N_i values were found by fitting experimental data with a standard minimum χ² approach. Since different experimental data contain different systematic uncertainties, we also allow normalization of experimental data, N_o, to vary during the fittings. The OM fittings were made using the computer code SFRESCO, which is a combination of FRESCO [33] and the search routine MINUIT [34]. The computer code JLM was adapted for calculations of the JLMB model nucleon–nucleus potentials [35]. A detailed description for procedures of data analysis can be found in [15]. Eventually, we were able to get the 'individual' parameters which give optimum fitting for each single set of data. Energy dependence of the potential parameters was then found by fitting these individual parameters with energy dependence functions in equations (8)–(13). Experimental data included in this study and their references are listed in table 1 together with the resulting t_ri, N_r, N_i and N_o values. All data were taken from the nuclear database EXFOR/CSISRS [36]. Experimental error bars were used in the fittings when they are available in the database; otherwise, uniform uncertainties of 10% were applied. Note, however, that the χ² values (averaged with number of data points) in table 1 were obtained with optical model calculations with the energy-dependent potentials derived in this work and uniformly 10% of error bars for all data sets.

Table 1. Experimental data analyzed in this work, their references and results of their individual fittings, where E_lab are in MeV, t_ri are in femtometers, J_r and J_i are volume integral per nucleon of these potentials in MeV fm³, N_o are normalization factors of these data obtained in the optical model fittings and χ² are calculated with the energy-dependent potentials with uniformly 10% of experimental error bars and with N_o taken into account.

	Elab	tri	Nr	Ni	No	χ2	Jr	Ji	Reference
α+12C	54.1				0.876	367			[37]
	65.0	1.13	0.691	0.730	0.811	15.0	−315.1	−95.1	[38]
	104	1.43	0.782	0.848	0.967	5.18	−330.5	−111.4	[39]
	120	1.47	0.806	0.877	1.05	4.27	−330.3	−115.8	[40]
	139	1.30	0.744	0.851	1.07	16.7	−293.4	−112.2	[41]
	145	1.50	0.786	0.835	1.12	2.79	−306.5	−110.0	[40]
	166	1.41	0.756	0.797	1.07	4.81	−282.4	−104.1	[42]
	172.5	1.52	0.766	0.777	1.18	2.51	−282.5	−101.2	[40]
	240	1.47	0.756	0.794	1.04	5.24	−242.0	−99.50	[43]
	386	1.48	0.927	1.34	0.880	2.99	−216.0	−163.6	[44]
d+12C	29.5	0.952	1.00	1.25	1.15	13.2	−459.6	−163.4	[45]
	34.4	1.12	1.08	1.23	1.17	1.85	−485.3	−157.6	[46]
	52.0	1.15	1.12	1.31	1.13	6.35	−472.0	−170.2	[47]
	56.0	0.905	1.02	1.13	1.13	12.4	−422.5	−148.0	[48]
	60.6	0.800	1.03	1.41	0.929	10.5	−418.5	−184.4	[49]
	77.3	0.909	1.02	1.30	1.07	1.47	−388.1	−169.0	[49]
	80.0	1.13	1.04	1.26	0.992	1.06	−392.9	−163.6	[50]
	90.0	0.972	1.08	1.31	1.49	1.85	−391.3	−168.7	[49]
	110	1.12	1.12	1.46	0.934	10.7	−371.3	−184.1	[51]
	120	1.12	1.14	1.50	0.825	16.2	−362.1	−186.3	[51]
	140	1.05	1.12	1.26	1.17	8.37	−327.6	−153.7	[52]
	170	1.39	1.49	1.63	1.59	8.43	−384.1	−194.7	[53]
	200	1.17	1.32	1.27	1.73	1.92	−297.7	−154.9	[54]

The individual parameter values of t_ri, N_r and N_i for α and deuteron are shown as symbols in figures 1 and 2, respectively. Their energy dependences, represented by equations (8)–(10) for the α particle and by equations (11)–(13) for deuteron, are shown as solid curves in figures 1 and 2. For comparison, the global values established in heavy-target region, which were reported in [15], were shown as dashed curves. As expected, the OMP parameters with a ¹²C target are systematically different from those with heavy targets.

$$\begin{equation} t_{\rm ri}^{\alpha }(e_{\rm c.m.})= 0.907 + 1.51{\rm e}^{-0.0112e_{\rm c.m.}}(1-{\rm e}^{-0.0238e_{\rm c.m.}}), \end{equation} \tag{ 8 }$$

$$\begin{equation} N_{r}^{\alpha }(e_{\rm c.m.})= 0.970-\frac{0.219}{1+{\rm e}^{(e_{\rm c.m.}-60.0)/10.0}} \end{equation} \tag{ 9 }$$

$$\begin{equation} N_{i}^{\alpha }(e_{\rm c.m.})= 1.47-\frac{0.685}{1+{\rm e}^{(e_{\rm c.m.}-60.0)/10.0}} \end{equation} \tag{ 10 }$$

$$\begin{equation} t_{\rm ri}^{{\rm d}}(e_{\rm c.m.})= 1.08, \end{equation} \tag{ 11 }$$

$$\begin{equation} N_{r}^{{\rm d}}(e_{\rm c.m.}) = 1.82- 0.815{\rm e}^{-e_{\rm c.m.}^2/11750} \end{equation} \tag{ 12 }$$

$$\begin{equation} N_{i}^{d}(e_{\rm c.m.})= 1.93-\frac{1.02}{1+{\rm e}^{(e_{\rm c.m.}-73.0)/80.8}}. \end{equation} \tag{ 13 }$$

