Particle-hole multiplets in 208Bi

Particle-hole multiplets in 208Bi

I 1.D.1 [ I Nuclear Physics A122 (1968) 325--342; (~) North-Holland Publishing Co., Amsterdam Not to be reproducedby photoprint or microfilmwithout w...

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I 1.D.1 [ I

Nuclear Physics A122 (1968) 325--342; (~) North-Holland Publishing Co., Amsterdam Not to be reproducedby photoprint or microfilmwithout writtenpermissionfrom the publisher

P A R T I C L E - H O L E M U L T I P L E T S IN 2°SBi T. T. S. KUOt

Palmer Physical Laboratory, Princeton University, Princeton, New Jersey and Physics Division, Argonne National Laboratory, Argonne, Illinois tt Received 6 September 1968

Abstract: The energy levels of Z°SBiare calculated with the reaction matrix elements deduced from the Hamada-Johnston potential, and compared with the recent experimental data of Alford, Schiffer and Schwartz. The calculated wave functions for most low-energy states show very little configuration admixture. The energy levels have been calculated with and without the inclusion of the core-polarization effects, and the results of the two calculations show no significant difference. Remarkable agreement with experiment is obtained for all 36 experimentally identified low energy levels. But a large discrepancy is observed for the isobaric analogue state of ~0sPb;the calculated position being about 4 MeV too low. A multipole analysis of the effective particle-hole interactions has been carried out.

1. Introduction Alford, Schiffer and Schwartz ttt have recently o b t a i n e d detailed experimental results o n the energy level structure of 2°SBi by studying the 2°Tpb(3He, d)2°SBi a n d 209Bi(d ' 0208Bi reactions, which m a i n l y excite the single-proton states coupled to the 2p~ n e u t r o n hole a n d s i n g l e - n e u t r o n hole states coupled to the 0h~ p r o t o n partMe. Their results are of great interest for studying the relation between the nuclear effective interactions a n d the free n u c l e o n - n u c l e o n interactions. In the past few years, there has been some progress in u n d e r s t a n d i n g the nuclear effective interactions, a n d it is n o w fairly well established that their d o m i n a n t parts can be deduced f r o m a free n u c l e o n - n u c l e o n potential such as the H J ( H a m a d a J o h n s t o n ) p o t e n t i a l by use of the r e a c t i o n - m a t r i x theory 3, 4). However, calculations of this type have been d o n e only for nuclei in the oxygen 3), calcium 5) a n d nickel 6) region, where the effects of n u c l e a r d e f o r m a t i o n can be very i m p o r t a n t . Thus, there is often a n a m b i g u i t y as to whether the discrepancy between theory a n d experiment is due to the effective interactions or due to an i n a d e q u a t e t r e a t m e n t of nuclear deformation, since the calculation is usually carried out in a spherical shell-model basis. A t Work supported in part by the U.S. Atomic Energy Commission and the Higgins Scientific Trust Fund of the Princeton University. The work also made use of the Princeton Computer Facilities supported in part by the National Science Foundation Grant NSF-GP579. tt Address after September 1, 1968 is Dept. of Physics, the State University of New York at Stony Brook. *it Ref. 1). The energy levels of 2°SBiwere also measured by Erskine 2). 325

326

T.T.S.

KUO

major advantage in studying the effective interactions in nuclei in the neighborhood of Pb (e.g., 208Bi ) is that the pure spherical shell model can be used more confidently in describing nuclei in this region, and hence a comparison between theory and experiment can provide a more direct test of the effective interactions. This advantage is further enhanced for 20SBi because of the abundance of experimental results. The energy levels of 2°8Bi have previously been calculated by use of phenomenological interactions *. Kim and Rasmussen 7) have calculated these levels by use of a Gaussian potential. The parameters in their potential were adjusted to fit the energy levels in 21°Bi and 2~°po. Their potential has a very strong tensor component, whose strength is about twice that of the Brueckner-Gammel-Thaler potential. Although the calculation of Kim and Rasmussen has given remarkable agreement ~) between theory and experiment for a°sBi, their calculation is questionable in the following respects. As just noted, their potential has a very strong tensor component, but they have not included the second-order tensor-force contributions which have been shown 3) to be very important. Secondly, they have set the offdiagonal tensor-force matrix elements to zero in the energy secular matrix, and as noted by Hughes et al. 8), these off-diagonal matrix elements have rather important effects. We also note that the many-body effects, such as those of core polarization, were not included in their calculation. Birbrair and Guman 9) have calculated Z°SBi and Z°ST1 by use of the Migdal theory. The effective interaction used by them is a spin- and isospin-dependent ffunction force whose parameters are adjusted to fit the levels in Z°8Bi and 2°ST1. Their results agree remarkably well with experiment, and thus indicate that a f-function force with suitable choices of strengths and mixing parameters is a good representation of the effective nuclear force for 2°SBi and 2°8T1. A more fundamental approach would be to start from a nucleon-nucleon potential based on the nucleon-nucleon scattering data. The purpose of this work is to perform such a calculation, and to see how well the observed energy levels can be reproduced. We shall use the HJ potential. Since the many-body effects such as those due to core polarization have been found to be important a-6) for nuclear-structure calculations with a free nucleon-nucleon interaction, these effects will be included and studied in the present work. 2. Formulation of the problem The low-lying states of 2°SBi wiU be described as a proton particle and a neutron hole populated on a virtually closed 2°spb core. Then, the eigenvalues and eigenfunctions of 20SBi will be simply obtained by diagonalizing the secular matrix (ep, -- ~h,)b, 2 + ( P l h-~ '[ Vaf ]p2 h~-~)',

(1)

where ep, and ehl are the single-particle and single-hole energies. The effective interf See refs. 7,,) and references quoted therein.

