A theoretical study on different cluster configurations of the 9Be nucleus by using a simple cluster model

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

A theoretical study on different cluster configurations of the ⁹Be nucleus by using a simple cluster model

M. Aygun，

Z. Aygun

Nuclear Science and Techniques

Vol.28, No.6

Article number 86

Published in print 01 Jun 2017

Available online 06 May 2017

DOI：10.1007/s41365-017-0239-2

46900

In this study, a comprehensive investigation on different cluster configurations of the ⁹Be nucleus is performed with a simple cluster approach. With this goal, the elastic scattering angular distributions of ⁹Be by ²⁷Al, ²⁸Si, ⁶⁴Zn, ¹⁴⁴Sm, ²⁰⁸Pb, and ²⁰⁹Bi target nuclei are reanalyzed for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He and n + ⁸Be cluster configurations of the ⁹Be projectile within the framework of the optical model. The theoretical results are compared with each other as well as the experimental data. The results provide an opportunity for a test of different cluster configurations in explaining the elastic scattering of ⁹Be nucleus.

Cluster structureOptical modelDouble folding modelElastic scattering

1 Introduction

Cluster structure is an important feature observed in both stable and unstable nuclei. For example, ¹¹Li, known as a halo nucleus, is considered as a core (⁹Li) and two-valence neutrons [1]. Or in stable nuclei, it is assumed that ¹²C and ¹⁶O nuclei have a α cluster structure. A large number of experimental and theoretical studies have been carried out to examine the cluster structures of nuclei [2-5]. Experimental techniques have been applied to display cluster cases in nuclei. For theoretical analysis, the Bloch-Brink α-cluster model (ACM), the antisymmetrized molecular dynamics (AMD), and the generator coordinate method (GCM) have been improved [6]. Therefore, it can be said that the cluster feature of a nucleus is an important parameter to examine the structure of a nucleus, to study cluster decay, break-up reactions, and stellar nucleosynthesis, to constitute different configurations with elements, and to understand the processes in nuclear astrophysics [7-9].

Recently, Aygun [10] applied a simple cluster method to a ¹²Be nucleus. He investigated different cluster structures of the ¹²Be nucleus within the optical model (OM). He reported the theoretical results in explaining the experimental data. We think that this simple method will be interesting in applying different nuclei. With this goal, in the present study, we focus on the theoretical analysis of the existing cluster models of the ⁹Be nucleus over literature.

In this work, we investigate α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be [11-13] structure models of the ⁹Be nucleus in terms of a simple cluster model. We obtain elastic scattering angular distributions of ⁹Be by ²⁷Al, ²⁸Si, ⁶⁴Zn, ¹⁴⁴Sm, ²⁰⁸Pb, and ²⁰⁹Bi target nuclei by using the double folding (DF) model based on the OM. We compare the theoretical results with the experimental data. Thus, the similarities and differences between cluster models applied in the calculations of ⁹Be nucleus are determined.

In the next section, a brief description of theoretical calculation is given. The results and discussion are define in Sect. 3. Section 4 is devoted to our summary and conclusions.

2 The calculation process

2.1 The optical model

The optical potential, which consists of real (V(r)) and imaginary (W(r)) potentials, is parameterized by

V optical (r) = V (r) + i W (r) .

(1)

To determine the real part of the optical potential, the DF model is used. The DF potential is obtained by means of the density distributions of projectiles and target together with an effective nucleon-nucleon interaction potential ν(_NN). In this manner, the DF potential is shown by

V_{DF} (r) = \int d r_{1} \int d r_{2} ρ_{P} (r_{1}) ρ_{T} (r_{2}) ν_{NN} (r_{12}),

(2)

where r₁₂=r-r₁+r₂, ν_NN(r₁₂) is the effective NN interaction, and ρP(r₁) and ρT(r₂), respectively, are the density distributions of projectile and target.

In order to make a comparative study, we have used five different density distributions for the ⁹Be nucleus. Each of these densities is explained in the following. On the other hand, for the densities of ²⁷Al, ²⁸Si, ²⁰⁸Pb, and ²⁰⁹Bi target nuclei, two-parameter Fermi (2pF) density distribution has been used, which is given by

ρ (r) = \frac{ρ_{0}}{1 + exp (\frac{r - c}{z})} .

(3)

ρ₀, c, and z parameters have been presented in Table 1.

The parameters of 2pF density distributions of ²⁷Al, ²⁸Si, ²⁰⁸Pb, and ²⁰⁹Bi nuclei.

