Revista Mexicana de Física 38, Suplemento 2 (1992) 43-52

Random Phase Approximation Calculations of theNuclear Response in the Continuum

A. GÓMEZ

Institnto de Ciencias Nue/eares, UNAMCircuito ExteriOl', C. U., A .P. 70-543

04510.,",léxico, D.F., MéxicoAND

T. UDAGAWA

Department of Physics, University of TexasA ustin, Texas 78712

ABSTRACT. A formulat.ion ofthe rontinuurn random phase approxilllation (CRPA) rquations w¡tha finite-range parlide-hole (ph) interaction is presented. Tite Tf'sulting equations can be applied lothe calculation of RPA wave fUllctions 1101only in the contilluum, but al50 in lhe bound region.These eRrA equations, whirh are inhomogeneous coupled-channcl integro-differential equationsw¡lh a large dimension and thus difficult lo salve, are modificd so that lhe Lanczos method can beapplied lo salve them relativcly easily. As examples we apply OUT method lo lhe giant quadrupoleresonance (GQR) in 160 and 40Ca. A nuclear matter G-matrix is used for the ph interaclion. Sincetreating the continuum exactly introduces in elfect an infinitely large shell model space, lhe phcorrelations indllced by the G-matrix interaclion are too strong, not only for the bound collectivesta les but for collective stales in the conlinuum as well. Rcnormalizing the exchange ph matrixelernents allows one to fit tite experimental dala in both tlle bound and conlinuum part of theexcitation spectrurn in a consistent manner. For bolh t.lte low-Iying and contillllUIn region the ringapproximalion is showll to be a good approximalion for dealing with tlle exchange part of tite phmatrix elcments that are rcsponsible for the ground statc correlalions.

RESUMEN. Se presenta una formulación de la aproximación de fases al azar en el continuo (CRPA)utilizando tina interacción partícula-agujero de alcance finito. Las ecuaciones resultantes se utilizanpara calcular las funciones de onda en la aproximación de fases al azar (RPA), no únicamente en laregión del cont.inuo, sino también en la de los estados ligados. Estas ecuaciones CRPA, constituyenun sistema in homogéneo de ecuaciones integrodiferenciales acopladas que son difíciles de resolver.Sin embargo, pueden modificarse apropiadamente y resolverse con el método de Lanrzos. Comoejemplo, se aplica el formalismo para describir las resonancias gigantes cuadrupolares (GQR) del160 y 40Ca. Para de.scribir la interacción partícula-agujero se utiliza la matriz G de materia nuclear.El tratamiento exacto del continuo introduce un espado del lIiodelo de capas infinitamente grande,las correlaciones partícula-agujero (ph) inducidas por la interacción de la matriz G son demasiadofuertes, tanto para los estados ligados como para estados colectivos del continuo. Renormalizandolos elementos de mat.riz ph de intercambio se permite ajustar los dalos experimentales en formaconsistent.e a la part.e ligada y del continuo del espectro de excitación.

rAes: 2I.GO.Fw

44 A. GÓMEZ ANO T. UOAGA\\'A

J. INTROOUCTION

Currently, the topic of the nuelear response in the continuulll is of great interest innuelear physics. A nueleus is excited prilllarely through partirle-hole (ph) excitationswhen an external force is applied to the nurleus. The external force can be either elec-troweak or hadronic in nature. When the energy transfered by the external field is abovethe partiele elllission threshold, the partiele which is excited from a hole state is in thecontinuulll. The interaction between partieles and holes creates rorrelations hetween theph pairs, which plays a critical role in deterlllining the characteristics of the excitationspectrum and the subsequent coincident partiele elllission from the nuelear continuumstates. Such excitation spectra in the continuum, ineluding the ph correlations along withthe roincident partiele elllission, can he calculated by CRPA lIlethod [1-5J.

