Collective effects in the damping width of giant resonances

Volume 101B, number 3

PHYSICS LETTERS

7 May 1981

COLLECTIVE EFFECTS IN THE DAMPING WIDTH OF GIANT RESONANCES R.A. BROGLIA and P.F. BORTIGNON 1 The Niels Bohr Institute, University o f Copenhagen, DK-21 O0 Copenhagen O, Denmark Received 13 November 1980 Revised manuscript received 23 February 1981

The low-lying 3- and 5 - collective modes seem to play a central role in the damping width of the giant resonances. The possibility of observing these couplings is suggested. In a recent paper [1] it was reported that a large fraction (~>50%) o f the damping width associated with the giant resonances arises from the coupling to intermediate two-particle-two-hole ( 2 p - 2 h ) states containing a p a r t i c l e - h o l e pair and a low-lying collective mode (cf. also ref. [2] ). It is the purpose o f the present study to suggest possible ways to directly observe these couplings. The two basic matrix elements responsible for the width o f the giant modes are displayed in fig. l a the corresponding analytic expressions being * 1

~k)((Jp)0 Jp ]Hpv ]Jp),

(1)

,].~-1; X) ((J{,)OJh IHpv I/h).

(2)

Vp = X(jp,/h-l;

and Vh = X(fp

The giant resonance has a RPA amplitude X ( / p , / h 1 ; 20 to be in the configuration determined b y the particle/p and the hole/'~ 1. The particle or the hole can bounce inelastically off the surface and excite a vibrational mode )~'. This process is governed by the particlevibration coupling hamiltonian [3]

Hpv=-R0

~au(r) C Xu

,~x u r x ~ ( r ) ' *

(3)

Vp

V.

J'P~/h

lp~V

jh

(AI

®X) Jh)

,~

--fix

gs A

A-1 (B)

Fig. 1. (a) Diagrammatic representation of the coupling of a vibrational mode with a particle-hole pair and another vibrational mode. Such couplings have also been considered in the study of the plasmon damping [ 10]. (b) Schematic description of the mechanism of population of the multiplet of states ((h'®j h-1 )Jh) after particle decay of the giant mode. with matrix elements

1 Permanent address: University of Padova, Istituto di Fisica Galileo Galilei, Padova and INFN, L.N. Legnaro, Italy. ~1 In ref. [4] it has been shown that the couplings (1) and (2) correctly account for the spreading widths of both giant resonances and of single-particle states.

<(/,x,)/rHpvl/> = (_0j+ln f ~j~/, ~-r au r 2 dr fin (X')

4~ 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company

0 -½ " (4) 135

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T h e q u a n t i t y ~ n ( X ' ) is t h e d e f o r m a t i o n p a r a m e t e r associated w i t h t h e n t h p h o n o n o f m u l t i p o l a r i t y X'. T h e p r o b a b i l i t y o f f i n d i n g t h e giant r e s o n a n c e p e r u n i t e n e r g y i n t e r v a l is d e t e r m i n e d b y t h e s p e c t r a l dist r i b u t i o n o f the m a t r i x e l e m e n t s

v/,

7 May 1981

V2a = V 2 + Vh2 + 2 V p V h X ( r e c o u p l i n g ) ,

(5)

w h e r e a i n d i c a t e s t h e giant m o d e a n d c~ s t a n d s for t h e labels o f t h e i n t e r m e d i a t e 2 p - 2 h s t a t e ( j p ® ]~- 1 ®X'). To t h e e x t e n t t h a t t h e s e m a t r i x e l e m e n t s are c o n s t a n t

(1)

(A)

0.5

PaIEI l (MO,t0.5 "~)I~-

0.4

(D)

(3) It I I

0.4 0.3

(2)

0.2

it) 0.2 f OJ

0 0

I 14

P

(E)(MeV)

IG

i

,

,[IkT 20 18 i

i

,

i

i

,

i I

22

24

i

i

I z

x-~16~

18

h,, 2G t

(s)

G 4 2

0

14

tG

18

20

I/.

16

18

20

22

24

i 2G

28

22

24

2

28

E(MeV) Fig. 2. The GQR in 4°Ca. (a) Coupling matrix elements Va2~between the (a) = (GQR(4°Ca)) and the intermediate two-particletwo-hole states (~) = ((~'®/~-1 )/h) in the energy region around the unperturbed energy of the giant resonance. The width r [cf. eq. (9)] is displayed in (b). Only the contributions of the 3- and 5 - lowqying collective modes are included. It shows a marked structure as a function of the excitation energy. It is correlated with the structure displayed by the matrix elements Va2a. This is because the main contributions to r arise from on-the-energy-shell transitions. In (c) the self-energy (8) is shown as a function of E calculated for A = 1 and including only the contributions arising from the lowest 3- and 5 - vibrations. (d) The strength function Pa(E) associated with the giant quadrupole resonance of 4°Ca, as a function of the energy. All contributions arising from states c~with an excitation energy < 60 MeV have been included (continuous line). The square of the associated standard deviation is ~z2= =fPa(E) En dE/fPa(E) dE. The centroid (E) is at 18 MeV. Also shown (dotted line) is the strength function Pa(E) calculated including only the contributions of intermediate states which contain the lowest 3 -(3.36 MeV) and 5-(4.18 MeV) surface mode of 4°Ca. 136

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and that the spacing between the different states cr is uniform, the damping width is given by the golden rule

