- Email: [email protected]
Three-body resonances with the complex scaling method: The case of 11Li
Nuclear
ELSEVIER
Physics
A722
(2003)
221c-226~ www.elsevier.com/locate/npe
Three-body E. Garrido”,D.V. “Institute
resonances Fedorovb,
de Estructura
bDepartment Denmark
of Physics
with the complex and AS.
scaling
method:
The case of “Li
Jensenb
de la Materia, and Astronomy,
CSIC, Aarhus
Serrano
123, E-28006
University,
DK-8000
Madrid, Aarhus
Spain C,
In this work we show how implementation of the complex scaling method into the hyperspheric adiabatic method permits simple computation of three-body resonances. The examples of 6He and the three-a’s system are shown. In both cases a good agreement with the experiment is obtained. The method is then used to investigate dipole excitations of llLi. The excitation energies of the l/2 +, 3/2+ and 5/2+ states are found to range from 0.6 to 1.0 MeV. 1. SUMMARY
OF
THE
THEORY
The hyperspheric adiabatic method was designed to compute bound state wave functions of three-body systems [1,2]. In principle the same procedure could be used to investigate resonance wave functions by use of the complex energy method and looking for the poles of the S-matrix in the lower half of the momentum plane [3]. However the resonance wave functions diverge exponentially, and an accurate calculation of them at large distances becomes soon a difficult task. The complex scaling method, introduced at the beginning of the 70’s, permits to solve this problem simply by rotating the radial coordinates into the complex plane [4]. Doing like this the rotated wave functions of the resonances fall off exponentially as soon as the complex scaling angle is larger than the argument of the resonance. Under these conditions it is then possible to compute resonance wave functions using the same numerical techniques as for bound states, in particular with the hyperspheric adiabatic method. 1.1. Hyperspheric adiabatic method To describe a three-body system we use the Jacobi coordinates {$, $} defined for instance in [l]. The three-body wave function is then written as a sum of three components $(i)(&,$), each of them written in terms of each of the three possible sets of Jacobi coordinates [1,2]. These three components satisfies the three Faddeev equations
where T is the kinetic energy operator, ticles j and k, and E is the three-body 0375-9474/03/$ - see front matter doi:10.1016/S0375-9474(03)01369-l
0 2003 Elsevier
is the two--body V~,C(CQ)
interaction
energy. Science
B.V.
All rights
reserved.
between
par-
222c
E. Garrido
et al. /Nuclear
Physics A722 (2003) 221c-226~
By rewriting the Faddeev equations (1) in terms of the hyperspheric coordinates (p = Jm, Qi = arctanZi/yJi, Cl,%, and n,) it is then possible to separate each Faddeev equation into angular and radial parts:
- y
+ $ (X,(p)
+ $
.fd(P) = 0
)I
(3)
where A is an angular operator [a], m is the normalization mass, n labels the angular eigenfunctions &) and the angular eigenvectors A,, and the.functions P,,, and Q,,, can be found for instance in [a]. The idea is then to expand each of the components of the wave function in terms of the angular eigenfunctions $2) (4) where Ri E {CX~,R,%, flYi} denotes the five hyperspheric angular coordinates. Therefore the procedure is then to solve the eigenvalue problem (a), and obtain the radial coefficients in the expansion (4) by solving the couple set of differential equations (3) where the eigenvalues X,(p) enter as effective potentials. Usually the expansion (4) converges rather fast, and only a few terms (typically no more than three) arc needed. 1.2. Complex scaling method The hyperspheric adiabatic method described in the previous subsection could in principle be used to compute not only bound, but also continuum wave functions. The procedure is the same, but one should look for radial solutions of eq.(3) with the proper asymptotics, that is given by
(5) where H(r) and H(‘) are the momentum associated to the S-matrix. -It is known that when the plex momenta (K = \n\e@R), lower half of the momentum asymptotically like
Hankel functions of first and second kind, effective potential X,,(p), K = dm,
K’ is the hyperand S,,, is the
