Fusion Reactions with Carbon Isotopes

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Fusion Reactions with Carbon Isotopes Henning Esbensen Argonne National Laboratory, Argonne IL, USA Problems: Large structures in 12 C+ 12 C fusion data. Large systematic uncertainties of 15% or more. Modified data: * σ f (mb) Analyze instead a smooth cross section, e. g. 13 C+ 13 C, and predict the cross section for fusion of 12 C+ 12 C. 1 3 1 2 1 1 1 1-1 1-2 1-3 1-4 1-5 1-6 12 C+ 12 C Work supported by U.S. Department of Energy, Office of Nuclear Physics. High* Becker Aguilera Mazarakis* Ch12sm 2 3 4 5 6 7

Overview. Coupled-channels technique and double-folding potentials. Applications to the fusion of calcium isotopes. Applications to low-energy fusion of carbon isotopes - fit 13 C+ 13 C; predict 12 C+ 12 C and 12 C+ 13 C fusion. Provide a constraint on or an upper limit for the extrapolation of 12 C+ 12 C fusion data to low energy. Structures in the high-energy fusion of 16 O+ 16 O, - a probe of the ion-ion potential and centrifugal barriers at high L. Structures in high-energy fusion of carbon isotopes.

Basic assumptions in coupled-channels description. Fusion can be simulated by ingoing-wave boundary conditions (IWBC) that are imposed at the minimum of the potential pocket. Coulomb barrier and pocket in the entrance channel potential: V(r) IWBC E IWBC are sometimes supplemented with a weak, short-ranged imaginary potential. Compound r Structure properties of nuclei do not change during the reaction. Excitation and transfer are independent degrees of freedom.

Standard approach to surface excitations. Include Coulomb couplings δv C to first order δr, δr = Rα λµ Y λµ(ˆr), nλ δr = β nλr 4π. Expand the nuclear interaction up to second order in δr, δv N = U (r) δr + 1 2 U (r) ( δr 2 δr 2 ), where U(r) is the standard empirical Woods-Saxon potential, U(r) = 16πγaR 1R 2 R 1 + R 2 1 1 + exp[(r R 1 R 2 R)/a]. Was developed to describe elastic scattering. It has the diffuseness a.65 fm, and an adjustable radius parameter R. It is consistent with the M3Y double-folding potential near and outside the Coulomb barrier.

The M3Y double-folding potential. U N (r) = dr 1 dr 2 ρ a (r 1 ) ρ A (r 2 ) v NN (r + r 2 r 1 ). The effective M3Y interaction produces a very realistic potential near and outside the barrier, and is consistent with the empirical Woods- Saxon potential. However, the entrance channel potential is too deep. Solution: Supplement the M3Y interaction with a repulsive term, v rep NN = v rep δ(r + r 2 r 1 ). Use a small, adjustable diffuseness of the densities, a r.3.4 fm, when calculating the repulsive part. The M3Y+repulsion potential. Calibrate the strength v rep, so that the total nuclear interaction for overlapping nuclei is consistent with the Equation of State, U N (r = ) = 2A a [ǫ(2ρ) ǫ(ρ)] A a 9 K, and nuclear incompressibility of K 234 MeV (Myers & Swiatecki).

Analysis of 4 Ca+ 4 Ca fusion data. Legnaro data: Montagnoli et al., PRC 85, 2467 (212). 1 3 1 2 6 5 σ f (mb) 1 1 1 1-1 1-2 4 Ca + 4 Ca Legnaro WS Ch-24 M3Y+rep Ch-24 Ch-1 5 55 6 U(r)+V C (r) (MeV) 4 3 2 1 8 Zr 4 6 8 1 12 14 r (fm) 4 Ca + 4 Ca M3Y+rep Woods-Saxon M3Y The Legnaro data are hindered with respect to the WS based calc. The M3Y+rep potential gives a better account of the data.

Advantages of using the M3Y+repulsion potential in coupled-channels analysis of heavy-ion fusion data. If the densities of the reacting nuclei are known, the ion-ion potential can be predicted. One can then focus on the structure input. Alternatively, one can extract the densities of the reacting nuclei from the analysis of fusion data. If the extracted density is poor, the nuclear structure input could be poor or incomplete. Applications to the fusion of calcium isotopes: extract the densities of 4 Ca and 48 Ca from the 4 Ca+ 4 Ca and 48 Ca+ 48 Ca fusion data. Predict the ion-ion potential and fusion of 4 Ca+ 48 Ca and compare the calculated fusion cross sections to data.

