- Email: [email protected]
Cluster structure in nuclei
ELSEWIER
Nuclear Physics A731 (2004) 329-338 www.elsevier.comllocate/npe
Cluster
Structure
in Nuclei
H. Horiuchi” “Department
of Physics,
Kyoto
University,
Kyoto
606-8502,
Japan
Two topics of the cluster structure study are discussed. They are (1) 01 cluster condened states, and (2) su p er d ef ormation and molecular states in 32S. In the first topic, I show that the 3a microscopic cluster model wave functions which were obtained long time ago for the 12C second Of state are almost completely the same as single 3cu condensate model wave functions. The a! condensate model wave functions we used for this study are obtained by projecting out good angular momenta from intrinsic cy condensate wave functions which are spatially deformed. It is found that the effect of deformation is not large for the present case of the 12C 0; states in the sense that the obtained wave functions contain the spherical condensate components more than 90 %. In the second topic, I show from the AMD + GCM study that the superdeformed excited rotational band in 32S is the same as the 160 + 160 lowest molecular band. I explain that the AMD + GCM calculation gives not only the 160 + 160 lowest molecular band which is identified as the superdeformed excited band but also the 160 + 160 second and third lowest molecular bands whose excitation energies are higher than the lowest band by about 10 MeV and 20 MeV, respectively. I point out that the excitation energies of these three lowest bands are almost the same as those of the three lowest bands generated by the so-called “unique optical potential” of the 160 + I”0 scattering.
1. INTRODUCTION Cluster structure is now seen widely both in stable nucleus region and neutron-rich unstable nucleus region. In stable nuclei cluster structure usually contains only clusterized nucleons but in neutron-rich nuclei it cannot be so because we have lots of excess neutrons which do not necessarily form clusters. Today due to the restriction of the talk time I skip the subjects on unstable nuclei and discuss two recent subjects in stable nucleus region. One is about alpha-cluster condensed states in self-conjugate 4n nuclei and the other is about the relation between superdeformation and molecular resonances in 32S. The first subject is studied under the collaboration work with Y. Funaki of Kyoto Univ., A.Tohsaki of Shinshu Univ., P. Schuck of IPN Orsay, and G. Rijpke of Restock Univ., while the second subject is studied under the collaboration work with M. Kimura of RIKEN. 0375-9474/s - see front matter 0 2004 Elsevier doi:10.1016/j.nuc1physa.2003,11.044
B.V.
All rights
reserved.
H. Horiuchi/Nuclear
330 2. ALPHA-CLUSTER
Physics A731 (2004) 329-338
CONDENSED
STATES
IN
NUCLEI
The interaction between clusters which form a cluster structure in a stable nucleus should be weak because if the interaction were strong the clusters would melt to form the mean field of the whole system. Weak inter-cluster interaction implies that the cluster structure shows up near the breakup threshold energy into the constituent clusters. Our present concern is the question : ” What kind of cluster structure is expected to be formed near the no breakup threshold in self-conjugate 4n nuclei ( 2 = N = 2n ) ? ” In this respect we recall the idea of Morinaga [l] who predicted the existence of the linear chain structure of alpha clusters. For example the second Of state in “C which is located slightly above the 3a breakup threshold was assigned to have a linear chain structure of 3a. About a quarter century ago, there were performed fully microscopic 3-body calculations of o clusters ( so-called 3a: RGM calculation ) by Kamimura et al. [3]. These two groups adopted different effective [2] and by U e g a k i and his collaborators mrclear forces and therefore their results are slightly different from each other. Their results reproduced the experimental data fairly well as shown in Table 1. What was remarkable in the results of their calculations was that the wave function of the second O+ state shows that the radius of the second O+ state is very large ( about 1 fm larger than the radius of the first O+ state) and the three alpha clusters interact weakly with each other via relative S waves in this 0: state. This result clearly does not support the idea of the linear chain structure of 301, because the linear chain structure demands large components of higher partial waves than S waves for the inter-a relative motion. The characteristic properties of the second Of state predicted by these two fully microscopic 3a calculations, especially its very large radius and the S-wave dominance of inter-o partial waves were considered to imply that the second Ot state has a gas-like structure of three Q clusters which interact weakly with each other predominantly in relative S waves. This understanding of the 0; state structure had been already reported also by a semi-microscopic 3cu calculation [4]. Recently, by combining the above-mentioned results of the microscopic 3a! calculations of the second O+ state of “C and the results of the investigations [5] by Ri5pke and coworkers on the possibility of a-particle condensation in low-density nuclear matter, Tohsaki, Schuck, REpke, and the present author proposed a conjecture that near the no threshold in self-conjugate 4n nuclei there exist excited states of dilute density, which are composed of a weakly interacting gas of self-bound cy particles and which can be considered as an no condensed state [6]. They expressed the no condensed state by a new o-cluster wave function of the form a,,(B)
= A(fi
exp(-%)4(m)
( ‘. 4(h)>.
i=l
Here X, and q5(a;) stand for the center-of-mass (C.M.) coordinate and the internal wave function of the i-th cy cluster, respectively. This wave function ana expresses the state where the C.M. motions of all the cy clusters occupy the same orbit exp(-2X2/B2); namely the C.M. motions of all the a: clusters are condensed into a single orbit exp(-2X2/B2). The wave function QrA,(B) becomes a shell-model wave function in the limit of I3 = b
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Physics A731 (2004) 329-338
331
Table 1 Good reproduction of 12C data by the full 3a calculations. Cal.1 and Cal.11 mean calculations by Kamimura et al. and by Uegaki et al., respectively. (Oi(a = 6 fm)),q
EXP.
