PHYSICAL REVIEW C VOLUME 37, NUMBER 2 FEBRUARY 1988

Application of Gamow resonances to continuum nuclear spectra

T. Vertse, ' P. Curutchet, O. Civitarese, L. S. Ferreira, ~ and R. J. LiottaResearch Institute of Physics, S-10405 Stockholm, Sweden

(Received 15 April 1987)

Neutron and proton Gamow resonances are evaluated in Pb and they are used together with

bound states as single-particle representation to calculate random phase approximation particle-hole excitations. A good account of the escape width corresponding to the particle-hole giant res-

onances is obtained.

The treatment of continuum spectra in nuclear physicsis an old and outstanding problem. The main reason forthis is that one has to somehow discretize the continuumto treat it numerically. One is thus left with an infiniteset of coupled equations which is diScult to solve. 'Yet, it is necessary to include the continuum in thestudy of many nuclear processes, e.g., the formation ofthe a particle in a decay or the building up of resonantstates lying high in the nuclear spectra.

The study of resonances and other processesinfluenced by the continuum is an actual problem notonly in nuclear physics but in other branches of physicsas well. An important step in the study of the con-tinuum in nuclear physics was given in the 1960'swith the introduction of the so-called Gamow reso-nances (GAR), i.e., a solution of the time independentSchrodinger equation with purely outgoing waves atlarge distances. Such a function (which diverges atinfinite) is not square integrable but one can still definethe norm of a GAR (Refs. 6 and 9) and an inner productbetween two GAR. '

Gamow resonances have recently been applied inmolecular and atomic physics also. ' Even from amathematical point of view the GAR are being studiedas probable powerful tools in spectral analysis. '

An important and rather peculiar property of theGAR is that although they are orthogonal to any realcontinuum state, they are generally not orthogonal towave packets formed from a superposition of the contin-uum states. Moreover, the GAR have a large overlapwith wave packets that are peaked at the resonance ener-gy." Therefore, from a physical point of view it makessense to use GAR to describe states which are immersedin the continuum, especially considering that boundstates, GAR, and an integral over a deformed path L inthe complex E plane provide a completeness relation.Recently it has been shown that an expansion on thisbasis can be defined in different ways both in spheri-cal' ' and in deformed systems. ' In this paper wewil1 use the completeness relation of Ref. 7, i.e.,

g (m )(m

( +g [n )(n

)+Int(L)=l,

where m (n) labels bound states (GAR) and Int(L) is thecontribution from the integral over the contour L. Dueto the fiexibility in choosing this contour one can include

in the basis the resonances which fit best the physicalproblem under consideration. The widths of observableresonances are usually smaller than their energy cen-troids. Therefore, one expects that rather narrow GARin the energy range of interest play an important rolewhile the others should have little effect on the results.In this paper we will use as basis vectors only boundstates and GAR with imaginary parts not larger (in ab-solute value) than the corresponding real parts, i.e.,GAR not broader than twice their energies, and weneglect the continuum integral in Eq. (1). In this waywe expect to include in a feasible manner the most im-

portant effects induced by the continuum on observablenuclear spectra. We shall see below that the resultsconfirm the soundness of this choice.

In this paper we will study resonances in ' 0 and in

Pb, but we will present details of the calculations onlyfor the more complex case of Pb. We will first ana-lyze the single-particle spectrum, i.e., the bound statesand GAR. For this we solve the one-particleSchrodinger equation with a Woods-Saxon potential, in-cluding the spin-orbit term, by using the computer codeGAMow. In ' 0 we obtained single-particle stateswhich fit the experimental spectrum rather well. ' Inparticular, a 96 keV wide d3/2 neutron state has beenobserved in the experiment at 5.085 MeV, while our cal-culation gives 5.201 MeV for the energy of the Od3/2state and 40 keV for its width. Since this is only the es-cape width, one can consider the agreement to be rathergood. In Pb we obtained the single particle statesshown in Table I. As one would have expected the abso-lute value of the imaginary part of the single particle en-ergies (decay widths) are smaller for protons than forneutrons due to the Coulomb barrier. In the same waystates with large orbital angular momenta have small de-cay widths due to the centrifugal barrier. Together withthe calculated states we give in Table I the experimentalvalues which are uncertain for high lying states. Whencomparing the experimental and calculated values inTable I one has to keep in mind that the experimentalvalues do not refer to pure single-particle states. Infact, the calculated bound single-particle energies inTable I are approximately the same as those obtained bysimilar previous calculations. ' From the point of viewof this paper, the interesting feature is the set of un-bound states. Although the proton resonances are nar-row, the width of the neutron resonances varies widely,

