Volume 165B, n u m b e r 1,2,3
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HOLE PAIRING GIANT RESONANCES M.W. H E R Z O G , R.J. L I O T T A Research Institute of Physics, S-10405 Stockholm 50, Sweden
and T. VERTSE Institute of Nuclear Research of the Hungarian Academy of Sciences, Pf 51, H-4001 Debrecen, Hungarr Received 8 July 1985; revised manuscript received 16 September 1985
A two-hole pairing collective (TDA) state built upon deep lying single-hole states is calculated to be strongly excited in the reaction 2°spb(p,t)2°6pb. This state lies at about 12 MeV of excitation energy above 2°6pb(gs). It is a "hole pairing giant resonance".
The collectivity of both particle-hole and twoparticle (pairing) vibrations is well understood in terms of microscopic degrees of freedom. In both cases all wavefunction components contribute with the same phase to the form factor corresponding to the reactions that probe the nuclear collectivity. In the particle-hole case that probe is given by inelastic scattering while in the pairing case it is given by twoparticle transfer reaction. This similarity between particle-hole and particle-particle excitations is not unique. In fact, it is known for a long time that there is a formal equivalence between particle-hole and pairing vibrations [ 1]. Yet, only recently two of the most important features of particle-hole vibrations have been generalized to the pairing case. While in particle-hole vibrations one can readily use a macroscopic picture to describe inelastic scattering (as done since at least three decades ) a similar description for the case of two-particle transfer reactions was proposed only recently [3 ]. Similarly, while the study of giant resonances in inelastic scattering within the framework of particle-hole excitations is an old subject , the corresponding study for two-particle transfer reactions was done only recently  (but see also ref. ). It was found in ref.  that the highlying single-particle states which give rise to a collec0370-2693/85/$ 0 3 3 0 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
tive particle-hole state (giant resonance) induce the building up of a collective pairing state. This collective state was called "pairing giant resonance" in ref.  to maintain the analogy with the particle-hole case. The collectivity of the pairing giant resonance would be manifested, e.g., in two-particle transfer reactions, where this state would be the most strongly excited state if the energy of the projectile is big enough. The presence of deep-lying single-hole states in nuclear spectra is a well established feature . One may then expect that these states would also give rise to a deep-lying pairing vibration, i.e. a "hole pairing giant resonance" (HPGR). In fact, some indications that the HPGR might exist was found in ref. . In this letter, we will study the question of whether the HPGR is a collective pairing state analysing two-hole states in 208pb and in ll4Sn. Including the five major shells shown in table 1 we calculated the two-hole wavefunction in 208pb within the TDA using a surface delta interaction. As usually we obtained the strength of the SDI by fitting the two-hole ground state energy. The "hole pairing giant resonance" is the first excited state after those corresponding to the first major shell, i.e. the state number 7 in table 2. The wavefunction configurations corre35
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Table 1 Single-hole states used in the calculations. Energies are in MeV. State
2pl ~ l~a 2pa ~ 0il~ m lfTa 0h9~ ld3~ 2sla Ohu~ 097~ lds~ 099~ lpl ~ Ofs~ lP3 ~ OfTn 0da~ lsln Ods~
-7.38 -7.95 -8.27 -9.01 -9.72 -10.85 -14.10 -14.48 -15.44 -15.68 -16.45 -20.00 -20.20 -21.00 -21.63 -24.30 -28.04 -28.91 -29.84
-10.79 -10.95 -15.00 -15.70 -16.50 -17.00 -20.10 -23.85 -24.58 -25.51
sponding to this state should contribute to the twoparticle transfer form factor all with the same phase, except the first six components (i.e. the configurations corresponding to the first major shell). However the absolute values o f these components are small and Table 2 Excitation enelgies and relative cross sections for final states 2°6pb(0~) up to i = 7. The energies under the label KH are from ref. . The results of our calculations are under the column labelled by SDI. The experimental ground state energy is 14.10 MeV against 13.80 MeV for the KH calculation (in the SDI case this energy is fitted to the experimental value). The experimental relative cross sections are taken from ref. [ 10]. The state i = 7 is the I-IPGR. State
Ep = 35 MeV
2 3 4 5 6 7
1.27 2.49 3.16 5.39 7.58 -
1.13 2.30 3.23 5.11 7.22 12.58
1.17 2.31 -
0.089 0.266 0.035 0.383 0.007 2.210
0.099 0.174 -
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their effect is not important (a similar feature occurs in the case o f the usual multipole p a r t i c l e - h o l e giant resonance). To investigate the collectivity o f the HPGR we first analysed the clustering features o f this state. As for the giant pairing resonance o f ref.  we found that the two-holes in the HPGR are strongly clustered. We than calculated the two-particle transfer reaction leading to the HPGR using the reaction 208pb(p, t)206pb (0 +) with the same optical model parameters as in ref. . To perform this calculation we made use o f the DWBA code DWUCK4 [ 11 ]. Although two-step processes may give important contributions to the absolute cross sections in these reactions  the relative cross sections (which will be analysed here) are not very much dependent on those processes [ 13]. The results o f our calculations are given in table 2, where one can see that the energies o f the low-lying states calculated using SDI agree rather well with those given by the K u o - B r o w n interaction (KH). In fact, the SDI results are closer to the available experimental data than the corresponding KH ones, both for the energies as well as for the relative cross sections. Since the Q-value (in absolute value) corresponding to the HPGR in 206pb is much larger then the one corresponding to the ground state (the HPGR Q-value is - 1 8 . 