Cal.1
Cal.11
0.72
0.98
0.68
B(E2,0,+
--f 2:)
13 f 4 e2fm4
5.6
3.5
B(E2,2? M-(0$
--f 0;) --t 0:)
7.8 e2fm4 5.4 f 0.2 fm2
9.3 6.7
8.0 6.6
2.40
2.53
3.47
3.50
Rm(O:)
2.43
fm
&ms(Oz+)
Table 2 Threshold states obtained by the GCM calculation by the use of QrL,(B) and 20Ne. Eth stands for the threshold energy of nQ: breakup. 2a 3a: 4cu Threshold state E - Eth (MeV)
where
b is the oscillator
lim anol(B)
= shell model
Bib
0: -0.17
size parameter
0: 0.38
of the a-cluster
= c
5a: 0: 1.8
wave function
4(a):
wave function
The GCM ( generator coordinate method ) calculation which diagonalizes Hamiltonian by superposing an,(B)% over various B was made: Q(O:)
in *Be, 12C, 160,
0; -0.44
internal
the
the microscopic
h(B)ka(B).
B
The GCM calculation succeeded to show the existense of a level of dilute density ( about one-third of ground state density ) in each system of 12C, 160, and 20Ne in the vicinity of the 3q 4q and 5a breakup threshold, respectively, which is shown in Table 2. Since the wave function ana is sph erically symmetric and can describe only a O+ state, it was extended so as to describe non-zero spin states [7]. It was made by introducing the a-cluster condensate with spatial deformation: @,,(B,,
B,, Bz) = A[fi exp{-(% ?,=I
+ 3
+ $)}O(W).
. $(~i~)]
In the case of the 3cu system, the GCM calculation which diagonalizes the microscopic Hamiltonian by superposing @sa(Bz = B,, B,)‘s over various sets of (B, = B,, B,) was made by assuming the axial symmetry of the deformation [8]:
H. Hoviuchi/Nucleav
,332
Physics A731 (2004) 329-338
Table 3 Comparison of the GCM calculation with the deformed-condensation wave function with the full 3cu calculations by Kamimura et al. (Force I) and by Uegaki et al. (Force II). Emirl stands for the minimum energy of the energy surface with good spin. Energies are Force I, ~!&(3~)
= -82.04
Emin -87.68
GCM
Full 3a:
Or:
-89.52
-89.4
G-
-81.55
-81.79
2;t
-84.65
-86.71
Force II, &h(3~)
= -81.01
Emin -86.09
GCM
Full 3a
-87.81
-87.92
-81.7
-79.83
-79.97
-79.3
-86.7
-83.61
-85.34
-85.7
Here PJ stands for the spin projection operator onto the spin J. As effective nuclear forces, the forces adopted by Kamimura et al. [2] (F orce I) and by Uegaki et al. [3] (Force II) were adopted, which enabled the comparison of the GCM results with the results by Kamimura et al. and by Uegaki et al.. The comparison of the GCM results with the full 3a calculations is shown in Table 3. Here E,in stands for the minimum energy of the energy surface with good spin. The energy surface for 0: is the energy surface by the wave function PIPJ’o@s,(BZ = BY, B,), where Pl is defined by
The wave function @r;J=o is the normalized wave function of Ps’o@3,(B2 = B,, B,) that gives the minimum energy of the energy surface of 0 ?. We see in this Table the following two remarkable points: (1) The GCM calculation reproduces almost the same results as the full ICY calculations for two choices of the effective nuclear force ( Force I and Force II). (2) The GCM wave function of the 0; state is almost the same as a single condensate wave function which is orthogonalized to the minimum-energy wave function of the 0: energy surface. This fact is seen from the result that the GCM energy of the 0; state is almost the same as the minimum energy of the 0: energy surface and also from the explicit calculation of the overlap of the orthogonalized condensate wave function with the GCM wave function which proved to be larger than 95 %. The second point implies that the 0; wave functions obtained more than a quarter century ago by the full 3a calculations by the use of two different effective nuclear forces are just equal to a single condensate wave functions of three CYparticles for the respective choices of the effective nuclear force. In Table 4, we compare the rms radii and the monopole matrix element M(O$ 4 0:) between the GCM calculation and the full 3a: calculations. We see here also that the GCM calculation reproduces very well the results of the full 3a calculations. This result gives us another proof of the equivalence between the GCM calculation with the deformedcondensation wave function and the full 3a: calculations. The amount of the deformation of the GCM 0: wave function was checked by calculating the amount of the component of the spherical 3a condensate wave function contained in the GCM 0: wave function. The
H. Horiuchi/Nucleav
Physics A731 (2004) 329-338
333
Table 4 Comparison of the rms radii R,,,, and the monopole matrix element M(Ozf + 0:) between the GCM calculation and the full 3a: calculations by Kamimura et al. (Force I) and by Uegaki et al. (Force II). R,,, are in fm, and M(O$ -+ 0:) are in fm2. Force I
hm(O:) &ns(O~) &n$:)
kqo: -+ 0:)
Force II Full 3cu
GCM
2.40 3.83 2.38 6.45
2.40 3.47 2.38 6.7
GCM
2.40 4.44 2.38 5.36
Full 3~
2.53 3.50 2.50 6.6
sphericity amount was 90.6 % for the Force I and 92.1 % for the Force II, which means that the contribution from the deformation in the GCM 0; wave function is small. In summary, Tables 3 and 4 imply that the full 3a calculations performed long time ago without any prejudices underline the fact that the second Of state of 12C in the vicinity of the 3a: breakup threshold has a gas-like structure of 3a clusters with “Bose condensation”
PI. 3. SUPERDEFORMATION
AND
MOLECULAR
RESONANCES
IN 32S
Microscopic studies of the 160 + 160 molecular states have been made by many authors but have not been able to give a conclusive answer. It is largely because of tha fact that the number of the molecular band, the excitation energy of the band head, and the moment of inertia strongly depend on the effective nuclear force. A rather conclusive answer was given recently by the macroscopic studies [9,10] by using the unique optical potential for the 160 - 160 system which was determined in 1990’s after the first discovery of the nuclear rainbow in 1989 [llj. Th ese studies gave the following answers for the 160 + ‘“0 molecular states: The unique optical potential places the band head Of state of the lowest Pauli-allowed rotational band with the principal quantum number N = 2n + L = 24 for the 160 - 160 relative motion at about 9 MeV in the excitation energy ( about 8 MeV below the 160 + 160 threshold ). The unique potential gives N = 26 and N = 28 bands about 10 MeV and 20 MeV above the N = 24 band, respectively. In Ref. [9] it was states correspond to the N = 28 band proposed that the observed 160 + “0 molecular of the unique optical potential. In these days, besides the cluster model, the superdeformed structure of 32S has been studied by many authors with the mean field theory [12-141. It is partly because many rotational spectra associated with superdefomed bands have been observed recently in bands have not been rather light mass regions [15-171 near A N 40. In 32S superdefomed observed yet, but the idea of the superdeformed double magic number N = 2 = 16 strongly suggests the existence of superdeformed excited states and the superdeformed structure of 32S is regarded as a key to understand the relation between the the superdeformed structure and the I60 + 160 molecular structure. Indeed, by the HF(B)