37 876 1988 The American Physical Society

37 BRIEF REPORTS 877

No. State

of7/2

2 Ofsn

4 1p ~/2

Of 9/2

6 Og 7/z

7 1d5/28 OA ]]/29 1d3/2

10 2$1/211 Oh 9/212 1f7n13 Oi&3/2

14 2p3/2lfsn

16 2p i/z

lf 9/z18 Ol &]/2

OJ is/220 2d 5/2

21 3s )/222 1f7/2

23 2d 3/224 2f7/2

25 1"il/226 2fs/227 Ok ~7/2

28 1h 9/2

OJ &3/2

30 1 l ]3/2

Ep

—22.67—20.17—18.32—17.33—16.23—12.37—11.04—9.26—9.10—8.71—3.78—3.54—1.84—0.69—0.52

0.494.03—i0.005.43 —i0.005.96—i0.006.75—i0.007.84—i0.048.09—i0.008.53—i0.03

Expt.

—11.48—9.68—9.35—8.36—8.01—3.80—2.90—2.19—0.68—0.98—0.17

—20.99—18.06—17.06—14.96—15.51—15.30—10.69—10.49—8.57—8.35—8.08—7.41—3.93—2.80—1.88—2.07—1.44—0.77—0.78

2.10—i0.872.25 —i0.032.70—l2.325.03—i0.005.40—i0.735.41—i0.017.66—i1.04

Expt.

—10.78—9.71—9.00—8.27—7.94—7.37—3.94—3.16—2.51—2.37—1.90—1.45—1.40

TABLE I. Single-particle states used in the calculations forPb. The proton (neutron) energies E~ (E„)are in MeV. The

experimental data are from Ref. 19. Note that all imaginaryvalues are negative, as it should be for outgoing (decaying) res-onant states (Ref. 11). The parameters in the Woods-Saxon po-tential are a =0.75 (0.70) fm, ra=1. 19 (1.27) fm, Vo ——66.0(44.4) MeV, and V =9.5 (8.25) MeV for protons (neutrons).

With the interaction matrix elements thus calculated wediagonalized the corresponding RPA matrix making useof the complex diagonalization routine F02BDF.

The calculated RPA energies are complex. The realpart corresponds to the position of the resonance whilethe imaginary part is related to its width. The experi-mental width consists of the escape width, produced byparticle emission, and the spreading width, due to mix-ing with more complicated configurations (including col-lisional damping ). It is usually the spreading widththat one calculates. The calculation of the escapewidth requires a proper inclusion of the continuum. 'In our case, however, the imaginary part of the energiescorresponds to the escape width as it is implied in thedefinition of the GAR.

In the rather simple case of ' 0 we calculated the gi-ant dipole resonance (GDR) adjusting the strength of theinteraction to obtain a zero-energy solution (the spurious1 state). We then obtained the GDR at an energy of(23.69—i0.28) MeV, exhausting 94% of the EWSR, infairly good agreement with experiment. ' We also testedthe reliability of our method by comparing with the clas-sical calculation of Ref. 1, where the continuum is treat-ed exactly. For this we chose the potential parametersas in Ref. 1 (case without absorption so that only pro-cesses that contribute to the escape width are included).The results thus obtained agree with those in Ref. 1

within 100 keV. Moreover, we also compared with thecontinuum RPA calculation of Ref. 27 (here we used theparticle-hole Migdal force of Ref. 27) and the results ofboth calculations agree within 150 keV.

In the case of Pb we calculated the quadrupolestates within the basis of Table I. In Table II we presentthe calculated correlated states up to an excitation ener-

gy of 13 MeV. There are some striking features in TableII worthwhile to be commented. The escape width of

from states with practically negligible widths up to thebroad state 2fs/2, for which the width (4.64 MeV) isnearly twice the energy.

With bound states and GAR as single-particle repre-sentation we are in a position to analyze more complexexcitations. We will present here random phase approxi-mation (RPA) particle-hole calculations done using a se-parable multipole-multipole interaction. ' The isoscal-ar strength of the interaction was fixed, as usual, byfitting the experimental energy of the first excited state.Thereby we calculated the isovector strength as in Ref.22. In order to calculate the RPA matrix and the ener-gy weighted sum rule (EWSR) for electromagneticoperators we had to determine the single-particle matrixelements of rBV/Br and r, respectively, between allcombinations of bound and resonant states. Due to thedivergent character of the GAR we had to apply a spe-cial procedure for the calculation of the radial integrals.The idea is the same as the one used in Ref. 9, whichlater became known as "exterior complex scaling. "

E,

4.09—i0.005.44 —i0.005.51 —i0.005.81 —i0.00

10.13—i 2.3310.18—i0.8710.44 —i0.0610.49—i0.8610.78—i2.3311.06—i2.3312.59—i0.8812.78 —i0.0312.79—i0.8712.94—i 0.03

S,

67—i510—i35 —l20+i 02—l20+ iO

349—i2310—i100+i 00+i 00+i00+i 00+i 00+ iO

TABLE II. Correlated particle-hole energy E, up to 13MeV and the corresponding contribution to the isoscalar ener-

gy weighted sum rule S, for the operator rBV/Br in SPb. Theenergies are in MeV and the sum rule is given in arbitraryunits. Ep (E„) are in MeV. The experimental data are fromRef. 19.

