The effects of magnetic dipole core polarization in 206Pb and 207Pb

Nuclear Physics A159 (1970) 209--221; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE EFFECTS OF MAGNETIC DIPOLE C O R E P O L A R I Z A T I O N I N 2°6pb AND '~°TPb K. H A R A D A t and S. PIq~I'EL tt

Bartol Research Foundation of The Franklin Institute, ~ Swarthmore, Pennsylvania 19081

Received 17 August 1970 Al~tract: The effects of magnetic dipole core polarization on dipole moments and MI transition rates in 2°6pb and 2°TPb are studied in the framework of the weak-coupling model. The coupling potential between a particle and the MI states of the core nucleus a°SPb is calculated microscopically and used to estimate the admixtures of these magnetic dipole states in the relevant states of 2°6Pb and 2°~pb. The qualitative agreement with experiment is good, except for the p~- x ~ p½-1 transition strength and the f~- x+dipole moment. The calculations also yield some general features of magnetic moments, MI decays and their interrelation based on purely geometrical considerations. Finally, exchange effects are studied and found to be rather small.

1. Introduction In the heavier nuclei, the observed magnetic dipole moments deviate appreciably f r o m the single-particle values 1-2). This trend has been nicely explained in terms of a spin-polarization effect of the dosed shells. This same effect has similarly been used to explain discrepancies in both M1 transition rates and dipole moments in the Cu isotopes 3). Recently some interesting data on dipole moments and transition rates in 2°6pb and 2°7pb have appeared. In most cases significant deviations f r o m the predictions of the single-particle model exist. It is the purpose of this investigation to see whether this data can be understood in a consistent fashion by including the effect of spin polarization of 2 ospb" Magnetic dipole states in 2°8pb have been calculated by Gillet et al. +) and Broglia et al. 5). Two 1 + states are predicted as mixtures of the (lh~-lhT~ 1) proton and (li~-liT~ t) neutron configurations. Though neither has been identified experimentally, the calculations of refs. +, 5) indicate that they probably lie near 7 MeV, with the upper state exhausting most of the M1 strength. It is expected that the effect of spin polarization in 2°6pb and 2°Tpb can be described by considering the small admixtures of these magnetic dipole states. I f we knew the phenomenological coupling potential which excites the core nucleus to the magnetic dipole states, we could calculate the mixing amplitudes using the particle-vibration t On leave from Japan Atomic Energy Research Institute, Tokai-mura, Japan. tt Present address: Department of Physics, University of Colorado Boulder, Colorado. 1; Work supported in part by the US Atomic Energy Commission. 209

210

K. HARADA AND S. PITTEL

weak-coupling model 6). Although a generalization of the phenomenological theory to the case of spin-flip is possible 7), not much is known about it at present. On the other hand, in the microscopic theory of the inelastic scattering process s), the form factor (which is equivalent to the coupling potential) is calculated microscopically. In view of the good agreement obtained in such calculations, we follow this procedure. A detailed description of our calculation including a discussion of this procedure is given in sect. 2. In a recent letter 9) we discussed in some detail the M1 decay of the 1.70 MeV 1 + state in 2°6pb. For this reason we focus here on 2°7pb, briefly noting the extensions needed for 2 oepb" The numerical results of our calculations are presented in sect. 3, together with a comparison with experiment. Since we follow the core-particle coupling model, the results in sect. 3 do not include exchange effects. Ir sect. 4 we describe a means of calculating these effects within the same formalism. 2. Description of the calculation The low-lying states in 2°Tpb are assumed to be primarily single-hole states with small admixtures arising from the coupling to M1 states in 2°spb. A state with angular m o m e n t u m j is written as lJ) = [2°7pb(J),

2°spb(O+);J)+e~,12°TPb(J'),2°spb(l+);J) •

(1)

Depending on the process under consideration, different e~, are important. For magnetic dipole moments, we need to evaluate only the single mixing amplitude 8~. To study the effects on magnetic dipole transitions, e.g., from [Jl) to lJ2), we must Jl and 8ji. J~ calculate both 8~, The calculation of the mixing amplitudes requires first a knowledge of the coreparticle coupling potential. In this analysis we follow a procedure first developed in the microscopic theory of inelastic scattering, described in detail in ref. 8). Adopting a similar notation, we write the core-particle interaction in the form

n' = E • (--)L+S+Jvs(JLs~(r,, A)" adJhs,r(P,,~,)).

