Self-consistent effective interactions in nuclei

Nuclear Physics ASOI (1989) 205-241 North-Holland, Amsterdam

S E L F - C O N S I S T E N T EFFECTIVE INTERACTIONS IN NUCLEI (I). Doubly-stretched multipole interactions in deformed nuclei Hideo SAKAMOTO FaculO, of Engineering, G(fu University, Yanagido, Gifu 501-11, Japan Teruo KISHIMOTO Institute.for Physics, University of Tsukuba, Ibaraki 305, Japan Received 28 December 1988 (Revised 18 April 1989)

Abstract: The self-consistent effective interactions in nuclei are studied in detail by putting strong emphasis on the concept of nuclear saturation and the self-consistency between the shape of an average potential and that of a density distribution (nuclear self-consistency). The conventional multipole interaction model is improved so as to satisfy the nuclear self-consistency rigorously even if the system is deformed. The resulting doubly-stretched multipole interaction model is then examined by applying to low-lying collective states and high-frequency giant resonances in deformed nuclei.

1. Introduction The s t u d y o f collective m o t i o n has b e e n a n d still is one o f the most exciting subjects in t h e o r e t i c a l as well as e x p e r i m e n t a l n u c l e a r physics. Since Bohr a n d M o t t e l s o n p r o p o s e d a p i o n e e r i n g work o f the unified m o d e l 1), there have been m a n y t h e o r e t i c a l w o r k s on the m i c r o s c o p i c d e s c r i p t i o n o f n u c l e a r collective motion. In p a r t i c u l a r , realistic n u m e r i c a l a p p l i c a t i o n s have b e e n p e r f o r m e d extensively by the use o f effective m u l t i p o l e - m u l t i p o l e interactions. A m o n g them, the q u a d r u p o l e q u a d r u p o l e i n t e r a c t i o n has f o u n d m a n y a p p l i c a t i o n s 2 7); e.g. to s p h e r i c a l v i b r a t i o n a l nuclei in r a n d o m - p h a s e a p p r o x i m a t i o n ( R P A ) , to d e f o r m e d r o t a t i o n a l nuclei in c o n s t r a i n e d H a r t r e e - F o c k m e t h o d s a n d in a d i a b a t i c t i m e - d e p e n d e n t H a r t r e e - F o c k m e t h o d s , a n d even to t r a n s i t i o n a l nuclei in b o s o n e x p a n s i o n theories. The origin o f such effective i n t e r a c t i o n s can be a t t r i b u t e d to the core p o l a r i z a t i o n 8,~). The o b s e r v a t i o n o f giant q u a d r u p o l e r e s o n a n c e s 10 is), which are n o t h i n g but the excitation o f the core p o l a r i z a t i o n , s t i m u l a t e d us a n d gave us rare m o t i v a t i o n to r e e x a m i n e a n d i m p r o v e the i n d u c e d effective interactions. O u r study o f effective i n t e r a c t i o n s d e v e l o p e d a n d c o n t i n u e d since then, a n d has n o w b r o u g h t out m a n y i m p o r t a n t c o n c l u s i o n s , s o m e o f w h i c h have b e e n r e p o r t e d in short d e s c r i p t i o n s elsewhere 16-1s). It is o u r p u r p o s e in this p a p e r , first o f all, to r e p o r t o u r s y s t e m a t i c s t u d y o f the effective i n t e r a c t i o n s in d e f o r m e d nuclei in full length. In o r d e r to clarify the b a s i c a s s u m p t i o n o f o u r t h e o r y , which will be extensively referred to in o u r f o r t h c o m i n g 0375-9474/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) Seplember 1989

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H. Sakamoto, 72 Kishimoto / Se(l-consistent ejJective interactions (I)

papers, we will emphasize some characteristic features of a nuclear system in the following. A nucleus can be viewed as a close packing system of nucleons bound together under the influence of strong interactions of nuclear forces ~). A nucleon inside such a system receives strong influences from the surrounding nuclear medium. Therefore, even a basic property of a nucleon will be completely modified or renormalized. In particular, renormalized effective charges can best be explained by the polarization of the system through the coupling between a particle motion and the collective vibration of polarization. Effective-charge phenomena should certainly affect also the interactions between nucleons in the nucleus, which may be quite different from the bare nucleon-nucleon interactions in free space. Such is the case, and the core polarization indeed is shown to be responsible for the appearance of the effective multipole-multipole interaction ~.v). Because of such circumstances, the success of the fundamental approach to the nuclear many-body theory of collective motion starting with bare nucleon-nucleon interactions has been very much limited. On the other hand, the study of effective interactions arising from the core polarization has become quite important, more than ever ~.v.~6:2). Another, yet quite important, feature of the nuclear many-body system finds its origin in the fact that the nucleonic density inside the nucleus is almost constant for most nuclei, and so is the binding energy per nucleon. Therefore, it is a spatially and energetically saturated system with a relatively sharp boundary. Because of the constant density with a sharp boundary, the important degree of freedom which characterizes the system can be chosen as a shape deformation. Due to short-range attractive forces, it can be concluded that the deformation of the density distribution and that of the potential are the same. Therefore, when the system undergoes collective excitations with a change in the density distribution, it must be accompanied by the same change in the potential. Then the nucleons moving in the potential will have to readjust their orbits. Such readjustments, of course, result in the change in the nucleonic density distribution. Since a nucleus is uniquely a self-sustained system, a collective motion itself is made out of nucleon degrees of freedom, and therefore the change in the density thus produced must be the very density change caused by the collective excitation to start with. The self-consistency between particle and collective aspects here is indeed the basic organizing agent responsible for inducing the effective interactions. With all the above picture of a nuclear system in mind, one may assume that the nuclear self-consistency is satisfied with quite good accuracy. Namely, the shape of a potential and that of a density must be the same even at the time when a nucleus, satisfying the saturation condition, undergoes collective excitations. We can utilize this remarkable property as an important guiding principle to derive the effective interactions which are reliable for describing the fundamental modes of motion ~ ~). In sect. 2 a rigorous application of the nuclear self-consistency, which constitutes the central theme in our present work, is performed specifically in a deformed

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

207

system to improve the conventional multipole interaction model, resulting in the so-called "doubly-stretched multipole interaction model". At the same time, the self-consistent values of the interaction strengths are determined. The conditions for separation of multipole modes from the translational modes and the existence of the zero-energy RPA modes are also discussed. In sect. 3, RPA analyses of low-lying collective states and high-frequency giant resonances are performed for some lower-multipole modes, where simple and powerful methods to evaluate ground-state matrix elements and partial sum rules for transition intensities in the harmonic-oscillator model developed in the appendices are extensively used. It is shown that the doubly-stretched multipole interaction model indeed represents a great improvement over the conventional multipole interaction model. The results are then summarized in sect. 4. 2. Nuclear self-consistency condition and effective multipole-multipole interactions in deformed nuclei The multipole-multipole interaction model has been extensively and successfully applied not only to low-lying collective multipole vibrational states but also to giant multipole resonances in spherical nuclei. However, it was found that the same model failed to reproduce the prominent features of the observed data in deformed nuclei J6-ts) and the main reason of the failure was traced back to the violation of the nuclear self-consistency condition. Based on such realization, the multipolemultipole interaction model will be improved significantly by imposing the selfconsistency condition more rigorously in a deformed system. 2.1. DOUBLY-STRETCHED QUADRUPOLE INTERACTION As an example, let us consider the quadrupole case making use of a deformed harmonic-oscillator potential model. The total energy is written as

E~o,=E(~l~-~ P-

_~_ 1 i At{

V

"~__

V

~,,,,~o~x-~-o~;,y

2

+o~zZ)l~)p,~,

(2.1)

,

(2.2)

with p,~ = 8~ ~> ~'F

w h e r e e,~ is the s i n g l e - p a r t i c l e e n e r g y o f a state

.,, z>o

w h i l e ev is the

Fermi energy. Using abbreviated notations, the total energy can be expressed as Etot

=

(2.3)

h(O)x.a~x-t-~Oy.a~y-l-Wz2z),

with Zx = • X~p~,

etc.

(2.4)

c~

=

+~)~,

etc.

(2.5)

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H. Sakamoto, T. Kishimoto / Serf:consistent effective interactions (I)

Let R~, R,. and R: be the extent of the potential in the x-, y- and z-direction, respectively, then a ratio representing the potential deformation is R2,: R~.: R2: = 1/w2, : 1/,o~. : 1/o92:,

(2.6)

while from the average extent o f the density distribution in the x-, y- and z-direction, a ratio representing the density deformation is given by a(x2) : a ( y 2 ) : a ( z 2) = ,,~ / w, : Xv/o) v : ,,~:/to:,

(2.7)

where A is the total n u m b e r o f particles in the system. N o w we assume that the shape of the potential and that of the density distribution are the same (nuclear self-consistency), namely the two ratios must be the same: R2,.: R~,: R2: = A(x2) • A ( y 2 ) : A ( z ~ ) .

(2.8)

From eqs. (2.6)-(2.8) we have a self-consistency relation as w~

= to, S,,.- w:X: = ~ (X~2;,.~) I/3 ,

(2.9)

where o3 is defined by = (toxto.,,to:)I/s .

(2.10)

It is worthwhile to notice that eq. (2.9) can also be derived by minimizing the total energy u n d e r the volume conservation condition w#o,,w: = o33 = const.,

(2.11)

the meaning of which will be discussed in the next subsection. Under this condition, the different oscillator frequencies which describe the shape o f the potential can be obtained as co~ =d~(~S,,.~Y:)~/;/S,,,

etc.,

(2.12)

3 h d ) ( X , X , Z : ) '/~ .

(2.13)

and the total energy of the system becomes Eo

=

Etot]

...... =

N o w we shall employ the Landau theory of Fermi liquid 23 25), in which the first and the second functional derivative of a total energy function with respect to occupation probabilities p can be identified as the Hartree single-particle energy and the matrix elements of an effective t w o - b o d y interaction, respectively. In the present case, the Hartree single-particle energy becomes ~,~=

6Eo =fi(w~X.+w,,Y,~+w:Z,~) 6p. = hco, (n~ +i),~ + hwv(nv +~),, + ho~(n: +~),~,

(2.14)

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions ( I)

209

while the matrix element of the effective two-body interaction is given by

(~#1v~2~1~/3)=_ ~ 2 E o _ 6p~,6pt3

3 t O ( Z x ~ , y Z z ) 1/3

× [(toxx~ + O~yY~ + o~z~)(,oxx~ +,oy Y~ + to~z~)

- 3(,o~xox,~ + to2yyo y~ + to2zzoz~)] v 4 M-tO O

-- 3/~o~(~x~y.,V )1/3 [ ( X ~ - F Y ~ + Z ] ) ( X ~ + - 3 ( X ~ X ~ + Y'~ Y~ +

Y~+Z'~)

Z]Z~)],

(2.15)

where the doubly-stretched coordinates and corresponding matrix elements are introduced and defined by -=

x,

etc.

(I 2 h~o~ X~-=(c~l(x")2la)= ,tOo,,W~ (c~lx2la)=~w__~w~X~,

(2.16) etc.

