Isospin asymmetry in the continuum of the A=14 mirror nuclei
We study the isospin asymmetry in the isoscalar (IS) excitations in the mirror nuclei O and C by using the Hartree-Fock(HF)+random phase approximation (RPA) linear response function theory with a Skyrme interaction to take into account the continuum effect properly. The asymmetry in the IS monopole, dipole responses is pointed out in the continuum near the particle threshold with respect to the excitation energy and the sum rule strength. On the other hand, no clear sign of the asymmetry is found in the giant resonance (GR) region. In the quadrupole case, the calculated strengths of the mirror nuclei show almost the same energy dependence from the threshold to the GR region. It is found that the transition densities of the monopole response show an extended halo structure near the threshold, while those of GR region show a typical radial dependence of the compressional collective mode without any halo effect. Contrary to the transition densities of the monopole response, those of quadrupole response do not show any sign of the extended feature of wave functions neither near the threshold nor the GR energy region. Calculated strength distributions of the IS multipole states are compared with recent experimental data obtained by the multipole decomposition analysis of inelastic scattering on O.
pacs:21.10.Hw,21.10.Sf,21.10.Pc,21.60.-n, 21.60.Jz, 27.20.+n
A fundamental feature of nuclear structure is associated with a basic symmetry between neutrons and protons in the nuclear interactions. The existence of a general symmetry between and interactions was first observed in the lowenergy nucleonnucleon scattering data. The charge independence of the interaction implies the conservation of the isospin symmetry in nuclei Heisenberg32 (); BM69 (). The charge symmetry assumption, based on the equality of the and interactions , might give almost the same excitation spectra in nuclei of the isospin partners. The isospin symmetry, however, is violated by the electromagnetic interaction. Additional symmetrybreaking effects arise from the neutronproton mass difference, and small charge symmetry and independence breaking forces in the strong interactions.
The isospin impurity was studied by using microscopic models in refs. HS93 (); HGS95 () in relation with GamowTeller and Fermi decays. It was pointed out that the isospin mixing increases rapidly as a function of proton number and becomes few % in the ground state of Sn. However it will be less than 0.1% in light nuclei with the mass less than A=16 and thus the isospin can be considered to be a good and useful quantum number in the light nuclei. For example, in the A=13 mirror nuclei N and C with the isospin T=1/2, all states in the spectra show one to one correspondence up to the excitation energy Ex=10MeV within few hundreds keV accuracy.
In heavier nuclei, it was expected that the isospin symmetry might be of little significance because of very strong Coulomb field. However, the experimental observation of the isobaric analog states (IAS) in medium heavy nuclei shows that the isospin is well preserved even under the influence of the strong Coulomb field Fox64 (). In light nuclei, the examples of the large isospin asymmetry were found in the first excited J=1/2 states of A=13 nuclei, C and N, and also in the J=1/2 states of A=17 nuclei O and F. This is called ThomasEhrman effect ET51 (). This large asymmetry is considered mainly due to the reduction of the Coulomb energy of looselybound low state with small centrifugal barrier. It should be noticed that a large anomaly was also found in the isoscalar (IS) magnetic moments in A=9 mirror nuclei experimentally Matsuta95 () and discussed in relation with the ThomasEhrman effect Utsuno04 ().
The Coulomb energy might be reduced not only for the loosely bound states but also for the unbound resonance and continuum states. In this paper, we will study the continuum states in the T=1 mirror nuclei O and C in order to extract the isospin asymmetry in the spectra. The A=14 isospin partners are particularly interesting because of a large difference in the threshold energies of the mirror nuclei. Namely, the neutron threshold energy of C is S=8.18MeV, while the proton threshold energy in O is S=4.63MeV. This large energy difference between the two nuclei may give rise to a substantial effect on the low multipole strength distributions with J=0 and 1, especially near the threshold. The lowest 1 state in O has been obtained much attention due to its importance on the astrophysical CNO cycle in the stars Moto91 (). The main aim of this paper is to study the isospin asymmetry of the continuum response functions in the A=14 mirror nuclei both near the threshold and giant resonance region. To this end, we use the HartreeFock (HF) random phase approximation (RPA) continuum response function with a Skyrme interaction to calculate the IS strength distributions of low multipole states in O and C. The two-body spin-orbit and Coulomb residual interactions are not included in the RPA response. This paper is organized as follows. A brief review of the RPA response function is given in Section II. Calculated results are shown in Section III and compared with available experimental data. A summary is given in Section IV.
