1.C I Nuclear Physics A152 (1970) 657--672; (~) North-Holland Publishing Co., A,nsterdam

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

M A N Y - P A R T I C L E M A N Y - H O L E STATES IN 160

N. DE TAKACSY and S. DAS GUPTA Institute of Theoretical Physics t, McGill Unirersity, Montreal, Canada

Received 23 January 1970 (Revised 27 April 1970)

Abstract: A method is proposed for the study of the energy spectra of collective many-particle many-hole states in nuclei near closed shells. In particular the 4p-4h and 3p-3h states of ~O are investigated. In the latter case, the mixing with the Ip-lh states is also calculated. The results are compared with the available experimental energy spectra.

1. Introduction

A considerable amount o f theoretical work has been done in recent years on the structure of some of the excited states of 160. Although in the simple shell model the ground state of 160 is doubly magic, so that the low-lying excited states are

expected to be particle-hole states 1), it has long been realised that many of these states are in fact collective many-particle many-hole ( N p - N h ) states which come low because o f the binding energy gained from deformation ~).

Now, a complete shell-model calculation o f these many-particle many-hole states is very complicated, but it can be done if one accepts a significant truncation of the single-particle basis states 3). Al though such a truncation is reasonable for spherical states, it is much more questionable for collective states. An alternative approach to these collective states is the following. The many-particle and the many-hole problems are solved separately by some approximat ion such as the deformed Hartree-Fock method 4). Then the basis states for the N p - N h system are constructed by vector coupling the lowest few eigenstates of the Np system to those of the Nh system. The eigenstates are obtained by diagonalizing the particle-hole interaction in this restricted

basis. The basic assumption involved in this approach is that thc particle-particle and the hole-hole parts o f the Hamil tonian (tl0~ and 11hh) give rise to strongly collective states, and the particle-hole part of the Hamil tonian (H~h) is not strong enough to disturb the correlations, but simply determines the relativc dynamics of the particles and the holes.

The weak coupling model o f Arima c t a l . 5) is a particularly simple example o f this approach, which assumes that H,h gives rise, at most, to a constant energy shift. Since the energies o f the many-part icle and the many-hole states can be taken from experiment, no calculation is required to predict, to within a constant energy shift, the spectra o f the many-part icle many-hole states in t 6 0 as well as in neighboring

t Work supported in part by the National Research Council of Canada.

657

658 N. DE TAKACSY A N D S. DAS G U P T A

nuclei. The fact that these predictions are reasonably successful indicates that the general approach is sound. Zamick 6) has suggested an isospin dependent monopole approximation to Hph which can be used to predict the energy shift as well. Although this model does not require an explicit description of the Np and the Nh states, the following picture can be suggested.

All the early sd shell nuclei with low isospins from I'~F up can be described in terms of a rotating intrinsic state 4). The same holds true of malay p-shell nuclei. It then follows that the weak coupling model describes the Np-Nh system in terms of two independant rotators.

The tri-axial rotator model which Stephenson and Banerjee 7) have proposed to describe the 4p-4h states of 160 involves the other extreme in relative particle-hole dynamics. They agree that the four particles and the four holes arc strongly deformed, as in Z°Ne and 12C respectively, but they suggest that these two intrinsic states are rigidly locked together into a single tri-axial intrinsic state from which rotational bands may be generated. The reason for the tri-axial structure is that it provides the best overlap between the oblate four particle and the prolate four hole densities.

In this paper, we present calculations done in an intermediate picture which assumes neither of the above extremes, but which can reproduce either of them depending on the effect of tlph. The Np and the Nh states are taken to be strongly deformed, so that collective model wave functions can be used to calculate the matrix elements of Hph. This at once simplifies what is otherwise a very complicated task. Our approach is very similar in spirit to that of Engeland and Ellis 8), who use an SU3 framework to calculate the matrix elements of llph. Although the techniques are different, very similar results are obtained in all cases by both methods.