878 BRIEF REPORTS 37

the GDR in heavy nuclei is small, only about 15% ofthe total width. The same value is usually assumed forthe isoscalar quadrupole giant resonance (SQGR) in

those nuclei. The experimental SQGR in Pb ex-hausts about 70% of the EWSR and is located at 10.6MeV with a total width of about 2.0 MeV. ' Thus, ac-cording to the previous argument the experimental es-

cape width would be 0.30 MeV.In our case, between 10 and 11 MeV of excitation en-

ergy in the uncorrelated spectrum all states are verybroad, either 1.74 or 4.64 MeV wide. Thus, it wouldseem that our calculation would predict a too wide giantresonance. Even more, up to 13 MeV in that spectrumthe values of the corresponding sum rule are so smallthat it is not clear whether there would be any collectivestate at all between 10 and 11 MeV. However, in thecorrelated spectrum of Table II a very collective stateappears (it exhausts 76% of the EWSR) at 10.44 MeVwith a width of only 0.12 MeV just engulfed by noncol-lective, but very wide, states. This is the SQGR. Al-though already this in itself is a surprising result, it isalso remarkable that the giant resonance is not thelowest state among the AN =2 excited states as it wouldbe within a real (bound) basis. To analyze this in moredetail we also calculated the quadrupole states using thecomputer code RFAFH (Ref. 32) within a harmonic oscil-lator representation with standard parameters. Wefound that bound states are about the same in both cal-culations, but the SQGR is indeed the first AN =2 excit-ed state in the harmonic oscillator case. It lies at 8.66MeV of excitation energy. This value is very low butagrees with the one given in Ref. 34. In our case, thefirst EN=2 excited state is the very broad one at 10.13MeV, which is only weakly excited by the external field.This indicates that our method differs considerably fromthe usual treatment of resonant states.

In general, the main features of the SQGR mentionedabove appear for the isovector case as well.

Just opposite to what one would have predicted fromthe uncorrelated spectrum, our calculated giant reso-nances turn out to be narrow. The main reason for thisis that the most important components of these statesare built upon high spin single-particle states, all of

which are either bound or narrow.Although the calculated position of the SQGR agrees

well with the corresponding experimental value, the cal-culated width seems to be too small. However, one hasto consider that the escape width extracted from the ex-perimental spectrum is rather uncertain. Moreover, onlya small mixing of the giant resonance with the near lyingwide states would enlarge it considerably.

Our strength function is also complex. While theimaginary part of the energy gives the width of the stateconcerned, it is not clear what meaning should be as-signed to the imaginary part of the strength functionand, in general, to the imaginary part of any transitionprobability. A reasonable interpretation of this quantityis that it is related to the interference between the reso-nance and the background of the process being studied.In our case, however, the EWSR has small imaginarycomponents.

We also calculated the sum rule corresponding to anelectromagnetic field to see if the long range componentsof this field would be unrealistically enhanced by thedivergent OAR but we found that the structure of thecorresponding strength function is very similar to ther 8 V /r)r case analyzed above.

To see the effect of the size of the basis upon the re-sults we increased the number of states in Table I by20% but the complex energies of the giant resonancesremained practically the same while the total EWSR in-creased by only 3%. We also verified that a knownproperty of the RPA energy weighted sum rule issatisfied, namely the total values of the uncorrelated andcorrelated EWSR exactly coincide.

Finally, it is worthwhile to point out that graphicalcalculations of the spreading width may be convenientlydone within the formalism presented in this paper.Since the energies can now be complex, one may avoidthe divergences associated with zero energy denomina-tors.

We would like to express our gratitude to T. Berggrenfor illuminating discussions. One of us (P.C.) was par-tially supported by Consejo Nacional de InvestigacionesCientificas y Tecnicas (CONICET), Argentina.

'On leave from Institute of Nuclear Research of the HungarianAcademy of Sciences, H-4001 Debrecen, Pf. 51, Hungary.

~Permanent address: Faculdad de Ciencias Exactas, Universi-dad Nacional de la Plata, Casilla de Correo 67, 1900 La Pla-ta, Argentina.

&Permanent address: Departamento de Fisica, Universidade deCoimbra, 3000 Coimbra, Portugal.

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