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.

Three Three -cluster description of the -cluster description of the 1212C C nucleus nucleus

A.V. MalykhA.V. Malykh

(JINR, BLTP)(JINR, BLTP)

The work was done in collaboration with The work was done in collaboration with

O.I. Kartavtsev, S.I. FedotovO.I. Kartavtsev, S.I. Fedotov

1. Introduction1. Introduction The The -particle is the most tightly bound nucleus-particle is the most tightly bound nucleus, therefore the , therefore the

description in the framework of the description in the framework of the -cluster model can be used for -cluster model can be used for many systems. many systems.

The simplest system isThe simplest system is 88Be which has a Be which has a near threshold resonancenear threshold resonance 88Be(0Be(0++

11)) (with energy E (with energy E22=92.04 keV and =92.04 keV and γγ == 5.47 5.47 ± 0.25 eV) and ± 0.25 eV) and therefore therefore hashas two-two--cluster structure.-cluster structure.

1212C also has C also has near threshold resonance near threshold resonance 1212C(0C(0++22 ) ) which leads to which leads to --

cluster structurecluster structure of this state. of this state. This state (Hoyle state) was This state (Hoyle state) was predicted to explain abundance of heavy elements in the predicted to explain abundance of heavy elements in the universeuniverse. .

Hoyle state Hoyle state 1212C(0C(0++22 ) ) and reaction mechanism and reaction mechanism

Resonance mechanismResonance mechanism The reaction of formation of the The reaction of formation of the 1212C nucleus in theC nucleus in the triple-triple- lowlow--energyenergycollisions 3 collisions 3 → → 88Be + Be + → → 1212C(0C(0++

22 ) ) → → 1212C + C + γγ is of key importance for stellar nucleosynthesis as ais of key importance for stellar nucleosynthesis as ann unique possibility unique possibility

for helium burning that allows further synthesis of heavier elements.for helium burning that allows further synthesis of heavier elements. Nonresonance mechanism Nonresonance mechanism

Experimental results show, that some of the lowest Experimental results show, that some of the lowest 1212CC states decay in states decay in -particles channels. -particles channels.

1212C lowest energy levelsC lowest energy levels 00+ + statesstates

Ground state is strongly boundedGround state is strongly bounded, , therefore it has non-cluster structure. therefore it has non-cluster structure. Fixing energy of the ground state gives a Fixing energy of the ground state gives a restriction on the behavior of the restriction on the behavior of the effective potentials at short distances.effective potentials at short distances.

EEgsgs = E(0= E(0++11) ) =-7.2747MeV,=-7.2747MeV,

R R (1)(1)expexp = 2.48 = 2.48 ±± 0.22 fm 0.22 fm

Excited state Excited state has has αα-cluster structure. -cluster structure. EErr = E(0= E(0++

22)) =0.3795 MeV =0.3795 MeV, , ГГ == 8.5 8.5 ±± 1.0 eV 1.0 eV MM12 12 = = 5.48 5.48 ±± 0.22 0.22 fm fm22

The wide resonanceThe wide resonance (0 (0++3 3 state) was state) was

found in some experimental and found in some experimental and theoretical works.theoretical works.

E(0E(0++33)) = =3.03.0 MeV MeV, ,

Г Г == 3 3.0 .0 ±± .7 MeV .7 MeV 11++ state state has a non- has a non-αα-cluster structure, but -cluster structure, but

can decay only to can decay only to αα-particles therefore this -particles therefore this decay is suppressed that lead to a very small decay is suppressed that lead to a very small width of this state despite large energy. The width of this state despite large energy. The interesting point is to study the angular and interesting point is to study the angular and energy correlation of three bosons being in energy correlation of three bosons being in the 1the 1+ + state.state.

E(E(11++11)) = =5.445.44 MeV MeV, ,

Г Г == 18 18.1 .1 ±± 2.8 eV 2.8 eV

α + α + α

2. Effective interactions2. Effective interactionsThe effective The effective − − potentialspotentials must be determined as an input must be determined as an input

for the for the -cluster-cluster modelmodelAll the effects сonnected with both the internal structure of All the effects сonnected with both the internal structure of --

particlesparticles and the identity of nucleons are incorporated in and the identity of nucleons are incorporated in the effective the effective − − potential. potential.