2. Formalism

The Hamiltonian is separated into pp, hh and ph parts:

H = Hpp+Haa+ Vph, (2.1)

and the Np-Nh basis states are written in a product representation:

I { N p - Nh)Jp Tp; Jh Th; JM TM,) = ]{Np}J. Tp)]{Nh)J a T.), (2.2) I - ~ I

JM, TMt

with the arrow indicating the appropriate angular momentum and isospin couplings. The states I{Np}JpTp) belong to the ground state band of the 16+N nucleus, which is assumed to be deformed and axially symmetric; they are, therefore, cigenstates of Hpp with eigenvalues Ej . and can bc approximately described by the collective model lo):

I{Np}JpMpTpM,p) = ~2Jp+l-] "~ L~.~-- j v/~{D~r,~,,(gp)l{ Np} Kp(ap)Vp Mtp)

+(-)s.-r.D~:_K.(ap)]{Np}K.p(ap)TpM,p)}. (2.3)

160 MANY-PARTICLE MANY-HOLE STAqES 659

If Kp = 0, then the 1,% 2 normalisation thctor and the second term are omitted. "the intrinsic state [{Np}Kp(g2p)Tp M,p) is a single determinant in a deformed representa- tion, with its axis of symmetry oriented in the Qp direction. The state I{Np}K'p(t2p) × TpMtp ) is obtained from [ . . . K p . . . ) by a rotation through 180 ° about the ),-axis. A similar description applies for the Nh states. The creation operators for a particle

+ or a hole in the deformed representation will be denoted by bku where k,lt are the z-components of angular momentum and isospin. They have the usual expansion in terms of spherical shell-model operators:

bk+u = 2 + Cjku bjku. J

In order to lighten the notation for the rest of this section, we omit the explicit use of Np-Nh specification. The matrix elements of the Hamiltonian are:

(J', T~; J~ Td; JM TMtIHIJ p Tp, Jh Th; JM TMt)

= 6,pa,,Ojhj,~,Sr,,.r,O.r:,h[E,,+E,h]+(... IVphl...), (2.4)

where the energies Ej~ and Ej. are taken from experiment 9) and the ph matrix element is:

t ~ t . v t . (Sp I p, Sh T~, JM TMt[ Vp.lSp rv; Jh Th ; JM JM,)

= ~ ~-~ [(2Jp+ 1)(2J~,+ 1)(2Jh+ 1)(2,/;,+ 1)] ~

x 2 (JpMpJhMh[JM)(J'pM'~JhMhIJM) MpM'pMhM'h

x ~ (TpMtpThMthITMt)(T~M;pT~M~hITMt) MtpM'tpMthM'th

x (Kp(Y2p)VpM;pl(Kh(Oh)V h M;h [ Vph I Kh(I2h)T h M,h)I Kp(Op) Tt i , p )

+ 15 other terms}. (2.5)

In practice, we need only consider the two cases Tp = 0 = Th and Tp = ~ = Tk, since in light nuclei only these are associated with collective states. In the lowest configuration with Tp = T h = 0, all the particles and all the holes are paired, so that K p = K h : 0 , while the lowest configurations with T o = T h : "12 contain one unpaired particle and one unpaired hole.

Case I: Tp = 0 = Th. We can greatly simplify eq. (2.5) by omitting all reference to isospin, the ¼ normalization factor, and all terms in the curly brackets except the first. The only remaining intrinsic matrix element can then be calculated in a straight-

660 N. DE TAKACSY AND S. DAS G U P T A

forward way:

(Kp = 0(t~p)l(Kh = 0(t2h)lVphlKh = 0((2h))lKp = 0((2p))

= ~. A,D~*o(Qp)D~, O(Qh) Im

= ~ A, P,(flph), (2.6)

where flph is the angle between the axis of symmetry of the particles and the holes. The coefficients At are given by:

At = ~ Z (--)k"-mh(jhmhJh--mhllO)(jhkhJh--khllO) kp/lpkh/~h jpj'pJhJ'hmh

X C (p) C (p) ¢-,(h) c(h) jpkplJp j'pkpjap ~,--" jhkh It h j'hkh~h

x (Olbj..k,,.,, b j..,.., h Vph bj..,~, h b+..,...lO). (2.7)

Eq. (2.6) is essentially an expression for the potential energy of relative orientation of the many-particle and the many-hole densities. The integrals over the D-function and the sums over the vector coupling coefficients can be done by using standard techniques ix), to yield a particularly simple expression for the total matrix element:

<j,p , . , Jh , J]Vph[Jp, J h ; J)

= [-(2Jp + 1)(2Sp + 1)(2J h + 1)(2J~ + 1 ) ] ½ ( - ) - s + s : s ' "

× ~, A,(2l+l)-'(JpOJpO[lO)(JhOJ~O[lO)W(JpJhJ'pJ~;J1 ). (2.8) l = O , 2

The time reversal symmetry of the many-particle intrinsic state for N even implies that "(P) _ )jp-kp t"'(P) ~" ' ]p--kp/ tp = ( '~"jpkp.up and that both kp = ±½ appear in the Zkp" A similar statement holds for the holes. It then follows that At = ( - ) tA t . Furthermore, since Jh,J£ <= ~ for oxygen, the possible values of l are restricted to I = 0,2 only.