327

2°spb CALCULATIONS

action Vcff will be discussed later. As usual, we shall take the values o f ep and eh f r o m experiments on 2°9Bi, 2°7pb and the empirical nuclear masses. The proton-particle and neutron-hole orbits f r o m 0h~ to 2p,r will be included in the present calculation. Table 1 shows the values of ep and ~h which we use. The particle-hole matrix elements in eq. (1) is calculated by (plh~- 11Vefflp2h~ 1) s = - ~ ( 2 , / ' + I ) ( J P l s' tjp~

Jh~

J,}

Jh2

x [(1 + 6jp2Jh,)(1 d- 6jp,Yh2)]'~(p 1 h 2 [V~ff[p2 hl)S',

(2)

TABLE 1 Proton-particle and neutron-hole energies 10,11) used in the present calculation Orbit label

n~j

15 13 14 16 12 11

2p~ lf~. 2p~ 0ia~ lf~ 0hsr

eproton

Eneutron

(MeV)

(MeV)

0.50 --0.99 --0.70 --2.20 --2.91 --3.80

-- 7.38 -- 7.95 -- 8.27 -- 9.01 -- 9.72 -- 10.85

The values for the 0hk proton orbit and 2p~_neutron orbit are the differences B.E. (2°9Bi)--B.E.(2°sPb) and B.E.(207pb)--B.E.(208pb), respectively, evaluated by use of the empirical nuclear masses 1~). Hence, our calculated eigenvalues for ~°aBiare measured from the ground state of 2°sPb. where the last matrix element is the p r o t o n - n e u t r o n particle-particle matrix element. The symbol enclosed by the curly brackets is the usual 64" coefficient. We note that p always refers to p r o t o n and h to neutron, and that they do not couple to g o o d T. Both the particle-particle and particle-hole matrix elements are antisymmetrized. In terms o f creation and annihilation operators, our particle-hole states are constructed by [atpah] operating on 10) which represents the d o u b l y closed shell 2°spb. F o r each shell-model orbital, we use the coupling scheme 3) l + s = j. Since the H J potential will be used for the nucleon-nucleon interaction, the first a p p r o x i m a t i o n to Vaf is Veff = G,

(3)

where G is the reaction matrix deduced f r o m the H J potential. The procedure described in ref. 3) has been followed exactly in calculating the G-matrix. Thus, the Gmatrix elements used in the present work can be specified by the constants ho9 = 7.0 MeV,

eeff = 240 MeV,

dse = 1.05 fm,

7 z = 2.0 f m - z ,

dte = 1.07 fm,

W(d) = 0.10 fm -~,

328

T. T. S. KUO TABLE 2 Radial m a t r i x elements

(nlSJIGJn'I'S.I)

in M e V [ref. 8)]

S

T

n

l

n'

1'

d

(G)

S

T

n

l

n'

0

0

0 0 0 1 1

1 3 5 1 3

0 0 0 1 1

1 3 5 1 3

1 3 5 1 3

0.606 0.116 0.034 0.841 0.170

1

0

0

1

0 0 0 0 0 0 1 2 3 4

0 0 0 0 2 2 0 0 0 0

0 1 2 3 0 1 1 2 3 4

0 0 0 0 2 2 0 0 0 0

0 0 0 0 2 2 0 0 0 0

--2.578 --2.529 --2.301 --2.040 --0.122 --0.143 --2.651 --2.433 --2.145 --1.853

0 0 0 0 0 1 2 3 4

0 0 0 2 2 0 0 0 0

2 2 3 0 1 1 2 3 4

0 2 0 2 2 0 0 0 0

1 1 1 2 2 1 1 0 0

--4.4t 1 --3.604 --4.081 --0.573 --0.659 --4.905 --4.668 --4.250 --3.778

1

1

0 0 0 0

0 0 0 0

0 0 1 1

0 2 0 2

1 1 1 1

--4.574 --1.945 --4.650 --2.917

0 0 0 0 0 0 1 1 1

1 1 1 3 3 3 1 1 1

0 0 0 0 0 0 1 1 1

1 l 1 3 3 3 1 1 1

0 1 2 2 3 4 0 1 2

--0.744 0.594 --0.191 --0.039 0.067 --0.006 --0.820 0.891 --0.407

1

0

G

Gph

1'

d

(G)

Gpp

S hS

ha

+

Ghh

G4

Fig. 1. D i a g r a m s representing the effective particle-hole interactions. E a c h wavy line represents a G interaction d e d u c e d f r o m t h e H J potential.

where

hco is t h e s p a c i n g

ds~ t h e s i n g l e t - e v e n

between

separation

the effective energy denominator

major

distance,

shells for a harmonic dt¢ t h e t r i p l e t - e v e n

for evaluating

oscillator-potential,

separation

the second-order

distance,

eel f

tensor force under

2°8pb CALCULATIONS

329

the closure approximation, ,,,2 the average decay constant for the defect wave function (~2 = 2 fm-2 corresponds to a gap of about 80 MeV between the occupied and unoccupied levels), and W(d) the average amplitude of the correlated D wave function at rite in the 3S 1 separation method. Some radial matrix elements 3) of G are tabulated ia table 2. The core-polarization effects have been shown to be important for the particleparticle effective interactions 3-6). Thus, one expects that these effects may also be important for the particle-hole effective interactions. The core-polarization diagrams and other diagrams to be included in the particle-hole effective interaction are shown in fig. 1. The diagrams Gp~, Gpp, Ghh and G 4 are all second order in G. For 2°8Bi, each of the incoming and outgoing external lines of these graphs consists of a proton particle and a neutron hole. The effects of Gph, Gyp, Ghh and G4 have been studied for other nuclei. De Takacsy [ref. 13)] studied their effects for the particle-hole excitation of 160, first using a simple central potential and then using the Tabakin potential. Their effects are generally not very important for 160. Dieperink et al. ~4) used the Tabakin potential in a similar calculation for 4°Ca. Again, these diagrams were found to be rather unimportant except in overdepressing the lowest RPA 3- state. Blomquist and Kuo a5), who used a G-matrix 3) deduced from the HJ interaction, have studied these graphs and the corresponding ones for the RPA diagrams in a TDA and RPA calculation of 4°Ca and 48Ca. The calculation done with G + Gph appears to give the best agreement with experiment, but the lowest 3- state is overdepressed when all these diagrams are included. In an extensive TDA and RPA calculation 16) of 2°8pb by use of the HJ potential, the effect of Gpa is found to be very important and desirable in raising the lowest 3- state whose RPA eigenvalue was imaginary when calculated with a bare G matrix alone. Thus, these calculations indicate that these renormalization diagrams are generally important only for the collective particle-hole excitations of closed-shell nuclei, and their overall effects are not as important as in the particle-particle case [refs. 3-6)]. In order to study these renormalizations for z 08Bi ' we shall make two calculations, first using Vaf = G and then using

Vat = G + Gph-b Gpp + Ghh"k G4 .