(2)

i LSJ

The index i runs over all valence nucleons. In the case of 207pb ' only a single term is necessary. To describe the coupling between the 0 + and 1 + states in 2°sPb, we need consider only two possible sets of (LSJ) values, either (011 ) or (211). The coupling potential is then written as

n' = E V,(YL,,(r, ,i).

(3)

L

We next define the form factor through the relation FLti(r) ---- Vl(2°spb(l+)llJLtt(r, A)ll2°spb(0+)),

(4)

and calculate it u.~irtg the shell-model wave functions of ref. 4). To be consistent we

211

2°6'2°TPbPOLARIZATION

Foil (r)

~

(u)

~ . . . . ~

"2 ~0

s.

o

Lt_

~lp.

r(fm)

-2.

-4.

Fig. 1. L = 0 f o r m factors for the two 1 + states in 2°ePb, calculated with the C O P a n d C A L forces o f GiUet. T h e indices (U) a n d (L) distinguish between t h e u p p e r a n d lower 1 + states.

2.

o.)

.=_ "2

F2ti (r)

~(L//

~ 0 . r(fm)

LL. N

Fig. 2. L = 2 f o r m factors for the two l + states in 2°Spb. T h e f o u r cases s h o w n h a v e t h e s a m e m e a n i n g as in fig. 1.

212

K. HARADA AND S. PITTEL

use the same interaction parameters (COP and CAL) and the same oscillator constant (v = 0.1849 l~m-2) as used by Gillet et al. In figs. 1 and 2 we show the resulting form factors for L = 0 and L = 2 respectively. The indices (U) and (L) distinguish between the higher and lower magnetic dipole states obtained by Gillet et al. Comparing figs. 1 and 2 we see that the magnitudes of the F21 t (r) are smaller than the F01 l(r) by about a factor of 3. We further see that the two particle-hole configurations making u p t h e 2°sPb 1 ÷ states add constructively in the C A L ( U ) and COP (L) cases but destructively in the others. The fact that the COP and CAL forces focus their strength in the opposite states is closely related to some specific features of these two forces. We can write the protonneutron hole and neutron-neutron hole force strengths in MeV for these two mixtures as follows: Vp~ = VN~ = I/],~ = VN~ =

[-0.5+6.5(c71. ~2)], [ - l Z . 0 - 5.0(if, • ~2)], [12.75+12.25(ffl " ~2)], [ - 1 1 . 6 2 5 + 4 . 1 2 5 ( ~ . ~2)].

in CAL (5) in COP

For the present problem only the ~ • 5 term contributes to the form factor. It is clear from the relative signs of the 6 • ~ terms in the PI~ and N ~ forces that the two mixtures lead to constructive interference for different states. Once the coupling potential is known, we can easily evaluate the mixing amplitudes. Anticipating them to be small, we use the following perturbation theory expression:

~, = - (2°7pb(J)' 2°8pb(O+); JlH'12°vPb(J')' 2°aPb(l+);J)

(6)

E(2°TPbj') - E(2°vPbj) + E(2°Spbl +) - E(2°sPb0 +) The energy denominators (denoted by A) involve two energy differences. The zo 7pb splittings are taken directly from the experimental spectrum as given in ref. a) while those in 2°8Pb are taken from the calculated energies of the same reference. The numerator is evaluated using standard Racah techniques. The resulting expression is j

1

1

Y
(7)

where the last factor is the overlap integral of the form factor for the relevant singleparticle orbitals. We see from eq. (7) that both the L = 0 and L = 2 terms contribute, but in a manner prescribed by the relative values of the reduced matrix elements . In table 1 we show the calculated values of
2o6, 2OTpb POLARIZATION

213

TABLE 1 Calculated values of J l J2

(jl[l~Lxallj~>

for v = 0.1849 fin - 2 i n M e V

CAL(L)

CAL(U)

COP(L)

COP(U)

P~rP½

L = 0 L = 2 sum

0.253 0.025 0.278

--0.679 --0.072 --0.751

1.144 0.119 1.263

0.060 0.008 0.068

P~rP~

L = 0 L = 2 sum

--0.283 0.023 --0.260

0.759 --0.064 0.695

--1.279 0.106 - - 1.173

--0.067 0.007 --0.060

p~p~.