334
H. Hoviuchi/Nuclear
Physics A731 (2004) 329-338
calculations it has been shown that the superdeformed minimum of the energy surface is well established for each angular momentum, and the wave function at the superdeformed local minimum shows the 160-160-like character. It is to be noticed that the mean-field calculations predict the location of the band head O+ state of the superdeformed band at around 10 MeV in the excitation energy which agrees with the band head energy of the N = 24 band predicted by the ‘so - 160 unique optical potential. Therefore it is enough conceivable that the superdeformed band and the I60 - 160 band with N = 24 are identical. The relation between the superdeformation and the 160 - 160 molecular structure was studied by AMD ( an t’is y mmetrized molecular dynamics ) [18]. Since the effect of the spinorbit force and the many-body dynamics to form the deformed mean field are expected to distort the pure 160 + 160 clustering, the use of AMD is very suitable for this purpose. AMD is known to be able to describe the mean-field structure and at the same time the cluster structure without assuming the existence of any clusters. The AMD framework used in this study was the deformed-base AMD [19,20] .m which the intrinsic basis wave function lint of the system is expressed by a Slater determinant of single-particle deformed wave packets ‘pi: @int = det I~i(ri)l The spatial part 4;(r) of ‘pi has th e f orm of deformed vector Z; and is given as q%(r) = exp{-uz(x
- &z)2
- U,(Y - Zi,)”
Gaussian
centered
at the complex
- uz(z - &)2},
and the spin part xi of ‘p; is the spin coherent state xi = (1 - &)x+ + &XJ with & being a complex number. The oscillator parameters v,, vr,, and v, are common to all the nucleons. The deformed-base AMD wave function has two limits: One is the deformedharmonic-oscillator limit which is reached when the centroids of all the single-particle Gaussian wave packets Zi are at the center of the nucleus and the Gaussian wave packets are deformed. The other limit is the cluster limit which is reached when the centroids of the single-particle Gaussian wave packets are separated into the centers of constituent clusters and the Gaussian wave packets are spherical. The deformed-base AMD wave function @int is parity-projected, @* = (l/2)(1 f P)Q)i,t, and then the energy variation is made with respect to the variational parameters, Z; (; = 1 N A),& (i = 1 N A), and uk (Ic = x, y, z), by using @* under the constraint on the quadrupole deformation p. The energy-minimum wave function under the constraint on the quadrupole deformation ,L3 is expressed as @*(,B). This a*(p) is projected onto the definite spin J, P”@*(p). The GCM calculation is performed by superposing P”@*(p) over various values of p for diagonalizing the microscopic Hamiltonian:
As the effective nuclear force the Gogny DlS force is adopted [21]. The energy surfaces of good spin-parity J+ which are calculated by P”@+(p) show the existence of the superdeformed local minimuma in addition to the normal-deformation minima which correspond to the ground and low-lying rotational band states, Before
H. Hoviuchi/Nuclear Table 5 Excitation 2) (e2fm4)
335
energy Ex. (MeV) and intra-band E2 transition probabilities B(E2; J --t J of the ground band (K” = 0:) and the first excited band (K” = 0:) in 32S. Ground
band (I(”
AMD J
Physics A731 (2004) 329-338
Ex.
= Or)
Excited
EXP B(E2)
Ex.
band
(K”
AMD
EXP
B(E2)
Ex.
31 88
3.78 4.28 6.85
98
9.78
0 2 4
2.3 5.75
66 109
2.23 4.46
6OJr6 72112
3.9 4.8 10.1
6
10.2
130
8.35
> 22.2
12.9
= 0;)
BP4
Ex.
BP4
35.4
+18X
-8.4
discussing the calculated superdeformed states, in order to show the reliability of the AMD calculation, we give in Table 5 the good reproduction of the properties of the ground and low-lying excited rotational band states. In the normal deformed states, the prolately deformed states and the triaxially deformed states (y = 6” N 30”) are energetically degenerate. After the GCM calculation, prolate wave functions mainly contribute to the ground band while the triaxially deformed wave functions to the first excited band. The behavior of the energy surface around the superdeformed local minimum is similar to that of the HF(B) calculations. In each angular momentum state, the super-deformed minimum is well developed and the excitation energy of the Of superdeformed minimum is around 10 MeV. The density distribution of the intrinsic AMD wave function @int at the superdeformed minimum is very similar to that of the HF calculation and shows the 160-160-like feature. In order to investigate the clustering character of the superdeformed states, the energy surfaces by the pure 160 + 160 cluster model wave functions ( Brink wave functions ) were calculated by using the same Gogny force. Compared to the energy of the superdeformed minimum, the minimum energy of the 160 + “0 cluster model is about 10 MeV higher. This fact implies a fairly large distortion of the 160 + 160 cluster structure in the superdeformed states. The expectation value of the two-body spin-orbit force by the superdeformed state is about -4.5 MeV which must be zero in the pure 160 + 160 wave function. Also the expectation value of the repulsive density-dependent force by the superdeformed state is about 6 MeV smaller than that of the pure 160 + 160 wave function. However the magnitude of the 160 + 160 component contained in the superdeformed Of state is calculated to be 57 %, which is a considerable amount of the cluster component ! Now we discuss the results of the AMD + GCM calculation in the excitation energy region of the superdeformed states. The calculation has yielded three rotational bands which are shown in Fig.1. All three bands have large 160 + 160 component, the magnitudes of which for the band head O+ states of the lowest, the second lowest, and the third bands are 42 %, 71 %, and 73 %, respectively. The principal quantum numbers N = 2n + J of the 160 + 160 components of the three bands are N = 24, N = 26, and N = 28, respectively. Hence they are referred to as N = 24, N = 26, and N = 28 bands,
H. Horiuchi/Nuclear
336
-
24
deformed
6
8
10
Physics A731 (2004) 329-338
base AMD+GCM
12
14
16
18
J
Figure 1. The excitation energies of the N = 24, 26, and 28 band members obtained by AMD + GCM ( solid lines ) and by the GCM with 160 + I60 Brink wave functions (dashed lines). Th e members of the N = 26 and 28 bands obtained by AMD + GCM are fragmented into several states and the averaged energies E&, (see text) of the fragmented states are shown for these bands. The excitation energies are measured from the 160 + 160 threshold energy which is displayed by the horizontal dotted line.
