The non-normal parity states of 209Bi and 207Pb

(1971) 97-123;

Nuclear Physics Al73

@ North-Holland

Publishing

Co., Amsterdam

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

THE NON-NORMAL

PARITY STATES OF *“Bi AND *“Pb

K. ARITA t and H. HORIE Department

of Physics,

Tokyo Institute

of Technology,

Oh-okayama,

Meguro,

Tokyo

Received 25 March 1971 The low-lying non-normal parity states of “‘Bi and 207Pb are studied by means of the shell model, assuming lh-2p and 2h-lp configurations, respectively. The observed energy spectra are well reproduced. The wave functions obtained are analysed in comparison with those defined in the weak-coupling model and existence of the octupole multiplet and the mixing with other states are discussed in detail. The conditions for the existence of the weak-coupling states are also discussed.

Abstract:

1. Introduction Recently, which

many

demonstrate

experiments that

have

been

the shell model

carried

works

out for nuclei

fairly

well and that

in the lead there

region,

are various

of simple and characteristic excitations ‘). In this paper we study several feawith tures of the low-lying states in odd-mass nuclei 209Bi and *“Pb in connection those of adjacent even nuclei. Inelastic scattering of various projectiles on these nuclei “-“) leads to populations of close multiplets, whose excitation energies are approximately equal to those of the lowest 3- and 5- states of neighbouring doubly closed shell nucleus *‘*Pb. The angular distributions and the sums of the transition strengths to these multiplets are almost identical with those to the corresponding states in “*Pb. All these experimental facts strongly suggest that these multiplets are formed by the weak-coupling ‘) of a single particle or a single hole to the excited states in 208Pb. It is well known that the 3- state is the prominent collective octupole state lo) and a fairly enhanced E5 strength has also been observed for the 5- state ‘*“). These core states are described in coherent particle-hole correlation. The spectrum above 2.7 MeV in *“Pb obtained by the (d, p) reaction on *O’Pb shows close similarity to that of the low-lying levels of *“Pb [refs. “,‘*)]. These states in “‘Pb have spectroscopic factors characteristically close to unity. So, it is expected that the addition of a single neutron in the particle-orbit does not disturb the ground state of the 2o 6Pb core, which may be dominantly in the two-hole correlation. The same kind of states in 209Bi , formed by the coupling of the single hole to the two-particle correlation, have not been observed experimentally, i.e., the reactions preferentially exciting to the states have not yet been achieved. kinds

t Present address: Department of Physics, College of Science and Engineering, University, Megurisawa, Setagaya-ku, Tokyo. 97

Aoyama Gakuin

98

K. ARITA AND H. HORIE

The energy spectra and electromagnetic properties of the octupole multiple& in odd-mass nuclei around “*Pb have been theoretically studied by several authors. Important effects of the pairing force and antisymmetrization for level structures of the multiplets are pointed out by Hafele 13). He assumes the product states of the single-particle and the 3- state, for which the wave function calculated by Gillet et al. [ref. “)I is adopted. They are also studied from the viewpoint of the collective model r4, “), assuming the energy and E3 transition probability of the 3- phonon. In this work, we investigate these states by the usual shell-model calculation without assuming weak coupling. We solve the secular equation constructed in a restricted shell-model space, assuming the residual interactions which have been made use of in the calculations for neighbouring nuclei. Then we can examine the energy spectra and the general features of the wave functions. There are, however, some difficulties in the analysis of the E3 transitions from the octupole states to the ground state by the shell model. It is shown by Pinkston 16) that the calculated value of the reduced transition probability in “aPb is too small by a factor of about 20 compared with the observed one and that any combination of his limited lh-lp configurations cannot reproduce it. Gillet et al. 17) point out that the disagreement can be improved by including all lh-lp configurations of Ihw and by taking into account mixing of excited configurations in the ground state. However, 3ho states can also affect it in spite of small mixing, because a number of the states with large transition matrix elements can contribute constructively, so a larger space should be required for the correct description of the E3 transition. Even if this kind of calculation which takes into account the 3ho excitations is possible in closed-shell nucleus, it will be almost impossible to extend the calculation to nuclei with a valence particle or hole. In the present calculation, orders of the shell-model matrices for “‘Bi and ” ‘Pb are considerably large, nevertheless the+corresponding configurations describing the E3 correlation in “*Pb are only 21. The difficulties pertaining to prediction of the E3 transitions remain in these odd-mass nuclei. In this sense, our shell-model space of “‘Bi and ‘07Pb is not sufficient. We can, however, admix the excluded configurations into the wave functions obtained in this calculation. The contributions of the excluded configurations to the transition probabilities are examined by perturbation theory in lower order under the assumption that the mixing probabilities of the configurations are small enough. The difficulties mentioned above are removed by this procedure and quantitative discussions concerning E3 transitions are made possible. We present here the discussions with respect to energy spectra and the weak-coupling aspect of the individual states. Their electromagnetic properties will be studied in the forthcoming paper. The assumptions in shell-model calculation are given in sect. 2. In sect. 3, model wave functions for weak-coupling states are defined to compare with the shell model. Numerical results and comparison of these two calculations and detailed discussions are summarized in sect. 4. Some remarks are given in sect. 5.

209Bi AND

zo7Pb

99

2. Method of calculation The simple spectra of single-particle or hole states 18-2o), the spectroscopic factors of which are almost of single-particle limits, are observed in 207Tl “‘Bi, 207Pb and 209Pb. Those spectra, together with the binding energy data 21) df these nuclei and of 208Pb, give the sing 1e-p article energies in the region. The ground states of 209Bi and ‘07Pb are a h, proton state and a pt neutron-hole state in good approximation, respectively. Except for these single-particle or hole states, the excited configurations of “‘Bi and ‘07Pb are obtained by raising particles from the doubly closed shell “‘Pb. We adopt all possible configurations with unperturbed energies up to 7 MeV, considering we are concerned with the low-lying non-normal parity states at about 3 MeV and that under the same assumption the spectrum of “‘Pb in the energy region is well reproduced, as will be shown. Two-particle excitations from the “‘Pb core of nonnormal parity can occur at above 8 MeV from the single-particle estimate, so the chosen configurations are those obtained by raising a particle from the closed shell and promoting the odd particle (hole) to excited single-particle (hole) orbits. Single-particle states with $+, q,’ and q?.’ are also found in “‘Bi. We assume 6.7 MeV and 7.0 MeV levels above h, orbit as 2g, and li, , respectively. The singleparticle energy of li, is taken higher than the observed one by 0.46 MeV, because of known admixtures of excited configurations into it 4*’ “). Consequently, the shell-model basis in the present calculation are single-particle states and lh-2p configurations in “‘Bi and 2h-lp ones in 207Pb, each with unperturbed excitation energy less than 7 MeV. The maximum dimension of shell-model matrices is 193 x 193 in the e+ state of “‘Bi. Coupling schemes of angular momenta in the lh-2p and 2h-lp configurations are taken as I~-‘P,P2

(4);

JM),

and K%‘(J&;

JM),

respectively, where h and p stand for single-hole and single-particle orbits. The harmonic oscillator function is assumed as the single-particle wave function for convenience in calculating a matrix element and separating a spurious state due to the c.m. motion. The configurations taken here involve more than two open major shells of proton or neutron, so attention should be paid to a spurious state due to excitation of the c.m. motion. We assume the harmonic oscillator model for the spurious state 22) and examine its mixing into physical states, though it is not clear whether the assumption is adequate in this region, where the one-body spin-orbit force ip so strong. Since the closed shell of 208Pb is in thejj-coupling scheme, the situation is complicated. Various kinds of the states can occur, the simplest one of which is generated by a c.m. excitation operator A +, acting on a negative-parity single-particle or single-hole state of