The e_c.m. in equations (8)–(13) are

$$\begin{equation} e_{\rm c.m.}= \frac{E_{\rm c.m.}}{A_{\rm P}}=\frac{A_{\rm T}}{A_{\rm P}(A_{\rm P}+A_{\rm T})}E_{\rm lab}. \end{equation} \tag{ 14 }$$

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3.2. Validity of the energy-dependent potentials

We check the energy-dependent potential parameters described in the previous section by examining their agreement with angular distributions of elastic scattering cross sections in figures 3 and 4, and with total reactions cross sections (σ_R) in figure 5. Note that these σ_R data were not included in the derivation of these potentials. The scattering angles in figure 3 are translated from the usual θ_c.m. to Θ_c.m., which was introduced in [15]:

$$\begin{equation} \Theta _{\rm c.m.}=\big[1+E_{\rm Lab}^{1/4}\times {}{\rm e}^{-\theta _{\rm c.m.}/w} \big]\times \theta _{\rm c.m.}, \end{equation} \tag{ 15 }$$

where w = 30° was used. In order to see the effects of these translations and more details in comparisons between the OM calculations and experimental data, we show in figure 4 the cases for α and deuteron elastic scattering from ¹²C at 26 and 60 MeV/nucleon without transitions of scattering angles.

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Comparisons between the present energy-dependent potentials and the systematic potentials by Dabrowski and Freindl for α [21] (the DF potential) and by An and Cai [27] for deuteron (the AC potential) were also made in figures 3–5, presented as dashed and dotted curves, respectively. The systematics of the DF potential was derived for target mass 12 ⩽ A_T ⩽ 208 and for incident energy 90 ⩽ E_lab ⩽ 172 MeV. Within this energy range, the DF systematics gives close angular distributions to the present potential but it is inferior to the latter in both of the low and high energy end. It also overestimates the σ_R data over the whole energy range studied here. The AC potential was derived for 12 ⩽ A_T ⩽ 208 and E_lab ⩽ 183 MeV. Within the whole energy range, the AC systematics gives close angular distributions as the present potential but it also systematically overestimates the σ_R data. Our energy-dependent potential parameters, on the other hand, account satisfactorily for both AD and σ_R data within the incident energy of 10–100 MeV/nucleon.

It is arguable whether the present prescription of local density approximation is suitable to light nuclei, say, A < 40. On the one hand, extensive studies have been made for scattering and reactions on light targets with the JLM model [59], including both stable nuclei, such as ⁷Li, ⁹Be, ¹⁰B, ¹²C, ¹³C and ¹⁶O [60–66], and unstable nuclei, such as ⁶He, ⁸He and ¹¹Be [61, 67, 68], which suggest that the present LDA is a sound hypothesis; on the other hand there is evidence that the LDA is not optimum for light targets [16]. In figure 6 we compare results of optical model calculations using the JLMB model potentials with experimental data for proton elastic scattering from ¹²C at 22.65 MeV [69] and 49.4 MeV [70]. Clearly the JLMB model does not account well for proton scattering from ¹²C. However, the agreement with experimental data can be much improved by renormalizing the real and imaginary potentials by N_r ∼ 1.1 and N_i ∼ 0.8, respectively. This N_i value is in close agreement with the systematics found in [60] for light targets with the JLM model.

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There are two approaches to construct α and deuteron potentials from nucleon potentials with a ¹²C target using the single-folding model: (1) find proton and neutron potentials that best reproduce the elastic scattering and total reaction cross-sectional data with ¹²C over the energy range in question, and use these potentials to construct the SF model potentials; (2) apply the JLMB model directly, knowing it is not satisfactory for ¹²C when unrenormalized but can be improved by appropriate renormalization, and search for t_ri, N_r and N_i values that best reproduce α and deuteron scattering data. In this paper we adopt the second approach, which has one procedure less than the first one. The price to pay is that the resulting renormalization factors are not directly connected to their fundamental basis, namely the effect of internal motions of nucleons in the projectile (or non-locality effect as discussed by Jackson and Johnson [71]), three-body terms and Pauli effects, as discussed by Perkin et al [32].

In summary, angular distributions of differential cross sections of α and deuteron elastic scattering from ¹²C with incident energies from 10 to 100 MeV/nucleon were analyzed with a single-folding model. Energy-dependent optical model potentials were obtained for the α+¹²C and deuteron+¹²C systems. These potentials account well for both elastic scattering angular distributions and total reaction cross sections within the energy range studied. The potentials reported in this paper have been used in systematic studies of breakup effects in ⁶Li elastic scattering from ¹²C in a CDCC approach [8].

DYP thanks Professor R S Mackintosh for helpful suggestions. This work is partly supported by the National Basic Research Program of China (grant no. 2007CB815002), the National Natural Science Foundation of China (grant nos 11035001, 10775003, 10735010 and 10975006) and China Postdoctoral Science Foundation (grant no. 20100470133).