respectively. The excitation energies of the N = 24 band members are lower by about 3 MeV than those of the band formed by the superdeformed minimum states of the energy surfaces for individual J. This may be due to the improvement of the description of the mean-field nature and the collective motion by the GCM calculation, because the amount of the 160 + 160 component in the N = 24 band is decreased from 57 % to 42 % by the GCM calculation. In Fig.1, for the sake of comparison, three rotational bands obtained by the GCM calculation using the superposition of the I60 + 160 Brink wave functions are shown. The principal quantum numbers N = 2n+J of the these bands are 24, 26, and 28, for the lowest, the second lowest, and the third bands, respectively. We see in Fig.1 the energy gain of the AMD+GCM band compared to the corresponding pure 160 + 160 band becomes smaller as the quantum number N increases from N = 24 to N = 28. This tendency is reflected to the increase of the amount of the 160 + 160 component contained in the AMD+GCM bands. In the cases of the AMD+GCM bands with N = 26 and N = 28, for each J there are several states which have the I60 + 160 component of the same principal quantum
H. Horiuchi/Nuclear
Physics A731 (2004) 329-338
331
number N. Namely the band members of the AMD+GCM bands with N = 26 and N = 28 are fragmented into several states. For example, the O+ state of the N = 26 band is fragmented into three states which contain the 160 + 160 component by 33 %, 13 %, and 25 %, respectively. These percentage numbers add up to 71 %, which is the number we quoted above when we said that the the 160 + 160 component contained in the band head O+ state of the N = 26 band is 71 %. The AMD+GCM bands with N = 26 and N = 28 shown in Fig.1 are drawn by using the averaged energies Eiv of these fragmented states for individual J where Eiv is defined as Eiv = (wfE{+wiEi+. ..)/(wf+wi+...) with WJ standing for the percentage of the 160 + 160 component contained in the i-th fragmented state. In the AMD+GCM calculation, the states along the AMD energy curve with respect to the deformation parameter /3 are superposed. Since the AMD energy curve is obtained to optimize mainly the N = 24 lowest band members in which the 160 + 160 structure is largely distorted, the states along the AMD energy curve may be inappropriate for the description of the N = 26 and N = 28 excited bands. In other words, the AMD+GCM calculation may underestimate the 160 + 160 structure in the N = 26 and N = 28 excited bands. Therefore the extended GCM calculation was performed in which not only the states along the AMD energy curve but also the I60 + 160 Brink wave functions are included in the basis states to be superposed. The modifications obtained by this calculation for the three rotation1 bands are as follows: (1) The percentage of the 160 + 160 component increased for all the three bands and they are 44010, 90%, and 98% for N= 24, 26, and 28, respectively. The increase of the percentage in the N= 26 and 28 are remarkable. (2) The energy gain is almost negligible for the N= 24 band while it becomes larger as N increases but it amounts to only a few MeV even for the N= 28 band. (3) The fragmentation of the band members are also existent in the N = 26 and 28 bands, and the percentage of the 160 + 160 component q uoted above for the N = 26 and 28 bands are the sum of the percentages of all the fragmented states. It is intersting that the N = 28 band members have almost pure 160 + 160 molecular structure when the fragmented percentages are summed up, and it looks plausible that this band members are assigned to correspond to the observed molecular resonances of 160 + 160 as was proposed in Ref.
PI.
To summalize, we have discussed that the superdeformed band obtained by HF(B) calculations and the Pauli-allowed lowest N=24 band within the 160 + 160 molecular bands are essentially identical. In this band, 160 + 160 structure is distorted by the effect of the deformed-mean-field formation and the spin-orbit force. This distortion is not small and lowers the excitation energy largely, but this band members still have considerable amount of the component of the 160 + 160 structure. In the excited N=26 and 28 bands, the distortion is less important and the band members have prominent molecular structure. The members of these bands are fragmented into several states. The assignment of the N=28 band members to the observed 160 + ‘“0 molecular resonances looks plausible.
REFERENCES 1.
H. Morinaga,
Phys. Rev. 101, 254 (1956);
Phys. Lett.
21, 78 (1966).
338 2.
3.