(6)

r =2~/D,

where lID is the density of states per unit energy. The spectral distribution of V 2 is shown in fig. 2a, for the case of the ,-,,-,n ,,,~K ot~ 40 Ca. asSingle-particle wave functions and energies from a H a r t r e e - F o c k calculation with a Skyrme III interaction were utilized. All RPA roots associated with the isoscalar normal parity states with 3,' ~< 5 were included in the calculation of V2a (cf. also ref. [1] ). They were obtained by diagonalizing a schematic particle-hole interaction with a strength fixed by the energy of the low-lying 3 - state. In particular the RPA energy E a of the GQR is predicted at 18,9 MeV. Modes with X' > 5, as well as isovector modes appear at rather high excitation energies. The associated 2 p - 2 h intermediate states will thus be so far off-the-energy-shell that they are not expected to contribute to the damping. Making use of (6) one roughly obtains F ~ 3 MeV [ ~ 2 ~ 0.1 meV 2 and D ~ (1 MeV/5)]. Note that the largest values of V2~ are associated with intermediate states containing the collective 3 i- (3.36 MeV) and 51 (4.18 MeV) low-lying modes. Because of the strong state dependence of the coupling matrix elements, a diagonalization is called for. This can be accomplished by calculating the strength function [5] (cf. also ref. [6]) 1

Pa(E) = ~

(La

+ AE

a

F+A - E) 2 + ¼ ( r + A) 2 '

(7)

which gives the probability of finding the state a as a function of the energy. The quantity E a is the unperturbed energy (RPA root) of the giant resonance, while the associated self energy is z2x15a = ~

a

2

Vae~(E

--

Ec~)2

( E - E a ) 2 + (A/2) 2"

(8)

r=a~

(t; - E ~ ) 2 + ( A / 2 ) 2

account, in some approximate way, the width of the 2 p - 2 h intermediate states. From systematic calculations of the damping widths of the monopole, dipole, quadrupole and octupole giant resonances [4], it seems possible to conclude that dx is of the order of 1 MeV. Assuming constant matrix elements and a uniform distribution of intermediate states a , AE a vanishes, F becomes equal to (6) and Pa(E) describes a Breit-Wigner resonance shape. The result for the GQR of 4°Ca is displayed in figs. 2 b - 2 d . The centroid o f P a ( E ) is at 18 MeV, and its standard deviation is o = 1 95 MeV. Assuming a gaussian distribution this value leads to a width P = (8 X In 2)1/2 a ~ 4.6 MeV, in agreement with the estimate above. If only the 3 i- and 5 i- low-lying collective modes are included one obtains a ~ 1A MeV (Pcoll 3.3 MeV), which accounts for ~ 70% of the total width (cf. fig. 2c). The corresponding centroid is now at E = 18.7 MeV. The above numbers are to be compared with the experimental values [7] ff ~ 17.8 MeV and P = (3.5 + 0.3) MeV. The agreement with experiment is typical. The couplings (1) and (2) seem to provide the correct mechanism of the damping of giant resonances (cf. also refs. [2,6,8 and references therein] ). The structure displayed by the matrix elements due to strong couplings associated with the low-lying collective vibrations will have consequences in the relative population of final states after particle emission, provided that the width of the intermediate 2 p 2h states is small compared to the spreading width F, that is, provided A ~ p (cf. also ref. [9]). This is because the largest values of V2a in fig. 2a are associated with the configurations (1) ~ ((f7/2 ® d5/12) ® 3 i - ) ' (2) -= ((f7/2 ® ds/1) ® 5 1 ) , O) -= ((P3/2 ® d5}2) ® 3 i - ) , (4) - ((P3/2 ® d5/2) ® 3 2 ) ,

The energy dependent width is defined as

vL

7 May 1981

(9)

The parameter A represents the energy interval around E~ over which averages are carried out. It takes into

for both proton and neutron particle-hole excitation. Thus, particle evaporation from the high-energy region ( > 2 0 MeV) of the GQR is expected to populate the multiplets (3 i- ® d~/2) and (5 i- ® d~/12) of 39K (cf. the schematic scheme shown in fig. lb). Localization in the 40Ca spectrum can be achieved 137

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exciting the GQR in, for example, an ( a , a ' ) reaction. By measuring in coincidence the 3'-rays emitted by 39 K, the final states populated after particle emission may be identified. A change in the multiplicity associated with the (X'®/'~- 1 ) multiplets as a function of the reaction Q-value is expected. Because the importance of collective modes in the damping width of giant modes seems to be universal [1,4], similar effects are expected for other multipolarities and different masses.

References [1 ] G.F. Bertsch, P.F. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Lett. 80B (1979) 161.

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[2] J.S. Dehesa, S. Krewald, J. Speth and A. Faessler, Phys. Rev. C15 (1977) 1858. [3] A. Bohr and B.R. Mottelson, Nuclear structure,Vol. II (Benjamin, New York, 1975). [4] P.F. Bortignon and R.A. Broglia, Nucl. Phys., to be published. [5] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. I (Benjamin, New York, 1969) Appendix 2D. [6] V.G. Soloviev, C.H. Stoyanov and A.I. Volovin, Nucl. Phys. A288 (1977) 504. [7] F. Bertrand, Ann. Rev. Nucl. Sci. 26 (1976) 457. [8] G.E. Brown, J.S. Dehesa and J. Speth, Nucl. Phys. A330 (1979) 290. [9] G.J. Wagner, Invited paper presented at the Intern. Syrup. on Highly excited states in nuclear reactions (Osaka, May 1980), to be published. [10] D.F. Du Bois, Ann. Phys. (NY) 8 (1959) 24.