2pF
Nucleus	c (fm)	z (fm)	ρ₀ (fm)^-3	Ref.
²⁷Al	2.84	0.569	0.2015	[14]
²⁸Si	3.15	0.475	0.175	[15]
²⁰⁸Pb	6.62	0.551	0.1600	[14]
²⁰⁹Bi	6.75	0.468	0.154887	[16]

The density distributions of ⁶⁴Zn and ¹⁴⁴Sm target nuclei are taken from the Hartree-Fock-Bogolubov (HFB) method based on the BSk2 Skyrme force [17].

For ν_NN, we have used the most common one, the M3Y nucleon-nucleon (Michigan 3 Yukawa) realistic interaction, which is formulated as

\begin{array}{l} ν_{NN} (r) = 7999 \frac{\exp (- 4 r)}{4 r} & - 2134 \frac{\exp (- 2.5 r)}{2.5 r} \\ + J_{00} (E) δ (r) MeV, \end{array}

(4)

where J₀₀(E) is the exchange term given by

J_{00} (E) = 276 (1 - 0.005 E_{Lab} / A_{P}) {MeV fm}^{3},

(5)

where E_Lab and A_P are the laboratory energy and mass number of the projectile, respectively. Finally, the imaginary part of the optical potential is assumed in Woods-Saxon (WS) form

W (r) = W_{0} f (r, R_{w}, a_{w}),

(6)

f (r, R_{w}, a_{w}) = [1 + exp (X {)]}^{- 1}, X = (r - R_{w}) / a_{w},

(7)

where $R_{w} = r_{w} (A_{P}^{1 / 3} + A_{T}^{1 / 3})$ and A_P and A_T are the mass numbers of the projectile and target nuclei, respectively. The code FRESCO has been used in OM calculations [18].

2.2 Simple parametrization of structure models of ⁹Be nucleus

Here, various cluster models of the ⁹Be are evaluated within a different approach. With this goal, it is assumed that ⁹Be nucleus consists of α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be systems. The density distributions of ⁹Be for these models are obtained and applied to produce the real potential in the DF model calculations based on the OM. However, the imaginary part of the optical potential is taken as the WS potential. The theoretical calculation of each system is conducted in the same form. We should say that we have not obtained a new density distribution. We have used the existing density distributions in the literature with only a simple approach.

2.2.1 α + α + n system

Firstly, we concentrate on the α + α + n cluster structure of the ⁹Be nucleus. In this manner, we assume the density distribution of ⁹Be in the following form

ρ_{^{9} Be} (r) = ρ_{α} (r) + ρ_{α} (r) + ρ_{n} (r) .

(8)

We use different density distribution for each α density. These densities, respectively, are

ρ_{α} (r) = 0.4229 exp (- 0.7024 r^{2})

(9)

and

ρ_{α} (r) = 4 (\frac{3 π b^{2}}{4})^{- 3 / 2} exp (- \frac{4 r^{2}}{3 b^{2}}),

(10)

where b=1.28 fm [15, 19]. The density distribution of 1n-halo is given by [20, 21]

ρ_{n} (r) = (\frac{1}{γ \sqrt{π}})^{3} exp (- r^{2} / γ^{2}),

(11)

where γ is adjusted to reproduce the experimental value for the rms radius of ⁹Be.

2.2.2 d + ⁷Li system

Secondly, the density distribution of ⁹Be projectile is obtained as the sum of densities of d and ⁷Li nuclei shown by

ρ_{^{9} Be} (r) = ρ_{d} (r) + ρ_{^{7} Li} (r) .

(12)

In this way, the density of d is in the following form

ρ_{d} (r) = ρ_{0} exp (- ϱ r^{2}),

(13)

where ρ₀=0.0992 fm^-3 and $ϱ$ =0.424 fm^-2 [22]. ⁷Li density is parameterized as [23]

ρ_{^{7} Li} (r) = 0.1387 (1 + 0.1673 r^{2}) exp (- 0.3341 r^{2}) .

(14)

2.2.3 ³H + ⁶Li system

The ⁹Be nucleus can be considered as a cluster form of ³H and ⁶Li nuclei. In this manner, ⁹Be density is taken as

ρ_{^{9} Be} (r) = ρ_{^{3} H} (r) + ρ_{^{6} Li} (r) .

