Nuclear structure calculations with a density-dependent force in 208Pb

Nuclear FIzysics A235 (1974) 315-351;

@ ~urth-~oliand

Publishjng Co., Amsterdam

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

NUCLEAR STRU~RE

CALCU~~ONS

WITH A DENSITY-DEPENDENT PETER

FORCE IN *‘*Pb

RING t

University of California, Lawrence Berkeley Laboratory, Berkeley, Ca. 94720 and JOSEF SPETH Institut fiir Kerttpitysik der Kernforschungsanlage Jiilich, D-517 Jlilich, West-Germany and Physik-Department der Universitist Bonn, D-53 Bonn, West Germany Received 20 June 1974 Abstract: Excitation energies and excitation probabilities in a”*Pb of the low lying states as well as of strongly collective high lying states (generalized multipole resonances) are calculated, using a large configuration space and a density dependent interaction. Within the extended theory of finite Fermi systems moments of excited states and transition probabilities between excited states are calculated. The theoretical results are in fair agreement with the experimental values.

1. Introduction

The great advances in nuclear physics during the last ten years provide us with a large amount of new experimental data. In addition to excitation energies, which one knows very accuratery, excitation probabilities and electric and magnetic moments of excited states have now been measured. These latter quantities give us much more information on the excited nucleus than the excitation energy, and the question arises whether one can find a microscopic nuclear model which allows calculation in a consistent way of all the known properties of the excited states. In this paper we present microscopic calculations of all of these quantities. Initially, in sect. 2, a short introduction to the Green function formulation is given, as far as we will need it in the following sections. In sect. 3 we report on the calculation of excitation energies and excitation probabilities using the renormalized random phase approximation. Several other papers on this subject have already been published ‘-‘f. None of these calculations, however, shows satisfactory agreement between theory and experiment for all angular momenta and parities. The main differences of our treatment compared to the previous calculations, e.g. ref. 2), are the following: t On leave of absence from the Physik-Department 315

der TU Miinchen.

316

P. RING AND J. SPETH

(i) Single-particle model. Present work: Woods-Saxon potential. Gillet et al. “): Harmonic oscillator. One gets large differences for the single-particle matrix elements rJ for the high lying single-particle levels. (ii) Conjiguration space. Present work: Two main shells above and below the Fermi surface. Ref. “): One main shell above and below the Fermi surface. This is of great influence for the positive parity states. In ref. “) e.g. only 5 p-h components contribute to the 2+ levels. In the present calculation 77 p-h components contribute. (iii) Efictive interaction. Present work: Density dependent J-force. Ref. ‘): Central force of Gaussian shape. It turns out in this investigation that the density dependence of the interaction influences not only strongly the Of results but it is also of great importance for simultaneously obtaining good results for the positive and the negative parity states. The differences with the other papers are more or less the same. In subsect. 3.1 a short derivation of the renormalized RPA is given which is of importance in connection with sect. 4. Numerical details of the calculation are given in subsect. 3.2. A comparison between the theoretical and experimental data of the low lying states is given in subsect. 3.3. We finally discuss the generalized giant multipole resonances in “*Pb in subsect. 3.4, some preliminary results of which are published in ref. “). Sect. 4 of this paper is concerned with the calculation of electric and magnetic multipole moments of excited states and the transition probabilities between excited states. In subsect. 4.1 we give a derivation of the extended theory of finite Fermi systems (EFFS) ‘) which allows one to calculate these quantities within the same formalism as the excitation energies and the excitation probabilities. In subsect. 4.2 we discuss the differences between this formulation and other theoretical approaches. Finally we compare our theoretical results with the experimental values. In the appendix some of our wave functions and FP matrices are given. 2. The Green function formalism

In this section we give a short introduction to the Green function formalism as far as it will be needed in the following sections. We define particularly the Green functions and the basic equations and we show how to derive these formulae in a unified manner using the functional derivative method of Kadanoff and Baym “). The Green functions from which we start are defined as the ground state expectation value of a N-particle system of the time-ordered product of pairs of creation and annihilation operators. It is assumed the expectation value is that of the exact ground state of the interacting N-particle system. Following Kadanoff and Baym, these Green functions may also be defined as functionals of a field q(12) which is a purely formal device to derive “higher” Green functions. One distinguishes the Green functions by the numbers of pairs of creation and annihilation operators. The single particle, two-particle and three-particle Green functions are defined in the following way:

317

2osPb STRUCTURE

G&12) = i


G&13,24) = i2

’

<~OIT{Sa,,(t,)a,,(t,)a,:(t,)a,:(t,)}lNO>

,


(2.1)

G,(135,246) = z.3 ~~‘,(~~)~~(~~)~~~(~Z)INO)


with

Here T is the Dyson time ordering operator, 1, 2, . . . denote time variable and quantum numbers and the integral involves integration over time variables and summation over quantum numbers. For q equal to zero one gets the usual definition of the Green function. In all the following formulae we drop the index q and we always assume that after performing the functional derivation q is set equal to zero. In connection with our theory the response functions are of special interest. The two-particle response function is defined by L(13,24) = ‘$$

= G(13,24) - G(12)G(34),

(2.2)

and in an analogous way we define the three-particle response function L(135,246) =

a2G(12)

= G(135,246)- G(13,24)G(56) M65&(43) - G(15,26)G(34)+2G(12)G(34)G(56) - G(12)G(35,46).

(2.3)

Starting from the equations of motion of the creation operator and the annihilation operator and including the external field q, one can derive the Dyson equation, which allows one in principle to calculate the single-particle Green function, -i-

a

+E,,

6(1,3)+q(l,

3)+M(l,

3) G(3,2) = 6(1,2).

(2.4)

at, Here, M( 1, 3) is the mass operator, E, denotes a single-particle energy and the generalized 6( 1,2) symbol includes a b-function in the time variables and Kronecker symbol in the quantum numbers. Using the functional derivation method, one gets from eq. (2.4) the well-known Bethe-Salpeter equation of the two-particle response function L(13,24) = -G(14)G(32)-iIdS.. where K is a generalized

. d8G(15)G(62)K(57,68&(83,74),

effective interaction

which is defined as

(2.5)

P. RING

318

AND

X(13,24)

J. SPETH

6M(12) = __. 66(43)

(2.6)

In an analogous way we can derive the equation of the three-particle response function from the equation of the two-particle response function. Using the definition (2.2), one gets the following equation: Z-(135,246) = -L(15,46)G(32)-G(14)L(35,26) -i

s

d5.. . d8{L(15,56)K(57,68)L(83,74)G(62)

+ G(13)K(y7,68)L(83,74)@5,26) +

s

-i

s

(2.7)

d~d6G(15)G(62)Z(57~,68@05,~6)L(83,74)} d5.. . d8G(l?)G(62)K(57,68)L(835,746).

There is one new quantity 1, an effective three-particle interaction which is defined by Z(135,246) = ‘=&4),

(2.8)

as all other quantities are already defined in the theory of the two-particle response function. Eqs. (2.5) and (2.7) are of the same structure, both being inhomogeneous integral equations. After transforming into the space of single-particle shell model wave functions, these equations reduce to a inhomogeneous system of linear equations. A more generalized version of eq. (2.7) which includes pairing correlations has been developed by Meyer and Speth in ref. ‘). As we are interested in solving the equations of the response functions, we have to know not only the effective interactions Kand I but also the single-particle Green functions G(12). These functions are the solutions of the Dyson equation (2.4) which includes the unknown mass operator M(12). All these quantities are well defined within the Green function formalism, but the equations are too complicated to be solved in an exact way. The semi-microscopic theory of “finite Fermi systems” of Migdal lo) starts with an ansatz for the single-particle Green function. One splits up this function into a “pole part” which strongly depends on the energy and a remainder g”’ which is a smooth function of the energy. For the Fourier-transformed version one gets GY,Vz(m)= Z,,

6YlYZ + gS:?,W, 0 iv sign (E,, -_cL) EVI

(2.9)

where 2, is the renormalization constant and /J is the Fermi energy. On the other hand, with a renormalization procedure proposed by Landau, eq. (2.5) can be transformed in such a way that only the “pole part” of eq. (2.9) appaars explicitly.

208Pb STRUCTURE

319

With eq. (2.9) one can split up the product of the two single-particle Green functions in eq. (2.5) into a “shell model part” S and a “regular part” R, %,(~

+3W-G,&

-34

= S,,,,, vm(w, a) + Rw.p&~

a),

(2.10)

with

where n, are the occupation probabilities 0 or 1 of the shell model quasi-particles. Using eq. (2.10), a renormalized equation for the quasi-particle response function z follows from eq. (2.5),

This quasi-particle response function z is connected with the two-particle response function L by the relation

where r0 is the static vertex function and FPh is a renormalized particle-hole interaction. From eq. (2.12) we will get in the next section the quasi-particle RPA equation and eq. (2.13) will give us the connection between the quasi-particle amplitudes and the transition probabilities, where zw give rise to effective single-particle operators. 3. Excitation energies and excitation probabilities in *‘*Pb 3.1. RENORMALIZED

RPA EQUATION

As already mentioned, eq. (2.12) allows to derive the renormalized RPA equation which we use in subsect. 3.3. Starting from the two-particle Green function one chooses the time order t3 t4 > t, tz and inserts a complete system of eigenfunctions of the N-particle system between the two particle-hole pairs, G(13,24) =V~0Gov(34)GYo(12),

(3-l)

with the generalized single-particle Green functions G’“(34) = i(NO]T{a(3)~+(4))1Nv), G”(12) = i(Nv~T{a(l)n+(2))~NO). Using the definition of the response function and putting t4 = t3 + E (where infinitesimal positive quantity), one gets L Y1Y3Y2V4(fl t, , t2 t, +E) = i f

x$,‘y, GY0(12)e-i”“3,

v=o

(3.2) E

is an

(3.3)

P. RING AND J. SPETH

320

where CO,= E,.-E,

and Xe3’vg= <~~I~~~~~~~~~.

(34

From eq, (3.3) one projects out the excited state p of the iV-particle system with an operator of the form ;“drJ exp i(w, -I- it& s Lim 4 3 0, i > rI , t2,

p::=q

,

W

and one gets PI”,J&3Y2”4(tt t3, tz tBfc) or in the Fourier-transformed

= i~:&G”~(12),

(3.61

version

KS &~V3V2&’ m’, s2$ = ill:;4 epy, (0, QXa+

e+)’

(3.7)

with do’ -.

pz, = f_ hm rIJ 2n Jj”OSl+fu,+iq

(3.8)

s 2n

If we multiply the Bethe-Salpeter equation (2.5) with Pg the unrenormal~zed equation of the generalized Green function follows, G”‘(12) = -i

s

d5 . . . dSG(U)G(62)K(57,

68)GP0(87).

(3.9)

From the generahzed Green functions GILOft,f,) one gets the transition density p”e by putting tz = tl i-8 or by integrating the Fourier-transformed expression with respect to CO,where the integral has to be closed in the upper half plane of CO, PO xyIy3 = (~~l~~~~~~i~~~ = 2

I

d~G~~~~~-~~~.