In calculating the nuelear response in the contiuuulll one usually uses the pseudo-potential or ring approxilllation [6J, where it is assullled that the exchange part of the phinteraction is zero-range in character. This approximation greatly silllplifies the numericalcalculations. If in addition the ph interaction is a b-force, for instance a Skyrme-like force,the calculations hecome even more simplified. It is desirable, however, to try to ineluderealistic finite-range ph effective interactions in CRPA calculations, for the same reasonsextensive efforts have been made to use realistic effective interactions in calculating theproperties of the low-lying collective states of nuelei.

There are two major difficulties of using a realistic finite-range interaction in a CRPAcalculation. First, it is difficult to calculate the contributions from the exchange terms ofthe ph interaction. This difficulty occurs hecause, unlike for the direct term, the partieleand hole that form a ph pair are laheled by different spatial coordinates. In a continuumcalculation where the ph interaction acts in general as an integral operator on ph wave-functions, this complicates matters considerably. When the continuum is diScretized thisdifficulty is less trouhlesome, since one simply constructs the RrA matrix Iiamiltonianthat is to he diagonalized using ph matrix elements which are constructed hy numeri-cal integration over known functions. Second, even though it is commou to refer to theG-matrix as a realistic effective interaction, it is well known that the G-matrix hy itselfis inadequate in descrihing the properties of the low-Iying collective states in a discreteRPA calculation [7-9]. Specifically, ph correlations created hy the exchange terms in theph matrix elements are too strong, causing the lowest collective state to hecome unstahle,when the size of the shell model space becomes large. Qne must inelude higher order termsin the G-lIlatrix, such as ph screening effects, in order to make a realistic calculation usingthe G-matrix. Prohahly the best microscopic discrete RPA caleulations of this type aremade by Brand el al. [lO]. Another approach is to use the G-matrix in higher RPA, forinstance second RrA [11J. Both approaches to discrete calculations have heen successfulin the low-lying as well as the giant resonance part of the nuclear spectrllll1.

An exact treatlllent of the continuum effectively introduces an infinitely large shellmodel space, and thus it is natural to expect problellls using the G-matrix in a CRPAcalculation to describe the properties of both the bound and continuum collective states.Ineluding higher order terms in the G.matrix or goiug to higher order RrA in a contin.\lum RPA calculation, is heyond the scope of the present research. In this paper we will

RANDOM PIIASE ApPROXIMATlON CALCULATIONS ••• 45

nevertheless refer to the G.matrix as a realistie interaetion, and simulate the effeet of phsereening in the effeetive interaetion by renormalizing the exehange ph matrix elements.

Several CRPA ealculations have now been made with a finite-range ph interaetion.Shigehara et al. [J2] eOlllputed the spin-isospin and longitudinal response in the quasielas-tic region for 40Ca using a nuelear matter G-matrix for the ph interaetion, ineluding a fullfinite.range treatment for the exehange terms. They found that only in the veetor.isovectorehannel is the ring approximation valid. A Iimitation to their approaeh is that beeause theydireetly compute the nuelear response funetion, as opposed to solving the CRPA equationsfor the excited partiele wave funetions, they can only ealculate the singles eross-section ofthe seattered partiele. Another limitation encountered in solving direetly for the responsefunetion is that one eannot obtain the partial ph strength eontributions to the totalstrength funetion. Uuballa ct al. [13] examined the response in the quasielastie region for12C by direetly solving the CRPA equations using a G-matrix for the ph interaction. Amajor eonelusion of their study is that a proper treatment of the exehange terms resultsin a large reduetion of the strength in the peak region of the longitudinal response.