(15)

³H density distribution is evaluated as the variational Monte Carlo (VMC) density obtained by using the Argonne v18 (AV18) two-nucleon and Urbana X three-nucleon potentials (AV18+UX) [24]. However, ⁶Li density is conducted as [25]

\begin{array}{l} ρ_{^{6} Li} (r) = & 0.203 exp (- 0.3306 r^{2}) \\ + (- 0.0131 + 0.001378 r^{2}) exp (- 0.1584 r^{2}) . \end{array}

(16)

2.2.4 ³He + ⁶He system

Another cluster structure of the ⁹Be nucleus is in the form of ³He and ⁶He nuclei. Thus, ⁹Be density can be written as

ρ_{^{9} Be} (r) = ρ_{^{3} He} (r) + ρ_{^{6} He} (r) .

(17)

³He density distribution is given by [26]

ρ_{^{3} He} (r) = 0.2201 exp (- 0.5505 r^{2}) .

(18)

⁶He density is evaluated in the following form

ρ_{^{6} He} (r) = ρ_{0} \exp (- β r^{2}),

(19)

where β is adjusted to reproduce the experimental value for the rms radius of the ⁶He=2.54 fm. ρ₀ can be obtained from the normalization condition

\int ρ (r) r^{2} d r = \frac{A}{4 π},

(20)

where A is the mass number.

2.2.5 n + ⁸Be system

Finally, the ⁹Be nucleus is thought as a n + ⁸Be cluster model. Thus, ⁹Be density is the sum of n and ⁸Be densities parameterized as

ρ_{^{9} Be} (r) = ρ_{n} (r) + ρ_{^{8} Be} (r) .

(21)

The density distribution of 1n-halo is the same as Eq. (11). The VMC density distribution for the ⁸Be nucleus is used [24].

3 Results and Discussion

We have investigated α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster configurations known for ⁹Be by using the DF model based on the OM. In this context, we have analyzed the elastic scattering data of six different nuclear reactions which consist of ²⁷Al, ²⁸Si, ⁶⁴Zn, ¹⁴⁴Sm, ²⁰⁸Pb, and ²⁰⁹Bi nuclei, in order to make a comparative study of the interactions with light, medium, and heavy target nuclei of ⁹Be. For a more comprehensive analysis, we have also investigated the elastic scattering data for different incident energies that can be obtained from the literature of the reactions analyzed with this work. While the real potential is obtained via the DF model, the imaginary potential has been assumed as WS potential. To acquire good agreement results with the experimental data, we have researched the normalization constant (NR) for the real part and W₀, rw, aw potential parameters for the imaginary part. In this context, the values of NR and W₀, rw, aw parameters have been listed in Table 2 and Table 3, respectively.

The NR values obtained with the DF calculations for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster structures of the ⁹Be nucleus in the analysis of the ⁹Be + ²⁷Al, ²⁸Si, ⁶⁴Zn, ¹⁴⁴Sm, ²⁰⁸Pb and ²⁰⁹Bi reactions.

System	Energy(MeV)	NR
		α+α+n	d+⁷Li	³H+⁶Li	³He+⁶He	n+⁸Be
⁹Be + ²⁷Al	22	0.66	0.96	0.67	0.56	0.40
	25	0.63	0.80	0.68	0.50	0.41
	32	1.00	1.00	1.00	0.84	0.82
	35	1.00	1.03	1.00	0.75	0.83
	13	0.82	1.09	1.00	0.66	0.55
⁹Be + ²⁸Si	17	0.67	1.06	0.865	0.53	0.46
	23	0.81	1.064	1.00	0.67	0.63
	26	0.95	1.07	1.03	0.86	0.74
	30	0.978	1.07	1.00	0.88	0.80
	21	0.66	0.90	0.70	0.50	0.40
⁹Be + ⁶⁴Zn	23	0.575	0.66	0.58	0.435	0.33
	26	0.50	0.56	0.50	0.36	0.29
	28	0.69	0.83	0.84	0.53	0.47
	39	0.716	0.85	0.73	0.53	0.40
⁹Be + ¹⁴⁴Sm	41	0.745	0.93	0.78	0.55	0.43
	44	0.733	0.92	0.79	0.54	0.443
	48	0.78	1.00	0.90	0.60	0.48
	40	0.950	1.04	1.000	0.78	0.650
⁹Be + ²⁰⁸Pb	42	0.680	0.78	0.715	0.525	0.425
	47.2	0.678	0.765	0.690	0.515	0.420
	50	0.77	0.83	0.810	0.58	0.500
	39	1.16	1.36	1.21	1.00	0.91
⁹Be + ²⁰⁹Bi	40	1.17	1.15	1.30	0.98	0.88
	42	0.94	1.00	1.00	0.69	0.63
	44	0.645	0.80	0.62	0.51	0.51

