Isoscalar monopole excitations in O: -cluster states at low energy and mean-field-type states at higher energy
Isoscalar monopole strength function in O up to MeV is discussed. We found that the fine structures at the low energy region up to MeV in the experimental monopole strength function obtained by the O reaction can be rather satisfactorily reproduced within the framework of the cluster model, while the gross three bump structures observed at the higher energy region ( MeV) look likely to be approximately reconciled by the mean-field calculations such as RPA and QRPA. In this paper, it is emphasized that two different types of monopole excitations exist in O; one is the monopole excitation to cluster states which is dominant in the lower energy part ( MeV), and the other is the monopole excitation of the mean-field type such as one-particle one-hole () which is attributed mainly to the higher energy part ( MeV). It is found that this character of the monopole excitations originates from the fact that the ground state of O with the dominant doubly closed shell structure has a duality of the mean-field-type as well as -clustering character. This dual nature of the ground state seems to be a common feature in light nuclei.
PACS numbers: 23.20.-g, 21.60.Gx, 27.20.+n
Isoscalar monopole excitation in nuclei provides important information on its underlying structure. In the collective liquid drop model, the isoscalar giant monopole resonance (ISGMR), which has been established in medium and heavy nuclei youngblood77 , corresponds to a breathing mode of the nucleus arising due to in-phase oscillations of the proton and neutron fluids. In heavy nuclei, the ISGMR is observed as a single peak in the inelastic scattering cross sections at small angles, and its excitation energy follows an empirical formula MeV, which is directly related to the compressibility of nuclear matter. A lot of work has been done to extract experimentally the nuclear compressibility by comparing it with microscopic calculations, for example, using the random phase approximation (RPA). Recently the isoscalar monopole distributions in Zr, Sn, Sm, and Pb were measured with greater precision than previously youngblood99 . The results indicated that the compressibility of nuclear matter is MeV.
It is interesting to study what happens for the ISGMR in lighter nuclei. When the nuclear masses decrease from medium nuclei to light nuclei, the surface-energy correction becomes more important and the excitation energy of ISGMR should become lower compared with the empirical formula, indicating a lower nuclear compressibility pandharipande70 ; blaizot76 ; lebrun80 . A lot of theoretical work has been so far devoted to the study of ISGMR in light nuclei, for example, within the RPA framework blaizot76 ; drozdz80 ; paar06 ; papakonstantinou07 ; papakonstantinou09 ; ma97 and others furuta10 .
The RPA calculations with the non-relativistic framework were performed in O, Ca, Zr, and Pb blaizot76 ; drozdz80 ; paar06 ; papakonstantinou07 ; papakonstantinou09 . According to the results with the Gogny force and Skyrme forces etc. blaizot76 , it was found that 1) the isoscalar monopole strength in O spreads out over some energy region of MeV with its centroid energy being MeV, the value of which depends on the interactions employed, 2) the monopole strength becomes more and more concentrated in a single peak as the nucleus becomes heavier, 3) the percentage of energy-weighted sum rule carried by the resonances increases with the mass number of the nucleus, and 4) the nuclear compressibility becomes smaller as the nucleus becomes lighter. On the other hand, Ma et al. investigated the isoscalar monopole modes in O, Ca, Zr, and Pb using the relativistic RPA (RRPA) method ma97 . They found that when going from heavy to lighter nuclei, the single-peak structure of ISGMR in Pb changes to a peak with several small humps in Ca and eventually the monopole strength spreads out widely to form a couple of peaks in O. Hence, it was pointed out that it becomes difficult, in a nucleus like Ca, to define theoretically the energy and width of the ISGMR.
The experimental isoscalar monopole strengths with a great precision were recently provided in C, O and Mg up to MeV, using inelastic scattering of particles, by the Texas group lui01 . They found that the isoscalar monopole strength in light nuclei does not concentrate on a single peak and the monopole strength spreads out in several regions of energies. The histogram in Fig. 1 shows the experimental isoscalar monopole strength function in O lui01 . It is compared with the RRPA calculation by Ma et al. ma97 . It was found that the centroid in the RRPA response function is at MeV, which is higher than the experimental data ( MeV). In order to match their calculation to the experimental centroid, the calculated strength function was shifted down in energy by MeV and furthermore they normalized it to approximately 30% of the isoscalar energy weighted sum rule (EWSR) by multiplying the RRPA curve by a factor of lui01 . Then, the normalized and shifted curve and the experimental result are in moderately good agreement with each other with respect to the shape of the gross three-peak structure. However, their calculation failed to reproduce the states found in the low energy region ( MeV), in particular, at , and MeV observed in inelastic scattering and electron scattering etc. lui01 ; ajzenberg93 . According to the O() experiments ajzenberg93 , the three states are excited rather strongly by the reaction, and their monopole matrix elements are , , and fm, respectively, comparable to the single-particle monopole strength yamada08 . The total percentage of the energy weighted strength to the isoscalar monopole EWSR (energy weighted sum rule) for these three states amounts to be as large as over ajzenberg93 ; yamada08 .