Case II: Tp = Th = -}. We can again drop all the isospin couplings and write the particle hole matrix element as a sum of normal and charge conjugate contributions:

r t ! t . ( d p J h , J M TOlVph[d p T p ; J M TO) = ( J p J h , JMlVph[JpJh; J M )

t t . ---+ (Jp Jh, JMI Vph exp (inTy)lJp Jh, J m ) , (2.9)

where the + , - refer to T = 0, 1 respectively and the states IJpJh; J M ) are m-scheme states as far as isospin is concerned (usually with Mtp = 3, M,h = --½).

All the intrinsic matrix elements are calculated as above. They all have the form l * 1 ~mAtD,,x~(~2p)D,,~.,(Qh) with CU~ = 0, +2Kp and '~h = 0, +_2Kh, so that the

integrals over Qp and Qh again factor into separate integrals over t?p and over t?h respectively, each involving three D-functions. In a completely straightforward manner we then obtain:

160 MANY-PARTICLE MANY-HOLE STATES 661

(J'pJh; JM[ VphlJpdh; JM> = [(2Jp+ 1)(2J'p+ 1)(2Jh+ I)(2Jh+ 1)] ½

x Z (2 /+ l)-lW(JpJhd'pJh; Jl){All)(Jp KpJ'p- KpllO)(J h KhJ' h - KhllO ) 1

+ A}2)( - )J'p-"P(Jp Kp J', K,l t. 2 Kp)(Jh Kh K,110) + al3)(-- )J'h + Kh(Jp KpJ'~-- Kpll, 0)(Jh Kh J~ KhIl, 2Kh) "~-AI4)(--)J'p-Kp+J'h+Kh(Jp gpJtp Kpi/, 2 K p ) ( J h KhJ h Kh[l , 2Kh) }. (2.10)

The coefficient AI 1) is identical to the At defined by eq. (2.7). It is a collective coeffi- cient in that all the particles and holes contribute. The AI 2), A[ 3) and AI '*) are spin- flip (or more accurately/,'-flip) coefficients which involve only the unpaired particle and hole. They are given in the appendix.

The charge conjugate matrix element is also given by eq. (2.10) if the AI ° are replaced by the corresponding B[ i~, which are also defined in the appendix. The B~ ~) only involve the unpaired particle and hole.

3. The 4p-4h band in ~60

The prolate 4p intrinsic state was obtained by an axially symmetric limited Hartree- Fock calculation "~) using the two-body potential of Inoae ct al. 12). The oblate 4h intrinsic state was obtained in the same way but using Cohen and Kurath's ~3) two- body matrix elements. Finally, the ph potential was taken from the work of Gillet and Vinh-Mau 1,~). "l-he coefficients AI obtained from cq. (12) are:

Ao - 8.75 MeV,

A2 = 8.51 MeV.

When these coefficients arc used to calculate the matrix elements of tlph, and the resulting Hamiltonian is diagonalised, the spectrum showq in fig. 1 results. For purposes of display, the whole calculated spectrum is shown shifted downwards by 2.57 MeV. It is, of course, reasonable that the 4p-4h states should be calculated to be somewhat high, since the mixing with 2p-2h states is not included.

The relative dynamics of the 4p and the 4h intrinsic states is governed by the coefficient A2. If A2 were very small, then the 4p and the 4h densities would be free to move independently of each other, and the weak coupling model of Arima et al. 5) would result. On the other hand, large values of A2 imply a strong correlation between the orientations of the 4p and the 4h densities, a negative value of A2 tending to align the two axes of symmetry, and a positive value of A2 tending to orient them 90 ~ apart. Since the value of A2 is in fact positive, the present model is intermediate between the weak coupling model of Arima et al. and the tri-axial strong coupling model of Stephenson and Banerjee a, 15). A comparison of the wave functions of these three models is shown in table 2. As expected because of the large A2 term, the tri-axial rotator model appears in this case to be a better approximation than the weak coupling model.