(2.17)

In this paper, we will parametrize the shape of the nuclear potential as tox=too 1-k~---~)

f l c o s ( 7 - 2 ~ -) =too[l+le2o-(])'/2e221,

toy=too 1 \~--g~/ ~ c o s ( ~ , - ~ ) toz =too 1 - \ ~ - ~ /

=too[1+I~2o+(~)'%22],

/3 cos y = tooE1-2e2o],

(2.18a)

(2.18b) (2.18c)

with e2o=~ ~ ) 5~ E22 =32 (\47r]

flcosy=ecosy,

'/2 ~/3 sin y = ~ es i n Y,

(2.19a) (2.19b)

where {/3, y}, {e2o, e22} and {e, y} are deformation parameters. Due to the saturation condition of eq. (2.1 1), the frequency too must depend on the deformation parameters as t o o = tOo(e, ")/) = a~(1 + ~ e 2 - + - 8 e 3 c o s 3 y + O(e4)).

(2.20)

Now let us introduce two types of multipole operators QA~, = Qx,(r) = rAYA~(O¢),

Q~, = QA~,(r") = (r")~YA~(O"qS"),

(2.21)

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H. Sakamoto, E Kishimoto / Se(lLconsistent effective interactions (I)

and define a scalar coupling as (QA" Q~,)=Z ( - ) ' Q A . Q A

(2.22)

~,

~t

where QA, is defined in terms of the ordinary coordinates while Q~, is defined in terms of the double-stretched coordinates. Then we can find (a/31(Q~

•

5

ii

Q2 ) afi) = ~

it

tl

[ 3( X , X t3 + YI'~ Y g + Z','~Z; )

- ( X , +1[

Y , + Z , ) (I1X ~ + ff

~f

II

Y~+Z~)] It

.

(2.23)

From eqs. (2.15) and (2.23), the two-body matrix element becomes (~fi[ V(E'la/3 ) = -x'~'(afil(O~ • O ~ ) l ~ > ,

(2.24)

where XI21=~

~eu

zX2

,

A((r,,)2) = ( ( r , , ) 2 ) i t

~u

X2 -

4~r

Mo) o

5 A((r")2) '

3hffJ

= T 7 - 5 (2.'~2;~.2: ) ~..,3. mw o "

(2.25) (2.26)

Thus, an improved form of the effective quadrupole interaction in deformed nuclei is dervied as V'2'(1, 2) = - x ' 2 ) ( Q ' ( l )

• Q~(2)),

(2.27)

which is a quadrupole-quadrupole interaction defined in terms of the doublystretched coordinates ~' 22). It must be noticed that the interaction strength X ~2~ is exactly twice the value of X~~", i.e. the self-consistent interaction strength for the (Q" • Q ' ) force. As will be explained in subsect. 3.3, the strength X (-~i must be used instead of X~~" when one renormalizes the core polarization effect of A N = 2 excitations. This is just a realization of the quadrupole effective-charge phenomena, as discussed by Mottelson, g) which claims that the mass quadrupole polarizability is one. It should be also emphasized here that the Hartree self-consistency can be achieved for any strength of the interaction, while the nuclear self-consistency is achieved only at the self-consistent strength. In this sense, the concept of the nuclear selfconsistency is much more stringent than the Hartree self-consistency. Finally in this subsection, we must mention that the improved interaction indeed is responsible for the fluctuation about the d e f o r m e d equilibrium because of the relation A ( Q ~ , ) = 0 for any #, owing to the self-consistency condition (2.9). 2.2. FORMULATION 1N THOMAS-FERMI THEORY The T h o m a s - F e r m i theory is k n o w n to be a useful procedure which gives global average quantities ~6,2v). Recently, it played an important role in separating a smooth

H. Sakamoto, T. Kishimoto / Self-consisten t effective interactions (I)

211

part and a quantal fluctuating part (shell effect) of the total nuclear binding energy 2s). The effective interaction which we derive should not depend sensitively on the detailed shell fillings, and rather should be expressed only in terms of global quantities. Using this semi-classical approximation, we can also investigate the relation between a mean field and a density distribution. It is our purpose in this subsection to examine the nuclear saturation condition and the self-consistency relation between a nuclear density and an average potential in the T h o m a s - F e r m i theory to prepare for general multipole modes in the next subsection. In the theory, a state density and a single-particle energy density in a classical phase space are defined by

n(r,p)={O0/h3

e(r,P) <~ev e(r, p) > ev,

(2.29)

2

e(r, p) = P-P--+ V(r)

(2.30)

2M

where .Q represents the number of intrinsic degrees of freedom (spin, isospin, etc.) and the Fermi energy is

2

ev=e(r, pv(r))= Pv + V(r). 2M Now a density in r-space,

p(r) =

I

n(r, p)

p(r),

can be related to

V(r)

d3p = ~ ~ 7"r{pF(r)}3 =

k{ev-

(2.31) as V ( r ) } 3/2 ,

(2.32)

with 04 k =~-3 ~ ¢r(2M) 3/2.

(2.33)

The Fermi energy is determined through a particle-number equation

A= f p(r) d3r=k f {eF- V(r)}3/2d3r,

(2.34)

where A is the total number of the particles in the system under consideration. On the other hand, the total energy is given by

E = f e(r,p)n(r,p)

d3r dap.

(2.35)

As an example, let us examine the relation between the nuclear saturation property and the volume conservation condition using a deformed harmonic-oscillator potential model. In this model, the potential V(r) is given as V(r)=~"~

2 2 +oovy 2 2 ) =~lVlOJo~r .2 2 + ('OzZ l~Ar 2z .~2 ~lvl~toxx ~ = V(r") ,

(2.36)

H. S a k a m o t o , T. K i s h i m o t o / Sell:consistent effective interactions ( I )

212

with (/,)2 = (x,,)2 + (3/,)2+ (z,,)2.

(2.37)

Therefore, the density is written as p ( r ) = k{e~ - V ( r " ) } 3/2=- p ( r " ) .

(2.38)

It must be noticed that when the potential V is spherical in doubly-stretched coordinates, then the density p also becomes spherical within the T h o m a s - F e r m i approximation. In this case, eq. (2.34) becomes A

47rw3

o(r")(r")Zdr"=i~

h(w~wv~o_)l/3j.

(2.39)

O) \-0) vO)z

Thus the Fermi energy is given by ev =

(2.40)

h(~o,~o#oz) '/3 .

At the same time, the total energy of the system is calculated as 4,3

I l/~ g = ~fi(wxw,.w:) -

3

(2.41)

=__Aev,

which can be identified as a volume-energy term since it is proportional to the total n u m b e r A. The higher-order terms in the T h o m a s - F e r m i theory would then be associated with a surface term (-A2/3), a curvature term ( - A ~'/3) and so on. From eqs. (2.40) and (2.41), we can observe that constancy of w:,w,.w: in the presence of a change in the shape guarantees constancy of a volume energy or constancy o f a Fermi energy. For this reason, constancy of ~o,w#o:, which is nothing but eq. (2.11) and is well k n o w n as the condition of the volume conservation within the equipotential surface introduced by Nilsson 29,3o), will be equally stated as "nuclear saturation c o n d i t i o n " in the present paper. In the T h o m a s - F e r m i theory, a change in a nuclear density and a change in an average potential due to collective excitations can be easily related. Our starting point is V(r)= V~(r)+6V(r)

,

O _~_

~v=~l:

~F,

(2.42)

where Vo(r) is an original nuclear potential which is assumed to be spherically symmetric and e~ is a Fermi energy corresponding to the case with 8 V = 0. In eq. (2.42) and in the remaining part of this subsection, r can be either the ordinary coordinate or the doubly-stretched coordinate. From eq. (2.32), the deviation in a nuclear density a c c o m p a n i e d by 6 V and 8e~- can be obtained as

p(r) =

k(

e ~o - vo) 3/~ - ~3k ( e , o-

V,,)'"2(~V-,%~-) + ~~k ( e Fo -

~- po(r) + 8 ( ' i p ( r ) + 6~21p(r) +" " " •

Vo)

b'2(SV--68F)2+

"" "

(2.43)

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

213

Since Vo is spherically symmetric in terms o f r, the resulting Po is also spherical. Therefore, we obtain

po(r) = k(e ° - Vo(r)) 3/2 , dpo

dr

3

o

(2.44)

d Vo

= -sk(ev-

Vo) ' / 2 dr '

(2.45)

d2po 3~, o ( d V o ) 2 3L, 0 d2Vo d r 2 = a K t e F - - Vo) 1/2 ~k d r / --SKkeF-- V°)I/2 dr 2 "

(2.46)

Thus, 8(~)p and 8(2/p are obtained as

8 ( , , p ( r ) = ~ r [(6 V - 8ev)/~-r] dVo] ,

(2.47)

8 ( 2 ) p ( r ) = { ~ d 2 p 2~ - - 2\ d° -r{2d/ 2d Vr ,°]/ d V ° ' r~d dr JP ° ~ [ ( 3 V - ( 3 e F ) / 9 ]

.

(2.48)

Notice that in order to impose the nuclear saturation condition, we should put 8ev= 0 in the above equations. 2.3. DOUBLY-STRETCHED MULTIPOLE INTERACTION NOW we will show, in general, that the effective interaction in d e f o r m e d nuclei satisfying the self-consistency condition rigorously can be written as a (Q~ • Q~) interaction, i.e. a multipole-multipole interaction defined in doubly-stretched coordinates ~7.18.22), which represents a great improvement over a c o m m o n l y used (QA " QA) interaction model. Let us start with a hamiltonian

H=

+½Mw~r'~'- -½ E EXzK 2 • .¢AK(i i=l

i

A~I

K

where

i=l

K(j),

)

o

QaK = Qf, K - ~

(2.49)

j=l

Q~K(i) -= Q~(K -(O~(K)o.

(2.50)

i=1

The Hartree field is then

V ( r ) ~ V(r")

J..

2. ,,-~_y,,-~+z,,-~)_

v A~I

=-

,~

,q.*

K

Vo(r") + ,sv(r"),

(2.51)

where collective coordinates are introduced as

° ) , Q;~K(t)

o~;,K = A(O'~K) =

for A ~> 1.

(2.52)

i=1

By using the T h o m a s - F e r m i theory, we can calculate the deviation in a nuclear density a c c o m p a n i e d by 8V with the result summarized in subsect. 2.2. For the

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H. Sakamoto, 12. Kishimoto / SelJ'-consistent efjective interactions (1)

present case of the d e f o r m e d harmonic oscillator potential Vo(r"), we obtain

e~ = e~o + 6eF,

(2.53)

p( r ) =- p( r") = po( r") + 6'1~p( r ") + 6(2~p( r ") + . . • ,

(2.54)

po(r,, ) = k(e(¢ -~_M~oor 1 2 ,2 ) 3/2 ,

(2.55)

1 dpo • 8 V(r") - 8eF} 8~I)P(r") -- Mw2r " dr . . . .

(2.56)

1

(d2po

1 d po,,) {6V(r,,)_Se~}2

8~2)P(r")- 2M2~oar '' \ d r "~

r" dr /

(2.57)

where e °- is a Fermi energy for the case with 8V = O. We will first impose the nucleon n u m b e r conservation condition. The first-order equation in { 8 V - 8 e F } is

fSl')p(r")d3r-wi~'

O.)xOOvO)~

fa("p(r)d3r=O,

(2.58)

and by substituting eq. (2.56), we have 8eF =

i r"i d.o. dr" 8V(r") d~r ''

r"

dr"

d3r ''.