(3.10)

With a renormafization procedure similar to that mentioned in sect. 2 one derives an equation for the quasi-particle transition density ipo of the following from: (3.11) This quantity zSo and the fuil transition density 11”’are connected by Ire XYIYZ -~~“I.Y4YIZC~’

(3.12)

The transition density xMoshows one to calculate transition Frobabiiities of a sir&eparticle operator UL from the excited state $1into the ground state of a N-particfe system, where N denotes an even number of nucleons,

rosPb

STRUCTURE

321

(3.13)

where aJ is a renormalized operator of the multipolarity J, (3.14) Thus the solution of eq. (3.11) which will be discussed in subsect. 3.2, gives not only the excitation energies but also the excitation probabilities of doubly even nuclei and of special cases of doubly odd nuclei. 3.2. NUMERICAL

DETAILS

To solve eq. (3.11) one needs a single-particle model which gives the energies ey, the single-particle wave functions (p”(r) and a residual p-h interaction Fph. We used a single-particle potential of the Woods-Saxon type of the following form:

Here, Vc is the Coulomb potential of a uniform charge distribution within the radius of the potential are different for protons and neutrons and are given in table 1. The radii Re,, and R,, are taken from ref. rl). The other parameters R,. The parameters

TABLE

The parameters

1

of the single-particle 1

potential

RLS (fm)

protons neutrons

-60.94 -49.40

7.52 7.0s

0.79 0.66

22.33 21.47

7.28 6.78

0.59 0.24

6.70

are determined by a least-squares fit to the experimentally known binding energies and single-particle energies of the nuclei 209Pb, ‘07Pb, ‘07T1 and “‘Bi. The eigenfunctions of this potential are obtained by diagonalizing in a truncated oscillator space, which procedure, first used in this connection by Klemt and Speth 12) gives also bound solutions for E > 0 whereas the states for E < 0 are practically identical with the solutions which one gets by numerical integration. In this way we are able to take into account also the second major shell above the Fermi energy which one needs for a quantitative description of the energies and especially of the transition probabilities of both, the even and the odd parity states in 2osPb. The configuration space and the single-particle energies used in this calculation are shown in table 2. The

322

P. RING AND J. SPETH TABLE2 Single-particle energies in (MeV) and configuration space

Proton particle ZOgBi

Neutron particle 20gPb

Proton hole 2O’Ti

nlj

energy

nQ

energy

nlj

energy

Ih% 2f; % 2ft 3p+ 3p+

0 0.89 1.61 2.82 3.12 3.64

3% *d* ‘b% 2d,F ‘8%

0 0.35 1.34 1.67 3.47

2gq lig lj* 3d4 2g; 3d3 2bK.

7.35 7.73 8.46 IO.01 10.54 11.31 14.07

‘8% *p* *p*

7.24 7.81 8.63 10 83 12.77

2ge lip lji 3d; 4% 2g; 3d+ 4p+ 4p* 2bq 3f;3ft

0 0.78 1.42 1.57 2.03 2.49 2.54 6.88 6.99 7.61 7.76 8.01

’ ft

lf;

Neutron hole zo7Pb no

energy

3pt 2fi 3p.3. till 2f$ lht

0 0.57 0.90 1.63 2.34 3.41

lhv 3% 2dt 2d3. 1g:.

9.17 9.81 10.01 11.81 13.37

energies are taken from excrement as far as possible 13), although in the cases where no experimental values are known we took the calculated ones. The p-h interaction which enters in eq. (3.11) is renormalized. It takes into account the truncaticn of the configuration space as well as the fact that the single-particle states are only approximately quasi particles. In the present calculation the density-dependent ansatz of Migdal has been used lo),

singfe-particle

Fph(r, r’) = C6(r-r’)(f(r)+f’(v)r

- z’+cr - a’(g(r)+g’(r)r

- d)].

(3.16)

The parameters f, f ‘, g and g’ are dimensionless parameters which depend on the centre of mass coordinate r in the following way: F(r) = FeX+ (P - P”)p(p), where p(r) is the density distribution distribution, P(r) =

(3.17)

of the nucleus. We used a spherical Fermi 1

1 fexp ((r-R&)

(3.18) +

Here, R and CLare of the order of the radius and the diffuseness of the nucleus, respectively. With this ansatz one interpolates between the interaction inside the nucleus and the interaction outside the nucleus which may be different in principle. The choice of the parameters is of crucial importance and they are further dependent on the configuration space and on the single-particle model which one uses. In literature there is some confusion on these parameters since often the authors given only the values of the parameters but not the configuration space and the single-particle

zosPb STRUCTURE

323

model. We used in the present calculation (except for the EO) the same parameters which have been used by the authors and others 6S14*’ “) to calculate the electric and magnetic properties of odd-mass nuclei around ‘OsPb. These parameters had been adjusted previously by fitting some selected electric and magnetic moments and transition probabilities in odd mass nuclei as well as the energies and the excitation probabilities of the first excited 3- and 2-+ level in “‘Pb. It is convenient to use the following combination of parameters: f”” = f+f’,

j-P” = f-f:

gpp= g+g', gP"= g-g'.

Only the f and f' parameters are different inside and outside the nucleus, whereas the spin-dependent g and g’ show no density dependence. One has to choose slightly different radii ROp and Ro, for the density distribution of neutrons and protons in eq. (3.18). The reason for this are the different single-particle potentials for protons and neutrons. For the interpolation of the p-h interaction we used the mean value of both. These interaction parameters, denoted by Kl, are shown in table 3. TABLE 3

Two different

sets of force parameters

f?P I" Kl K2

0.6 0.4

F ex -1.9 -1.7

used in this calculation isotopic shifts) f?”In -0.75 -0.263

(K2 gives good results also for

fP" cx

SP

P

-2.0 -2.63

1.30 1.30

-0.15 -0.15

During the present investigation, Zamick and the present authors 16) have shown that the parameter set Kl badly reproduces the experimental isotopic shifts. They showed also that a second set of the four parameters f ppand f pnexists, not very different from the first one which, gives good agreement between the experimental and theoretical isotopic shifts, leaving all other results published so far by our group practically unchanged. The magnetic moments and transition probabilities in odd mass nuclei are not at all influenced by the f-parameters. The two parameter sets differ mainly in their density dependence. In K2 the density dependence of the proton-proton interaction is increased while the density dependence of the protonneutron interaction is decreased compared with set Kl. This importance of the density dependence for isotopic and isomeric shifts has been pointed out already by several authors. A more detailed discussion of that point may be also found in ref. 16). The results depend very sensitively on the radius of the density distribution (3.17) but weakly on the diffuseness parameter CCIt turned out that the singleparticle density of the neutrons is slightly larger than that of the protons. To compensate for this effect we took R, > R,, R, = 6.80 fm,

R,, = 6.92fm,

R, = 7.04 fm,

c1= 0.6Ofm.

324

P. RING

AND

J. SPETH

We used the same values for both parameter sets Kl and K2. For comparison we show in table 4 some typical results &c&ted with the parameter sets Kl and K2. There is practically no difference between these two calculations except for the 6(r2> which one deduces from the isotopic shift. As a matter of convenience we did not repeat all our calculations with the new set K2. For that reason all the present results, except the EO, are calculated with the old set Kl. TABLE4 Comparison

3-

of some typical results calculated with the two different parameter

9 (MeV) B(E3; O+ -+ 3-)(e2.

fm6)

Q (e *b) AEP,+(keV) 5- 9 (MeV) B(E5; O+ + 5-)(e”

2+

. fm“‘)

9 (MeV)

* fm4) 3p+ Wcharne (fmz) 20gPb 2gs Q(e * b) ‘OgBi lhq Q(e * b) B(E2; 24 + lh+)(e* * fm4) B(E2; O+ + 2+)(e*

““Pb

sets Kl and K2

Kl

K2

EXP

2.63 546 x lo3 -0.168

2.64 538 x lo3 -0.182

6.50

5.87

3.39 2.85 x lo8 4.49 3070 -0.012 -0.241 -0.425 18

3.36 3.13 x 108 4.46 3172 -0.063 -0.255 -0.424 18

2.61 (54Of30) x 103 -1.3rtO.6 6.78hO.45 16.25hO.35 3.19 (4.62*0.55)x 10” 4.08 2965 -0.072f0.013 -0.290 -0.379 30+3

The numerical calculation is highly simplified by coupling the state IEti,rn,) and the hole state I&m& to the particle-hole state IIIL,lJM) with the definition IAm) = In&z). We used the following convention for the single-particle state and the particle-hole state, respectively, (3.19a) II,IZ;lJM)

= c (-)‘I’*(->“-“‘c(j,

jzJ~m,-m,M)~l,m,)~~,m,)-‘.

(3.19b)

The coupled version of the RPA equation follows from eq. (3.1 l), (3.20) and the excitation probability is given by W)

= I12

(3.21) where bJ denote the renormalized electric or magnetic operator of multipolarity J. Using Ward identities lo) one can show for the electric operators that oJ is identical

325

*OSPb STRUCTURE

with the bare electric operators. In that case no effective charges occur and the summation in eq. (3.21) runs only over proton states. For magnetic transitions we used the effective operators of ref. ’ ‘) which include the effects of the backflow current and of mesonic corrections,

with the parameters gt, = 1.119, IC= 0.00723 fm-‘,

gl, = -0.031, free

g_, = 2.793,

gs = 0.887 gf”“, gz

= -1.913.

(3.23)

3.3. THE LOW LYING SPECTRUM OF aosPb

Solving the RPA equation (3.20) we get as many solutions for a given Jrc combination as there are particle-hole con~gurations. In the case of 2+ one obtains in our configuration space 77 particle-hole configurations; that means one has also 77 2+ levels. The classica sum rule of the transition strength however is exhausted by only a few states whereas the transition strength of all other states is about two orders of magnitude smaller. The states are not coupled to a good isospin, nevertheless one is able to decide whether a state is predominantly T = 0 or T = 1 by the following procedure: We calculated the proton transition amplitudes and the neutron transition amplitudes of a given operator. If the amplitudes had the same sign we interpreted this state as a T = 0, if the amplitudes differ in sign we assumed T = 1. It turned out that the most collective states in general have only little isospin mixing. The results of our calculation for the low lying states in 2O*Pb are shown in table 5. The theoretical (FFS) energies and transition probabilities in the second and fourth column are compared as far as possible with the experimental ones in columns three and five. In the last column the major configurations are shown. In nearly al1 cases the theoretical energies lie somewhat too high, nevertheless we get fair agreement with the experimental results. ft is of great importance that the theoretical and experimental transition probabilities are also in good agreement since they are the best test for the wave function f which we will also need in the following section. We wish to point out that no effective charges are used : the charge of the protons is one and the charge of the neutrons zero. This good agreement of the negative and positive parity states is mainly due to the large configuration space but it is also due to the density dependence of the force as we found in our calculation. In the appendix in tables 11, 13 and 15 some of the wave functions of the lowest 3-, 5- and the most cohective 2+ states are given. In addition we also give in tables 12, 14 and 16 the quantities Fcyz which are defined by (3.24)

P. RING AND J. SPETH

326

TABLE5 Energies and excitation probabilities

B(OJ)f

E (MeV)

35456435654572+ 54364+ 28+ 76+ 4564638+

FFS 2.634 3.395 3.611 3.825 4.090 4.208 4.212 4.275 4.354 4.361 4.364 4.368 4.388 4.490 4.556 4.574 4.629 4.671 4.691 4.700 4.764 4.774 4.716 4.178 4.841 4.882 4.892 4.977 5.022 5.03

of the low lying states in losPb Major configuration

exp

FFS

exp

2.614 3.197 3.475 3.710 3.920

546 x 10s 285 x lo6 967 x 10s 301 x 106 615~10~ 175 x 106 118 535 x 10s 626 x 10s 877 x lo3 3930 200x10” 201 x 109 3070 391 x 10s 199 x 104 4386 292 x IO* 757 x 104 13.45 281 x 10” 502 x lo* 210 x 108 326 x lo3 330 x 102 502 x lo6 199 x 102 105 x 106 3895 344 x 10’0

(540f30) x 10s (462+55)x lo6

(4.180) (4.204) (4.296) (3.961) (4.204) (4.038) 4.086 (4.258) (4.357) (4.698) (4.480) 4.323 (4.231) 4.608 4.425 (4.258) (4.382)

collective n(2gq3p+-‘) n(2g*3pt.-i) collective n(2gp2ft-r) n(2g+2fs-‘) n(2g$f+-i) n(2g~3p~-‘)+n(2g~2f~-‘) n(li+.3p+-I) p(lhg3s+)-f)+n(2gp3p+-‘) p(lhe3s+-‘) p(lhe3s~-‘)+n(lig3p~)-‘) n(2g$f+-i) collective p(lhe2d3-i) n(2ge3p+-‘)

330 x 106

2965

p(lhe2d3-‘)+n(2gt3P~-‘) n(2g+3p+-1) 1287~10~

collective n(2g$f+-‘) n(ljy3p+-‘)

230 x 1Oa

collective p(lhs2d+-i) n(li+2f+-i) p(lhs2dg-‘) n(li+2ft-‘) n(li+2f+-I) p(lhe2db-‘)+coll.