4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
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Physics A731 (2004) 329--338
Y. Fukushima and M. Kamimura, Proc. Int. Conf. on Nuclear Structure, Tokyo, 1977, ed. T. Marumori (Suppl. of J. Phys. Sot. Japan, Vo1.44, 1978), p.225; M. Kamimura, Nucl. Phys. A 351, 456 (1981). E. Uegaki, S. Okabe, Y. Abe, and H. Tanaka, Prog. Theor. Phys. 57, 1262 (1977); E. Uegaki, Y. Abe, S. Okabe, and H. Tanaka, Prog. Theor. Phys. 59, 1031 (1978); 62, 1621 (1979). H. Horiuchi, Prog. Theor. Phys. 51, 1266 (1974); 53, 447 (1975). G. Rijpke, A. Schnell, P. Schuck, and P. Nozieres, Phys. Rev. Lett. 80, 3177 (1998); M. Beyer, S. A. Sofianos, C. Kuhrts, G. Rijpke, and P. Schuck, Phys. Lett. B 488, 247 (2000). A. Tohsaki, H. Horiuchi, P. Schuck, and G. Rijpke, Phys. Rev. Lett. 87, 192501 (2001). Y. Funaki, H. Horiuchi, A. Tohsaki, P. Schuck, and G. Rijpke, Prog. Theor. Phys. 108, 297 (2002). Y. Funaki, H. Horiuchi, A. Tohsaki, P. Schuck, and G. Riipke, Phys. Rev. C 67, 051306(R) (2003). S. Ohkubo and K. Yamashita, Phys. Rev. C66, 021301 (R) (2002). Y. Kondo, B. A. Robson, and R. Smith, Journ. Phys. Sot. Japan, 58, Supplement, 597 (1989). E. Stiliaris et al., Phys. Lett. B223, 291 (1989). M. Yamagami and K. Matsuyanagi, Nucl.Phys. A672, 123 (2000). H. Molique, J. Dobaczewski, and J. Dudek, Phys. Rev. C61, 044304 (2000). R.R.Rodriguez-Guzmin, J.L.Egido and L.M.Robledo, Phys. Rev. C62, 054308 (2000). C. E. Svensson et al., Phys. Rev. Lett. 85, 2693 (2000). E. Ideguchi et al., Phys. Rev. Lett. 87, 222501 (2001). C. D. O’Leary et al., Phys. Rev.C 61, 064314 (2000). M. Kimura and H. Horiuchi, submitted to Phys. Rev. C. M. Kimura, Phys. Rev. C (2003), to be published. M. Kimura and H. Horiuchi, Prog. Theor. Phys. 107, 33 (2002). J. Decharg& and D. Gogny, Phys. Rev. C 21, 1568 (1980).
Nuclear Physics A731 (2004) 329-338 www.elsevier.comllocate/npe
Cluster
Structure
in Nuclei
H. Horiuchi” “Department
of Physics,
Kyoto
University,
Kyoto
606-8502,
Japan
Two topics of the cluster structure study are discussed. They are (1) 01 cluster condened states, and (2) su p er d ef ormation and molecular states in 32S. In the first topic, I show that the 3a microscopic cluster model wave functions which were obtained long time ago for the 12C second Of state are almost completely the same as single 3cu condensate model wave functions. The a! condensate model wave functions we used for this study are obtained by projecting out good angular momenta from intrinsic cy condensate wave functions which are spatially deformed. It is found that the effect of deformation is not large for the present case of the 12C 0; states in the sense that the obtained wave functions contain the spherical condensate components more than 90 %. In the second topic, I show from the AMD + GCM study that the superdeformed excited rotational band in 32S is the same as the 160 + 160 lowest molecular band. I explain that the AMD + GCM calculation gives not only the 160 + 160 lowest molecular band which is identified as the superdeformed excited band but also the 160 + 160 second and third lowest molecular bands whose excitation energies are higher than the lowest band by about 10 MeV and 20 MeV, respectively. I point out that the excitation energies of these three lowest bands are almost the same as those of the three lowest bands generated by the so-called “unique optical potential” of the 160 + I”0 scattering.
1. INTRODUCTION Cluster structure is now seen widely both in stable nucleus region and neutron-rich unstable nucleus region. In stable nuclei cluster structure usually contains only clusterized nucleons but in neutron-rich nuclei it cannot be so because we have lots of excess neutrons which do not necessarily form clusters. Today due to the restriction of the talk time I skip the subjects on unstable nuclei and discuss two recent subjects in stable nucleus region. One is about alpha-cluster condensed states in self-conjugate 4n nuclei and the other is about the relation between superdeformation and molecular resonances in 32S. The first subject is studied under the collaboration work with Y. Funaki of Kyoto Univ., A.Tohsaki of Shinshu Univ., P. Schuck of IPN Orsay, and G. Rijpke of Restock Univ., while the second subject is studied under the collaboration work with M. Kimura of RIKEN. 0375-9474/s - see front matter 0 2004 Elsevier doi:10.1016/j.nuc1physa.2003,11.044
B.V.
All rights
reserved.
H. Horiuchi/Nuclear
330 2. ALPHA-CLUSTER
Physics A731 (2004) 329-338
CONDENSED
STATES
IN
NUCLEI
The interaction between clusters which form a cluster structure in a stable nucleus should be weak because if the interaction were strong the clusters would melt to form the mean field of the whole system. Weak inter-cluster interaction implies that the cluster structure shows up near the breakup threshold energy into the constituent clusters. Our present concern is the question : ” What kind of cluster structure is expected to be formed near the no breakup threshold in self-conjugate 4n nuclei ( 2 = N = 2n ) ? ” In this respect we recall the idea of Morinaga [l] who predicted the existence of the linear chain structure of alpha clusters. For example the second Of state in “C which is located slightly above the 3a breakup threshold was assigned to have a linear chain structure of 3a. About a quarter century ago, there were performed fully microscopic 3-body calculations of o clusters ( so-called 3a: RGM calculation ) by Kamimura et al. [3]. These two groups adopted different effective [2] and by U e g a k i and his collaborators mrclear forces and therefore their results are slightly different from each other. Their results reproduced the experimental data fairly well as shown in Table 1. What was remarkable in the results of their calculations was that the wave function of the second O+ state shows that the radius of the second O+ state is very large ( about 1 fm larger than the radius of the first O+ state) and the three alpha clusters interact weakly with each other via relative S waves in this 0: state. This result clearly does not support the idea of the linear chain structure of 301, because the linear chain structure demands large components of higher partial waves than S waves for the inter-a relative motion. The characteristic properties of the second Of state predicted by these two fully microscopic 3a calculations, especially its very large radius and the S-wave dominance of inter-o partial waves were considered to imply that the second Ot state has a gas-like structure of three Q clusters which interact weakly with each other predominantly in relative S waves. This understanding of the 0; state structure had been already reported also by a semi-microscopic 3cu calculation [4]. Recently, by combining the above-mentioned results of the microscopic 3a! calculations of the second O+ state of “C and the results of the investigations [5] by Ri5pke and coworkers on the possibility of a-particle condensation in low-density nuclear matter, Tohsaki, Schuck, REpke, and the present author proposed a conjecture that near the no threshold in self-conjugate 4n nuclei there exist excited states of dilute density, which are composed of a weakly interacting gas of self-bound cy particles and which can be considered as an no condensed state [6]. They expressed the no condensed state by a new o-cluster wave function of the form a,,(B)
= A(fi
exp(-%)4(m)
( ‘. 4(h)>.