2 MeV against - 5 . 6 MeV for the ground state) one expects that rather high proton energies would be needed to reach the HPGR. We studied the two-particle transfer cross sections for all 0 + states in 206pb as a function o f the projectile energy. The dependence o f the optical model parameters upon the projectile energy was taken into account as in ref. . Surprisingly, we found that the angle-integrated HPGR cross section relative'to the corresponding ground state cross section increases strongly with the projectile energy, as shown in fig. 1. One can see in this figure that there is a wide range o f proton energies for which the HPGR is much more strongly excited than the ground state. The relative cross sections corresponding to all the other states are small. F o r the low lying 0 + states all these cross sections would have approximately the same angular distribution because the differences in Q-values are small compared to the projectile energy . But due to the large (negative) Q-value o f the HPGR the angular distribution corresponding to this state is shifted b y a few degrees c o m -
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Fig. 1. Relative angle-integrated cross section Orel = a(0/+)/o(0~) corresponding to the reaction 2°SFo (p, t)2°6Pb (0~) as a function of the proton energy Ep (in MeV). The optical model parameters are from ref. [ 10].
pared to the low lying states. The maxima occur in all cases at 0 °. At this angle the HPGR relative cross section is even larger than the one shown in fig. 1. To be sure that the large cross section o f fig. 1 is not just an
accidental product o f the optical parameters o f ref. [ 10], we changed these parameters within reasonable limits and in all cases the essential features o f fig. 1 remained. One may then expect that the I-IPGR in 37
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206pb should easily be seen experimentally using the reaction 208pb(p, t)206pb (even at the proton energy of 35 MeV used in ref. , as seen in table 2). There may be several reasons why the HPGR has not been observed yet in 206pb. It might be that the experimental spectrum has not been searched carefully enough in the high energy region. This is plausible since the appearance of the HPGR was not expected (in particular the Q-value in the data shown in ref.  goes up to - 1 3 MeV only). But, perhaps more likely, other excitations (like two-particle-twohole states, the isoscalar E0 giant resonance, etc.) may strongly mix with the HPGR and this state might then be strongly fragmented. In this case, it might not be possible to resolve the individual components of the HPGR in the high level-density region where these components would lie. In particular, states with angular momenta other than ~ = 0 may have large twoparticle cross sections in this energy region, as it happens at low energy . We performed the calculation o f cross sections corresponding to the states 2 + and 4 + and in no case found a relative cross section larger than the one leading to the HPGR. It is also possible that the HPGR is too wide to be observed. A rough estimate of the HPGR width can be made assuming that the damping of the HPGR proceeds approximately in the same way as it does for the quadrupole particle-hole giant resonance  (this is consistent with the equivalence between the two modes). In this case the HPGR width would be about 3 MeV. One can also consider other nuclear regions where the HPGR may be expected. One interesting possibility is the Sn region, especially considering the experimental data of ref. [ 15]. We performed the calculation of the isotope ll4Sn assuming the shells 0g7/2 and lds/2 filled. Including these two shells plus three major shells as shown in table 1, we obtained the HPGR at an excitation energy of 9.4 MeV above 112Sn(gs). However, in this case the HPGR is mainly built upon the shell 0g9/2 and its collectivity is very weak. As a result, the angular integrated l l 4 S n ( p , t ) l l 2 S n ( H P G R ) cross section relative to the ground state cross section is OreI = 0.94 at the proton energy of ref.  i.e. 42 MeV. In fact, the
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maximum value of this cross section is OreI = 0.99 for a proton energy of 50 MeV (we performed these calculations using optical model parameters as in ref. ). Moreover, there are a number o f states 2 + and 4 + lying at around 9 MeV of excitation energy (where the HPGR lies) with Crre1 > 1. Therefore in this case the HPGR would hardly be seen, in agreement with the experimental results of ref. . In conclusion one can say that our calculations suggest that in the 208pb(p, t)206pb reaction one should see a strong 0 + resonance at about 12 MeV above 206pb(gs) if a proton energy of 4 0 - 5 0 MeV is used, as shown in fig. 1. The corresponding cross section should show the typical angular distribution corresponding to light ion reactions leading to 0 + states [1,10]. This resonance represents the hole pairing giant resonance and as such it might give important information about the structure of deep-lying states in nuclei.
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