100

K. ARITA AND H. HORIE

*OgBi or ‘*‘Be, for example in ‘09Bi, y,, = [A+ x al]:12’*Pb

core),

and p is single-particle orbit, a, being the creation operator for the orbit. Here, v is defined as mm/R, where hw is the quantum of the harmonic oscillator, This kind of state consists of a single-particle state and 1h-2p configurations. The spurious states related to the p = h, state have their intensities projected into the restricted shellmodel space of about 0.5 in J = $+, 4’ and 9” states and are the only ones with appreciable intensities in the space. The states not having full intensities in the space, it is impossible to separate the true states from them exactly and some approximation is required. The calculations have been carried out in two different ways, one without any caution about the spurious states and the other with taking only the linear combinations orthogonal to the projected components of the spurious states into our space, i.e.,

where P is the projection operator into the space and the normalization given as N-2 = (YS,lPl Ysp),

factor N is

which provides at the same time the projected intensity into the restricted shell-model space. In the first method of calculation, the states can mix into the physical states and in the second, some of the physical states are removed. From the calculated result of “‘Bi using the first method, the states with large intensities of the spurious states appear at about 2.2 MeV commonly in $‘, 4’ and y’ states. Whether they are spurious or not is known by examining the wave functions and by observing if the states disappear in the second calculation. The energies and wave functions of other lowlying solutions scarcely change in the two calculations. Consequently it can be concluded that mixing of the spurious states to the physical states is so small and the unduly removed physical components are unimportant for low-lying states in this case. The same argument holds also for 207Pb, where the spurious states generated by A + acting on 3p+, 2f+ and 3p, single-hole states are considered. Only the results of the calculations by the second method are shown in both nuclei. Most configurations are in a pure isospin state with T = *(N-Z), and exceptional ones are in the $+ and y+ states of ‘09Bi, where gS and i, single-particle states and lji ‘j,&(J,); J = 4) with],, =&, appear. Isospin has been coupled in the 9’ state. From the calculated result, however, the effect in both energies and wave functions of the low-lying states is found to be very small. This is because the states with higher isospin are few and they have higher energies.

209Bi AND

The energy matrix element <~-‘~I

P&?)IWr’-

between

*O’Pb

101

lh-2p configurations

is given by

‘Pi IW;>>.7 hY(I-5 3P;j%,

= (%I +sP* -%)W, 1

3P;w,

YJ;)

[42J,+1)(25;+1)

+ J(1+6(P,,Pz))(l+6(P;,P;))

x { g (2Jc+ 1)wp,

JP, ; J, JPj wh’p; * PzJ)WPi

+ ~(2Jc+wTPlJpJc~~ - ;(-1)

P1+Jc-J(2Jc+l)W(pI

J,J,h;

JP; ; J, J;)+P,

1uh’~;)J~

a(~, ypi)

JLJC h’; Pi JKhPA ~‘l~‘P;>.d(PI pzJ)W(h’p;

Jp;;

JcJ;)

x
pj

J,J,)W(p;

P”+J=J(2J,+1)W(hpIJpz;

3 Pi)

>Jc

6(P,

J;J&p;J) x #PI IV’~h’~;>JcG(~2

+

Jp6(k

h’ja(J,,

9 P;j

3 Pi))

J;)l,

where E is single-particle energy of each orbit and (p1p21 Vlp;p; > and (hpl Y’lh’p’) are antisymmetrized matrix elements of particle-particle and hole-particle interactions, respectively, defined by J

J

and {hpliqh’p’),

E (h-‘plVlh’-‘p’)J = _( _ l)h+p+h’+p’ 7 (2J’+

l)W(hpp’h’;

JJ’)(hp’lVlh’p)J,.

The matrix element between 2h-lp configurations can be written in an analogous but the particle-particle interaction should read hole-hole interaction. The matrix element between the lh-2p configuration and the single-particle

(~~IVI~-‘P,PZ(J~)>J=,

( - 1)2’s = dl + 6(p, , pz>

25 +l fz

(4s hl ‘h

form, state

pz)J, *

Two-body interactions are referred to the studies of neighbouring nuclei. The central potential used by Carter et al. 23) which acts only in an even state of relative motion is assumed as a hole-particle interaction except between the neutron hole and the proton particle. The states of “‘Pb are calculated with the potential and Coulomb force, taking Ih-lp configurations with energies less than 7 MeV. The calculation is almost the energy of same as the one done by True and Pinkston ‘“) except the single-particle

102

K. ARITA

AND

H. HORIE

the 2d, proton-hole orbit is taken higher by 0.6 MeV than their assumed value. The obtained spectrum and wave functions are almost identical with theirs. The energy spectrum is quite well reproduced as shown in fig. 1. Neutron and proton configurations of these states have been studied in various experiments 26-28), whose results are qualitatively consistent with the calculated wave functions. 6-

(67

3.961 3.749

5-)

m

4075

7_

w

5-

3700

4-

3475

4-

3639

5-

3198

'-

3277

3-

2615

3-

2.705

0’

0'

Expt.

Cal

2G8Pb Fig. 1. Energy spectra of 208Pb. The experimental data are taken from ref. 25). The calculation is based on the shell model, taking all lh-lp configurations with unperturbed excitation energies less than 7 MeV and assuming the potential of Carter et al. 23) and the Coulomb force as the residual two-body interaction.

In 209Bi all other matrix elements are calculated using the Kim-Rasmussen force [ref. “)I ihich is obtained from analysis of states in “‘PO and ‘loBi and also describes well those states 3oS31) in 208Bi . The Coulomb force between protons is also taken into account. True and Ford 32) explain the levels of “‘Pb with a pure singlet-even potential, which we assume as the neutron hole-hole interaction in 2o ‘Pb. The Kim-Rasmussen force is also assumed as the interaction between the neutron-hole and the proton-hole. Ratios I of the force ranges r. to the range constant of harmonic oscillator well (A = r,JG) are taken to be the same as those used by these authors. In our preliminary report 33), all ;1 are determined using the value L’= 0.1842 fmm2 which is the value used by True et al. 23) and the Coulomb force is neglected in hole-particle and particleparticle interactions.

‘09Bi

AND

103

*“‘Pb

Elements of shell-model matrices are calculated with these interactions, to eliminate the spurious states and isospin impurity and diagonalized final results. They are shown in figs. 2 and 8 and discussed in sect. 4.

3. The weak-coupling

transformed to obtain the

model

In order to study the structures of the calculated spectra and wave functions in the last section, we will assume the model wave function of the weak-coupling multiplet and calculate the energies with the wave function. Under the assumption that the low-lying negative-parity state lJ, MC) of the doubly closed shell nucleus “‘Pb can be described by lh-lp configurations, namely IJCM,)

= c x(hP)lh-%; hp

the following product wave function vibration coupled state: IJ, 4; JM) Ih-‘p(J,)q;

JM)

of a core and a particle is defined for the particle-

= NJ c x(b)lh-‘dJ,)q; b = ((bl

JCM,),

JW,

x u~)‘“~’ x a:)g)1208Pb

(3.1) closed shell),

where q is single-particle orbit of the odd particle and b+ and a+ denote the creation operators of a hole and a particle, respectively. The phase convention used here is the usual one, bh’, = (-l)h+muh_m.

The normalization

factor N;’

NJ is given by = l-(2J,+l)

1 x2(hq)W(qJ,J,q;

Jh).

h

The wave function of the hole-vibration coupled state is defined quite analogously and, henceforth, we write the expressions for various quantities only of the particlevibration coupled state. Description starting from the wave function defined by eq. (3.1) is called the weak-coupling model. It might be noted that the definition of the product wave function is not unique. In eq. (3.1) the state is assumed as the superposition of non-normalized bases and normalized as a whole. Instead, it is also possible to assume it as the superposition of normalized bases. The normalization factor as a whole is again necessary in this definition, since some bases can possibly vanish due to the Pauli principle. When the effect of antisymmetrization is not large, these two wave functions give similar results and when the effect is large, coupling between the particle (hole) and the core is not weak, so that the weak-coupling wave function has little significance. We employ the first definition and do not discuss the difference any more.

104

K. ARITA AND H. HORIE

The product

wave functions

are not orthogonal
with different core states and/or different particle (hole)

to each other. The overlapping

integrals

are given as

= N., N;[G(c, c’)6(4,4’) - J(2J,

+ 1)(2J:. + 1) 7 &(h4’)&04)

W(4J,J:.

