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Effects of Transitional Shape and Shape Admixture and of Anti-symmetrization on Sub-Barrier Fusion
Nuclear Physics A 787 (2007) 190c–197c
Effects of Transitional Shape and Shape Admixture and of Anti-symmetrization on Sub-Barrier Fusion N. Takigawa a , Zakarya Mohamed Mohamed Mahmoud a , Muhammad Zamrun F. a , Nyein Wink Lwin a , T. Takehi a , and K. Hagino a a
Department of Physics, Tohoku University, Sendai 980-8578, Japan
We discuss two separate issues. The first is to examine whether the fusion cross section can resolve debates on the structure of Ge isotopes. The second is to propose the effect of anti-symmetrization as an origin of the high energy problem of fusion reactions, i.e. the problem that the double folding potential overestimates the fusion cross section at energies above the Coulomb barrier. 1. Introduction There exist debates concerning the structure of Ge isotopes. Fig.1 shows the low lying energy levels of Ge isotopes [1]. Let us focus on 74 Ge. The first excited state is a + + 2+ state and there exist excited 0+ 2 ,22 and 41 states at energies nearly twice as high as + that of the first 2 state suggesting two phonon members. There also exists a low lying 3− state at 2.54MeV. These facts indicate that 74 Ge is a spherical nucleus, which is soft against deformation. On the other hand, a series of experiments of Coulomb excitation [2–4] provide an alternative point of view. The key words of the alternative point of view are shape coexistence, shape transition and shape admixture. As indicated in Fig.1, the experimental data of the Coulomb excitation suggest that two different shapes coexist in low energy region in each Ge isotope, and that the shape of the ground state changes from a spherical shape to a deformed shape as the neutron number increases. The 72 Ge is the transitional nucleus, where a large shape admixture is expected [4]. The idea of the shape transition has been tested through the analysis of the fusion excitation functions of 27 Al+70,72,73,74,76 Ge reactions [5]. Based on simple analyses using the CCFUS, the authors claimed the structural change from a spherical (or oblate) shape in 70,72,73 Ge to a prolate shape in 74,76 Ge. Recently, Esbensen performed a more detailed coupled channels calculations of the fusion excitation functions for the 16 O+70,72,74,76 Ge and 27 Al+70,72,74,76 Ge reactions [6], and obtained puzzlingly the results that the phonon coupling model yields a smaller χ2 value than the rotational coupling model for 74 Ge. He also concluded that the quadratic coupling is required to obtain a good agreement between the coupled-channels calculations and the experimental data and that the inclusion of multi-phonon excitations improves the agreement between theory and experiments. In [7,8], Esbensen analyzed the fusion excitation function of 74 Ge+ 74 Ge reaction, and again claims that the vibrational coupling model gives a smaller χ2 value. As we show in the next section, the data of Coulomb excitation function show a transitional property of 74 Ge. The first subject of 0375-9474/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.nuclphysa.2006.12.031
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E/A [MeV]
this contribution is to examine whether one can probe the transitional structure of 74 Ge through the fusion barrier distribution analysis [9]. We also discuss whether one can probe the shape admixture in 72 Ge. Our calculation is based on the CCFULL once the transition matrix is specified in each model. Incidentally, we have performed RMF calculations to learn the shape of Ge isotopes. The result using the NLSH parameter set [10] and a constant G approximation for the pairing interaction is shown in Fig.2. The figure suggests that Ge isotopes are very soft against deformation indeed, and that there exist two energy minima with nearly equal energy for each isotope. Ref. [11] also had obtained nearly the same results as those in Fig.2 based on the Skyrme Hartree Fock calculations.
Figure 1. Low lying energy levels of Ge isotopes.
-8.30 -8.35 -8.40 -8.45 -8.50 -8.55 -8.60 -8.65 -8.70 -8.75 -0.6
68 Ge 70 Ge 72 Ge 74
Ge
-0.4
-0.2
0.0 β
0.2
0.4
0.6
Figure 2. Energy surface of Ge isotopes as a function of the quadrupole deformation parameter β obtained with RMF calculations.
The second subject of this contribution is to discuss the role of anti-symmetrization. It is known that the calculations based on the double folding potential obtained by convoluting the M3Y nucleon-nucleon interaction with the realistic densities of the target and projectile nuclei overestimate the fusion cross section at energies above the Coulomb barrier once the nucleus-nucleus interaction is normalized to reproduce the experimental fusion cross section at energies near the Coulomb barrier [12,13]. On the other hand, the coupled channels calculations using a phenomenological Woods-Saxon potential require a much larger value for the surface diffuseness parameter a than the value accepted to fit the data of elastic scattering and also the surface diffuseness of the double folding potential in order to reproduce the fusion excitation function at high energies [14]. We propose that this high energy problem originates partly from the effect of anti-symmetrization by showing the fusion excitation function obtained with an effective potential which approximately takes the normalization kernel in the resonating group method(RGM) into account.
