Examinations of the consistency of the quasiparticle random-phase approximation approach to double- decay of Ca
The nuclear matrix elements (NMEs) of the neutrinoless and two-neutrino double- decays of Ca are calculated by the quasiparticle random-phase approximation (QRPA) with emphasis on the consistency examinations of this calculation method. The main new examination points are the consistency of two ways to treat the intermediate-state energies in the two-neutrino double- NME and comparison with the experimental charge-exchange strength functions obtained from Ca and Ti reactions. No decisive problem preventing the QRPA approach is found. The obtained neutrinoless double- NME adjusted by the ratio of the effective and bare axial-vector current couplings is lowest in those calculated by different groups and close to one of the QRPA values obtained by another group.
If the neutrinoless double- () decay is observed, one can conclude that the neutrino is a Majorana particle. In this case, the effective neutrino mass can be determined by the half-life of the decay, expected to be measured by the experiments, the phase-space factor, and the nuclear matrix element (NME). Recently the study of the decay has obtained a stronger motivation than before by the discovery of the neutrino oscillation Fukuda et al. (1998); Ahmad et al. (2002); Eguchi et al. (2003); Aliu et al. (2005) proving the existence of the finite neutrino mass. The phase-space factor and NME are the quantities that the theory should supply, and the latter is more difficult than the former because the accurate nuclear many-body wave functions are necessary. As is well-known, the calculated values of the NMEs are distributed in the range of a factor of 23 Engel and Menéndez (2017), and this range is not reduced in spite of the effort of many theorists. For now there is no perfect calculation because all of the candidate nuclei for the decay are heavy so that the exact nuclear wave functions cannot be obtained. In addition, effective strength of the spin-isospin transition operators is necessary for reproducing the related experimental data.
One of the tasks that the theorists need to do is to examine the consistency of their calculations for clarifying the reliability. The purpose of this paper is to examine the consistency of the QRPA approach to the decays of Ca in detail. There are two main check points not yet investigated. One is the treatment of the intermediate-state energy in the two-neutrino double- () decay. The QRPA approach has two sets of the intermediate states defined by the QRPA calculations based on the initial and final states. I clarify the validity of using the two sets of intermediate-state energies in the -NME calculation. Another new check point is the comparison of the Gamow-Teller (GT) strength function between the experimental data Yako et al. (2009) and the calculation. This check point includes a question of whether theory can explain the quenching of the experimental GT strength.
My motivation to investigate Ca is based on a fact that this mother nucleus is not always discussed in the papers of the systematic application of the QRPA to the decays. I clarify in this paper whether Ca Ti is particularly difficult for the QRPA approach. Several experimental projects searching for the decay of Ca are in progress, or have been finished, see Refs. Brudanin et al. (2000) (TGV), Ogawa et al. (2004) (ELEGANT VI), Barabash and V. B. Brudanin (NEMO Collaboration) (2011) (NEMO-3), Arnold et al. (2010) (SuperNEMO), Umehara et al. (2008) (CANDLES), Zdesenko et al. (2005) (CARVEL), and You et al. (1991). The advantage of Ca is the large Q value (4.7 MeV), and this nucleus is one of the major candidates for the decay. Thus, it is worthy of investigating theoretically in detail.
My calculation method is explained in Sec. II specifically for Ca Ti including the technical aspects. The method to examine the use of two sets of the intermediate-state energies in the NME is described in Sec. III. The results of the calculations are shown in Sec. IV, and the comparison with the results of other groups is made. The GT strength function is discussed in Sec. V, and this study is summarized in Sec. VI.
Ii Calculation method
ii.1 Hartree-Fock-Bogoliubov calculation
Ca is not often discussed in the QRPA approach. This may be because the pairing gaps of the ground state of this nucleus are not as certain as those of other nuclei. I explain how the pairing gaps are determined in my calculation. The three-point formula Bohr and Mottelson (1969) is used for obtaining the experimental pairing gaps from the experimental nuclear masses nnd (). The formula for the neutron pairing gap is
where denotes the binding energy of the nucleus with the neutron number and proton number , and that for the protons, , is obtained analogously. The presumption is that the odd-even mass staggering in the systematics occurs solely by the pairing correlations. Thus, this method is usually not used for the magic nuclei. The pairing gaps of Ti deduced from the masses are MeV and MeV. I reproduced approximately these pairing gaps by the Hartree-Fock-Bogoliubov (HFB) calculation Blazkiewicz et al. (2005); Oberacker et al. (2007); Terán et al. (2003) using the Skyrme interaction SkM Bartel et al. (1982) and the like-particle contact pairing interactions with the strengths adjusted for the protons and neutrons separately: MeVfm (protons) and MeVfm (neutrons) with the active range of the pairing interaction up to 30 MeV of the effective single-particle energy Blazkiewicz et al. (2005). The pairing gaps obtained by my calculation are MeV and MeV (average pairing gaps). I also use this pairing interaction for Ca assuming that the pairing-interaction strength does not change significantly by the small change in the proton and neutron numbers. In order to check this assumption, I performed the HFB calculation for Ar and obtained MeV and MeV. The corresponding experimental values are MeV and MeV. This result justifies my procedure to treat the pairing interaction.
