In the preceding Comment, our calculations based on the shell model and on the anisotropic harmonic oscillator model for (8)Ee are criticized. In this Reply we argue that our calculations and remarks on intruder states are relevant.This work was supported by Department of Energy Grant No. DE-FG02-95ER40940, by DGICYT (Spain) grants under Contract Nos. PB95/0123 and SA95-0371.Peer reviewe

Fayache, M.S.

Moya de Guerra, Elvira

Sarriguren, Pedro

Sharon, Y.Y.

Zamick, L.

Physical Review C

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PHYSICAL REVIEW C MAY 1999VOLUME 59, NUMBER 5Reply to ‘‘Comment on ‘Question of low-lying intruder states in 8Be and neighboring nuclei’ ’’M. S. Fayache,1,2 E. Moya de Guerra,3 P. Sarriguren,3 Y. Y. Sharon,2 and L. Zamick2,31De´partement de Physique, Faculte´ des Sciences de Tunis, Tunis 1060, Tunisia2Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 088553Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientı´ficas, Serrano 119, 28006 Madrid, Spain~Received 22 December 1998!In the preceding Comment, our calculations based on the shell model and on the anisotropic harmonicoscillator model for 8Be are criticized. In this Reply we argue that our calculations and remarks on intruderstates are relevant. @S0556-2813~99!01905-6#PACS number~s!: 27.20.1n, 21.10.Pc, 21.60.Cs, 21.60.FwThe main criticism of the preceding Comment @1# is thatthe models we used in Ref. @2# are not sufficiently realistic tosettle the question of the existence of intruder states in 8Be,formerly predicted in Ref. @3# to be at 8 and 9 MeV. Admit-tedly, the anisotropic harmonic oscillator model is a sche-matic model and one may not expect it to be accurate at aquantitative level. However, at the qualitative and semiquan-titative levels the predictions of this model should not bedismissed. Gross features of realistic self-consistent meanfield calculations can be well understood and accounted forwith this model, especially for states which are dominantlyspin S50 states. We therefore feel that we made in Ref. @2#a very solid case to the effect that there are no low-lyingintruders in 8Be despite the fact that they are present in 10Be,12C, and 16O. Not only do we present calculations with re-alistic and schematic interactions, but we also provide physi-cal arguments to support this. Our basic point is that 8Bediffers from the other nuclei ( 10Be, 12C, and 16O) because toget intruder states in 8Be one must excite particles from alower Nilsson orbit, and this costs a lot more energy. In whatfollows we respond to specific points raised in Ref. @1#.In the opening paragraphs of the preceding Comment,Barker implies that he did not suggest any analogy betweenthe energies of low-lying intruder states in 10Be, 12C, and16O and his proposed low-lying intruders ~at 6 and 9 MeV!in 8Be. But in 1988 @3# he does present a table of all theseenergies, and we are sure that any reader would conclude, aswe have, that by so doing he is lending support for his ideasin 8Be.We feel that our conclusion that ‘‘the presence of a low-lying intruder state in 10Be does not imply that there shouldbe a low-lying intruder state in 8Be’’ is irrefutable and standsbeyond any argument on the realistic character of the mod-els. Indeed, the main role of the models used was to help tounderstand that this indeed happens: we find intruder statesin 10Be, but not in 8Be. At no point in Ref. @2# do we say orimply that the question of the existence of low-lying intruderstates is settled by our results. It is our view, however, thatthose calculations contain a great deal of physical insight andthe results are to be taken into account. This view is sup-ported by several facts: ~a! whereas intruder states advo-cated by Barker as R-matrix states in 8Be at 6 and 10 MeVappear as possible ~dashed line states! in the seventh editionof Table of Isotopes ~1978!, they have been removed fromthe latest edition ~1996!. ~b! Recent more realistic shell-PRC 590556-2813/99/59~5!