(4)

For calculating Gph, Gpp, ahh and G4, the equations, basically the same as those in refs. 13,1s), are (Pl h~" ~[GphlP2 h~- 1)J

=(-1)JE X 1 /jp, ph Job

Jh~

J }

~P--ph tJh2 Jp2 Jph

x [(-- 1)J.2 +J~- sp~x/(2jp ~+ 1)(2jh2 + 1)L(phJphp,, p2)L(phJphhl, h2) + ( - 1)iv' +J%-Sphx/i2jm + 1)(2jh~ + 1)t(phJphP 2 , pl)L(phJphh2, h,)],

(5)

330

T. T. S. KUO

(p~ h~- 1[Gpplp2 h~ ~)s = Z

Z 1

PaPbdpp AEpp

(2Jpp+l)(Jm

Jh,

J } (1-1-6papb)4(1 -}-~p,h2)(l "}-~pah,)

~Jp2 Jh2 Jpp x (p2hllG[papb)SpP(paPblGlplh2) s~p,

(6)

(Pl h~- l lGhh[p2h; 1)s

= Z Z 1 ~ (2Jhh"~l) [jp. Jhl J } (1 + 6hoh~)[(1+ 6p~h~)(1+ 6p~h,)]½ hahbJhhAEhh ~Jp2 Jh2 Jhh x (plh2lGlhhb)Shh(hh~[GlP2P~) shh,

--paSha~ --,~h, ~ ~

(7)

(plhll[G[paha 1)J(paha I[G]p2h2 x)J 1

(0lGlpxh[~' pbhb-X; J)(p2h21' pbh;X;JlGlO)"

(8)

The L-matrix element in eq. (5) is given by

L(phJph a, b ) = (--1) j~+jb-JphV ~Jph+ l ~

ljp Jh

J;~} w

x x/(1 + Jap)(1 + Jbh)(bh[G[pay',

(9)

and the delta function 6ab has the value 1 if (nalaja) = (nblb.~) and zero otherwise. For the present calculation, px and P2 are always proton and h~ and h 2 are always neutron; and hence Pa, Pb in eq. (6) must be a proton-neutron combination and so are ha, hb in eq. (7). The intermediate states Pa, ha in eq. (8) must be a proton particle and neutron hole, but Pb, hb must be neutron particle and proton hole. In eqs. (5) and (9), p and h must be both proton or both neutron. The RPA matrix element appearing in eq. (8) is given by (0[Glp~ h~- 1 P2 h ; x; J ) = ( - 1)jp2+Jh2+ ~X/(1+ 5p,p2)(1 + t~h,h2)(Pl h I ~[Glh2 P2 ~).(10) In the calculation of Gph, Gop, Ghh and G4, the single-particle and single-hole orbits included as intermediate states are: proton hole: proton and neutron hole: proton particle and neutron hole: proton and neutron particle: neutron particle:

0fk, lpk, 0f~, lp~, 0g~, 0g~, ld~, 0h~, ldi, 2s~, 0hl, lf~, 0i~, 2p~, lf~, 2p~, lg~, 0i÷, 0j~t, 2d~, 3s½, lgk, 2d~, 0j~/, lh¥, lh~, 2f~, 2f~, 2p~, 3p~.

All allowed p-h excitations within the above ranges and outside the model space (table 1) in which the matrix (1) will be diagonalized will be included as intermediate

2°spb CALCULATIONS

331

states in evaluating Gph , Gpp, Ghh , and G 4. Thus, for example, in Gph we may have p = lg~ (neutron) and h = 0g~ (neutron). In Gpp we may have p, = 0h~ (proton) and Pb = lh~ (neutron). In G4 we may have p, = lg~ (proton) and h, = 0g~ (neutron) but not p, = 0h~_ (proton) and ha = 2p~ (neutron) since this excitation is already included in diagonalizing matrix (1) in the model space of table 1. The intermediate states Pb and hb are related to the energy levels in 2°ST1; for example, we may have Pb = 0i~ (neutron) and hb = 2s~ (neutron). The energy denominators AEph, AEpp, AEhh, AEa and AEb, which are all positive for eqs. (5)-(8), should be the difference between the sum of the intermediate-state single-particle and single-hole energies and the initial-state single-particle and singlehole energies. Since accurate values of these single-particle and single-hole energies are not available, we shall take these denominators all as 2h~o or lhco (hco = 7.0 MeV), depending on the excitations in major oscillator quantum number (2n+ l). There is the problem of double counting for Gpp. If, in the calculation ot the Gmatrix, the Pauli operator has been accurately constructed so that all intermediate states with both particles above the Fermi sea (outside the closed shell of 208pb ) are included in G, then Gpp should not be included again since it is in G already. The treatment 2) of Pauli operator in the present work includes primarily intermediate states of very high energy, and hence the dominant part of Gpp should not be included in G. Their contributions will be tabulated separately in the following section, so that their effects can be studied individually. 3. Results and discussion