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2. Probe of a transitional structure of
74
Ge
We start from the discussion of the fusion between two 74 Ge nuclei. We assume that the nuclear interaction between the projectile and target nuclei can be well represented by a Woods-Saxon shape. As often done, we introduce the no-Coriolis approximation to reduce the dimension of coupled-channels calculations [15]. The relevant expression for the nuclear interaction then reads V0 V (r; {αλ0 }) = − (1) (P ) (T ) (T ) 1 2λ+1 (P ) 1 + exp( a r − R0 (1 + Σλ 4π α ˆ λ0 ) − R0 (1 + Σλ 2λ+1 α ˆ λ0 ) ) 4π where r is the distance between the centers of mass of the projectile and target nuclei, (P,T ) α ˆ λμ the deformation parameters and the indices P or T stand for the projectile or the target nucleus, respectively. We take the radii of the projectile and the target nuclei to be (P,T ) 1/3 R0 = r0 AP,T with r0 =1.2 fm. In the fusion of two even-even nuclei, only states with the azimuthal quantum number being zero couple in the no-Coriolis approximation. One therefore needs to specify only the coupling matrix J1 M1 = 0|ˆ αλ0 |J2 M2 = 0 within the model space and diagonalize to perform the full order coupled channels calculations. We consider only the quadrupole coupling λ=2 and determine the matrix elements either from the data of Coulomb excitation or based on either vibrational or rotational coupling modˆ λμ by Q ˆ λμ ∼ 3 eZRλ α els. In the first case, we relate α ˆ λμ to the quadrupole operator Q 0 ˆ λμ . 4π The Coulomb coupling is included up to the second order.
Table 1 Experimental E2 coupling matrix among 5 low lying levels in 74 Ge.
Table 2 Comparison with the vibrational coupling matrix, where CJ = 2020|J0.
√ α20 |J2 M2 = 0 given by the Coulomb excitation [3] Table 1 shows 5 × J1 M1 = 0|ˆ + + + + among 5 low lying 0+ 1 , 21 , 22 , 41 , 02 states. Table 2 compares Table 1 with the vibrational coupling model by focusing on the coupling matrix elements between the one phonon state and two phonon states. The underlined numbers multiplying the Clebsch-Gordan coefficients C0 , C2 and C4 are the hindrance factors suggested by the data of Coulomb excitation. Table 3 compares Table 1 with the rotational coupling model by focusing + + on the reorientation term in the 2+ 1 state and on the coupling between the 21 and 41 states. The relevant matrix elements are represented as the product of the value in the pure rotational coupling and a modification factor. In making Tables 2 and 3, we divided each matrix element in Table 1 by the value of 0+ α20 |2+ 1 =0.284, which corresponds gs |ˆ to the value of the deformation parameter β2 in order to clearly see the characteristics of the coupling matrix. Tables 2 and 3 clearly show the transitional property of 74 Ge [16].
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Table 3 Comparison with rotational coupling matrix
Fig.3 compares the fusion excitation function and the fusion barrier distribution calculated by using the coupling matrix given by the Coulomb excitation (the solid line denoted as TS-5, standing for the Transitional Structure with 5 levels) and in the pure vibrational(the dashed line denoted as V2-P) and pure rotational(the dot-dashed line denoted as Rot 4+ ) models. In the second and third calculations, we took up to two phonon states and up to 4+ member of the ground state rotational band, respectively, into account.
Especially, the reorientation matrix is about 40% of the pure rotational model [16]. The dotted line is the result in the absence of channel coupling. The calculations suggest a clear difference in the barrier distribution between the rotational coupling and the other two cases, and some quantitative difference between the latter two. It would be interesting if experiments are performed to compare with these results.
102
Ge + 74Ge
d2(Eσ)/dE2 [mb MeV-1]
103 74
σ [mb]
1
10
0
10
no-coup TS-5 V2-P Rot 4+ exp-data
-1
10
10-2 -3
10
110
120 130 140 Ec.m. [MeV]
150
700 TS-5 600 74Ge + 74Ge V2-P Rot 4+ 500 400 300 200 100 0 -100 -200 110 115 120 125 130 135 Ec.m. [MeV]
140
Figure 3. Comparison of the fusion excitation function and fusion barrier distribution for 74 Ge+74 Ge reaction calculated in different models.
3. Probe of shape admixture in the ground state of
72
Ge
We now discuss the effects of a shape admixture in 72 Ge. Ref.[4] claims that the ground state and the first excited 0+ state at 0.70 MeV, and the first excited 2+ state at 0.83 MeV and the third excited 2+ state at 2.40 MeV are admixtures of two different shapes, √ √ + + 1 − α2 |0+ |0+ 1 − α2 |0+ (2) |0+ 1 = α|0n + i , 2 = n − α|0i , + |2+ 1 = β|2n +
1 − β 2 |2+ i ,
|2+ 3 =
+ 1 − β 2 |2+ n − β|2i
(3)
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where the indices n and i stand for normal and intruder, respectively. According to [4], α=0.784, β=0.996 and the normal and the intruder states are deformed and almost spherical, respectively.