/2958~2!/$15.00model calculations using the Arizona interaction @4# give re-sults that completely agree with ours. ~c! There is still muchcontroversy about how the R-matrix parameters should bechosen @5# and, in particular, a thorough analysis by Humbletet al. @6# comparing R and K parametrizations for the elastica-a scattering finds no evidence for the existence of a reso-nance near 9 MeV.It seems that our argument based on the Nilsson diagramwas not transparent enough and needs further clarification.Since the argument was qualitative, we drew the Nilssondiagram as a function of ubu to indicate that the main effect ofdeformation ~whether prolate or oblate! is to mix l wavessplitting the Vp levels. It is well known that this splitting isdifferent on the oblate and on the prolate sides, in particularin the prolate case the levels with larger V go up and thosewith smaller V go down in energy, while the reverse is truein the oblate case. This is so well known that we could notexpect anyone to think that we implied that the Nilsson dia-gram is symmetric about b50. Clearly the qualitative argu-ment given in Ref. @2# is equally valid for b.0 as for b,0. There is no reason to say that this argument is valid onlyfor the prolate side. Our main point in Fig. 1 of Ref. @2# wasto show graphically why, a priori, there is no analogy in thebehavior of 8Be and that of 10Be or 12C. To form an intruderstate in 8Be one must remove nucleons from the lowest Nils-son orbit in the p shell. This costs a lot of energy. In 10Beand 12C one can remove them from higher Nilsson orbits.This is the crux of the reason why there are no low-lyingintruders in 8Be.Further insight as to why there are no low-lying intrudersin 8Be has been given to us by Vogt @5#. In the a-particlemodel the ground state of 12C would consist of three a par-ticles in a triangle. The intruder state is formed by movingone of the a particles so as to form a linear chain. In 8Be onehas only two a particles. One can rotate them around eachother, but this would only give us higher angular momentumstates of the ground state rotational band. To form an in-truder state, we have to excite one of the a particles. But thelowest excitation energy in 4He is at 20.1 MeV.We strongly disagree with the paragraphs on ‘‘errors andomissions in Table VIII of Fayache et al.’’ In the precedingComment it is said that ‘‘for 10Be^Jy2& for the ~0p-0h!triaxialstate should be 6.35 ~rather than 2.3!.’’ Here is the proof thatthe value we give in Table VIII ~2.3! is correct. The explicitcalculation of ^Jy2& in the ~0p-0h!triaxial configuration (Sx57, Sy55, Sz59) gives2958 ©1999 The American Physical SocietyPRC 59 2959REPLY TO ‘‘COMMENT ON ‘QUESTION OF LOW-LYING . . .^Jy2&54u^101ulyu000&u212u^100ulyu001&u214u^102ulyu001&u212u^201ulyu100&u254FSxSz 1 SzSx22 G1 12 FSxSz 1 SzSx 12 G52.3, Q.E.D.Concerning the factor of 1/2 in Eq. ~16! of Ref. @2#, it maylook reasonable at first sight. However, we have counterar-guments that we will explore in more detail elsewhere. Abrief comment on our ideas is as follows. Expression ~16! ofRef. @2# was derived using quantum mechanical methodsbased on variation after angular momentum projection ~seein particular Ref. @7#!, taking into account that (2IiYoc)21.(Iicr)21 for the anisotropic harmonic oscillator, where IiYocand Iicr denote Yoccoz and cranking moments of inertia, re-spectively. With this factor one gets a continuous transitionfrom the triaxial to the axial case. Since the zero-point en-ergy is a pure quantum mechanical correction, our classicalintuition may fail. Thus, a discontinuity in going from thetriaxial to the axial case may have to do with the fact that inthe axial case rotations around x and y axes are equivalentquantum mechanically ~lead to the same rotational states!.One possible way of seeing that there could be a discontinu-ity in going from the axial