L = 0 L = 2 sum

0.089 --0.071 0.018

--0.240 0.203 --0.037

0.405" --0.335 0.069

0.021 --0.021 0.000

f~f~

L = 0 L = 2 sum

0.490 0.049 0.540

--1.283 --0.136 -- 1.419

2.180 0.227 2.407

0.103 0.013 O. ! 16

f~f~r

L = 0 L = 2 sum

0.274 --0.089 O. 186

--0.717 0.243 - - 0.474

1.218 --0.407 0.811

0.058 --0.023 0.035

f~.f~

p~zf~

L = 0

--0.424

1.111

-- 1.888

- - 0.089

L = 2 sum

0.057 --0.367

--0.157 0.954

0.262 - - 1.625

0.015 --0.074

L = 0 L = 2 sum

0. 0.084 0.084

0. --0.262 -- 0.262

0. 0.442 0.422

0. 0.035 0.035

structively, whereas for M1 transitions (j :/:j') they add constructively. Thus we expect in general larger effects on magnetic dipole transition rates. (ii) The signs of t h e j = l - ½ a n d j = l+½ terms for dipole moments are opposite. Thus one will be increased relative to its Schmidt value and one decreased. We will see shortly that the j = l - ½ moments are reduced. (iii) F o r j = j ' = p~, the L = 0 and L = 2 terms almost completely cancel. Thus the correction to the p~ moment will be small. (iv) For l ~ l' (i.e.,/-forbidden M 1 transitions), there are no L = 0 contributions. However the "tensor-like" @'211(r, t~) F211 (r) term in the coupling potential can induce weak M 1 transitions in such cases. Since we use harmonic oscillator wave functions to evaluate the radial integrals. the wave functions for the j = 1_½ state are identical. Thus these general features result solely from the relative signs and magnitudes of the "geometrical" reduced matrix elements. For this reason, we expect these features to appear throughout the entire periodic table and, in fact, the second and third do. Arima et al. 1), in a survey of magnetic dipole moments, noted that all lie between the Schmidt lines and also that the moments of p~ nuclei are always very close to their Schmidt values. The necessary expressions for magnetic dipole moments and transition strengths

214

K. HARADA AND S. PITFEL

can also be evaluated with standard techniques. The resulting expressions are

=

2

j

~/~-~(2°sVb(O+)ll.,#'(M1)ll2°SVb(l+ )),

(8)

in units of eh/2Mc and B(MI,j~ ~ J z ) = 1 2A+l

i

+ ( 4 ~ - - ~ + 1 ) ~ ; - 4~(2jz + 1)e~;)12,

(9)

in units of (3/4n)(eh/2Mc) z. Eqs. (8) and (9) involve in addition to the 8~, some reduced matrix elements of d / ( M 1 ) . The Schmidt moments and the matrix elements for single-hole transitions are well known and given in many references 9). The matrix elements between the two I + states in Z°sPb and the ground state were calculated for both the CAL and COP wave functions of ref. a) and are shown in table 2. As mentioned before, the upper state in both case exhausts most of the M1 strength. TABLE 2

MI matrix elements <2°SPb(0+)]l..&'(M1)Hz°sPb(l+)) in units of COP(L) COP(U) CAL(L) CAL(U)

(eh/2Mc)V'3~

-- 7.76 12.14 -- 3.74 13.97

To calculate the effects of M1 core polarization in ~°6Pb we follow essentially the same procedure as for 2°7pb. The states are now written as lJ+> = I2°6pb(J+), 2°sPb(O+); J+>+n~,12°6pb(J'+), 2°spb(l+); J+>.

