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Three -cluster description of the 12 C nucleus A.V. Malykh (JINR, BLTP) The work was done in collaboration with O.I. Kartavtsev, S.I. Fedotov

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1. Introduction The -particle is the most tightly bound nucleus, therefore the description in the framework of the -cluster model can be used for many systems. The -particle is the most tightly bound nucleus, therefore the description in the framework of the -cluster model can be used for many systems. The simplest system is 8 Be which has a near threshold resonance 8 Be(0 + 1 ) (with energy E 2 =92.04 keV and γ = 5.47 ± 0.25 eV) and therefore has two- -cluster structure. The simplest system is 8 Be which has a near threshold resonance 8 Be(0 + 1 ) (with energy E 2 =92.04 keV and γ = 5.47 ± 0.25 eV) and therefore has two- -cluster structure. 12 C also has near threshold resonance 12 C(0 + 2 ) which leads to -cluster structure of this state. This state (Hoyle state) was predicted to explain abundance of heavy elements in the universe. 12 C also has near threshold resonance 12 C(0 + 2 ) which leads to -cluster structure of this state. This state (Hoyle state) was predicted to explain abundance of heavy elements in the universe. Hoyle state 12 C(0 + 2 ) and reaction mechanism Hoyle state 12 C(0 + 2 ) and reaction mechanism Resonance mechanism Resonance mechanism The reaction of formation of the 12 C nucleus in the triple- low-energy collisions 3 8 Be + 12 C(0 + 2 ) 12 C + γ is of key importance for stellar nucleosynthesis as an unique possibility for helium burning that allows further synthesis of heavier elements. is of key importance for stellar nucleosynthesis as an unique possibility for helium burning that allows further synthesis of heavier elements. Nonresonance mechanism Nonresonance mechanism Experimental results show, that some of the lowest 12 C states decay in - particles channels. Experimental results show, that some of the lowest 12 C states decay in - particles channels.

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12 C lowest energy levels 0 + states 0 + states Ground state is strongly bounded, therefore it has non-cluster structure. Fixing energy of the ground state gives a restriction on the behavior of the effective potentials at short distances. E gs = E(0 + 1 ) =-7.2747MeV, R (1) exp = 2.48 ± 0.22 fm R (1) exp = 2.48 ± 0.22 fm Excited state has α-cluster structure. E r = E(0 + 2 ) =0.3795 MeV, Г = 8.5 ± 1.0 eV Г = 8.5 ± 1.0 eV M 12 = 5.48 ± 0.22 fm 2 M 12 = 5.48 ± 0.22 fm 2 The wide resonance (0 + 3 state) was found in some experimental and theoretical works. E(0 + 3 ) =3.0 MeV, Г = 3.0 ±.7 MeV Г = 3.0 ±.7 MeV 1 + state has a non-α-cluster structure, but can decay only to α-particles therefore this decay is suppressed that lead to a very small width of this state despite large energy. The interesting point is to study the angular and energy correlation of three bosons being in the 1 + state. 1 + state has a non-α-cluster structure, but can decay only to α-particles therefore this decay is suppressed that lead to a very small width of this state despite large energy. The interesting point is to study the angular and energy correlation of three bosons being in the 1 + state. E( 1 + 1 ) =5.44 MeV, E( 1 + 1 ) =5.44 MeV, Г = 18.1 ± 2.8 eV Г = 18.1 ± 2.8 eV α + α + α

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2. Effective interactions The effective potentials must be determined as an input for the -cluster model All the effects сonnected with both the internal structure of -particles and the identity of nucleons are incorporated in the effective potential. The effective potential V (x) is a sum of the Coulomb interaction and local short-range Ali-Bodmer-type potentials V s (x) The effective potential V (x) is a sum of the Coulomb interaction and local short-range Ali-Bodmer-type potentials V s (x) Besides, the additional three-body potential V 3 (ρ) as a simple Gaussian function of the hyper-radius ρ Besides, the additional three-body potential V 3 (ρ) as a simple Gaussian function of the hyper-radius ρ is introduced to describe the effects beyond the three-cluster approximation. The studies of the three- scattering allow one to reduce the uncertainty in the two-body effective potential which can be hardly determined only from the two-body data.

