New giant resonances

I ~

NuclearPhysics A217 (1973)

182--188;

(~North-Holland Publishin# Co.,Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

NEW GIANT RESONANCES TOSHIO SUZUKI t

Cyclotron Laboratory, Institute of Physical and Chemical Research, Wako-Shi, Saitama, 351 Japan Received 25 June 1973 Abstract: The excitation energies for giant monopole and quadrupole resonances are estimated

using self-consistent field theory. The mass parameters for these oscillations are determined by using classical energy weighted sum rule values. It is predicted that both the isoscalar monopole and quadrupole states have an eigenfrequency of 1.41 Ogo[e~0 ~ 41/AX~MeV] and that the isovector monopole and quadrupole resonances will appear at Ex g 4.35 COo and 3.16 ~Oo, respectively. The systematics of the excitation energies for the observed new giant resonances are well explained by the simple model. 1. Introduction

Recently, a new g i a n t r e s o n a n c e has been systematically o b s e r v e d at E x ~ 60fA ~ M e V in ( e , e ' ) , ( p , p ' ) a n d (3He, aHe ') experiments a n d it is suggested to be due to isoscalar m o n o p o l e a n d / o r q u a d r u p o l e oscillations 1' 2). The b r o a d r e s o n a n c e at Ex g 120/.4 ~ M e V is also o b s e r v e d in (e, e ' ) experiments ' ) , a n d is considered to be an isovector m o n o p o l e a n d / o r q u a d r u p o l e resonance. It is the p u r p o s e o f this p a p e r to investigate the excitation energies for these collective m o n o p o l e a n d q u a d r u p o l e states, whose excitation strengths exhaust the classical energy weighted s u m rule values. T h e nuclear p o t e n t i a l for these oscillations will be o b t a i n e d f r o m the h a r m o n i c oscillator shell model, t a k i n g into a c c o u n t the conditions for a self-consistent field as p r o p o s e d b y B o h r a n d M o t t e l s o n 3). O n the o t h e r h a n d , the mass p a r a m e t e r s will be d e t e r m i n e d by the classical energy weighted s u m rule values. F o r simplicity, we will use n a t u r a l units t h r o u g h o u t the calculations.

2. I s o s c a l a r m o d e

2.1. ISOSCALAR MONOPOLE OSCILLATION A s s u m i n g the nuclear p o t e n t i a l for isoscalar m o n o p o l e oscillation ( I S M O ) to be a h a r m o n i c oscillator p o t e n t i a l with a m o n o p o l e - m o n o p o l e interaction, mcoo2 2 v =

•

Z--

1 moo 2

r, + - - 2

2

E i

E j

( = l f po(r)r2 dr) , t On leave from the Department of Physics, Tohoku University, Sendai, Japan. 182

(1)

NEW GIANT RESONANCES

183

the one-body potential is described as

V(r) -= rnm2 r2(1 + Z~o~o)+ 1 rne~ Z~oA
5).

(2)

i

When we consider ~ as a dynamic variable for small monopole oscillations, the above equation represents the fact that the point r in the harmonic oscillator potential 1 2 (~mCOo r2) goes into the point r ~ r(1-½X~ ~ ) , (3) by the monopole-monopole interaction. Then, correspondingly, the variation of the density due to the monopole-monopole interaction may be of the form

3p(r)

1 s s 1 d ~Zo % -~ d-r {rap°(r)}'

=

(4)

where we take into account the conservation of mass number:

f3p(r)dr = 0.

(5)

By making use of eq. (4) in the condition for the self-consistent field, s ~0

= f 3p(r)r 2 dr,

(6)

we obtain the strength of the monopole-monopole interaction:

Z~o = -1/A(r2).

(7)

Hence, denoting the mass parameter and excitation energy for ISMO as B~ and E~), respectively, the potential for ISMO can be expressed as / s s Bo(Eo)

2

s

2

=

l

2

4A(r2) 1I (~o) z,

(8)

which gives E o = o~o 4

2A B~o( r 2

.

(9)

In the above expression, we take 20~o for the frequency of ISMO in the harmonic oscillator potential. The mass parameter B~ is determined by the relation between the zero point energy and the classical energy weighted sum rule value, 1

s s 2 s 2 Bo(Eo) <(%) )

= gEo,l s

s s 2) = -2 a(r2), Eo((%) m

(10)

184

T. S U Z U K I

yielding

BSo = m/4A.

(11)

From eqs. (11) and (9), the excitation energy for ISMO is obtained as E; = x/?coo ~ 58/A ~ MeV

(co o ~ 41/A ~ MeV).