The energy levels of 2°8Bi were first calculated by use of the bare reaction matrix, as indicated in eq. (3). The resulting wave functions for low-lying states were found to have indeed very little configuration admixture. Thus, the low-energy levels of 20SBi can be grouped into multiplets by coupling the proton particle to the neutron hole of the dominant component in the wave function. Several such multiplets are compared with experiment in figs. 2a-2d. The general agreement between theory and experiment is indeed quite satisfactory. To study the effects of renormalization, we shall recalculate the levels of 2°8Bi, u s i n g Veff = G+Gpu+G;p+Ghh+G4, as discussed in sect. 2. Some particle-hole matrix elements of these levels are shown in table 3. It is seen that the dominant contribution usually comes from G. The calculation of Gph is quite complicated and thus needs a rather large amount of computer time. Hence, in the calculation with the renormalized G, the renormalization is included for the diagonal matrix elements only, that is, Gph ,Gpp, Ghh and G4 have been ignored for all off-diagonal matrix elements. This approximation may be reasonable, since the wave functions calculated with bare G show very little configuration admixture. Also, for all the diagonal matrix elements and a few off-diagonal matrix elements which we have calculated, the magnitudes of Gph , Gpp, ahh and G4 are small compared with G.

332

T.T.S. KUO

The results o b t a i n e d with the r e n o r m a l i z e d G-matrix are also shown in figs. 2a-2d. A l t h o u g h they give s o m e w h a t better agreement with experiment t h a n the bare Gmatrix, the overall effect of r e n o r m a l i z a t i o n is clearly rather small. This is c o n t r a r y to the situation in the particle-particle 3 - 6 ) case, for which its effect is very significant.. TABLE 3

Particle-hole matrix elements (in MeV) as defined by eqs. (2), (3) and (5)--(8) Jp

Jh

J~

(Ph-ll Vtph-1) s

G h~r

p~

h~_

f~

hgr

p.,}

hz~

i~

f~_

p~

4+ 5+ 2+ 3+ 4+ 5+ 6+ 7+ 3+ 4+ 5+ 6+ 2345678910113+ 4+

0.266 0.086 0.747 0.249 O.193 O.123 0.070 O.117 0.182 0.139 0.065 0.315 1.525 0.414 0.485 0.300 0.300 0.272 0.221 0.307 0.165 0.667 0.238 0.348

Gph

Gpp

Ghh

G4

--0.011 --0.001 0.130 --0.016 --0.048 --0.036 --0.076 0.077 0.075 --0.038 --0.055 0.018 0.375 0.116 0.042 0.004 --0.033 --0.027 --0.058 --0.019 --0.068 0.095 0.041 0.040

--0.000 0.014 --0.005 0.012 --0.005 0.013 --0.003 0.027 0.023 0.005 0.006 0.003 0.024 --0.010 0.010 0.003 0.011 0.006 0.008 0.009 0.003 0.023 0.027 0.005

--0.001 0.021 --0.011 0.026 --0.014 0.029 --0.008 0.057 0.023 0.004 0.008 0.004 0.025 --0.028 0.007 0.007 0.015 0.012 0.012 0.011 0.006 0.015 0.009 0.002

--0.025 --0.005 --0.098 --0.020 --0.020 --0.008 --0.007 --0.005 --0.014 --0.017 --0.002 --0.025 --0.086 --0.042 --0.051 --0.028 -- 0.023 --0.020 --0.012 --0.020 --0.006 --0.050 --0.013 --0.029

Except for the particle-hole doublet, the observed 1) general p a t t e r n for each multiplet, as shown by the figures, is the following. The states of Jmax = Jp+Jh and Jmin = lJp-Jh[ lie the highest at each end. I n accordance with the coupling rule of B r e n n a n a n d Bernstein 17), the spin of the lowest state is always J = Jp+Jh- 1. This p a t t e r n is well r e p r o d u c e d by o u r calculation. As shown by the figures, the renorm a l i z a t i o n effects tend to e n h a n c e this p a t t e r n ; they raise the Jm, x a n d Jmln states and depress the state of J = jp + J h - 1 for all o u r calculated multiplets. The individual effects of G, Gp~, Gpp, Ghh a n d G 4 can be seen in table 3. Except for the particle-hole doublet, Gph is repulsive for Jmin a n d Jmax a n d attractive for 3" = jp + J h - - 1. The m a g n i t u d e s of Gph a n d G , are generally larger t h a n those of Gpp a n d Ghh- It is clearly seen that the effective particle-hole interactions come m a i n l y f r o m G.

333

Z°sPb C A L C U L A T I O N S

(b)

Bi 2 ° s

4.7

\

Expt. o H - J Bore u H - J Renorm. ×

(a)

\

×\ \\ 45

5.3

.rhg/zfs/z -1 ])1, 13

5.19

5.2

\ \ \ \

I

I

6-j~ r 7-

4.8

>=

/

o- //x / / ×

4.6 41

Bi 2°e

"-~/

x Expt

\0/

4 15

[,.2 ..%],2

0

45-

w z bJ

o H - J Bore ° H- d Renorm.

o

[ h9/2 PI-/]1,, 15

o

// //

o

4.7

3.8 5.7

'\ 4.5

3.58

.,

// [h./z P3.z],, ,.

x~u

3.6

448

xO

~,,

~ o . ,//// / "~/

4.47 Ep l~ h

~P-(h I 2+

, 3+

I , 4+ 5* J~

I 6+

_~+

6.9

I 3+

I 4+

I 5"

i 6+

(c)

Bi zo8 Expt o H-J Bore o H-d Renorm. x

6.~

o

6.6 -

x~x

n

if0...-%] 13,15

,

I

[h9/2 iI-31/2 ]1), 16

o/ x

~

f

6.0

~

6.39

6.4 I

u

?

