PHYSICAL REVIEW C

VOLUME 54, NUMBER 6

DECEMBER 1996

Two-particle–one-hole excitations in the continuum S. Fortunato, 1,2,3 A. Insolia, 1,2 R. J. Liotta, 1,3 and T. Vertse 1,4 1

Royal Institute of Technology at Frescati, S-104 05 Stockholm, Sweden Department of Physics, University of Catania and INFN, I-95129 Catania, Italy 3 Royal Institute of Technology at Frescati, S-10405 Stockholm, Sweden 4 Institute of Nuclear Research of the Hungarian Academy of Sciences, Pf 51, H-4001 Debrecen, Hungary ~Received 8 August 1996! 2

The coupling between single-particle resonances and two-particle–one-hole excitations are studied within a representation that includes correlated bound and resonant states. Good agreement with available experimental data is obtained. @S0556-2813~96!02312-6# PACS number~s!: 21.10.Jx, 21.60.Cs, 24.30.Gd, 25.70.Ef

Nuclear excitations lying in the continuum part of nuclear spectra are difficult to study, both experimentally and theoretically, because their formation and decay are time dependent processes. However, new experimental facilities and methods have made it possible during the last years to measure such excitations, particularly single-particle states lying high in the nuclear spectrum @1–4#. As with the excitation of giant resonances, at such high energies the single-particle states are strongly coupled to more complicated configurations, mainly two-particle–one-hole ~2p1h! states. These give rise to the spreading width of the resonance. Attempts have been made to analyze the formation of 2p1h excitations in the framework of the quasiparticlephonon model @5,6#, but only recently has such a study been performed by including the continuum explicitly @7#. In this paper we will present a formalism that also includes the continuum explicitly and is, therefore, an alternative to the method of Ref. @7#. We will show also the importance of the continuum to understand some characteristics of the experimental data. We will describe the coupling of the single-particle resonance ~1p! with the nearby lying 2p1h states in 208Pb by using the single-particle representation described in Ref. @8# ~Berggren representation!. Our aim is to analyze the fragmentation of states in 209Bi measured in Ref. @1# as well as the neutron decay of high lying states in 209Pb, which has recently been observed @4#. The Berggren representation consists, in principle, of bound states, outgoing ~Gamow! resonances, and a limited number of scattering states along a path in the complex energy plane corresponding to a general one-body potential. Using such a basis one would obtain the same results as those obtained by any of the standard methods ~e.g., as described in Ref. @9#! used to calculate quantities along the real energy axis. However, quantities that are evaluated on the complex plane, like, e.g., partial decay widths ~which are defined as the residues of the S matrix @9,10#!, can provide complex physical quantities. An interpretation of the imaginary part of these quantities was given recently by Berggren @11# as the uncertainty in the determination of the mean value of the corresponding operator when the system is in a resonant state. In the case of the partial decay widths from giant resonances the large imaginary parts were found in Ref. @12# to be a manifestation of the fact that the resonance is not 0556-2813/96/54~6!/3279~4!/$10.00