Using this pairing interaction, I obtained MeV and MeV for Ca. The neutrons have more difficulty in getting the pairing gap than the protons because there are shell gaps above and below the neutron Fermi surface at the first (1) orbital. Let us see the low-lying spectra for seeking possible reflection of the proton pairing gap. The first experimental excited state of Ca is at 3.831 MeV nnd () with . Since the proton one-particle-one-hole (1p-1h) excitation with the positive parity needs so called the jump, the main components of the excitation are those of the neutrons in the QRPA state. The second experimental excited state of Ca is at 4.283 MeV (). Again the 1p-1h excitations with the jump are necessary for having the excited state in the QRPA. This condition applies for both the protons and neutrons, so that the QRPA cannot create that low-lying state. The third and fourth excited states are at 4.503 MeV () and 4.506 MeV (), respectively, and the corresponding QRPA excitation energies are 3.956 MeV () and 5.550 MeV (). No clear indication is obtained on the proton pairing gap from the corresponding energies. If enhancement of the two-proton transfer is seen experimentally, it would be the encouraging indication. However, there is no experiment of that reaction for Ca currently. Below I use the HFB solutions with the finite proton pairing gap but no neutron one. The uncertainty of the pairing gaps of Ca is minimized by the self-consistent calculation of the HFB approximation.
ii.2 QRPA calculation and technical parameters
The calculation scheme of the QRPA is the same as that used in Refs. Terasaki (2015, 2016). Here I note some technical parameters related to the accuracy of the calculation. The single-particle basis for representing the QRPA Hamiltonian matrix is constructed by the diagonalization of the one-body density matrix obtained from the HFB solutions (the canonical basis Ring and Schuck (1980)) with the axial and parity symmetries (the symmetry axis is ). This basis is identical to the HF basis, if there is no pairing gap. The number of the single-particle states used in the calculations of this paper is around 16001700 including those with both the positive and negative (the -component of the angular momentum) for each of the protons and neutrons. The maximum is 19/2. That dimension of the single-particle space approximately corresponds to harmonic-oscillator shells. The wave functions are expressed with the B-spline mesh Blazkiewicz et al. (2005); Oberacker et al. (2007); Terán et al. (2003) in a cylinder box with the radius of 20 fm in the plane and fm. The root-mean-square radius of Ca is 3.531 fm in the HFB solution. The number of mesh points is 42 for the region of 20 fm. The spherical symmetry of the spherical nuclei can be satisfied accurately with this geometrical preparation. It was confirmed by the HFB calculation that the ground states of Ca and Ti are spherical. Many of the single-particle states are in the discretized-continuum region. The density matrix and pairing tensor are calculated using the HFB wave functions in the active energy range mentioned in the previous section.
The dimension of the two-quasiparticle basis for representing the QRPA excitation is truncated by the cutoff scheme used in the previous calculations Terasaki (2015, 2016). The cutoff criteria in those calculations were determined so as to obtain the convergence of the final NMEs with respect to the dimension of the two-quasiparticle space and to satisfy the geometrical symmetries of the Hamiltonian accurately in the calculation. The same criteria are used for the calculation of Ca and Ti of this paper. That dimension decreases as the quantum number (total of the nucleus) increases; it is approximately 24000 for and 13000 for .