72
Ge +
72
1000 800
Ge
no-coup Normal-1 P SA
-1
1200
d (Eσ)/dE [mb MeV ]
(a)
2
600 400 200
2
d2(Eσ)/dE2 [mb MeV-1]
1400
0 110
115
120 125 Ec.m. [MeV]
130
135
700 72 SA 72 Ge + Ge SA-V(2 +,4 +) 600 1 2 V2-P 500 V2-LPExp (b) 400 300 200 100 0 -100 110
115
120 125 Ec.m. [MeV]
130
135
Figure 4. Comparison of the fusion barrier distribution calculated in different models.
We have calculated the fusion cross section between two 72 Ge by including these 4 levels and by using the coupling matrix obtained from the experiments of the Coulomb excitation [4]. The obtained fusion barrier distribution is shown in Fig.4a by the solid line. It significantly differs from the fusion barrier distribution obtained by assuming a vibrational model truncated at the one phonon excitation (the dashed line). The dotted line is the fusion barrier distribution in the absence of channel coupling. Though Fig.4a seems to suggest that the analysis of fusion reactions is promising to probe the shape admixture, one has to examine whether the significant difference of the fusion barrier distribution among different models persists even after one took higher levels into account. In order to learn the situation, Fig.4b compares the fusion barrier distribution obtained in various ways to incorporate the effects of high lying states. The thick solid line is the same as the solid line in Fig.4a. It is the result when we take only 4 levels into account and respect the shape admixture as claimed in [4]. The thin solid line was obtained by respecting the shape admixture, but adding the coupling of the 2+ 1 state + to the 2+ 2 and 41 states according to the vibrational model. The dashed line was obtained by assuming the pure vibrational model up to 2 phonon states. The dot-dashed line was obtained based on the vibrational coupling model, but by using the experimental level positions. Our calculations suggest that the clear effect of the shape admixture shown in + Fig.4a becomes obscured by the coupling to 2+ 2 and 41 states as long as the vibrational scheme is employed to fix the off-diagonal coupling matrices in order to incorporate these high lying states.
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4. Effects of anti-symmetrization: a possible origin of the high energy problem We now turn to discuss the effects of anti-symmetrization. In [9], we showed that the knock-on exchange term in the double folding potential strongly affects the properties of the Coulomb barrier by comparing the double folding potentials obtained with and without the knock-on exchange term in the M3Y nucleon-nucleon interaction. This urges us to develop a theory which takes the effects of nucleon exchange into account either exactly or in an approximate way. A proper way is to develop a reaction theory based on the resonating group method (RGM). For a single channel problem, the RGM assumes that the total wave function is given by Ψ = A{ΦP (xP )ΦT (xT )χ (r)Ym (θr , ϕr )}
(4)
where ΦP , ΦT are the wave functions of the ground states of the projectile and target nuclei, respectively, and χ (r) the wave function for the radial part of the relative motion with partial wave . The A is the anti-symmetrization operator among the nucleons in the target and projectile nuclei. It is known that the physical wave function which is free from the redundant states and which is normalised in the same way as the wave √ function in the phenomenological analysis using a local potential is defined by χ˜ = 1 − KN χ , KN being the exchange non-local kernel originating from the normalization. It obeys the Schr¨odinger equation ˆ ef f χ˜ = E χ˜ H 1 1 ˆ ef f = {TD + VD + KT + KV } H (1 − KN )1/2 (1 − KN )1/2
(5) (6)
where TD , VD , KT and KV are the local direct kinetic energy operator, local direct potential, the non-local exchange kernel originating from the kinetic energy operators and the non-local exchange kernel originating from the potential energy, respectively. It is not easy, however, to exactly follow this procedure except for light systems such as the scattering between two α particles. Here we wish to mention that a practical way with reasonable accuracy can be obtained by using the connection between the RGM and the generator coordinate method (GCM), and by introducing several approximations relevant to heavy ion collisions [17]. One of the simplest approximations reads [17] [TD + VD + EKN ]χ˜ ≈ E χ˜
(7)
Notice that the anti-symmetrization introduces an explicitly energy dependent potential. Notice also that if we identify VD in eq.(7) with the double folding potential VDF with an M3Y nucleon nucleon interaction, then it includes a part of KV in eq.(6), i.e. the knock-on exchange term, in addition to the direct term. Fig.5a compares the fusion excitation function for the 16 O+208 Pb reaction calculated by the folding potential with M3Y force (the dashed line) and by including the EKN term in eq.(7)(the solid line). We have calculated the normalization kernel by converting the GCM normalization kernel to the RGM normalization kernel using several approximations relevant to heavy-ion collisions and by converting the original non-local normalization kernel into its Wigner transform [18]. The details of the procedure will be published in [17].
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16
O+208Pb V(r)[MeV]
σ [mb]
1000 800
(a)
600 400 200
N’VDF+EKE=120MeV N’VDF+EKE=VB N’VDF
80
N’VDF+EKN NVDF Exp.