• • The effective The effective − − potentialpotential V (x) is a sum of the Coulomb V (x) is a sum of the Coulomb interaction and interaction and local local short-range Ali-Bodmershort-range Ali-Bodmer-type-type potential potentialss VVss(x) (x)

• • Besides, the additional Besides, the additional three-body potentialthree-body potential V V33(ρ) as a (ρ) as a simple Gaussian function of the hyper-radius ρsimple Gaussian function of the hyper-radius ρ

is introduced to describe the effects beyond the three-is introduced to describe the effects beyond the three-clustercluster approximation.approximation.

The studies of the three-The studies of the three- scattering allow one to reduce the scattering allow one to reduce the uncertainty in the two-body effective uncertainty in the two-body effective − − potential which potential which can be hardly determined only from the two-body data. can be hardly determined only from the two-body data.

3. Aim3. Aim Calculate the fine Calculate the fine characteristics of thecharacteristics of the 1212CC(0(0++)) states states (energies (energies

and root-mean-squareand root-mean-square (rms) radii of the ground (0(rms) radii of the ground (0++11)) and excited and excited

(0(0++22)) states, an extremely narrow widthstates, an extremely narrow width ГГ of the 0of the 0++

22 state and state and monopole transition matrix element monopole transition matrix element MM1212)) . .

Study dependence on the effective two- and three-body potentialsStudy dependence on the effective two- and three-body potentials

AdjustAdjust the parameters of the the parameters of the two-body effectivetwo-body effective Ali-Bodmer- Ali-Bodmer-type potentials to type potentials to fix the position and width of fix the position and width of 88BeBe at the at the experimental values and to experimental values and to fit the s-wave phase shiftfit the s-wave phase shift at low at low energyenergy Adjust the parameter of the Adjust the parameter of the three-body effectivethree-body effective interactions interactions to to fix the energies of the ground fix the energies of the ground 00++

11 and excited and excited 00++22 states and states and

the rms radius of the ground state the rms radius of the ground state RR(1)(1) to known experimental to known experimental data.data.

Study the reaction mechanism Study the reaction mechanism of formation of formation 1212C at low energiesC at low energies above the two-body resonanceabove the two-body resonance ( (88Be).Be).

4. 4. MethodMethod

The SchrThe Schröödinger equation (dinger equation (ħħ = m = e = 1) in the = m = e = 1) in the scaled Jacobi coordinatesscaled Jacobi coordinates x, yx, y for three for three --particles readsparticles reads

In the following it is convenient to use the In the following it is convenient to use the hyperspherical coordinates 0 ≤hyperspherical coordinates 0 ≤ ρ < ∞, 0 ≤ ρ < ∞, 0 ≤ i i ≤ ≤ ππ/2/2, and 0 ≤ θ, and 0 ≤ θi i ≤ ≤ ππ defined defined as as

4.1 4.1 Eigenfunctions on the Eigenfunctions on the hyperspherehypersphere

In order to solve both the eigenvalue and scattering In order to solve both the eigenvalue and scattering problems for Eq. (3) the total wave function is problems for Eq. (3) the total wave function is expanded in a seriesexpanded in a series

on a discrete set of normalized eigenfunctions Φon a discrete set of normalized eigenfunctions Φnn of of the following the following equation on the hypersphereequation on the hypersphere

wherewhere

4.2 System of HRE4.2 System of HRE

Given the expansion (5) of the total wave function, Given the expansion (5) of the total wave function, the Schrthe Schröödinger equation (3) is reduced to the dinger equation (3) is reduced to the system of hyper-radial equations (HRE)system of hyper-radial equations (HRE)

wherewhere

4.3 4.3 Numerical procedure Numerical procedure toto solvesolve equation equation on the hypersphereon the hypersphere

The The eigenvalueseigenvalues λλnn((ρρ)) and the eigenfunction and the eigenfunction ФФnn((ρρ, , , , θθ)) are are calculated calculated by by using the using the vvariationalariational methodmethod. Bas. Basisis consistsconsists of of

a set of the a set of the symmetric hyperspherical harmonicssymmetric hyperspherical harmonics (SHH) H(SHH) Hnmnm

a set of the a set of the ρρ-dependent -dependent symmetrisymmetrizedzed functions functions which are which are chosen to describe the chosen to describe the ++88Be configuration of the Be configuration of the wave wave functionfunction

where where ii(x)(x) is a is a few Gaussian functionsfew Gaussian functions and function and function

allow toallow to describedescribe the two-body wave the two-body wave functionfunction within the range within the range oof f thethe

nuclear potential nuclear potential VVss and iand in the under-barrier regionn the under-barrier region..Matrix elements Matrix elements QQnmnm((ρρ), and P), and Pnmnm((ρρ)) are calculated are calculated byby