(12)

2.2. ISOSCALAR QUADRUPOLE OSCILLATION The nuclear potential for the isoscalar quadrupole oscillation (ISQO) is assumed to be a harmonic oscillator potential with a quadrupole-quadrupole interaction: mco2 2 V = ~ -~r,+

1 mco2 z ~ E E (r,2Y2~)(rj , 2 Y2.). - --

•

2

2

(13)

i,j.

In considering fl~, = ~ i ri2Y2~ as the dynamic variable for small quadrupole oscillation, the variation of the density caused by the Q-Q interaction may be described as

5p(r) =

½Z~~ fl~2, Y*~(P)rdpo/dr,

(14)

It

corresponding to the variation of the one-body potential, a v ( r ) =½mco or 2 2Zs 2 ~ f l ~ #

* A

(15)

Since the condition for the self-consistent field, fl~ = ~6pr2Y2udr, gives the strength of the Q-Q interaction,

Z~2 = - 8rc/5A(rZ),

(16)

the nuclear potential for ISQO can be written in the form

r ½B~(2coo)2 - 27zmco°2]~-~ns. ns

s s 2 ~-~as* ns = ½B2(E2)

L

Z~/XP'2/z/J2'u

5A(r2)] ~u /JEgP2.U"

(1 7)

On the other hand, using the classical energy weighted sum rule value, a

s

2> =

5A

4~zm

(18)

we c a n o b t a i n the mass p a r a m e t e r

Bs2 = 2rrm/5A(r2).

(19)

Utilizing eqs. (19) and (17), one finds the excitation energy for ISQO,

E"2 = ~/2coo ~ 58/A~ MeV.

(20)

NEW GIANT RESONANCES

185

3. Isoveetor mode

The potential for the isovector mode is estimated by the nuclear symmetry potential 3). 3.1. ISOVECTOR MONOPOLE OSCILLATION

Provided that the nuclear potential for the isovector monopole state (IVMO) is given as ~ m o 9 2 r E + 1 me) 2 v V = . -7 ~ - ~ - Zo .~. {( r 2 - (rZ>)T~}{(r]-- (r2>)'~), (21) l~J

the one-body potential is of the form:

"2- ) 3

V(r) = ½,.o9 ,: + ½mo9

( ~ = ~ (r 2 - (r2))z~),

(22)

i

where 33 = + 1 for a neutron and - 1 for a proton. By comparing eq. (22) with the symmetry potential, ~sym = ¼Vl N -~Z

~'3 = IVl Pn-Pp --"~'3' PO

(Vz

= 100 MeV),

(23)

the variation of the isovector component of the density due to the isovector monopole-monopole interaction is obtained as 6(pn _pp) = "Z"tv-' ~....2 01 ZoV0~{r 2 _
(24)

f g(p°-pp)dr = o.

The normalization condition, ct~ = ~6(p.-pp)r2dr, gives the strength of Z~: 2 V~ 175 Zo - mo924A 12R 4'

(25)

where we use the uniform density distribution with radius R for po(r). Then, in the same way as for the isoscalar mode, the excitation energy for IVMO is expressed as ( E~ = 2 o 9 0 1 +

I/'i 175 ~ + 16o92AB~o 12R4 ] .

(26)

When we take the mass parameter B~ for IVMO, ~ .__5m B o ,~ B o = 12AR z

(R = 1.2A ~ fm),

(27)

the value of the excitation energy is Eo = 2o90 \1 + 16mo9---~g2]

"~ 4"350)0 ~ 178/A~ MeV.

(28)

186

T. SUZUKI

3.2. ISOVECTOR QUADRUPOLE OSCILLATION In the case of the isovector quadrupole oscillation (IVQO), the effective nuclear potential may be described as

rE-y-=

'~o0~O r ? + _ _1_ mo0o Z ; E2

22

~

~ •

~

,J _ (,., r~. ~3)(rj r2, ~).

(29)

Then the one-body potential is given as V ( r ) = :mo00r 1 2 2 + 12mo00)~2 2 v E fl2, v , r 2 Y2.(P)Za

(30)

,u

F r o m the relation between eqs. (30) and (23), we obtain for IVQO ~ ( P n - - P p ) = ½trio02 )~v2E flY2:r2112, #

4p0/1/1.

(31)

By considering fl21x " = ~iri 2 Y2~z3i as the dynamic variable for IVQO, the strength of the X~ is estimated by the normalization

fl~2. = f 6(pn-- pp)r z Y2. dr, d

(32)

and has the value v )~2 -

2

Vl

4~z

mo0g 4 A(r4) "

(33)

Because the quadrupole transition with excitation energy 2o0o is taken in the harmonic oscillator potential, we can obtain the excitation energy for IVQO from eqs. (33) and (29),

[

_Vl _q+

E~ = 2(o0 1+ 4o0~ AB~(r4)A 2o00 FI + L

7V1 7 + 8rneo~ R ZJ

3.16O)o ~ 130/A + MeV.