The excitation probabilities B(OJ) areelectric B(EJ) values for the states 2+, 3-, 4+, . . ., and magnetic B(MJ) values for the states l+, 2-, 3+, 4- . . .. The units are e2. fmzJ and px* * fm2(J-1) respectively. The last column shows the dominant p-h configuration of the wave function. The experimental values are taken from refs. 22*23).

which may be of some importance 3.4. GENERALIZED

GIANT

for further

MULTIPOLE

applications

RESONANCES

of the wave functions.

IN *OsPb

In addition to the low lying states which can be compared with experimental results we also obtain high lying levels with a rather collective structure. Only few of them have been observed experimentally. The most collective of these states are shown in table 6. The first two states are the magnetic giant dipole resonances which have been found experimentally between 7.55 and 8.23 MeV [ref. I’)]. Since, in our approach,

BPb STRUCTURE

327

TABLE6 Energies and excitation J” 1+ 1+ 22+ 2+ Of 4+ 0+ 0+ 2+ 0+ 0+ 3333-

probabilities

E 7.50 8.31 7.51 11.02 11.06 11.97 11.42 13.38 14.97 16.98 18.20 21.75 20.112 20.401 20.536 20.698

of some rather collective states at higher energies

B(OJ)f 5.68 11.00 11.0x103 2370 4370 326 780 x 104 1049 4002 1200 1789 2535 0.15xlO~ 0.65 x 10s 0.68 x 105 1.37 x 105

T

0 0 0 0 0 0 1 1 1

Where possible, the isospin T is determined by the contributions of the state to the sum rule with 7’ = 0 and T = 1. The B(OJ) values are determined as in table 5. In the case of J = 0 they are given by B(OO)f = ~<,u~~~~~O)~~ (2 * fm4).

only one-particle and one-hole configurations are taken into account one gets only two l+ levels in that region. The main components are rrlh,, rclh;l and vii,, vlilpl ; for the lower energy case the amplitudes of these two configurations have the same sign, for the higher energy state the sign is opposite. These two states practically exhaust the sum rule. At 7.5 MeV we find a 2- state which has a M2 transition probability of B(M2) = 11000 fm’ *pi, which is larger by two orders of magnitude than for other 2- states in this energy region. This “magnetic giant quadrupole resonance” should be detectable by electron scattering experiments at backward angles. Around 1I MeV we get a couple of 2+ states with predominantly T = 0,which account for more than 60 % of the theoretical B(E2) strength. The most strongest of these are shown in table 6. The integrated theoretical reduced transition probability of all 2+ states in that energy region is B(E2)f = 8.2 x IO3 ez * fm4. This can be compared with the most recent experimental results of ref. I*) which gives a B(E2)“TP = (3.8 kO.4) x lo3 e2 * fm4 at the energy of EZf = 10.5kO.2 MeV. At about the same energy we found also a strongly collective 4+ level. The T = 1 giant quadrupole resonance is predicted in this model at 17 MeV. Of special interest in connection with the nuclear compressibility are the high lying O+ transitions, the so called “breathing modes”. Unfortunately there exist no low lying O+ states which can be described by the RPA which will allow to check the interaction for the O+ selection rule. For that reason we calculated the isotopic shifts in the neighbouring nuclei ’ 6). As already mentioned we found strongly different

328

P. RING AND J. SPETH

0

0

10

20

30 E(MeV1

Fig. 1. The O+ vibrations

calculated with the two different parameter sets Kl and K2. The latter gives good results also for the isotopic shifts.

results for the parameter sets Kl and K2. The first one underestimates the isotopic shifts by nearly one order of magnitude whereas the results calculated with K2 are in fair agreement with the experiments. The O+ levels shown in table 6 are calculated with K2. The differences in the O+ excitations using the two parameter sets are shown in fig. 1. From that we may learn that it is absolutely necessary to check the force parameters by calculating isotopic shifts. Unfortunately this has not been done in the work of Jakubassa I’) wh o used our programs for her microscopic calculation of the breathing modes (e.g. for “‘Pb she used the parameter set Kl). A summary of all the RPA results is shown in fig. 2. From this we may expect, in addition to that what has been discussed previously, a strong concentration of 3’ transition strength around 21 MeV. We will also deduce from this calculation that there are no very strongly collective states in the case of the 5- and 6+ from 10-20 MeV. A qualitatively similar result for the 2+ and 3- states has been published recently in ref. “). 4. Static moments of excited states and transition probabilities between excited states of “‘Pb 4.1. EXTENDED

THEORY

OF FINITE

FERMI

SYSTEMS

In the last section transition probabilities between an excited state and the ground state of “‘Pb have been calculated within the theory of finite Fermi systems which is equivalent to the renormalized RPA. However there is no possibility within the framework of Migdal’s original theory to calculate static moments of excited states or transition probabilities between two excited states of a doubly even nucleus. To calculate such expectation values we need matrix elements of the following form:

329

208Pb STRUCTURE

05”

.,.....,.,,

to

20

~j,.l,

E(MeVl30

G

=

w al

.L..d

0

I’

10

1-

z p” : -6 21

T

--Ii!

-L

i

2 i

*

;

OO

10

P

E&&V) 30

20

E(MeV) 30

00*

?d

‘I

OO

10

20

EWVf

30

0

lo

P

E(Mer

Fig. 2. Summary of the present RPA calculation.

where N denotes a nucleus with an even number of nucleons and P, v excited states of that nucleus. Recently one of the authors published ‘) an extension of the theory of finite Fermi systems which allows the calculation of the quantities of eq. (4.1). We will give here a short recapitulation of that derivation.

P. RING

330

AND

J. SPETH

First of all we will show that p’” is included in the three-particle Green function G(135, 246) defined in eq. (2.1). The procedure which we use is similar to that described in the preceding section. One chooses a special time order t4 = t3 +E and t5 = t6 +E and assumes that t3

e

t, ,

t,

2

t,.

(4.2)

With this assumption the pairs of operators a+(4)a(3) and a+(6)a(5) can be taken out of the T-product. After inserting a complete system of eigenfunctions of the Nparticle system between the three particle-hole pairs we project out the special states p and v. The operator P&, which projects out these states from the complete system is the product of two operators P/ defined in the preceding section, --r m PV _ 2 ei(~p+L$3dt P tjfj - ? ‘1-o s t

3

e

-i(O,+

i&5&5

.

s -U3

(4.3)

At the end of the derivation one can put ItI + 00, which means one can choose t3 -P + 03 and t5 -+ - co implying [eq. (4.2)] no restriction to tl and t,. Using eq. (4.3) and eq. (2.1) one gets (t 1 t 3 t 5 9 t 2 t 3 +Et5+E) P”w5 G Y1Y~Yg,V2VqYg

= i3q2 C AI,.b ‘1-o

exp [i(Ei-EF,+q,+iq)t]

exp [i(E:-ETz+w,+iq)t]

i’(o, - EyI + Ei + iq)(o, - Et2 + Ei + iv)

(4.4)

x (~Ol~,+,~“,l~~,)(~~,I~j~(l)~+(2)}1~~2)<~vla,+,~,,l~O>.

If q + 0, only those terms of the sum remain whose denominators are zero, i.e. 1, = p and i, = v. On the other hand one notices that after performing the limit q + 0 one can also choose It I + + 00. Finally one gets from eq. (4.4) P!s;s G(1359246) = ~y”,“y,x:;~~ G”‘(12),

(4.5)

where we define a generalized Green function G”‘(12) = i(Np~T{u(l)a+(2))~Nv).

(4.6)

Since one possesses an equation for the three-particle response function but not for the three-particle Green function one has to apply Ppy to eq. (2.3). From the different terms of the right hand side only the first and the last term remain for which one finds, in the same way as in eq. (4.4), P,“,‘;,G(12)Gw,,

vog( t3 t, , t3 + E t, + 8) = -xt&

x::v, G(193,,,

(4.7)

where G(12) is the usual single-particle Green function. Therefore the result reads P&L(135,246)

= -&&,{G’Y(12)-G(12)6,,}

= --x~~~J;~JG~“(~~).

(4.8)

Because of the Kronecker symbol at the single-particle Green function, the threeparticle response function only includes the change of the density matrix between

208Pb STRUCTURE

331

the excited state and the ground state of the doubly even nucleus. Therefore it is indeed possible to calculate the transition probability between two excited states. However one cannot calculate the absolute value of the moments of excited states but only the change of the moments between excited states and the ground state. This, however, is no strong restriction in our case because all magnetic moments and all electric moments except EO are zero in the ground state of *“Pb. Following the procedure of sect. 3 one has to apply the projection operator on the equation of the three-particle response function (2.7) which will give us an equation for dGNY.After some analytical calculations the final result reads dG”“(12) = jd3 d4{G0’(13)G- ‘(34)G”(42)+ -i

G’0(13)G-‘(34)G0’(42))

d3..

. d8 G(13)1(357, 468)G0’(87)Gc1”(65)G(42)

d3..

. d6 G(13)G(42)K(35,46)46““(65).

s -i

f

(4.9)

This equation is of the type of the Bethe-Salpeter equation of the two-particle response function and all quantities, but the effective three-particle interaction I is already known from the linear response theory. This new term involves all correlations higher than the two-particle correlations. In the present calculation this contribution is not taken into account. As shown in ref. ‘) eq. (4.9) can be approximately renormalized in the same way as the equation of the two-particle response function. The generalized density matrix p”’ of eq. (4.1) follows from the generalized Green function by (4.10) In the same way as in the usual theory of finite Fermi systems we define a generalized quasi-particle density p”““, (4.11) and the renormalized equation of this quasi-particle density has the following form: (4.12) with the inhomogeneous

term iQv,

P. RING AND J. SPETH

332

The other quantities are defined by

r:,,,

(4.14)

s2,” = a,--St,. We omitted the index zero at the amplitudes x0’ and xv0 because of the relation xvY,VZ= x&

=

X,‘py, .