Iii nuclear matrix element
The NME, see Eq. (25)(28) in Ref. Terasaki (2016), can be written
with the GT NME , the Fermi NME , the vector-current coupling , and the axial-vector current coupling . The formulation for the axially-symmetric nuclei is applied to the spherical nuclei Ca and Ti in the calculation. The initial and final states of the decay are states. Under these conditions, the GT NME is given by
, , and are the exact initial, final, and intermediate states with , respectively, and is the charge-change operator from a neutron to proton.111I have noted this operator as previously Terasaki (2012, 2013, 2015, 2016). In this paper, I change it to the convention of nuclear physics Suhonen (2007) because the GT strength functions are discussed below. The spin-Pauli matrix is denoted by , and is the Hamiltonian. and are the nuclear masses of the initial and final states, respectively, and is the energy of the exact intermediate state. The electron mass is denoted by . An abbreviation for the one-body operator
The NME of the decay is sensitive to the energy denominator , thus, the closure approximation is not applied.
Let us introduce the QRPA by replacing the nuclear states and with the corresponding QRPA states and . The intermediate states are defined two ways; one is by the QRPA calculation using the initial ground state, and another is that using the final one. The former (latter) intermediate states are denoted by (), with which I have two sets of equations:
The operator includes the higher-order many quasiparticle, or many-particle-many-hole, components beyond the QRPA, thus, Eqs. (10) and (12) [Eqs. (11) and (13)] do not coincide exactly. However, if the QRPA is a good approximation, the effect of those higher-order components would be small. Then, the following equations are derived:
The energy of the intermediate state is calculated using the proton-neutron (pn) QRPA excitation energies and as
where and are the proton and neutron chemical potentials of the initial state, and and are those of the final state; those of the HFB ground states are used, and and are the proton and neutron masses. For and the experimental data are used. The accuracy of Eq. (20) is a consistency check point of the QRPA approach. This is shown numerically below.
The overlap is calculated using the equations developed in Ref. Terasaki (2013). However, there is a difference from the calculation of Nd Sm Terasaki (2015, 2016). The norm of the unnormalized QRPA ground state, and in Eqs. (14) and (15) in Ref. Terasaki (2013), diverges, therefore, the norm was renormalized by truncating the contribution of the QRPA solutions so that the semiexperimental correlation energy222This is obtained from the experimental binding energy and the HFB ground-state energy. is reproduced by the QRPA Terasaki (2015). I used the correlation energy because this is sensitive to the QRPA correlations, and the QRPA-correlation energy diverges without the truncation Terasaki (2013). The unnormalized overlap does not diverge because the bra and ket states are created by the charge change from the nuclei with and , and many components of the unnormalized overlap vanish which keep the configuration around the Fermi surface of the HF(B) ground states (see Fig. 1) Terasaki (2015).
Usually the Skyrme interaction (energy density functional) is constructed so as to reproduce experimental physical quantities including the binding energies of the doubly-magic nuclei by the HF ground states (Kohn-Sham states). Therefore, the HF(B) ground state replaces the (Q)RPA ground state of Ca in the overlap calculation, and the HFB ground state is also used in the same manner for the ground state of Ti in the overlap approximately. In the calculation of NdSm, the product of the norms of the unnormalized QRPA ground states played a role to decrease the NME through a factor of , and because of this the NME close to the semiempirical value was obtained without very strong pn pairing interaction causing the near-instability of the QRPA solutions. Therefore it is a check point of my QRPA approach to see whether the very strong pn pairing interaction is unnecessary for CaTi.
Iv Calculation result
The strength of the isovector pn  pairing interaction is determined to be the average of the proton-proton  and neutron-neutron  pairing interactions for satisfying approximately the isospin invariance of the pairing interaction. denotes the isospin, and is its -component. Two QRPA calculations are performed for the NME; one is the pnQRPA, and another is the like-particle (lp) QRPA Terasaki (2015, 2016). The latter can be used under the closure approximation. The calculation by the lpQRPA corresponds to the virtual decay path via the two-particle transfer. The Hamiltonian used for the HFB calculation is used for the two QRPA calculations. However, the important interaction components are different for the two QRPA calculations. Thus, the equivalence of the two paths is a theoretical constraint to the effective interactions used in the QRPA. The strength of the pairing interaction is determined so as to have the GT NMEs obtained by the two methods to be equal because other interactions are established. Note that the pn pairing interactions have no contribution to the lpQRPA.