In the nonrelativistic calculation for O blaizot76 a significant discrepancy is also revealed as compared with the experimental data, in particular, in the low energy region ( MeV), although the gross structures at the higher energy region ( MeV) in the RPA calculations are in rather good agreement with the experimental data. This discrepancy in the low-energy region can also be seen in Fig. 2 obtained by the recent second random-phase approximation (SRPA) calculations with a Skyrme force for O gambacurta10 , in which the coupling between and as well as between configurations among themselves are fully taken into account. In particular, their calculation fails to reproduce the monopole transition strength to the state at MeV observed by the O experiment. Thus, the monopole strengths in the lower energy region ( MeV) are likely to be out of scope in the mean field theory. These results mean that the monopole strength function of O is not fully understood in the mean-field theory at the present stage, and other degree of freedoms beyond the mean field should be taken into account.
The and levels of O including its ground state, together with their monopole strengths, have in the past nicely been reproduced with a semimicroscopic cluster model, i.e. the +C orthogonality condition model (OCM) suzuki76 . The OCM is an approximation of the resonating group method (RGM) saito68 . Many successful applications of OCM are reported in Ref. ptp_supple_68 . The +C OCM calculation as well as the +C generator-coordinate-method one libert80 demonstrates that the state at MeV and the state at MeV have +C structures, where the particle orbits around the C() core in an wave and around the C() core in a wave, respectively. The 14.1-MeV state, however, could not be explained by the +C model calculations suzuki76 ; libert80 .
Recently the structure study of O has made a great advance up to MeV around the disintegration threshold. The six lowest states of O, up to MeV, including the ground state, have for first time been reproduced very well with the OCM funaki08 . The OCM shares 68 %of the energy weighted sum rule value of the isoscalar monopole transition of O, while the +C OCM does 31 %, as will be discussed below. Thus, it is interesting to investigate whether the OCM can reproduce the experimental isoscalar monopole strength function in the low energy region up to MeV in O, a region which is difficult to be treated in the mean-field theory. As will be discussed below, the five excited states of O up to MeV have -cluster structures funaki08 ; suzuki76 ; libert80 ; ptp_supple_68 .
The purpose of the present paper is two fold: first is to show that the isoscalar monopole strength function calculated with the OCM is in good correspondence to the experimental one in the low energy region up to MeV shown in Fig. 1, and the second is to emphasize two features in the isoscalar monopole excitation of O, i.e. that the monopole excitation to cluster states is dominant in the lower energy part ( MeV) of the monopole strength function, whereas the monopole excitation of the -type contributes to the higher energy region ( MeV). We will show that the two features arise from the fact that the ground state of O originally possesses a dual nature allowing -type excitations as well as -type ones, as will be discussed below. In this paper, a shell model calculation with the model space of -, -, -, and -shells for O is also performed to investigate the extent to which the shell model works for describing the low-lying states.
In Sec. II, the monopole excitation function with the OCM is formulated after a brief explanation of the OCM framework together with the shell-model framework for O. Results and discussions are devoted to Sec. III, together with the energy weighted sum rule of the isoscalar monopole transition. Finally we present a summary in Sec. IV.
First we formulate the isoscalar monopole strength function within the framework of the OCM. Then, the formulation of the shell model analysis is presented for O within the model space of , , , and shells.
ii.1 Monopole strength function
The strength function of the monopole excitation from the O ground state is defined with use of the isoscalar monopole operator as follows,
where () are the coordinates of nucleons, is the c.o.m. coordinate of O, and denotes the excitation energy of the state of O. On the other hand, the response function for the transition operator is defined as
with representing an infinitesimal positive number. Then, is related to through
When the state is a resonance state with the complex energy , the strength function is expressed as
where represents the width of the state. The isoscalar monopole transition matrix element, , has a relation with the transition matrix element for the and states with the total isospin as follows,
The energy weighted sum rule (EWSR) of the isoscalar monopole transition yamada08 reads
where and represent the r.m.s radius of the ground state and nucleon mass, respectively. Here, we assume that the interaction has no velocity dependence. Employing the experimental charge radius of O ( fm ajzenberg93 ), the value of in Eq. (8) is estimated to be fm, in which the effects of the charge radius of proton ( fm) and that of neutron ( fm) ajzenberg93 are subtracted from the charge radius of O (): fm. Then, the total EWSR value, , is fmMeV.