(4.15)

From the solution of eq. (4.12) we get the change of electric or magnetic moments between the excited state and ground state and transition amplitudes by using eq. (4.11) (N~CLIOINV)-(NOI~INO)S,,

=

c

r&&L,P& (4.16)

In the following subsection we will discuss the physical meaning of this equation and we will give numerical results for the low lying states of “‘Pb. 4.2. INTERPRETATION

AND NUMERICAL

RESULTS

In discussing eqs. (4.12) and (4.13), we first give an interpretation of the first two terms of the inhomogeneous part i”‘. It is easy to see that the first two terms of eq. (4.13) are the usual RPA result which one receives for instance from a boson expansion ““). The main property of this part is that it gives a contribution to eq. (4.16) only if both single-particle states vi and v2 are on the same side of the Fermi surface. The first term of eq. (4.13) is shown in fig. 3a, from which the second term follows by interchanging the arrows. Taking into account the homogeneous part of eq. (4.12), we get additional contributions of the form shown in fig. 3b. This however can be interpreted as a renormalization of the operator V. Therefore eq. (4.12) is the mathematical proof of the well known empirical fact that the theoretical results of moments in even nuclei are highly improved if one uses, instead of the shell model matrix elements, the measured matrix elements of the neighbouring odd mass nuclei. There exists however one difference. Comparing eq. (4.12) with the corresponding equation for moments and transitions in odd mass nuclei, e.g. ref. lo), one notices a difference in the energy denominator of the homogeneous term. In the odd mass nuclei Sz,, is equal to the transition energy of the corresponding single-particle states, whereas in eq. (4.12) Jz,, denotes the transition energy between the two states p and v of the even nucleus. This energy dependence gives rise to a resonance behaviour of eq. (4.12) if the transition energy a#, is near to a ground state excitation. ln fig. 4 this energy dependence is shown for the B(E2) transition strength between the 5; and 3;

333

z08Pb STRUCTURE

bl

a)

cl

Fig. 3. Three different contributions to the moments of excited states and transition probabilities between excited states. In (a) the RPA contribution is shown, (b) is the RPA contribution including the renormalization of the operator V and (cc) shows one example of the “backward graphs”.

2

01

lo-

o

0

:

058

c I

2

3

4

R IMeW Pig. 4. Dependence

of the E2 (Ji- + 3,-) transition amplitude on the transition actual transition energy is 0.58 MeV.

energy. The

levels in “*Pb. If the energy a is near to the 2: excitation energy (Eth’ = 4.49) the B(E2) is strongly enhanced. The actual transition energy is 0.58 MeV, therefore one obtains only a very small enhancement. The additional terms in eq. (4.13) follow from a consistent treatment of the two-particte correlations. One typical

2.61

3.19

3.47 4.08

7.50

8.31 0.59

3-

5-

42+

1+

5- + 3-

-

&:E2)

Y p

;:

Z* c1

p

~rZckl.r.c

QZ

Moment (transition)

-1.212 0.758

-34.29 7.531 0.113 -0.569 0.418 3.078

-0.001 0.387 0.367 2.617

-36.25 4.67

3.43

0.005 1.724 0.003 0.026 0.889 -0.019 0.476 -0.337 0.614

0.541 5.868 0.151 - 34.27 8.807 0.461 2.525 -1.130 4.450

0.006

0.022 -0.003 0.311 0.149

1.217 0.539 3.832 -0.002

1.246

-16.777 0.057

EFFS

-6.891 0.016

Backw. graphs

-5.765 0.004

Polarization

-4.111 0.038

RPA

13.73

0.140

- 14.68

e Pot --1

of excited states

value dipole shows 5 to 7

!Gc PK e* * fm2

PK

%

5 4

3 PK fm* fms

keV

? ?rf

fm2 fm2 !%

Unit

The third column shows the moment: Q2 is the electric quadrupole moment, 6r” (charge and mass) means the difference in the expectation of the mean squared radius between the excited state and the ground state for the charge and the mass distribution. ,u is the magnetic moment, as+ is the magnetic h.f.s. constant for a muon in the Is+ state. AE,+ is the isomeric shift due to a muon in the Is+ state. Column 4 the RPA value calculated with an effective charge 2 for protons and 1 for neutrons (which means a polarization charge e,,, = 1). Column give the different contributions of fig. 3 to the full theoretical value of the extended theory of finite Fermi systems (EFFS).

1+

Energy

State

TABLE 7

Some moments and transitions

208Pb STRUCTURE

335

contribution together with the homogeneous part of eq. (4.12) is shown in fig. 3c. These additional terms in eq. (4.13) have been simultaneously derived in a somewhat different way in ref. 2’). To avoid confusions we point out that the name “backward graphs” is not related to the “small” components of the RPA wave functions. The quantities which one needs to solve eq. (4.12) are all defined in sect. 3. With the calculated density matrix p”’ we obtain from eq. (4.16) the electric and magnetic moments of an excited state p and the electric and magnetic transition probabilities between two excited states ~1and v in “sPb. Table 7 shows the contributions of the different terms to the quadrupole moment (Q2), magnetic moment (p), muonic isomer shift and magnetic hyperfine splitting constant in the muonic Is+ state (dE,+, as+) and the change of the mean squared charge radius (6r2). In columns five, six and seven the three possible contributions shown in fig. 3 are given. RPA denotes the contributions of the form 3a, polarisation denotes the contributions of the form 3b and backward graphs denote the contributions of the form 3c. The sum of the three columns gives the final theoretical results shown in the eighth column and denoted by EFFS. Comparing the three different contributions one notices that in the electric case the contribution of the backward graphs is as large as the RPA contribution or even larger, whereas in the magnetic case the backward graphs give usually only a small correction. The polarization contribution on the other hand is in both cases of great importance. For comparison we also show in the fourth column the RPA results calculated with an effective charge 2 for protons and 1 for neutrons (which means a polarisation charge epo, = 1). In table 8 a comparison between the theoretical and experimental values is given. With the exception of the quadrupole moment all theoretical results are in fair agreement with the experiment. Comparison

of calculated

State

Energy

3-

2.61

moments

TABLE8 and transition

Moment (transition)

Q2 &+

55- -+3-

3.19 0.59

EFFS

0.17 5.868 1.25 0.15 19.81

probabilities

with experimental

values

Exp.

Unit

Ref.

(l.l-0.9)f0.4 6.78f0.45 1.47*0.23 ‘) 16.25&0.35 b, 0.1 kO.05 28*2

b k”,”

25 24: 28)

PK e2 . fm4

23) 23)

‘) Recalculated for the new half-life 2z). b, Deduced from 2opBi.

In table 9 we show quadrupole moments, magnetic moments, muonic isomeric shifts, magnetic hyperfine splitting constants and the change of the mean squared charge and mass radius of some low lying states in 208Pb , calculated within the frame

336

P. RING AND J. SPETH TAELE9

Moments of the low lying states of ‘OaPb calculated

within the extended theory of finite Fermi

systems State

E;,, (MeV)

3545642+ 4+ 6+ 8+ 21+ 1+

2.61 3.19 3.47 3.71 3.92 3.96 4.07 4.32 4.42 4.61 7.51’) 7.50 “) 8.31 “)

Q2 (b)

-0.17 -0.20 -0.34 -0.30 -0.15 -0.55 0.09 0.12 0.06 -0.23 -0.14 0.02 0.02

P Gun)

a,+ (kev)

0.54 0.12 -0.24 0.41 -0.19 1.43 0.19 0.18 0.37 0.22 0.93 0.64 -0.22

1.25 0.15 -1.37 0.77 -0.55 2.63 0.46 0.25 0.64 0.27 2.32 2.52 -1.13

The symbols of the different expectation ‘1 Theoretical values.

Srz (10T2 fm2)

dE,,

charge

mass

6.65 2.31 1.12 2.53 1.51 5.73 -0.28 -0.24 1.46 4.81 5.49 -3.93 -4.35

5.81 2.02 1.37 2.28 3.00 4.72 0.91 1.09 2.06 3.68 3.09 -2.33 -3.77

(keV)

6.50 2.57 1.25 3.18 1.48 9.19 -0.93 -0.43 1.60 4.72 4.96 -3.64 -4.04

values are given in the text of table 7.

of the extended theory of finite Fermi systems. In table 10 the results of the calculated transition probabilities and transition rates between two excited states in “‘Pb are shown. The same results are also shown in fig. 5 where in addition the lifetimes are state energy

IM@Vl I+-

tramtim trarwtm

i psec)

rate Iscc“)

(psecf

3% Ml

6-

392

5-

371

L-

367

5-

320

3-

261

0’

0

Fig. S. Lifetimes and transition

rates of 2osPb calculated

within the EFFS approximation.

zosPb

STRUCTURE

337

TABLE 10 Transitions between the low lying states of roaPb Q(MeV)

OL

5- (3.19) -+ 3- (2.61) 4- (3.47) + 3- (2.61)

0.583 0.860

*

5- (3.19)

0.277

5- (3.71) + 3- (2.61) -+ 5- (3.19)

1.093 0.511

+ 4- (3.47)

0.233

6- (3.92) + 3- (2.61)

1.306

E2 Ml E2 Ml E2 E2 Ml E2 Ml E2 M3 E4 Ml E2 E2 Ml E2 Ml E2 Ml E2 Ml E2 Ml E2 E2


Ii>

+ 5- (3.19)

0.630

+ 4- (3.47) + 5- (3.71)

0.460 0.211

4- (3.96) + 3- (2.61)

1.347

+ 5- (3.19)

0.763

+ 4- (3.47)

0.486

-+ 5- (3.71)

0.253

+ 6- (3.92)

0.041

B(OL) 19.80 3.17x10” 0.19 0.178 7.00 13.32 0.500 27.53 0.164 13.50 0.174 41.89 6.989 x 1O-3 39.57 14.68 5.238 x 1O-2 91.92 3.8 x lo-’ 0.320 0.37 9.74 4.63 x lo-’ 3.6 x lo- 5 0.126 29.21 1.3 x lo-*

Rate (set-‘) 1.635 x log 3.555 x 10’0 1.099x10* 6.699 x 1O’O 1409x10 2.545 x 1O’O 1.173x10’2 1.173 x 109 3.683 x 1O’O 1.148 x 10’ 7.110 0.078 3.073 x 10’0 4.441 x lo9 3.704 x lo* 8.715 x lo9 4.766 x 10’ 1.639 x 10’ 1.731 x 109 2.894 x 10” 3.089 x lo9 9.341 x 10s 1.23 x103 3.604 x 10’0 3.68 x 10’ 0.018

The first and the second column give the initial and the final states and their energies in MeV. The third column gives the transition energy. The type of the transition is shown in column four. Column five shows the B(EL) or B(ML) values. The units are e2 * fmzL and ~~~~~fm2(L-1), respectively.

also given. As far as possible the theoretical results are compared with the experimental values. Unfortunately only the lifetimes of the first two excited states are known. We hope that these theoretical results might stimulate some further experiments. 5. Conclusion In this paper the excitation probabilities and excitation energies of the low lying levels as well as the most collective high lying levels are calculated simultaneously within the renormalized RPA. A large configuration space and a density dependent residual p-h interaction are used which allow one to describe quantitatively positive and negative parity states. We got fair agreement not only for the low lying states of “‘Pb but also for the experimentally known high lying states within this calculation.

P. RING

338

AND

J. SPETH

Moreover we predict strong concentrations of transition strength of several Jz combinations between 10 and 20 MeV. In the second part of the paper we calculated within the framework of the extended theory of finite Fermi systems moments of excited states and transition probabilities between excited states of “*Pb . The different contributions to these quantities are discussed in detail. It is shown that the additional terms which follow from that approach are of great importance for a quantitative agreement between theory and experiment. Thus we are now able to calculate in a consistent way moments and transition probabilities in odd- and even-mass nuclei. We thank L. Zamick for some valuable discussions and J. Morrison for reading the English manuscript. Appendix Here we give some examples of the calculated RPA amplitudes x” and of the quantities F;” = y’ defined by eqs. (3.24) and (4.13), respectively. The phases are determined accordingly to Edmonds 27) and our convention for the radial wave functions is as follows 28) R,,(&

= A,t P’~,,%P’) exp ( - +P”),

(A-1)

with 2”+‘++4y

1

pI(*) k

=

1 ivr” = 1/*

An1 = (2n+2&l)!!J&; i

h

’

W+m+l)

r=oT(r+m+l)

E’

p =

--.