In this paper I introduce a practical modification to my method used in Ref. Terasaki (2016); the Fermi component of the NME is not used for the constraint to the effective interaction. The fundamental reason is that the effective interaction Skyrme SkM plus the Coulomb interaction does not have the isospin invariance. The Fermi NME is sensitive to the pairing interaction. Thus, if the equality of the Fermi NME is required between the two different-path calculations, can be determined. However, then, the pairing interaction does not satisfy the isospin invariance. The value of is determined so as to reproduce the experimental half-life of the decay. Those three parameters , , and can be determined separately in this order. I also use this to the NME calculation of decay because the very large single-particle space is used. If this space is not enough large, different ’s would be necessary for the two decays because the neutrino potential of the decay has a divergence. The value of is always 1 throughout this paper.
The value of is MeVfm, and was found to be MeVfm according to the above method. The experimental half-life of the decay of Ca is = (6.41.2)10 yr Arnold et al. (2016). The corresponding theoretical half-life is calculated by Kotila and Iachello (2012)
with the phase-space factor = 15550 yr Kotila and Iachello (2012). was reproduced by = 0.48 with = 0.138 and = 0.49 with = 0.133. The relative difference of and is 4 %, thus the consistency of the QRPA approach discussed above is approximately satisfied. The GT and Fermi NMEs are shown in Table 1. The absolute value of is less than 5% of , thus, the isospin invariance of is also approximately satisfied.
By using the result of Ref. Terasaki (2015) for Nd, it turns out that reproduces yr Barabash (2013) of that nucleus. The value of Ca is 58% of that of Nd. One of the causes for this difference is apparently the normalization factors of the QRPA ground states. The product of the two normalization factors of Nd and Sm is 1.860 Terasaki (2015), but the corresponding product of Ca and Ti is 1.0; see the previous section. If the product of the normalization factors of 1.860 is applied artificially to the Ca calculation, the is reproduced by . This value is larger than 0.49 as expected, although still smaller than 0.84 of Nd. Other reasons may be the differences in the nuclear structure of those nuclei, however they are not obvious. The of my calculation is consistent with a recent tendency to accept , even (not for Ca), e.g., Suhonen (2017). For the decays of Ca, the bare value of 1.25 or 1.27 is usually used by other groups; see the comparison below.
I performed a reference calculation using a usual method; was determined so as to reproduce the . For , an effective value of 1.0 was used, and I used already determined. The -dependence of is shown in Fig. 2. The difference in input for the two sets of result is whether ’s are used (connected by solid line) or ’s are used (connected by dotted line). The mean value of the two results at MeVfm (the rightmost points) is close to the experimental value of (6.4 yr. However, the discrepancy of the two points is too large, therefore the QRPA is not a good approximation. This method to determine cannot be used. Generally, the instability of the mean-field or HFB ground state occurs in relation to the symmetry breaking, if the strength of attractive interaction increases significantly. The QRPA is usually not used near this instability.
|( MeV yr)|
Subsequently, I calculated the NME (for the equation, see Ref. Terasaki (2015)) using , and = 3.054 was obtained. My effective is much smaller than the usual ones 1.0. In Table 2, , (Gamow-Teller NME), (Fermi NME), ( NME of the tensor transition operator, shown if used), and of the different groups are shown. For comparison of the results with different ’s, I also show in the table the reduced half-life
where is the value of not including the many-body effect or compensation of approximation, and = 1.27 is used, e.g., Engel and Menéndez (2017). The methods of Calculations 1 and 2 are the (Q)RPA (the latter is my calculation), those of 35 are the (interacting) shell-model, that of 6 is the interacting-boson model, and those of 79 are the generator-coordinate method, see the caption for the references. is the quantity used for deriving the effective neutrino mass in
with the half-life of the decay expected to be available experimentally. and are unique; if all calculations using different approximations are correct, would be identical. Therefore is better than for comparison of different calculations. of my calculation, Calculation 2, is close to the largest one of another (Q)RPA calculation. The QRPA calculations show the largest in the calculations by the different methods. The predicted by my calculation is larger than that by the shell model Iwata et al. (2016) by a factor of five.
It is noted that I used the latest value of , which is nearly 50 % larger than the previous values Barabash (2013). However, the old value is used in some of the other calculations. If the old one, 4.410 yr, is used for the fitting, my is 16.29710 MeVyr [ = 0.53, Eqs. (14)(16)] and 15.57410 MeVyr [ = 0.54, Eqs. (17)(19)]. Thus, there is no qualitative influence on the comparison.