It is instructive to see a characteristic feature of the isoscalar monopole operator in Eq. (5), which can be decomposed into two parts, internal parts and relative parts, with respect to clusters in O (as well as and C clusters in O). Since the operator in Eq. (5) has a quadratic form with respect to the coordinates of nucleons, the following interesting identities are realized,
where is the c.o.m. coordinate of the -th cluster, and ( are Jacobi coordinates with respect to the c.o.m. coordinates of clusters (, ): , , and . In Eq. (10), and stand for the c.o.m. coordinates of and C clusters, respectively. Here we should recall the useful identity of in Eq. (9), where (, , and ) correspond to the reduced masses with respect to the Jacobi coordinates .
Equation (9) shows that the monopole operator consists of two parts: 1) internal parts [first term in the right hand of Eq. (9)] composed of the internal coordinates of each -cluster, and 2) relative parts [second term in the right hand of Eq. (9)] acting on the relative motions of the clusters with respect to the c.o.m. of O. On the other hand, Equation (10) shows that the monopole operator can also be decomposed into two other parts: 1) two internal parts, i.e. first and second terms in the right hand of Eq. (10), composed of the internal coordinates of the and C clusters, respectively, and 2) relative part acting on the relative motion between the and C clusters. The fact that the isoscalar monopole operator consists of the two parts, the internal part and the relative part, plays an important role in the monopole excitation of O, see Sec. III.
The total wave function of the system with total angular momentum in the OCM framework is expressed by the product of the internal wave functions of clusters and the relative wave function among the clusters
The relative wave function is expanded in terms of Gaussian basis functions as follows,
where , and are the Jacobi coordinates describing internal motions of the system. stands for the symmetrization operator acting on all particles obeying Bose statistics. denotes the set of size parameters and of the normalized Gaussian function, , and the set of relative orbital angular momentum channels depending on either of the coordinate type of or funaki08 ; GEM , where , and are the orbital angular momenta with respect to the corresponding Jacobi coordinates. Equation (14) represents the orthogonality condition that the total wave function (12) should be orthogonal to the Pauli-forbidden states of the system, ’s, which are constructed from Pauli forbidden states between two -particles in , and states horiuchi_77 . The ground state with the dominant shell-model-like configuration can be described properly in the present OCM framework, as discussed below.
The Hamiltonian for is given as follows:
where , , , and stand for the operators of kinetic energy for the -th particle, two-body, Coulomb, three-body and four-body forces between particles, respectively. The center-of-mass kinetic energy is subtracted from the Hamiltonian. The effective - interaction is constructed by the folding procedure from an effective two-nucleon force. Here we take the Modified Hasegawa-Nagata (MHN) force mhn as the effective force, which is constructed based on the -matrix theory. It is noted that the folded - potential reproduces the - scattering phase shifts and energies of the Be ground state and of the Hoyle state. The three-body force is phenomenologically introduced so as to fit the ground state energy of the C with the framework of the OCM. The same force parameter set as used in Ref. yamada05_3a_ocm is adopted in the present calculation. In addition, the phenomenological four-body force is adjusted to the ground state energy of O. The three-body and four-body forces are short-range, and, hence, they only act in compact configurations. The coefficients in Eq. (12) are determined according to the Rayleigh-Ritz variational principle.
The isoscalar monopole matrix element is evaluated as follows:
where is the r.m.s. radius of particle, yamada05_3a_ocm ; funaki08 . It is important to study the EWSR of the isoscalar monopole transition within the framework of the OCM. We call it the OCM-EWSR, and its definition reads
where is given in Eq. (15) and [see Eq. (17)], and denotes the r.m.s. radius of the O ground state given in Eq. (8). It is noted that the OCM can describe the shell-model-like structure of the O ground state, as shown later. Then, the ratio of the OCM-EWSR to the total EWSR in Eq. (7) is
Here we use fm and fm, which are estimated from the experimental charge radii ( fm and fm, respectively ajzenberg93 ) with subtracting the effects of the charge radius of proton and that of neutron from them, the method of which is the same as that shown in previous section. This result means that the OCM framework shares about of the total EWSR value (the OCM-EWSR is also discussed in Appendix). This is one of the important reasons that the OCM works rather well in reproducing the isoscalar monopole transitions in the low-energy region of O as shown later.