i=l
Here X, and q5(a;) stand for the center-of-mass (C.M.) coordinate and the internal wave function of the i-th cy cluster, respectively. This wave function ana expresses the state where the C.M. motions of all the cy clusters occupy the same orbit exp(-2X2/B2); namely the C.M. motions of all the a: clusters are condensed into a single orbit exp(-2X2/B2). The wave function QrA,(B) becomes a shell-model wave function in the limit of I3 = b
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Physics A731 (2004) 329-338
331
Table 1 Good reproduction of 12C data by the full 3a calculations. Cal.1 and Cal.11 mean calculations by Kamimura et al. and by Uegaki et al., respectively. (Oi(a = 6 fm)),q
EXP.
Cal.1
Cal.11
0.72
0.98
0.68
B(E2,0,+
--f 2:)
13 f 4 e2fm4
5.6
3.5
B(E2,2? M-(0$
--f 0;) --t 0:)
7.8 e2fm4 5.4 f 0.2 fm2
9.3 6.7
8.0 6.6
2.40
2.53
3.47
3.50
Rm(O:)
2.43
fm
&ms(Oz+)
Table 2 Threshold states obtained by the GCM calculation by the use of QrL,(B) and 20Ne. Eth stands for the threshold energy of nQ: breakup. 2a 3a: 4cu Threshold state E - Eth (MeV)
where
b is the oscillator
lim anol(B)
= shell model
Bib
0: -0.17
size parameter
0: 0.38
of the a-cluster
= c
5a: 0: 1.8
wave function
4(a):
wave function
The GCM ( generator coordinate method ) calculation which diagonalizes Hamiltonian by superposing an,(B)% over various B was made: Q(O:)
in *Be, 12C, 160,
0; -0.44
internal
the
the microscopic
h(B)ka(B).
B
The GCM calculation succeeded to show the existense of a level of dilute density ( about one-third of ground state density ) in each system of 12C, 160, and 20Ne in the vicinity of the 3q 4q and 5a breakup threshold, respectively, which is shown in Table 2. Since the wave function ana is sph erically symmetric and can describe only a O+ state, it was extended so as to describe non-zero spin states [7]. It was made by introducing the a-cluster condensate with spatial deformation: @,,(B,,
B,, Bz) = A[fi exp{-(% ?,=I
+ 3
+ $)}O(W).
. $(~i~)]
In the case of the 3cu system, the GCM calculation which diagonalizes the microscopic Hamiltonian by superposing @sa(Bz = B,, B,)‘s over various sets of (B, = B,, B,) was made by assuming the axial symmetry of the deformation [8]:
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,332
Physics A731 (2004) 329-338
Table 3 Comparison of the GCM calculation with the deformed-condensation wave function with the full 3cu calculations by Kamimura et al. (Force I) and by Uegaki et al. (Force II). Emirl stands for the minimum energy of the energy surface with good spin. Energies are Force I, ~!&(3~)
= -82.04
Emin -87.68
GCM
Full 3a:
Or:
-89.52
-89.4
G-
-81.55
-81.79
2;t
-84.65
-86.71
Force II, &h(3~)
= -81.01
Emin -86.09
GCM
Full 3a
-87.81
-87.92
-81.7
-79.83
-79.97
-79.3
-86.7
-83.61
-85.34
-85.7
Here PJ stands for the spin projection operator onto the spin J. As effective nuclear forces, the forces adopted by Kamimura et al. [2] (F orce I) and by Uegaki et al. [3] (Force II) were adopted, which enabled the comparison of the GCM results with the results by Kamimura et al. and by Uegaki et al.. The comparison of the GCM results with the full 3a calculations is shown in Table 3. Here E,in stands for the minimum energy of the energy surface with good spin. The energy surface for 0: is the energy surface by the wave function PIPJ’o@s,(BZ = BY, B,), where Pl is defined by
The wave function @r;J=o is the normalized wave function of Ps’o@3,(B2 = B,, B,) that gives the minimum energy of the energy surface of 0 ?. We see in this Table the following two remarkable points: (1) The GCM calculation reproduces almost the same results as the full ICY calculations for two choices of the effective nuclear force ( Force I and Force II). (2) The GCM wave function of the 0; state is almost the same as a single condensate wave function which is orthogonalized to the minimum-energy wave function of the 0: energy surface. This fact is seen from the result that the GCM energy of the 0; state is almost the same as the minimum energy of the 0: energy surface and also from the explicit calculation of the overlap of the orthogonalized condensate wave function with the GCM wave function which proved to be larger than 95 %. The second point implies that the 0; wave functions obtained more than a quarter century ago by the full 3a calculations by the use of two different effective nuclear forces are just equal to a single condensate wave functions of three CYparticles for the respective choices of the effective nuclear force. In Table 4, we compare the rms radii and the monopole matrix element M(O$ 4 0:) between the GCM calculation and the full 3a: calculations. We see here also that the GCM calculation reproduces very well the results of the full 3a calculations. This result gives us another proof of the equivalence between the GCM calculation with the deformedcondensation wave function and the full 3a: calculations. The amount of the deformation of the GCM 0: wave function was checked by calculating the amount of the component of the spherical 3a condensate wave function contained in the GCM 0: wave function. The
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Physics A731 (2004) 329-338
333
Table 4 Comparison of the rms radii R,,,, and the monopole matrix element M(Ozf + 0:) between the GCM calculation and the full 3a: calculations by Kamimura et al. (Force I) and by Uegaki et al. (Force II). R,,, are in fm, and M(O$ -+ 0:) are in fm2. Force I
hm(O:) &ns(O~) &n$:)
kqo: -+ 0:)
Force II Full 3cu
GCM
2.40 3.83 2.38 6.45
2.40 3.47 2.38 6.7
GCM
2.40 4.44 2.38 5.36
Full 3~
2.53 3.50 2.50 6.6