S--matrix is analytically continued into the region of comtl ren resonances show up as poles of the Smatrix in the plane. This means that the resonance wave functions go
lrclpsin(0~),i(l~lpcos:(BR)-K~/2+3n/4) 1
(6)
and therefore the resonance wave function diverges exponentially. This divergence gives rise to delicate numerical problems, since very large numbers need to be compared to the
E. Garvido et al. /Nuclear
Physics A722 (2003) 221c-226~
223~
very small ones coming from Hg?+a in eq.(5) in order to determine the precise three-body energy at which a resonance is present. This problem is solved by use of the complex scaling method, that consists simply in rotating into the complex plane the radial coordinates. In our particular case rotation by an arbitrary angle 0 of the Jacobi coordinates 5 and y leads to the transformation p + peis, while the five hyperangles remain unchanged. After this transformation the radial wave function of the resonance behaves asymptotically like .fnn, +
,-lr;lpsin(O--B~)~z(l~~pcos(8-8~)--K?r/2+3n/il)
1
(7)
and therefore, as soon as 0 is larger than the argument of the resonance HR, then the radial wave function falls off exponentially, exactly as a bound state. Thus, after complex scaling, the same numerical techniques used to compute bound states can be used for resonances, in particular the hyperspheric adiabatic method. In fact, both of them, resonances and bound states, can simultaneously be found following this technique [5]. Continuum states are rotated by an angle 20 in the energy plane [5]. 2. EXAMPLES
Figure 1. Inner part: Energies for the 2+ states in ‘He obtained after solving eq.(3) for scaling angles t)=O.l and 8=0.2. Outer part: Rotated radial wave function for the 2+ resonance for /3=0.2 (thick lines) and the asymptotic function (7) (thin lines).
P (fm) Figure 2. Inner part: Energies for the 0+ states in a 3a’s system after solving eq.(3) for scaling angles 0=0.05 and Q=O.lO. Outer part: Rotated radial wave functions for the lowest O+ resonance for 8=0.05 and corresponding to the two most contributing X’s in eq.(3).
2.1. 2+ resonance in 6He 6He can be considered as a test case for all three-body calculations, since the interactions involved are very well known. We have investigated the 2+ resonance by using the method described above. The interactions used are the ones given in [3], such that
224~
E. Garrido et al. /Nuclear
Physics A722 (2003) 221c-226~
the experimental a-neutron phase shifts are properly reproduced up to an energy of 15 MeV for the sl/z, p,lz, pa/z, da/z, and d5/2 waves. These interactions give rise to a binding energy and a root mean square radius for the “He ground state in agreement with the experimental data. As explained, calculation of the 2+ resonance is completely analogous to the calculation of the ground state, but adding on top the complex scaling and of course including now the components consistent with the new quantum numbers. In particular, for the 2+ resonance in 6He, the total angular momentum has to be 2 and 1,+1, must be even (LZ and 1, are the orbital angular momenta associated to the Jacoby coordinates Z and y’, respectively). In the inner part of Fig.1 we show first the energies obtained after solving eq.(3) with the asymptotic condition of f(p,,,) = 0, where pmaZ is an arbitrary large value of p. In this way the continuum spectrum is discretized, and the different levels are along a line rotated by an angle 20 in the energy plane. In the figure the calculations for 0 = 0.1 and 0 = 0.2 are shown. As seen in the figure, one of the energies obtained from eq.(3) does not change with the rotation angle and is out of the continuum lines. This is the 2+ resonance. This is confirmed by the radial wave function (in the outer part of the figure we show the one corresponding to the most contributing X,-potential), that falls off exponentially. In the figure we also show the asymptotic expression (7), that already matches the computed curves at a distance of around 40 fm. The energy and width of the computed 2’ resonance is 0.84 MeV and 0.09 MeV, that agrees well with the experimental values of 0.82O~tO.025 MeV and 0.113~tO.020 MeV, respectively.
2.2.