662 N. DE TAKACSY AND S. I)AS GUPTA

The values of Ao and A2 depend on the dcformation of the particle and hole densities. This is shown in fig. 2 with the expansion coefficients for the dcformed orbitals taken from the work of Chi 16). As the particles and holes become more nearly spherical, the coefficient A2 decreases while Ao remains roughly constant, so that the weak coupling model becomes an increasingly good approximation.

12

2 + 3 + 11.52 11.26 0 + 11.36

11 11.08 3 +

4- 10.36 4

10.12 2 4-

10 9 .85 24- 10.02 4 +

9

8

7.52 2 +

7 6.92 2 ÷

6 .06 0 + 6 6 . 0 6 0 ÷

0 4-

EXP'T ( 4 - p . 4 - h )

Fig. 1. The lowest 4p-4h energy levels in 160. The calculated spectrum has been lowered by 2.57 MeV.

TABLE I

Energies in the ground state bands of 2°Ne and 12C

0 2 4 6 8

2°Ne 0.0(--40.70) 1.63 4.25 8.79 11.99

t2C 0.0(F42.01) 4.43 14.08

The indicated ground state energies are measured with respect to the t60 ground state with Coulomb energies subtracted out.

160 MANY-PARTICLE blANY-HOLE STATES 663

TABLE 2

A compar i son o f the squares o f thc overlaps o f the wave funct ions obta ined in the present calculat ion with those obta ined in a weak coupl ing model (co lumn 1) and those obta incd in a rigid rotor model

(co lumn 2)

1 2

first 0 + 0.860 0.954

first 2* 0.820 0.951

second 2 + 0.565 0.950

first 3 + 0.884 0.972

1 i

13 X... (~p= 0.3 rq...~p = 0o1

12 o. ~p=0.2 ,,x_ ..t~p =0.0

11

/x _,x 1

1C A - - - - - - - - - - - ~ - - 9 ~ - - m - - -~' Ao

7 / ' " A~_

3

2

1

0 I T 0 . 0 - 0 . 1 - 0 . 2 - 0 . 3 - 0 . 4

~h Fig. 2. The s t rength o f the monopo le and quadrupo le te rms in the relative potential energy o f the

4p and 4h densities as a funct ion o f deformat ion.

4. Negative parity states in 160

The product basis for the 3p-3h states of 160 is constructed by vector coupling the wave functions of the prolatc K = ~ + ground state band of 19F to those of the oblate K = ½- ground state band of 13C. The energies /:), and I:'j, [refs. 9, 1T)] are listed in table 3. The intrinsic states of these two bands arc again obtained by a de- formed Hartree-l-ock calculation with the forces mentioned in sect. 3.

The coefficients A} ~) and BI ~, calculated with the effective ph force of Gillet and Vinh-Mau, are shown in table 4. It can be observed that the collective coefficients A(o l) and A~2 ~), to which all the particles and all the holes contribute, are substantially larger than any of the other coefficients which arise from the interaction of only the

664 m. DE TAKACSY AND S. DAS GUPTA

unpaired particle and hole. The fact that A(o I) and A~ I) are only half as large as the

corresponding 4p-4h coefficients can be understood since the magnitude of these

collective terms is expected to be p ropo r t i ona l to the number o f part icles times the

n u m b e r o f holes. The substant ia l B} i) coelficients lead to a large spli t t ing between

the T = 0 and the T = 1 3p-3h states.

TABLE 3

Energies in the ground state bands of 19F and 13N

J

19 F -q- 0.0(-- 23.68) 1.50 0.197 4.39 2.79

13N 4-0.0(+37.07) 3.51 7.38 10.36

The indicated ground state energies are measured with respect to the 160 ground state with Coulomb energies subtracted out.