[,9,J;,2 ],,. ,2

I I 5.5

(d)

208

Bi x Expt. o H-d Bore n H - J Renorm.

o

i

xII

~'\,~--~

°

I~\

~'-~-"

~

LtJ Z bJ

~u

6.2

I

/

\o

~"'-%

o

o

/

\

o

o

]

o /I

\ ol

x---~-

\~//

~.xu/

6.0 5.21

5.2

5.92

~'p -- Eh 1 , 2-

1

, 4-

1

, 1 6jr

i

8-

i

I0-

I I÷

I 2+

I 3÷

I I 4÷ 5" jrr

I 6+

I 7+

I 8+

Ep -E h

Figs. 2a-2d. C o m p a r i s o n between the calculated and experimental energy levels. The calculation was done first with Vetr = G and then with Vet r = G - F G p h + G p p q - G h h q - G 4 ' denoted respectively by " H J b a r e " and " H J r e n o r m " . All energies are measured f r o m the ground-state energy o f 20spb. The unperturbed energies for each multiplet are obtained f r o m table 1. The lines joining the levels o f various J values are merely for the sake o f clearness in presenting the results.

334

T.T.S. KUO

The wave functions calculated with the renormalized matrix elements are tabulated in appendix 1. The wave functions calculated with the bare matrix elements are very similar. Alford et al. 1) have given some wave function structures of the h ~ f f ~ and f~p~-1 multiplets. In table 4 their experimental results are compared with our wave functions from appendix 1. As shown, the agreement for the h ~ f f I multiplet is reasonable, but there are some discrepancies for the f~p~ t multiplet. The calculated wave functions seem to have less admixtures from other multiplets than indicated by experiment. TABLE 4

Comparison of wave functions or

Admixture ( ~ ; squared amplitude) hezp~ 1 and h~p~ -1 h~f~r-1

Dominant configuration

expt. 3+ 4+ 5+ 6+ 3+

h.~fk -1 ,, ,, f~:p~-i

4+

,,

9 12 9 4 < 2

9

theory 4 13 2 2 2

0.2

expt.

2.6 0.1

theory

0.02 0.1

The largest discrepancies in energy seem to occur for the hlf~-1 multiplet (fig. 2a); the calculated 2 + and 7 + states are much higher than the observed values. The calculated spread of energy levels (about 600 keV) is considerably larger than the observed value (about 400 keV). In view of the fits for other multiplets, such as the h c f f 1 multiplet (fig. 2d) and for other nuclear regions 3-6), one would expect that our matrix elements should give a better fit for the h ~ f f 1 multiplet. In order to see whether the fit can be improved, it m a y be worth while to investigate the approximation introduced by ignoring the off-diagonal matrix elements of Gph, Gpp, Ghh and G,. This approximation could be studied by carrying out an exact calculation, but this has not been done, mainly because computer time was not available. If the wave functions of 2°8Bi are indeed of pure single-particle and single-hole nature, then the observed energy levels (after being corrected by the unperturbed energies) are directly related to the proton-neutron matrix elements, and from them we can derive the multipole components of the proton-neutron interaction. Schiffer et al. t) have carried out such an analysis for their data and observed some noticeable regularities. It is thus interesting to make a similar decomposition of our calculated results. The multipole moment ek is related to the particle-hole matrix elements by ~k =

(--l)k+l X

(iv J h

1

2(2J+l)(2k+l)(-1)s-JP-~h

[(2L+ 1)(2gh+ 1)1 J

1[ Veff] j p Jh 1)J.

Jh Jp (11)

2°spb CALCULATIONS

335

TABLE 5 Comparison of the multipole moments ~k in MeV [ph -1 ] k

0 1 2 3 4 5 6 7 8 9

h~i~ -~

h}f{:- t

h~rf~r-:

h~rPl.-:

--0.330 (--0.365) 0.056 (--0.010) --0.258 (--0.249) 0.042 (--0.017) --0.152 (--0.159) --0.001 (--0.034) --0.110 (--0.099) --0.051 (--0.036) --0.073 (--0.057) --0.028 (--0.042)

--0.216 (--0.227) 0.025 (0.012) --0.120 (--0.141) 0.004 (0.002) --0.062 (--0.080) --0.008 (--0.004) --0.023 (--0.041) --0.018 (--0.013)

--0.140 (--0.164) --0.042 (--0.041) --0.074 (--0.149) --0.009 (--0.005) --0.057 (--0.076) --0.003 (0.016)

--0.190 (--0.180) 0.025 (0.015) --0.086 (--0.109) 0.016 (0.028)

h~rP½-: --0.104 (--0.118) --0.030 (--0.019)

f l p } -1 --0.176 (--0.228) 0.050 (0.011)

iyp½-:

f{_pt_-1

--0.122 --0.153 (--0.110) (--0.200) 0.019 --0.027 (0.015) (--0.008)

The numbers in the upper row for each value of k are the experimental :) values. The calculated values (in parentheses) are deduced from the eigenvalues tabulated in appendix t after subtraction o f (~o-~h). TABLE 6 Multipole moments ~k in MeV calculated from the particle-hole matrix elements of G, Gph and Gsu m = G + G p h + G p p + G h h + G 4 Jp

Jh

k

Gsum

G

Gph

h¢~

i~

0 1 2 3 4 5 6 7 8

h~r

f~

--0.392 --0.035 --0.266 --0.034 --0.171 --0.041 --0.110 --0.041 --0.066 --0.042 --0.194 --0.070 --0.171 --0.024 --0.080 --0.001

--0.390 --0.027 --0.205 --0.025 --0.131 --0.031 --0.085 --0.033 --0.053 --0.039 --0.187 --0.119 --0.106 --0.051 --0.051 --0.029

--0.012 --0.019 --0.076 --0.012 --0.040 --0.005 --0.022 --0.002 --0.011 0.001 +0.002 0.009 --0.063 0.001 --0.027 --0.000

9

0 1 2 3 4 5

336

T.T.S.