Ii Continuum strength in O and C
The RPA linear response theory is based on the timedependent HF (TDHF) theory BT75 (). The RPA Green’s function is obtained as a small amplitude approximation of TDHF and expressed as
where is the residual particlehole interaction. The unperturbed Green’s function is defined as
where and are the eigenfunction and the eigenenergy of the HF hamiltonian . The operator equation in the r.h.s. of (2) is nothing but the one-body Green’s function in the coordinate space representation. We can use the standard technique to solve the Green’s function taking into account the coupling to the continuum LG76 ().
The transition strength for the states above the threshold can be obtained from the imaginary part of as
where is the transition operator with multipolarity and isospin . Below the threshold, the strength is calculated from the residue of the poles in the real part of the response function. The transition operators are given by
The IS dipole operator excites the spurious state corresponding to the center-of-mass motion. Thus, we consider the next order term in the expansion of the spherical Bessel function for ,
The modified IS dipole operator is also defined as
The energy weighted sum rule (EWSR) is defined to be
where is the hamiltonian for HF and RPA calculations. For the IS excitations, the Skyrme two-body interactions do not give any contributions for the EWSR BT75 (). Thus the EWSR can be expressed for the operators (4), (5) and (7) to be
The transition density for an excited state, ,
can be obtained from the RPA response, since the imaginary part of near the resonance is proportional to . The reduced transition probability is calculated using the transition density (12) as
where we use an identity of the one-body transition operator
The radial transition density, , is defined by
The Skyrme interaction SkM is used to calculate HF potentials and RPA response functions. The HF calculations are performed with the filling approximation putting all nucleons in the HF orbits from the bottom of the mean field potentials. In O, the proton number Z=8 corresponds to the closed shell while the neutron number N=8 in C is the closed shell. Moreover, the energy difference between the and states for neutrons in O and protons in C are more than 6MeV in the HF calculations. Thus, the pairing correlations might not play any important role in the RPA response in these nuclei Khan ().
The Skyrme interaction SkM gives the neutron separation energy S=8.95MeV in C and the proton separation energy S=5.70MeV in O. These values are somewhat larger than the empirical ones S(exp)=8.18MeV in C and S(exp)=4.63MeV in O. In order to perform quantitative study of the threshold effect in the response, we enlarge the spin-orbit strength of the SkM interaction to be =150MeVfm in C and =155MeVfm in O from the original value =130MeVfm. These modifications give S=8.11MeV in C and S=4.66MeV in O which are very close to the experimental ones.
The existing Skyrme parameters do not reproduce the empirical separation energies of the two nuclei C and O with the accuracy which is good enough to study the threshold behavior of excitations in loosely-bound nuclei. It has been known that the particle-vibration coupling has a substantial effect on the single-particle energy , especially around the Fermi surface Colo (). For more sophisticated calculations, one should make an effort to readjust the Skyrme parameters including the particle-vibration coupling. Since it is not the aim of the present study to readjust all the Skyrme parameters, we take a conventional approach to obtain the empirical separation energies of A=14 nuclei.
The unperturbed and RPA isoscalar monopole responses of C and O are given in Fig. 1. There are two bound particlehole excitations , (1s2s) ( and stand for neutron and proton, respectively, hereafter) at Ex31MeV in C, while the proton particle state of the (1ss) configuration is in the continuum in O and gives a sharp peak at Ex27.3MeV in the unperturbed response. The discrete states of the unperturbed response , which are not shown in Fig. 1, couple with the continuum in the RPA response and appears as a sharp peak at Ex30MeV in C and also a peak at Ex28.5MeV in O. The unperturbed peak at Ex27.3MeV in O is integrated in the giant resonance peak in the RPA response.