C-4

r!(k-r)!

All other definitions are given in the text of table 11.

(A-2)

339

11 Particle-hole amplitudes ofthethree lowest 3- states TABLE

PARITY = -1

JOT = 3 E =

2.634

f

=

4.212

E =

4.629

TNLK TNLK

Xilrlo YIIvKf

X(ltKt YlltK)

X(I,K) YII,K1

1155 1145 1155 1144 1155 1223 1155 1222 1234 1145 1234 1144 1234 1223 1234 1222 1234 1301 1167 1134 1167 1156 1233 1145 1233 1144 1233 1223 1233 1222 1233 1301 1312 1145 1312 1144 1312 1223 1312 1222 1311 1144 1311 1223 1245 1134 1245 1133 1245 1212 1245 1156 1166 1134 I.166 1133 1166 1156 1178 1145 1323 1134 1323 1133 1323 1212 1323 1211 1323 1156 1244 1134 1244 1133 1244 1212 1244 1211 1244 1156 1322 1134 1322 1133 1322 1212 1256 1145 1256 1144 1256 1223 0245 0156

-0.035 -0.150 -0.077 -0.391 0.067 -0.050 0.158 -0.121 0.257 0.078 -0.291 -0.021 -0.067 -0.077 -0.111 -0.137 0.047 -0.027 0.068 -0.099 -0.046 -0.074 0.021 -0.009 0.035 -0.072 -0.012 -0.045 0.030 -0.093 0.008 -0.003 O.OPO -0.011 -0.038 -0.007 -0.015 -0.016 -0.027 0.017 -0.005 -0.005 -0.013 -0.019 0.007 -0.041 -0.070

-0.000 -0.001 0.001 0.003 -0.001 0.000 -0.013 0.022 -0.013 -0.000 0.001 0.000 0.000 0.003 O.OOR 0.001 -0.001 0.002 0.003 -0.006 0.002 -0.003 -0.000 0.000 -0.002 0.001 -0.000 -0.000 0.000 0.000 -0,oot O.OOl 0.002 -0.002 0.001 0.000 0.000 0.000 0.001 -0.000 0.000 0.001 -0.001 0.000 -0.000 0.001 Q.000

0.010 -0.005 0.005 -0.001 0.044 -0.009 -0.580 -0.014 O.CO3 0.004 -F.025 0.004 -0.018 -0.003 0.024 0,001 -0.123 -0.006 0.007 -0.003 -0.018 0.001 0.006 -0.005 -0.011 -0.002 0.004 0.002 0.007 0.001 c1.021 0.004 -0.001 0.003 -0.007 0.001 -0.012 -0.003 0.014 0.006 -0.008 -0.000 0.014 0.003 0.003 0.001 -0.oc4 0.001 -0.002 -0.001 -O.OOf -0.002 0.003 -0.002 0.003 -0.002 -0.010 0.006 -0.008 0.003 -0.000 -0.000 0.001 -0.000 -0.002 -0.002 0.002 0.002 0.002 0.002 0.002 -0.003 -0.001 -0.002 0.001 0.000 0.001 0.000 -0.004 0.003 -0.000 0.000 0.000 -0.000 0.003 0.001 -0.001 -0.000 0.002 -0.001 0.007 0.003 -0.004 -0.002

-0.025 -0.074 -0.039 -0.117 0.045 -O.Olff 0.072 -0.050 0.088 0.056 -0.133 -0.020 -0.039 -0.035 -0.051 -0.057 0.034 -0.014 0.037 -0.050 -0.026 -0.041 0.016 -0.004 0.029 -0.048 -0.011 -0.037 0.024 -0.068 0.006 -0.004 0.008 -0.008 -0.026 -0.009 -0.013 -0.011 -0.020 O.OL6 -0.004 -0.004 -0.011 -0.015 0.003 -0.035 -0.047

0.000 -0.000 o.wo O*OOQ -0.000 0.000 -0.003 0.001 -o.oco 0.000 o.oco 0.000 0.006 0.002 0.003 0.001 -0.FC1 0.000 0.001 -0.001 0.001 -0.001 -0.000 0.000 -0.OOf o.oof. -0.000 -0.000 -0.000 0.000 -0.001 0.000 0.001 -0.001 0.001 0.000 0.000 0.001 o.ogr -0.000 0.000 0.000 -0.001 0.000 -0.000 0.000 0.000

.

TABLE 11 (continued)

340 TNLK

0245 0245 0245 0245 0166 0166 0166 0164 017% 0323 0323 0323 0323 0323 0323 0401 0401 0244 0244 0244 0244 0244 0244 0322 0322 0322 0322 04F2 0412 0412 0411 0411 0256 0256 0256 0334 0334 0334 0334 0334 0333 0333 0333 0333

tNl.K 0155 0234 0312 0233 0256 0155 0234 0233 0167 0156 0155 0234 0312 0233 0311 0234 0233 0156 0155 0234 0312 0233 0311 0155 0234 0312 0233 0144 0223 0222 0144 0223 0144 0223 Olbt 0144 0223 0222 0301 0‘163 0144 0223 0222 0301

XlfrK)

0.061 -0.222 -0.413 O.lb3 0.025 0.130 0.047 0.375 0.336 -0.047 0.023 -0.093 -0.100 0.084 0.148 -0.063 0.078 C.016 0.082 C.072 0.095 0.170 0.202 0.055 0.049 0.106 0.086 -0.003 0,003 -0.004 -0.005 -0.OO4 0.007 -0.030 0.069 -0.001 -0.004 0.002 -0.005 0.020 OtOOl 0.002 0.003 0.004

YlI,Kb O*Of4 -0.OE8 -0.117 0.054 0.016 0.067 0.031 0,122 06141 -0.032 0.011 -0.043 -0.040 0.034 0,053 -0.031 0.031 0.016 0.043 0.024 0.033 01068 O-069 0,030 0.022 0.042. 0.037 -0.000 o.co2 -0.003 -0.002 -0.003 0.002 -0*028 0.046 O*OOl -0.003 0.001 -0.004 OtOlO O.OOf. 0.002 0.002 0.003

X(I,K)

-0.012 -0.042 -0.387 -0.913 -0.007 -0.009 0‘005 0.010 0.002 0.005 0.012 0.002 -0.008 -0.055 0.062 -0.013 -0.05C -0.000 -0.002 -0.047 -0.022 -0.Ol6 -0.035 0.010 0.007

-0.020 -0.010 0.002 -0.000 -0.003 0.001 0.002 -0.002 -0.003 -0.001 0.003 0.002 -0.006 0.003 -0.003 0.001 01004 0.000 0.003

Ylf,Kl

0.003 0.005 0.009 -0*022 0.003 -0.000 -0.001 -0.003 -0,003 -0.003 -0.003 0.003 0.001 0.005 -0.005 0,007 0.006 -0,000 0.000 0.016 0.004 0.005 0.004 -0.003 -0.004 0.005 -0,000 -0.001 0.000 0.002 -0.001

-01002 0.001 0.005 o*ooo -0.002 -0.000 0.003 -0.002 0*002 -0.000 -0.004 -o.ocz -0,002

X(I,K)

Y(1.K)

-0.009 -0.076 0.681 -0.279 -0.ot4 O.Cl9 -0.003 0.257 OclOl 0,003 -0.001 -0.004 0.001 0.090 o.oss 0.000 0.01x -0.001 o.aar -0.046 -0.048 -0.006 -0.056

0.003 0.002 O.Ofl -0.017 0.012 0.012 0.004 0.012 0.007 -0.001 -0.000 -0.004 0.002 -0,005 -0.004 -0.002 -0.000 0.002 0.003 0.015 0.013 0.006 0.014

-0.001

-0.001

-0.014 -0.019 0.019 0.000 o.oot -0.006 -0.000 0.002 -0.003 -0.008 0.000 -0.000 0.000 -0.001 0.002 -0.006 -0.001 0.00~ 0.000 0.001

0.007 0.002 0.001 -0.001 0.001 0.001 -0.002 -0.003 0.002 0.004 0.00x -0.001 -0.000 0.000 -0.000 -0.002 -O.OOL -O*OOO -0.000 -0.001

The first and the second cohnnn show the quantum numbers 7’, N, L, X for the particle and the hole of the correspondent ph pair: T = 1 means proton, T = 0 means neutron; N is the radial quantum number; L is the orbital angular momentum; K = if) Iabels the sir&e-particle angular momentum. The third and the fourth column give the ph atnpiitude X(1, K) = ;ym and F@, K) = kr of the towest 3- state at tke theoretical energy of 2.634 MeV. The phases are determined accordingly to ref. )‘I) with the underlying coupling scheme: ](s&> for single-particle states and

341

zosPb STRUCTURE TABLE12 The matrix elements Fi [eq. (3.24)] of the three lowest 3- states

JOT = 3 E = TNLK

TNLK

1155 1155 1155 1155 1155 1155 1234 1234 1234 1234 1234 1234 1167 1167 1233 1233 1233 1233 1233 1312 1312 1312 1312 1311 1311 1245 1245 1166 1166 1323 1244 0245 0245 0245 0245 0245 0166 Of66 0166 0166 0323 0323 0323 0323 0323

1167 1245 1166 1323 1244 1322 1167 1245 1166 1323 1244 1322 1178 1256 1245 1166 1323 1244 1322 1245 1323 1244 1322 1323 1244 1178 1256 1178 1256 1256 1256 0178 0412 0256 0334 0333 0178 0256 0334 0333 0412 0411. 0256 0334 0333

2.634

E

PARITY

=

= -1

4,212

E

I:

4.629

F(I,K) F(K,I) -0.286 -0,183 -1.020 -0.127 -0.618 -0.367 1.217 0.815 -0,308 0.472 -0.336 -0.297 -1.570 -0.712 -0,363 -1.082 -0.2?2 -0.712 -0.370 0.848 0.396 -0.375 -0.504 -0,445 -0.672 -1.306 -0.826 0.293 0.154 -0.759 0.294 -1.240 -0.090 -0.747 -0.252 0.104 0.244 0.123 0.053 0.124 -0.143 0.154 -0.762 -0.295 0.168

-0.339 -0.251 -1.007 -0.110 -0.638 -0.357 1.289 0.791 -0.238 0.465 -0.417 -0.277 -1.592 -0.705 -0.288 -1.001 -0.299 -0.731 -0.375 0.826 0.396 -0.393 -01506 -0.445 -0.689 -1.351 -0.806 0.358 0.186 -0.181 0.231 -1.284 -0.047 -0.698 -0.246 0.088 0.255 0.196 0.007 0.079 -0.141 0.145 -0.704 -0.291 0.184

-0.000 0.002 -0.000 0.002 0.008 0.012 -0.010 -0.013 0,005 0.031 0.014 -0.021 0.006 0.011 0.001 0.011 -0.012 0.019 -0.020 0.017 -O.@Ll -0.010 0.014 0.010 -0.014 O*Oll -0.003 0.000 -0.001 -0.020 0.002 0.053 0.050 0.041 0.036 -0,072 -0.059 -0.001 -0.032 -0.027 -0.002 0.005 -0.001 O.OOf 0.063

0.001 0.006 -o*oor 0.009 0.009 0.018 -0.015 -0.016 0.001 0.026 0.001 -0.011 0.007 0.009 0.013 0.008 -0.025 0.015 -0.022 0.021 [email protected] -0.007 o-011 0.012 -ct.012 0.017 -0.002 -0.003 -0.006 -0.029 -0.002 -0.044 -0.047 -0.036 -0.009 0,061 0.052 0.003 0.036 0.039 0.000 -0.004 0.043 -0.026

-0.052

0.081 0.043 -0.036 0.000 -0.@18 0.014 -0.050 -0.075 -0.079 -0.038 0.044 0.008 0.016 0.003 0.017 -0.126 0.047 0.060 0.039 -0.089 -0.001 0.027 -0.007 0.008 0,055 0.064 0.089 -0.094

-0.090 -0.067 0,011 0.015 -0.052 0.018 0.109 -0.084 0.056 -0.049 0.022 0.044 -0.050 0.018 0.043 0.035 0.009 0.056 0.031 -0.075 0.003 0.044 0.010 -0.007 0.070 -0.052 0.096 0.103 -0.026 0.028 0.075 0.022 -0.Ol.f -0.041 0.017 -0.141 0.078 -0.015 0.027 -0.075 0,010 0.005 -0,009 0.003 -0.132 0.189 0.019 -0.018 -0.017 -0.004 -0.041 -0.030 0.002 0.012 0.014 -0.029 0.067 -0.108 0.028 0.019 0.002 -0.03x

.