4.4 Results of the variational calculation4.4 Results of the variational calculation4.4 (a) Eigenpotentials4.4 (a) Eigenpotentials

The eigenpotentials The eigenpotentials Un= Un= [[44λλnn((ρρ) + ) +

1515//4]4]//ρρ22 of the first, second and third of the first, second and third channels are plotted with red, green, channels are plotted with red, green, pink lines, respectively. The blue line pink lines, respectively. The blue line shows the two-body asymptotic shows the two-body asymptotic expression expression EE22αα+q+q//ρρ . .

• EE22 = 92.0 = 92.04 ± 0.054 ± 0.05 keV keV• q=13.30 KeVq=13.30 KeV··fmfm

The inset shows the effective potential The inset shows the effective potential near the turning point near the turning point ρρtt..

4.4 (b) The first channel eigenfunction 4.4 (b) The first channel eigenfunction ФФ11

LargeLarge ρρ ( (ρρ =45fm) =45fm) The hyperradial function has the two-cluster The hyperradial function has the two-cluster structure that structure that confirms the sequential mechanism of confirms the sequential mechanism of 00++

2 2 state decaystate decay with formation of with formation of αα++88Be at the first Be at the first step.step. IntermediateIntermediate ρρ ( (ρρ =15fm) =15fm) The two-cluster structure widens; and the most The two-cluster structure widens; and the most important are the equilateral-triangle and the linear important are the equilateral-triangle and the linear configuration.configuration.SmallSmall ρρ ( (ρρ =5 fm) =5 fm) The most important is the equilateralThe most important is the equilateral-triangle triangle configuration.configuration.

4.5 4.5 Boundary conditions for HREBoundary conditions for HRE

PProperties of the ground 0roperties of the ground 0++11 state and the excited state and the excited 00++

22 resonance resonance are determined by solvingare determined by solving the eigenvalue problem (at the eigenvalue problem (at E < E < 0) 0) and scatteringand scattering problem (at problem (at E > E > 0) for HRE (8), respectively.0) for HRE (8), respectively.

For the eigenvalue problemFor the eigenvalue problem the hyperradial functions for the the hyperradial functions for the groundground state state ffnn (1)(1) ((ρρ) have to ) have to be be normalized and thereforenormalized and therefore

For the scattering problemFor the scattering problem at energy above the two-body at energy above the two-body resonance (E >Eresonance (E >E22αα), in view of 2-cluster asymptotic expression), in view of 2-cluster asymptotic expression of the eof the effffective potential in the ective potential in the fifirstrst channel channel UU11((ρρ) = [) = [44λλ11((ρρ) + ) + 1515//4]4]//ρρ22 ≈≈ E E22αα+q+q//ρρ (as shown in Fig. 1) the hyper-radial (as shown in Fig. 1) the hyper-radial functionfunction ff11

(E)(E) ( (ρρ) can be written as) can be written as

in the range in the range of hyperof hyper-ra-radius values dius values ρρtt.. Here the wave number in Here the wave number in the the fifirst channelrst channel kk = = √√EE-E-E ,, FF00((ηη k) and G k) and G00((ηη k) k) are the are the Coulomb functions with the parameterCoulomb functions with the parameter ηη==88//((√√33kk)) and and δδEE is the is the scatteringscattering phase shift. All other boundary conditionsphase shift. All other boundary conditions equal to equal to zerozero..

Characteristics of Characteristics of 1212C statesC states Energy of the ground state EEnergy of the ground state Egsgs

The resonance position EThe resonance position Err and width Гand width Г as well as the as well as the non-resonant non-resonant phase shift phase shift δδbgbg are deare defifined by ned by fifitting the calculated neartting the calculated near resonanceresonance phase shift phase shift δδEE to the Wignerto the Wigner dependencedependence on energyon energy

Root-mean-square (RMS) radiiRoot-mean-square (RMS) radii of the ground of the ground (i=1) and excited (i=2) (i=1) and excited (i=2) states readstates read

The monopole transition matrix elementThe monopole transition matrix element takes the form takes the form