0
75
(b)
70 16
65
O+
208
Pb
60 65 70 75 80 85 90 95 100 105 110 Ec.m. [MeV]
9
10
11
12 13 r [fm]
14
15
16
Figure 5. (a) The fusion excitation function and (b) the effective potentials for 16 O+208 Pb reaction. The N and N ’ are appropriately chosen normalization factors.
We observe that the high energy problem, i.e. the overestimate of the fusion cross section at energies above the Coulomb barrier by the double folding model, is cured by taking the effects of anti-symmetrization through the EKN term in eq.(7). In order to examine whether this is a fluke which holds only for 16 O+208 Pb reaction, we examined also 32 S+208 Pb fusion reaction and confirmed a similar effect of the EKN term in eq.(7). In order to understand the origin of the change of the fusion excitation function by the EKN term, we show in Fig.5b the effective potential VDF + EKN in the region of the Coulomb barrier for two collision energies. We observe that the anti-symmetrization shifts the barrier position more inside and makes the barrier height higher with increasing bombarding energy. We wish to comment that the major contribution around the potential barrier arises from the single nucleon exchange terms, though the norm kernel KN we used includes all the exchange terms. 5. Summary We have discussed whether the analysis of fusion cross section can probe the characteristic properties of the structure of Ge isotopes suggested by the experiments of Coulomb excitation. We have shown that the barrier distribution for the 74 Ge+74 Ge fusion reaction calculated by using the transition matrix given by the Coulomb excitation noticeably differs from that in the rotational model, though it qualitatively resembles that in the vibrational coupling model. Our interest in 74 Ge stemed not only from the data of Coulomb excitation, but also from the abnormally large asymptotic energy shift reported in [19]. Though our re-estimate gives a somewhat smaller value, it still significantly deviates from the systematics as discussed in [19]. The coupled channels calculations using the transition matrix elements from the experiments of Coulomb excitation give a noticeably smaller asymptotic energy shift, though it is much larger than the systematic value suggested in [19] indicating the importance of other effects beyond the present coupled channels calculations. In this connection, we wish to remind Ref.[20], which pointed the important role played
N. Takigawa et al. / Nuclear Physics A 787 (2007) 190c–197c
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cooperatively by the shell and deformation effects. We analysed also the 72 Ge+72 Ge fusion reaction, and have shown that the shape admixture in the ground state of 72 Ge suggested by the experiments of Coulomb excitation yields a significantly different barrier distribution from that of the vibrational coupling + model including up to the one phonon state. However, the coupling to the 2+ 2 and 41 states leads to the fusion barrier distribution very similar to that obtained in the vibrational model including two phonon states and makes it difficult to probe the effects of shape admixture as long as one uses the vibrational model to assess the coupling strength of the 2+ 1 state to these high lying states. In the second part of this contribution, we have argued that the effect of anti-symmetrization provides an origin of the high energy problem, i.e. the problem that the standard double folding potential overestimates the fusion cross section at energies above the Coulomb barrier, or that a value of the surface diffuseness parameter which is much larger than that accepted by elastic scattering or obtained from the double folding potential is required to fit the fusion excitation function in that energy region. REFERENCES 1. Table of Isotopes, 8th edition, eds. R.B.Firestone and V.S.Shirley, John Wiley and Sons, New York, 1996. 2. R. Lecomte et al., Phys. Rev. C22 (1980) 1530; 2420. 3. Y. Toh et al., Eur. Phys. J. A9 (2000) 353. 4. M. Sugawara et al., Eur. Phys. J. A16 (2003) 409. 5. E.F. Aguilera et al., Phys. Rev. C41 (1990)910. 6. H. Esbensen, Phys. Rev. C68 (2003) 034604. 7. H. Esbensen, Phys. Rev. C72 (2005) 054607. 8. H. Esbensen and S. Misicu, Proc. of the Fusion06, eds. L. Corradi et al., AIP Conference Proceedings 853 (2006), p. 13. 9. N. Takigawa, Proc. of the Fusion06, AIP Conference Proceedings 853 (2006), p. 479. 10. M.M. Sharma, M.A. Nagarajan and P. Ring, Phys.Lett.B312(1993)377. 11. J. Dobaczewski et al., Phys. Rev. Lett. 60 (1988) 2254. 12. I.I. Gontchar et al., Phys. Rev. C69 (2004) 024610. 13. N. Takigawa et al., Proc. of the Fusion03, Prog. Theor. Phys. Suppl. 154 (2004) 192. 14. J.O. Newton et al., Phys.Rev. C70 (2004) 024605. 15. N. Takigawa and K. Ikeda, ANL-PHY-86-1, p.613-620; Proc. of the Intl. Symposium on the Many Facets of Heavy Ion Fusion Reactions, March 24-26, 1986, ANL. 16. The matrices inserted in Figure 2 in [9] should be corrected according to Table 3 of the present √ contribution, i.e. 0.11 should be replaced by 0.38 and the minus sign in front of 2 7 5 should be removed. The error occurred by the misinterpretation of the intrinsic quadrupole moment for the observed quadrupole moment. 17. T. Takehi, K. Hagino and N. Takigawa, to be published. 18. N. Takigawa and K. Hara, Z. Physik A276 (1976) 79. 19. C.E. Aguiar et al. Phys. Lett. B201 (1988) 22. 20. A. Iwamoto and N. Takigawa, Phys. Lett. B219 (1989) 176.