TABLF 4

The AI (~) and B/~) coefficients for the 3p-3h states calculated with the Gillet potential

0 1 2 3

Az (11 3.91 0.34 2.66 0.04 AI (2~ 0.0 0.19 0.0 0.03 Al (~J 0.0 0.49 0.0 0.03 A~ (4~ 0.0 1.06 --0.13 0.03

B t ( t ) - - 1.85 0.44 --0.36 0.04 B~ ¢z) 0.0 0.17 0.0 0.05 B~ (3) 0.0 0.42 0.0 0.05 Hi ('~) 0.0 1.09 --0.23 0.06

The energies o f the lowest 3p-3h states ob ta ined by d iagonal iz ing the Hami l ton ian

o f eq. (2.4) are shown in fig. 3. Immedia te compar i son with exper iment would not

be meaningful , since it is expected that the lowest negative par i ty states o f ~60 are

l p - l h states and consequent ly , it is necessary to calculate the mixing between these

l p - l h and th~ 3p-3h states. The l p - l h basis states can be writ ten as I(Aa)JM, TMt) where A and a represent

sets o f hole and par t ic le quan tum numbers respectively. The matr ix elements o f the

Hami l t on i an are given by:

= ' ' r <(A'a')JT[HI(Aa)JT> ,Saa,5~,(e,~--~A)+<(A a )3TIVI(Aa)J >, (4.1)

where the s ingle-part ic le energies are taken f rom the exper imenta l spectra o f ~70

and ~ SO, and the poten t ia l mat r ix element can be ca lcula ted by wel l -known methods

160 MANY-PARIICLE MANY-HOLE STATES 665

[refs. " t4)]. The calculation of the mixing matrix element

< {3p-3h}Jp Jh ; JM" TM r[ VI(Aa)JM; TMr>

is described in the appendix. The lp- lh basis contains a completely spurious J~, T = 1-, 0 state which is

eliminated by the method of Baranger and gee ~8). It is unfortunately not possible to use this method to eliminate any possible spurious 3p-3h states since the states which we include in the present calculation arc far from providing a complete basis; on the other hand, thc very fact that the 3p-3h basis is so restricted militates against the presence of dominantly spurious states. This can be checked by calculating the number of spurious quanta in each eigenstate.

18.70 3- 1 8 5 2 . • 2 - 18.42 " 1 -

18.05 3- 17.89 .. 1- 1 7 . 7 0 " . 4 -

17.41 2 -

17.17 • 5 -

16.61 0 -

15 .48 1-

15-06 2 -

14-42 3 - 14.24 1-

( 3 - p , 3 - h )

Fig. 3. The 3p-3h s t a t e s c a l c u l a t e d wi th the G i l l e t po ten t i a l .

We must now choose the interaction potentials which are to be used in evaluating the different types of matrix elements. The diagonal l p-1 h and 3p-3h matrix elements, which both involve the diagram

are calculated using the potential of Gillet and Vinh-Mau 14), while the matrix elements which mix the lp- lh and 3p-3h configurations, and which involve the dia- gram

)

are calculated with the potential of lnoue ~t al. ~z). The fact that the same potential

666 N. I)E TAKACSY AND S. DAS GUPTA

cannot generally be used to describe both the effective pp type and the ph type matrix elements has bccn noted before 1,), 20).

The spectrum obtained by including both the lp-lh and 3p-3h states in the calcula- tion is compared to the experimental spectrum of 160 [refs. ~).21)] in fig. 4. The

1Z3 1- 16.99.17.17 5- 271

16.9 5 - 16-88.. : 0 - 16.72 . . . . 2- 16.62 2-

16.3 O- 16.58 " 1@.26 3 - "2- 16.58 4 - 16.2 4 - 16.15 4 - 15.7 3- 15.94 . . . . 1-

15.82 2- 15.4 (1-,3-) 15.30 3-

15.21 2- 15.01 1-

14.63 2- 14.44 3 -

14-26 1- 13.97 2-

13.69 0~,1

13.26 3~1 13.14 2- 13.56 1"-;1 1310 . . . . .1,1 13.11. • 1- 12.99 2~1 12.97- 271 12.86 0~1 12.79-" 12.52 2-"071 12.62 1-:! 12.77 3~,1

12.43 1- 12.21 2-~1

11.63 3- 11.99 3~1

11.17 0- 10.95 0 -

10.55 2-

9.81 0 - 9.58 1 -

8 .88 2- 8.71 2-

7.99 1-

7.42 3- 7.12 1-

6.79 1 -

6.13 3-

5.02 3- EXP' T. (1- p,l-h) ® (3-p,3-h) (1-13,1-;",)