A sums is taken over A sums is taken over NNtt nuclenucleoons and over ns and over NNpp protons, Rprotons, Rcmcm is the center-is the center-of-mass position vector and of-mass position vector and ρρ22

ii equals to equals to

5. 5. Numerical resultsNumerical results5.1 Two-body effective potentials5.1 Two-body effective potentials

Calculations have been performed with Calculations have been performed with −− potentials potentials which which reproduce the experimental value of the reproduce the experimental value of the −− resonance ( resonance (88Be) Be) energy Eenergy E22=92.04 keV =92.04 keV

Modified Ali-Bodmer potentials Modified Ali-Bodmer potentials S. Ali and A. R. Bodmer. Nucl. Phys., S. Ali and A. R. Bodmer. Nucl. Phys., 80:99, 196680:99, 1966.. A set of A set of potentials 1-11potentials 1-11 with parameters with parameters μμrr

-1-1 = 1.53 = 1.53 fm andfm and μμaa-1-1 = =

2.852.85 fmfm is constructed to study the dependence onis constructed to study the dependence on the the 88Be width Be width γγ, which vary within the interval from, which vary within the interval from 5.1 eV to 8.53 eV (this 5.1 eV to 8.53 eV (this interval corresponds to earlier experimentalinterval corresponds to earlier experimental measurements of measurements of γγ = = 6.8 6.8 ±± 1.7 eV 1.7 eV))..

The The potential 12potential 12 with parameters with parameters μμrr= 0.7 fm= 0.7 fm-1-1 and and μμaa= 0.475= 0.475 fmfm--11 is is usedused to illustrate the dependence on the potential range.to illustrate the dependence on the potential range.

The potentials 13-15, that fit the two-body experimental dataThe potentials 13-15, that fit the two-body experimental data• EE22=92.04 keV, =92.04 keV, γγ == 5.47 5.47 ± 0.25 eV± 0.25 eV• Fit the experimental phase shift up to the energy 12 MeVFit the experimental phase shift up to the energy 12 MeV

5.2 Parameters of the two-body potentials5.2 Parameters of the two-body potentials

Parameters of the Parameters of the two-bodytwo-body effectiveeffective potentialpotentialss pproviding theroviding the - - resonance resonance position and position and widthwidth γγ = 6.8 = 6.8 ± ± 1.7 eV 1.7 eV (potentials 1-12) (potentials 1-12) γγ== 5.47 5.47 ± 0.25 eV (potentials 13-15).± 0.25 eV (potentials 13-15).

Parameters of the Parameters of the two-bodytwo-body effectiveeffective potentialpotentialss pproviding theroviding the - - resonance resonance position and position and widthwidth γγ = 6.8 = 6.8 ± ± 1.7 eV 1.7 eV (potentials 1-12) (potentials 1-12) γγ== 5.47 5.47 ± 0.25 eV (potentials 13-15).± 0.25 eV (potentials 13-15).

5.3 Two-body phase shift5.3 Two-body phase shift

The experimental and calculated The experimental and calculated αα--αα s-wave elastic-scattering phase shift s-wave elastic-scattering phase shift δδ versus versus the center-of mass energy E (MeV) for the two-body potentials 1, 2, 6, 9, 10, and 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 11 (top to bottom, left panel) and for the two-body potentials 2, 13, 14, and 15 providing the providing the 88Be width within the range of the experimental uncertainty 5.57 eV < Be width within the range of the experimental uncertainty 5.57 eV < γγ < 5.82 eV (right panel) < 5.82 eV (right panel)

The experimental and calculated The experimental and calculated αα--αα s-wave elastic-scattering phase shift s-wave elastic-scattering phase shift δδ versus versus the center-of mass energy E (MeV) for the two-body potentials 1, 2, 6, 9, 10, and 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 11 (top to bottom, left panel) and for the two-body potentials 2, 13, 14, and 15 providing the providing the 88Be width within the range of the experimental uncertainty 5.57 eV < Be width within the range of the experimental uncertainty 5.57 eV < γγ < 5.82 eV (right panel) < 5.82 eV (right panel)

5.4 Results of the solution HRE 5.4 Results of the solution HRE (a) The one-term three-body effective potential (V(a) The one-term three-body effective potential (V11=0)=0)

Fixed at the experimental valuesFixed at the experimental values EEgsgs = = -7.2747 MeV-7.2747 MeVEErr == 0.3795 MeV0.3795 MeV Not fixed at the experimental Not fixed at the experimental

valuesvaluesRR(1)(1)= 2.48= 2.48 ±± 0.22 fm 0.22 fmГ Г == 8.5 8.5 ±± 1.0 eV 1.0 eVMM1212= = 5.48 5.48 ±± 0.22 0.22 fm fm22