The W₀ values of the imaginary potential for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster configurations of the ⁹Be nucleus in the analysis of the ⁹Be + ²⁷Al, ²⁸Si, ⁶⁴Zn, ¹⁴⁴Sm, ²⁰⁸Pb, and ²⁰⁹Bi reactions. In all the calculations, the Coulomb radius (R_c), rw, and aw values have been fixed as 1.25, 1.27, and 0.75, respectively.

System	Energy(MeV)	W₀ (MeV)
		α+α+n	d+⁷Li	³H+⁶Li	³He+⁶He	n+⁸Be
⁹Be + ²⁷Al	22	18.70	15.00	15.70	17.6	19.00
	25	18.90	15.30	15.80	20.6	20.00
	32	19.00	15.50	16.00	21.6	30.00
	35	23.40	18.50	20.75	22.6	30.70
	13	7.00	12.90	9.90	9.00	7.70
⁹Be + ²⁸Si	17	16.90	14.00	15.90	16.90	16.90
	23	17.00	14.10	16.20	17.60	19.10
	26	20.00	14.20	16.30	25.50	24.00
	30	22.00	14.30	16.80	26.80	27.10
	21	10.10	12.20	12.50	11.30	11.45
⁹Be + ⁶⁴Zn	23	10.20	12.30	12.60	8.80	11.50
	26	10.40	13.00	13.00	10.80	12.00
	28	10.70	13.50	15.50	12.80	12.50
	39	10.00	13.00	12.80	11.30	13.00
⁹Be + ¹⁴⁴Sm	41	10.94	15.40	14.00	12.00	14.00
	44	10.96	16.00	14.50	12.50	15.00
	48	12.00	16.10	15.00	13.00	16.00
	40	12.00	16.50	14.85	13.00	12.50
⁹Be + ²⁰⁸Pb	42	15.50	18.00	17.00	16.10	16.10
	47.2	18.00	21.00	21.00	19.00	18.50
	50	19.00	21.70	22.30	20.30	20.30
	39	10.00	16.00	13.90	10.40	8.90
⁹Be + ²⁰⁹Bi	40	10.20	18.10	14.00	10.80	9.00
	42	12.20	18.30	15.40	14.20	11.90
	44	16.50	19.90	20.10	16.90	13.90

Elastic scattering of the ⁹Be + ²⁷Al reaction has been investigated for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster systems at 22, 25, 32, and 35 MeV. The results shown in Fig. 1 have been compared with each other as well as the experimental data. It has been seen that all the theoretical results are in very good agreement with the data.

Fig. 1.

(Color online) The elastic scattering angular distributions for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster configurations of ⁹Be in the analysis of the ⁹Be + ²⁷Al system in comparison with the experimental data at 22, 25, 32, and 35 MeV. The experimental data are from Refs. [27-29].

Angular distributions of elastic scattering of ⁹Be on ²⁸Si have been studied for five different cluster configurations at incident energies of 13, 17, 23, 26, and 30 MeV. The theoretical results have been plotted comparatively in Fig. 2. It has been observed that the results have displayed an agreement behavior with the data.

Fig. 2.

(Color online) The elastic scattering angular distributions for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster cases of ⁹Be in the analysis of the ⁹Be + ²⁸Si system in comparison with the experimental data at 13, 17, 23, 26, and 30 MeV. The experimental data are from Refs. [29-31].

Elastic scattering of the ⁹Be + ⁶⁴Zn reaction has been analyzed for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster cases at 21, 23, 26, and 28 MeV. In this context, the results have been shown in Fig. 3. Agreement between theoretical results and experimental data is very good. Also, our results display very similar behavior to each other.

Fig. 3.

(Color online) The elastic scattering angular distributions for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster structures of ⁹Be in the analysis of the ⁹Be + ⁶⁴Zn system in comparison with the experimental data at 21, 23, 26, and 28 MeV. The experimental data are from Refs. [29, 32].

⁹Be elastic scattering by ¹⁴⁴Sm has been examined by using the DF model at 39, 41, 44, and 48 MeV. As seen from Fig. 4, agreement of the theoretical results with the data is almost excellent.

Fig. 4.