(34)

The mass parameter B~ ~ B~ = 4rmt/lOA ( r 2) was calculated from the classical energy weighted sum rule value in subsect. 2.2.

4. Comparison with experiment and discussion In fig. 1, we compare our results with the experimental values summarized in ref. 4). The black circles stand for the experimental excitation energy values for giant dipole resonances. The crosses and open circles are for monopole or quadrupole resonances, the former being obtained from (p, p') experiments and the latter from electron scattering. The straight lines refer to the calculated excitation energy values.

NEW GIANT RESONANCES

187

This figure also shows the excitation energy for isovector dipole resonances estimated by Bohr and Mottelson 3).

t

E~

160t"~ 140 120 100 80 •

60 _

X X

".

~x

o

o ~ % O

X×

o

o

Eo~ ~ d

40

56

100

--'150

"200

E~

A--

Fig. 1. Systematics of excitation energies for giant multipole resonances. The experimental data are taken from ref. 4). The black circles are for El. The crosses and open circles are for E2 or E0, the former being obtained from (p, p') reactions and the latter from electron scattering. The straight lines represent the calculated values of the excitation energies.

It is not yet determined experimentally whether the new giant resonance observed at about 60/A ~ and 120/A -~ MeV is of monopole or quadrupole type. In our model, it is predicted that the resonance observed near 60/A + MeV is due to both the isoscalar monopole and quadrupole states, while the broad resonance at about 120/,4 + MeV is due to the isovector quadrupole state. The isovector monopole resonance will appear at a higher excitation energy of 178/A ~ MeV. It is interesting to compare our results with the previous one obtained by the hydrodynamic model and RPA calculations. The excitation energy for isoscalar monopole oscillation is estimated to be 56/A ~ MeV ( ~ 1.4~Oo) by the compressible liquid drop model with a uniform compressibility of 100 MeV [ref. s)], and in the hydrodynamic model the isovector monopole and quadrupole resonance energies are predicted at 4.3o) o and 3.2o)o, respectively, if the energy for thegiant dipole resonance is normalized to be 2o) 0 ~ 82/A ~ MeV [ref. 6)]. Hamamoto 7) has performed an RPA calculation based on the model proposed by Bohr and Mottelson 3) and obtained excitation energies of 1.4o) o and 3.3o) o for isoscalar and isovector collective quadrupole states, respectively. It should be noted that these values are all of the same order of magnitude as the present results.

188

T. SUZUKI

5. Summary W e estimate the excitation energies for m o n o p o l e a n d q u a d r u p o l e states whose excitation strengths exhaust the classical energy weighted s u m rule value. T h e p o tential for those states is o b t a i n e d f r o m the h a r m o n i c oscillator shell m o d e l with a self-consistent i n t e r a c t i o n 3). T h e mass p a r a m e t e r s are d e t e r m i n e d b y the classical energy weighted s u m rule value. A c c o r d i n g to o u r model, b o t h the isoscalar m o n o p o l e a n d q u a d r u p o l e states have an excitation energy o f 58[A ~ MeV. T h e isovector m o n o p o l e r e s o n a n c e will a p p e a r at a b o u t 180//1 ~ MeV, while the isovector q u a d r u p o l e r e s o n a n c e will a p p e a r at a b o u t 130/A t MeV. T h e systematics o f the excitation energies for the o b s e r v e d new g i a n t m u l t i p o l e resonances are well e x p l a i n e d by the simple model. T h e a u t h o r w o u l d like to t h a n k Professors H. Ui, G. T a k e d a , Y. T o r i z u k a a n d H. K a m i t s u b o for useful discussions a n d e n c o u r a g e m e n t .

References 1) Proc. Conf. on nuclear structure studies using electron scattering and photoreaction, Tohoku University, Sendai, Japan, 1972 2) M. B. Lewis and F. E. Bertrand, Nucl. Phys. A196 (1972) 337; M. B. Lewis, Phys. Rev., in press 3) A. Bohr and B. R. Mottelson, in Proc. of neutron capture gamma-ray spectroscopy, Studsvik (IAEA, Vienna, 1969) 4) Y. Torizuka, in Proc. of the Int. Conf. on photonuclear reactions and applications, Asilomar, 1973 5) J. D. Walecka, Phys. Rev. 126 (1962) 653 6) M. Danos, Nucl. Phys. 5 (1958) 23; A. Bohr, J. Damgaard and B. R. Mottelson, Nuclear structure, ed. A. Hossain (North-Holland, Amsterdam, 1967); J. M. Eisenberg and W. Greiner, Nuclear models vol. 1 (North-Holland, Amsterdam, 1970) 7) I. Hamamoto, in Proc. Conf. on nuclear structure studies using electron scattering and photoreaction, Tohoku University, Sendai, Japan, 1972