Fig. 4. Negative parity states in 160. The diagonal 3p-3h and Ip- lh matrix elements are calculated with the Gillct potential, while the mixing matrix elements are calculated with the lnoue potential.

purely lp-lh spectrum is also shown. Table 5 gives the relative importance of lp- lh and 3p-3h configurations in the different states. It is evident that all the lp-lh stat~s below 14 MeV are strongly depressed by the mixing with the 3p-3h states, there is therefore, a serious difference between this calculation and those based on the RPA

160 MANY-PARTICLE MANY-HOI.E STATES 66"I

[ref. 14)] which generally predict that only the lowest 3 - ; 0 state is seriously affected.

The fact that the lirst negative parity states of each spin and isospin now come too low is an immediate consequence of our choice of potentials. Since Gillet and Vinh- Mau had fitted the parameters of their potential so as to get a best fit to the negative parity states of 160 in a purely lp-1 h calculation (using the RPA), it is clear that this

TABLE 5

The summed probability of l p - l h configurations in the negative parity states of ~60 below 18 MeV

Gillet-Inoue-Gillet potential set Tabakin-Tabakin-Gillet potential set

ytr; T E ~ ( l p - l h ) c.m. E ~ ( l p - l h ) c.m. quanta quanta

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

0- ; 0 9.81 89 0.01 10.00 86 0.02 16.88 3 0.04 17.00 5 0.04

0 - ; 1 12.86 88 <,0.01 13.64 92 <0.01

1 - ; 0 6.79 77 0.04 7.63 76 0.04 13.11 54 0.08 14.68 13 0.18 15.01 73 0.04 15.69 20 0.05 15.94 19 0.05 17.82 67 0.03 17.70 31 0.15

1 - ; 1 12.62 92 <0.01 12.76 91 0.01 17.68 93 <0.01 16.99 92 <0.01

2- ; 0 8.71 81 0.02 9.48 75 0.03 13.14 77 0.03 14.48 67 0.03 15.82 75 0.02 16.42 25 0.04 16.58 30 0.04 17.41 63 0.02 16.72 82 0.02

2- ; 1 12.21 92 <0.01 12.85 93 <0.01 16.99 85 0.01 16.65 95 <0.01

3 - ; 0 5.02 82 0.04 6.37 82 0.04 14.44 77 0.02 15.18 39 0.06 16.26 23 0.06 16.61 66 0.03

3 - ; I 11.99 93 <0.01 12.83 93 <0.01

4 - , 0 16.15 92 0.01 17.70 <1 0.08 17.71 <1 0.08

5 - ; 0 17.17 0 0.13 17.17 0 0.13

The number of spurious c.m. quanta is also shown.

potential must to some extent include the renormalisation effects due to the 3p-3h states. Since we include the latter explicitly, there is a certain amount of double counting. What is more disturbing is the fact that including the 3p-3h states does not provide us with additional T = 0 levels below the T = 1 quartet at 12.5 MeV. In particular, no states are calculated to lie anywhere near the experimentally observed 1-(9.58 MeV) and 3-(11.63 MeV) levels that have been generally considered to be

668 Y. I)E TAKA( 'SY A N D S. DAS ( ; U P T A

3p-3h states 2). It is of course possible that a different choice of potentials would correct this situation; it is however difficult to envisage a modification that would lower the second calculated 1 - and 3- levels by some 3 or 4 MeV without also lower- ing the tirst 5- and second 0- states which are now obtained just above 16 MeV in excellent agreement with the corresponding experimental levels. Furthermore, we have observed that the coeflicicnts AtJ ) and A~2 t~ are very roughly proportional to the number of particles times the number of holes; a modification of the potentials that would affect the 3p-3h states would therefore be expected to affect the 4p-4h states even more seriously, by lowering them to an unacceptable position.

We can, in principle, avoid the problem of double counting which arises when the Gillet potential is used to calculate the diagonal lp - lh matrix elements by using a realistic force, in this case, the Tabakin potential 22, 23). The details of the calculation of the ph matrix elements, including all second order corrections which are not of the 3p-3h type, is given in ref. 23). The mixing matrix elements ( { 3 p - 3 h } . . . • . - I V I { I p - l h } . . . ) are also calculated with the Tabakin potential, but in this case, renormalized only to include the second order ladder graphs. All the other quantities that enter the calculation are the same as before. In particular, the diagonal 3p-3h matrix elements are again calculated using the Gillet potential. The reason for this is that in this case there is no double counting since the 5p-5h states are not explicitly included in the calculation, so that this effect must be included in the effective inter- action itself.