4’; Jh)],

(3.2)

where c and c’ specify quantum numbers of the core states including the angular momenta J, and J,,, respectively. The non-orthogonality appears again as the effect of the antisymmetrization between the odd particle (hole) and the core. The energy matrix element between the wave functions is given,
= N, N; G

x,(hp)x,(h’p’)(h-‘p(J,)qlHlh’-‘p’(J:)q’),

,

(3.3a)

h’p’

where

(h-‘p(J,)qlHlh’-‘P’(J~)~‘)~ (1) = (Eq+Ep-E&(h,

(11) = 6(p, p’)J(2J,+

= (I)+(II)+(III)+(IV),

(3.3b)

h’)

1)(2J;+

1) c (2J’+ l)W(qJ,J’p; J’

Jh)W(q’J:J’p;

Jh’)

x (h4l V’lh’4’),*, (III) = -J(2JC+1)(2Jf+1)(6(4,

p’)W(qJ,J:q’; +d(&

(IV) = 6(h, h’)J(2J,+1)(25:+

1) X(25’+ J’

Jh')(hplV'lh'4')Jc

4’)w(4&&4’;

l)W(hpJq;

Jh)
J,J’)W(hp’Jq’;

.CJ’) x
The third term (III) measures modification of the hole-particle interaction in the core resulting from the Pauli principle and we call this the blocking term. The last term (IV) is the particle-particle interaction between the odd particle and the one excited in the core. All terms receive the effect of the antisymmetrization through the normalization factors. Assuming (3- x h%)J and (5- x h,) J states in 209Bi and (pi 1 x 3-) J and (Pi ’ X 5-) J 207Pb, we calculate their spectra by taking the expectation values of energy, e.g., for 2ogBi 2 in

EJ =
-

The core states and interactions are assumed as those mentioned in the previous section. It may be clear from the restriction of the shell-model spaces in 209Bi, 207Pb and “*Pb that all configurations in this weak-coupling model calculation are involved in the shell-model space assumed in the last section. The resulting spectra are shown in the next section.

2osBi AND

105

207Pb

The wave functions obtained in the shell-model calculation are studied, taking overlap with a product state.

4-l)

P1+PZ-JpQw2)

WbzJp,;

J,J,)el,

(3.4)

PI)).

The product state of single-hole to the two-particle correlation, e.g. d; 1 x ” ‘Po(O+), is built as Ih-rc; JW

= c xc(P1 p&h-‘p,

pz(J,);

(3.5)

JM).

PIP2

The overlapping integral of this state to the wave function (3.1) will be obtained straightforwardly from eq. (3.4). If several weak-coupling states or some other configurations lie inclose spacing, they generally mix with each other. The energy spectrum resulting from the mixing is obtained by solving a secular equation constructed by the states. But, in contrast with the orthogonal bases in the shell model, these states form a non-orthogonal set, as mentioned above. So, the states should be orthogonalized with the methods developed by Schmidt or others 34). Th e secular equation is constructed in the orthogonalized space. From now on, the kind of calculation is sometimes called the intermediate-coupling model. A coupling matrix element of the state described by eq. (3.1) to a single-particle state qEis given by (3.6) A matrix element between the states (3.1) and (3.5) is calculated from the following equation W1p(J,)qWIK’~,

~z(Jp)h

=

(2J,+

1)(2J,+

1)

1 -I-d(P, 9 Pz)

x Cd@, ho)W(hpJq;

J, Jp)

x {(&q+Ep-%)(qPY

P&%4, Pz)-(-

1) pl+pz-Jp~(p~ p2)6(4,

PI))+

@.d

vh

pZ)J,

+%4, Pz)W(hop~ JPZ; J,J,)
p1+p2-Jp~(4, PI)W@OPZ JPI; JcJ,)
-a(~,

~2) 2

+(-I)

t2J:.+l)W(~JcJ:q; J’C

hJ)W(hop,

P1+P*-Jp%~, PI)~(~J:+~)~+‘(PJ,J:,~; E

V’1hOp2)J, JP,; J:J,)~~ hJ)W(ho~zJp,;

J:Jp)

x 041 V’lhO pZ>J’,1*

(3.7)

106

K. ARITA

AND H. HORIE

The (3- x h+), and (5- x h%), states can couple to each other in J = $‘-“,‘-’ states and the gt, i, and i, single-particle states can also couple to them. The secular equations with these states are solved. The normalization factors of the states and their mutual overlapping integrals are summarized in table 1 and the calculated spectra are shown and discussed in the next section.

Normalization

TABLE1 factors and overlaps of the product wave functions in *OgBi and 207Pb

209Bi J

4’

t+

9’ .- _ .-..-.

~.__

N,‘(3-

xht)

N,‘(5-

x ht)

I .oo

<3-xh*15-xhg>

8’

8’

y+

q+

+P+ -

y+

1.00 0.99

1.02 0.96

1.07 0.90

1.21

1.01 0.99

1.02 0.97

1.04 0.95

1.08 0.91

-0.03

0.04

-0.06

0.06

‘+ t

3’

8’

%?+

1.03

1.02 1.00

2.59

-0.07

0.06

y+

1.17

-0.05

=“Pb J N,‘@+-‘x3-)

N,‘@+-’

x5-)

The expansion of the wave function Y in the non-orthogonal coupling states {4i} is carried out as follows: Y = Caitpi+!P’, I where Y’ is orthogonal to the set {&}. If (4i} IS ’ 1inearly a, are determined by solving a set of linear simultaneous

C i

U,jaj

=

Oi,

set of the weak-

(3.8) independent, equations,

the coefficients

(3.9)

where and

The total intensity

G expanded

in {+i} is given as G = C aiOi, I

(3.10)

which is less than or equal to unity. The equality G = 1 holds, only if Y’ = 0. The coefficients a, are not necessarily less than unity. If only one weak-coupling state is concerned, G is reduced to a square of an overlap Oi.

zogBi AND

107

*O’Pb

4. Results and discussious 4.1. THE

NUCLEUS

zosBi

The calculated spectrum of zo9Bi due to the shell model described in sect. 2 is compared with the experimental one in fig. 2, in which the known single-particle states are omitted.

-

S-

3198

s-

328

-4

:1/z*

272

,Gzi

TA

712.'

'269

3-

I

271

-I 26 \ I

9/z-

0’

‘“Pb

2ogBi Expt

_r

0’

912‘

‘“‘Bi

“*Pb

Cal

Fig. 2. Experimental and calculated spectra of 2ogBi. The lowest 3- and 5- levels in zosPb are also shown for the sake of comparison. All the energy values are in MeV. The spacing to the ground state is not to scale and the energy scale of the calculated spectra is fixed so as to align the calculated 3level with the observed one. The experimental data of the seven levels below 2.74 MeV are taken from ref. 2, and those of the next three levels are from ref. ‘). Tentative spin assignments to higher levels are given in ref. 4). The calculation is based on the she11 model, taking all possible lh-2p configurations with unperturbed excitation energies less than 7 MeV and some single-particle states. But, in 4’ states, the shell-model space is enlarged as mentioned in the text.