342

P.RING AND J.SPETH TABLE

12 (continued)

TNLK TNLK 0401 0401 0244 0244 0244 0244 0244 0322 0322 0322 1134 1134 1134 1134 1134 1133 1133 1133 1133 1133 r212 1212 1212 1212 1211 1211

0334 0333 0412 0411 0256

0334 0333 0612 0334 0333 1145 1144 1223 t222 1301 1145 1144 1223 1222 130x 1145 1144 1223 1222 1144 1223

If45 1144

1156

1223 0144 0144 0144 0144 0144 0144 0223 0223 0223 0223 0223 0223 0222 0222 0222 0222 0301 0301 0155 0155 0234

1156

1156 01'36 a155 0234 0312 0233 0311 0156

0155 0234 0312 0233 0312 0155 0234 0312 0233 023% 0233 0167 0161

Or67

-0.329 0.256 0.019 0.030 0.309 0.110 O*Zfl OIltO 0.179 0.217 0.748 -0.310 0*53t -0.351 0.502 -0.274 -0.5TJ6 -0.231 -0.341 -01305 0,737 -0.343 0.460 -0.543 -0.559 -0.51% -1.023 0.279 -1.009 0.221 0.473 0.135 0.159 0.393 0.324 -0.828 0.272 -0.604 -0.418 0.335 0.461 0.637 0.363

0.515 0.435

-0.313 0.273 0.048 0.057 0.170 0.124 0.213 0.179 0.165 0.221 0.753 -0.304 0.505 -0.262 0.435 -0.281 -0.537 -01336 -0.368 -0,385 01814 -0.271 0.452 -0.577 -0.498 -0.481 -1.036 0.312 -1.096 0.182 0.491 0.281 0.243 0.426 0.398 -0.914 0.180 -0.579 -0.430 0.420 0.514 0.552 0.289 0.470 0.450

-0,647

-0.563

0.434 1.024 -0,229 1.084

0.570 2.025 -0.221 1.165

0.042 0.060

0.039 0.035 -0.120 0.069 0.006 -0.002 -0.044 0.016

0.001 -o*ooo 0,007 -0,007 -0.006 -0.001 -0.002 -0.002 -O.OO? 0.005 0.005 -0.007 -0.033 0.055 -0.010 0.030 0.000 o.no1 0.005 -0.058 0.004

0.071 -0,003 0,008 0.002 -0.002 -0.044

0.039 0.003 0.237 0,030 -0.056 -0.188 -0.041 0.062 0.125 0.15f -0.033 0.053 -0.029

-0.058 -0.047 -0.041 -01037 0.119 -0.080 -0,026 0.004 0.054 -0.004

0,001 -0.001 0.005 -0,002 -0.006 -0,000 -0.002 -0.009 -0.009 0.004 0*008 -0,002 -0.038 0.034 -0.007 0.049 O*OOf -o*ooo 0.006 0.050

-0.029 -0.080 0.009 -0.029 0.006 0.032 0.034 -0.079 0.017 -0.212 -0.050 0.026 0.213 0,018 -0.028 -c1.138 -0.145 0.002 -0.045

0.012

0.048 0.022 0.023 0.008 -0.144 o.oor -0.009 -0,032 -0.029 -0.015 -0.018 -0.042 0.065 -0.098 0.101 O.#4b 0.003 0.043 -0.020 0.012 -0,075 -0.085 0.010 -0.038 -0.087 -0.004 0.0'12 -O,Cb4

0.038 -0.116 0.128 0.049 o.t\30 0.047 0.033 -0.034 0.052 0.009 -0.072 0.154 0.180 0.110 -0.102 -0.081 0.046 0.088 0.189 0.073 0.222 0.017

-0.017 -0.048 -0.054 -0.062 0.X56 -0.011 -0.011 0.012 0.002 -0.021 0.012 0.051 0.025 0.047 -0.002 -0.040 0.022 -0.109 -0.052 -0,107 0,114 0.047 0.016 -0.004 0.025 -0.033 -0.034 0.058

-0.136 0.179 0.041 -0.010 -0*0x7 0.040 -0.004 -0.154 0.000 -0.065 -0.014 -0.lf8 -0.083 0.041 0.131 0.184 -0.005 -0.178 -0.110 0.174 -0.182 0.155

The meaning of the quantum numbers and the coupling scheme is the same as in table 11.

*08Pb STRUCTURE

343

TABLE 13 Particle-hole

amplitudes of the three lowest 5- states

JOT E =

3.396

=5

PAFITY = -1 E =

3.826

E =

4.275

TNLK TNLK

X(I,KJ Y(I,KJ

X~I,KJ YtI,KJ

X(I,KJ YtIrKJ

1155 1155 1155 1155 1155 1234 1234 1234 1234 1167 1167 1167 1167 1233 1233 1233 1312 1312 1311 1245 1245 1245 1245 1245 1166 1166 1166 1166 1166 1178 1178 1178 1323 1323 1323 1244 1244 1244 1244 1322 1322 1256

-0.023 -0.058 -0.058 -0.145 -0.209 0.021 -0.031 0.060 -0.135 0.023 -0.012 0.036 -0.116 -0.013 -0.019 -0.066 0.011 -0.021 -0.013 0.006 -0.005 0.009 -0.011 -0.024 -0.008 -0.013 -0.009 -0.015 0.018 -0.027 0.013 -0.047 0.002 -0.002 -0.011 -0.004 -0.004 -0.013 0.010 -0.004 0.007 -0.006

0.019 0.051 0.057 0.205 0.381 -0.022 0.041 -0.056 0.138 -0.019 0.010 -0.035 0.112 0.011 0.022 0.063 -0.011 0.027 0.012 -0.007 0.008 -0.008 0.010 0.024 O.OC6 0.010 0.007 0.017 -0.015 0.024 -0.012 0.046 -0.002 0.003 0.010 0.003 0.004 0.012 -0.008 0.004 -0.006 0.007

0.001 -0.000 0.000 0.000 0.004 -0.003 -0.017 -0.002 -0.184 -0.003 0.001 0.001 -0.009 c.001 -0.013 -0.003 0.045 0.004 0.000 -0.000 -0.000 0.000 0.003 -0.000 -0.002 -0.000 0.001 -0.003 -0.003 -0.001 0.010 0.004 -0.001 0.000 -0.002 0.001 0.001 -C.OOl 0.001 0.000 -0.002 0.000 -O.Oc)l -0.001 0.002 0.001 -0.001 -0.001 0.000 -0.000 o*ooo 0.000 0.000 -0.001 0.000 -0.001 -0.001 0.001 -0.001 0.000 0.001 -0.000 -0.003 0.001 -0.000 -0.000 0.001 0.001 0.001 0.000 0.000 -0.001 -0.001 -0.001 0.002 0.001 -0.001 0.001 0.001 0.000 -0.001 0.000 -0.001 -0.000

1145 1144 1223 1222 1301 1145 1144 1223 1222 1134 1133 1212 1156 1145 1144 1223 1145 1144 1145 1134 1133 1212 1211 1156 1134 1133 1212 1211 1156 1145 1144 1223 1134 1133 1156 1134 1133 1212 1156 1134 1156 1145

-0.014 -0.023 -0.021 -0.024 -0.029 0.013 -0.008 0.020 -0.036 0.015 -0.007 0.020 -0.041 -0.011 -0.009 -0.025 o.oc7 -0.009 -0.010 0.004 -0.002 0.007 -0.009 -0.014 -0.007 -0.010 -0.008 -0.012 0.013 -0.018 0.007 -0.026 0.002 -0.002 -0.007 -0.005 -0.003 -0.009 0.009 -0.003 0.005 -0.004

0.011 0.017 0.022 0.023 0.030 -0.012 0,007 -0.016 0.031 -0.012 0.005 -0.016 0.033 c.012 0.009 0.020 -0.008 0.008 0.011 -0.005 0.001 -0.006 0.008 0.014 0.005 0.008 0.007 0.011 -0.011 0.015 -0.005 0.022 -0.002 0.002 0.006 0.006 0.004 0.008 -0.009 0.003 -0.004 0.004

P.RING AND J.SPETH

344

TABLB

13(continued)

TNLK TNLK

X(I,K) Yfl ,Kl

Xti,K)

Yf1.K)

XtT,K) YII,Kf

1256 1256 1256 1256 0245 0245 0245 0245 0245 0245 0166 0165 0166 0166 0166 0166 0178 0178 Olf8 0323 0323 0323 0323 0401 0401 0244 0244 0244 0244 0244 0322 0322 0322 0412 0412 0256 0256 0256 0256 0256 0334 0334 0334 0334 0333 0333 0333

0.004 0.001 -0.013 -0.010 0.008 0.007 -0.017 -0.012 -0.025 -0.015 0.060 0.004 -0.068 -0.022 -0.081 -0.024 0.054 0.036 0.871 0.044 0.022 0.001 0.045 0.014 0.032 0.018 0.059 0.020 0.152 0.029 0.234 0.033 0.001 0.007 -0.028 -0.021 0.118 0.040 -0.013 -0.009 0.024 0.005 -0.024 -0.007 0.017 0.024 -0.010 -0.006 0.014 0.006 0.010 0.010 0.032 0,013 0.060 0.004 0.099 0.006 0.055 0.015 0.008 0.006 O.Ol-8 0.007 0.047 0.004 -0.004 -0.000 -0.001 -0.003 0.008 -0.000 -0.006 -0.007 0.002 0.006 -0.003 -0.010 0,025 0.015 -0.000 0.001 -0.000 -0.000 -0.001 0.001 0.006 0.003 0.000 0.000 0.001 -0.000 -0,003 -0.001

-0.006 0.009 -0.006 0,011 0.023 -0.055 0.088 0.277 -0.487 0.431 -0.012 -0.050 -0.026 -0.065 -0.186 -0.413 -0.007 0.033 -0.142 0.011 -0.017 0.027 -0.115 0.009 -0.010 -0.007 -0.030 -0.049 -0.073 -0.058 -0.007 -0.014 -0.026 0.003 0.002 -0.006 0.006 -0.003 0.007 -0.022 0,001 0.001 -0.004 -0.004 0.000 0.001 0.002

-0.001 0.007 -0.006 0.009 0,033 -0.004 0.020 0.018 -0.026 -0.026 -O.OOR -0.017 -0.018 -0.016 -0.027 -0.028 -0.004 0.016 -0.036 0.007 -0.005 0.009 -0.010 0.005 -0.005 -0.010 -0.011 -0.009 -0.020 -0,014 -0.005 -0.005 -0.016 0.000 0.003 -0.000 0.006 -0,005 0.007 -0.012 -0.001 0.000 0.001 -0.002 -0.000 -0.003 0.001