(2.59)

Therefore, it is easily verified that our 8V(r") defined in eq. (2.51) automatically satisfies the constancy of a Fermi energy, i.e. 6eF = 0, during the collective change of a nuclear shape, which is nothing but a nuclear saturation condition. Notice here that if 8 V =f(r")YAK (r") is taken in eq. (2.59), then 8e~ = 0 for any h ~> 1 due to ~ YAKt"~"'J d~" =O, w h i l e i f 8 V = f ( r ) Y A K ( ~ ) , t h e n t h e c o n d i t i o n S ~ v = O is not always guaranteed. Notice also that the condition for a center-of-mass, i.e. rSp(r) d3r = 0 , is automatically satisfied for 6 V = f ( r " ) Y A K ( ¢ " ) with A/>2, in particular with A = 3. Thus if 8V is written in doubly-stretched coordinates as n Z X~K O~AKQa~,

8V=-5" h:,l

(2.60)

K

then the conditions 6el = 0 ,

I rSp(r) d3r = O,

for 8VAK

(2.61) with

A >12,

(2.62)

are satisfied. The last relation indicates already that translational spurious modes are exactly removed if po(r") is indeed spherically symmetric. Decoupling of the A = 1 and A = 3 modes will be discussed in sect. 3.

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions ( I)

215

N o w let us determine the self-consistent strength t"AK se~rxor each K - c o m p o n e n t for d e f o r m e d nuclei. From eqs. (2.52) and (2.54), we have t~'hK ~

A(0~(K) =

f O~,Kp(r)d3r

= f Q~(K6B d3r,

f {Q~K-(Q'~K)o}{po + 6p)d3r

=

for A ~> 1,

(2.63)

where we used the saturation condition ~ 6p d 3 r = 0 and the definition o f ( )o:

i=1

0

From eqs. (2.56) and (2.63) we obtain a first-order self-consistency relation as

c~aK =

f

Q~(KS(1)p

3;

Wo

d3r_

Q~K6(I)p(r,, ) d3r, ,

O)xO)vO)z

1 w3° fj Q~(K mw~r"

dpo(r")

dr"

(,O~0) v~Oz

6V(r")

d3r '',

for A ~> 1 . (2.65)

Using eq. (2.60), this relation becomes 2A + 1

X~K

A((r,,)2A

2)0,

for A/> 1

(2.66)

with

a((r,,)2A_2)o=

f

(r,,)2~ 2p° d3 r -

3I

Wo

(r")2A-2po(r") d3r " .

(2.67)

O)x O)y(-Oz

Therefore, the interaction strengths t'~K are determined as seJ¢__ 4~" XAK=XAK=23.+ 1

Mw~

A((r,,)2 a 2)o,

forAy>l,

(2.68)

which are K - i n d e p e n d e n t . The reason for the K - i n d e p e n d e n c e can be traced back to the sphericity o f Po in the T h o m a s - F e r m i approximation. The slightly improved results will be rederived in the next subsection. We finally conclude that the t w o - b o d y multipole-multipole interaction in a deformed system should be expressed in terms o f doubly-stretched coordinates to insure the conditions o f saturation, self-consistency and separation of multipole modes from the translational modes.

2.4. F U R T H E R

IMPROVEMENT

In the previous subsection, we derived a self-consistent strength X~K sell for a (Q~( • Q~() interaction in d e f o r m e d nuclei, which turned out to be K - i n d e p e n d e n t

216

H. Sakamoto, T. Kishimoto / Self consistent effective interactions (I)

due to our use of the T h o m a s - F e r m i approximation. Here we will determine XAK ~" by the m e t h o d of the displacement and c o m p a r e the two results. Let us start with the Hartree field of eq. (2.51) V(r) ~ V(r') =l

2 ~Mwo(X

.2

+ y " - •+

Z n2)

g~tr* - ~ ~XA~C~KV~K

A~I K

-= Vo(r")+ E 6VAK(r").

(2.69)

AK

The displacement r " ~ r " + 6r" to produce 6VxK (r") can be determined as Vo(r")-,

V(/')

= 2Mw0{(x ' ~ . + . a . x .

) ~ + ( y " + a y " ) ~ + ( z " + a z " ) ~}

=- Vo(r") + 6VAK (r"),

6VAK (r") = M w o ( x " 6 x t ' + y " 6 y " + z " 6 z

(2.70) ") = --XAKaAK~AK'"*.

(2.71)

By using the relation ( r . ~7)QAK = AQAK, we can d e c o m p o s e eq. (2.71) as 8r"=

X~KaaK 1 ~,,tq,,* Mw~ h -- v x K •

(2.72)

Notice here that this decomposition is not unique but rather we assume eq. (2.72) as one o f the possible and plausible solutions. It is worthwhile to note that the collective velocity field associated with this displacement 8r" becomes v(r)=-

XAK6AK 1 ~"ta"* Mw~ A - - "eAK ,

(2.73)

and in a spherical limit, it satisfies V x v = 0 and (~ • v) -- 0. Therefore the velocity field of eq. (2.73) is a natural extension of the irrotational and incompressible velocity field of Bohr's liquid-drop model ~) to a d e f o r m e d system. In this case, the change in a density caused by the displacement is given by 8 p a K ( r , , ) = ( 6 r , , . V,,)po(r,,)_

XAKaaK 1

. . . . )~] , 7 ~i~, 1 [OQ~K .m~-,, Potr

(2.74)

where the relation A"Q~K = 0 for the laplacian operator is used. It should be noted here that f 8pAK d 3 r = 0 for any A and K, and ~ rSp~K d 3 r = 0 for A = 3 due to the q u a d r u p o l e self-consistency relation o f eq. (2.9) and for any even 3. trivially. The self-consistency condition corresponding to eq. (2.63) is now written as aaK =

I

Q~K6paK(r") d3r

/~AKOIAK

'I

wi~

Q~K6paK(r") d3r ''

W,:O9vtOAIA'[I'~I

= 23.Mo)o " ~

tt

~1#~,"

l\

~te~,K'.ZAK~/O,

(2.75)

where c

It rig< __ [ a nl ~tt ~tt~ x A(A t¢ (QzKQxK))o= j Pokz r ttx1~ t~XKtdaK)d 3r.

(2.76)

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions ( l )

217

By using the following relations QaKQ,K = ~ 1 + ( - ) ' (2l+1"] '/2 ,=o

2

\-7~

/

(aOlOlaO)(aKlOlaK)r=~Y'°(~')'

A(rkyt,,)={k(k + l ) _ l ( l + l)}r k 2y~,,,

(2.77) (2.78)

we obtain 4¢r 2h(2h + 1)

n i-ittg< ~

A " ( Q A K '~ aK J =

2a-2 l + ( _ ) t _(I) E

0

2

gaK

( r2a 2191)"

(2.79)

,

where Pt is the Legendre polynomial of rank 1 and 6aK"(t)is given by g(t) AK--

2h(2h + 1 ) - / ( 1 + 1)

2a (2a + 1)

(2t+ l ) ( a O t O l A O ) ( a m O l A K ) .

(2.80)

Thus the strength gaK is determined as

~o,f_4~Mo'o [2~-2 l+(-)tg XaK=Xa~= 2 A + l Lt=o

2

~'~A((r 2a 2P,)")o

]-'

.

(2.81)

It is quite interesting to observe that the strength XAK here is generally Kdependent. Also notice that if p0(r") is spherically symmetric with respect to the doubly-stretched coordinates as in the T h o m a s - F e r m i approximation, then eq. (2.81) exactly coincides with eq. (2.68) because of the relation A((r 2* 2Pl)")0~ 6,oa((r")2*-2)o,

when po(r") ~ po(r").

(2.82)

Furthermore, the two methods give identical results for 3, = 2 because of the nuclear self-consistency condition of eq. (2.9), or A((rZPY)o = 0. The interesting difference in fact appears only for A/> 3, and the h = 3 case will be discussed in detail in subsect. 3.3, where the two results will be compared. It should be noted that 6p = ~AK 8p~K rather than 8PAK should appear, in principle, in eq. (2.75), which should result in the eigen-equation for XxK. It can be easily shown, however, that the lower multipole modes up to A = 3 do not couple each other. Since all the applications in sect. 3 are limited to such cases, we will not go into further details in the present work.

2.5. THE C O N D I T I O N FOR SELF-CONSISTENCY A N D ZERO-ENERGY MODES

The self-consistency condition of eq. (2.63) in principle should mean that the system has no restoring forces against multipole shape oscillations. Therefore it may be quite instructive to examine the condition in detail in relation to the occurrence of zero-energy modes.

218

H. Sakamoto, T Kishimoto / Se!f-consistent effective interactions (1)

If the field coupling 8VA~( of eq. (2.71) is weak, the perturbed nuclear state is given by (IQhKO t )" .

Io)) = Io)+x

(2.83)

where i labels the excited nuclear states with energies G , and 10) represents the unperturbed nuclear ground state. The quantum analogue of the self-consistency condition of eq. (2.63) can then be written as

1

= x~K,~,~ E

E,- E,~,{l(ilo~l°)l~-

+l(ilo'~l°)~} "

(2.84)

Therefore an interaction strength would be obtained microscopically, which we may call XAK, mi~ as mic (XAK)

1

= R(,o = 0 ) ,

(2.85)

with R(co)

E i -

E 0

--.wY =T (f,Eo)--,o

~ {](il Q~,(Io) 2 + [(il Q~*

~

Io)12/

(2.86)

Since R(w) is nothing but the RPA response function for the (Q~ Q~) interaction, we have to conclude that the quantum analogue of the self-consistency condition always guarantees that there must exist a zero-energy RPA mode. Let us now investigate the relation between XAK ~" obtained semi-classically and mic XAK obtained microscopically in the deformed harmonic-oscillator-potential model: Ho = T + Vo(r"),

Ho[i) = E~li).

(2.87)

Let us introduce one-body operators S~K

1

_

g).*

2Mw~h [d", YAK],

UAK=[T, SAK]=

(2.88)

- - ~ - A , SAK -

(2.89)

) n ;~: [ H o - T, SA~ ] = [ Vo(r"), SA~(] = g~'AK

(2.90)

Then we can easily verify that

(i]Q'~* [0) = ( E , - Eo)(i[SzK [0)- (i[ UAK[0),

(2.91 a)

(0[ Q;* [i) = - ( E ~ - Eo)(O[SAK ]i)- (0[ U,~K[i).