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3. Aim Calculate the fine characteristics of the 12 C(0 + ) states (energies and root- mean-square (rms) radii of the ground (0 + 1 ) and excited (0 + 2 ) states, an extremely narrow width Г of the 0 + 2 state and monopole transition matrix element M 12 ). Calculate the fine characteristics of the 12 C(0 + ) states (energies and root- mean-square (rms) radii of the ground (0 + 1 ) and excited (0 + 2 ) states, an extremely narrow width Г of the 0 + 2 state and monopole transition matrix element M 12 ). Study dependence on the effective two- and three-body potentials Study dependence on the effective two- and three-body potentials Adjust the parameters of the two-body effective Ali-Bodmer-type potentials to fix the position and width of 8 Be at the experimental values and to fit the s-wave phase shift at low energy Adjust the parameters of the two-body effective Ali-Bodmer-type potentials to fix the position and width of 8 Be at the experimental values and to fit the s-wave phase shift at low energy Adjust the parameter of the three-body effective interactions to fix the energies of the ground 0 + 1 and excited 0 + 2 states and the rms radius of the ground state R (1) to known experimental data. Adjust the parameter of the three-body effective interactions to fix the energies of the ground 0 + 1 and excited 0 + 2 states and the rms radius of the ground state R (1) to known experimental data. Study the reaction mechanism of formation 12 C at low energies above the two-body resonance ( 8 Be). Study the reaction mechanism of formation 12 C at low energies above the two-body resonance ( 8 Be).

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4. Method The Schrödinger equation (ħ = m = e = 1) in the scaled Jacobi coordinates x, y for three -particles reads In the following it is convenient to use the hyperspherical coordinates 0 ρ <, 0 i π /2, and 0 θ i π defined as

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4.1 Eigenfunctions on the hypersphere In order to solve both the eigenvalue and scattering problems for Eq. (3) the total wave function is expanded in a series on a discrete set of normalized eigenfunctions Φ n of the following equation on the hypersphere where

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4.2 System of HRE Given the expansion (5) of the total wave function, the Schrödinger equation (3) is reduced to the system of hyper-radial equations (HRE) where

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4.3 Numerical procedure to solve equation on the hypersphere The eigenvalues λ n (ρ) and the eigenfunction Ф n (ρ,, θ) are calculated by using the variational method. Basis consists of a set of the symmetric hyperspherical harmonics (SHH) H nm a set of the symmetric hyperspherical harmonics (SHH) H nm a set of the ρ-dependent symmetrized functions which are chosen to describe the + 8 Be configuration of the wave function a set of the ρ-dependent symmetrized functions which are chosen to describe the + 8 Be configuration of the wave function where i (x) is a few Gaussian functions and function allow to describe the two-body wave function within the range of the nuclear potential V s and in the under-barrier region. Matrix elements Q nm (ρ), and P nm (ρ) are calculated by

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4.4 Results of the variational calculation 4.4 (a) Eigenpotentials The eigenpotentials Un= [4λ n (ρ) + 15/4]/ρ 2 of the first, second and third channels are plotted with red, green, pink lines, respectively. The blue line shows the two-body asymptotic expression E 2α +q/ρ. E 2 = 92.04 ± 0.05 keV E 2 = 92.04 ± 0.05 keV q=13.30 KeV·fm q=13.30 KeV·fm The inset shows the effective potential near the turning point ρ t.

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4.4 (b) The first channel eigenfunction Ф 1 Large ρ (ρ =45fm) The hyperradial function has the two-cluster structure that confirms the sequential mechanism of 0 + 2 state decay with formation of α+ 8 Be at the first step. The hyperradial function has the two-cluster structure that confirms the sequential mechanism of 0 + 2 state decay with formation of α+ 8 Be at the first step. Intermediate ρ (ρ =15fm) Intermediate ρ (ρ =15fm) The two-cluster structure widens; and the most important are the equilateral-triangle and the linear configuration. The two-cluster structure widens; and the most important are the equilateral-triangle and the linear configuration. Small ρ (ρ =5 fm) Small ρ (ρ =5 fm) The most important is the equilateraltriangle configuration. The most important is the equilateral-triangle configuration.

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4.5 Boundary conditions for HRE Properties of the ground 0 + 1 state and the excited 0 + 2 resonance are determined by solving the eigenvalue problem (at E 0) for HRE (8), respectively. For the eigenvalue problem the hyperradial functions for the ground state f n (1) (ρ) have to be normalized and therefore For the eigenvalue problem the hyperradial functions for the ground state f n (1) (ρ) have to be normalized and therefore For the scattering problem at energy above the two-body resonance (E >E 2 α ), in view of 2-cluster asymptotic expression of the effective potential in the first channel U 1 (ρ) = [4λ 1 (ρ) + 15/4]/ρ 2 E 2 α +q/ρ (as shown in Fig. 1) the hyper-radial function f 1 (E) (ρ) can be written as For the scattering problem at energy above the two-body resonance (E >E 2 α ), in view of 2-cluster asymptotic expression of the effective potential in the first channel U 1 (ρ) = [4λ 1 (ρ) + 15/4]/ρ 2 E 2 α +q/ρ (as shown in Fig. 1) the hyper-radial function f 1 (E) (ρ) can be written as in the range of hyper-radius values ρ t. Here the wave number in the first channel k = E-E, F 0 (η k) and G 0 (η k) are the Coulomb functions with the parameter η=8/(3k) and δ E is the scattering phase shift. All other boundary conditions equal to zero.