The large threshold strength can be seen just above the proton threshold in O and also above the neutron threshold in C. The energy of the threshold peak in O is more than one MeV lower that that in C due to the lower proton separation energy S. One can see that the RPA correlations do not make any appreciable changes on the unperturbed response around the threshold peaks as was pointed out in ref. HSZ97 (). The monopole strength below 14MeV in O exhausts 10.1% of the energy weighted sum rule value (EWSR), while the monopole strength below 14MeV in C has only 8.1% of the EWSR value. The giant resonance peaks appear at Ex22MeV both in O and C where RPA correlations enhance the monopole strength. The integrated monopole strength up to GR region (Ex25MeV) is similar in the two nuclei 49.6% in O and 48.5% in C, respectively. The transition densities of the threshold peaks are shown in Figs. 2 (a) for C and (c) for O. The dominant contribution comes from the neutron excitation to the continuum in C, while the proton continuum excitation has the major contribution in O. The neutron density shows a halo-type long tail extended more than 10fm in the continuum excitation of C, while the proton density is rather compact and gives a minor contribution. The same halo effect appears in the proton transition density in O near the threshold, while the neutron one is very small and compact. Contrarily, the transition densities of GR region of C and O in Figs. 2 (b) and (d), respectively, have substantial contributions from both protons and neutrons. The radial dependence of the two contributions is very similar showing the peak and the node at almost the same radii. The ratio between the proton and the neutron contributions around the peak at r4fm is almost the same as that of the proton and the neutron numbers of each nucleus. This is one of the typical feature of the collective IS GR Sagawa02 (). In general, one can see in Fig. 2 that the proton contribution to the transition densities in O is almost identical to the neutron one in C both in the radial dependence and in the ratio to another contribution; the ratio of proton to neutron contributions in O is very close to that of neutron to proton ones in C. The IS monopole strength in O was observed recently by the multipole decomposition analysis of inelastic scattering experiments Baba (). The IS peak is observed at Ex7MeV and the integrated strength up to 13.8MeV exhausts about 8% of the EWSR. These experimental data can be compared with the calculated peak at Ex8MeV which exhausts 10% of the EWSR value. The observed strength in the GR region is about 38% of the EWSR up to Ex=25MeV in the inelastic scattering. This value is close to the integrated strength of RPA response up to Ex=25MeV, i.e., 49.6% of the EWSR.
The isoscalar compressional dipole responses of C and O are given in Fig. 3. In order to subtract accurately the spurious component, I took a small mesh size 0.075fm and the reference radius 15fm. Then more than 95% of the spurious center of mass motion was eliminated from the physical states HS02 (). In Fig. 3, we adopted the dipole operator (7) where the spurious component is further subtracted from the operator. The dipole response of O shows a sharp peak just above the threshold at Ex6.1MeV and a broad bump peaked at Ex9MeV. In C, a discrete 1 state is found at Ex=7.1MeV together with a broad peak at Ex11MeV. In the two nuclei, substantial strengths are found also above Ex=15MeV.
The observed lowest 1 state is at Ex=5.17MeV in O and at Ex=6.09MeV in C, respectively Firestone (). It is remarkable that the calculated energy difference E=1MeV between the lowest 1 RPA states in the two nuclei is as large as the experimental one E(exp)=0.92MeV, although the observed 1 states are found at somewhat lower excitation energies than the calculated ones. The strength of the threshold 1 peak in O has 2.5% of the EWSR value, while the discrete lowest 1 state in C has only 1.1% of the EWSR strength. The low energy peaks above the first excited states show also large differences both in energy and in transition strength. Namely, the peak energy in O is 8.9MeV, while that is shifted to about 2MeV higher in C with Ex(peak)11MeV. The integrated strength up to Ex=12MeV is 17.3% of the EWSR in O which is much larger than that of the value in C with 10.7% of the EWSR value. On the other hand, in the GR region, the two nuclei have almost the same transition strength, i.e., 23% of the EWSR between 12MeV Ex25MeV both in O and C.
The transition densities of several 1 states are shown in Fig. 4; (a) and (d) for the lowest 1 states, (b) and (e) for the threshold peaks, and (c) and (f) for the GR peaks. In order to subtract the spurious component from the transition density, we define a modified transition density
where is the total density of the ground state. The factor is determined to satisfy the condition
which is equivalent to evaluate the strength function with the operator (7)HSZ98 (). The transition densities for the lowest 1 states show similar radial dependence in Figs. 4 (a) and (d) for C and O ,respectively, although the larger contribution comes from neutrons in C and from protons in O. The transition density of the peak around Ex=11.0MeV in C (Fig. 4 (b)) is dominated by a large neutron contribution (1p2s) with a long tail extended until r=10fm, while that of the peak around Ex=9.0MeV in O (Fig. 4 (e)) with the dominant configuration (1ps) does not show any long tail. This difference might be caused by the Coulomb barrier for the dominant proton wave functions in O. It is interesting to notice in Fig. 4 (f) that the proton and the neutron contributions have no node in the transition densities, but the IS one shows the node because of the different extensions of the radial wave functions.