Reeently, a powerful new method of solving the CRPA equations has been introdueed,whieh can ealculate the eoineidenee particle emission eross-seetion [14.15). The approaehineludes the deeay of the continuum states by the dampiug (speading) of the single pheonfigurations into more eomplieated nuelear states. The essential idea of this approaeh isto modify the basÍc RPA equations, anel then use the Lanezos method to sol ve the modifiedequations. Tbe formalislll presented in Refs. [14-15] was based, however, on the ringapproximation. The aim of this papel' is to first present the formulation of the modifiedCRPA equations, when tbe exehange part of the effeetive ph interaetion is finite-rangein eharaeter, and then apply our approaeh to both the bound and continuum regionsof the nuelear excitation spectrum. For the bound states, the CRPA approaeh ineludesthe coupling with the continuum exaetly, thus treating both the bound and continuumregions of the nuelear excitation speetrulll in a eonsistent manner. It is important to notethat tbe major differenee of the present forlllalism and the one presented in Ref. [13)liesin the method used to solve the CRPA equations.

A brief formulation of the CRPA approaeh is presented in Seetion 2. In Seetion 3 wesho\\' the ealeulation of total and partial strength funetions, finally in Section 4 we presentthe ealeulations exeeuted fol' the 2+ strength funetion in the GQR region for 160 and 40Ca.

2. FORMAL!SM

In the present seetiou, we derive the inhomogetwous CHPA equations in the coordi-nate-spaee for a finite-range ph interaetion, by extending the approaeh of Cavinato et al.[16}, who have previously derived the homogeneous coorelinate-spaee CRPA equations fortite special case uf a zcro-range intcractioll. This inhomogcncolls coordinate-spacc CRPAe'luations thus del'iveu are then Illouified in a manller that allows the use of the Lanezosmethod in their solution. Explicit expressiolls are then given for the total alld partial phstrellgth fUlletions.

46 A. C:ÓMEZ AND T. UDAGAWA

2.1 Coutiuuum RPA iu Coufiguratiou-Spacc

Qur starting point is the following well knowlI set of illholllogeneolls conpled-channel(CC) CRrA equations [17], given ill configllfation-space as

u,J )-E(I[+ vJ O ( 1 )

!lere [ is the unperturbed ph energy lIlatrix d"fined as

while vJ and wJ are the ph interaction matrices, whose elelllellts are obtaill"d as

and

J r' '-IJ/[' '-I)/I~1110tl1ph,p'h' =< JpJ/¡ . Jp'Jh" A >

(2)

(3)

(4 )

In Eqs. (1 )-( 4), the subscript ph refers to a ph configuration cOlIsisting of apartide abovethe Fermi ellergy level with '1uantum nUlllbers l' = (L",jp, /lp, alld (p) and a hole state withquantulll ntllllbers h = (h,jh,I/J¡, and (h) that are coupled lo lotal angular mOllleutUIllJ1\!. Further, (p(fh) and jp(jh) are the orbital ami total angular momentulll of the partide(hole) state. AIso (p and (h are the single partirle and hole energies, respectively. Ingeneral, (p call he either ahove or helow the parlirle emission tllfesl,old energy. Ahove lhethreshold energy, ip takes rontinuous \TaIues (ronlillllouS sp('ctrulll). Since (p can acc¡uirccontinllolls vallles, the RrA matrix llamiltonian is of infinite order.Further, X alld X are column vectors, whose ph components are the forward and

backward going RrA amplitudes Xph(p) and X hp(p) respectively, and p(pph(p)) andfiC1ihp(p)) are the corresponding amplitlldes (components) of the external field. vJh 'h' isp ,pthe ph matrix elemellt betw('"n the forward-forward and hackward-backward ampliludes,while wJh '1' is the ph matrix elemellt betweell tl,e forward-hackward and backward-p ,p ,

forward amplitudes. Since 11,1 is the ph intcraction matrix e!C'lIlcnt responsiblc for theground state condaliolls lakell into account by HI'A, we will refer lo ",J as the grollndstate corrdalion (gsc) ph matrix element, while we will refer to vJ as simply the ph matrixelemen!. Finall)", VA is the alltisymmetrized ph illteraction defined as

(5)

where V is the non-antisymmetrized ph illteraction and Pr ['" In is the exchange operatorwhich exchanges space, spin, ane! isospiu coordinales. In litis stlldy we assullIe lile two-body ph illteraction to be local.