Fixed at the experimental valuesFixed at the experimental values EEgsgs = = -7.2747 MeV-7.2747 MeVEErr == 0.3795 MeV0.3795 MeV Not fixed at the experimental Not fixed at the experimental

valuesvaluesRR(1)(1)= 2.48= 2.48 ±± 0.22 fm 0.22 fmГ Г == 8.5 8.5 ±± 1.0 eV 1.0 eVMM1212= = 5.48 5.48 ±± 0.22 0.22 fm fm22

(b) The three-body effective potential (V(b) The three-body effective potential (V11≠≠0)0)

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.

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 M12-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 M12= 5.48 ± 0.22 fm2

The calculated M12-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 M12= 5.48 ± 0.22 fm2

Fixed at the experimental valuesFixed at the experimental valuesEEgsgs = = -7.2747 MeV-7.2747 MeVEErr == 0.3795 MeV0.3795 MeVRR(1)(1)= 2.48= 2.48 ±± 0.22 fm 0.22 fmNot fixed at the experimental valuesNot fixed at the experimental valuesГ Г == 8.5 8.5 ±± 1.0 eV 1.0 eVMM1212= = 5.48 5.48 ±± 0.22 0.22 fm fm22

Fixed at the experimental valuesFixed at the experimental valuesEEgsgs = = -7.2747 MeV-7.2747 MeVEErr == 0.3795 MeV0.3795 MeVRR(1)(1)= 2.48= 2.48 ±± 0.22 fm 0.22 fmNot fixed at the experimental valuesNot fixed at the experimental valuesГ Г == 8.5 8.5 ±± 1.0 eV 1.0 eVMM1212= = 5.48 5.48 ±± 0.22 0.22 fm fm22

ConclusionConclusion ConfirmedConfirmed that at low energies 0 that at low energies 0++

22-state decays by means of the-state decays by means of the sequential mechanism of decay sequential mechanism of decay 1212C C → → ++ 88Be Be → → 3 3

Determined the fine characteristics of the 0Determined the fine characteristics of the 0++ states states 1212CC nuclei nuclei (an (an

extremely narrow width extremely narrow width ГГ, the rms radius R, the rms radius R(2)(2) of the 0 of the 0++22 state and state and

monopole transition matrix element Mmonopole transition matrix element M1212)) for a set of the two- and three-for a set of the two- and three-body effective potentialsbody effective potentials

AdjustedAdjusted the parameters of the two-body effective Ali-Bodmer-type the parameters of the two-body effective Ali-Bodmer-type potentials to potentials to fix the position and width of fix the position and width of 88BeBe at the experimental values at the experimental values and to and to fit the s-wave phase shiftfit the s-wave phase shift (the fit of d-wave and p-wave phase shifts (the fit of d-wave and p-wave phase shifts still need to be added)still need to be added)

AdjustedAdjusted the parameters of the three-body effective potentials the parameters of the three-body effective potentials to fix the to fix the energies of the 0energies of the 0++

states states 1212CC nuclei nuclei ( (EEgsgs, E, Err), ), andand rms radii of the 0 rms radii of the 0++ 2 2 state state

((RR(1)(1)) at the experimental values ) at the experimental values Study the Study the dependencedependence of the characteristics of of the characteristics of 1212C on the effective C on the effective two- two-

and three-body potentialsand three-body potentials The calculation of characteristics of the The calculation of characteristics of the 00++

3 3 state is still neededstate is still needed Some results discussed in presentation have been published inSome results discussed in presentation have been published in

Phys. Rev. C 70, 014006Phys. Rev. C 70, 014006 (2004)(2004) Eur. PhysEur. Phys. . JJ. . AA 2626, , 201-207201-207(200(20055))

and reported at and reported at UNISA-JINR Symposium ``Models and UNISA-JINR Symposium ``Models and MMethods in Few- and Many-Body ethods in Few- and Many-Body

Systems'' (6--9 February 2007, Skukuza, Systems'' (6--9 February 2007, Skukuza, KKruger National Park, South Africa).ruger National Park, South Africa). 1212CC system is intensively investigating by other groups system is intensively investigating by other groups

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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/0703001R. Alvarez-Rodriguez at al. Nucl-th/0703001

Thank you for attentionThank you for attention