Two groups of closely spaced levels at about 2.6 MeV and 3.2 MeV are reproduced and it is easy to find the correspondences between the calculated levels and the observed ones. First, the states in the vicinity of 2.6 MeV are discussed. In order to examine the structure of the wave functions, we take overlapping integrals with the product wave functions (3- x h9)$ defined in eq. (3.1). Spectrum of the product states is shown in fig. 3, in which the successive sums of the contributions of the terms, (I)-(IV), in eq. (3.3b) to E, are shown separately. The squares of the overlaps are listed in table 2 together with the observed energy shifts from the 3- state in 208Pb, GE_,, and those in the shell model SE,,,i, and the weak-coupling model SEW,, the last two of which

108

K. ARITA

AND H. HORIE

i”

7

c

2.5

I

r

(a)

(b)

(d)

Fig. 3. Energy spectra of the (3”’ x ha) septuplet in zopBi based on the weak-coupling model. The calculated 3- IeveI in 2osPb is shown on right-hand side for reference. Successive sums of the contributions of the terms in eq. (3-3b), (I)-(IV), to eq. (3.3s) are given in the columns. (a): (I), (b): (I+II), (c): (I+II+III), (d): (I+II+III+IV). The 3- core state taken here is obtained in the calculation whose resulting spectrum is shown in fig. 1. Two-body interactions are assumed to be the same as those used in the shell-model calculation. TABLE 2 Summary of the calculated results for the octupole septuplet in 2opBi

SE.,,

-126 0 -35 -55 -18 -18 124

@sw

-130 -10 -20 -40 10 -120 240

G

tic (4

fb)

-80 10 20 -20 120 -140 310

-80 10 0 -20 30 -70 260

GSZ 0.95 0.99 0.97 0.99 0.96 0.95 0.87

Gb 0.96 0.99 0.99 0.99 0.99 0.98 0.99

The quantity dE.,, is the observed energy shift of each member of the septuplet from the lowest 3- level observed in 2osPb. The data are taken from ref. 2). The energy shifts SE,,,, and 6.E,. are those corresponding to the spectra due to the shell model (fig. 2) and the weak-coupling model (figs. 3 and 5), respectively, measured from the calculated 3- level. All the energy values are in keV. Intensity G is evaluated by expanding the wave function due to the shell model in one or several product states according to eq. (3.10). The energy shifts in column (a) and the intensity G. are calculated, taking a product state (3- x h%), i.e., G, = <3- x hqjY>, 2. The energy shifts in (b) correspond to the spectrum shown in column (c) of fig. 5. Gr, is the total intensity expanded in (3 - x hs)J, (5 - x hp )J and single-particle states with #+, 9 + and 8 +.

zOsBi AND

207Pb

109

are measured from the calculated energy of the 3- state. The energies are generally well reproduced, especially in the shell model. The spectra predicted by the shell model and the weak-coupling model fairly resemble each other in overall features and the overlaps are very close to unity in almost all the states. These results enable us to regard these states as the members of the octupole septuplet and to discuss the level structures from the weak-coupling model in good approximation. Possible important effects which are responsible for energy splittings of the levels in the multiplet are particle-particle interactions between like nucleons and between unlike nucleons, antisymmetrization of total system and mixing of other states, first two of which can be seen explicitly in the spectrum of the product states. Hafele 13) calculates it qualitatively and he stresses the importance of the pairing and blocking effects between protons in the h, orbit and it is shown that the 3’ state is pushed down and the y’ state is pushed up as their effects. Here we take into account the effect of the neutron-proton force in addition to the above mentioned effects and consider the effects of antisymmetrization in more detail. It is clear from fig. 3 that essential features of the pairing effect in the 3’ state and the blocking effect in the -$K’ state are similar to those in ref. 13), but that there are two qualitative differences in the spectrum. One is the appreciable depression of the q’ state by the particleparticle interaction as is shown in column (d) of the figure. This can be ascribed to the neutron-proton force which works strongly between particles with spins parallel in addition to the pairing force between identical nucleons. Another one is that the blocking effect works alternatively repulsively or attractively for successive J-values as is seen in column (c), while it is always repulsive in Hafele’s calculation. Main contribution of the effect comes from the (d; ’ hg) configuration in the “‘Pb core. The alternative contribution of the blocking effect appears also in the studies by the collective model 14*15). It is common that the energy shift due to the effect increases with increasing J in the 209Bi septuplet. There are relatively large disagreements with experiment concerning the energies of the y’ and y’ states in both the shell-model and weak-coupling model calculations as is seen in table 2. These suggest that our treatment of the blocking effect is an overestimate due to the restriction of the shell-model space for the description of the E3 correlation. For, if the correlation is fully collective and a larger space is assumed, the intensities will be distributed among all configurations excited from the closed shell without spin-flip and the blocking term becomes less effective in accordance with the decrease of the probability of the (d; ’ h%) configuration in the 3- state. This can be checked by the additional weak-coupling model calculation. The negative-parity states in “‘Pb are calculated again in lh-lp configurations, each with unperturbed energy less than 8 MeV, instead of 7 MeV in the previous calculation. The strength of the hole-particle interaction of Carter et al. ‘“) is adjusted so as to reproduce the 3; state at the observed energy, i.e., the strength is reduced to 74 % of the original one. Then, the weak-coupling states are constructed, taking the 3- state just obtained as the core wave function, and the spectrum is calculated by

110

K. ARITA

AND H. HORIE

eq. (3.3), all other interactions being assumed the same as before. The result is shown in fig. 4. By the comparison of figs. 3 and 4, it seems that the differences between the two calculations are largest concerning the blocking effect, column (c). Thus, better agreement with the experiment for the 9’ and y’ states is obtained in this weakcoupling model under the restriction of unperturbed energies less than 8 MeV. The space becomes too large to carry out the general shell-model calculations under the same assumption, i.e., taking all lh-2p configurations with unperturbed energies less than 8 MeV. We do not try it, because disagreements are not so large and it is expected that the essential features will not change from those in the present calculation.

L

(al

(b)

Cc)

(d)

239&

Fig. 4. Energy spectra of the (3- x he) septuplet based on the weak-coupling model. The column labels are explained in the caption to fig. 3. The 3- core state taken here is calculated, taking all Ih-lp configurations with unperturbed energies less than 8 MeV and reducing the strength of the hole-particle interaction to 74 % of the original one of Carter etal. 23). The resulting energy of the 3state is shown to the right. All other interactions are the same as those used in the shell-model calculation.

Although the calculated results suggest the weak-coupling aspect, there are still some admixtures of other states to the septuplet. One of the possible admixtures is the (5- x h+)J states which are expected to lie at about 600 keV above the octupole septuplet. The above-mentioned shell-model wave functions are expanded in the (3- x hS)J and (5- x hS)J states and the possible single-particle states, the non-orthogonality between these states being taken into account. The resulting expanded intensities are shown in the last column of table 2. The effect of the admixture can be seen also in the spectrum (b) of fig. 5, which is obtained in the intermediate-coupling model calculation, assuming the coupling between the (3- x h*) and (5- x h+) states. It is seen that the effect is appreciable only in the y’ state. The effect is, however, much smaller, when larger space is assumed for the core wave functions, as is shown in column (b) of fig. 6.

zOsBi AND “‘Pb

111

Admixtures of single-particle states are detected in (3He, d) and (c(, t) reactions ’ ‘) on “*Pb 3 in which weak excitation is at about 2.6 MeV. A DWBA analysis of the angular distribution determines the I,., = 6 transition and the spectroscopic factor about 10 y0 of i, at 1.62 MeV. It is believed that the single-particle states i, and i, mix with the unresolved y’ and -‘$’ states in the octupole septuplet 14). The predicted values by the two models are summarized in table 3. The energy shifts due to the admixtures are calculated by the intermediate-coupling model, the spectra of which are shown in columns (c) of figs. 5 and 6. The g, single-particle state is admixed very little into the low-lying 4’ states.

1512’

2.97

912’ .

2.64 m

1512’

2.97

2.F

(a)

(b)

Fig. 5. Effects of admixtures into the (3- x h*) septuplet. The column labels denote the states included in secular equations. (a): (3- x he), (b): (3- x hs) and (5- x ha), (c): (3- x ht), (5- X hs) and single-particle states with t+, q + and 8 +. The 3- and 5- core states taken here are those obtained inthe calculation whose resulting spectrum is shown in fig. 1. The spectrum in (a) corresponds to the one in column (d) of fig. 3. The non-orthogonality between the (3- x hg)J and (5- X hg)J is taken into account. Only the states involving dominantly the (3- x h%) states are shown.

r

1 1512'

L ‘-Pb

286

1512.