In the present paper, the energies and isoscalar monopole matrix elements in Eq. (4) are obtained by the OCM calculation. As for the widths , we estimate the -decay widths with the -matrix theory lane58 ,
where , and are the wave number of the -C relative motion, the channel radius, and the reduced mass, respectively, and , , and are the regular and irregular Coulomb wave functions and the corresponding penetration factor, respectively. The reduced width of is related to the reduced width amplitude or overlap amplitude as, , and the definition of is presented as
Here, is the wave function of C, given by the OCM calculation yamada05_3a_ocm , and is the relative distance between the center-of-mass of C and the particle. The spectroscopic factor of the +C() channel in the state of O, defined as
is useful to analyze the obtained wave functions.
In the present study, we perform more careful analyses than the previous ones funaki08 , in particular, for identifying the states around the threshold. The calculation of the resonant state in the bound state approximation is usually done by diagonalizing the Hamiltonian with use of a finite number of square-integrable basis wave functions. The positive-energy eigenstates obtained by the diagonalization are divided into resonant states and continuum states, and many methods for carrying out the division are proposed kukulin_book89 . In the present study, a pseudopotential method is adopted to divide the resonant states and continuum states, as shown below.
Let’s us first consider a repulsive pseudopotential that is added to the original Hamiltonian , yielding
where is a constant used to vary the strength of the pseudopotential. As increasing into negative values the constant from the physical value, , the eigenenergy of this new Hamiltonian decreases for any resonance state, which is eventually transformed into a bound state. On the contrary, continuum states show almost no change in their eigenvalues as increases into the negative region. In the present OCM framework, it is important to study the eigenenergies with changing the constant but with no change in the threshold energies of the +C and decay channels, even though we introduce the pseudopotential . Here, we take the four-body potential in Eq. (15) as the pseudopotential , because the choice is convenient for practical reasons in the present numerical calculation. This pseudopotential method is simple but helpful to identify the resonant states under the bound state approximation. As a result, we obtained almost the same results as the previous ones funaki08 , as will be shown below.
ii.3 Shell model calculation
The shell model Hamiltonian of O adopted here is presented as follows:
where denotes the kinetic energy of the -th nucleon, and () represents the central () force of the effective interaction. The c.o.m. kinetic energy is subtracted from the total kinetic energy. The model space adopted here covers all configurations of and within the , , , and shells. The spurious states of the c.o.m. motion are eliminated with the Lawson’s method lawson80 .
In the present study, we take the Volkov No. 2 force volkov65 and G3RS force tamagaki68 for and , respectively. The Majorana parameter () in the Volkov No.2 force, the multiplying factor () of the G3RS, and the nucleon size parameter () are chosen so as to reproduce the total binding energies of the ground states of O and O, the -splitting between and in O, and the r.m.s. radius of the ground state of O as well as possible. The following two parameter sets are adopted: case A for and case B for . The present shell-model code is based on the code in Refs. myo07 ; myo11 .
Iii Results and discussion
iii.1 OCM calculation
The energy levels of states in O obtained by the present OCM calculation are shown in Fig. 3 and Table 1. One can make the one-to-one correspondence of the six lowest states observed up to MeV in the OCM calculation. It is reminded that the +C OCM cluster model suzuki76 can reproduce only the lowest three states. We obtained almost the same results as the previous OCM calculation funaki08 . The six states have the following characteristic structures funaki08 : 1) the ground state () has dominantly a doubly-closed-shell structure, 2) the state at MeV and the state at MeV have mainly C structures horiuchi68 where the -particle orbits around the C core in an -wave and around the C core in a -wave, respectively, the results of which are consistent with the previous studies with C OCM suzuki76 and the C Generator Coordinate Method (GCM) libert80 , 3) the ( MeV) and ( MeV) states mainly have C structure with higher nodal behavior and C structure, respectively, where in the latter the particle moves around the C core (corresponding to the first state at MeV having an intermediate structure between the shell-model-like structure and cluster structure ptp_supple_68 ) in -orbit, and 4) the state at MeV is a strong candidate of the condensate, , with the probability of .