In ref.Baba (), the IS dipole strength in O is observed both in the low energy region and the high energy GR region. The experimental data show two peaks in the low energy region below Ex=10MeV. The integrated strength exhausts about 24% of the EWSR value below Ex=13.8MeV. The two peak structure and the similar sum rule value (18.5% of EWSR up to 13.8MeV) are found in the calculated results of Fig. 3. On the other hand, the observed dipole strength up to GR energy region (up to Ex=25MeV) amounts to be 70% of the EWSR value which is much larger than the calculated value of 40.5%.
In the isoscalar quadrupole response of Figs. 5 (a) and (b), the discrete states are found in both nuclei at Ex=7.1MeV with B(IS)=94.6 fm in C and at Ex=7.2MeV with B(IS)=52.8 fm in O, respectively. As far as the excitation energy is concerned, the isospin asymmetry is much smaller in the quadrupole state than the dipole states. This may be traced back to the difference of the particle configurations between the 1 and 2 states in the two nuclei. For the 2 states, the main configurations are (1p1p) for O and (1p1p) for C, respectively, and the 1p states are bound in both nuclei. The bound particle states for for 2 excitations cause only a small energy difference between O and C. On the other hand, the main configurations for the 1 states are (1ps) for O and (1p2s) for C, respectively. The particle state (2s) is bound in C, while the state (s) in O is the continuum state. This difference in the particle configurations is the origin of the large difference in the excitation energies of 1 states between O and C. The experimentally observed lowest 2 state is at Ex=6.59MeV in O and at Ex=7.01MeV in C. These data are consistent with the calculated smaller isospin asymmetry in the 2 states than that of the lowest 1 states. Namely, the observed energy differences are E=0.92MeV for the 1 states, while that is E=0.42MeV in the case of 2 states. The calculated results show also the same trend.
As is seen in Fig. 5, the threshold strength does not show any substantial enhancement in the quadrupole response, but the strong GR peaks are clearly seen around Ex=20MeV in the two nuclei. The EWSR of IS quadrupole response below Ex =25MeV exhausts 85.3% in C and 85.5% in O. As far as the GR is concerned, the two nuclei show almost the same excitation spectra both in energy and in strength distribution. The transition densities of the lowest 2 and GR states are shown in Fig. 6. It is interesting to notice that the proton amplitude is larger in the lowest 2 state of C, while the neutrons have the dominant contribution in O. This is because the proton excitation (1p1p) is allowed in C, while the neutron excitation (1p1p) is allowed in O. In Figs. 6 (b) and (d), the transition densities of quadrupole GR show a typical collective IS nature, i.e., the radial dependence is surface peaked, so called, Tassie-type and the neutron (proton) contribution is larger than the proton (neutron) one in the neutron-rich (proton-rich) nucleus C (O). One can not see any substantial long-tail in the transition densities of the quadrupole response even in the low energy states. This is due to a larger centrifugal barrier for the quadrupole response than the monopole and the dipole responses Sagawa92 ().
We studied the isospin asymmetry in the IS multipole responses of A=14 mirror nuclei C and O using the HF+RPA theory with Skyrme interaction to take into account the continuum coupling properly. We found that the IS strength distributions near the threshold show a large asymmetry between the two nuclei for the monopole and the dipole responses. On the other hand, the strength distributions in the GR region are very similar in both the peak energy and the sum rule value. It is seen a clear sign of the halo-type extension of the transition densities in the threshold monopole response. Contrarily, those of GR peaks do not show any sign of the halo effect, but give a typical IS collective features. For the quadrupole response, the two nuclei show very similar features not only in the GR region, but also in the lower energy region near the threshold. The calculated results of O are compared with recent experimental data obtained by the multipole decomposition analysis of inelastic scattering. The calculated monopole and the dipole strength distributions show good agreement with the empirical ones near the threshold. The empirical sum rule strength of the monopole states is also close to the calculated one in the GR region below Ex=25MeV, while the empirical dipole strength is reported to be larger than the calculated one in the high energy region between 12MeV Ex 25MeV. In order to confirm further the isospin asymmetry of A=14 nuclei, we need the experimental data of C in the continuum region.
Acknowledgements.I thank T. Shimoura for enlightening discussions. I thank also H. Baba for informing his data prior to a publication. This work is supported in part by the Japanese Ministry of Education, Culture ,Sports, Science and Technology by Grant-in-Aid for Scientific Research under the program number (C(2)) 16540259.
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