KUO

Here ek is a measure of the kth multipole strength of the proton-neutron interaction. Note the similarity between this equation and eq. (5) for calculating Gph. The monopole moment ~0 is just the centroid energy, namely (2J + 1) (jp Jh 1[VefflJpJh- 1)s ~0

~

--

(12)

y(2J+l) J

Since the wave functions are not entirely of pure single-particle and single-hole nature, the experimental ak should be compared with the ek derived from the calculated eigenvalues. Thus we replace the particle-hole matrix elements in eq. (11) by the eigenvalues given in appendix 1 after subtraction of the unperturbed energies (Sp-- •h)' The calculated e k are compared with experimental in table 5. The agreement is satisfactory except for the quadrupole moment of the h~f~-1 m ultiplet. We recall that the deviation in energy was much larger for this multiplet than for others. Now this deviation seems to concentrate on the quadrupole moment. It is seen that the dominant contributions come from the monopole and the even multipoles. The monopole moments are more or less of the same magnitude, and are all attractive. This means that the average proton-neutron particle-hole interactions are all repulsive. We have seen that the core-polarization effects are not very significant for the energy levels of Z°SBi. The multipole decomposition may be a clearer and better way to study the core-polarization effects, since a k will pick up only the kth multipele of Gph (i.e., Jph = k) as is obvious from eqs. (5) and (11 ). Table 6 compares the values of % obtained with G, Gph, and G + Gph + Gpp + Ghh+ G4. It is seen that the contributions of Gph are, in fact, rather important for several multipoles. The fact that it does not have comparable importance for the energy levels is probably due to some cancellation of various multipoles within each Gph matrix element. The monopole moment ~0 of Gph has been found to be very small. Hence, Gph contributes very little in shifting the centroid energy. The monopole contribution from Gph is just the Jph = 0 part of Gph, as can be seen from eqs. (5) and (11). A rather serious discrepancy has been found between the theoretical and experimental positions of the isobaric analogue of the ground state of 2 ospb" The highest 0 + state given in appendix 1 corresponds to this analogue state. As shown in table 7, the admixtures of this state closely follow the (2.]+ 1) proportions for a pure isobaric analogue state. The calculated value of 14.23 MeV for the energy of this state is about 4 MeV lower than the empirical value (about 18 MeV) deduced from the Coulomb displacement energy is) Ac(ZOSpb ) ~ 18.8 MeV and the neutron-proton mass difference. This rather large deviation is difficult to explain. Its main source does not seem to be the matrix elements, since they have given quite satisfactory results for many other states of 2°8Bi. In fact they have already raised the isobaric analogue state by about 7 MeV with respect to the unperturbed energies. The inclusion of Gpu raises this state by about 1 MeV, but the inclusion of Gph, Gpp, Ghh and G4 gives practically the same

2°8Pb CALCULATIONS

337

result for this state as was obtained with G alone. A similar difficulty was also found for the giant dipole states 16) in 2°apb. The calculated position of the giant dipole resonance was about 2 MeV too low, but quite satisfactory results were obtained for the low-lying states. These deviations pose a major problem for the theory; a new mechanism m a y be needed for their explanation. TABLE 7 A d m i x t u r e s (squared amplitudes) o f t h e 14.23 M e V 0 + state

jdh-i

Calculated (2j+ 1)/10

~-I

~-1

~-1

~-1

~-1

~ 3~-1

1 1

0.62 0.8

0.49 0.6

0.4 0.4

0.23 0.2

1.06 1.40

It is interesting to note that the spectrum of 2°8Bi can also be satisfactorily reproduced by use of a simple spin-dependent f-function force, as shown by Birbrair and G u m a n 9) and by Kurath 19). This indicates that the main part of the effective force for 2 08Bi can be represented by a 6-function force with parameters properly adjusted. This is supported by the fact that the present calculation, which employs a nucleonnucleon potential based on the scattering data and includes the many-body corrections, gives results very similar to those from a 6-function force (without many-body corrections). The results of K i m and Rasmussen 7) have been compared with our results, and found to be quite similar. The reasons for the close similarity are obscured by the various approximations made in their calculation. A significance of the present calculation is that the effective interactions deduced from the H J potential has given satisfactory results for many other nuclei 3- 6). Hughes et al. 8) has pointed out that the Kim-Rasmussen force gave a rather poor result for 2 06T1" However, rather good results for this nucleide have been obtained 2 o) by use of the same effective interactions. Equally good results were also obtained 2 o) for 2 t OBi' 2 ~opb ' 210po and 2 o6pb by calculations with the same effective interactions. As mentioned in sect. 1, the pure spherical shell model can be used with more confidence for 208Bi than for lighter nuclei. This can be checked by looking to see if some observed states of rather low energy are completely missing in our calculation. Any such missing states m a y be composed predominantly of other components. A detailed comparison of our calculated results (appendix 1) with experiment 1) shows that all the lowest 36 observed levels with spins identified (or tentatively identified) are well reproduced by our calculation. The four lowest observed states * with weak intensities and unidentified spins are at 4.295, 5.126, 5.186 and 5.218 MeV, the first and last of which are observed in the (3He, d) reaction and the middle two in the (d, t). Thus these states should have small admixture of the 2p~ hole for the (3He, d) reaction and 0h~ particle for the (d, t). As shown by fig. 3, the 4.795 MeV state is clearly missing t Energies measured from 2°aPb.

338

T.T.S.

KUO

from our calculation. The higher three states may correspond to the calculated states in the ftf~- 1 multiplet. These states may be weakly excited by the (3He, d) and (d, t) reactions since f~ and f~- 1 are the first excited single-proton and single-neutron orbits in 2°9Bi and 2°7pb. It is not certain whether the 4.795 MeV state belongs to 2°8Bi or not; but if it does, this will be a major defect of our present model. Thus, except for the few rather uncertain weakly-excited states, we may conclude that the low-energy levels are remarkably well described by the pure spherical shell model and an effective interaction deduced from the H J free nucleon-nucleon potential. This success of the Bi 5.5

208

-

6 +

(H~,d)-{d,t)--

2*--

(d, t ) - -

4~

_

_

5: .>5.0 (.9 or"

I.IJ

(He3,d) -

4.7

EXPT.