Analysis of 48 Ca+ 48 Ca fusion data. Experiment: Stefanini et al., PLB 679, 95 (29). Calculations: Esbensen et al., PRC 82, 54621 (21). 1 3 8 1 2 σ f (mb) 1 1 1 1-1 48 Ca + 48 Ca σ f (mb) 6 4 48 Ca + 48 Ca 1-2 1-3 1-4 Legnaro WS Ch-24 M3Y+rep Ch-24 Ch-1 45 5 55 6 2 Legnaro WS Ch-24 M3Y+rep Ch-24 Ch-1 5 55 6 65 The data are hindered at low energy and suppressed at high energy with respect to the Woods-Saxon based calculation. The shallow M3Y+repulsion potential provides much better agreement.

Extracted density parameters for calcium isotopes. Nucleus a r (fm) R (fm) a (fm) rms (fm) 4 Ca.42 3.47.56 3.4 48 Ca.4295 3.798.54 3.56 Compare the RMS radii to the point-proton and point-nucleon RMS radii, obtained from electron and proton scattering experiments, or to the predictions Hartree-Fock (HF) calculations. Nucleus Fusion point-proton point-nucleon HF (Negele) 4 Ca 3.4 3.383(1) 3.44(5) 3.39 48 Ca 3.56 3.387(1) 3.53(3) 3.58 The RMS radii extracted from fusion look quite reasonable, so the nuclear structure input must have been reasonable.

Analysis of 4 Ca+ 48 Ca fusion data. Legnaro data: Jiang et al., PRC 82, 4161 (21), MIT data: Aljuwair et al., PRC 3, 1223 (1984). σ f (mb) 1 3 1 2 1 1 1 1-1 1-2 1-3 1-4 4 Ca + 48 Ca 2N Exc+ptr Excitations no coupling Aljuwair Jiang 45 5 55 6 65 8 6 4 2 4 Ca + 48 Ca noc Excitations Exc+ptr Aljuwair Jiang 5 55 6 65 7 75 The prediction based on excitations alone is too low at low energies and too high at high energies. Include couplings to transfer channels with positive Q-value.

Isotope dependence of Ca+Ca fusion cross sections, Montagnoli et al., PRC 85, 2467 (212). σ f (mb) 1 3 1 2 1 1 1 1-1 1-2 1-3 1-4 48 Ca + 48 Ca 4 Ca + 4 Ca 4 Ca+ 48 Ca 45 5 55 6 65 The 4 Ca+ 48 Ca cross sections exceed the 48 Ca+ 48 Ca data at low energy, but are suppressed compared to the 4 Ca+ 4 Ca data at high energy.

Fusion of carbon isotopes. Best fit to Trentalanges 13 C+ 13 C fusion data. Esbensen, Tang, & Jiang: PRC 84, 64613 (211). Excellent fit to the shape of the data is achieved by adjusting the ion-ion potential. Best fit to Trentalange s data for S c =.85. Best fit to Charvet s data for S c =.96. σ f (mb) 1 3 1 2 1 1 1 1-1 1-2 1-3 13 C+ 13 C cxtr Charvet/.96 Trentalange/.85 Ch-26 No coupl 3 4 5 6 7 8 9 1

Irregular normalization of fusion data. Esbensen, Tang, & Jiang: PRC 84, 64613 (211). 1.5 σ exp /σ cal (13C+ 13 C) 1.2 1.8.5 12 C+ 13 Dayras 12+13 13 C+ 13 C Trenta 13+13 2 4 6 8 1 The two data sets are normalized to the same 13 C+ 13 C calculation. Why are the data are lower for the larger systems?

Coupling Scheme for 13 C+ 13 C reactions. Ch-13 calculations: include all states below 1 MeV. 16 14 12 E x (MeV) 1 8 6 4 5/2-3/2-5/2 + 5/2 + 5/2-3/2-2 1/2-1/2-13 C 13 C Ch-26 calculations: include in addition one-neutron transfer. Q 1n =3.2 MeV. Assume independent modes of excitation and transfer.