(Color online) The elastic scattering angular distributions for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster models of ⁹Be in the analysis of the ⁹Be + ¹⁴⁴Sm system in comparison with the experimental data at 39, 41, 44, and 48 MeV. The experimental data are from Refs. [29, 33, 34].

As heavy target nuclei, elastic scattering angular distributions of ⁹Be + ²⁰⁸Pb (at 40, 42, 47.2, and 50 MeV) and ⁹Be + ²⁰⁹Bi (at 39, 40, 42, and 44 MeV) reactions have been investigated for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster configurations of ⁹Be by using the DF model. The theoretical results have been plotted in Fig. 5 for ⁹Be + ²⁰⁸Pb and in Fig. 6 for ⁹Be + ²⁰⁹Bi. The results for both ⁹Be + ²⁰⁸Pb and ⁹Be + ²⁰⁹Bi systems are in good agreement with the data.

Fig. 5.

(Color online) The elastic scattering angular distributions for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster cases of ⁹Be in the analysis of the ⁹Be + ²⁰⁸Pb system in comparison with the experimental data at 40, 42, 47.2, and 50 MeV. The experimental data are from Ref. [35].

Fig. 6.

(Color online) The elastic scattering angular distributions for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster configurations of ⁹Be in the analysis of the ⁹Be + ²⁰⁹Bi system in comparison with the experimental data at 39, 40, 42, and 44 MeV. The experimental data are from Refs. [29, 35].

It is well known that NR, when applied to obtain good agreement results with the data, shows the success of the DF model [36]. With this goal, to evaluate the results of α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster configurations from another angle, we have shown the NR values for all the systems investigated in Table 2. Also, changes of NR values for the systems have been plotted comparatively in Fig. 7. We have observed that NR values of d + ⁷Li and ³H + ⁶Li cases are close to each other and are better than the NR values of the other systems. However, the worst NR values have been found for the n + ⁸Be cluster case. As a result of this, we can say that α + α + n, d + ⁷Li, and ³H + ⁶Li cluster cases are more suitable within different cluster configurations of the ⁹Be nucleus.

Fig. 7.

(Color online) The NR changes as a function of incident energy for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster cases of each system examined with this work. The solid line shown with NR=1 is to guide the eye.

In the present research, we have given the reaction cross-sections (σ) of the systems analyzed in Table 4. When we have examined the results, we have observed that d + ⁷Li and ³H + ⁶Li cluster cases have given very close values to each other. Also, we have noticed that d + ⁷Li and ³H + ⁶Li cluster structures have a larger cross-section than the other systems.

The σ values obtained for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster cases in the analysis of the ⁹Be + ²⁷Al, ²⁸Si, ⁶⁴Zn, ¹⁴⁴Sm, ²⁰⁸Pb, and ²⁰⁹Bi reactions.

System	Energy(MeV)	σ (mb)
		α+α+n	d+⁷Li	³H+⁶Li	³He+⁶He	n+⁸Be
⁹Be + ²⁷Al	22	1413.3	1377.9	1361.8	1400.5	1412.9
	25	1536.4	1488.3	1487.2	1561.2	1554.3
	32	1758.7	1694.7	1706.4	1802.2	1921.8
	35	1876.2	1798.6	1834.0	1859.5	1984.6
	13	366.4	437.2	412.4	399.1	384.0
⁹Be + ²⁸Si	17	965.8	934.2	956.9	963.5	968.8
	23	1377.9	1334.8	1372.6	1390.6	1425.9
	26	1558.9	1458.7	1498.7	1639.8	1626.7
	30	1709.7	1579.0	1623.1	1780.0	1792.1
	21	384.3	414.5	412.7	398.2	400.6
⁹Be + ⁶⁴Zn	23	593.4	623.9	628.9	558.1	610.7
	26	860.8	910.9	910.1	863.1	891.4
	28	1046.9	1102.4	1146.8	1088.9	1094.1
	39	762.0	818.1	812.8	784.7	816.7
⁹Be + ¹⁴⁴Sm	41	944.3	1028.7	1000.9	960.7	1001.6
	44	1147.6	1251.8	1222.6	1177.5	1234.0
	48	1411.0	1500.0	1480.6	1432.0	1497.5
	40	297.3	335.3	322.3	308.5	304.8
⁹Be + ²⁰⁸Pb	42	511.7	533.3	522.6	514.2	515.1
	47.2	1021.8	1053.4	1055.9	1030.6	1024.0
	50	1264.2	1286.8	1303.2	1276.3	1282.4
	39	164.0	207.6	192.4	167.2	155.6
⁹Be + ²⁰⁹Bi	40	239.3	304.7	274.3	245.7	228.1
	42	428.4	488.6	461.2	449.7	421.1
	44	659.5	694.5	696.9	661.3	624.7

As another comparison parameter of α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster cases, we have investigated χ²/N values. With this goal, we have calculated χ²/N values for each system according to the experimental error of around 10% and have listed the results in Table 5. We have observed that the χ²/N values are rather small in a general sense.

