Continuum random-phase-approximation study of direct decay of giant monopole resonances in90Zr and208Pb

Volume 230, number 1 2

PHYSICS LETTERS B

26 October 1989

C O N T I N U U M R A N D O M - P H A S E - A P P R O X I M A T I O N STUDY OF DIRECT DECAY OF GIANT M O N O P O L E R E S O N A N C E S IN 9°Zr A N D 2°Spb T UDAGAWA

Department of Phystcs, Umverstty of Texas, Austin, TX 78712, USA and BT KIM

Department of Phystcs, Sung Kyun Kwan Umverszty, Suwon 440-746, Korea Received 30 May 1989, revised manuscript received 11 August 1989

Using a method recently developed by us, continuum random-phase approxxmaUon calculations are performed for the d~rect decay of grant monopole resonances m 9°Zr and 2°Spb, including spreading effects via the imaginary part of the optical potential for the particle excited by the external monopole field The calculated results are discussed and compared with experimental data

G i a n t resonances ( G R ) are collective states described as a coherent s u p e r p o s m o n o f m a n y one-particle-one-hole ( p h ) configurations Since these states a p p e a r in the c o n t i n u u m where the level density is h~gh, the states decay etther by e m m i n g particles or by spreading ( d a m p i n g ) Into more c o m p l i c a t e d c o m p o u n d nuclear states Through a n u m b e r o f studies m a d e m the past [ 1 ], the collective nature o f G R has been well estabhshed The decay properties, however, have not been well u n d e r s t o o d to date, and currently the subject is under intensive study from both experimental [ 2 - 7 ] and theoretical [ 8-17 ] points o f view In our recent p u b h c a t i o n [ 18 ], we reported a new m e t h o d for solving the basic RPA equations and calculating nuclear responses xn the c o n t i n u u m It utahzes the Lanczos m e t h o d [ 19] As an example, the m e t h o d was applied to calculate the 2 + strength functmn S for J60, 4°Ca a n d 2°SPb in the giant quadrupole resonance ( G Q R ) region [18] The spreadmg ( d a m p i n g ) effects were taken into account vm the imaginary part o f the optical potential used for the particle p excited by the external force These calculations successfully reproduced the experimental spectra o f S, including the width The m e t h o d we have d e v e l o p e d is a p p h c a b l e not

only to the total S, but also to the partial contributions from various p a r t i c l e - h o l e ( p h ) c o m p o n e n t s (Sph) or equivalently the p a r h a l widths ( F p . ) This is possible both for the spreadmg and direct particle emission parts, S* and S t, respectively, o f S, a n d / n a n d / ' ~ , respectively, o f F The aim o f the present article is to report on the results o f calculatmns m a d e for these partial contributions, particularly o f the dtrect neutron emtssion from the grant m o n o p o l e resonance ( G M R ) in 2°Spb [6] and 9°Zr [7], on which much attention has recently been focused [3,5-7 ] A discussmn is also given on the present m e t h o d of treating the damping in view o f arguments [ 20 ] based on mtcroscop~c calculations We start w~th g~vmg a b r i e f o u t h n e o f the approach o f r e f [ 18 ] There we introduced a quantity 2 ( r ) that satisfies the following m h o m o g e n e o u s integral equatton

)~(r) =p(r) + f drl dr'l V(r, rt )Ro(rl, r'l, E)2(r'l )

( 1)

In ( 1 ), p ( r ) , V(r, r' ) and Ro(r, r', E) are the external field ( m o n o p o l e in the present case, 1 e , p(r) = ~, r 2 ), the ph interaction, and the free optical

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model response function, respectively. In RPA, Ro (r, r', E) is given as [21 ]

Ro(r,r',E)= ~ O'~(r)gt+)(r,r',E)Oh(r ' ) h

+ O'~(r' )gt +)(r, r', --E)Oh(r) ] ,

(2)

with

g~+)(r,r',E)=(rl

Q

E+Eh--Hp+IE

It')

(3)