284

1112'

2 73

lli2

2.72

712'

2.68

7,2'

2.67

5/2'

261 2.56

512' j I', 2 91 : I

’ ’ 9/2&

uw

261 254 u&.y

1512'

284

7/2Tu2

267

512:11/Z' z*

261 258

bw

15/Z'

2%

72' 13/Z' 1112. m

61 &u

312'

(a)

(b)

(c)

251

-

(d)

Fig. 6. Effects of admixtures into the (3- x h$ septuplet. The column indices (a)-(c) are the same as explained in the caption to fig. 5. The 3- and 5- core states are obtained, taking lh-lp configurations with excitation energies less than 8 MeV and reducing the strength of the hole-particle interaction to 74 % of the original one. The spectrum in (a) corresponds to the one shown in column (d) of fig. 4. In the last column (d) of this figure, mixings of the product states, (d+-l x 210Po(O+)) and (p+-’ X zloBi(J,,)), (.I,, = 11--81-), are also taken into account in addition to the states included in (c). The wave function calculated by Kim and Rasmussen is used for 210Po(O+) and those for *“Bi are explained in the text. The non-orthogonalities are taken into account.

112

IL ARITA AND H. HOBIE

Mixing of (3- x hf)J, p into the i, single-particle state is suggested in the (d, d’) experiment “) and the probability is estimated to be 0.054, extracting from the inelastic intensity the contribution of single-particle excitation and the effect of mixing (3- xif)J_S into the ground state “). The i, single-particle state is pushed down by 0.40 MeV in the shell-model calculation and the observed ip single-particle state at 1.62 MeV is reproduced. The calculated mixing probabilities by the shell model and the weak-coupling model are 0.04, and 0.10, respectively. TABLE 3 Calculated strength distributions of iv and i+ single-particle states to low-lying states of ‘OgBi

Single-particle state

Ii*

Ii*

Level (Z)

(Z)

(K)

(%2t)

(Z)

Shell model

0.86

0.03

0.01

0.01

0.01

Intermediate-coupling model

0.87

0.10

0.03

0.01

0.01

Numbers in parentheses are the excitation energies due to the shell model in MeV. The calculation of the strengths in the last row is based on the ~te~~ate-~upling model, taking into account mixing of (3- x h+), (5- xb+) and single-particle states.

Possible states which can mix with the 3” state are (d; ’ x 210Po,,d) and (pi ’ x 210Bignd),whose excitation energies are roughly estimated to be 3.37 MeV and 2.74 MeV, respectively, from the binding energies, assuming that the states in “‘PO and 21013ican be described by the two-particle configurations and the hole-particle interaction can be ignored. The wave functions of the states in “‘PO and 210Bi have been calculated by Kim and Rasmussen ~2’). But we calculate them again, taking configurations (j,j,; JM), where j, and j, are those orbits which are involved in our lh-2p configurations of “‘Bi as ld;lj,j,( J p),- JIM) or /pi’jl jz(Jp); .Jlw). In that calculation (ii; J+) in 2’oPo is excluded from the space they assumed and (ip j,; J-) in ‘loBi is included. The product wave functions (3.5) are constructed with these states and the 3’ state of 2ogBi is expanded as 0.94(208Pb[3-)x

h%)--O.l3(d; l x 210Po(O+))+0.12(p;1

x z”“Bi(lc)),

with the total intensity of 0.98. Inclusion of (5- x h,) in the expansion increases the intensity only by 0.04 %. Considering the (ig,0’) configuration is important for the description of the ground state of “‘PO, the shell-model calculation of the 3’ states is again carried out, adding the configurations (d; ’ i$(Jp); s+), (J, =I 0,2), to the shell-model space given in sect. 2. The resulting energy of the 3: state is 2.51 MeV and (3- x hg( ul>:=, is 0.88. Using the wave function of Kim and Rasmussen 2g) for the ground state of 210Po, the 3’ state is expanded as 0.86(20sPb(3-)

x ht)-0.29(d,

’ x z10Po(O~))~O~13(p~f x “‘“Bi(l-)),

nogBi AND

*O’Pb

113

with the total intensity of 0.98. Both results show that these states give rise to important effects. In order to see the effects in another way, a calculation starting from the weak-coupling model taking a larger space for the excited states of the “*Pb core is carried out, assuming as the admixed states the (d; ’ x 2’oPo(O+)) and (pi 1 x ‘loBi (J,,)) states in addition to the (5- x h%) and single-particle states, where J,, means the lowest l- - 8- states in 210Bi. The wave function of Kim and Rasmussen is used for the O+ state in ‘r”Po and the wave functions obtained earlier are used for the states in ‘loBi. The result is shown in column (d) of fig. 6. The agreement of the spectrum in this calculation with experiment is better than that in fig. 5.

3%

5-

1

3.28

*“Pb

I 30 i

I 30

(a)

(b)

Cc)

Cd)

Fig. 7. Energy spectra of the (5- x ha) multiplet based on the weak-coupling model. The calculated 5- level in *O*Pb is shown to the right for reference. The column labels are explained in the caption to fig. 3. The 5- core state taken here is obtained in the calculation whose resulting spectrum is shown in fig. 1.

Secondly, the states near 3.2 MeV are discussed. So far, levels in this energy region are resolved only in the (d, d’) experiment of Diamond et al. 4), where tentative spin assignments based upon the intensity rule are given. Coupling of the 5- state at 3.198 MeV in “sPb to a h, proton, which might be responsible for these levels, produces the states with the spins ranging from 3’ to q+, but the 3’ and 3’ states are not identified in ref. “). The levels of the 3+--y’ states are again very closely spaced and the shell-model calculation can satisfactorily reproduce them as is shown in fig. 2. It is experimentally known that the 5- state has predominantly the configuration pi 1 g+ [refs. 26*“)I. Only weak excitation leading to the 5- state is found in the reactions 209Bi(t, z) or (d, 3He) [ref. ““)I suggesting a small probability of proton excitation. Therefore, if the product wave functions of a h, proton and the 5- core state give good approximation to these states, both the pairing force and the blocking term which are the most important in the spectrum of the octupole septuplet will not give large contributions. The neutron-proton force is important in this case and the situation is reflected in a somewhat different manner for the calculated spectrum of a (5- x hp) multiplet from that of (3- x 11%)as is seen in fig. 7.

114

AND

K. ARIT-A

H. HORlE

The expanded intensities of the wave functions by the shell-model calculation in the (5- x h+) states and in the (5 x h*) and (3- x h*) states are shown in table 4. The importance of the neutron-proton force for these states is also found in appreciable mixing of the (pi ’ x “‘Bi(J P )) states into some of them as shown in the last column are taken into acof table 4. Eight of the ten lowest states in 210Bi, (J, = 2--9-), count, using the wave functions mentioned above. These states below 0.6 MeV in 210Bi are populated in 209Bi(d, p) [ref. “‘)I and have the dominant configuration (hl gi). Therefore, the overlaps between the (5- x h+), and (pi ’ x 210Bi), states are sometimes very large. For example, <208Pb(5-)~

h,]p;’

x 210Bi(9-))J=4 TABLE

Summary

of the calculated J”

8’ 4’ 8’ 9+ !2+ 3. + 9’ 9’

results E IhCIl

for the (5-

x

h*)

= 0.84.

4

multiplet

E WC

except

for the )’

G

and

GC .___.-

3.19

3.26

0.87

0.87

0.95

3.21

3.27

0.84

0.87

0.97

3.24

3.28

0.97

0.97

0.97

3.22

3.27

0.89

0.93

0.97

3.26

3.27

0.95

0.95

0.96

3.25

3.26

0.77

0.90

0.97

3.19

3.23

0.95

0.91

3.17

3.28

0.86

0.97

3’

members

.~

The quantities Eacll and E.,, are excitation energies in MeV due to the shell-model and the weakcoupling model, respectively. The energy EWE corresponds to the spectrum in fig. 7. The intensities expanded in product wave functions are shown in the last three columns. G.: square of the overlap . Gb: total intensity expanded in (S- x ht_) and (3- x hi). G,: total intensity expanded in (S- x hs), (3- x ht) and (pt-’ XzroBi(Jp)), where rr”Bi(Jp) is the lowest 2- --9states in t‘“Bi.