These characteristic features of the structures of the six states can be verified from the analysis of the spectoscopic factors defined in Eq. (29). The results are shown in Fig. 4. Since the ground state has a closed shell structure with the dominant component of SU(3) elliott58 , the values of the spectroscopic factors for in Fig. 4 can be explained by the SU(3) nature of the state. This SU(3) character was confirmed by the recent no-core shell model dytrych07 . As mentioned above, the structures of the and states are well established as having the C and C cluster structures, respectively. These structures of the and states are confirmed by the OCM calculation. In fact, one sees in Fig. 4 that the factors for the C and C channels are dominant in the and states, respectively. In the state, however, the factors of the +C channel in the OCM calculation is rather large as compared with the result of the +C OCM one suzuki76 . This is due to the following facts: 1) The calculated excitation energy of the C state in the present OCM calculation is underestimated by MeV, while it is set to the experimental value ( MeV) in the +C OCM calculation, and 2) thus in the OCM calculation a coupled-channel effect of the +C channel with the +C channel is reinforced and consequently, the factor of the +C channel becomes larger. We expect that the factor of the +C channel will be smaller when the excitation energy of C is properly reproduced in the OCM calculation.
Table 1 lists the transition matrix elements . The value of the state is reproduced well, while that of the state underestimates the experimental result, and this trend is similar to the result of the +C OCM model suzuki76 . On the other hand, the and states mainly have the C structure with higher nodal behavior and an C structure, respectively. The transition matrix element of the state is reproduced nicely within the experimental error (see Table 1). In Table 1, the largest r.m.s. radius is about 5 fm for the state, the wave function of which has a large overlap amplitude with the C channel (see Fig. 3 in Ref. funaki08 ). Hence the factor of the C channel is dominant in the state (see Fig. 4), whereas those in the other channels are much suppressed. This dominance of the factor of the C channel is one of the evidences for the state being the -condensate-state, , because the Hoyle state has the main configuration, tohsaki01 ; funaki03 ; yamada05_3a_ocm , and the overlap amplitude of becomes large.
As for the decay widths of the and states, the results are shown in Table 1. The calculated width of the state is keV, which is quite a bit larger than that found for the state keV. Both are quantitatively consistent with the corresponding experimental data, keV and keV, respectively. On the other hand, the decay width of the state is very small, keV, in reasonable agreement with the corresponding experimental value of 166 keV, indicating that this state is unusually long lived. We should note that our calculation consistently reproduces the ratio of the widths of the , , and states, i.e. about , respectively (see Table 1).
Comparing the energy levels of the six states with the experimental monopole response function of O shown in Fig. 1, one notices that the energy positions of the fine structures in the low energy region ( MeV) of the experimental response function seem to be in good correspondence with the energy levels of , , , and in Fig. 3 and Table 1. It should be noted that the peak corresponding to the state at MeV is not visible in Fig. 1, because of an energy cut in the experimental condition lui01 . Thus, it is important to study the isoscalar monopole strength function within the framework of the OCM calculation.
iii.2 Isoscalar monopole excitation function with the OCM calculation
Figure 5(a) shows the calculated isoscalar monopole strength function of O defined in Eq. (4), where we use the calculated monopole matrix elements and the calculated decay widths for the six states up to MeV obtained by the OCM calculation (see Table 1), also the experimental excitation energies for the six states are employed. We take into account the experimental energy resolution of keV lui01 for the width in Eq. (4) through , where denotes the calculated decay width of the -th state of O given in Table 1. The calculated strength function is normalized so as to match the calculated strength of the 12.1-MeV peak to the experimental one. We can see a rather good correspondence with the experimental data. The fine structures in the calculated strength function, i.e. one peak at MeV (corresponding to the state), one shoulder-like peak at MeV (), two peaks at MeV () and MeV (), are well reproduced. As mentioned above, the fine structures in the energy region of MeV as well as the sharp peak at MeV (corresponding to the state) are difficult to be reproduced by any mean-field calculations blaizot76 ; ma97 ; drozdz80 ; paar06 ; papakonstantinou07 ; papakonstantinou09 ; gambacurta10 , as far as the present authors know. The calculated values of at each states () in Fig. 5 are approximately proportional to the squared values of the respective calculated monopole matrix elements.
It is instructive and interesting to discuss the mechanism of why the five cluster states (, , , , and ) of O are excited relatively strongly from the ground state with the doubly-closed-shell-like structure yamada08 . Their monopole matrix elements shown in Table 1 are comparable to the single particle strength ( fm yamada08 ) and share about of the total EWSR value. Since the mechanism is closely related to the property of the ground state of O as shown below, we first demonstrate its interesting properties with the use of the microscopic wave function and then discuss the monopole matrix elements in the OCM calculation.
The wave function of the O ground state has dominantly the doubly closed shell model configuration with the nucleon size parameter (: nucleon mass), corresponding to the SU(3) wave function elliott58 . This doubly closed shell model wave function is mathematically equivalent to a single cluster model wave function of +C with the total harmonic oscillator quanta yamada08 ,