Entrance Channel Potential for 13 C+ 13 C fusion. M3Y+repulsion, double-folding potential. Density parameters: R = 2.285 fm, a =.44 fm, a r =.31 fm. V CB = 6.5 MeV. V min = -12. MeV. Coupled-channels calc.: use ingoing-wave boundary conditions at the pocket minimum. Fusion is determined by the ingoing flux. U(r) (MeV) 15 1 5-5 -1-15 -2-25 Closed channels: treated as decaying states when E c.m. E x (Whittaker function.) 2 4 6 8 1 12 r (fm) 13 C + 13 C 26 Mg M3Y+rep WS M3Y

Large sensitivity to two-phonon excitations. S factor = E c.m. σ f exp(2π η) S factor (MeV b) 1 17 1 16 Trento/.85 no coupl. one-phon Ch-13 Ch-26 13 C+ 13 C cxtr 2 3 4 5 6 7 The one-phonon calculation rises too slowly. Ch-13: The two-phonon calculation gives an excellent fit. Ch26: Small effect of transfer. Minor structures in the data.

Poor fit for Woods-Saxon potential. S factor = E c.m. σ f exp(2π η) S factor (MeV b) 1 17 1 16 Trenta no-coup One-phon Ch-26 13 C+ 13 C cxtr 2 3 4 5 6 7 The one-phonon calculation rises too slowly. Ch26: not much different from one-phonon calculation.

Ratio of the measured and calculated cross sections 2 13 C+ 13 C Trentalange s data 1.5 σ exp /σ cal 1.2 1.8.5 WS Ch-26 M3Y+rep Ch-26 2 4 6 8 1 Woods-Saxon potential best fit: S c = 1 and χ 2 /N=9. M3Y+repulsion best fit: S c =.85 and χ 2 /N = 1.

Matter densities of carbon isotopes. Nucleus R (fm) a (fm) RMS point-particle 12 C 2.155.44 2.337 2.337(2) 13 C Charge 2.332(3) 13 C 2.228.44 ESTIMATE 2.378 13 C (*) 2.285.44 2.41 2.41 (*) extracted from fit to Trentalange s fusion data (a r =.31 fm). 13 C matter ESTIMATE: where r 2 n = 8.66 fm 2. r 2 13 = 12 ( r 2 12 + r2 ) n, 13 13

Nuclear Structure Input for 12 C to coupled-channels calculations. Properties of E2 and E3 transitions to the low-lying states. State E x (MeV) Transition B(Eλ) (W.u.) β λ 2 + 4.439 E2: + 1 2 + 4.65(26).57 + 2 7.654 E2: 2 + + 2 8.(11).236 3 9.641 E3: + 1 3 12(2).9(7) Note that β 3 =.9(7) is extremely large, and E x (3 ) = 9.64 MeV is quite high. Note that all inelastic channels are closed for E cm < 4.4 MeV.

Coupling Scheme for 12 C+ 12 C reactions Assume independent modes of excitation. β 2 =.57 β 3 =.9(7) Ch-1: all red states. Ch-12SM: Add the mutual (2 +, 3 ) excitation in each nucleus but only with half E x (MeV) 2 18 16 14 12 1 8 6 4 2 + 3-2 + + (2 +,3 - ) 12 C (3 - ) 2 (3 -,3 - ) strength. Suggested by shell model calculation of the (2 +, 3 ) 4 state (Alex Brown). (2 +,3 - ) (3 -,2 + ) (2 +,2 + ) (3 - ) 2 (3 -,2 + ) 3-12 C + 2 + + Ignore the two-phonon (3 ) 2 and mutual (3,3 ) excitations. The excitation energy is very high.

Prediction of 12 C+ 12 C fusion using the point-proton density and a r =.31 fm. Error band from β 3 =.9 ±.6 S factor (MeV b) 1 17 1 16 1 15 S factor = E c.m. σ f exp(2π η) 12 C+ 12 C Ch12SM No coupl β 3 =.9(7) The Ch-12SM calculations is consistent with the maxima of the peaks. 1 2 3 4 5 6 7 Note: the no-coupling limit is not the background.

Coupling Scheme for 12 C+ 13 C reactions Assume independent modes of excitation and transfer. Ch-7: all red states. Ch-15sm: Add the blue and green mutual excs. sm means half strength of mutual exc. in the same nucleus. E x (MeV) 16 14 12 1 8 6 4 2-3 + 2 + (2 +,3 - ) + 1/2-12 C 5/2 + 13 C 5/2-3/2 - Ch-3sm: include also one-neutron transfer with Q 1n =.