sphericity amount was 90.6 % for the Force I and 92.1 % for the Force II, which means that the contribution from the deformation in the GCM 0; wave function is small. In summary, Tables 3 and 4 imply that the full 3a calculations performed long time ago without any prejudices underline the fact that the second Of state of 12C in the vicinity of the 3a: breakup threshold has a gas-like structure of 3a clusters with “Bose condensation”
PI. 3. SUPERDEFORMATION
AND
MOLECULAR
RESONANCES
IN 32S
Microscopic studies of the 160 + 160 molecular states have been made by many authors but have not been able to give a conclusive answer. It is largely because of tha fact that the number of the molecular band, the excitation energy of the band head, and the moment of inertia strongly depend on the effective nuclear force. A rather conclusive answer was given recently by the macroscopic studies [9,10] by using the unique optical potential for the 160 - 160 system which was determined in 1990’s after the first discovery of the nuclear rainbow in 1989 [llj. Th ese studies gave the following answers for the 160 + ‘“0 molecular states: The unique optical potential places the band head Of state of the lowest Pauli-allowed rotational band with the principal quantum number N = 2n + L = 24 for the 160 - 160 relative motion at about 9 MeV in the excitation energy ( about 8 MeV below the 160 + 160 threshold ). The unique potential gives N = 26 and N = 28 bands about 10 MeV and 20 MeV above the N = 24 band, respectively. In Ref. [9] it was states correspond to the N = 28 band proposed that the observed 160 + “0 molecular of the unique optical potential. In these days, besides the cluster model, the superdeformed structure of 32S has been studied by many authors with the mean field theory [12-141. It is partly because many rotational spectra associated with superdefomed bands have been observed recently in bands have not been rather light mass regions [15-171 near A N 40. In 32S superdefomed observed yet, but the idea of the superdeformed double magic number N = 2 = 16 strongly suggests the existence of superdeformed excited states and the superdeformed structure of 32S is regarded as a key to understand the relation between the the superdeformed structure and the I60 + 160 molecular structure. Indeed, by the HF(B)
334
H. Hoviuchi/Nuclear
Physics A731 (2004) 329-338
calculations it has been shown that the superdeformed minimum of the energy surface is well established for each angular momentum, and the wave function at the superdeformed local minimum shows the 160-160-like character. It is to be noticed that the mean-field calculations predict the location of the band head O+ state of the superdeformed band at around 10 MeV in the excitation energy which agrees with the band head energy of the N = 24 band predicted by the ‘so - 160 unique optical potential. Therefore it is enough conceivable that the superdeformed band and the I60 - 160 band with N = 24 are identical. The relation between the superdeformation and the 160 - 160 molecular structure was studied by AMD ( an t’is y mmetrized molecular dynamics ) [18]. Since the effect of the spinorbit force and the many-body dynamics to form the deformed mean field are expected to distort the pure 160 + 160 clustering, the use of AMD is very suitable for this purpose. AMD is known to be able to describe the mean-field structure and at the same time the cluster structure without assuming the existence of any clusters. The AMD framework used in this study was the deformed-base AMD [19,20] .m which the intrinsic basis wave function lint of the system is expressed by a Slater determinant of single-particle deformed wave packets ‘pi: @int = det I~i(ri)l The spatial part 4;(r) of ‘pi has th e f orm of deformed vector Z; and is given as q%(r) = exp{-uz(x
- &z)2
- U,(Y - Zi,)”
Gaussian
centered
at the complex
- uz(z - &)2},
and the spin part xi of ‘p; is the spin coherent state xi = (1 - &)x+ + &XJ with & being a complex number. The oscillator parameters v,, vr,, and v, are common to all the nucleons. The deformed-base AMD wave function has two limits: One is the deformedharmonic-oscillator limit which is reached when the centroids of all the single-particle Gaussian wave packets Zi are at the center of the nucleus and the Gaussian wave packets are deformed. The other limit is the cluster limit which is reached when the centroids of the single-particle Gaussian wave packets are separated into the centers of constituent clusters and the Gaussian wave packets are spherical. The deformed-base AMD wave function @int is parity-projected, @* = (l/2)(1 f P)Q)i,t, and then the energy variation is made with respect to the variational parameters, Z; (; = 1 N A),& (i = 1 N A), and uk (Ic = x, y, z), by using @* under the constraint on the quadrupole deformation p. The energy-minimum wave function under the constraint on the quadrupole deformation ,L3 is expressed as @*(,B). This a*(p) is projected onto the definite spin J, P”@*(p). The GCM calculation is performed by superposing P”@*(p) over various values of p for diagonalizing the microscopic Hamiltonian:
As the effective nuclear force the Gogny DlS force is adopted [21]. The energy surfaces of good spin-parity J+ which are calculated by P”@+(p) show the existence of the superdeformed local minimuma in addition to the normal-deformation minima which correspond to the ground and low-lying rotational band states, Before
H. Hoviuchi/Nuclear Table 5 Excitation 2) (e2fm4)
335
energy Ex. (MeV) and intra-band E2 transition probabilities B(E2; J --t J of the ground band (K” = 0:) and the first excited band (K” = 0:) in 32S. Ground
band (I(”
AMD J
Physics A731 (2004) 329-338
Ex.
= Or)
Excited
EXP B(E2)
Ex.
band
(K”
AMD
EXP
B(E2)
Ex.
31 88
3.78 4.28 6.85
98
9.78
0 2 4
2.3 5.75
66 109
2.23 4.46
6OJr6 72112
3.9 4.8 10.1
6
10.2
130
8.35
> 22.2
12.9
= 0;)
BP4
Ex.