Effects of Transitional Shape and Shape Admixture and of Anti-symmetrization on Sub-Barrier Fusion N. Takigawa a , Zakarya Mohamed Mohamed Mahmoud a , Muhammad Zamrun F. a , Nyein Wink Lwin a , T. Takehi a , and K. Hagino a a
Department of Physics, Tohoku University, Sendai 980-8578, Japan
We discuss two separate issues. The first is to examine whether the fusion cross section can resolve debates on the structure of Ge isotopes. The second is to propose the effect of anti-symmetrization as an origin of the high energy problem of fusion reactions, i.e. the problem that the double folding potential overestimates the fusion cross section at energies above the Coulomb barrier. 1. Introduction There exist debates concerning the structure of Ge isotopes. Fig.1 shows the low lying energy levels of Ge isotopes [1]. Let us focus on 74 Ge. The first excited state is a + + 2+ state and there exist excited 0+ 2 ,22 and 41 states at energies nearly twice as high as + that of the first 2 state suggesting two phonon members. There also exists a low lying 3− state at 2.54MeV. These facts indicate that 74 Ge is a spherical nucleus, which is soft against deformation. On the other hand, a series of experiments of Coulomb excitation [2–4] provide an alternative point of view. The key words of the alternative point of view are shape coexistence, shape transition and shape admixture. As indicated in Fig.1, the experimental data of the Coulomb excitation suggest that two different shapes coexist in low energy region in each Ge isotope, and that the shape of the ground state changes from a spherical shape to a deformed shape as the neutron number increases. The 72 Ge is the transitional nucleus, where a large shape admixture is expected [4]. The idea of the shape transition has been tested through the analysis of the fusion excitation functions of 27 Al+70,72,73,74,76 Ge reactions [5]. Based on simple analyses using the CCFUS, the authors claimed the structural change from a spherical (or oblate) shape in 70,72,73 Ge to a prolate shape in 74,76 Ge. Recently, Esbensen performed a more detailed coupled channels calculations of the fusion excitation functions for the 16 O+70,72,74,76 Ge and 27 Al+70,72,74,76 Ge reactions [6], and obtained puzzlingly the results that the phonon coupling model yields a smaller χ2 value than the rotational coupling model for 74 Ge. He also concluded that the quadratic coupling is required to obtain a good agreement between the coupled-channels calculations and the experimental data and that the inclusion of multi-phonon excitations improves the agreement between theory and experiments. In [7,8], Esbensen analyzed the fusion excitation function of 74 Ge+ 74 Ge reaction, and again claims that the vibrational coupling model gives a smaller χ2 value. As we show in the next section, the data of Coulomb excitation function show a transitional property of 74 Ge. The first subject of 0375-9474/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.nuclphysa.2006.12.031
191c
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E/A [MeV]
this contribution is to examine whether one can probe the transitional structure of 74 Ge through the fusion barrier distribution analysis [9]. We also discuss whether one can probe the shape admixture in 72 Ge. Our calculation is based on the CCFULL once the transition matrix is specified in each model. Incidentally, we have performed RMF calculations to learn the shape of Ge isotopes. The result using the NLSH parameter set [10] and a constant G approximation for the pairing interaction is shown in Fig.2. The figure suggests that Ge isotopes are very soft against deformation indeed, and that there exist two energy minima with nearly equal energy for each isotope. Ref. [11] also had obtained nearly the same results as those in Fig.2 based on the Skyrme Hartree Fock calculations.
Figure 1. Low lying energy levels of Ge isotopes.
-8.30 -8.35 -8.40 -8.45 -8.50 -8.55 -8.60 -8.65 -8.70 -8.75 -0.6
68 Ge 70 Ge 72 Ge 74
Ge
-0.4
-0.2
0.0 β
0.2
0.4
0.6
Figure 2. Energy surface of Ge isotopes as a function of the quadrupole deformation parameter β obtained with RMF calculations.
The second subject of this contribution is to discuss the role of anti-symmetrization. It is known that the calculations based on the double folding potential obtained by convoluting the M3Y nucleon-nucleon interaction with the realistic densities of the target and projectile nuclei overestimate the fusion cross section at energies above the Coulomb barrier once the nucleus-nucleus interaction is normalized to reproduce the experimental fusion cross section at energies near the Coulomb barrier [12,13]. On the other hand, the coupled channels calculations using a phenomenological Woods-Saxon potential require a much larger value for the surface diffuseness parameter a than the value accepted to fit the data of elastic scattering and also the surface diffuseness of the double folding potential in order to reproduce the fusion excitation function at high energies [14]. We propose that this high energy problem originates partly from the effect of anti-symmetrization by showing the fusion excitation function obtained with an effective potential which approximately takes the normalization kernel in the resonating group method(RGM) into account.