Three-alpha
system
In [6] the excited states of the 3a’s system was investigated by use of the hyperspheric adiabatic method and the complex energy method. A slightly modified version of the Ali-Bodmer potential [7] was used. The experimental energies of the bound state and the first O+ resonance were well reproduced. Use of the same interaction and tie complex scaling method gives the results shown in Fig.2. Again in the inner part the dkretized continuum spectrum is shown, and lies along a line whose slope changes for different scaling angles. In the figure the cases of 0 = 0.05 and 0 = 0.10 are shown. Also a bound state at -6.78 MeV and two resonances with (E~,I~)=(0.36,8 . 10e5) MeV and (4.38,0.45) MeV are found. The bound state and the lowest O+ resonance agree with the ones obtained in [6] and the experimental energies of -7.27 MeV and (0.38,8.3 & 1.0. 10P6) MeV. The second O+ resonance was not found in [6], and could correspond to the experimentally found O+ resonance at (ER, I’R)= (3.1 + 0.3,3.0 * 0.7) MeV. The lowest 0’ resonance in the figure could be mixed with the continuum spectra. However, the facts that the energy is independent of the rotation angle, and that the radial wave functions fall off to zero, guarantee that the energy found corresponds to a resonance. In the outer part of the figure the rotated radial wave functions corresponding to the two most contributing Xn’s are shown.
E. Gawido et al. /Nuclear
1 /Q,/2, J.1) (a)
2,
Physics A722 (2003) 221c-226~
5/2f
(0.51, 0.31)
(0.32, 0.15)
1/2+ 3/2+
(0.44, 0.29) (0.52, 0.39)
(0.70, 0.65)
5/2+
(0.63, 0.46)
(0.42, 0.25)
(0.62, 0.52)
Cbl
Figure 3. l”Li spectra consistent with the experimental “Li and ‘iLi data and with the momentum distributions obtained after fragmentation of r1Li [la].
Table 1. Energies and widths corresponding to the three with the “Li possible l- excitations of ‘iLi obtained spectra (a), left part, and (b), right part, of Fig.3. In the lower part the l+ energy in i”Li is 0.35 MeV.
3. DIPOLE
IN “Li
EXCITATIONS
Although several investigations of the excited states of 6He by use of the scaling method can be found in the literature [S-lo], only [ll] is presently found concerning the halo nucleus IlLi, where the O+ excitations are investigated. This is mainly due to the fact that the neutron-gLi interaction is not fully known, and also the spin 312 of the core is enlarging the number of components needed in the calculation. For the “Li subsystem a pi/;! neutron can couple to two different pstates, l+ and 2+, while an slj2 neutron gives rise to a l- and a 2- states. Contrary to what happens for the O+ excitations, dipole excitations require one of the halo neutrons in one of the s-states, while the other one is in one of the p-states. Therefore not only the averaged s and p energies are relevant, as it happens for the ground state or the O+ excitations, but the individual energies of the two s and the two p states are needed. In [la] we have developed a ‘Li-neutron interaction consistent at the same time with the ‘OLi experimental data and the known properties of llLi, binding energy, r.m.s. radius, pwave content, and the observables obtained after fragmentation of llLi. We have found that the most likely interaction corresponds to the “Li spectra shown in Fig.3. The energies of the l+ and 2+ p-resonances are 0.25 MeV and 0.54 MeV, respectively, consistent with the experimental data given in [13]. Furthermore, one of the two s-states is at a very low energy, around 50 keV, although it is not possible to establish which one. Use of this neutron-gLi interaction, together with the neutron-neutron interaction given for instance in [3], gives rise to the l- excitation energies shown in the upper part of Table 1. The left and right parts of the table correspond to the neutron-gLi interaction producing the “Li spectra in the left and right parts of Fig.3, respectively. The excitation energies are obtained after adding to the resonance energy ER the two-neutron separation energy (0.3 MeV for ‘lLi). Therefore the computed excitation energies range between 0.62
226~
E. Gavrido et al. /iV~de~r
Physics A722 (2003) 221c-226~
MeV and 0.95 MeV, that, are below the experimental energy, that is found to be 1.2510.15 MeV or 1.02&0.07 MeV [14,15]. To appreciate how sensible are the results to possible changes in the neutron-“Li interaction we show in the lower part of Table 1 the resonance energies when the l+ resonance in “Li is at 0.35 MeV instead of 0.25 MeV. This new interaction is still maintaining the good agreement with the experimental data for the ground state of iiLi and the observables after fragmentation reactions. Of course the resonance energies are now slightly pushed up, but still the computed excitation energies are a bit below the experiment,al ones. Nevertheless, apart from possible experimental uncertainties, on the theoretical side some unknowns remain, as for instance the role played by a three-body force, that is usually needed in three-body calculations to take into account effects, like polarizations, that can not be incorporated into the model. These three-body effects have not been included in the results shown in this work.