CALC.

Fig. 3. States which are weakly excited in (SHe, d) and (d, t) reactions. The calculated states are mainly fk proton and f~r neutron hole.

present calculation suggests that a more careful calculation of 2°SBi may be very useful in revealing some detailed aspects of the effective nuclear forces. The author is very grateful to J. P. Schiffer and D. Kurath for many helpful discussions, and for making their results available before publication. He deeply appreciates the warm hospitality of M. Peshkin and other colleagues at Argonne during his stay there as a summer visitor. The computer program used in the present calculation was constructed in collaboration with J. Blomqvist for the calculation of 4°Ca [ref. 15)] and 2°Spb [ref. 16)].

Appendix This is a tabulation of the wave functions and eigenvalues of 2 0 8Bi calculated with Veff of eq. (4). The eigeuvalues are measured from the ground state of 2°Bpb. The configurations are labeled by numbers as explained in table 1 ; the first number refers to the proton particle and the second one to the neutron hole. The configurations with amplitudes less than 0.1 for all States included in the table are omitted. All orbitals of table 1 have been included in the calculation.

E(MeV)

6.395 6.868 7.429 7.905 9.183 14.228

5.769 6.685 7.175

4.744 5.236 5.810 6.207 6.600 6.887 6.976

4.366 4.696 4.753 5.405 5.573 6.133 6.583

J~

0+

1+

2+

3+

--0.200 --0.006 0.027 0.949 0.044 --0.097 --0.135

11,12 0.022 0.025 0.102 0.007 0.100 0.977 0.105

0.032 --0.031 0.078 --0.055 --0.017 --0.149 --0.662

11,11 --0.038 --0.018 0.007 --0.006 --0.023 --0.045 --0.054

11,12

11,11

12,12

0.156 --0.034 --0.139

0.091 0.966 0.028

11,12

12,12

0.217 --0.634 --0.218 --0.574 --0.099 0.405

11,11

--0.654 --0.235 0.043 0.288 0.410 0.513

13,13

11,13 0.975 0.015 --0.210 0.004 --0.001 0.004 0.021

--0.967 0.014 --0.037 --0.159 --0.046 0.068 0.069

11,13

0.929 --0.057 --0.125

12,13

--0.286 0.675 --0.046 --0.56t --0.122 0.361

11,14 --0.212 0.145 --0.958 0.050 0.032 --0.087 0.031

--0.040 --0.007 0.037 --0.129 --0.029 --0.008 --0.011

12,11

--0.162 --0.026 0.140

13,13

0.031 --0.083 0.719 0.160 --0.588 0.322

14,14

15,15

12,12 0.017 0.055 --0.024 0.051 0.109 --0.020 0.075

--0.004 --0.032 0.075 --0.022 0.270 --0.562 0.119

12,12

0.085 --0.193 --0.198

13,14

--0.038 0.066 --0.656 0.432 --0.563 0.245

12,13 0.004 --0.136 0.032 0.974 0.101 0.006 --0.094

0.014 0.973 --0.114 --0.005 0.015 --0.024 --0.011

12.13

0.087 0.018 --0.211

14,14

0.664 0.274 0.023 0.246 0.380 0.528

16,16

12,14 0.002 0.119 --0.006 0.129 --0.969 --0.099 0.014

0.048 --0.144 0.934 0.037 --0.204 --0.082 0.020

12.14

--0.227 0.080 --0.911

14,15

12,15 0.012 0.968 0.152 0.109 0.110 0.065 0.002

0.009 --0.067 0.085 --0.003 0.163 --0.410 0.414

13,13

0.061 0.002 --0.134

15,14

Wave functions

A p p e n d i x (continued)

13,13 --0.018 --0.040 0.026 --0.034 --0.085 0.036 --0.169

--0.031 --0.031 0.013 --0.029 --0.179 --0.074 0.245

13,14

13,15 --0.024 --0.038 0.035 0.082 --0.014 0.120 0.966

0.058 0.042 0.241 0.087 0.891 --0.133 0.140

13,15

--0.013 --0.055 0.134 --0.061 --0.050 0.048 --0.080

14,13

--0.004 --0.104 0.063 --0.022 --0.005 0.400 --0.203

14,15

16,16 0.117 0.003 0.038 0.192 0.143 0.527 0.470

~D

o z

c

4.418 5.972 6.965

6.346 6.784

6.856

5.593 7.058

7+

8+

2-

3-

3.680 4.275 4.488 5.393 5.666 6.026 6.885

5+

4.103 4.755 5.293 6.095 6.768

3.719 4.213 4.615 4.718 5.111 5.486 6.081 6.776

4+

6+

E(MeV)