The resulting spectrum is shown in fig. 5. It can be seen that the dominantly lp - lh states are in very good agreement with experiment, as are also the first 5- ; 0 and the second 0 - ; 0 levels. The second 1 - ; 0 and 3- ; 0 levels are now calculated to lie even higher than before, thus making the identification with the 1-(9.58 MeV) and 3-(11.63 MeV) levels even more questionable.

The calculations of Engeland and Ellis 8) which were done in a SU3 framework, give results which are very similar in all respects to those presented here. This is, of course, to be expected since the physics in both cases is essentially the same although the techniques are different.

We conclude this section with a warning about the possibility of some double counting in these calculations. This comes about because the mixing between lp - lh and 3p-3h states includes contributions which can be represented by graphs such as the following:

(,a) (bl (c)

while diagram (a) represents a proper interaction term, diagrams (b) and (c) and others like them represent contributions that have already been included by the use of experimental energies for co and ca in eq. (4.1) and by the use of the experimental

160 MANY-PARTICLE MANY-IIOLE STATES 669

ground state energy of 16 0 as the rcfcrenc~ with respect to which all the energies are measured 24). It is hoped that any double counting is minimized by the fact that only

a very restricted set of 3p-3h states is included in the calculation.

1Z3 1- 17"70"17'41 2 - ' " 4 - 17 OO 17.17 5 -

16.9 5 - - • = = = _ , ~ , . . - ' 0 - 16-99"~6.65 271"'.1";1 16.61 . . . . 3 - 16.39 3-

16.3 O- 16.42 . . . . 2 - 16.30 1- 16.2 4 - 16.28 2 - 15.7 3- 15.69 1- 15.4 (1-,3-)

15.18 3 - 15.21 2 -

14.68 1-

14.46 2- 14.15 0'~1 13.97 2 - 13.~0 1~1

13.64 071 13-70 3,1 13.26 371 13.57 2~,1

13.10 . . . . . . . 1-1 12.97 . . . . . 23"1 12.85 271 '"371 12.79...12.52 2_..0,1 12.83 72-76 1~"~'~1

12-43 1-

11.63 3 - 11.71 0 -

11.68 2 - 10.95 0 -

10 .00 0 -

9 .58 1- 9.48 2 - 9.37 1-

8 . 8 8 2 -

8 .45 3 -

7.63 1-

7-12 1 -

6.37 3 - 6.13 3-

EXP'T. (1-p,l-h) (~ (3-p,3-h) (1-p,l-h)

Fig. 5. Negat ive pari ty states in ~60. The diagonal 3p-3h matr ix e lements and the mixing matr ix e lements are calculated with the Tabak in potential , whilc the diagonal l p t l h matr ix elements arc

calculated with the Gillct potential .

5. Conclusions

We have shown that a particularly simple method can bc used to calculate the excitation energies of collective many-particle many-hole states. Indeed, the matrix elements of eq. (2.10) can be calculated by hand, if necessary, and the calculation of

670 N. DE TAKACSY AND S. DAS GUPTA

the coefficients AI i) and B~ i) is only as difficult as the calculatior of the usual two-body matrix elements. Apart from simplicity, the method also has the advantage that the various quantities that enter call all be discussed in terms of the relative dynamics of the many-particle and the many-hole densities.

The application of this method to the 4p-4h states of ~ 60 yields results which are closer to the tri-axial rotator model of Stephenson and Banerjee than to the weak coupling model of Arima et al. A calculation of the negative parity states of 160 using the vector space spanned by the lp- lh states and the 3p-3h states constructed according to the prescription of sect. 4 yields very good agreement with those cxperi- mental states that have been traditionally interpreted as l p-ih states. This agreement is much better than that which can be achieved with a basis consisting only of lp- lh states, and it indicates that the RPA does not account for the 3p-3h admixture suf- ficiently well. In addition, the first 5- and second 0- levels are also correctly calcu- lated. On the other hand, it seems very difficult to bring the second 1- and 3- levels sufficiently low to identify them with the experimentally observed levels at 9.58 and 11.63 McV.