RANDaM PIIASE ApPRaXIMATION CALCULATIONS ••• 47

The quantities vJ, 'h' and wJ, 'h' are more explieitly expressed, respeetively, asp ~,p p t,p

(6)

and

(7)

<P" (with fi = P and h) is the single partide wave functian, and is given as

where

Yio"'o(f) = ¿>lo(f"m~qlj"m")Ylo",(í')Xq"'q

(8)

(9)

In these expressians, ¡¡; represents the state related lo fi by lime-reversa!. The lwo lermsappearing in Eq. (6) are usually refered lo as lhe direcl and exehange parl of lhe ph malrixelemenl respeelively. Similarly in Eq. (7) are lhe direel and exchange conlriblllions lo lhegsc ph malrix elemenls. In Fig. 1 lhese ph and gsc ph malrix dements appearing in thestandard RPA are shown. Note thal for lhe direct parl of vJ¡ 'h' and wJ¡ 'h" lhe particle

p L,p P l,pand hale of a ph pair share lhe same coordinale r, whereas for the exchange part lheparticle and hale have differenl eaordinales, i.e. r and r'. The latler fael lhat lhe parlicleand hale in lhe exchange diagrams carry differenl coordinates r and r', makes il diffiellltto calculate lheir effecl. This is parliclllarly lhe case far ",J, 'h' as we shall discuss later.p l,p

For a realislic finite-range inleraction we consider lhe local nuclear matter G-matrix,considering only the natural parity states [Uf which the non-central part of lhe int<>ractionis unimportant

48 A. GÓMEZ ANO T. UOAGAWA

20.040

Ca GQR

l.L

15.0

10.0

5.0

0.010

G-Motrllt

Shlomo- B.r',ch

15

E( MeV)

20

" . .......

25

FIGURE l. Comparison of ealculated energy-weightcd quadrupole transition strength dist.ributions[oc G-Matrix and Shlorno-BertsC'h intcractioll in units of lhe sum-fule I¡mil. with experimento

3. TOTAL ANO PARTIAL PII STRENGTII rUNCTIONS ANO lJ(EL;O+ - f)

As seen in Hefs. [14-1.5] once lA > is introdueed the strength f"netion S can be obtainedvia

1S = -/m(- < rigolA »

"S can be decomposed into two eomponents, SI and SI, as

S = SI + SI .

(lO)

( 11)

SI describes the contrib"tion from the damping proeess, i.e., the absorption dne to theimaginary potential Wp in the optieal potential Up( = Vp + iWp) introdueed for p, andSI is due to direet particle emission. As diseussed in !ter. [151 the eontribution to thestrength from damping is given by

(12)

RANDOM PUASE ApPROXIMATION CALCULATIONS ... 49

20.0

160 GQR

G-Motrlll

15.0 Shlomo- B•• tlch

~~ 10.0LL

5.0

0.010 15 20

E (MeV)

25 30

FIGURE 2. Histograrns with a 1 MeY hin are experirnental data dedueed frorn hadronie inelastieseattering experirnents of Refs. (18-20].

where

(13)

Wp is a diagonal matrix whose diagonal element is IVp• The contribution SI from thepartide emission may then simply be obtained frolll Eq. (37) as

S' = S - SI . (14)

In addition, each partial ph strength function can be separated into emission and dampingcom ponen ts as

This allows \lS to write SI as

SI = ¿S~hph

( 15)

(16)

50 A. GÓMEZ ANO T. UOAGAWA

Por the low-Iying bound states tbe widths are extremely small, since tbe dampiugpotential, IVp, for tbese states sbould be small, and tbus in order to locate these statesin the present CRrA calculation we add a small artificial damping potential Wp• TheB(EL; 0+ ~ 1) value for tbese states is then given as

lE,+t>.B(EL;O+ ~ 1) = (2L + 1) S(E)dE

Ec-A( 17)

lIere E, is tbe centroid energy whicb corresponds to tbe actual excitation energy and óis tbe range of integration on eitber side of Ee, taken to he a sumciently large value sothat the strength of tbe state is exhausted.