(2.9 lb)

From eqs. (2.85), (2.86) and (2.91), we obtain mic (XA~)

I

n =(01[QA~,SA~][0)--IAK

(2.92)

14. Sakamoto, T. Kishimoto / Self-consistent effective interactions (1)

219

with 1

IAK =~ E;- Eo{<°IUA':Ii)tiIQ'f'KI°)÷(°[Q'J'Kli)(itU;'KI°)}"

(2.93)

On the other hand, from eq. (2.75), the self-consistent strength XAK self of eq. (2.81) can be rewritten as self'~

XAK)

1

2

1

=(2MwoA)

ct

¢t

rt~<

l!

a ( A ( Q A K Q A K ) ) o = A ( [ Q A K , SAK])O,

(2.94)

where we have used the identity A , , ,t~r vA, K ' cv,* . ' A , , ,~ = - [ Q b , , E a " ,

QAK]]. "*

(2.95)

Thus the relation between XAK ~e~f and XAK m~c can be obtained as mic

1

/

s e l f x -- 1

(XAK) =tXAK)

--IAK-

(2.96)

Consequently, if IAK 0 holds then the (Q~( • QA) interaction with the strength XAKself gives a zero-energy RPA mode. As will be shown in several examples in the next section, however, XAK mic does not always coincide with XAK self due to the shell structure in the single-particle excitation spectra, but can deviate from XAK ~ f by a value with a discrete jump. This is the main reason why one has to employ a semi-classical method to obtain the interaction strengths, which do not sensitively depend on the detailed shell fillings. =

3. S o m e applications of ( Q ~ • Q~)-interaction model

Here we will examine the results of the (Q~ • Q:)-interaction model in RPA calculations I~). To make a comparison between the models with different interactions, we introduce three types of multipole operators QAK, Q~,K and Q~K which are defined in terms of original, stretched and doubly-stretched coordinates, respectively. The definitions of these coordinates are summarized in eq. (3.6) and more details may be found in appendix B. Consider the hamiltonian f

H=

P

--

1 m ar

1~-2~v'ooot

2z

tt~,2/

I

r ) ~ i - 2 ~XAK 2 0 * K ( i ) O a K ( j )

i=1

AK

(3.1)

i,j

where OAK can be either QAK, Q~K or Q~K- Then the RPA dispersion equation for each AK mode to determine the eigenvalue ~o for this hamiltonian becomes (2XAK)-' = 7£ ( E , ;

Eo)l12

(E_Eo)2_w

2

,

(3.2)

where IO) and [i) are the ground state and the particle-hole excited state for the unperturbed hamiltonian, (i.e. deformed harmonic-oscillator wave functions), while

H. Sakamoto, T Kishimoto / Se!l'-cons'istent qffective interaction,s (I)

220

Eo and Ei are corresponding unperturbed energies. In this section, h ~ 1 is employed to simplify the expressions. Let us introduce the following notation •(AE;AK)=--

2

(3.3)

](i]OAK]O)I 2,

.Jr:)

where the r.h.s, is a partial sum of particle-hole strengths with a definite excitation energy AE = E~ - Eo. By using this notation, eq, (3.2) can be written as (2XAK) 1= 2 A E 2 ( A E ; A K ) a~

(AE) 2

(3.4)

co2

To obtain an analytic expression of the r.h.s, o f eq. (3.4) for each AK mode, we have developed simple and powerful methods to evaluate ground-state expectation values and partial sum rules for transition intensities in the harmonic-oscillator model (see appendices). A sum-rule technique was also developed in ref. ,9), but unfortunately it can be used only for the case when there is a single collective state that exhausts the whole strength. The self-consistent interaction strength X ~ for a (Q~ • Q~) interaction is derived in subsect. 2.4, and we have from eq. (2.81) as 47rMwo

21+1

~lf -=A(I / 4~X~ \2t(21+1) 21 2 l + ( _ ) l V

--g~,a(tr

I =0

a"~tq" ~',* ) ~ ~','AK'¢A~) o (I) - - ~ J

21

(3.5)

2Pi)")o,

2

where A is a laplacian operator, P~ is a Legendre polynomial and g~K ~" is given in eq. (2.80). For the rest of this section, the system under consideration is assumed to be axially symmetric. The definition of the coordinates and the scaling factors for each case is summarized: (i) original;

x,

etc.,f~ = w ~ ~2,

(ii) stretched;

x, =

(O.),~/W0)I/2x,

(iii) doubly-stretched; x " = ( w ~ / w , ) x ,

etc., J~

wo 1 / 2 ,

etc.,f;=

CA) 1/'21

.[: = w _ I,'2 ' ,

j~-- wo

~ /Ooo,f:=o2:

I,'2

/2

(3.6a)

,

(3.6b)

/wo,

(3.6c)

where w~(=~o~ = w,) and oo. are given by putting y = 0 in eqs. (2.18) with eq. (2.19). N o w we are ready to solve eq. (3.4) for each AK mode.

3.1. D I P O L E

MODE

For the case of A = 1, the self-consistent interaction strength becomes ~;ei[1

~v~K

4"7T M¢~o 0

---~(~ 3 A(1)o

~,elf

(K-independent).

(3.7)

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

221

Then if we put X~K = ~X]e~r for both K = 1 and K = 0 cases and by using the dipole sum rules o f eqs. (B.28), eq. (3.4) can be solved as 2 O) 11 =

~'

2

2

co 2 - ~ W o w ± f ± ,

2 £010 =

2 OJz -

2

2

s¢COoCO~f~,

(3.8)

where the scaling factors f± and f : take different values corresponding to the three types of coordinates. By substituting the c o r r e s p o n d i n g values o f f ± and f~ given in eqs. (3.6), we finally obtain the eigenvalue WlK for each o f the three cases (1) O , K = Q l h : ,

~ ~ WlI=W±

-

-~,,,~,

~ = Wl0

o , ~~-

s¢O);,

(3.9a)

(2) 0 , , , = 0 ' 1 , , ',

,0:ll=W±--sqO0W~, 2

W~l o : o , ~2 _ sqOoW,,

(3.9b)

(3) O1K

O)~l = (1 - ~:)co~ ,

CO,o=Z(1 _ sc)w2~.

(3.9c)

= Q[K,"

It must be emphasized that for the self-consistent strength (~:--1), translational spurious modes, c o r r e s p o n d i n g to cola = 0 for K = 0, ±1, are exactly obtained only for the case o f OiK = Q~'K. This result can also be understood by calculating UAK and IAK defined by eqs. (2.89) and (2.93), respectively. In fact we obtain UtK = 0 , I~K = 0 and consequently XxK ~¢~r=XAK m~¢ for K = 0 , ±1, which ensure the existence of the zero-energy R P A modes.

3.2. ISOVECTOR DIPOLE MODE Since it is shown that a (Q~" Q~) interaction model represents an important i m p r o v e m e n t over a (QA" QA) interaction model for a shape oscillation mode, it may be expected that even for an isovector oscillational mode, the interactions in terms o f doubly-stretched coordinates are favourable. It is our purpose in this subsection to c o m p a r e two interaction models, namely a c o m m o n l y used (<-QA • r-QA) interaction model and our i m p r o v e d (rzQ'~ • r:Q~)-interaction model, where rz = - 1 for protons and rz = +1 for neutrons. Here, as an example, we will investigate the results o f the application o f a (r,Q'[ " r~Q~')-interqaction model in the R P A calculation. Consider the hamiltonian

H= ~ ,=,

p f +½Mco2r,2

eM

:I

y=, Tz( i ) T z ( j ) ]O*l K( i) O1K ( j ) .

j , - ~ Ey[xT~°+x'K Kq

(3.10)

The R P A dispersion equation for each K - m o d e for this hamiltonian is given by T=I)(Rp+Rn)__. 1 - - tlX 1 TK= 0 - - -rX~K

T=0 T=ln "-'~XIK X1K a p an , = 0 ,

(3.11)

with

Eo)ll 2 (Ei-Eo)2-w 2

2(E,R~=-R~(w)~

.T~ ~

r=porn.

(3.12)

Thus if we put R p = Rn as is the case for a nucleus with Z = N, eq. (3.11) splits into two i n d e p e n d e n t equations. One is eq. (3.2) for an isoscalar m o d e with the interaction

222

H. Sakamoto, T. Kishimoto / Se![:consistent qO'ective interactions ( I )

strength X[K ~u now denoted as X,K r~o , and the other for an isovector mode is given by (2X(K ') ' = 2 ( E , ;

E,,)[(iIOxKIOY

( E ~ - Eo) 2 - o ) 2

(3.13)

'

which has a similar structure as eq. (3.2) except that X~K ~ in eq. (3.13) takes a negative value because o f the fact that the interaction is repulsive for an isovector mode. Thus by putting X~K r:~ =-¢~X~ ~ ~H, the solution o f eq. (3.13) can be obtained from eqs. (3.9) just by replacing ( with - - E l , i.e. (1) O[K = Q[K,

(2)

O, ~ = 0'l ~,

1

wll=oSoj)~

lq 3 ( 1 + ( [

Wlo=thODR

1

-

-

/=;} ,

3(1+([)

2+~,

- OJ[j=o3Ol)R 1-~6(1+~1

E},

2+~[ m[o = ~O(;DR 1 (3) O,K = Q ' I ' K ,

(3.14a)

3.14b)

3(1 +~1

W,,=OSGDR(I÷~e), Wm = OS(n)R(l --~e),

(3.14C)

where ~SGUR--=~ ~[W0. Thus the theoretical estimation for a total energy splitting of an isovector giant dipole resonance ( I V - G D R ) is given by 1

a'O~;,,R/'5~;DR-- - -

1+([

e

-

2+(~ -

' 2(1 +~[)

~, ~ ,

for O,K = Q[K, Q',~, Qi'K,

(3.15)

respectively. To make a comparison with experimental data for isovector giant dipole resonances, we must determine the value of ~:~ in eq. (3.15). Since, from the experimental systematics ,~-[5), the average resonance energy is expressed by - c x p -~O~j~)R_80 A ,,,3 [ M e V ] ,

(3.16)

we take the value ~[ = 3, which gives ..... i gO C,[)R

= , / l + ~ j.w o ~ 2 w o ~ 8 2 A

l/3[MeV]

(3.17)

Therefore, if we put ~ = 3, eq. (3.15) becomes cal 6WGI)R/WGDR

I 5 = 4F,, glS', e,

for

O1K = QIK, Q'[K, O'[K ,

(3.18)

respectively. On the other hand, an experimental splitting of the I V - G D R in a d e f o r m e d nucleus satisfies a simple classical geometrical relation as ~O.)~xPR/O~GI)R__ I / R ~ - l / R _ °~ w ± 1/ R ,

~o,

w .-

e,

(3.19)

H. Sakamoto, T. Kishimoto / Self consistent effective interactions (I)

223

where R~ ( = R ~ = R v) and R. express the extent of the nuclear density in each direction, and Ro is the average value 3~-34). Thus it is clear that the total energy splitting o f the I V - G D R can be naturally explained only by the use of O~K = Q[K-

3.3. QUADRUPOLE MODE

For the case o f A = 2, eq. (3.5) becomes 4

9

1

~TrMw6) Te~f = g(2~ A (r"-)o+ g(2~ A ( (r2p2)")o •

(3.20)

X2K

Since A((r2p2)")o=O due to eq. (2.9), we obtain self 4 M(-°2 =_ ~elf ( K - i n d e p e n d e n t ) X2K =~7rA(r,,:)o Xe

(3.21)

By using the q u a d r u p o l e sum rules of eqs. (B.29), we can solve the R P A dispersion equation to obtain the eigenenergies. The results, corresponding to the case of X2K =X2sell" for all the K - c o m p o n e n t s , are summarized in table 1. Thus for the self-consistent strength, the rotational spurious m o d e characterized by w2~ = 0 is exactly obtained only for the case of O2K = Q~K. TABLE 1 The RPA eigenenergies for three types of quadrupole interactions Interaction type

(Q2" Q2)

( Q " Q~)

(Q~ " Q")

K-O

AN-2

x/2~oo{1-e+O(e2)}

.f2w,,{l

3e +O(e2)}

x/2too{l-~e+O(e2)}

K - 1

A N =0 AN-2

O(e ~/~) .,~wo{l-½e+O(e2)}

O(e 3/2) x/2 0)0{1 --//7 + O(E2)}

xf2¢o,){l--~,Eq-O(82)}

AN=2

x/2wo{l+e+O(e2)}

x/2wo{l+~e+O(e2)}

-,/2 too( 1 + ~e)

K =2

0 (exact)

In each case, the interaction strengths are fixed as X2K = X~c'r for all K-components.