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isolated and, therefore, one cannot associate its decay to the corresponding Breit-Wigner parametrization of the S matrix, which is only valid if the resonance is isolated. The complex probabilities obtained in Ref. @12#, where the Berggren representation was applied, were not due to limitations of the representation. Rather, it indicated that one cannot obtain reliable information about a non-isolated resonance without considering that it interferes with other resonances. By using the Berggren representation one can perform shell-model calculations in the continuum in a similar way as within the standard shell-model formalism. There are, however, two differences. Since the Gamow resonances diverge at infinity one has to use special regularization procedures to treat them. Besides, one has to introduce a metric which is not the same as the Hilbert metric. The difference is only in the radial part, where the scalar product of two functions f and g is not the integral of f 3g * but simply of f 3g. For bound states or scattering states on the real energy axis this metric coincides with the Hilbert metric, since in this case one can always define the phases such that the radial functions are real. Details of the normalization of functions belonging to the Berggren representation and the corresponding scalar product can be found in @12# and references therein. The inclusion of the scattering states in the Berggren representation may cause that the corresponding shell-model dimensions become too large. This point was probed in Ref. @12# in the calculation of giant resonances within the continuum RPA. It was found that the results provided by the calculations agree with those obtained with the truncated Berggren representation, i.e., without scattering states, within 10% in the worst cases, but usually within 5%. We will use this approximation here and neglect all the scattering states along the path. We will thus use as the Berggren representation the eigenvectors of the Woods-Saxon potential of @13# which reproduce rather well the available experimental data corresponding to the low lying ~bound! states. We included in this set of states all eigenvectors that we could find with l<11 and energies up to 150 MeV high and 110 MeV wide. We thus found 244 neutron and 185 proton states @12#. Using this representation we will study the coupling of 1p and 2p1h excitations within the framework of the multistep shell-model method ~MSM! @14#. The MSM solves the shellmodel equations in terms of correlated bases. In the case of 209 Bi the basis elements can be written schematically as 3279

© 1996 The American Physical Society

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u c 1 & 5 u 209Bi& ,

while in

u c 1 Q 1 & 5 u 209Bi ^ 208Pb& ,

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u c P 1 & 5 u 207Tl ^ 210Po&

~1!

u c P 1 & 5 u 207Pb ^ 210Pb& .

~2!

209

Pb the basis states are of the form u c 1 & 5 u 209Pb& ,

u c 1 Q 1 & 5 u 209Pb ^ 208Pb& ,

The bases defined by Eqs. ~1! and ~2! are overcomplete due to violations of the Pauli principle as well as overcounting of states. This problem is solved in the MSM by introducing the overlap ~metric! matrix. Details of this procedure can be found, for the case of the standard shell-model representation, in Ref. @15#. Since our representation contains resonances as well as bound states we call the method used here resonant multistep shell-model method ~RMSM!. A similar basis, but excluding the ~2p!~1h! elements, was recently applied in Ref. @7#. One convenient feature of the MSM is that one can check the results of the calculation with the corresponding experimental data at each step. Thus, in the first state of our case we chose the single-particle states such that the corresponding ~bound! energies agree with experiment, as mentioned above. In the second step one has to evaluate the correlated 2p and 1p1h states. As in Refs. @7,12# we adopt for this a separable interaction with a radial part which is the derivative of the Woods-Saxon potential used to generate the basis. The strength of the interaction is determined by adjusting the energy of a given state. In the two-particle case we always fitted the energy of the corresponding yrast state, while in the particle-hole case we did the same, except for the monopole and dipole states, for which we used the same procedure as in @12#. The dimension of the shell-model equations are very large, but the dispersion relation that is provided by the separable interaction can be solved without major difficulties as the dimension of the two-particle or particle-hole basis increases. The energies and transition probabilities thus obtained reproduce surprisingly well the available experimental data for the particle-hole states @7,12#, including giant resonances @12#, as well as for the two-particle states @16#. Moreover, the wave functions of the two-particle states agree well with the ones given in Ref. @17#, where the Kuo-Brown interaction was used. Finally, in the third step one solves the RMSM equations by using the wave functions and energies of the singleparticle, particle-hole, and two-particle states previously evaluated. The equations of the RMSM can be easily obtained by using the graphical method presented in Ref. @14#. We will not go into detail of this here because for the particular case of 2p1h excitations the formalism, and the corresponding equations, are as in Ref. @15#. To construct the RMSM basis we included only the five lowest 2p and 1p1h states of each angular momentum l @notice that the corresponding parity is (21) l , since separable forces give only natural parity states#. Of the solutions that one obtains by solving the RMSM equations one has to decide which are the physical ones. That is, in a number of cases the energies will have a large imaginary part. One knows that only narrow resonances have a definite physical meaning @12#. Moreover, wide resonances can be considered