In table 4, the intensities <5- x h,]Y,)’ are not so large as in the case of the octupole septuplet but are still rather large particularly in the states with the spins satisfying(-I)‘-+ = 1. The absence of the f’ and 3’ states in the (d, d’) spectrum “) is recognizable from the presumption that the excitations leading to these states might be relatively weak because of their small statistical factors, even if the members of the (5- x h4) multiplet are unfragmented. The product states with J = _1’ or 3’ expected in this region are (s; ’ x ‘t”Po(O+)), (pi ’ x 2’oBi(2-)), (di ’ x (Pi ’ x 210Bi(l -)), (pi ’ x “‘Bi(O-)), 2’oPo(O+)), (“sPb(3-) x h,) and (20*Pb(5-) x h,), w h ose excitation energies are estimated to be 2.74, 2.79, 3.02, 3.06, 3.31, 2.62 and 3.20 MeV, respectively. Twoparticle correlations outside the 20ePb core are very important for these states except for the last two states. Our restriction of the shell-model space has no clear physical ground. For example, the configurations with higher unperturbed energies are also

209Bi AND

z07Pb

115

necessary to describe the pairing correlation correctly. Our space involves the configurations, (j- ’ hi(O)) and (j - ’ f:(O)), but the (j-r ii(O)) configuration is not involved, because its unperturbed energy is larger than 7 MeV. Inclusion of the configuration gives appreciable effects on the 3: state, as was shown previously, but very is also important small effects on the +1,3 s tates. It is expected that the configuration for the 3’ states. The shell-model calculation of the 4’ states is carried out, including configurations with higher energies up to 8 MeV, all two-body interactions remaining unchanged from those mentioned in sect. 2. Now, the (s;’ i$(O)) configuration is involved in the space. Comparison of the calculated results in the original space and in the enlarged space shows that for one of the 4’ states enlargement of the space provides a significant decrease in energy from 3.14 MeV to 2.66 MeV and an increase of the intensity of (s; 1 x 210Po(O+)) from 0.50 to 0.82. Other $’ states do not change their energies and wave functions and do not have large intensities of the (si 1 x ‘l”Po (O+)) state. Thus, it may be right to say that the different results in two calculations are provided by different treatments of the pairing correlation. In fig. 2, we show the energies of the -)’ states obtained by the calculation in the enlarged space. It is also checked that other states with higher spins are not affected by including the (j-r i$ (0)) configuration. TABLE 5 Analysis of the 4’ states obtained in the shell model E (MeV)

<+I IW

2.66 2.99 3.06

0.00 0.36 0.57

($2 Iv 0.05 0.51 0.31


c4.S IW

0.82 0.04 0.02

0.39 0.40 0.00

G 0.96 0.94 0.90

The excitation energy of each state is shown in the first column. In the next four columns, the squares of the overlaps of the states with product states are shown. &: p&-’ x’r”Bi(l-). Cz: pt-l X”“Bi(0-), 43: s&-l x’~~Po(O+), I$~: 208Pb(5-) X hs. It should be noted that the states &, is not orthogonal to other states. The quantity G in the last column is the total intensity expanded in these four product states according to eq. (3.10). TABLE6 Analysis of the 8” states obtained in the shell model

2.91 3.30

0.83 0.05

0.03 0.70

0.01 0.04

0.02 0.00

0.65 0.22

0.95 0.86

4,: p+-l x2roBi(l-), &: p+-r X 210Bi(2-), $3: d+-r X”‘Po(O+), &: z08Pb(3-) X hp. 4s: 2osPb(5-) x ht. The quantity G is the total intensity expanded in these five product states.

Below 3.3 MeV in the spectrum of fig. 2, there are three 3’ states at 2.66, 3.06 MeV and three 3’ states at 2.91, 3.30 and 2.58 MeV, the last of which to the octupole septuplet as already discussed. The expanded intensities states in the above-mentioned product wave functions are shown in tables

2.99 and belongs of these 5 and 6,

K. ARITA

116

AND

H. HORIE

which exhibit strong mixing of those product states especially in the 3’ states, so that we cannot give a definitive property of each state in terms of the weak-coupling model. The 4’ and +’ members of the (5- x h*) multiplet are shared by several states. There are little experimental data available with respect to these states. In neutron inelastic scattering ‘) from “‘Bi some incompletely resolved levels are found at 2.82,2.94 and 2.98 MeV. But therd is no evidence that these should correspond to the calculated +’ and 3’ states discussed above. The 2.82 MeV level might possibly correspond to the 26 single-particle state in view of its energy, though the nature of the level is not reported in ref. ‘). The single-particle pick-up reaction on “‘PO or the two-nucleon stripping reaction on “‘Pb may help as significant experiments to study these states if possible. 4.2. THE

NUCLEUS

=“Pb

The resulting spectrum of ‘07Pb from the shell-model calculation is compared with the experimental one in fig. 8, in which the known single-hole states are omitted. The +’ and 3’ states do not appear below 3.5 MeV in either spectra.

1 5-

3380"

32zaC' 3.186=)

3198

9/Z’

273 '

7/Z

2655." 26 a,

512'

l/T

““Pb

34

1112'

363

912'

329

1112

318

912'

273

512'

266

712'

263

1/z-

‘07Pb Expt

712'

1

35

5-

3.28

i -l

0’ 20BPb

‘07Pb Cal

Fig. 8. Experimental and calculated spectra of s07Pb. The lowest 3- and 5- levels in zosPb are also shown for the sake of comparison. The known single-hole states are omitted. The spacing to the ground state is not to scale and the energy scale of the calculated spectra is fixed so as to align the calculated 3- level with the experimental one. The experimental data for the levels marked with (a) are taken from the (p, p’) study by Vallois et al. 3), (b) from ref. r2) and (c) from ref. ‘). The calculation is based on the shell model, taking all possible 2h-lp configurations with unperturbed excitation energies less than 7 MeV. See the text in sect. 2.

There is various experimental evidence that the octupole excitation also occurs in the vicinity of 2.6 MeV in this nucleus as a closely spaced doublet formed by coupling

117

209Bi AND ‘O’Pb

to a single hole in the p+ orbit. The states in the doublet, therefore, have the spins s and 2. They are reproduced in our calculation and the energy shifts from the octupole state in “*Pb and spacing between the states are very small in accordance with observation. The energy shifts and square of the overlaps of the wave functions of these states with the (pi ’ x 3-)J product wave functions are tabulated in table 7. The overlaps are quite close to unity and then, the energies of the product states which are shown in fig. 9 are quite similar to those obtained by the shell model. There are no other Q’ and 8’ states which can couple to the states in the doublet and are expected to lie in the range of excitation energy below 3 MeV. TABLE7 Summary of the calculated results for the octupole doublet in *“‘Pb

8’ %’

-3 38

-60

-20

0.98

-90

-60

0.99

The energy shifts 6E are the same as explained in the caption to table 2. The shift 6E,, is calculated taking the expectation value of energy of the product state (p*-r x 3-). Square of the overlaps of the calculated wave functions due to the shell model with the product states are shown in the last column.

A5

(a)

(b)

(cl

Cd)

Fig. 9. Energy spectra of the (p+-’ x 3-) doublet based on the weak-coupling model. The calculated 3- level in 208Pb is shown to the right for reference. The spectrum of the hole-vibration coupled states is calculated according to the formula equivalent to eq. (3.3) for particle-vibration coupled states, but the term (IV) should read the hole-hole interaction. The column labels are explained in the caption to fig. 3. The 3 - core state taken here is obtained in the calculation whose resulting spectrum is shown in fig. 1.