It is a pleasure to thank Professor B. Margolis for discussions and encouragement.

Appendix A

The coefficients AI 2), AI a) and A~ 4) that appear in eq. (2.10) are given by the following expressions in which kppp and khp h refer to the unpaired particle and the unpaired hole respectively.

AI 2)-- ~. (--)J'P+kP+mh-kh(jhmhJh--m£[l--2kp)(jhkhj~--khllO) jpj'pmh Jhj'hm'h

F ( P ) [ ' ( P ) f - (h) f , (h ) )< ~"~jpkpBp ~'~j'pkp~up "'~jhkh~Uh "Jj'hkh//h

+ + × (Olbj'o-k~u, bj'hm'hUh Vph bYhmhUh bjpkpop[O)'

AI 3 )= ~. (--)J'h+"h(jhmhj£--mhllO)(jhkhJhkh]12kh) jpj'pmh

JhJ"h

r ( p ) / - (p) f , (h) f , (h) X k.~jpkp/~ p v...~j,pkp//p ~...,jhkhr/h ~...,j,hkhllh

X (01 bj, kpup bj,hrnhUh Vph bj+rahUh bfpk~uplO),

AI 4) = Z ( -)J 'P + k~ + j". +,.~(jh mh Jh-- mhll-- 2kp)(jh kh Jh kh112kh) jpj'pmh jpJ'hm'h

C(p) C(p) c ( h ) f , (h) X jpkpp, p j ' p/¢p/./p jhkh/~h ~'j'hkh/~h

+ + x (0lbj,p_kpUp bj,h,,,,hu,, Vph bjh,,~u ~ bj~q, Upl0 ).

160 MANY-PARTICLE MANY-HOLE STATES 671

The charge conjugate matrix elemcnt of eq. (2.9) is also given by eq. (2.10) if the coefficients AI ° are replaced by the corresponding charge conjugate coefficients B} °. These can be calculated using eq. (2.7) and the three equations of this appendix with any summations over kolt~khlth omitted and the operator Vph replaced by Voh e x p ( - , n Ty).

Appendix B

We now want to calculate the matrix elements that mix the Ip-1 h and 3p-3h states. The lp- lh states do not contain the collective coordinates O,, and -Qh so that the col- lective model wave functions cannot be used to calculate the mixing matrix elements. Instead, the 3p wave functions and the 3h wave functions are obtained by angular momentum projection from the same intrinsic states that were used previously. Thus

and a similar expression describes the 3h states. That part of the two-bedy potential which creates two particles and two holes can

be written

v = I E ; + + (B.2) - Vgd.n[(2a + I)(2T + 1)]~b~ " b2 b. bo, b d B D J T l ~ . . . I I ~ _ . 1

J T J T L_.~-- . - I

0; 0

where the lower case subscripts represent the quantum numbers of a particle orbital,

the upper case subscripts label hole orbitals, and s r V&on is an antisymmetrized potential matrix element. When this operator acts on the lp- lh states, it yields 3p-3h states whose overlap with the collective 3p-3h states [eqs. (2.2) and (B.I)] determines the mixing matrix elements. The result is

( {3p -3h}Jp Tp, Jh Th, JM TMT[V](Aa)JM TMT)

_ N ( p ) j~.(h) f ' ~ J p + T p + j +~[(2Jp+l)(2jt~+l)(2Tp+l)(2Th+l)]~ - - J p K p z v J h K h ~ ]

x ~,.W(j, jaJpJh; JJ)W(.}, ½, Tp, Th; T~) Z ¼ V ~ J T b d B D

x ({3p}Kplbfb;bflO)({3h}Kh[b f + + bn bolO) (B.3)

I . . . .

L~.--I

J p T p J l . ,Th

The calculation of the overlap coefficients only involves the expansion coefficients of the deformed orbitals in the spherical basis and some vector coupling coefficients. Both these expansion coefficients and the normalization factors ,Ar~P).j~K~ and ~vs~,~'(h) are obtained from a limited Hartree-Fock calculation o n 19F with the [noue force and on ~3N with the Cohen-Kurath matrix elements.

672 N . DE T A K A C S Y A N D S. DAS G U P T A

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