as forming part of the continuous background @13#. We therefore decided that only the resonances narrower than 1 MeV @i.e., with 2Im(E n )< 500 keV, notice that the imaginary part of the energy is always negative# have physical meaning and the others are considered as part of the continuum. This reduces drastically the dimension of the RMSM basis. The RMSM basis elements are ordered according to increasing values of the real parts of the energies. We found that increasing the dimension above 280 the results do not change appreciably. Therefore we call the case with dimension 280 ‘‘full calculation.’’ We first calculated states in 209Bi. This case is interesting because a strength fragmentation of a high lying state has been observed @1#. We found that the calculated energies corresponding to the low lying states agree rather well with the corresponding experimental data @19#. Even the structure of the states agree, in all cases, with experimental assignments when they are available. Thus, the first excited state 3/21 is built up mainly on the RMSM configurations u 209Bi(h 9/2) ^ 208Pb(3 2 1 ) & and 210 u 207Tl(d 21 Po(g.s.) & , which agrees with experiment 3/2 ) ^ @19,20#. For high lying states we calculated the corresponding strength functions @1#. To be able to compare with experiment, we constructed a histogram using 1 MeV windows. In Fig. 1 we show the results of the calculation together with the corresponding experimental data @1#. To see the influence of the size of the basis upon the results we also show in Fig. 1 a calculation including only 80 basis states. It is important to notice that by using a bound ~e.g., harmonic oscillator! representation the number of states would increase as one increases the dimension of the basis. In our case, at high energies most of the basis states have large imaginary parts. Therefore, after a certain dimension the number of physical states are not increased because we neglect broad basis states. On the other hand, if the dimension is not large enough the distribution of states may be deficient ~concentrated in a too low energy region!. This can be seen in Fig. 1, where with the reduced dimension 80 one gets a distribution of states that is below the one obtained with the full calculation. One can say that the agreement between the full calculation and experiment is good, particularly if one compares with other calculations @18#. Since most of the states in Fig. 1 correspond to basis elements of the form u c 1 Q 1 & and u c P 1 & one can say that the spreading width is in this case mostly due to the coupling of the single-particle state with the 2p1h configurations. A puzzling feature in Fig. 1~b! is the experimental state between 4 MeV and 5 MeV. This seems to be too low to be the corresponding single-particle state j 15/2 , which lies at 9.74 MeV ~see Table II of Ref. @13#!. When ‘‘dressed’’ by the 2p1h excitations this state moves down to 7.6 MeV, as

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FIG. 1. Strength function in 209Bi calculated within 80 basis elements ~dashed line! and within 280 basis elements ~full line!. Full squares are the experimental results of Ref. @1#. ~a! State i 11/2 , ~b! state j 15/2 .

can be seen from Fig. 1~b!. Given the good agreement between theory and experiment, it is likely that the experimental single-particle state is the one lying at 7.5 MeV in Fig. 1~b!. We calculated also the spectrum of 209Pb, mainly to analyze recent experimental data concerning neutron decay of high angular momentum states @4#. However, even in this case the calculated RMSM low lying states agree well with experimental data @19# regarding both the energies and wave function components. To derive the partial escape widths G i n (E) from the state 209 Pb(i) to the state 208Pb( n ) within the RMSM one has to consider that there are two ways of decay in this case. From the wave function component u c 1 & @see the basis elements ~2!# the calculated state u 209Pb(i) & in the mother nucleus can only decay to the ground state, i.e., to the state u 208Pb( n 5g.s.) & . Instead, from the other two components, i.e., u c 1 Q 1 & and u c P 1 & , the mother nucleus can only decay to a correlated 1p1h state l, i.e., u 208Pb( n 5l) & . The corresponding derivations are rather tiresome, but straightforward. Since in this case there is a strong overlap among all resonances one can only measure the branching ratios between the decay and excitation cross sections. The decay cross section for a given resonance i to a state n is given by

FIG. 2. Branching ratios corresponding to neutron decay from all states in 209Pb to the state 208Pb( n ) for the cases ~a! n 5 ground 2 state, ~b! n 5 3 2 1 , ~c! n 5 5 1 . The full squares with error bars are the experimental results @7#.

s i n ~ E ! 5Nn ~ E !