From table 1, the theoretical estimation for the total energy splitting of a giant q u a d r u p o l e resonance ( G Q R ) is given by 60.)GQR/0~GQ

R =

2e, re, 4 2 re,

for O2K = Q2K, Q~K, Q~K ,

(3.22)

respectively, where O3oQR-----=x/2Wo is the centroid energy o f the G Q R . If we employ the c o m m o n l y used value of wo as tOo~ ~3 ~ 4 1 A

~/3[MeV],

(3.23)

we obtain O-°"~ )GQ R =

x/2w0 ~ 58A 1/3 [MeV]

(3.24)

224

H. Sakamoto, T. Kishimoto / Se!f-consistent eJfective interactions (I)

which is in good agreement with experimental systematics lO 1_~)expressed by -~P ~ I/3[MeV] -- 6 3 A

(3.25)

O) ( ; Q R

-cal

-exp

Based on such agreement between ¢.DGQR and O2GQR,we can compare the calculated splitting of eq. (3.22) with experimental data. The observed narrow energy splitting of the G Q R in deformed nuclei seems to be naturally explained oy . . . O. O. ) G Q R I = 3R~ 7 0-2) G Q , i.e. only for the case of 02K = Q'2'K, which is ~ of the value for 02~ - Q2K. For the case of 02K = Q~K, our result coincides with that obtained by Suzuki and Rowe ~9). Now let us investigate the electromagnetic properties of the GQR. For a transition operator MAx (=QAK, Q~K or Q2K), a transition probability from the RPA ground state 10)) to a RPA excited state Iw,,)) is given by

B(AK"

w.)-

I(<'°"IM~"Io))I= - l+SKO 1'~('°)12

1 + 6Ko

~

R(w) . . . . ,,'

(3.26)

with R(w)

Y

E, -

E,,

~ (1(/I 0 ~ IOY+ I12) ,

(3,27)

"7 ( E, - E o ) 2 - ,o ~

Ei - Eo

(E _Eo)2 o~{(o[o*~,,[i)(ilM~,~lo)+(OlM~Kli)(ipO*~,,Io)}. (3.28)

~(~)--~

It is convenient to introduce an abbreviated notation for a partial sum of particle-hole strength with a fixed energy AE - Ei - Eo as

5 ( ~ E ; O,, O~)= V .

(3.29)

i(B/:)

Then eqs. (3.27) and (3.28) are rewritten as AE R ( w ) = JE2 ( 3 E ) 2 _ w _ AE ,~ ( , o ) =

J~Z ( ~ E ) - " - ,o 2

{2(AE" O~,,, O A K ) + 2 ( A E ; O~,~, O~K)}

(3.3O)

{X(BE.' O*K ' MA~)+~Y(AE;M,~K,O~K)} "

(3.31)

At the same time, the energy-weighted sum rule (EWSR) for the transition operator M,~K becomes S;'K-V(E,-Eo)I(iMAK[O)]

= r/. A E £ ( A E ; M ~ K , M A K )

i

.

(3.32)

if'

The partial sums appearing in eqs. (3.30)-(3.32) can be easily calculated by using the powerful technique developed in appendix B. For example, we obtain 20

Svw

~

~ I

•

4

Soo)~3{J±(b ) + 2f(b)4},

(3.33a)

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I) s~l=So02~½(~+~)fl(b)2f~(b)

2,

225 (3.33b)

SEW22= So02gf± ( b )4 ,

(3.33c)

with

So

15 Ko =---5, 47r 020

(3.34)

Ko = w ± a ( n + +½) = w z a ( n _ +½)= w~A(n~ +½),

(3.35)

w h e r e f ~ ( b ) a n d f ~ ( b ) , the scaling factors for the t r a n s i t i o n s o p e r a t o r MaK, are i n t r o d u c e d as A ( b ) = 02o'/2(1 +3~) ,_x(z-h)/2 ,

f ~ ( b ) = o2ol/2(1 --3~j2 ~(2

b)/2 ,

(3.36)

with b = 3 , 2, 1,

for MAK = QaK, Q I K , Q~K -

(3.37)

The c o r r e s p o n d i n g scaling factors, f ~ ( a ) a n d f ~ ( a ) , for the i n t e r a c t i o n o p e r a t o r will also be i n t r o d u c e d j u s t like those o f eq. (3.36) with a = 3, 2, 1 for OaK = QAK, Q~K a n d Q~K- It s h o u l d be n o t e d that eq. (3.35), which is e q u i v a l e n t to eq. (2.9), is the n u c l e a r s e l f - c o n s i s t e n c y c o n d i t i o n for the q u a d r u p o l e m o d e . Similarly, we can o b t a i n explicit e x p r e s s i o n o f eq. (3.26). The resulting electrom a g n e t i c p r o p e r t i e s o f the G Q R for K = 0, 1 a n d 2 c o m p o n e n t s are s u m m a r i z e d in t a b l e 2, where all the q u a n t i t i e s are given u p to first o r d e r in the d e f o r m a t i o n p a r a m e t e r e. It is interesting to observe that u p to this o r d e r o u r result for the case o f 02K = Q~K with M2K = Q2K c o i n c i d e s with that o b t a i n e d in ref. 19) since the w h o l e strength o f the q u a d r u p o l e E W S R is e x h a u s t e d by the G Q R state for each K - c o m p o n e n t . H o w e v e r , w h e n we go to h i g h e r o r d e r in e, the s u m - r u l e t e c h n i q u e TABLE 2 E l e c t r o m a g n e t i c p r o p e r t i e s of G Q R for K = 0, 1 and 2 c o m p o n e n t s

A

B

C

WCI,:)R/~b(.,QR

B ( 2 K ; mGQR)/ B 0

2K SLw/ S0

g =0

1-~aE

1-~a~

1 -5(4-2b)e

K =1

l-hoe

2(1 -kae)

1-~(4-2b)e

K =2

l+~ae

2(l+~ae)

i 1 -,~(4-2b)e

Results for three types of interaction operator (O2K ~ Q2K, Q'K and Q~K for a =3,2, 1) and for three types of transition operator (M2K = Q2K, Q~K and Q~K for b - 3, 2, 1) are summarized. Quantities in columns A, B and C are given in units of O3(;OR=~2W0, Bo=So/O3GQ R and So=15Ko/4rrw~, respectively, and a = 4 2b a.

226

H. Sakamoto, 72 Kishimoto / Self consistent effective interactions (I)

for ref. ~9) is not sufficient a n d we s h o u l d use the p a r t i a l - s u m - r u l e t e c h n i q u e b e c a u s e the G Q R states no l o n g e r e x h a u s t the w h o l e strength. Here we will c o m m e n t on the origin o f r e n o r m a l i z e d interaction strength X ~2) a p p e a r e d in eq. (2.25). C o n s i d e r the a d i a b a t i c limit o f to << A E . Then, by using eq. (B.29), we can verify that the restricted s u m m a t i o n over A E = 2hw excitations in the r.h.s, o f eq. (3.4) is equal to a h a l f o f the 1.h.s. o f it. Therefore, we must use X ~2) (=2X~ elf) i n s t e a d o f X f ~f when we r e n o r m a l i z e the effect o f A N = 2 excitations. Finally in this subsection, we will discuss the p r o p e r t i e s o f the R P A lowest-energy q u a d r u p o l e m o d e for each K - c o m p o n e n t for the case o f O2K = Q~K. Notice here that only the K = 1 m o d e has a z e r o - e n e r g y soljtion with sc = 1 (table 1). By use o f the t r a n s f o r m a t i o n o f the m u l t i p o l e o p e r a t o r s given in table 4 below, we can calculate U_~ defined by eq. (2.89) as U,~ = -

- % P+ tP+ I ,

--

~/15 U~1 = -

-

g~o

~

(3.38a)

Mw;

~o2 +,o~ (3.38b)

., , /~p+Jpo, 16~" lv~oJov w=¢ol

--

(3.38c)

M w ; ~ (to ~p+~p ~ + to,_popo)

where p , ~ a n d Po are the spherical c o m p o n e n t s o f the m o m e n t u m o p e r a t o r defined by eq. (B.6) in a p p e n d i x B. Then by e m p l o y i n g the p a r t i a l - s u m - r u l e t e c h n i q u e e x p l a i n e d in a p p e n d i x B, we can calculate I2K defined by eq. (2.93). The result is 15 1 16~" M 2 w ~ o) L A ( n ~ + n + 1) = (2X ~tt) i

(3.39a)

l~l

15 1 ~o~+~o~ ~ ~(w~A(n++n 16~- M : t o 4 0 ) 2 - 0 ) :

(3.39b)

I>

5 1 167r M2to,,a(w~A(n++n + l ) + 2 w - A (~2 n ' + l )_) =~ ( 2 X ~ e ~ c )

I~

_

+ 1 ) - to:A(2n: + 1)) = 0

,

(3.39c)

where we used the n u c l e a r self-consistency c o n d i t i o n o f eq. (3.35). Substitution o f eqs. (3.39) into eq. (2.96) gives mic

)(22

sell

=

2X2e ,

mic

XzJ

sell"

X21 ,

mic

~

sell

X2o =zX2o -

(3.40)

It is quite interesting to observe the K - d e p e n d e n t q u a n t u m j u m p in the final results of I2K. T h e r e f o r e we can see that the d o u b l y - s t r e t c h e d q u a d r u p o l e interaction with the self-consistent strength X~~" g u a r a n t e e s the existence o f the R P A zero-energy m o d e for K = 1 c o m p o n e n t c o r r e s p o n d i n g to the rotational s p u r i o u s mode. F o r K = 2 a n d K = 0 c o m p o n e n t s , on the o t h e r h a n d , the excitation energy o f the RPA

H. Sakamoto, T. Kishimoto / Self consistent effective interactions (I)

227

l o w e s t - e n e r g y m o d e for the strength X~e" b e c o m e s x/2o9o (1 + ~ e ) a n d x/2o9o ( 1 - ~ e ) , respectively, reflecting the fact that there is no A N = 0 e x c i t a t i o n m o d e for these K - c o m p o n e n t s d u e to the c l o s e d subshell structure o f the a x i a l l y s y m m e t r i c d e f o r m e d system u n d e r c o n s i d e r a t i o n . F u r t h e r m o r e , from eq. (3.40), we can see that if the self i n t e r a c t i o n strength is artificially set equal to zX2 , a z e r o - e n e r g y m o d e can be o b t a i n e d as the R P A lowest energy m o d e b o t h for K = 2 a n d K = 0 c o m p o n e n t s . self This f e a t u r e is c o n s i s t e n t with the fact that if we p u t X2K = ¢XzK, the excitation energy becomes RPA = ,/2-(2 -- S¢) tO. 0)22 RPA [ w l2+2 {ogz o92,

=

RPA o9 o

=

( 2 - s ¢)

2o9~

[,/

(3.41a)

~ 1+

1

~- o g ± o g z q

c±

o) 6 1

2 - Wz)22~11/2 8(1 - ff)(og± ----~;--7.-5S, too, ( 2 - ~)o9o(o9~+ og:)JJ 1/2

o9o,

(3.41b)

(3.41c)

with

c=½(6-s~)(w--!l]2+2(3-s¢)(og---{~] 2 , \ O9o/ \Wo/

(3.42)

F o r K = 2, 1 a n d 0, respectively.