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2. Probe of a transitional structure of
74
Ge
We start from the discussion of the fusion between two 74 Ge nuclei. We assume that the nuclear interaction between the projectile and target nuclei can be well represented by a Woods-Saxon shape. As often done, we introduce the no-Coriolis approximation to reduce the dimension of coupled-channels calculations [15]. The relevant expression for the nuclear interaction then reads V0 V (r; {αλ0 }) = − (1) (P ) (T ) (T ) 1 2λ+1 (P ) 1 + exp( a r − R0 (1 + Σλ 4π α ˆ λ0 ) − R0 (1 + Σλ 2λ+1 α ˆ λ0 ) ) 4π where r is the distance between the centers of mass of the projectile and target nuclei, (P,T ) α ˆ λμ the deformation parameters and the indices P or T stand for the projectile or the target nucleus, respectively. We take the radii of the projectile and the target nuclei to be (P,T ) 1/3 R0 = r0 AP,T with r0 =1.2 fm. In the fusion of two even-even nuclei, only states with the azimuthal quantum number being zero couple in the no-Coriolis approximation. One therefore needs to specify only the coupling matrix J1 M1 = 0|ˆ αλ0 |J2 M2 = 0 within the model space and diagonalize to perform the full order coupled channels calculations. We consider only the quadrupole coupling λ=2 and determine the matrix elements either from the data of Coulomb excitation or based on either vibrational or rotational coupling modˆ λμ by Q ˆ λμ ∼ 3 eZRλ α els. In the first case, we relate α ˆ λμ to the quadrupole operator Q 0 ˆ λμ . 4π The Coulomb coupling is included up to the second order.
Table 1 Experimental E2 coupling matrix among 5 low lying levels in 74 Ge.
Table 2 Comparison with the vibrational coupling matrix, where CJ = 2020|J0.
√ α20 |J2 M2 = 0 given by the Coulomb excitation [3] Table 1 shows 5 × J1 M1 = 0|ˆ + + + + among 5 low lying 0+ 1 , 21 , 22 , 41 , 02 states. Table 2 compares Table 1 with the vibrational coupling model by focusing on the coupling matrix elements between the one phonon state and two phonon states. The underlined numbers multiplying the Clebsch-Gordan coefficients C0 , C2 and C4 are the hindrance factors suggested by the data of Coulomb excitation. Table 3 compares Table 1 with the rotational coupling model by focusing + + on the reorientation term in the 2+ 1 state and on the coupling between the 21 and 41 states. The relevant matrix elements are represented as the product of the value in the pure rotational coupling and a modification factor. In making Tables 2 and 3, we divided each matrix element in Table 1 by the value of 0+ α20 |2+ 1 =0.284, which corresponds gs |ˆ to the value of the deformation parameter β2 in order to clearly see the characteristics of the coupling matrix. Tables 2 and 3 clearly show the transitional property of 74 Ge [16].
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Table 3 Comparison with rotational coupling matrix
Fig.3 compares the fusion excitation function and the fusion barrier distribution calculated by using the coupling matrix given by the Coulomb excitation (the solid line denoted as TS-5, standing for the Transitional Structure with 5 levels) and in the pure vibrational(the dashed line denoted as V2-P) and pure rotational(the dot-dashed line denoted as Rot 4+ ) models. In the second and third calculations, we took up to two phonon states and up to 4+ member of the ground state rotational band, respectively, into account.
Especially, the reorientation matrix is about 40% of the pure rotational model [16]. The dotted line is the result in the absence of channel coupling. The calculations suggest a clear difference in the barrier distribution between the rotational coupling and the other two cases, and some quantitative difference between the latter two. It would be interesting if experiments are performed to compare with these results.
102
Ge + 74Ge
d2(Eσ)/dE2 [mb MeV-1]
103 74
σ [mb]
1
10
0
10
no-coup TS-5 V2-P Rot 4+ exp-data
-1
10
10-2 -3
10
110
120 130 140 Ec.m. [MeV]
150
700 TS-5 600 74Ge + 74Ge V2-P Rot 4+ 500 400 300 200 100 0 -100 -200 110 115 120 125 130 135 Ec.m. [MeV]
140
Figure 3. Comparison of the fusion excitation function and fusion barrier distribution for 74 Ge+74 Ge reaction calculated in different models.