G i n ~ E ! /2 , ~ E2E i ! 2 1 ~ G i /2! 2

~3!

where Nn is a quantity independent of i ~it cancels out in the branching ratio!, E i is the real part of the energy of the calculated state in 209Pb and G i is minus twice the corresponding imaginary part. For a given energy E the decay cross section to a state n is given by @7#

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s n~ E ! 5

(i p i s i n~ E ! ,

~4!

where p i is the probability of populating the state i in the reaction process, which in the case of Ref. @4# is the transfer reaction ( a , 3 He). We took the values of p from Ref. @7#. The branching ratio is then B n~ E ! 5

s n~ E ! . ( i p is in~ E !

~5!

Again to compare with experiment we integrated the quantity s within a window of 50 keV and calculated the branching ratios ~5! corresponding to the decay to different states in 208 Pb. In Fig. 2 we show the cases for which there are experimental data. The results of this figure are similar to those obtained in Ref. @7#, except perhaps for the ground state, where for the highest energy transitions our calculation understimate the experimental data. One can say therefore that, in general, the RMSM gives a good account of the experimental data. Finally, it is worthwhile to point out that the basis elements of the form u c P 1 & do not play an important role for

@1# S. Gales, C. P. Massolo, S. Fortier, J. P. Schapira, P. Martin, and V. Comparat, Phys. Rev. C 31, 94 ~1984!. @2# S. Gales, Ch. Stoyanov, and A. I. Vdovin, Phys. Rep. 166, 127 ~1988!. @3# D. Beaumel et al., Phys. Rev. C 49, 2444 ~1994!. @4# S. Fortier et al., Phys. Rev. C 52, 2401 ~1995!. @5# A. I. Vdovin, V. V. Voronov, V. G. Soloviev, and Ch. Stoyanov, Sov. J. Part. Nucl. 16, 245 ~1985!. @6# V. G. Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonos ~Institute of Physics Publishing, Bristol, 1992!. @7# N. Van Giai, Ch. Stoyanov, V. V. Voronov, and S. Fortier, Phys. Rev. C 53, 730 ~1996!. @8# R. J. Liotta, E. Maglione, N. Sandulescu, and T. Vertse, Phys. Lett. B 367 1 ~1996!. @9# C. Mahaux and H. A. Weidenmu¨ller, Shell Model Approach to Nuclear Reactions ~North-Holland, Amsterdam, 1969!. @10# R. G. Thomas, Prog. Theor. Phys. 12, 253 ~1954!; A. M. Lane

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excitations into the continuum, although they can be of fundamental importance at low energy, as seen for the nucleus 209 Bi above. In conclusion, we have presented in this paper a method to solve the many-body shell-model equations in the continuum by using a basis consisting of correlated elements. This method is based on the multistep shell-model method @14#, but uses as single-particle states the Berggren representation @8#, which is suitable for describing processes in the continuum. We therefore called this method the ‘‘resonant multistep shell-model method’’ ~RMSM!. We have applied here the RMSM to study decaying proton states in 209Bi as well as neutron states in 209Pb. The calculated quantities agree well with available experimental data, both for bound and resonant states. We would like to express our gratitude to N. Van Giai for clarifications regarding the experimental data as well as the formalism of Ref. @7#. This work was supported in part by the Hungarian Research Fund ~OTKA! through Contract T17298 and the exchange program between the Royal Swedish Academy of Engineering Sciences ~IVA! and the Hungarian Academy of Sciences. S. Fortunato was supported by the Swedish Institute.

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