As seen from Eqs. (24)(26), is also a quantity used for obtaining . Other necessary input for obtaining are the experimental data and constants other than , thus, can also be compared between calculations with different . This NME is less affected by the uncertainty of than because the dependence of is not higher order than . The most significant difference between and is seen in Calculations 14. The results of these four calculations seem close in , but the differences in are significantly larger. Difference as this can occur because of the relation
When ’s are 0.5 or smaller than this, the difference between them is magnified in . Both and are important. One can concentrate on the nuclear property in discussing . On the other hand, the half-life is the physical quantity measured, and is more directly related to that than because of the proportionality.
The bare value of is usually used in the Ca calculations. Note, however, that in the larger set of samples of other nuclei including the single- decays, the effective value of 1.0 is historically more usual, e.g., Brown and Wildenthal (1985). My and are close to those of Calculation 6 (the interacting-boson model), however are quite different because of the difference in the used . In my calculation, that is necessary for reproducing the . It is seen from the comparison of the first two rows of Calculations 1 and 2 (both are the QRPA calculations) that and are quite different, however, and are close. Both calculations use for determining parameter, but the methods to determine and are different. The fluctuation of seen in Table 2 is larger than those of and .
For a reference, I also made a comparison under as unified conditions as possible; and calculated with the same = 1.0 and without are shown in Table 3 for those calculations in which and are available. The method dependence of is similar to that of , as seen from the comparison of my result and the others in Tables 2 and 3. An analogous tendency (but the inverted method dependence) is seen for . My values of and change much more than those of other groups between the two tables because my value of is much smaller than those used by other groups.
|( MeV yr)|
A possible difference between the QRPA calculation of Ref. Šimkovic et al. (2013) and mine is the pairing-interaction strength because my original method is used for determining that strength. Since the different operators are used for defining that interaction, the interaction strength can only be compared in terms of the position on the curve of the NME versus (this information of Ref. Šimkovic et al. (2013) is not available). I show in Fig. 3 the plots of , , and versus of my calculation. The adopted value of is 180.0 MeV fm, as mentioned above, thus, it is seen that my calculation is in the safe region of the QRPA. This is reasonable because is determined referring to obtained by the lpQRPA Terasaki (2016), for which the HFB ground state is stable. The specialty of the overlap of Ca does not cause a problem.
V Gamow-Teller strength function
v.1 Brief review
The GT strength function is calculated using the GT transition matrix, of which transition density is also an ingredient of the NMEs of the decay. Naturally the GT strength function obtained from the experiments of the and reactions are important information for checking a part of the calculated NMEs. This strength function and the half-life of the 2 decay are the most direct experimental data helping the calculation of the NME because the decay of Ca is suppressed by the very small Q value of 279 keV nnd (), and Ti does not have the decay or electron capture.
The above charge exchange occurs in the hadron knock-on reaction, therefore, the mechanism is independent of . Thus, the charge-exchange reaction seems to be a quite adequate method for checking the GT transition matrix elements. However, the extraction of the GT strength function from the experimental cross section is not straightforward, in addition, there was a historical problem of the quenching of the measured GT strength. I briefly review those discussions.
The excitation by the spin-isospin operator of is effectively induced by the and reactions with the incident energy of 200 MeV or larger at the forward angles. The basic equation for extracting the GT transition strength (GT) from the cross section is
where is the unit cross section determined experimentally, and is a function depending on the momentum transfer and energy loss (variables) , see, e.g., Ref. Wakasa et al. (1997) for these factors. For the derivation of Eq. (28), sometimes called the proportionality relation, see, e.g., Ref. Ichimura et al. (2006). The limit of is used for extracting from the exeprimental cross section Wakasa et al. (1997); Ichimura et al. (2006). The presumption for Eq. (28) is that the reaction is induced by a one-body field (the impulse approximation) Osterfeld et al. (1985); Taddeucci et al. (1987) and a single-step reaction Watanabe and Kawai (1993). There is also another method for determining by Eq. (28) and the experimental data of the -decay for pairs of mirror nuclei Molina et al. (2015) (not applied to Ca and Ti).
The quenching factor of the sum of the experimental charge-exchange strengths corresponding to the GT sum rule (Ikeda sum rule) Ikeda et al. (1963) is defined by
where and are the sums of the experimental transition strengths of the - and - decay type, respectively. In the early days, this was 0.400.65 systematically in a broad mass region Ichimura et al. (2006). This problem stimulated the discussion on the contribution of the -isobar nucleon hole; for this see the references in, e.g., Ref. Ichimura et al. (2006). Below is the history of the studies on the basis of the nucleon degrees of freedom.