3.4. OCTUPOLE MODE In recent years, an a p p r e c i a b l e a m o u n t o f e x p e r i m e n t a l evidence 3s 37) has b e e n a c c u m u l a t e d which i m p l i e s that the stable o c t u p o l e d e f o r m a t i o n ( o c t u p o l e instability) m a y o c c u r in the g r o u n d state o f s o m e nuclei w h e n s i n g l e - p a r t i c l e orbits close to the F e r m i surface can form o c t u p o l e pairs. Since the onset o f the instability o f the H a r t r e e - F o c k g r o u n d state is i n t i m a t e l y r e l a t e d to the o c c u r r e n c e o f the i m a g i n a r y R P A solutions, the study o f the effective i n t e r a c t i o n is o f great i m p o r t a n c e . F o r this p u r p o s e , in this s u b s e c t i o n , we will carefully e x a m i n e the p r o p e r t i e s o f l o w - e n e r g y o c t u p o l e m o d e s in a d e f o r m e d system by use o f the d o u b l y - s t r e t c h e d o c t u p o l e interaction. F o r the case o f A = 3, eq, (3.5) can be written as 47rMo9 2 , self = g(3~A((r')")o + g 3(2) ~ : a ( ( r 4 Pz) tt) o + g 3(4) K a ( ( r 4P4) r,)o /X3K

=A((r4)')o+2(4-Ke)A((r4pe)")o~a(Ke(7Ke-67)+72)A((r4p4)')o.

(3.43)

Since A((r4P2)")o a n d A((r4p4)")o do not vanish in general, X3K self,n e c o m e s K - d e p e n dent. H e r e we will investigate the K - d e p e n d e n c e Ol" X3K self in detail. In the a x i a l l y - s y m m e t r i c d e f o r m e d h a r m o n i c - o s c i l l a t o r p o t e n t i a l m o d e l u n d e r c o n s i d e r a t i o n , the f o l l o w i n g g r o u n d - s t a t e c o n f i g u r a t i o n is a s s u m e d : the shells are

H. Sakamoto, 7". Kishimoto / Se!f-consistent effective interactions (I)

228

completely filled up to the N - 1 major shell, and as for the major shell N, the subshells with N >~ n_ ~> no are also completely filled while those with n o - 1 > n: ~>0 are completely open. By putting y = 0 in eq. (2.18) with eq. (2.19) and by use of the nuclear selfconsistency condition of eq. (2.9), the equilibrium deformation of the nucleus can be calculated as E

3(w±-wz) 2w± + to:

--

3(-Yz-Xl) 2X: + X±

lZN/

IX2(1- - X)q-O(l//~2),

(3.44)

where x = (_M - no)/.M with ~r = N + 1 is introduced to parametrize the contribution from the particles outside of the closed core. Here, and in the following, the ground-state expectation value of an arbitrary operator will be calculated by employing the powerful method explained in appendix A, and then we expand any quantities in powers of 1/N, and neglect the terms of O(1/]~2). Namely, we consider the mass number to be sufficiently large while, to investigate the shell structure effect, the value of x is kept finite and fixed. For example, we obtain /5

A(( r4)")o = 8 M2too { 1 + (} N ')x2(5x: - 8x + 18)}, ~/5

(3.45a)

2

A((r4p2)")° 8M2wo 15 /~ 'xe(5x2-8x+3)'

(3.45b)

/~5

A((r4P4)")"-8Mhoo2N 'x2(5x2-8x + 3).

(3.45c)

Substitution of eqs. (3.45) into eq. (3.43) gives the K-dependent octupole strengths as self~ X33 }

1

sell"

I

(X32)

=r/{l+(]~)

Ix2(5x-~-8x+8)},

5

=r/{1-3 N lx3(5x-8)}

(X;]'r) ~ = r / { l + l ' S N self

(X3o)

1

1 =r/{1 + ~ N

1

(3.46a) (3.46b)

'x2(55x2-S8x+lO8)},

(3.46c)

~ 2 x-(55x - 8 8 x + 58)} ,

(3.46d)

with 7

/~/5

~ / - 47rMwo 8M2~oo "

(3.47)

For comparison, the K-independent octupole strength X;~r of eq. (2.68) becomes (X; elf) ' = r / { l + ~ N

Ix2(5x2-8x+18)}.

(3.48) self

sell"

In the closed-shell limit of x = 0 or 1, of course, all the five strengths X33 , X32 , sell" self X31 , X3o and X3s e l l have the same value, but in general they have different values characteristic to the shell fillings.

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions ( I )

229

We can now solve the R P A equation for each K - m o d e by use o f the formulas (B.30)-(B.33). Results actually are simple for K = 3 and 2, but less transparent for K = 1 and 0. Since we are particularly interested in the octupole instability, it is more instructive to c o m p a r e the above K - d e p e n d e n t self-consistent strengths with the microscopically derived strengths X3K m~c defined in subsect. 2.5. From eqs. (2.85) and (2.86), and by using the partial-sum-rule values of octupole operators listed in a p p e n d i x B, we obtain mic (X33)

1

=

'r]{l+(~[)-lx2(5X2--8X-~8)},

(3.49a)

(X~c) -1 = rt{1 - ~ r - l x 3 ( 5 x - 8)},

(3.49b)

mic (X31)

(3.49C)

1

micx- 1

,)(30 )

= T ? { l + l ] ~ 1X2(85X2 _ 1 3 6 x + 7 6 ) } , ~-

r/{1 +l]~-IX2(155X2- - 2 4 8 X + 118)}.

(3.49d)

By substituting eqs. (3.46) and (3.49) into eq. (2.96), we get I33 = 0 ,

(3.50a)

I32 = 0 ,

(3.50b)

131 = - .~]Q-lx2(5x2 - 8x + 3) r/,

(3.50c)

13o = - 4 ~ t - l x 2 ( 5 x e - 8x + 3)r/.

(Y50d)

Notice here that these values o f I3K c a n also be obtained by calculating the r.h.s. o f eq. (2.93) directly. N o w along the lines o f subsect. 2.5, we can see that the doubly-stretched octupole interaction with the K - i n d e p e n d e n t self-consistent strengths X3K self guarantees the existence o f the R P A zero-energy modes for K = 3 and K = 2 c o m p o n e n t s for any deformation. For K = 1 and 0 components, however, the zero-energy modes do not a p p e a r in this model except for the cases of x = 0, and 1, where both 131 and 13o vanish. As can be seen from the definition of x and also from eq. (3.44), the cases o f x = 0 and 1 correspond to the spherical closed-shell nuclei. On the other hand, from eqs. (3.45), x =3 corresponds to the case where both A((r4p2)")o and A((r4p4)")o vanish, expressing the fact that the system does not have such higher-multipole moments. In fact, eq. (2.96) tells us that if IAK is negative (positive), then the R P A lowest energy AK m o d e is unstable (stable). Therefore, from eqs. (3.45) and (Y50), we can see that when a nucleus lies in the region o f 0 < x < ~ 3 ( 33 < x < 1), both o f the R P A lowest energy AK = 3 1 and 30 modes are unstable (stable) due to the fact that both A((r4P2)")o and A((r4p4)")o are positive (negative). It must be noticed that if the K - i n d e p e n d e n t strength X; ejc o f eq. (3.48) is used instead of the K - i n d e p e n d e n t strength X3K ~etf of eq. (3.46), then the exact zero-energy octupole m o d e can not be obtained in general not only for K = 1 and 0 modes but also for K = 3 and 2 modes except for the spherical cases o f x = 0 and 1. It is thus quite important to choose a proper effective interaction for the discussion o f the o c t u p o l e instability in d e f o r m e d nuclei.

230

H. Sakamoto, T. Kishimoto / S e l f consistent effective interactions ( 1 )

Finally, the c o u p l i n g between the dipole a n d octupole mode can be studied in the R P A . T h e c o u p l i n g t e r m s i n v o l v e d are s u m m a r i z e d in eqs. (B.34), w h i c h in fact v a n i s h d u e to the q u a d r u p o l e n u c l e a r s e l f - c o n s i s t e n c y o f eq. (3.35).

3.5.

SOME TESTS IN REALISTIC CALCULATIONS

Finally, we will comment on some results o f our realistic calculation for low-lying collective states in deformed nuclei ~s) just to substantiate the general conclusions made in the previous subsections. Figs. 1 and 2 show the strength XaK o f the effective 2a-pole interaction necessary to reproduce the lowest collective state of each Ki

(xl() a) 4

(xlO 2) 4

Sm-isotopes K = 2

Sm-isotopes

> ,K=0 ,% x~

~a c-i

2

×2

Z

Q.Q I

I

148

152

I •(D,"

I

156

(b)

I

L

I

148

152

156

A

A

Fig. 1. The K-dependence of the interaction strengths required to fit the low-lying quadrupole states in Sm isotopes: (a) (Q2' Q2) model, (b) (Q~. Q~) model. The deformation parameters are fixed as 0.0, 0.19, 0.30, 0.35 and 0.35 for 148Sm, I~°Sm, IS2Sm, IS4Sm and 156Sm, respectively. The theoretical value of the self-consistent strength )()~f is denoted by a cross point for each nucleus. In the unit of )¢2K, % is defined by uo =- M d / h.

,

(xl0-3) A) 15°Sm

b

,

,

B) 'S%m

<,lo l

c>

'

'

6 /

5

/

P

/

iI

iiI

/ "

~2

/'

/

21

Q3- Q3

II1¢*

11 o

6

5 K

o,

6

i K

K

Fig. 2. The K-dependence of the interaction strengths required to fit the low-lying octupole states: (a) IS°Sin, (b) '54Sm and (c) 23°Th. The dashed lines correspond to the (Q3' Q3) model while the full lines correspond to the (Q~. Q'~') model. The deformation parameters are fixed as 0.19, 0.35 and 0.24 for tS%m 15%m, and >°Th, respectively. The theoretical values of the self-consistent strengths X3K'~'u(for K - 0 , 1,2 and 3) are denoted by cross points for each nucleus. In the unit of )¢3K, ~, is defined by ,,,~=- M~/ h.

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

231

component for A = 2 and 3, respectively, based on the RPA. In this calculation, we used the Nilsson hamiltonian 29,3o) for single-particle energies, and a monopolepairing interaction is treated by means of the BCS theory. Full quasiparticle states are included in the model space. The general conclusion we are going to draw here does not depend sensitively on the particular choice of the parameters. As can be clearly seen in these figures, if we were to use the ordinary (QA. QA) interactions, different values for XAK would be required for each K-component. For the quadrupole mode, our results are consistent with the result of the analyses of • ~/;' /3- and y-vibrational states 38) and the reported deformation dependence 19,39) of the effective force strengths. Now such difficulty is drastically removed by using the improved ( Q ~ . Q~) interactions, whose strengths are almost constant for each K - c o m p o n e n t and are quite close to the predicted self-consistent values. Essentially the same difficulty of the K - d e p e n d e n c e in the interaction strength was already pointed out by Neerg~rd and Vogel 40) for the octupole mode by use of their modified octupole interaction, though the reason of it was not so very clearly explained at that time. From our present point of view, however, it can naturally be understood in terms of the doubly-stretched octupole interaction. It should be emphasized here that only about 5% increase of the octupole force strength for each K - c o m p o n e n t in the present case would give imaginary RPA solutions, i.e. the octupole instability.