3. Probe of shape admixture in the ground state of
72
Ge
We now discuss the effects of a shape admixture in 72 Ge. Ref.[4] claims that the ground state and the first excited 0+ state at 0.70 MeV, and the first excited 2+ state at 0.83 MeV and the third excited 2+ state at 2.40 MeV are admixtures of two different shapes, √ √ + + 1 − α2 |0+ |0+ 1 − α2 |0+ (2) |0+ 1 = α|0n + i , 2 = n − α|0i , + |2+ 1 = β|2n +
1 − β 2 |2+ i ,
|2+ 3 =
+ 1 − β 2 |2+ n − β|2i
(3)
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where the indices n and i stand for normal and intruder, respectively. According to [4], α=0.784, β=0.996 and the normal and the intruder states are deformed and almost spherical, respectively.
72
Ge +
72
1000 800
Ge
no-coup Normal-1 P SA
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Figure 4. Comparison of the fusion barrier distribution calculated in different models.
We have calculated the fusion cross section between two 72 Ge by including these 4 levels and by using the coupling matrix obtained from the experiments of the Coulomb excitation [4]. The obtained fusion barrier distribution is shown in Fig.4a by the solid line. It significantly differs from the fusion barrier distribution obtained by assuming a vibrational model truncated at the one phonon excitation (the dashed line). The dotted line is the fusion barrier distribution in the absence of channel coupling. Though Fig.4a seems to suggest that the analysis of fusion reactions is promising to probe the shape admixture, one has to examine whether the significant difference of the fusion barrier distribution among different models persists even after one took higher levels into account. In order to learn the situation, Fig.4b compares the fusion barrier distribution obtained in various ways to incorporate the effects of high lying states. The thick solid line is the same as the solid line in Fig.4a. It is the result when we take only 4 levels into account and respect the shape admixture as claimed in [4]. The thin solid line was obtained by respecting the shape admixture, but adding the coupling of the 2+ 1 state + to the 2+ 2 and 41 states according to the vibrational model. The dashed line was obtained by assuming the pure vibrational model up to 2 phonon states. The dot-dashed line was obtained based on the vibrational coupling model, but by using the experimental level positions. Our calculations suggest that the clear effect of the shape admixture shown in + Fig.4a becomes obscured by the coupling to 2+ 2 and 41 states as long as the vibrational scheme is employed to fix the off-diagonal coupling matrices in order to incorporate these high lying states.
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4. Effects of anti-symmetrization: a possible origin of the high energy problem We now turn to discuss the effects of anti-symmetrization. In [9], we showed that the knock-on exchange term in the double folding potential strongly affects the properties of the Coulomb barrier by comparing the double folding potentials obtained with and without the knock-on exchange term in the M3Y nucleon-nucleon interaction. This urges us to develop a theory which takes the effects of nucleon exchange into account either exactly or in an approximate way. A proper way is to develop a reaction theory based on the resonating group method (RGM). For a single channel problem, the RGM assumes that the total wave function is given by Ψ = A{ΦP (xP )ΦT (xT )χ (r)Ym (θr , ϕr )}
(4)
where ΦP , ΦT are the wave functions of the ground states of the projectile and target nuclei, respectively, and χ (r) the wave function for the radial part of the relative motion with partial wave . The A is the anti-symmetrization operator among the nucleons in the target and projectile nuclei. It is known that the physical wave function which is free from the redundant states and which is normalised in the same way as the wave √ function in the phenomenological analysis using a local potential is defined by χ˜ = 1 − KN χ , KN being the exchange non-local kernel originating from the normalization. It obeys the Schr¨odinger equation ˆ ef f χ˜ = E χ˜ H 1 1 ˆ ef f = {TD + VD + KT + KV } H (1 − KN )1/2 (1 − KN )1/2
(5) (6)
where TD , VD , KT and KV are the local direct kinetic energy operator, local direct potential, the non-local exchange kernel originating from the kinetic energy operators and the non-local exchange kernel originating from the potential energy, respectively. It is not easy, however, to exactly follow this procedure except for light systems such as the scattering between two α particles. Here we wish to mention that a practical way with reasonable accuracy can be obtained by using the connection between the RGM and the generator coordinate method (GCM), and by introducing several approximations relevant to heavy ion collisions [17]. One of the simplest approximations reads [17] [TD + VD + EKN ]χ˜ ≈ E χ˜
(7)
Notice that the anti-symmetrization introduces an explicitly energy dependent potential. Notice also that if we identify VD in eq.(7) with the double folding potential VDF with an M3Y nucleon nucleon interaction, then it includes a part of KV in eq.(6), i.e. the knock-on exchange term, in addition to the direct term. Fig.5a compares the fusion excitation function for the 16 O+208 Pb reaction calculated by the folding potential with M3Y force (the dashed line) and by including the EKN term in eq.(7)(the solid line). We have calculated the normalization kernel by converting the GCM normalization kernel to the RGM normalization kernel using several approximations relevant to heavy-ion collisions and by converting the original non-local normalization kernel into its Wigner transform [18]. The details of the procedure will be published in [17].
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Figure 5. (a) The fusion excitation function and (b) the effective potentials for 16 O+208 Pb reaction. The N and N ’ are appropriately chosen normalization factors.