The cross sections and deduced strength functions are reported by several experiments for mother nuclei Zr Yako et al. (2005); Prout et al. (2000); Wakasa et al. (1997); Raywood et al. (1990) and Pb Prout et al. (2000); Raywood et al. (1990); Gaarde et al. (1981). The strength function of Zr(p,n) Yako et al. (2005) consists of a sharp peak around MeV, the giant resonance in 20 MeV, and a broad and low strength distribution in MeV. The transition strength of Zr(n,p) is much smaller, as anticipated for the neutron-excess nucleus, but has a non-negligible broad distribution. That of Pb(p,n) has a similar structure of the giant resonance and broad distribution.
It has been found Prout et al. (2000) that the broad and low distribution in the high-energy region was seen with the incident energy of 795 MeV but not seen with 200 MeV. The authors of that paper argue that the projectile with the higher energy can be absorbed more efficiently than that with the lower energy, thus the high-energy broad distribution is due to the isovector spin monopole excitation. That is a compression mode and induced by the transition operator . Theoretically, this possibility was discussed in, e.g., Ref. Auerbach et al. (1989).
The authors of Refs. Dang et al. (1997) and Bertsch and Hamamoto (1982) showed independently that the distribution of the transition strength was shifted substantially to the high-energy side by the 2-particle-2-hole (2p-2h) correlations of the nuclear states. It is noted that the charge-exchange transition strengths due to the above two mechanisms have similar high-energy broad distributions if scaled to the same height. It has been pointed out Hamamoto and Sagawa (2000) that induces not only the compression mode but also the GT transition, and a method to separate these two components was suggested.
The experimental GT strengths in which the isovector spin monopole component has been subtracted were derived Yako et al. (2005) for Zr(n,p) and Zr(p,n) reactions up to MeV, and the quenching factor was found to be . This is a value much close to the unity compared to those in the early days. Summarizing the status of the Zr studies, the quenching problem of the experimental GT strength seems solvable by extending the measured energy region. The high-energy ( MeV) broad and low distribution of the strength in the original data contains both the 2p-2h and the isovector spin monopole excitations (this is 1p-1h excitations).
v.2 Gamow-Teller strength functions of Ca and Ti
Now I examine my transition strengths of Ca and Ti. I obtained the calculated value of the GT sum rule of
and are the GT strength sums calculated by the QRPA corresponding to and , respectively. The exact sum-rule values are 24 (Ca) and 12 (Ti). Thus the QRPA calculation satisfies the sum rule accurately; this is a technical check of the QRPA calculation.
The experiments of Ca(p,n)Sc and Ti(n,p)Sc reactions have been made Yako et al. (2009), and the charge-exchange strength functions have been obtained. Figure 5 shows the measured and calculated GT strength functions for Ca Sc. These results have two common structures; one is the low-energy peak ( MeV in the experiment and 1.3 MeV in the calculation) and the giant resonance in 13 MeV. There is no major structure above this energy region in the calculated result, but the tail of the experimental data is higher than the calculated one (see the inset). The width parameter of 0.2 and 1.0 MeV are used in the Lorenzian folding for the states below MeV and above this energy, respectively, for simulating the discrete and continuum states. The overall feature of the experimental data is reproduced. Figure 5 illustrates the strength function for . The same structure as that of the GT strength function is seen, in addition, a broad low distribution of the strength is seen in 40 MeV. It can be speculated from the common feature of the two figures that the strength function for also includes the GT strength, although the dimension is different. This speculation can be confirmed by Fig. 7, in which the transition operator is used Hamamoto and Sagawa (2000). The is the mean square radius of one of the excess neutrons in 1. The GT structure is almost removed, thus, the effectiveness of the separation method of Ref. Hamamoto and Sagawa (2000) is confirmed. On the basis of this physical interpretation, the strength function of Fig. 5 and that of Fig. 7 are denoted as and , respectively. Note that and have the different dimensions. Figures 79 are those for Ti Sc corresponding to Figs. 57. The calculated lowest-energy peak is lower than the corresponding experimental peak, and the calculated giant resonance (8 MeV) has more strength than the lowest-energy peak has. The corresponding experimental giant resonance around 6 MeV seems to be a shoulder. The operator of yields the broad strength distribution in MeV corresponding to that for Ca Sc.