dg

0.158 --0.984

16,11

0.984 0.150

11,16

16,11

0.235

11,16

--0.972

0.968 --0.182

11,12

11,11

0.032 --0.591

11,12 0.029 --0.999 0.039

11,I1 0.020 --0.021 --0.153

11,12

11,11

--0.041 --0.167 0.022 0.975 --0.071

11,13 0.128 --0.990 --0.035 0.006 --0.011 --0.007 0.026

11,12 0.022 0.007 0.050 --0.003 0.159 --0.984 0.040

0.012 0.001 0.059 0.028 --0.299

11,12 --0.046 --0.058 --0.138 0.007 0.018 --0.001 0.976 --0.058

11,11 --0.002 0.021 0.008 0.022 0.035 0.049 0.012 --0.198

0.248 0.784

16,16

11,13 --0.999 --0.028 0.026

--0.989 --0.132 --0.014 --0.058 0.018

11,13

11,14 0.041 0.041 --0.997 0.005 0.021 --0.045 0.020

11,13 0.221 --0.924 --0.291 --0.025 --0.003 --0.029 --0.075 0.027

12,12 --0.016 --0.005 --0.332

--0.142 0.976 0.023 0.151 --0.024

11,14

11,15 --0.990 --0.125 --0.044 0.008 0.016 --0.021 0.024

11,14 --0.184 --0.342 0.914 --0.040 --0.022 0.006 0.093 --0.016

16,16 0.024 0.038 0.929

0.000 0.014 0.130 0.010 --0.642

12,12

12,12 --0.012 --0.021 --0.011 0.012 0.112 0.006 --0.812

11,15 --0.956 --0.143 --0.235 --0.014 0.003 --0.006 --0.079 0.014

0.013 0.025 --0.985 0.037 --0.052

12,13

12,13 0.010 0.004 0.009 0.981 0.186 0.027 0.012

12,12 0.009 0.002 0.014 0.020 0.069 0.076 0.016 --0.614

0.024 0.035 0.075 0.127 0.699

16,16

12,14 --0.014 0.008 --0.014 0.190 --0.958 --0.156 --0.086

12,13 --0.003 0.008 --0.010 0.230 --0.938 --0.197 0.013 --0.056 13,13 0.013 0.025 0.013 0.030 --0.056 0.026 0.455

12,14 --0.003 0.018 0.031 0.206 0.263 --0.933 0.003 --0.015

Wave functions

Appendix (continued)

16,16 0.006 0.003 0.012 0.003 0.069 0.053 0.341

12,15 0.014 0.048 --0.022 --0.948 --0.161 --0.237 0.010 --0.023

13,13 0.006 0.003 0.012 0.012 0.068 0.060 0.010 --0.381

14.13 --0.001 0.000 0.000 0.047 0.094 0.113 --0.044 0.087

16,16 0.014 0.041 0.047 0.019 0.028 0.060 0.121 0.648

c o

12,16 0.031 --0.052 0.880 --0.468

12,16 0.126 0.939 --0.314

12,16 --0.056 0.997

11,16 --0.997 0.050 0.029 --0.012

11,16 0.989 --0.127 0.004

11,16 0.998 0.055

5.373 5.780 6.159 6.291

5.474 6.028 6.161

5.306 6.420

5.957

8-

9-

10-

11-

11,16 --0.999

12,16 0.033 --0.119 --0.003 0.890 --0.430

11,16 0.115 --0.978 --0.099 --0.127 0.010

5.314 5.429 5.845 6.101 6.188

7-

12,16 --0.047 0.006 --0.130 --0.500 0.852

11,16 0.006 --0.994 0.090 --0.006 0.015

5.284 5.463 5.879 6.120 6.241

6-

12,16 0.131 0.076 0.979 --0.056

11,16 0.977 0.140 --0.139 --0.038

5.455 5.802 6.254 6.482

5-

12,16 --0.022 --0.594 0.798

11,16 --0.986 0.114 0.048

5.652 6.216 6.402

4-

16,12 --0.014 --0.009 --0.108

16,13 0.052 0.990 --0.011 --0.124

16,12 --0.036 0.007 --0.045 --0.015 --0.126

16,13 --0.180 0.091 0.960 0.067 0.172

14,16 0.008 0.010 --0.100 --0.008

13,16 0.082 --0.031 --0.106

16,13 0.035 0.313 0.941

16,14 0.007 0.116 0.474 0.869

16,13 --0.232 --0.136 0.855 0.184 0.391

16,14 0.071 0.011 0.160 --0.861 --0.472

16,12 0.002 0.039 0.028 --0.221

16,12 --0.003 0.045 0.116

16,14 --0.146 --0.100 --0.484 0.379 0.771

16,15 0.979 0.023 0.157 0.050 0.103

16,13 0.159 --0.952 0.066 0.231

16,13 0.134 0.791 0.575

16,15 0.953 0.087 0.144 0.087 0.223

16,14 0.011 0.254 0.039 0.944

z

,-]

342

T.T.S.

KUO

References 1) W. P. Alford, J. P. Schiffer and J. J. Schwartz, Phys. Rev. Lett. 21 (1968) 156; M. Moinster, J. P. Schiffer and W. P. Alford, preprint (Argonne National Laboratory) 2) J. R. Erskine, Phys. Rev. 135 (1964) B l l 0 3) T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 4) G. E. Brown and T. T. S. Kuo, Nucl. Phys. A92 (1967) 481 5) T. T. S. Kuo and G. E. Brown, Nucl. Phys. A l l 4 (1968) 241 6) T. T. S. Kuo, Nucl. Phys. A90 (1967) 199 7) Y. E. Kim and J. O. Rasmussen, Phys. Rev. 135 (1964) B44 8) T. A. Hughes, R. Snow and W. T. Pinkston, Nucl. Phys. 82 (1966) 129 9) B. L. Birbrair and V. N. Guman, Sov. J. Nucl. Phys. 1 (1965) 693 10) B. H. Wildenthal, P. M. Preedom, E. Newman and M. R. Cares, Phys. Rev. Lett. 19 (1967) 960 11) V. Gillet, A. M. Green and E. A. Sanderson, Nucl. Phys. 88 (1966) 321 12) J. H. E. Mattauch, W. Thiele and A. H. Wapstra, Nucl. Phys. 65 (1965) 1 13) N. de Takacsy, Nucl. Phys. A95 (1967) 505; and Shell-model calculations in the x60 region with the Tabakin potential, preprint (McGill University, Montreal) 14) A. E. L. Dieperink, H. P. Leenhouts and P. J. Brussaard, Nucl. Phys., to be published 15) J. Blomquist and 1L T. S. Kuo, to be published 16) T. T. S. Kuo, G. E. Brown and J. Blomquist, to be published 17) M. H. Brennan and A. M. Bernstein, Phys. Rev. 120 (1960) 927 18) G. M. Temmer, Proc. Int. Conf. nucl. phys. (Academic Press, New York, 1967) p. 223 19) D. Kurath, private communication 20) T. T. S. Kuo, unpublished