4. CALCULATIONS

In tbis section we apply tbe formalism presented in tbe previous section to calculate tbetotal strengtb function S for hotb bound and continuum states. Specifically we considertbe 3- octopole strength function to ohtain information on the lowest collective state andthe 2+ strengtb fundion in tl,c giant '1uadrupole resonanee (CQR) region for 160 and40Ca. In addition to S we consider, for tbe CQlt states, SI, Sph, and S~h. lt is importantto note that our calculations are not self-consistent in tbat we use empirically obtainedoptical potentials as opposed to deriving potentials using the C-matrix interaction in aIlartree-Pock-type calrulation. The optical potentiaJs used in tbis study are tbe same asthose uscd in Rer. [14] except for tbe IVp introduced for tbe bound states. Tbe sameoptical potential, wbicb is real for energics below tbe Permi-surface, is used to generatethe proton and neutron single partirle wave functions and energies for tbe occupied states.Por the bound state ca1culations we are intNested in calculating tbe B(E3) values andthus for tl,ese calculations we use an electromagnetic octupole field for tbe external fieldwhicb can be expressed as

(18)

where tbe effective charges foc tbe proton and neutron aT(' defined in tbe usual manner as

and

cp=((A-I)jA)3-(Z-1)jA3, (1a)\.

(20)

Por the continuum ralculations of tbe CQR we compare our results to (I',¡/) experimentaldata, cboosing a mass <¡uadrupole field for the external field expressed as

(21 )

RANDOM PIIASE ArrROXIMATION CALCULATIONS ••. 51

2520

.....

""" " .. - ~.:::..-.~ ~:;:_._._.:a--.~.:.:.:.:

15

.............

.' /.... /..' /

.... /..... ;'

...:.:.'~..;.:..::...'--

40Ca GQR G-Matrht

-199!l d¥2

9 d -17/2 3/2

- Id5/2 '1/2

d5/2 d5/2-1

0.6~>(l) 0.52'-..:Jel. 0.4(J)~.c- 0.3CJle(l)

"--(J) 0.2.2-"- 0.1oel.

0.010

E (MeV)FIGURE 3. Partial proton quadrupole strenglhs Sgid~ISgIdJ,ISd:t,ll and Sd{¿d>l. roc 40Ca as a

,;¡ J;¡ :I;¡ ;¡ J

function of excitat.ion energy ror G-J\1atrix and Shlomo-nertsch interaction.

For lhe purpose of comparison we shall show for lhe CQR slales, in addilion lo calcu-lalions using the C-malrix inleraclion, lhose oblained fram lhe calculations made in Ref.[14] where a Shlomo-llertsch version of lhe Skyrme force (momenlum and spin dependentlerms in VA sel lo zero) was used. \Ve shall dislinguish lhe S, Sph, and S~h calculaledwith lhe G-malrix (GM) and Shlomo-llertsch (SO) interactions by allaching to them thesuperscripts GM and SO respec.tively.

In Figs. 1 and 2, we show our calculations for the quadrupole strength in 40Ca and160 compared to lhose of Shlomo and ¡¡ertsch and lhe experimental data of Refs. 18-20.In Fig. 3 we show lhe main partial ]1 - h componenls for lhe 2+ slrenglh in 40Ca as afunclion of the exci lalion energy.