(10)

We assume that the states of 2°6pb and 2°Spb are described by the wave functions calculated by True 11) and Gillet et al. 4) respectively. Since the states calculated by True already include holes in the i V and h~ neutron orbits, the magnetic dipole states built on 2°6pb will certainly be different from those in 2°spb. However because of the high degeneracies of these two orbits and their generally small vacant probabilities, we expect only a slight error to arise from this approximation. The mixing amplitudes sss, are again calculated in perturbation theory and related directly to the coefficients ~(j[l~t~adij' ) (jIFLll(r)Ij'). Finally, expressions analogous to (8) and (9) can be derived giving the corrections to magnetic moments and B(MI) values in terms of the calculated 8sJ,.

215

206,207pb POLARIZATION 3. Numerical results

3.1. 2°Tpb MAGNETIC DIPOLE MOMENTS Experimentally, the magnetic dipole moments of the ½- ground state and the 0.57 MeV ½- state have been measured. The values obtained were 0.59 and 0.65 nuclear magnetons respectively ~o), to be compared with the Schmidt predictions of 0.64 and 1.36. While the moments of the p ~ l and f~ ~ levels have not been measured, it is expected that they should lie above the Schmidt value of - 1.91 nuclear magnetons in both cases. The calculated mixing amplitudes and dipole moments are given in table 3. The TABLE 3

Mixing amplitudes and dipole moments for states in 2°Tpb

j

e~ (L)

~c11¢

(U)

l/'/©zp

/lSchmldt

(in units of ehl2Mc)

P~"

COP CAL

--0.0043 -- 0.0013

0.0000 0.0021

0.61 0.61

0.59

0.64

f~r

COP CAL

--0.0291 -- 0.0073

--0.0011 0.0157

1.15 1.12

0.65

1.37

P~r

COP CAL

0.0515 0.0126

0.0024 --0.0282

-- 1.58 -- 1.52

--1.91

l~.

COP CAL

0.0506 0.0126

0.0021 --0.0274

-- 1.54 -- 1.48

1.91

symbols (U) and (L) again distinguish between the " u p p e r " a n d "lower" M 1 states in 2°spb. The first feature to note is that the small change in the p~r 1 dipole moment is nicely reproduced. This was due to the almost exact cancellation between the L = 0 and L = 2 contributions as shown in table 1. The f~ 1 dipole moment is not reproduced very well, although the fact that the calculation gives a result smaller than the Schmidt value is encouraging. An analysis based on the method of Arima et al. also yielded a value of about 1.1 nuclear magnetons. Finally, for the remaining two levels, we obtain the expected trend f o r j = / + ½ levels, namely that the dipole moments should lie above the Schmidt values. 3.2. 2°TPb MI TRANSITIONS Two magnetic dipole transitions in 2°Tpb have been studied experimentally. Hausser and W a r d 12) recently reported measurements of the 0.89 MeV ½- ~ ½- M 1 transition. By measuring the B(E2) value f r o m Coulomb excitation and the M l-E2 mixing ratio from the ?-ray angular distribution, they concluded that the M1 part of

216

K. HARADA AND S. PITYEL

the transition, is retarded by a factor of 18 relative to the Weisskopf estimate. This implies a B(M1) of 0.055 s.p.u, as compared to the single-hole value of 0.651 s.p.u. Chilosi et al. 13) studied a number of transitions in 2°TPb. By making what they considered a reasonable guess for the lifetime of the 2341 keV ~r- level, they concluded that the "effective" g-factor for the ~ - --, ~ - MI transition should be approximately ½ the free-neutron g-factor. This would imply a B(M1) for that transition of 0.21 s.p.u., the single-particle model giving 0.84. The calculated results for these two transitions are presented in table 4, together TABLE 4

Mixing amplitudes and B(M1) values for transitions in ~°TPb (L)

(U)

(L)

(U)