We observe that the high energy problem, i.e. the overestimate of the fusion cross section at energies above the Coulomb barrier by the double folding model, is cured by taking the effects of anti-symmetrization through the EKN term in eq.(7). In order to examine whether this is a fluke which holds only for 16 O+208 Pb reaction, we examined also 32 S+208 Pb fusion reaction and confirmed a similar effect of the EKN term in eq.(7). In order to understand the origin of the change of the fusion excitation function by the EKN term, we show in Fig.5b the effective potential VDF + EKN in the region of the Coulomb barrier for two collision energies. We observe that the anti-symmetrization shifts the barrier position more inside and makes the barrier height higher with increasing bombarding energy. We wish to comment that the major contribution around the potential barrier arises from the single nucleon exchange terms, though the norm kernel KN we used includes all the exchange terms. 5. Summary We have discussed whether the analysis of fusion cross section can probe the characteristic properties of the structure of Ge isotopes suggested by the experiments of Coulomb excitation. We have shown that the barrier distribution for the 74 Ge+74 Ge fusion reaction calculated by using the transition matrix given by the Coulomb excitation noticeably differs from that in the rotational model, though it qualitatively resembles that in the vibrational coupling model. Our interest in 74 Ge stemed not only from the data of Coulomb excitation, but also from the abnormally large asymptotic energy shift reported in [19]. Though our re-estimate gives a somewhat smaller value, it still significantly deviates from the systematics as discussed in [19]. The coupled channels calculations using the transition matrix elements from the experiments of Coulomb excitation give a noticeably smaller asymptotic energy shift, though it is much larger than the systematic value suggested in [19] indicating the importance of other effects beyond the present coupled channels calculations. In this connection, we wish to remind Ref.[20], which pointed the important role played
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cooperatively by the shell and deformation effects. We analysed also the 72 Ge+72 Ge fusion reaction, and have shown that the shape admixture in the ground state of 72 Ge suggested by the experiments of Coulomb excitation yields a significantly different barrier distribution from that of the vibrational coupling + model including up to the one phonon state. However, the coupling to the 2+ 2 and 41 states leads to the fusion barrier distribution very similar to that obtained in the vibrational model including two phonon states and makes it difficult to probe the effects of shape admixture as long as one uses the vibrational model to assess the coupling strength of the 2+ 1 state to these high lying states. In the second part of this contribution, we have argued that the effect of anti-symmetrization provides an origin of the high energy problem, i.e. the problem that the standard double folding potential overestimates the fusion cross section at energies above the Coulomb barrier, or that a value of the surface diffuseness parameter which is much larger than that accepted by elastic scattering or obtained from the double folding potential is required to fit the fusion excitation function in that energy region. REFERENCES 1. Table of Isotopes, 8th edition, eds. R.B.Firestone and V.S.Shirley, John Wiley and Sons, New York, 1996. 2. R. Lecomte et al., Phys. Rev. C22 (1980) 1530; 2420. 3. Y. Toh et al., Eur. Phys. J. A9 (2000) 353. 4. M. Sugawara et al., Eur. Phys. J. A16 (2003) 409. 5. E.F. Aguilera et al., Phys. Rev. C41 (1990)910. 6. H. Esbensen, Phys. Rev. C68 (2003) 034604. 7. H. Esbensen, Phys. Rev. C72 (2005) 054607. 8. H. Esbensen and S. Misicu, Proc. of the Fusion06, eds. L. Corradi et al., AIP Conference Proceedings 853 (2006), p. 13. 9. N. Takigawa, Proc. of the Fusion06, AIP Conference Proceedings 853 (2006), p. 479. 10. M.M. Sharma, M.A. Nagarajan and P. Ring, Phys.Lett.B312(1993)377. 11. J. Dobaczewski et al., Phys. Rev. Lett. 60 (1988) 2254. 12. I.I. Gontchar et al., Phys. Rev. C69 (2004) 024610. 13. N. Takigawa et al., Proc. of the Fusion03, Prog. Theor. Phys. Suppl. 154 (2004) 192. 14. J.O. Newton et al., Phys.Rev. C70 (2004) 024605. 15. N. Takigawa and K. Ikeda, ANL-PHY-86-1, p.613-620; Proc. of the Intl. Symposium on the Many Facets of Heavy Ion Fusion Reactions, March 24-26, 1986, ANL. 16. The matrices inserted in Figure 2 in [9] should be corrected according to Table 3 of the present √ contribution, i.e. 0.11 should be replaced by 0.38 and the minus sign in front of 2 7 5 should be removed. The error occurred by the misinterpretation of the intrinsic quadrupole moment for the observed quadrupole moment. 17. T. Takehi, K. Hagino and N. Takigawa, to be published. 18. N. Takigawa and K. Hara, Z. Physik A276 (1976) 79. 19. C.E. Aguiar et al. Phys. Lett. B201 (1988) 22. 20. A. Iwamoto and N. Takigawa, Phys. Lett. B219 (1989) 176.