4. Summary and discussion A systematic method to derive the effective interactions in deformed nuclei is devloped. By imposing the nuclear self-consistency condition rigorously, the conventional multipole interaction model has been improved so as to obtain the doublystretched multipole interaction model (i.e. the multipole interaction model with doubly-stretched coordinates). The model then has been tested by applying it to the collective vibrational states in axially symmetric deformed nuclei. We summarize here the main results of the doubly-stretched 2A-pole interaction model, which differs crucially from the conventional multipole interaction model. (i) A = 1 (with a K-independent strength ~(~K =X]e~f): The translational spurious modes tO1K = 0 (for K = 0 and 1) are exactly obtained. (ii) A = 1, T = 1 (with a K-independent strength X(K ~= -~X~~eJr,.~. By use of the (~Q~'. ~'~Q~') model, the total energy splitting of an I V - G D R is predicted as eaSGDR, i.e. 4 times as large as the value with the (r:Q1 • rzQ~) model, and is consistent with experimental observation. (iii) A = 2 (with a K-independent strength X2K = X~elf): - The rotational spurious mode w21 = 0 is exactly obtained. - The reason for different strengths for each K - c o m p o n e n t in the conventional (Q2' Q2) model can be naturally explained by the (Q~ • Q~) model. - The total energy splitting of a G Q R is predicted as 2eoScQR, i.e. ~ of the value with the (Q2" Q2) model, and is consistent with experimental observation.

232

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

selfx. (iv) A = 3 (with K-dependent strengths )¢3K =•3KJIn a large-mass-number limit, the RPA zero-energy modes are obtained both for K = 3 and K = 2 components irrespective of the shell fillings. For K = 1 and K = 0 components, on the other hand, the excitation energy of the lowest-energy mode depends on the shell fillings and the zero-energy mode can exist only for the spherical closed-shell cases or for the case with A ( ( r 4 p 2 ) ' ) o = A((r4p4)")o = O. - The difficulty of the conventional (Q3" Q3) interaction model, where different values of XBK is required for each K-component, i.e. about a factor 6 between the K = 0 and K = 3 components in a well deformed nucleus, is drastically removed. In summary, the doubly-stretched multipole interaction model has been successfully applied to both low-lying vibrational states and high-frequency giant resonances. The model has been shown to represent a great improvement over the conventional multipole interaction model. T h u s the model seems to be quite reliable at least for the construction of a fundamental mode of excitation, or in other words for the description of a system with a single mode of excitation. However, there has been an appreciable amount of accumulated evidence which implies that such a (two-body) multipole interaction model may not be accurate enough for the description of the couplings between different modes of motion since it satisfies the nuclear self-consistency only in lowest order 22). -

The rigorous application of the nuclear self-consistency will be presented in our forthcoming paper. There, it will be shown that higher-order effective interactions (e.g. a three-body interaction, etc.) must be induced so as to satisfy the nuclear self-consistency in higher-order accuracy especially when more than one mode is simultaneously excited in the system. The approach used in this paper has its origin in the Copenhagen philosophy ~"~). On the other hand, there are much more fundamental approaches for the origin of the shell-model effective interaction. In particular, based on the Brueckner theory 4~), it was pointed out by several authors 42,43) that the core polarization effect is important for the origin of the Q - Q interaction in spherical nuclei. Since there exists state dependence in a reaction matrix (G-matrix), it is an interesting problem to demonstrate whether the Q " - Q " interaction can be derived when one renormalizes the core polrization effect on the G-matrix in deformed nuclei. Through the course of the present work, we are very much indebted to stimulating discussions with many colleagues. One of the authors (T.K.) is grateful to Drs. T. Tamura, T. Udagawa, and S. Yoshida for their helpful comments at the early stage of the present work. He owes very much to Drs. T.T. Sugihara and D.H. Youngblood for their hospitality and nice experiments on giant resonances at Texas A&M, where the present work actually was initiated, and also to Dr. A. Bohr for his hospitality and the discussion on the octupole instability. We are grateful to Drs. T. Kammuri and T. Marumori for their continuous encouragement.

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

233

Appendix A DIAGONAL MATRIX ELEMENTS IN HARMONIC OSCILLATORGROUND STATE Diagonal matrix elements in the ground state of a harmonic-oscillator potential model most often end up with such quantities as 2 1, 2 n~, ~ n~ny, ~n 2, etc., where the summation is over all the occupied states. The calculation involved is elementary but tedious. Therefore a simple and useful method to evaluate them in an axiallysymmetric deformed harmonic-oscillator-potential model is given here. Let us define, A

A

2 q(nx, ny, nz)=- E qi(nxnyn~)=-- E (nxnyn~lqln~nynz)i, occ

i=1

(A.1)

i=1

where A is a total number of orbits occupied. The following three cases will be discussed: No

n +n +n z = N

(I1; L =- E occ

f(xyz)=- ~ ( l + x ) " ' ( l + y ) " ' ( l + z ) " : ,

E (n•,n~,n:)

N=O

n +n

(II); E'=occ

=N--n

E

E

no~
(n•n~)

,

n•+n ~N

(III);

E"--occ

0~ n:~

f'(xyz)~'

(l+x)"'(l+y)",(l+z)":,

no-I

(A.3)

occ

n

f"(xyz)=-2"(l+x)"'(l+y)n'(l+z)

E

(A.2)

oct

(nvn ~)

n: .

(A.4)

occ

Namely, the sum in the case (I) involves all the major shell N<~ No which is completely occupied, that in the case (II) only the states in the major shell N which are described by N ~> nz ~> no for a prolate deformation and that in the case (III) the remaining part in the shell N. The explicit summations for f ( x y z ) , f ' ( x y z ) and f " ( x y z ) defined can be carried out with the results given in terms of a binomial coefficient by (I); f ( x y z ) =

No

E CkFk(XyZ)

k=o

(II); f ' ( x y z ) = ( l + z )

N °,,

"o 2

k=O

Ck =

'

C'kFk(XyZ),

C'k =

( N o + 3'] k+3 /

,

(A.5)

(S-n0+2~ k+2

] '

(IlI); f " ( x y z ) = [f'(xyz)],,, o - [ f ' ( x y z ) ] ....

(A.6) (A.7)

where

(z - y ) x k + 2 + (X &(xyz) -

=

z)yk+2+

(y - z)z k+~

(x-y)(y-z)(z-x)

Y~ a+b+c=h

xaybz c all possible combination of rank k.

(A.8)

234

H. Sakamoto, T. Kishimoto / SelJ@onsistent effectit, e interactions ( I )

Based on the above expressions, the following examples show bow to evaluate the quantities in eq. (A.1); V l=[f]o, .........

f";1 ,

v n,

LTx J0

r,']

n,(n,-1)=Li~x~ f

oc%

0

,

Y n,n~.=

o~-%

["Z] f

o

,

where[ ]0stands for the value a t x = y = z - 0 . General results are given b y (I): ,V~ ( n " ) ( ; ' ) ( ; ~ ) =

( kNo+3~ +3 /'

......

(A.9)

c-

(Ill): ~ ' : ( n ' ) ( b ~ ) ( ~ / ) = ( k + 2

+ b+ m +2/

,,~o(c-

/

'

(A.10)

(a+b+m+2/"

(A.11)

where k = a + b + c and No=n,+n,.+n_ in eq. (A.9), while N=n~+n,.+n_ in eqs. (A.10) and (A.1I). Explicit expressions which appear up to the quadrupole mode will be summarized in table 3, although the method becomes more powerful in the higher mode. TABLE

3

T h e v a l u e s o f '?.~oc,- q( n, , n,, n: ) C a s e ( 1) •u

k

q ( n , , n,, n )

n~

( \' )

~v q

v,:..' q

ocl

,,c,

~33) ~:+3)

1

C a s e (l 1 ): n =- N - n,)

("~) (,,32)

17

~,,/3) (n42) N,,;3) (,,4-~; N.~3) n.(n32)+(n42) N,,;3) (~,)(n+2)+(:o)(n+32)+(n42)

½n,(n, 1)

In ( n _ - - I )

Case(Ill):

.... (N+2~ o,:~\~'' q k+2J

~5~'/ ,,,~ q"

H. Sakamoto, 7". Kishimoto / Self-consistent effective interactions (I) Appendix

PARTIAL

SUM

RULE

FOR

HARMONIC-OSCILLATOR

TRANSITION

235

B

INTENSITIES

1N

MODEL

A simple m e t h o d to evaluate a partial sum rule, e.g.

E Ir,

y ,

i(AE)

i(~E)

etc.,

in an axially-symmetric d e f o r m e d harmonic-oscillator potential model is given here, where the sum in the above expression is a partial and restricted sum over particlehole excited states [i) with a definite excitation energy AE = E~ - E0. First o f all, we introduce three types o f coordinates: (i) original:

x~ = f ~

(ii) stretched:

;

x" = f ~ : ~ ;

(iii) doubly-stretched: x~ =f~(~ ;

f~ = 1/x/~,

(B.la)

Z~ = 1/x/~Wo,

(B.lb)

f~ =~---wxJtoo.

(B.lc)

In the above expressions, h = c = M = l ; K = 1 , 2 , 3 ; {x~} -- x, y, z; {~:~}= ~:, "q, ~'; {to~.} = tOx, tO,,, tO~. Furthermore, to± -= ~o~ = ~Oy; f± -=f~ = f y , in the axially symmetric case. The dimensionless coordinate operators ~, ~, ~ with p = (~2+ 2+~2)1/2 will also be expressed as t + a¢.), ~ = x/~7 (ae~

pe~ = i~/~ (a~. - ae.),

with [p¢~, #~] = - i , etc.,

(B.2)

Then the following operators are defined:

tan),

-

q+,=-~/~(~+,n)

•

~/~ (a e - ia~),

=~/~(a+-a ), 7

+

• T p+, = -tx/~ (a_t + a + ) ,

az = a'~ ,

q_l = - ( q + , ) t ,

P-, = - ( P + l ) +,

and

their

b.c.,

q o = ~ = x / ~7 ( a zt+ a z ) , Po = i~/~ ( a ~ - az).

(B.3) (B.4)

(B.5)

Thus q2=(-)"q-u,

P2=(-)UP.,

[p,, %,] = -i6,,,

etc.

for/~ = 1, 0, - 1 .

(B.6)

The basic properties satisfied by the above operators are summarized below: The hamilton±an: H0 = to~(a~a~+l)+ , tO,( a nt a , +½) + tO.( a~a~ +½)

= w ± ( a + a + + a ~ a + 1)+ toz(a~az +½)

(B.7)

with [ H0, q, ] = - t w" , p *u , and eigenstate: In+, n , nz).