The χ²/N values calculated for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster structures in the analysis of the ⁹Be + ²⁷Al, ²⁸Si, ⁶⁴Zn, ¹⁴⁴Sm, ²⁰⁸Pb, and ²⁰⁹Bi reactions.

System	Energy(MeV)	χ²/N
		α+α+n	d+⁷Li	³H+⁶Li	³He+⁶He	n+⁸Be
⁹Be + ²⁷Al	22	0.7106	0.4661	0.5072	0.5526	0.8924
	25	1.2267	0.3399	0.3888	1.1110	1.0784
	32	0.7702	2.8317	0.8422	0.7360	1.2528
	35	2.7976	3.7862	2.8590	2.5997	3.2359
	13	0.3192	0.1815	0.1896	0.2674	0.2681
⁹Be + ²⁸Si	17	0.1268	0.3231	0.1644	0.1236	0.1160
	23	6.5501	7.5186	3.1606	3.2760	3.4991
	26	2.8418	2.5614	0.9783	1.9139	1.8879
	30	21.423	10.984	9.5054	13.561	12.561
	21	0.3289	0.5345	0.3735	0.3862	0.3564
⁹Be + ⁶⁴Zn	23	0.3722	0.3136	0.3317	0.3823	0.2922
	26	1.5801	1.7245	1.6326	1.7803	1.6315
	28	1.7144	1.9992	2.0108	1.6673	2.5068
	39	0.4279	0.4839	0.4126	0.4696	0.2974
⁹Be + ¹⁴⁴Sm	41	1.9833	1.9756	1.9868	2.2124	1.7976
	44	0.4520	0.4675	0.4194	0.5638	0.3572
	48	0.8970	0.9888	0.7446	0.8173	0.8738
	40	0.0358	0.0568	0.0450	0.0553	0.0727
⁹Be + ²⁰⁸Pb	42	0.1251	0.2156	0.1847	0.1461	0.1605
	47.2	0.4213	0.6075	0.4933	0.4574	0.5412
	50	3.0902	2.2716	2.9785	2.7373	4.5775
	39	0.0167	0.0464	0.0306	0.0108	0.0165
⁹Be + ²⁰⁹Bi	40	0.0284	0.0987	0.0417	0.0516	0.0897
	42	0.1852	0.1645	0.2286	0.1973	0.3453
	44	0.3487	0.6098	0.3950	0.4912	1.2496

4 Summary and Conclusions

We have reported the study of different cluster configurations of ⁹Be investigated by terms of a simple cluster method. With this goal, we have performed a different and new study to determine which cluster configurations of ⁹Be are more valid. The theoretical analysis has been carried out for α + α + n, d + ⁷Li, ³H + ⁶Li, ³He + ⁶He, and n + ⁸Be cluster cases by using the DF model within the scope of the OM. The elastic scattering results for each system have been plotted in figures. Also, NR values, the optical potential parameters, cross sections, and χ²/N values have been listed in tables. It has been seen that our results are in very good harmony with the experimental data in general sense. However, the α + α + n, d + ⁷Li, and ³H + ⁶Li results are close to each other and are better than the results of the other cluster cases. Additionally, we have noticed that the α + α + n, d + ⁷Li, and ³H + ⁶Li results are more compatible with the experimental data. However, we should say that the ³H + ⁶Li results are in more harmony with the data according to χ²/N values.

Consequently, we have applied a different and simple approach to the analysis of the internal structure of the ⁹Be nucleus within the framework of the DF model. We should say that we do not claim very precise results. We have observed that this method has given important results. We consider that this method would be useful and interesting in applying to cluster configurations of both different nuclei and ⁹Be.

References

[1]

K. Ikeda, T. Myo, K. Kato, H. Toki,

Di-neutron clustering and deuteron-like tensor correlation in nuclear structure focusing on 11Li

. arXiv:nucl-th/1007.2474v2