4. SUMMARY In this work we first describe how the use of the complex scaling method permits to investigate three-body excited states by use of the complex scaled hyperspheric adiabatic method. The cases of the 2+ resonance in 6He and the O+ excitations in the 3a’s system are used as illustrations. The good agreement in these two cases with the experimental data and previous calculations with the complex energy method leads us to apply the method to investigate the dipole excitations of ilLi. An appropriate ‘Li-neutron interaction is used. The excitation energies for the three possible 1- excited states (l/2-, 3/22, and 5/a-) range between 0.65 MeV and 1.0 MeV.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
D.V. Fedorov, AS. Jensen and K. Riisager, Phys. Rev. C50 (1994) 2372. E. Nielsen, D.V. Fedorov, AS. Jensen and E. Garrido, Phys. Rep. 347 (2001) 374. A. Cobis, D.V. Fedorov and A.S. Jensen, Phys. Rev. C 58 (1998) 1403. J. Aguilar and J.M. Combes, Commun. Math. Phys. 22 (1971) 169. Y.K. Ho, Phys. Rep. 99 (1983) 1. D.V. Fedorov and A.S. Jensen, Phys. Lett. B 389 (1996) 631. S. Ali and A.R. Bodmer, Nucl. Plhys. 80 (1966) 99. A. C&6, Phys. Rev. C 49 (1994) 3035. S. Aoyama, S. Mukhai, K. Kato and K. Ikeda, Progr. Theor. Phys. 94 (1995) 343. T. Myo, K. Kato, S. Aoyama and K. Ikeda, Phys. Rev. C 63 (2001) 054313. S. Aoyama, K. Kato, T. Myo and K. Ikeda, Progr. Theor. Phys. 107 (2002) 543. E. Garrido, D.V. Fedorov and AS. Jensen, Nucl. Phys. A 700 (2002) 117. H.G. Bohlen ct al., Prog. Part. Nucl. Phys. (1999) 17. A.A. Korsheninnikov et al., Phys. Rev. C 53 (1996) R537. M.G. Gronov et al., Phys. Rev. Lett. 81 (1998) 4325.
ELSEVIER
Physics
A722
(2003)
221c-226~ www.elsevier.com/locate/npe
Three-body E. Garrido”,D.V. “Institute
resonances Fedorovb,
de Estructura
bDepartment Denmark
of Physics
with the complex and AS.