In (2) 0 h ( r ) IS the wave function for the (occupied) single hole state h, while g~ +) (r, r', E) is the optical model Green's function for p, Q being the Pauh-operator that projects onto the space spanned by the unoccupied particle states only Since both 0 h ( r ) and g <+ ) ( r, r', E) can be evaluated without any problem, Ro(r, r', E) is a known quantity We may call 2(r) the correlated source function, since 2(r) plays a role similar to that played by p ( r ) , except that 2(r) includes the ph correlation. In terms of 2(r), the strength function S can be given as s=llm(-~z -

fdrdr'p*(r)Ro(r,r',E)~(r')) (4)

Eq ( 1 ) with (2) and (3) is our basic equation. In solving It, we reduced it to a set of equations for the

radial functions 2~h (r) and X~p (r), defined, respectively, as (yplJ. 10h-)= ~ Jg

(Oh [21Y0) = ~ JM

(JpmpJhmhlJM)-1AJph(r)

,

(5)

1~

(JhmhJpmoIJg)

1A-h3p(r) , ¢

(6)

where yp is the spin-angle wave function for p, ~ being the time reversal state corresponding to p Further, the symbols ( II ) and ( II ) were used to indicate that the integrals are carried out over only the spinangle variables, thus leaving the resultant integrals as functions of the radial distance r Similarly, we may define the radial functions, P~h (r) a n d / ~ p (r) for p(r) Note that/~Jh (r) and XhJp(r) correspond to the forward and backward RPA ph amplitudes [21 ] In the present case, these amplitudes are functions of r, and satisfy a set of coupled integral equations that are derivable from eq. ( 1 ). The coupled equations were derived and presented in ref [ 18 ], where we also de-

26 October 1989

scribed a rapid method for their numerical solution. Since details of the method along with the equation were given there, we shall not recapitulate them here After this brief outline of the method of ref. [ 18 ], we now proceed to present several formulas for the calculation of the partial contributions to S and F. We first consider S, which can be decomposed into a sum of the ph components, as S:

E Sp h ' ph

(7)

where

Sph = l I m ( - f dr [p~h(r)Oph(r)+P~P(r)Ohp(r)]) (8)

O,(r) = j dr' g~,+)(r, r' )2~(r')

(9)

The above relations ( 7 ) - (9) can be obtained simply if the integration over the spin-angle variables involved in the RHS ofeq. (4) is performed. Sph, obtained in this way, may further be decomposed into two parts S~h and S~h, the damping (spreading) and direct particle decay parts, respectively The explicit form for S~h may be obtained much the same way as Sph IS obtained by casting the total S ~ (given in ref [ 18 ] ) into a sum ofph components. One obtains then S£ph -~- -- f dr 0ph(r)

Wp(r)Oph(r

)

(10)

and 1 __ ,t Sph --Sph --Sph

( 11)

In terms ofS~h, Fh of G M R may be expressed as T F1" Ff ERWAE ER_AEXpSph(E ) dE

h=

r~ +~-S~ 3 E R -- A E ~-3 k

.t

~--/~

,

(12)

where ER and F are the energy and the width of the resonance, respectively The values of ER and F may be extracted from numerically calculated spectra of S(E) AE defines the range of the integration involved, for which we choose as AE= 2 MeV With this chosen value, the integration In eq (12) can cover almost the entire region of the resonance (see fig 1 given below) Based on the formalism described above, numerical calculations are performed for S, S~ph and F~ Use is made of the same theoretical parameters (the sin-

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gle particle h a m l l t o n t a n for the hole and particle, and the ph interaction, etc ) as those e m p l o y e d in our previous study [ 18 ], where the parameters were taken from Shlomo and Bertsch (SB) [8 ] Here, we make the following two remarks ( 1 ) F o r 9°Zr, n o calculation was done m ref [ 18 ], and therefore no p a r a m eter lS available there for thts nucleus Here, we simply adopt, for thts nucleus, the p a r a m e t e r s used for 4°Ca (U) To study the d e p e n d e n c e o f theoretical calculations on the ph interaction, the calculations are m a d e with two different ph interactions, 1 e , those o f SB and o f Ring and Speth ( R S ) [22] In all the calculations, the strengths o f the ph mteractlon are readj u s t e d so as to reproduce the observed resonance energy ER In fig 1, we present the calculated distributions F(E) o f the energy-weighted m o n o p o l e transition strength defined as