The observed spectrum indicates that the centre of gravity of the doublet is shifted upwards from the 3- state of “*Pb and that the 3’ state lies lower than the 3’ state. In the calculated result, these states are shifted downwards and the 5’ state lies higher. The spin assignments of 3 and 4 to the 2.610 and 2.655 MeV levels are given from the statistical rule applied to the excitation intensities ‘* “) in the inelastic scattering and are confirmed by other independent evidences in the isobaric analog resonance study [ref. ““)I. The calculated spectrum is not improved in the weak-coupling model,

K. ARITA

118

AND

H. HORIE

taking the same larger space for the core wave function as that in the octupole septuplet of 209Bi, but the differences from the experimental values are always small. In the reaction 206Pb(d, p) [refs. I’, ‘“)I, the prominent transition to the 2.73 MeV state is observed. The DWBA calculation implies an 1, = 4 transfer and the state is assigned as the $+ state formed by the addition of a g4 neutron to the ground state of “‘Pb. The experimental presumption is also supported by another experimental result that in 2o ‘Pb(p, p’) [refs. 36S“‘)I, the inelastic yield to the $+ state is strongly enhanced at the bombarding energies corresponding to the formation of the analogs of 5; and 4; of 208Pb, which have the dominant configuration pi 1 g+. Assuming dominance of two-hole correlation in the state of “‘Pb and ignoring the holeparticle interaction, the energy of the ( 206Pb(O+) x gf),,% state is estimated from the

L

_I

(a)

(b)

(cl

Cd)

Fig. 10. Energy spectra of the (pa - 1x 5-) doublet based on the weak-coupling model. The calculated 5- level in *OsPb is shown to the right for reference. The column labels are explained in the caption to fig. 3. The 5- core state is obtained in the calculation whose resulting spectrum is shown in fig. 1. This doublet does not appear in the shell-model calculation. See the text.

binding energies 21) as 2.80 MeV in fairly good agreement with the experimental energy. In our calculation of “‘Pb, the energy of the 3’ state is reproduced at 2.73 MeV and the intensity x g4]Yg)2 is found to have the value 0.93. The ( “‘Pb(O+) wave function of the Of state used here is obtained within the two-hole configurations IhW2; O’), where the single-hole orbits h are involved in the ]h-‘(0)gg; 4’) configurations in the shell-model calculation of 207Pb. The same hole-hole interaction as used in ‘07Pb are assumed. The large value of the overlap gives plausible support to the proposed configuration assignment. The e+ and y’ states appear in the vicinity of 3.2 MeV in the calculated spectrum. In view of their energies, these states seem to be the members of a 5- doublet (pi 1 x 5-), but examination of the calculated wave functions of these states sug-

““‘Bi

AND

gests the breakdown of the weak-coupling grals are calculated and the results are

119

“‘Pb

model for the doublet.


x 5-1YJ$

= 0.21,

(pi’

x 5-lY,)2

= 0.75,

The overlap inte-

where Y, and Yy, are the calculated wave functions of the 4’ state at 3.29 MeV and the ‘f’ state at 3.18 MeV, respectively. The spectrum of the weak-coupling model shown in fig. 10 is also quite different from that of the shell model. The source of the breakdown does not seem to be the existence of some other states strongly coupled to the doublet, but to be the disappearance of the 5- weak-coupling wave functions themselves as discussed below. The component which is involved in the weak-coupling state Ipi I x .I,), and is disturbed by the effect of the Pauli principle, is JW,

IP; ‘3 P; ‘j,(J,);

wherej, is a particle orbit. The p+ orbit being vacant in this component, equation clearly holds,

lpi’, p;‘j,(J,); JW =

0

for

J # j,.

the following

(4.1)

It can be understood from eq. (4.1) why the contribution of the blocking term is larger in the lower J state of the octupole doublet of 207Pb, while it increases with increasing J in the octupole septuplet of “‘Bi. According to eq. (4.1), the blocking term in “‘Pb works only in a state with J # jp, so the magnitude of the contributions depends only on the probabilities of pi ’ j, configurations in the core, not on J-value itself. The probability of the configuration pi ’ g; is larger than that of pi ’ d, in the octupole core, then the blocking term works stronger in the 3’ state of the doublet as seen in fig. 9. Both probabilities being small in magnitude, the energy splitting of the doublet is also small. The dominant configuration in the 5- state of ‘08 Pb is pi ’ g% as mentioned earlier. According to eq. (4.1) it is impossible to form a q-’ state by coupling a p+ hole to the configuration. In the -‘$’ state, therefore, addition of the single hole in the p+ orbit makes the main component of the core vanish and disturbs the core very much. The effect appears as the extremely large value of the normalization factor of the _‘i-’ weak-coupling state in table 1 and as a large overlap with other state, i.e.,
x 5;>,=

y = 0.92,

where 5, is the 5- state at 3.70 MeV shown in fig. I. The value of (pi ’ x 5; lY,))* is not so small, but
120

K. ARITA

AND

H. HORIE

The other member of the doublet, the 4’ state, is not affected by the Pauli principle as much as the y’ state, since the pi 1 i, configuration in the 5- state is only a small component. On the other hand, we have, for J =_ip, ip;l,

p;~&@,);

JM)

= ( -l)++jp-Jc

IJ

5

lp;*(O)j,;

JM),

(4.2)

P

instead of eq. (4.1). The statistical factor appears on the right-hand side, because is not defined to be normalized. Therefore, it is easy to see IP~‘~P;‘_l,(Jc);JM) that the weak-coupling state has a large overlap with (*06Pb(0+) x &),+. Actually, it is large, i.e., <206Pb(O+) x g%lp;’ x 5-),+

= 0.65.

So, this portion of the intensity of the (pi r x 5-)J=4 state is exhausted in the 2.73 MeV state. The remaining intensity is fragmented among several states, and the state with the concentrated intensity is not found in the calculation. In the (p, p’) experiment of Vallois et al. 3), E5 excitations are observed at the excitation energies 3.200,3.225 and 3.380 MeV, but the intensities are smaller by an order of magnitude than that observed at 3.198 MeV in *‘sPb. On the other hand, in the (a, c(‘) experiment of Alster ‘), the E5 excitations are found at 3.4 and 3.7 MeV with the sum of the transition strengths almost equal to that in *‘*Pb. The experimental situations appear not so clear in this energy region. Some higher-lying states are observed experimentally 3*r’s ‘*, 3**3g), but the calculated results in the region are not discussed, since the calculation is not expected to give a good approximation for higher-lying states because of our restriction of the shell-model space. In order to see the sensitivity of the obtained results of *“Bi and *O’Pb to the choice of parameters, the two-body interactions are changed in the calculations. First the proton particle-particle interaction used by Glendenning and Harada 40) is employed instead of that of Kim and Rasmussen *‘). The resulting spectrum of *“Bi differs little from the previous one. Secondly, the strength of the hole-particle interaction is adjusted to reproduce the octupole state in *‘sPb at the observed energy and the one of the neutron-hole proton-particle interaction is also slightly adjusted to reproduce the ground state energy of *OsBi. Furthermore, the particle-particle interactions between protons and between the neutron and the proton and the hole-hole interactions between neutrons and between the neutron and the proton are modified by adjusting their strengths to reproduce the ground state energies of 210P~, *loBi, 206Pb and 206T1 respectively. Then, the energies of the octupole multiplets in *“Bi and 207Pb shift down as a whole by the same amount as the shift of the octupole state of *“Pb without appreciable changes of relative spacings between the members. The state with a large intensity of (*06Pb(O+) x g+)J=9 is rather sensitive to the choice of the strength of hole-hole interaction. But there is not an essential change in the energy spectra and the wave functions.