5. CONCLUSION

In conclusion, we have inlroduced a formulalion of lhe CRrA equalions wilh a finile-range residual interaction, where the exchange terms are treatf'd exactly. The eRrAequalions are modified in a manner lhal allows for lhem lo be sol ved ralher easily bylhe Lanczos melhod. The CH PA melhod allows for a consislenl lrealmenl of boundand conlinuum slales. Using a nuclear maller G-malrix for lhe residual inleraclion, we

52 A. CÓMEZ AND T. UOAGAWA

applied the CRPA approach to the 3- bound colleetive states and the CQR of 160and 40Ca. It was found that in order to obtain a good fit to the experinlPntal data,it was neeessary to renormalize the exehange part of the ph matrix elements. The reasonfor this renormalization is that the use of the CRPA approaeh effeetively introd ueesan infinitely large shell model spaee, eausing the ph correlations to beeome too strong.These renormalizations simulate higher order terms in the effeetive interaetion su eh aspartide-hole sereening of the exehange terms.It was found that it was possihle to fit both the properties of the 3- bound collee-

ti ve states and the GQR continuum states consistently. This eannot be aehieved with aSkyrme-like foree [:3,4J. Using a Skyrme foree for the ph interaetion allows one to obtainreasonable fits to the experimental data in the eontinuum regions, but it works poorly inthe bound regions. lt is well known that one eannot fit both excitation energy and B(EL)values simultaneously with a Skyrme interaetion. In additiou it was found that the ringapproximation for the ground state correlation exehange ph matrix elements gave resultsdose to the exac.t results.In the ease of the GQR it was fouud that the G-matrix and Shlomo-Bertseh iuteraetion

lead to siguifieantly different results in the detail of the partial emission strengths. Thesedifferenees ean be resolved by future measurements of the coineidenee partide emissioncross-sectioH.An improvement to the pr •.sent ealeulation wOllld be to explieitly indude higher order

terms in th •. C-matrix in the e1[•.etive interaetioll, sueh as the ph sereening of the cxehangeterms. Beeause ollr formalism exposcs the simplieily of the exehange ph matrix clcmenls,ph sereening for these tCrlns may be simple to indude.

RE,ERENCES

1. S. Shlomo alld C.F. Oerlseh, Nud. Phys. A243 (1975) 5072. C.F. Berteh ano S.F. Tsai, I'hys. Re/l. 18<:(1975) 1253. S.F. Tsai, I'hys. Rev. C17 (197B) 18624. S. Krewalo, V. Klemt, J. Speth ami A. Falssler, Nud. Phys. A281 (1977) 1665. S. Krewalo, Phys. Rev. Lelt. 33 (1974) 13866. F. Petrovich, H. ~lc~lanns, V.A. ~laosen ano J. Atkinson, Phys. Rev. Leli. 22 (1969) 8957. T.T.S. Kno,.1. Blomqvisl allo e.E. Orown, I'hys. Lelt. 31D (1970) 938. r.J. Ellis, E. Osnes, Rev. Mod. [,hys. 40 (1977) 7779. W. Hengevelo, \V.H. Dirkholf allo K. Allaarl, Nud. Phys. A451 (1986) 26910. M.C.E. ilrano, K. Allaarl ami \V.H. Dickhoff, I'hys. Lelt. D214 (1988) 48311. S. Drozdz, S. Nishizzki,.1. Spelh anu J. Wambach, [,hys. Re/l. 107 (1990) I12. T. Shigehara, K. Shimizll ano A. ,\rima, Nud. [,hys. A492 (1989) 38813. ~1. Buballa, S. Drozoz, S. Krewalo aml.1. Spelh, Auu. oll'hys. 208 (1991) (in press)14. T. Uoagawa ano 0.1'. Kim, I'hy. Rev. C40 (1989) 227115. T. Uoagawa ano B.T. Killl, [,hy.v. Lell. D230 (1989) 616. ~1. Cavinato, M. ~larangoni, P.L. Ottaviani ano A.M. Saruis, Nud. Phys. A373 (1982) 44517. P. Ring ami P. Schuck, The Nuclear Many Body Problem (Springer-Verlag, Berlin)lB. K.T. Kno"I'f1e el al., Phys Rev. Lelt. 35 (197;»)77919. Y.\V. Lni el al., Phys. Rev. C24 (1981) 88,120. J. Lisantii el al., },hy.v. Rev. C40 (1989) 211