F(E)=

ES(E) × 100 f~ E'S(E')dE'

(13)

The corresponding experimental data, reduced from the measured ct-parttcle inelastic scattering cross secI

I

I

(o) 2ospb 4o

20

(b)

~ 40

9OZr klJ

20

I

10

15 Eex(MeV)

20

Fig 1 Comparison of calculated dlstnbuUons of the energyweighted monopole transition strength w~th experiment for (a) 2°Spb and (b) 9°Zr

26 October 1989

tlons [3,23] are also presented there by histograms with an 1 MeV bin As seen, the calculated F(E) fits the experimental d a t a fatrly well F r o m the calculated spectra, one can deduce the wxdths o f F°al= 3 0 and 2 8 MeV for 9°Zr and 2°8pb, respectively These values agree very well with the experimental values o f Fexp=3 0 and 2 5 MeV [ 3,23 ] It should be remarked at this stage that the calculated F ' s with two different ph interaction are almost ~dentlcal Th~s is not the case, however, for F i , as will be seen below In table 1, we s u m m a r i z e the calculated and experimental values o f F i for neutron emission leading to various hole states h in 2°Tpb and 89Zr / ' i sB and F i Rs shown there are those o b t a i n e d by using the SB and RS ph interacttons, respectively The experimental values ( F i c-p) hsted there are taken from Borghols [5] for 9°Zr and Brandenburg et al [3,6] for 2°8pb Although a great deal o f effort has been m a d e in extracting these values, the best values avadable now, particularly for 2°8pb, are still upper h m l t values Th~s results from the fact that to extract these values from the measured (coincidence cross sect i o n ) data lS extremely difficult, because in doing so one has to subtract from the measured d a t a large background and statistical components, neither o f which is very well known The values hsted m table 1 are those d e t e r m i n e d by assuming the lower limit o f the statistxcal components, and therefore should be viewed as the upper limit As seen in table 1, the two theoretical values are fatrly different from each other, indicating the sensitivity o f F ~ ca~ on the ph interaction Such a s e n s m v l t y was n o u c e d earher in ref [ 5 ] Note that F i ca~ is generally smaller than F~ cxp This is particularly the case for F i sa, F i sB is smaller by a factor o f 4 - 8 m 2°8pb and 3 m 9°Zr than is F i exv The situation is better in F i Rs, but it still underestimates F i exp (except the t"7/2 case) Note also that the FIRS-values are very stmilar to those calculated by Bracco et al [ 5 ] This should be the case, since they used the RS ph lnteractton A difference, though minor, ~s nevertheless seen in the results o f the two calculations, which m a y be ascribed to the difference in the m e t h o d s used and also to some details of the calculattons The result that the theoretical calculations are sensitive to the ph interaction ~mplies that we m a y use the d a t a of Fiex° for testing the interaction At this m o m e n t , however,

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26 October 1989

Table 1 Calculated and experimental F~-values Nucleus

Hole

F~hexp ( keY )

F/,sB (keY)

/mhRS ( keY )