[Ho, p~] = u" o , q ,~ ,

t o ± ~ (.t)± ~

tO o z tOz

(B.8)

H. S a k a m o t o ,

236

7]. K i s h i m o t o

/ Se!l-consistent

effective interactions

(I)

Then the time-dependent operators are also introduced: Generally O(t)

q,(t)=./'~(a+e"°"-a q_,(t)=v/~(a; e "°,'

=e'""O

e ;o,,),

;"'~',

p,(t)=-i,/~(a[

a+ e ;,o,),

qo(t)=v/~(a:ei,O,+a_e

e

(B.9)

e'~°~'+a, e ' " " ' ) ,

p ~(t)=-i./~(ai!~ e " ° ' + a

io. )

po(t) = tv2 -/7 ( a :: e i....

e ;'""),

(B.10) (B.11)

a: e " ' :) ,

(B.12)

and all other commutators are 0.

(B.13)

with [ p . ( t ) , q,,(t)] = -i6.,,,

Non-equal-time c o m m u t a t o r s will be made best use of in the present technique:

[p*~,p~(t)] =~(e , ,,o," - e - ,o., )8~,

[ q l , q , , ( t ) ] =~(e ~ ; "-' - e

[p.. q,,(t)] = - l v•~;',( e

[ p . ( t ) , q,] = - t 2 ( e

.~

i,.. , ) 8 , , . ,

i,o,, ' + e

i¢o t

'"'V )6.,,,

,' + e

(B.14)

i.~ t

- )6.,,,

(B.15) [ Ho, q.(t)]

-k%p:.(t)=-i~t

d

q.(t),

[Ho,p.(t)]=iw.q'.(t)

d

- i ~ t P.( t) , (B.16)

or generally [Ho, O ( t ) ] = - i ( d / d t ) O ( t ) .

(B.17)

Now, we are ready to calculate

Z I
Let us a* and with a The

Z
etc.

i(JE)

introduce the notations as O: an operator written in terms of p . and q;,, or a; O ( A E ) : a part o f the operator O, which can only cause such transitions given energy J E , i.e. O =~.JE O ( A E ) , then O ( t ) = V j ~ O ( A E ) e ij~'. function /2(t) is defined by

/2(t)~'(ol[o,,o~,(0]lo)---

Z [/2,(aE)e "E' JE

/2 (~E)e ;~F']÷/2o,

(B.18)

"O

where the second equality defines ,Q+(AE), /2 (AE) and -(2o. On the other hand, we can calculate /2(t) as

s~(t)

~35{(ilo,[o>} i

= ~E E {(o[o,li)(ilo=(aE)D) e;~'-(OlO~(-AE)[i>(ilO,[O> e 'J"} i A E -O

Z JE

E ~{e 'j~'-
+ E ~{-
(B.19)

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

237

Therefore we obtain / 2 + ( A E ) = ~1 2 i(JE)

.O ( A E ) = ~

1

2

(O[Ot]i)(i[O~[O),

(B.20)

<~lo,lo>,

(B.21)

i(JE)

no=½ E {
(B.22)

It must be noticed that we can actually put ~o = 0, if the ground state is assumed to have no degeneracy as in the present case. In this way, we can evaluate the partial summation without performing the direct summation. As an example, let us calculate the partial sum rules for the multipole operator defined by

E (AE; A~,)~ E I(irQ~,,[O)[~,

(B.23a)

i(JE)

E (alE; Av, A'v)-= E i(JE)

(OlQ;~.li)(ilQ~,.Io),

(B.23b)

which appear frequently in the RPA calculation just as in sect. 4. For a dipole matrix element of Au = 11, for example, I2(t)--= '~(0[[ql,I q,(t)][0) = ¼(1){e i'°~' - e i'°l'}

(B.24)

is obtained by carrying out the commutator, where (O)-(01010). Therefore, from the time dependence, we get a+(w±)=½ •

(Olq~li)(ilql[O)=¼(1),

(B.25)

(Olq,li){i[q~[O)=~4(1).

(B.26)

i(~o L

O (w±)=½ • i(oJ±)

By using Q,±, = (3/47r)'/2f~q~, (see table 4 below), these equations give the final result for the partial dipole sum rule as Z (to±" 11) = • (toi; 1 - 1) = ½ ( 1 ) 4 ~ / i .

(B.27)

Similarly, the following partial sum rules can be derived: Dipole sum rules: 3

2

(w, ; 1 + 1) =½(1)~-~f±,

(B.28a)

3 2 E (w_., 1 0 ) = ½ ( 1 ) T f ~ ,

(B.28b)

4~

Quadrupole sum rules: 15 4 E (2to± ; 2 + 2 ) = ½ { n ~ + ½ ) T f ± , 47r

(B.29a)

H. Sakamoto, 72 Kishimoto / SelJ@onsistent effective interactions (I)

238

TABLE 4 The t r a n s f o r m a t i o n of the m u l t i p o l e o p e r a t o r

A

v

1

0

±1 2

\4rc/

.f~q*l

0

4rr

ml

\4~-/

±v

( 1 5 ] '/2

-

3

,fqo

('f~q°+flq+'q ') .f_l~qoq:,

\8rr/

.l~(q:t) 2

0

]~ (f~qo+3f?,q+,q ,)qo ft(2f~q~+flq+lq ,)q#l

±1

±2

(,os

\87r/

±3 4

(2f4q4+12f~f2q~q+lq t + 3fq+~q 4 2 2 ~) . . . . . f f ~(2f ~q[~+3f ~q+,q l)qoq+,

±1

±4

f'[~q"(qtl)2

,1"~~(q=L)

0

±2

.....

(45]l/!f2(gf~q~+[2q+lq

\8rr/

l)(q, 1) 2

. . . .

\8rr/

f=f3q°(q*~)~

\32~/

" '~q:')

15 2 (oai + w: ; 2+ 1) =¼(n: + n ~ + l ) ~ f ~ f ~ , 15

2 ( w i - w z ; 2± 1) = ¼ ( n ~ - n~) ~ f : f i ,

2 v

E (2w_. ; 20)= (n~ +½) 4 ~ f 4 , 5 4 Y (2~o~ ; 20) = ¼(n++ n + I ) ~ A .

(B.29b) (B.29c) (B.29d) B.29e)

H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

239

Octupole sum rules: ~ (3wl", 3±3) =~(3n±+an±+2) 3 2 35f 6 87r '

(a.30a)

~ (o9~" 3+3) =~(4n+n_-n±+n±+2n.;) 9 2 o3~ 5 f6,

(B.30b)

,

O77"

5~ (2w±+w." ~, 3+2) = ~~( n2± + 4 n z n < + 3 n ± + 2 n ~ + 2105 ) ~ f z 2f ~ 4,

(B.31a)

5~ (2toi - w~ ; 3 ± 2) = ½(4nzn±- n~ + n± + 2n~) ~ f z f 4 ,

(B.31b)

1

E(to~;3+2)=5(n+n

-n:n±+n~n~

+m-

105

2

4

)-8-~f~f±,

(B.3 lc)

21 6 E (3w±; 3± 1) = ~ ( n 2 + 4 n + n _ + 3 n ± + 2 n ~ + 2 ) - ~ f ± ,

(B.32a)

21 2 4 E (2w~ + to~ ; 3+ 1 ) = ½ ( n 2 + 4 n ~ n ± + 3 n ~ + 2 n ± + 2 ) ~ f ± f ~ ,

(B.32b)

Y. (2o9: -to±; 3+ 1) --!/-n2+4n.m:+n~+2n~)~-~f~f4,2\ z _

(B.32c)

21 4 Z (wz; 3+ 1)=l(4n~+4n~+l)2T~-lf~f4--((Zn:+l)(n++n~Tr + l ) ) ~ f z / . +1 + 2 21 6 g(8n+n_ 3n±+4n2 + 7 n ± + 8 m _ + 4 ) ~ f ; , 2 (3to~ ; 30) =3(3n2=+3n.+Z)-~f6,

2

(B.32d) (B.33a)

Z(2w±+w~;30)=9(n.m++n~n +n+n +n~+n++n + l ) ~ f72 f ±~, 4

(B.33b)

~2 (2w±- w~ ; 30) =~(n;n++ 9 n.n - n+n + n ~ ) -7- ~ f ~2 f i4,

(B.33c)

7 6 2 (o9~; 3 0 ) = 9 ( 3 n ~ + 3 n ~ + l ) ~ f ~ 7 f J4 ~2 -](2n~n+ + 2n.n + 2n~ + n+ + n_ + l)--~

+~(n++n~+2n+n_ +2n++2n +1)

7

f~f4.

(B.33d)

240

H. Sakamoto, 7". Kis'himoto / Se(f-con~istent effective interactions ( I )

Dipole-octupole coupling sum rules;

v~(wj'3±l,l±l)=(.f~(2n=+l)-f~(n++n + 1)) 8 ~ ~/~./2 ~c2 ~(wz;30, lO)=(f2:(2n~+l)--f~(n+ +n + 1 ) ) 8 ~3 ~/,~ -4.i~

(B.34a)

(B.34b)

The diagonal matrix elements appearing in the r.h.s, of these sum rules can be calculated by using the powerful technique explained in appendix A, where several basic matrix elements are summarized in table 3. Finally, the transformation of the multipole operator are given in table 4.

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H. Sakamoto, T. Kishimoto / Self-consistent effective interactions (I)

241

23) L.D. Landau, JETP (Sov. Phys.) 3 (1956) 920; 5 (1957) 101; 8 (1959) 70 24) P. Nozieres, Theory of interacting Fermi systems, (Benjamin, New York, 1964) 25) G.E. Brown, Rev. Mod. Phys. 43 (1971) 1; G.E. Brown, Many-body problems (North-Holland, Amsterdam, 1972) Ch. 5 26) L.H. Thomas, Proc. Camb. Phil. Soc. 23 (1927) 542 27) E. Fermi, Z. Phys. 48 (1928) 73 28) B.K. Jennings and R.K. Bhaduri, Nucl. Phys. A237 (1975) 149; P. Ring and P. Schuck, The nuclear many-body problem, (Springer, New York, 1980) Ch. 13 29) S.G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955) No. 16 30) S.G. Nilsson et al., Nucl. Phys. AI31 (1969) 1 31) M. Danos, Nucl. Phys. 5 (1958) 23 32) K. Okamoto, Phys. Rev. 110 (1958) 143 33) B.R. Mottelson and S.G. Nilsson, Nucl. Phys. 13 (1959) 281 34) A. Bohr and B.R. Mottelson, Nuclear structure, vol. II (Benjamin, New York, 1975) Ch. 6, p. 490 35) W. Kurcewicz et al., Nucl. Phys. A356 (1981) 15 36) 1. Ahmad et aL, Phys. Rev. Lett. 49 (1982) 1758; 52 (1984) 503 37) R.K. Sheline and G.A. Leander, Phys. Rev. Lett. 51 (1983) 359 38) D.R. Bes, Nucl. Phys. 49 (1963) 544; D.R. Bes, P. Federman, E. Maqueda and A. Zuker, Nucl. Phys. 65 (1965) 1 39) Z. Bochnacki, Phys. Lett. B31 (1970) 175 40) K. NeergArd and P. Vogel, Nucl. Phys. A145 (1970) 33 41) K.A. Brueckner and C.A. Levinson, Phys. Rev. 97 (1955) 1344; K.A. Brueckner and J.L. Gammel, Phys. Rev. 109 (1958) 1023; K.A. Brueckner, J.L. Gammel and H. Weitzner, Phys. Rev. ll0 (1958) 37 42) T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966) 40 43) H. Bando, Prog. Theor. Phys. 38 (1967) 1285