scaling
method:
The case of “Li
Jensenb
de la Materia, and Astronomy,
CSIC, Aarhus
Serrano
123, E-28006
University,
DK-8000
Madrid, Aarhus
Spain C,
In this work we show how implementation of the complex scaling method into the hyperspheric adiabatic method permits simple computation of three-body resonances. The examples of 6He and the three-a’s system are shown. In both cases a good agreement with the experiment is obtained. The method is then used to investigate dipole excitations of llLi. The excitation energies of the l/2 +, 3/2+ and 5/2+ states are found to range from 0.6 to 1.0 MeV. 1. SUMMARY
OF
THE
THEORY
The hyperspheric adiabatic method was designed to compute bound state wave functions of three-body systems [1,2]. In principle the same procedure could be used to investigate resonance wave functions by use of the complex energy method and looking for the poles of the S-matrix in the lower half of the momentum plane [3]. However the resonance wave functions diverge exponentially, and an accurate calculation of them at large distances becomes soon a difficult task. The complex scaling method, introduced at the beginning of the 70’s, permits to solve this problem simply by rotating the radial coordinates into the complex plane [4]. Doing like this the rotated wave functions of the resonances fall off exponentially as soon as the complex scaling angle is larger than the argument of the resonance. Under these conditions it is then possible to compute resonance wave functions using the same numerical techniques as for bound states, in particular with the hyperspheric adiabatic method. 1.1. Hyperspheric adiabatic method To describe a three-body system we use the Jacobi coordinates {$, $} defined for instance in [l]. The three-body wave function is then written as a sum of three components $(i)(&,$), each of them written in terms of each of the three possible sets of Jacobi coordinates [1,2]. These three components satisfies the three Faddeev equations
where T is the kinetic energy operator, ticles j and k, and E is the three-body 0375-9474/03/$ - see front matter doi:10.1016/S0375-9474(03)01369-l
0 2003 Elsevier
is the two--body V~,C(CQ)
interaction
energy. Science
B.V.
All rights
reserved.
between
par-
222c
E. Garrido
et al. /Nuclear
Physics A722 (2003) 221c-226~
By rewriting the Faddeev equations (1) in terms of the hyperspheric coordinates (p = Jm, Qi = arctanZi/yJi, Cl,%, and n,) it is then possible to separate each Faddeev equation into angular and radial parts:
- y
+ $ (X,(p)
+ $
.fd(P) = 0
)I
(3)
where A is an angular operator [a], m is the normalization mass, n labels the angular eigenfunctions &) and the angular eigenvectors A,, and the.functions P,,, and Q,,, can be found for instance in [a]. The idea is then to expand each of the components of the wave function in terms of the angular eigenfunctions $2) (4) where Ri E {CX~,R,%, flYi} denotes the five hyperspheric angular coordinates. Therefore the procedure is then to solve the eigenvalue problem (a), and obtain the radial coefficients in the expansion (4) by solving the couple set of differential equations (3) where the eigenvalues X,(p) enter as effective potentials. Usually the expansion (4) converges rather fast, and only a few terms (typically no more than three) arc needed. 1.2. Complex scaling method The hyperspheric adiabatic method described in the previous subsection could in principle be used to compute not only bound, but also continuum wave functions. The procedure is the same, but one should look for radial solutions of eq.(3) with the proper asymptotics, that is given by
(5) where H(r) and H(‘) are the momentum associated to the S-matrix. -It is known that when the plex momenta (K = \n\e@R), lower half of the momentum asymptotically like
Hankel functions of first and second kind, effective potential X,,(p), K = dm,
K’ is the hyperand S,,, is the
S--matrix is analytically continued into the region of comtl ren resonances show up as poles of the Smatrix in the plane. This means that the resonance wave functions go
lrclpsin(0~),i(l~lpcos:(BR)-K~/2+3n/4) 1
(6)
and therefore the resonance wave function diverges exponentially. This divergence gives rise to delicate numerical problems, since very large numbers need to be compared to the
E. Garvido et al. /Nuclear
Physics A722 (2003) 221c-226~
223~
very small ones coming from Hg?+a in eq.(5) in order to determine the precise three-body energy at which a resonance is present. This problem is solved by use of the complex scaling method, that consists simply in rotating into the complex plane the radial coordinates. In our particular case rotation by an arbitrary angle 0 of the Jacobi coordinates 5 and y leads to the transformation p + peis, while the five hyperangles remain unchanged. After this transformation the radial wave function of the resonance behaves asymptotically like .fnn, +
,-lr;lpsin(O--B~)~z(l~~pcos(8-8~)--K?r/2+3n/il)
1
(7)
and therefore, as soon as 0 is larger than the argument of the resonance HR, then the radial wave function falls off exponentially, exactly as a bound state. Thus, after complex scaling, the same numerical techniques used to compute bound states can be used for resonances, in particular the hyperspheric adiabatic method. In fact, both of them, resonances and bound states, can simultaneously be found following this technique [5]. Continuum states are rotated by an angle 20 in the energy plane [5]. 2. EXAMPLES
Figure 1. Inner part: Energies for the 2+ states in ‘He obtained after solving eq.(3) for scaling angles t)=O.l and 8=0.2. Outer part: Rotated radial wave function for the 2+ resonance for /3=0.2 (thick lines) and the asymptotic function (7) (thin lines).