The sensitivity to 1n transfer is modest. S factor (MeV b) 1 17 1 16 1 15 WS Ch3sm Ch-3sm Ch-15sm Ch-1 12 C+ 13 C 1 2 3 4 5 6 7 The Ch-3sm calculation is is slightly enhanced compared to the Ch-15sm calculation (no transfer). Note hindrance compared to the Woods-Saxon Ch3sm calculation. Data by Notani et al.: PRC 85, 1467 (212), and Dayras et al.: NPA 265, 153 (1976).

Systematics of Carbon Fusion. Error band: β 3 =.9 ±.7 in 12 C. S factor (MeV b) 1 18 1 17 1 16 1 15 12 C+ 13 C Spillane Coupled-Channels 12 C+ 12 C 13 C+ 13 C (data/.85) 1 2 3 4 5 6 7 8 E (MeV) Coupled-channels calculations based on Ingoing-wave boundary conditions provide a fairly consistent description of the carbon fusion. They provide an upper limit for the 12 C+ 12 C fusion data. Spillane s lowest point is too high.

Predicted fusion of 12 C + 12 C for β 3 =.9 ±.7. σ f (mb) 1 3 1 2 1 1 1 1-1 1-2 1-3 1-4 1-5 1-6 12 C+ 12 C High* Becker Aguilera Mazarakis* Ch12sm 2 3 4 5 6 7 Coupled-channels calculations based on a shallow M3Y+repulsion potential provides an upper limit of the data. Esbensen, Tang, & Jiang: PRC 84, 64613 (211).

1 17 12 C+ 12 C fusion S factor (MeVb) 1 16 1 15 1 14 1 13 σ σcc σ WS Satkowiak Becker Patterson Kovar 2 4 6 8 1 E cm (MeV) The suppression is caused by the low level density in 24 Mg. σ fus (E) = π (2J + 1) P k 2 fus (E,J) P CN (E,J). J Moldauer (1967): P CN = 1 exp( 2πΓ J /D J ). Jiang et al., PRL 11, 7271 (213).

Conclusions. Excellent fit to the re-normalized 13 C+ 13 C data (divided by.85.) Large uncertainty in the prediction of the 12 C+ 12 C fusion - due to the uncertainty in the large β 3 -value. Consistent description of all carbon data is achieved by using the Shell Model prediction of the mutual excitations in 12 C, i.e., half strength of the 2 + (2 +, 3 ) and the 3 (2 +, 3 ) transitions. Conjecture: Coupled-channels calculations and ingoing wave boundary conditions provide in principle an upper limit for fusion. Structures in the low-energy 12 C+ 12 C fusion data are not resonances. They are canyons that are caused by the hindrance due to the low level density of the compound nucleus, 24 Mg.

Evidence of a shallow potential in high energy fusion. σ f (mb) 12 1 8 6 4 2 16 O+ 16 O Kovar Tserruya Woods-Saxon M3Y+rep. L max =8-18 1 15 2 25 3 35 Coupled-channels calculations based on a shallow M3Y+repulsion potential provides a good description of the data by Tserruya et al., PRC 18, 1688 (1978). Calculations by Esbensen: PRC 77, 5468 (28).

Explanation of structures in high-energy fusion. σ HW f = π k 2 L max L (2L + 1) exp(x L ) 1 + exp(x L ), where x L = [E V B (L)]/ǫ. d(eσ HW f ) de = π h2 2µ L max L 2L + 1 ǫ exp(x L ) (1 + exp(x L )) 2. 16 O + 16 O. HW: VCB =9.9, R CB =8.4, ε=.4 MeV 6 Hill-Wheeler Wong L=2 d(eσ f )/de (mb) 4 2 L=12 L=16 1 15 2 25 3 See also Poffee, Rowley & Lindsay, NPA 41, 498 (1983).

Effective centrifugal barriers are revealed in d(eσ f )/de plot. d(eσ)/de [mb] 8 6 4 2-2 Tserruya Thomas Kondo Elastic Scatt.: L=12 L=16 16 O+ 16 O 1 15 2 25 3 35 Calculations by Esbensen: PRC 85, 64611 (212). Peaks are consistent with the elastic scattering resonances found by Kondo, Bromley, & Abe, PRC 22, 168 (198).