0.001 -0.000 0.002 0.001 -0.001 -0.000 0.001 0.000 -0.001 -0.000 -0.007 0.002 -0.036 0.000 -0.608 0.003 -0.734 -0.019 -0.102 -0.010 -0.008 0.005 -0.001 0.002 O*OOl 0.001 -0.020 0.003 0.007 0.001 0.197 0.004 0.004 -0.002 -0.006 0.002 0.015 o*ooo O.001 -0.002 0.012 -0,003 0.002 0.001 -0.041 0.002 0.003 -0.002 0.012 -0.003 -0.001 0.001 0.000 0.000 -0,039 0.016 -0.030 0.008 0.000 0.004 -0.003 0.002 0.004 -0.000 o.ot"2 -0.002 0.001 -0,001 -0.001 -0.000 -0.002 0.001 -0.004 0.001 c.007 -0.003 -0.007 0.003 0.000 0.001 0.003 -0.001 0.001 0.000 -0.007 0.003 -0.001 0.001 0.000 0.000 0.003 -0.004 0.002 -0.001

1144 1223 2222 1301 0156 0155 0234 0312 0233 0311 0156 0155 0234 0312 0233 0311 0144 0223 0167 0156 0155 0234 0233 0156 0155 0156 0155 0234 0312 0233 0156 0155 0234 0144 0167 0144 0223 0222 0301 0161 0144 0223 0222 01167 0144 0223 0167

345

zosPbSTRUCTURE TABLE 14

The matrix elements F$ ofthe three lowest 5- states JOT E f

= 5

3.396

E

PARITY I

3.E’26

TNLK

TNLK

F(I,K)

FlK,I)

F(1.K)

F(K,I)

1155 1155 1155 1155 1155 1155 1234 1234 1234 1234 1234 1234 1167 1167 1167 1167 1233 1233 1233 1233 1312 1312 1312 1311 1311 1245 1245 1166 1166 1178 1178 1323 1244 1322 0245 0245 0245 0245 0245 0245 0166 0166

1167 1245 1166 1323 1244 1322 1167 1245 1166 1323 1244 1322 1233 1312 1178 1256 1245 1166 1323 1244 1245 1166 1244 1245 1166 1178 1256 1178 1256 1323 1244 1256 1256 1256 0178 0412 0411 0256 0334 0333 0178 0412

-0.168 -0.089 -0.338 -0.077 -0.179 -0.088 0.318 0.287

-0.205 -0.136 -0.333 -0.076 -0.186 -0.089 0.343 0.282

-0.181

-0.131

0.156 -0.233 -0.269 -0.185 0.334 -0.523 -0.230 -0.245 -0.274 -0.215 -0.229 0.229 -0.143 -0.341 -0.284 -0.235 -0.351 -0.296 cI.167 0.075 -0.281 0.167 -0.199 0.186 0.131 -0.337 -0.050 0.078 -0.205 -0.072 0.050 0.204 0.005

0.151 -0.265 -0.229 -0.144 0.305 -0.532 -0.225 -0.213 -0.249 -0.260 -0.233 0.23 -0.120 -0.329 -0.290 -0.210 -0.372 -0.292 0.214 0.105 -0.271 0.132 -0.209 0.164 0.146 -0.338 -0.019 -0.002 -0.156 -0.074 0.063 0.048 -0.041

0.133 0.070 0.279 0.068 0.177 0.081 -0.282 -0.214 0.194 -0.103 0.155 0.178 0.187 -0.319 0.446 0.214 0.204 0.269 0,142 0.174 -0.163 0.149 0.223 0.212 0.237 0.305 0.208 -0.130 -0.058 0.238 -0.163 0.145 -0.153 -0.097 0.281 0.026 -0.038 0.191 0.072 -0.084 -0.103 0.002

0.179 0.162 0.273 0.075 0.191 0.086 -C.327 -0.203 0.104 -0.100 0.221 0.156 0.115 -0.263 0.459 0.205 0.139 0.226 0.171 0.181 -0.153 0.102 0.245 0.186 0.186 0.341 0.197 -0.196 -0.118 0.224 -0.103 0.149 -0.092 -0.102 0.327 0.009 0.004 0.176 0.060 -0.017 -0.150 0.033

1

=

-1 E

I

0.008 0.013 -0.003 0.009 -0.008 0.007 -0.001 -0.024 -0.026 0.023 0.033 -0.034 -0.020 0.016 -0.003 -0.006 0.011 -0.020 -0.031 0.022 0.006 -0.012 -0.004 -0.009 -0.014 0.003 0.013 -0.012 -0.011 -0.003 0.016 -0.013 0.001 o.co7 0.015 0.021 -0.048 0.006 0.020 -0.057 -0.070 -0,009

4.275

-0.011 -0.027 -0.001 0,002 -0.014 0.003 0.019 -0.028 0.013 0.022 0.012 -0.027 0.011 -0.011 -0.009 -0.002 0.032 -0.002 -0.041 0.020 0.003 0.010 -0.011 0.000 0.010 -0.014 0.016 0.016 0.017 0.007 -0.011 -0.015 -0.019 0.009 -0.042 -0.011 0.035 -0.036 0.004 0.039 0.081 0.007

P.RING AND J. SPETH

346

TABLE 14 (continued)

TNCK

TNCK

F(I,K)

0166 0166 0166 0166

0411 0256 0334 0333

0178 0178 0323 0323 0323 0401 0244 0244 0244 0244 0322 0322 1134 1134 1134 1134 1133 t133 1133 1212 1212 1211 1145 1144 1223 1156 1156 0144 0144 0144 0144 0144 0223 0223 0223 0223 0222 0222 0222 0301 0301 0156 0155 0234 0167 0167

0323 0244 0256 0334 0333 0256 0412 0256 0334 0333 0256 0334 1145 1144 1223 1222 1145 1144 1223 1145 1144 1145 1156 1156 1156 1222 1301 0156 0155 0234 0312 0233 0156 0155 0234 0233 0156 0155 0234 0156 0155 0167 0167 0167 0312 0233

-0.011 0.039 0.031 0.032 -0.275 0.126 -0.199 -0.102 0.153 -0.200 0.012 0.245 0,073 0.057 0.176 0.157 0.247 -0.197 0.142 -0.254 -0.183 -0.168 -0.144 0.173 -0.255 -0,199 -0.339 0.172 -0.253 0.198 -0.278 0.1% 0.085 0.058 0.099 0.118 -0.215 0.162 -0.t71 0.191 0.138 0.142 0.306 -0.215 0.229 c.303 -0.198 0.291 0.331 -0.144

FfK,It

-0.056 0.156 0.005 0.021 -0.303 0.141 -0.120 -0.102 0.153 -0.132 0.087 -0.013 0.055 0.055 0.041 0.154 0.249 -0dl93 0.134 -0.182 -0.189 -0.168 -0.223 0.192 -0.197 -0.246 -0.344 0.195 -0.275 0.154 -0.239 -0.013 0.422 0.214 0.272 0.132 -0.221 0.116 -0.154 0.336 0.133 0.133 0.181 -0.258 0.140 0.273 -0.020 0.303 0.303 -0.137

FfI,Kb

fo(,Ii

0.016 0.049 -0.034 -0.124 -0.025 0.002 -0.023 -0.011 0.270 0.233 -0.156 -0.087 0.150 0,166 0.077 0.071 -0.086 -0.135 0.129 0.175 -0.004 -0.046 -0.152 -0.075 -0.021 -0.093 -0.046 -0.054 -0.094 -0.117 -0.135 -0.092 -0.194 -0.194 0.141 0.154 -0.141 -0.128 0.266 0.150 0.146 0.137 0.122 0.125 0.119 0.250 -0.148 -0.178 0.245 0.159 0.161 0.231 0.279 0.284 -0.138 -0.159 0.226 o.zs9 -0.200 -0.125 0.283 0.211 -0.111 -0.095 -0.124 -0.130 -0.040 -0.197 -0.102 -0.190 -0.104 -C.l22 0.182 0.233 -0.187 -0.075 0.152 0.135 -0.152 -0.309 -0.090 -0.173 -0.153 -0.112 -0.285 -0.137 0.173 0.259 -0.216 -0.212 -0.290 -0.297 0.105 0.124 -0.245 -0.298 -0;319 -0.245 0.171 0.090

F(l,Kt

FfK.18

-0.009 -0.002 -0.025 -0.009 -0.067 0.051 0.009 -0.002 0.076 -0.006 0.034 -0.112 0.041 -0.003 -0.014 -0.046 -0.004 0.007 O.G22 -0.052 0.002 0.006 -0.009 0.008 -0.039 -0.003 -0.000 -0.004 -0.004 0.@24 -0.024 -0.064 0.018 O.C5I O.GO3 O.C@3 -0.001 -0.019 -0.004 0.291 -0.002 -0.OC8 -0.176 0.003 -0.008 0.000 0.062 -0.007 0.061 -0.032

0.006 0.008 0.030 0.015 0.043 -0.041 0.002 -0.016 -0.049 0.021 -0.048 0.129 -0.060 -0.013 0.009 0.071 -0.003 0.002 0.017 -0.007 0.006 0.005 -0.060 0.019 -0.006 -0.030 -0.002 0.004 -0.022 -0.007 0.006 0.075 -0.004 -0.056 0.016 -0.009 -0.006 0.019 -0.054 -0.187 0.006 0.007 0.265 -0.016 0.014 0.027 -0.072 0.028 -0.031 0.022

347

zOsPb STRUCTURE TABLE 15

Particle-hole

amplitudes of three very collective 2” states

JOT = 2 E =

4.489

P4A!TY = +1

E = Il.065

E = 16.981

TYLK TNLK

X(I,K) Y(IvK)

X(I,Kb Y(I,K)

X(I,K) Y(I,K)

1155 1155 1155 1234 1234 1234 1234 1167 1233 1233 1233 1233 1312 1312 1312 1312 1311 1311 1245 i245 1245 1166 1166 1178 1323 1323 1323 1323 1323 1244 1244 1244 1244 1322 1322 1322 1322 1256 1256 0245 0245 0245

d.008 -C.G37 -0.939 -r?.o22 O.lCb G.rt19 0.037 r! . ,3 22 -G.?18 -S.C>5 0.352 (3.026 -c).411 -r?.384 -3.114 -1?*051 -3.Gr39 -0.1709 -0.029 -C.017 -a.017 -C.@lG -0.r335 -0.019 0.025 O.rj18 -,3.013 -0.~05 3.321 G.012 -0.024 -G.OlZ -0.922 -O.r?lZ -C.620 -0.013 -c,.GciG-0.025

3.539 -9.003 c.125 0.011 C.352 Q.r)GZ -G.G45 -rJ.311 0.'?92 O.GG6 -0.114 -cl.017 -0.154 3.027 0,285 ?*G3B 0.024 G.GGl 1?.(?35 C+.\308 c1.048 O.GO3 0.087 cl.\>09 -3.033 -0.00' C.GQ6 F.OC4 -0.032 -O.V)h C.G27 O.GO7 G.Gl8 C.007 G.337 6.004 0.047 F.G?2 -3.002 -G.OG6 Q.192 O.G22 -n.c!35 0.001 -0.159 -g*l)Iti -0.251 -r1.046 0.322 G.W? -0.'117 -?.002 0.044 3.008 -0.023 -r).GC5 3.c7-f 0.011. -0.C18 -13.00G -0.042 -3*011! -ci.338 -C.GC3 -0.115 -,I.014 -0.019 -5.006 -0.023 -c*cv?3 -0.c34 -0.C06 -c1.056 -'3.@37 0.011 n.Oe7 -c).r)43-,1.Gl.2 -?.CGO -0.CO5 0.119 O.GZ5 0.138 -0.036