P (fm) Figure 2. Inner part: Energies for the 0+ states in a 3a’s system after solving eq.(3) for scaling angles 0=0.05 and Q=O.lO. Outer part: Rotated radial wave functions for the lowest O+ resonance for 8=0.05 and corresponding to the two most contributing X’s in eq.(3).
2.1. 2+ resonance in 6He 6He can be considered as a test case for all three-body calculations, since the interactions involved are very well known. We have investigated the 2+ resonance by using the method described above. The interactions used are the ones given in [3], such that
224~
E. Garrido et al. /Nuclear
Physics A722 (2003) 221c-226~
the experimental a-neutron phase shifts are properly reproduced up to an energy of 15 MeV for the sl/z, p,lz, pa/z, da/z, and d5/2 waves. These interactions give rise to a binding energy and a root mean square radius for the “He ground state in agreement with the experimental data. As explained, calculation of the 2+ resonance is completely analogous to the calculation of the ground state, but adding on top the complex scaling and of course including now the components consistent with the new quantum numbers. In particular, for the 2+ resonance in 6He, the total angular momentum has to be 2 and 1,+1, must be even (LZ and 1, are the orbital angular momenta associated to the Jacoby coordinates Z and y’, respectively). In the inner part of Fig.1 we show first the energies obtained after solving eq.(3) with the asymptotic condition of f(p,,,) = 0, where pmaZ is an arbitrary large value of p. In this way the continuum spectrum is discretized, and the different levels are along a line rotated by an angle 20 in the energy plane. In the figure the calculations for 0 = 0.1 and 0 = 0.2 are shown. As seen in the figure, one of the energies obtained from eq.(3) does not change with the rotation angle and is out of the continuum lines. This is the 2+ resonance. This is confirmed by the radial wave function (in the outer part of the figure we show the one corresponding to the most contributing X,-potential), that falls off exponentially. In the figure we also show the asymptotic expression (7), that already matches the computed curves at a distance of around 40 fm. The energy and width of the computed 2’ resonance is 0.84 MeV and 0.09 MeV, that agrees well with the experimental values of 0.82O~tO.025 MeV and 0.113~tO.020 MeV, respectively.
2.2.
Three-alpha
system
In [6] the excited states of the 3a’s system was investigated by use of the hyperspheric adiabatic method and the complex energy method. A slightly modified version of the Ali-Bodmer potential [7] was used. The experimental energies of the bound state and the first O+ resonance were well reproduced. Use of the same interaction and tie complex scaling method gives the results shown in Fig.2. Again in the inner part the dkretized continuum spectrum is shown, and lies along a line whose slope changes for different scaling angles. In the figure the cases of 0 = 0.05 and 0 = 0.10 are shown. Also a bound state at -6.78 MeV and two resonances with (E~,I~)=(0.36,8 . 10e5) MeV and (4.38,0.45) MeV are found. The bound state and the lowest O+ resonance agree with the ones obtained in [6] and the experimental energies of -7.27 MeV and (0.38,8.3 & 1.0. 10P6) MeV. The second O+ resonance was not found in [6], and could correspond to the experimentally found O+ resonance at (ER, I’R)= (3.1 + 0.3,3.0 * 0.7) MeV. The lowest 0’ resonance in the figure could be mixed with the continuum spectra. However, the facts that the energy is independent of the rotation angle, and that the radial wave functions fall off to zero, guarantee that the energy found corresponds to a resonance. In the outer part of the figure the rotated radial wave functions corresponding to the two most contributing Xn’s are shown.