BP4
35.4
+18X
-8.4
discussing the calculated superdeformed states, in order to show the reliability of the AMD calculation, we give in Table 5 the good reproduction of the properties of the ground and low-lying excited rotational band states. In the normal deformed states, the prolately deformed states and the triaxially deformed states (y = 6” N 30”) are energetically degenerate. After the GCM calculation, prolate wave functions mainly contribute to the ground band while the triaxially deformed wave functions to the first excited band. The behavior of the energy surface around the superdeformed local minimum is similar to that of the HF(B) calculations. In each angular momentum state, the super-deformed minimum is well developed and the excitation energy of the Of superdeformed minimum is around 10 MeV. The density distribution of the intrinsic AMD wave function @int at the superdeformed minimum is very similar to that of the HF calculation and shows the 160-160-like feature. In order to investigate the clustering character of the superdeformed states, the energy surfaces by the pure 160 + 160 cluster model wave functions ( Brink wave functions ) were calculated by using the same Gogny force. Compared to the energy of the superdeformed minimum, the minimum energy of the 160 + “0 cluster model is about 10 MeV higher. This fact implies a fairly large distortion of the 160 + 160 cluster structure in the superdeformed states. The expectation value of the two-body spin-orbit force by the superdeformed state is about -4.5 MeV which must be zero in the pure 160 + 160 wave function. Also the expectation value of the repulsive density-dependent force by the superdeformed state is about 6 MeV smaller than that of the pure 160 + 160 wave function. However the magnitude of the 160 + 160 component contained in the superdeformed Of state is calculated to be 57 %, which is a considerable amount of the cluster component ! Now we discuss the results of the AMD + GCM calculation in the excitation energy region of the superdeformed states. The calculation has yielded three rotational bands which are shown in Fig.1. All three bands have large 160 + 160 component, the magnitudes of which for the band head O+ states of the lowest, the second lowest, and the third bands are 42 %, 71 %, and 73 %, respectively. The principal quantum numbers N = 2n + J of the 160 + 160 components of the three bands are N = 24, N = 26, and N = 28, respectively. Hence they are referred to as N = 24, N = 26, and N = 28 bands,
H. Horiuchi/Nuclear
336
-
24
deformed
6
8
10
Physics A731 (2004) 329-338
base AMD+GCM
12
14
16
18
J
Figure 1. The excitation energies of the N = 24, 26, and 28 band members obtained by AMD + GCM ( solid lines ) and by the GCM with 160 + I60 Brink wave functions (dashed lines). Th e members of the N = 26 and 28 bands obtained by AMD + GCM are fragmented into several states and the averaged energies E&, (see text) of the fragmented states are shown for these bands. The excitation energies are measured from the 160 + 160 threshold energy which is displayed by the horizontal dotted line.
respectively. The excitation energies of the N = 24 band members are lower by about 3 MeV than those of the band formed by the superdeformed minimum states of the energy surfaces for individual J. This may be due to the improvement of the description of the mean-field nature and the collective motion by the GCM calculation, because the amount of the 160 + 160 component in the N = 24 band is decreased from 57 % to 42 % by the GCM calculation. In Fig.1, for the sake of comparison, three rotational bands obtained by the GCM calculation using the superposition of the I60 + 160 Brink wave functions are shown. The principal quantum numbers N = 2n+J of the these bands are 24, 26, and 28, for the lowest, the second lowest, and the third bands, respectively. We see in Fig.1 the energy gain of the AMD+GCM band compared to the corresponding pure 160 + 160 band becomes smaller as the quantum number N increases from N = 24 to N = 28. This tendency is reflected to the increase of the amount of the 160 + 160 component contained in the AMD+GCM bands. In the cases of the AMD+GCM bands with N = 26 and N = 28, for each J there are several states which have the I60 + 160 component of the same principal quantum
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331
number N. Namely the band members of the AMD+GCM bands with N = 26 and N = 28 are fragmented into several states. For example, the O+ state of the N = 26 band is fragmented into three states which contain the 160 + 160 component by 33 %, 13 %, and 25 %, respectively. These percentage numbers add up to 71 %, which is the number we quoted above when we said that the the 160 + 160 component contained in the band head O+ state of the N = 26 band is 71 %. The AMD+GCM bands with N = 26 and N = 28 shown in Fig.1 are drawn by using the averaged energies Eiv of these fragmented states for individual J where Eiv is defined as Eiv = (wfE{+wiEi+. ..)/(wf+wi+...) with WJ standing for the percentage of the 160 + 160 component contained in the i-th fragmented state. In the AMD+GCM calculation, the states along the AMD energy curve with respect to the deformation parameter /3 are superposed. Since the AMD energy curve is obtained to optimize mainly the N = 24 lowest band members in which the 160 + 160 structure is largely distorted, the states along the AMD energy curve may be inappropriate for the description of the N = 26 and N = 28 excited bands. In other words, the AMD+GCM calculation may underestimate the 160 + 160 structure in the N = 26 and N = 28 excited bands. Therefore the extended GCM calculation was performed in which not only the states along the AMD energy curve but also the I60 + 160 Brink wave functions are included in the basis states to be superposed. The modifications obtained by this calculation for the three rotation1 bands are as follows: (1) The percentage of the 160 + 160 component increased for all the three bands and they are 44010, 90%, and 98% for N= 24, 26, and 28, respectively. The increase of the percentage in the N= 26 and 28 are remarkable. (2) The energy gain is almost negligible for the N= 24 band while it becomes larger as N increases but it amounts to only a few MeV even for the N= 28 band. (3) The fragmentation of the band members are also existent in the N = 26 and 28 bands, and the percentage of the 160 + 160 component q uoted above for the N = 26 and 28 bands are the sum of the percentages of all the fragmented states. It is intersting that the N = 28 band members have almost pure 160 + 160 molecular structure when the fragmented percentages are summed up, and it looks plausible that this band members are assigned to correspond to the observed molecular resonances of 160 + 160 as was proposed in Ref.
PI.
To summalize, we have discussed that the superdeformed band obtained by HF(B) calculations and the Pauli-allowed lowest N=24 band within the 160 + 160 molecular bands are essentially identical. In this band, 160 + 160 structure is distorted by the effect of the deformed-mean-field formation and the spin-orbit force. This distortion is not small and lowers the excitation energy largely, but this band members still have considerable amount of the component of the 160 + 160 structure. In the excited N=26 and 28 bands, the distortion is less important and the band members have prominent molecular structure. The members of these bands are fragmented into several states. The assignment of the N=28 band members to the observed 160 + ‘“0 molecular resonances looks plausible.
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