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Characteristics of 12 C states Energy of the ground state E gs Energy of the ground state E gs The resonance position E r and width Г as well as the non-resonant phase shift δ bg are defined by fitting the calculated near resonance phase shift δ E to the Wigner dependence on energy The resonance position E r and width Г as well as the non-resonant phase shift δ bg are defined by fitting the calculated near resonance phase shift δ E to the Wigner dependence on energy Root-mean-square (RMS) radii of the ground (i=1) and excited (i=2) states read Root-mean-square (RMS) radii of the ground (i=1) and excited (i=2) states read The monopole transition matrix element takes the form The monopole transition matrix element takes the form A sums is taken over N t nucleons and over N p protons, R cm is the center-of- mass position vector and ρ 2 i equals to

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5. Numerical results 5.1 Two-body effective potentials Calculations have been performed with potentials which reproduce the experimental value of the resonance ( 8 Be) energy E 2 =92.04 keV Modified Ali-Bodmer potentials S. Ali and A. R. Bodmer. Nucl. Phys., 80:99, 1966. Modified Ali-Bodmer potentials S. Ali and A. R. Bodmer. Nucl. Phys., 80:99, 1966. A set of potentials 1-11 with parameters μ r -1 = 1.53 fm and μ a -1 = 2.85 fm is constructed to study the dependence on the 8 Be width γ, which vary within the interval from 5.1 eV to 8.53 eV (this interval corresponds to earlier experimental measurements of γ = 6.8 ± 1.7 eV). A set of potentials 1-11 with parameters μ r -1 = 1.53 fm and μ a -1 = 2.85 fm is constructed to study the dependence on the 8 Be width γ, which vary within the interval from 5.1 eV to 8.53 eV (this interval corresponds to earlier experimental measurements of γ = 6.8 ± 1.7 eV). The potential 12 with parameters μ r = 0.7 fm -1 and μ a = 0.475 fm -1 is used to illustrate the dependence on the potential range. The potential 12 with parameters μ r = 0.7 fm -1 and μ a = 0.475 fm -1 is used to illustrate the dependence on the potential range. The potentials 13-15, that fit the two-body experimental data The potentials 13-15, that fit the two-body experimental data E 2 =92.04 keV, γ = 5.47 ± 0.25 eV E 2 =92.04 keV, γ = 5.47 ± 0.25 eV Fit the experimental phase shift up to the energy 12 MeV Fit the experimental phase shift up to the energy 12 MeV

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5.2 Parameters of the two-body potentials Parameters of the two-bodyeffective potentials providing the - resonance position and width Parameters of the two-body effective potentials providing the - resonance position and width γ = 6.8 ± 1.7 eV (potentials 1-12) γ = 5.47 ± 0.25 eV (potentials 13-15). Parameters of the two-bodyeffective potentials providing the - resonance position and width Parameters of the two-body effective potentials providing the - resonance position and width γ = 6.8 ± 1.7 eV (potentials 1-12) γ = 5.47 ± 0.25 eV (potentials 13-15).

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5.3 Two-body phase shift The experimental and calculated α-α s-wave elastic-scattering phase shift δ versus the center-of mass energy E (MeV) for the two-body potentials 1, 2, 6, 9, 10, and 11 (top to bottom, left panel) and for the two-body potentials 2, 13, 14, and 15 providing the 8 Be width within the range of the experimental uncertainty 5.57 eV < γ < 5.82 eV (right panel)

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5.4 Results of the solution HRE (a) The one-term three-body effective potential (V 1 =0) Fixed at the experimental values Fixed at the experimental values E gs = -7.2747 MeV E r = 0.3795 MeV Not fixed at the experimental values Not fixed at the experimental values R (1) = 2.48 ± 0.22 fm Г = 8.5 ± 1.0 eV M 12 = 5.48 ± 0.22 fm 2 Fixed at the experimental values Fixed at the experimental values E gs = -7.2747 MeV E r = 0.3795 MeV Not fixed at the experimental values Not fixed at the experimental values R (1) = 2.48 ± 0.22 fm Г = 8.5 ± 1.0 eV M 12 = 5.48 ± 0.22 fm 2