E. Gawido et al. /Nuclear
1 /Q,/2, J.1) (a)
2,
Physics A722 (2003) 221c-226~
5/2f
(0.51, 0.31)
(0.32, 0.15)
1/2+ 3/2+
(0.44, 0.29) (0.52, 0.39)
(0.70, 0.65)
5/2+
(0.63, 0.46)
(0.42, 0.25)
(0.62, 0.52)
Cbl
Figure 3. l”Li spectra consistent with the experimental “Li and ‘iLi data and with the momentum distributions obtained after fragmentation of r1Li [la].
Table 1. Energies and widths corresponding to the three with the “Li possible l- excitations of ‘iLi obtained spectra (a), left part, and (b), right part, of Fig.3. In the lower part the l+ energy in i”Li is 0.35 MeV.
3. DIPOLE
IN “Li
EXCITATIONS
Although several investigations of the excited states of 6He by use of the scaling method can be found in the literature [S-lo], only [ll] is presently found concerning the halo nucleus IlLi, where the O+ excitations are investigated. This is mainly due to the fact that the neutron-gLi interaction is not fully known, and also the spin 312 of the core is enlarging the number of components needed in the calculation. For the “Li subsystem a pi/;! neutron can couple to two different pstates, l+ and 2+, while an slj2 neutron gives rise to a l- and a 2- states. Contrary to what happens for the O+ excitations, dipole excitations require one of the halo neutrons in one of the s-states, while the other one is in one of the p-states. Therefore not only the averaged s and p energies are relevant, as it happens for the ground state or the O+ excitations, but the individual energies of the two s and the two p states are needed. In [la] we have developed a ‘Li-neutron interaction consistent at the same time with the ‘OLi experimental data and the known properties of llLi, binding energy, r.m.s. radius, pwave content, and the observables obtained after fragmentation of llLi. We have found that the most likely interaction corresponds to the “Li spectra shown in Fig.3. The energies of the l+ and 2+ p-resonances are 0.25 MeV and 0.54 MeV, respectively, consistent with the experimental data given in [13]. Furthermore, one of the two s-states is at a very low energy, around 50 keV, although it is not possible to establish which one. Use of this neutron-gLi interaction, together with the neutron-neutron interaction given for instance in [3], gives rise to the l- excitation energies shown in the upper part of Table 1. The left and right parts of the table correspond to the neutron-gLi interaction producing the “Li spectra in the left and right parts of Fig.3, respectively. The excitation energies are obtained after adding to the resonance energy ER the two-neutron separation energy (0.3 MeV for ‘lLi). Therefore the computed excitation energies range between 0.62
226~
E. Gavrido et al. /iV~de~r
Physics A722 (2003) 221c-226~
MeV and 0.95 MeV, that, are below the experimental energy, that is found to be 1.2510.15 MeV or 1.02&0.07 MeV [14,15]. To appreciate how sensible are the results to possible changes in the neutron-“Li interaction we show in the lower part of Table 1 the resonance energies when the l+ resonance in “Li is at 0.35 MeV instead of 0.25 MeV. This new interaction is still maintaining the good agreement with the experimental data for the ground state of iiLi and the observables after fragmentation reactions. Of course the resonance energies are now slightly pushed up, but still the computed excitation energies are a bit below the experiment,al ones. Nevertheless, apart from possible experimental uncertainties, on the theoretical side some unknowns remain, as for instance the role played by a three-body force, that is usually needed in three-body calculations to take into account effects, like polarizations, that can not be incorporated into the model. These three-body effects have not been included in the results shown in this work.
4. SUMMARY In this work we first describe how the use of the complex scaling method permits to investigate three-body excited states by use of the complex scaled hyperspheric adiabatic method. The cases of the 2+ resonance in 6He and the O+ excitations in the 3a’s system are used as illustrations. The good agreement in these two cases with the experimental data and previous calculations with the complex energy method leads us to apply the method to investigate the dipole excitations of ilLi. An appropriate ‘Li-neutron interaction is used. The excitation energies for the three possible 1- excited states (l/2-, 3/22, and 5/a-) range between 0.65 MeV and 1.0 MeV.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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