209Bi AND

5. Summary

121

207Pb

and concluding

remarks

For the low-lying non-normal parity states of “‘Bi and 2o ‘Pb, simple shell-model calculations have been performed. The two-body interactions adopted have been referred to the neighboring nuclei and, without adjustment of the interaction parameters, good agreement of the energy levels below 3.3 MeV with the experiments has been obtained. For a few levels in “‘Bi , further experimental evidence seems to be anticipated. The weak-coupling model wave functions have been defined for the product of the core and particle or hole and by taking their overlap with the wave functions calculated by the shell model, the characteristic features of the weak-coupling has been investigated, which are expected from experiments. Actually, it has been confirmed that the without large fragmentation. It is octupole multiplets appear in 209Bi and “‘Pb to be noted that the existence of the octupole multiplets does not mean the holeparticle interactions are assumed to be quite strong in comparison with the particleparticle or hole-hole interactions. It may be seen from the situation that the weakly coupled states of the single-hole (or particle) to the two-particle (or hole) states in particle-particle (or hole-hole) correlations appear also in these nuclei. The effects which cause the splittings of the multiplets have also been investigated. These effects do not always give rise to the large splitting cooperatively but, as is seen in the octupole multiplets of “‘Bi , they can produce rather small splittings due to the cancellation with each other. It has been pointed out that the effect of the antisymmetrization is very important in the level structure of the multiplets and in the fact that the weakly coupled states exist. The effect is especially important and the core suffers quite a strong disturbance, as is expected, when the degeneracy of the orbit of the odd particle (or hole) is small and the probability of the excitation to (or from) this orbit is large. In some extreme cases, the weak-coupling states do not hold completely because those states vanish or come to have large overlapping with other states of different character. One of such examples is given by the non-existence of the (pi’ x 5-) states in “‘Pb. One of the reasons of the empirical knowledge that the weakly coupled states are found for the core states which have collective nature, might be reduced to this effect of antisymmetrization. The configurations assumed in our shell-model calculations have simple structure but the numbers of the configurations are rather large and thus the wave functions obtained as the final results are complicated. It has been shown, however, that the wave functions can be expanded in terms of a few product wave functions with good approximations, except for a few examples. Although the quantitative discussions of the mixing probabilities need some further experimental evidences, it suggests that, by assuming some weak-coupling states and by solving the mixing coefficients, we can obtain a good description for some excited levels of the odd-A nuclei. It is to be noted that we must pay attention for the non-orthogonal&y between those states in this case.

122

K. ARITA AND H. HORIE

The basis space adopted in our calculation is not enough to interpret the E3 correlation. This defect has appeared as the overestimate of the blocking effect in the octupole multiplets, but the agreement between the calculated and experimental level structure is satisfactory. The effect of the omitted configurations in the E3 transition probabilities are now investigated.

The authors wish to thank Professor H. Taketani for helpful discussions and advice. Their thanks are also due to Professors T. Komoda and H. Nakamura for their interest and encouragement. Numerical calculations were carried out by the HITAC 502OE at the Computer Centre, University of Tokyo and by the IBM 7040 at the Computer Center, Aoyama Gakuin University.

References 1) D. A. Bromley and 3. Weneser, Comm. Nucl. Part. Phys. 2 (1968) 151; 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

20)

21) 22) 23)

N. Stem, Proc. Int. Conf. on properties of nuclear states (Les Presses de I’Universite de Mont&l, 1969)p. 337 J. C. Hafele and R. Woods, Phys. Lett. 23 (1966) 579 G. Vallois, J. Saudinos, 0. Beer, M. Gendrot and P. Lopato, Phys. Lett. 22 (1966) 659; E. Grosse, G. F. Moore, J. Solf, W. R. Hering and P. von Brentano, Z. Phys. 218 (1969) 213 R. M. Diamond, B. Elbeck and P. 0. Tjom, as reported by Mottelson i4), p. 97 J. Aister, Phys. Rev. 141 (1966) 1138 J. W. Hertei, D. G. Fleming, J. P. Schiffer and H. E. Gove, Phys. Rev. Lett. 23 (1969) 488; R. A. Broglia, J. S. Lilley, R. Perazzo and W. R. Phillips, Phys. Rev. Cl (1970) 1508 L. Cranberg, T. A. Oliphant, J. Levin and C. D. ZaBratos, Phys. Rev. 159 (1967) 969 J. F. Ziegler and G. A. Peterson, Phys. Rev. 165 (1968) 1337 and the references cited therein A. de-Shalit, Phys. Rev. 122 (1961) 1530 A. M. Lane and E. D. Pendlebury, Nucl. Phys. 15 (1960) 39 P. Mukherjee and B. L. Cohen, Phys. Rev. 127 (1962) 1284 W. Darcey, A. F. Jeans and K. N. Jones, Phys. Lett. 25B (1967) 599; W. R. Hering, A. D. Achterath and M. Dost, Phys. Lett. 26B (1968) 568 J. C. Hafele, Phys. Rev. 159 (1967) 996 B. R. Mottelson, Proc. Int. Conf. on nuclear structure, Tokyo, 1967, ed. J. Sanada [J. Phys. Sot. Japan Suppl. 24 (1968)] p. 87 I. Hamamoto, Nucl. Phys. Al26 (1969) 545; A135 (1969) 576; A141 (1970) 1; A148 (1970) 465; A155 (1970) 362 W. T. Pinkston, Nucl. Phys. 37 (1962) 312 V. Gillet, A. M. Green and E. A. Sanderson, Nucl. Phys. 88 (1966) 321 S. Hinds, R. Middleton, J. H. Bjerregaard, 0. Hansen aud 0. Nathan, Nucl. Phys. 83 (1966) 17 J. S. Lilley and N. Stein, Phys. Rev. L&t. 19 (1967) 709; J. Bardwick and R. Tickle, Phys. Rev. 171 (1968) 1305; C. Ellegaard and P. Vedelsby, Phys. Lett. 26B (1968) 155 G. Muehllehner, A. S. Poltorak, W. C. Parkinson and R. H. Bassel, Phys. Rev. 159 (1967) 1039; G. J. Igo, P. D. Barnes, E. R. Flynn and D. D. Armstrong, Phys. Rev. X77 (1969) 1831; C. A. Whitten, Jr., N. Stein, G. E. Holland and D. A. Bromley, Phys. Rev. 188 (1969) 1941, and references therein L. A. K&-rig, 3. H. E. Mattauch and A. H. Wapstra, Nucl. Phys. 31 (1962) 18 J. P. Elliott and T. H. Skyrme, Proc. Roy. Sot. A232 (1955) 561; S. Gartenhaus and C. Schwartz, Phys. Rev. 108 (1957) 482 J. C. Carter, W. T. Pinkston and W. W. True, Phys. Rev. 120 (1960) 504

z09Bi AND ““Pb 24) 25) 26) 27)

28)

29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40)

123

W. W. True and W. T. Pinkston, unpublished C. M. Lederer, J. M. Hollander and I. Perlman, Table of isotopes (Wiley, New York, 1967) J. Bardwick and R. Tickle, Phys. Rev. 161 (1967) 1217 C. F. Moore, J. G. Kulleck, P. von Brentano and F. Rickey, Phys. Rev. 164 (1967) 1559; S. A. A. Zaidi, J. L. Parish, J. G. Kulleck, C. F. Moore and P. von Brentano, Phys. Rev. 165 (1968) 1312; P. Richard, P. von Brentano, H. Weiman, W. Wharton, W. G. Weitkamp, W. W. McDonald and D. Spalding, Phys. Rev. 183 (1969) 1007 J. H. Bjerregaard, 0. Hansen, 0. Nathan, R. Chapman and S. Hinds, Nucl. Phys. A107 (1968) 241; E. A. McClatchie, C. Glashausser and D. L. Hendrie, Phys. Rev. Cl (1970) 1828 Y. E. Kim and J. 0. Rasmussen, Nucl. Phys. 47 (1963) 184 Y. E. Kim and J. 0. Rasmussen, Phys. Rev. 135 (1964) B44 W. P. Alford, J. P. SchitTer and J. J. Schwartz, Phys. Rev. Lett. 21 (1968) 156 W. W. True and K. W. Ford, Phys. Rev. 109 (1958) 1675 K. Arita and H. Horie, Phys. Lett. 30B (1969) 14 P.-O. Lowdin, Rev. Mod. Phys. 39 (1967) 259 J. R. Erskine, W. W. Buechner and H. A. Enge, Phys. Rev. 128 (1962) 720 E. Grosse, P. von Brentano, J. Solf and C. F. Moore, Z. Phys. 219 (1969) 75 G. H. Lenz and G. M. Temmer, Nucl. Phys. All2 (1968) 625 N. Stein, C. A. Whitten, Jr. and D. A. Bromley, Phys. Rev. Lett. 20 (1968) 113 W. P. Alford and D. G. Burke, Phys. Rev. 185 (1969) 1560; R. A. Moyer, B. L. Cohen and R. C. Diehl, Phys. Rev. C2 (1970) 1898 N. K. Glendenning and K. Harada, Nucl. Phys. 72 (1965) 481