0.183 -0.005 3.146 -0.014 0.003 O.CQ9 0.019 0.001 -0.013 -r).cJtto -3.081 d.002 D.Ci)O -0.008 0.146 -0.010 -3tcM3 -O.COl -c.o30 -O.COl -0.7'35 -3.C36 -0.013 -0.CO6 3.030 -0.D32 -C*.Oll. 0.232 -2.lSl 0,303 G.fB9 0.305 c1.009 G.c\02 -(3.3ql -O.Cil G.012 0.001 -C.?14 -0.co2 0.1;)s 0,002 -3.018 O.GC’b -0.199 G.Ql5 -0.214 a.009 0.040 -0.001 0.799 -r).oo4 ?.Q55 -0,002 -cI.G34 -O.CGl -3.1331 -a*003 -C*G13 0.332 -C+.oo4 -0.COG 0.331 G.008 5.026 0.511 3.GG7 -03,334 -3.125 3.093 -c.o43 0.301 -0.D68 -0.003 it007 0.034 -3.025 -0.302 C.lS(! 0.301 -0.214 -O.ZDl -0.OQ6 -13.013

1134 1133 1156 1134 1133 1212 1156 1145 1134 1133 1212 1211 1134 1133 1212 1211 1133 1212 1145 1144 1223 1145 1144 1156 1145 1144 1223 1222 13Gl 1145 1144 1223 1222 1144 1223 1222 13Gl 1134 1156 Cl44 0223 Cl67

0.017 -0.4>62

G.035

-0.331 C.GC8 C.338 0.058 2.534 1?.134 *3.x3 -0.013 -0.913 O.CO6 c.cc3 -3.022 -0.n12 ,?.rz13 G.r>*M -Q.r326 -6.913 7.307 0.038 3.935 0.921 3.014 0.308 0.047 0.326 3.G19 c.n11 7.013 O.CO7 3.318 O.ClG O.r)lst O.Gll -0.019 -C.Q12 O.C)36 0 .rs 23 ci.027 -C.318 -3.1?355-C)r027 ~I.876 2.106

P. RING

348

TABLE

AND J. SPETH

1S (continued)

TNLK TFJLK 0166 ol66 0178 0323 0323 0323 0323 0401 0401 0244 0244 0244 0322 0322 0322 0322 0412 0412 0412 0412 0411 0411 0256 0256 0256 0334 0334 0334 0334 0334 0333 0333 0333 0?33 0333

QL44 0167 Cl56 0144 C223 0222 0301 0223 0222 @l/*4 0223 0222 0144 0223 0222 0301 0234 0312 0233 C311 C312 0233 0256 Cl55 0234 Cl56 Cl55 C234 c1312 c233 0155 C234 t\312 C233 0311

-?*I49 -3.930 7.044 0.930 '3.417 -5.001 3.068 -O.COl -C,436 0.002 t*025 -0.0~0 -0.029 -i),DciO 0*008 -Q.OOl -r5,02a -0.000 -0.012 -0.001 *?.215 -Q.018 -3.027 -3.01r) 5,053 '3,002 0.313 -0.002 0.347 -0.001 -C.O25 -0.032 -C'.r)'"lS 0.006 -0.003 0.3OL 0.020 t3,DOl 0.001 ia*3f0 -C.O62 -r)*OO2 c.024 -Q.CfJl 3.023 ').CO2 0.030 r).c??i P.192 -C*330 o.oro 0,307 3.059 3,302 -0.007 r)*C31 -3.0;)s 0.032 @*D12 (3.OQl 7.108 -0.001 -0.0'113-O.c!Ol o.:m5 -D*CGO '3,006 -D*CDf. Q*w36 -cl*nor

349

TABLE 16 The matrix ebments

Fg of three very cdlective

JOT f~:2 E =

4.489

2f states

P4RITY = +l

E = 11.065

FfI,K) FfK,IB

TNLK TNtK 1155 1155 1234 1234 1167 1167 1233 1233 1312 r3t2 1245 1245 1166 1166 1178 1178 1323 1323 1244 1244 1322 x322 1256 1256 0245 0245 0166 0166 0178 01713 0323 0323 0244 0244 0322 0322 0412 0412 0256 0256 0334 c334 0333 0333 1134 1134 1133 1133 1212 1212 1145 1145 1144 1x44 1223 1223 1156 1156 1222 1222 0144 Cl44 9223 0223 0222 c222 0156 0156 0155 0155 0234 0234 0167 0167 tt312 Q312 13233 0233 1155 1234 1155 1233 1155 1256 1234 1233 1234 1312 1234 1256 1167 1245 1167 1166

f = lb*951

-0.550 G-498 -1.:326 -0.453 0.277 -0.5OO 0,768 1.236 -0.271 0.508 3,217 0,443. -3.6X0 0.64X 1:'187 -0.347 0.573 0,272 Q.078 0.576 il.1913 -5.154 5.335 -0.2% +1*280 -C.555 Lt.359 -0.424 3.793 0.323 0.342 -0,392 0,276 O.TQti -0.475 .3.534 -0.898 0.3tt -0.446 -0.116 -0.223 -0.659 -0.759 0.071 G.142 -0.173 -0.134 0,471 0,462 -0.537 -3.511 -rj,980 -0.357 0.223 r3.119

0,471 -0.440 0.801 0.379 -0.305 0.506 -0.420 -0.957 0.311 -0.475 -C.246 -t3.509 0.630 -3.729 -1.129 0.335 -01589 -0.264 -0.102 -0.595 -0.238 0.197 -0.322 0.222 -0.233 0.472 -#.33A 0.341 WC.637 -0.255 -C*4?1 (3.429 -0,308 -0.785 Q.569 -0.56J 0.961 -G,287 Cf.483 0.218 O*?x?4 0.551 0.36R -0.109 -0.021. Q.C96 '1.220 -G.370 -0.462 0,573 0.491 C.559 3,762 O.f.?62-0*33x

-0.342 r?.18", -0.381 -3,155 0,~21 -0.2c.6 0.366 3.380 -0.020 n* 177 3.015 n.200

0.233 -3.516 -0.537 0.040 -0.151 -0.92Q -c.c\20 -0.213 -3,037 !?.026 0.277 -3.243 1?,050 -0.341 5.303 -0.124 2.368 5.093 -c*4'33 3.152 -0.114 -0,516 3.475 -2.206 0,538 -c!*062 0,175 3.050 -01078 -0.041 -0.152 -O.lCO 0.059 -0.314 0.194 3.319 0.042 -13,038 -r).380 -3.185 0.092 3.229 -3.337

350

TNCK L233 r233 1312 1245 1245 1245 iL66 1178 1323 1323 1244 0245 3245 3245 Cl66 3178 C323 0323 0323 0401 0244 0412 0412 0412 0411 0256 0334 1134 1134 ll34 1133 1133 1212 1145 1145 1144 1144 1223 1223 1222 0144 0144 0223 0223 0222 0156 0156 0155 0155 0234 0234 0312 G312 0233

TABLE16 (continued)

TNLK 13t2 1311 L3lf 1166 1323 1244 1244 l25& 1244 I.322 2322 Cl66 0323 0244 5244 0256 0401 0244 0322 0322 0322 04ll C334 0333 0333 0334 0333 1133 1212 lL56 12L2 s211 1231 1144 l223 1223 1222 1222 1302 13OL 0223 0222 0222 0301 0301 0155 6235 0234 Q233 03L2 0233 C233 C311 0311

F(IfK) F(K,I) -0.188 -G*ZG1 -0137L -0,381 -6.287 -3.281 a.199 C.104 -0.453 -0.427 0.133 0.153 3.782 Q.377 0.892 0.826 &I40 OelZ4 C.141 t3*13E a.390 C.411 Il.372-Q.i54 -r).547

-C.

506

3.161 c.173 5.381 0,327 'S,Q5r7 c.5f5 -*3.3:x -r?,289 3.19(? 2.114 O.ZA7 Q.164 0.261 9.231 33.432 C.504 -O.3HO -?*773 C.laq 0.153 -C*clrjft -c.'35c -0.,1("9- ,J. 'I 99 0.346 ?.28C -0.072 -II.?359 -C*lCili-.3.355 c.470 c.410 -0.b24 -3,675 -i3.11?.h -?.I?5 -rj.266 -r.315 -(f.zf35-0.264 0.147 3.107 -3,737 -0.638 13-124 0,223 43.472 0,555 0.214 0.2?)9 -0.379 -0.374 0.302 0,311 -0.327 0.259 0.242 0.447 93.232 0.143 -r)r463 -111.335 0.243 0.329 -0.131 -3.131 3.897 0.436 C.110 -3.341) -i1,3r)?-0,666 0.543 5*49? -0.209 -r).157 -01245 -0.169 -Q.332 -c.300 -0.36? -0.426

flItK)

F(K,II 0.215 C.119 0,358 12.282 9.265 0*351 -0.310 -or194 Q.478 014Q4 -0.C80 -0.198 mC.671 -01466 -??.581 -O*?l9 -C*cJ87 -q.L67 -0,120 -O*l9& -c.433 -3.369 o.c1>1 -6.1?'3 0.423 C.566 -ct.065 -0.279 -0.678 -t).46f; -0,555 -2.747 0.276 fI.338 -0.073 -0,213 -3.101 -r).1?39 -0.214 -0*29C -0.506 -3.383 c.072 0*129 -01210 -'3,180 C.C63 c9,cmz 0.131 n,152 -0.271 -0,349 ci.o4(? C.Xlf? c.014 r).186 -C*237 -0.412 p.687 7+42') o.200 c1034 0.278 *J.155 (I.159 .I.300 a.013 -r).239 0,357 0.553 -"1*22t7-C*t112 -0.444 -;3,251, -0 .c97 -‘).240 C.238 r1.329 -0.282 -0.167 -0.216 -r),C2rl -0,445 -0,273 -c'.112 -0.303 0.356 13.463 -0.384 -ij*261') -0.022 0,324 -0.516 -0,775 0.216 0,004 0.585 (3.376 -0.455 -fJ3,602 G.W2 0.3co 0.139 0.204 0.188 0~398 C.464 0.351

F(I,KI F(K,I) C.079 -0.221 -0,320 -".258 0.097 -0.143 n.113 -0.110 [email protected] -0.110 9.275 -9.168 -C.O48 0.165 cr.132 -0.155 *C*B40 i3.EilS -2.1516 0.036 C.c)72 3.116 -c.127 0.113 Z.288 9.032 -C,334 (3.19G ',i?!sB-0.279 -O.'l64 3,'384 r1,1,39-0.024 -'.I48 0.872 -3.112 cl.093 -0.136 3.350 -‘3**IIl6 -0.212 0*059 -r)*O18 -C,Cr)7 -0,098 0.364 -0.023 0.071 0.005 -0.1CC 0.308 0.080 -0.057 -C.f88 -5.015 0.175 0.164 -n.295 -0.501 -or075 -0.053 -C.l38 -C.!22 -C.Q5C -0.047 0.187 -0.co5 -0*191 -3.100 c.021 0.044 OIll6 0,130 C.245 -3.123 -C.3"'4 C.G46 -n*127 0.391 C.160 -0.289 -G*OIZ -0.435 -C.321 0.172 i7.359 C.Cl4 3.059 -3.357 C.h46 -0.421 -2.441 0.044 -0.157 0.220 -3.028 0,400 -11,400 -0,351 0.335 -O*LV5 0,256 -0.082 0.205 -0.377 0.034 a.301

208Pb STRUCTURE

351

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29)

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