2°8pb

P l/2 fs/2 P3/2 113/2 f7/2 hg/2

< 35 75 75 < 140 -

5 10 16 8 20 6

19 63 43 18 102 1

45 20

56 59

9°Zr

g9/2 Pl/2

150

It 1S not possible to do such a test, sine the data available now are only the upper h m l t values It is thus highly desirable that more accurate data will be taken In the present study, we included the d a m p m g o f the excited particles, but not of the holes We also ignored the interference effect, whose importance has been indicated in recent stu&es [20] The neglect o f the d am p i n g o f the hole states may be justified, since the holes o f the mare p h - c o m p o n e n t s in G M R are those in the u p p er m o s t occupied shells, whose spreading width ( d a m p i n g ) is rather small Microscopic calculations done in the past have indicated the importance o f the interference effect, but such calculations still significantly underestimate the width o f G M R [20] O n e possible source may be t-aced back to the fact that the calculations introduce a rather dramatic a p p r o x i m a t i o n to replace real final nuclear states by pure 2 p - 2 h states, m t o which the collective m o n o p o l e state damps Such an approximation leads to a large interference effect, which cancels out contributions from the d a m p i n g o f the partlcle and the hole i n v o l v e d It is highly probable that the interference effects will be reduced if the approximation is removed, a strong mixing o f 2 p - 2 h and more complicated states with r a n d o m phase nature will then be induced, which in turn causes cancellations between contributions from various 2 p - 2 h components, thereby reducing the net contribution At this m o m e n t , ~t is very difficult to confirm what was argued above by carrying out a reahstic calculation In the present study, we have adopted a phenomenolog~cal approach, neglecting the effect o f the hole d a m p i n g and the interference effect with the reason discussed above, but treating the effect o f the da mp i n g o f the excited particle m a realistic m a n n e r

The successes o b t a m e d in the present study and also in our previous study [ 18 ] support our view, though much has to be done before reaching a more concrete understandmg o f the problem The authors wish to thank Professor M N Harakeh for his valuable c o m m e n t s and Professor W R Coker for his careful readmg o f the manuscript The work is supported in part by the U S D e p a r t m e n t of Energy and by the Ministry o f Education, Korea.

References [1]K Goeke and J Speth, Annu Rev Nucl Part Scl 32 (1982) 65, A vanderWoude, Prog Part Nucl Phys 18 (1987)217 [2] W Eyrlch et al, Phys Rev C 29 (1984) 418, K Fuchs et al, Plays Rev C 32 (1985) 418 [ 3 ] S Brandenburg et al, Nucl Plays A 466 ( 1987 ) 29 [4]K Okadaetal,Phys Rev Lett 48 (1982) 1382, H Ohsuml et al, Phys Rev C 32 (1985) 1789 [ 5 ] A Bracco,J R Beene,N Van Glal, P F Borhgnon, F Zardl andRA Brogha, Phys Rev Lett 60 (1988)2603 [6] S Brandenburg et al, Phys Rev C, to be pubhshed [7] W Borghols,Ph D Thesis (Gronmgen, 1988), unpubhshed [8] S ShlomoandG F Bertsch, Nucl Phys A243 (1975) 507 [ 9 ] G F Bertsch and S F Tsaa, Phys Rep 18 ( 1975 ) 125, S Tsal, Phys Rev C 17 (1978) 1862 [10] KF LluandN Van Glal, Phys Lett B65 (1976)23 [I1]N VanGmlandH Sagawa, Nucl Phys A371 (1981) 1 [12] R de Haro, S Krewald and J Speth, Nucl Phys A 388 (1982) 265 [ 13] S Yoshlda and S Adachl, Nucl Phys A 457 (1986) 84 [ 14] S Adachl and S Yoshxda, Nucl Plays A 462 (1987) 61 [ 15 ] T Vertse et al, Plays Rev C 37 ( 1988 ) 876 [16]M Cavmato, M Marangom, PL Ottavxam and AM Saruls, Nucl Phys A 373 (1982) 445

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[17] M Ichlmura, K Kawahlgashl, TS Jorgensen and C Gaarde, to be pubhshed [ 18 ] T Udagawa and B T Kam, Phys Rev C, to be pubhshed [ 1 9 ] R R Whitehead, A Watt, BJ Cole and I Mornson, Advances m nuclear physics, Vol 9 (1977) p 123 [20] See e g the following revaew articles G F Bertsch, P F Bortlgnon and R A Brogha, Rev Mod Phys 55 (1983) 287,

10

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J Wambach, Rep Prog Phys 51 (1988)989 [21 ] See e g P Ring and P Schuck, The many body problem, (Sprmger, Berlin, 1980) p 324 [22] P RmgandJ Speth, Phys Lett B 44 (1973) 477 [23] C M Rozen, D H Youngblood, J D Bronson, Y-W Lm and U Garg, Phys Rev C 21 (1980) 1252