The shallow M3Y+repulsion potential. 4 3 L=18,2 L=1-16 L= V L (r) (MeV) 2 1 16 O+ 16 O M3Y+repulsion 4 6 8 1 12 r (fm) The pocket disappears for L =16. Need an imaginary potential to make the nuclei stick at higher Ls and higher energies.

High energy fusion of 12 C+ 12 C. Kovar et al., PRC 2, 135 (1979). The data are suppressed. 14 12 12 C+ 12 C σ f (mb) 1 8 6 4 2 Kovar Ch12sm L max =1-18 no coupl. 5 1 15 2 25 3 35 4 The data were recently confirmed by REHM (MUSIC). Vandenbosch, PLB 87, 183 (1979), explained the suppression of the data by the low level density of I= 8, 1, 12... states in 24 Mg.

Effective centrifugal barriers in 12 C+ 12 C fusion are revealed in d(eσ f )/de plot of Kovar s data. d(eσ)/de [mb] 6 5 4 3 2 1-1 -2 L=12,14,18 Ch12sm 12 C+ 12 C 12 14 18 5 1 15 2 25 3 35 E cm (MeV) See recent MUSIC measurements by Rehm et al.

High energy fusion of 13 C+ 13 C. Both even and odd values of L. Good norm; data 4% below. 12 1 Charvets data σ f (mb) 8 6 4 2 13 C+ 13 C cxtr No coupl Ch-26 L max =1-14 5 1 15 2 25 Charvet et al., NPA A 376, 292 (1982). Even values of L: weight=25%. Odd values of L: weight=75%.

Effective centrifugal barriers in 13 C+ 13 C fusion. d(eσ)/de [mb] 3 2 1 13 C+ 13 C cxtr Trent/.85 Charvet Ch-1 Ch-26 L=11,13 5 1 15 2 E cm (MeV) Even values of L: weight=25%. Odd values of L: weight=75%.

High energy fusion of 12 C+ 13 C. Both even and odd values of L. Excellent norm. 12 σ f (mb) 1 8 6 4 2 12 C+ 13 C cxtr Dayras Kovar Ch-1w2 Ch-3 5 1 15 2 25 Dayras et al., NPA 265, 153 (1976). Kovar et al., PRC 2, 135 (1979).

Effective centrifugal barriers in 12 C+ 13 C fusion. d(eσ)/de [mb] 3 2 1 12 C+ 13 C Kovar L m =17 Ch-3 5 1 15 2 25 E cm (MeV)

Conclusions. The hindrance of fusion far below the Coulomb barrier is a general phenomenon, which has been observed in many heavy-ion systems. It can be explained by coupled-channels calculations that are based on IWBCs and a SHALLOW potential in the entrance channel. The SHALLOW potential resolves the problem of a suppression of high energy fusion data and may explain the observed structures. The 12 C+ 12 C data are suppressed both at low and high energies, possibly due to the low level density of states in 24 Mg. However, it appears that the assumptions: 1) the system is trapped after barrier penetration, 2) the structure of the reacting nuclei does not change, 3) transfer and excitations are independent modes, work quite well and can be used in many cases to explain the data.

L dependence of 12 C + 12 C fusion. Coupled-channels calculations based on a shallow M3Y+repulsion. σ f (mb) 1 3 1 2 1 1 1 1-1 1-2 1-3 1-4 1-5 1-6 12 C+ 12 C cxtr L= L=2 L=4 L=6 L=8 Ch12SM 2 3 4 5 6 7 Cross section is dominated by L =, 2 below 6 MeV.

Entrance Channel Potentials for Carbon fusion. 15 15 15 1 (A) 13 C + 13 C 1 (B) 12 C + 13 C 1 (C) 12 C + 12 C U N (r) + V C (r) (MeV) 5-5 -1-15 -2-25 2 26 Mg M3Y WS M3Y+rep 4 6 8 1 12 5-5 -1-15 -2-25 2 25 Mg M3Y WS M3Y+rep 4 6 8 1 12 5-5 -1-15 -2-25 24 Mg 2 4 M3Y M3Y+rep 6 8 1 12 r (fm) r (fm) r (fm)

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