to the triaxial case is as follows.In the axial case the intrinsic state will correspond to the K50 band in 10Be, the members of which have angular mo-menta J50, 2, and 4. The triaxial intrinsic state containsboth this K50 band but also a K52 band with angularmomenta J52, 3, and 4.In any case we note that the consequence of putting in the1/2 factor in Eq. ~16! would be that the ground state of 10Bewould not be pushed down so much. This could result inabout 5 MeV lower excitation energies of the ~2p-2h! and~0p-0h! axial configurations. To summarize, putting in thisfactor would reduce by 5 MeV the excitation energies givenin Table VIII for 10Be, But none of these will have any effecton the intruders in 8Be, which is the main point of our paperand of these comments. On the contrary, it would make thepoint even more dramatically that one can have low-lyingintruders in 10Be, but not in 8Be.Barker claims that other 01 low-lying nonintruder statescan be found with the anisotropic harmonic oscillator modelin 12C and 8Be that are in disagreement with experiment. Wethink that if spin and isospin symmetries are taken into ac-count, there are no such states at these low energies. In anyevent, we would like to stress that the shell-model calcula-tions in Ref. @2# stand by themselves and are independent ofthe deformed oscillator model. Indeed, we introduced thismodel in an effort to look for intruder states in 8Be within acomplementary scheme. We would add that we have previ-ously studied the deformed oscillator model and compared itwith the Nilsson model and Skyrme-Hartree-Fock model, es-pecially in the context of intruder states. We find that thesemodels track very nicely in light nuclei @8#.In conclusion, Barker and we should focus on the mainpoint: are there low-lying intruder states in 8Be? As far aswe can tell he describes our calculations as ‘‘unrealistic’’simply because he does not like our conclusions. Indeed,Barker had been sent calculations by other physicists, usingthe Arizona interaction which gives results that completelyagree with ours @4#. Perhaps something good will come outof this controversy. Barker has been correct over the years inemphasizing the importance of using the R-matrix theory forvarious problems involving the continuum. But one is nowrealizing that one cannot take the R-matrix theory ‘‘off theshelf.’’ No one is more suited than Barker to lead the way toshowing what changes need to be made in the application ofthis theory so that it can become a reliable tool in dealingwith fundamental problems in nuclear physics. But the use ofthe R-matrix theory cannot be separated from nuclear struc-ture at both the technical and intuitive level.This work was supported by Department of Energy GrantNo. DE-FG02-95ER40940, by DGICYT ~Spain! grants un-der Contract Nos. PB95/0123 and SA95-0371. M.S.F. kindlyacknowledges travel support from the Laboratoire de la Phy-sique de la Matie`re Condense´e at the Universite´ de Tunis,Tunisia. Y.Y.S. acknowledges financial support from Stock-ton State College.@1# F. C. Barker, Phys. Rev. C 59, 2956 ~1999!, the precedingpaper.@2# M. S. Fayache, E. Moya de Guerra, P. Sarriguren, Y. Y.Sharon, and L. Zamick, Phys. Rev. C 57, 2351 ~1988!.@3# F. C. Barker, Aust. J. Phys. 42, 25 ~1989!; 41, 743 ~1988!; 22,293 ~1969!; F. C. Barker, G. M. Crawley, P. S. Millener, andW. F. Steele, ibid. 29, 245 ~1976!; F. C. Barker, H. J. Hay, andP. B. Treacy, ibid. 21, 239 ~1968!.@4# S. Karataglidis ~private communication!.@5# Erich Vogt, Phys. Lett. B 389, 637 ~1996!; and ~private com-munication!.@6# J. Humblet, A. Csoto, and K. Langanke, Nucl. Phys. A638,714 ~1998!.@7# A. K. Kerman and N. Onishi, Nucl. Phys. A281, 373 ~1977!;F. Villars and Shmeing-Rogerson, Ann. Phys. ~N.Y.! 63, 443~1971!.@8# L. Zamick, D. C. Zheng, S. J. Lee, J. A. Caballero, and E.Moya de Guerra, Ann. Phys. ~N.Y.! 212, 402 ~1991!.

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