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(b) The three-body effective potential (V 10) Relation between the parameters of the three-body effective potentials for the two-body potentials 13, 14 and 15 plotted by solid, dashed, dotted lines, respectively. The calculated M 12 -Г and R (2) -Г dependences for the two-body potentials 6, 7, 9, 13, 14, and 15. The point with errorbars shows the experimental data Г = 8.5 ± 1.0 eV M 12 = 5.48 ± 0.22 fm 2 Fixed at the experimental values Fixed at the experimental values E gs = -7.2747 MeV E r = 0.3795 MeV R (1) = 2.48 ± 0.22 fm Not fixed at the experimental values Г = 8.5 ± 1.0 eV M 12 = 5.48 ± 0.22 fm 2 Fixed at the experimental values Fixed at the experimental values E gs = -7.2747 MeV E r = 0.3795 MeV R (1) = 2.48 ± 0.22 fm Not fixed at the experimental values Г = 8.5 ± 1.0 eV M 12 = 5.48 ± 0.22 fm 2

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Conclusion Confirmed that at low energies 0 + 2 -state decays by means of the sequential mechanism of decay 12 C + 8 Be 3 Confirmed that at low energies 0 + 2 -state decays by means of the sequential mechanism of decay 12 C + 8 Be 3 Determined the fine characteristics of the 0 + states 12 C nuclei (an extremely narrow width Г, the rms radius R (2) of the 0 + 2 state and monopole transition matrix element M 12 ) for a set of the two- and three-body effective potentials Determined the fine characteristics of the 0 + states 12 C nuclei (an extremely narrow width Г, the rms radius R (2) of the 0 + 2 state and monopole transition matrix element M 12 ) for a set of the two- and three-body effective potentials Adjusted the parameters of the two-body effective Ali-Bodmer-type potentials to fix the position and width of 8 Be at the experimental values and to fit the s-wave phase shift (the fit of d-wave and p-wave phase shifts still need to be added) Adjusted the parameters of the two-body effective Ali-Bodmer-type potentials to fix the position and width of 8 Be at the experimental values and to fit the s-wave phase shift (the fit of d-wave and p-wave phase shifts still need to be added) Adjusted the parameters of the three-body effective potentials to fix the energies of the 0 + states 12 C nuclei (E gs, E r ), and rms radii of the 0 + 2 state (R (1) ) at the experimental values Adjusted the parameters of the three-body effective potentials to fix the energies of the 0 + states 12 C nuclei (E gs, E r ), and rms radii of the 0 + 2 state (R (1) ) at the experimental values Study the dependence of the characteristics of 12 C on the effective two- and three- body potentials Study the dependence of the characteristics of 12 C on the effective two- and three- body potentials The calculation of characteristics of the 0 + 3 state is still needed The calculation of characteristics of the 0 + 3 state is still needed Some results discussed in presentation have been published in Some results discussed in presentation have been published in Phys. Rev. C 70, 014006 (2004) Phys. Rev. C 70, 014006 (2004) Eur. Phys. J. A 26, 201-207(2005) Eur. Phys. J. A 26, 201-207(2005) and reported at and reported at UNISA-JINR Symposium ``Models and Methods in Few- and Many-Body Systems'' (6--9 February 2007, Skukuza, Kruger National Park, South Africa). UNISA-JINR Symposium ``Models and Methods in Few- and Many-Body Systems'' (6--9 February 2007, Skukuza, Kruger National Park, South Africa). 12 C system is intensively investigating by other groups 12 C system is intensively investigating by other groups D. V. Fedorov at al. Phys. Lett. B, 389, 631, (1996) N. N. Filikhin. Yad. Fiz., 63, 1612, (2000) N. N. Filikhin at al. J. Phys. G, 31, 1207, (2005) C. Kurokawa at al. Phys. Rev. C 71, 021301, (2005) Y.Funaki at al. Eur. Phys. J. A 24, 368, (2005) K.Arai Phys. Rev. C 74, 064311, (2006) Y. Suzuki at al. Nucl-th/0703001 R. Alvarez-Rodriguez at al. Nucl-th/0703001 D. V. Fedorov at al. Phys. Lett. B, 389, 631, (1996) N. N. Filikhin. Yad. Fiz., 63, 1612, (2000) N. N. Filikhin at al. J. Phys. G, 31, 1207, (2005) C. Kurokawa at al. Phys. Rev. C 71, 021301, (2005) Y.Funaki at al. Eur. Phys. J. A 24, 368, (2005) K.Arai Phys. Rev. C 74, 064311, (2006) Y. Suzuki at al. Nucl-th/0703001 R. Alvarez-Rodriguez at al. Nucl-th/0703001

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