B(MI; j~-x ...j~- 1) in s.p.u. calc exp single particle

P~P~r

COP CAL

--0.0643 --0.0158

--0.0031 0.0348

0.0691 0.0165

0.0033 --0.0381

0.40 0.36

0.06

0.65

fif~

COP CAL

--0.1024 --0.0262

--0.0044 0.0541

0.0680 0.0164

0.0031 --0.0376

0.45 0.40

0.21

0.84

P~f~

COP CAL

--0.0195 --0.0043

--0.0015 0.0111

0.0144 0.0031

0.0011 --0.0083

0.0029 0.0048

0.

with the predicted strength for the/-forbidden p~ ~ ---, f~-I transition. For the two A1 = 0 transitions, the effect of core excitation was to lower the B(M 1) value, but in neither case are they lowered sufficiently. The discrepancy in the p~ 1 --, p~- 1 transition strength is particularly large. Compared to the measurement of the B(E2), the determination of the mixing ratio for this transition is rather indirect. It would be interesting to have a more direct measurement of the lifetime of the p~ t level, in order to have a check on the experimental B(M 1) value. The agreement for the f~ ~ --, f~ t transition strength is comparable to that obtained in our other calculations. It should however be noted that the "experimental" value was arrived at indirectly. Furthermore, it is known t t ) that the f~": state is not a p u r e singre-hole state, containing about 26 ~ of the configuration (i~1× 3 - ) i _ . We did not consider this in our calculation. We predict the/-forbidden p~l --, f~-i M1 transition to be hindered by a factor e f about 100 relative to the two AI --- 0 transitions. A measurement of this decay although difficult would be very interesting. 3.3. 2°6pb MAGNETIC DIPOLE MOMENTS

The g-factor of the 803 keV 2 + state in 2°6pb was recently measured by Zawislak and Bowman 15). They obtained a value of g(2 +) = - 0 . 0 1 + 0 . 0 7 , which they

206, 2o7pb POLARIZATION

217

compared with various model predictions. They noted that if one uses any reasonable shell-model wave functions and the free-neutron g-factors, then the resulting g(2 +) agrees with experiment. On the other hand, using the same wave functions, but with "effective" g-factors obtained from 2°~pb single-hole measurements, leads to values in disagreement with their experimental result. Our results for the magnetic moment and g-factor for this level are.presented in table 5. The two-hole wave functions of True st) were used to describe the 2°6pb TABLE 5 Mixing amplitudes and magnetic moments for 803 keV 2+ state in 2°~Pb

e~

COP CAL

t~o.,o

(L)

(U)

0.0023 0.0005

0.0003 --0.0013

a(2 +)

calc 0.056 0.051

0.028 0.025

exp

--0.01 +0.07

states before coupling to the MI states in 2°spb. Both the CAL and COP mixtures give results in agreement with the experimental value. The interesting point to note is the smallness of the mixing amplitudes in this case. This occurs as a result of a cancellation between the p~ 1 f~ 1 and p~ 1 p~l components in the True wave function. It is because of these small mixing amplitudes that the calculation using freeneutron g-factors gave reasonable results. By using "effective" g-factors, one loses this cancellation effect. This example illustrates quite clearly the possible dangers in using "effective" g-factors. Zawislak and Bowman also reported a measurement of the g-factor for the 2385 keV 6- state in 2°6Pb. Since the shell-model wave function for this state is predominantly p~l iT t, our assumptions concerning the use of Gillet's wave functions are not valid in this case. This level can therefore not be properly treated with our approach. 3.4. 2°6pb M1 TRANSITION The M1 decay of the 1.70 MeV 1+ state in 2°6pb has been studied by Metzger 14) using the resonance fluorescence technique. He obtained a B(M1) value for this transition of 0.095+0.018 s.p.u., a factor of 3 smaller than predicted by True. In a recent letter 9), we discussed this decay in detail. We first noted that by replacing the True wave functions by experimental wave functions 17), the B(MI) could be reduced to 0.21 s.p.u. We then considered the influence of magnetic dipole core polarization on the 1.70 MeV 1+ state and on the ground state. With rather small mixing amplitudes, the transition strength was reduced by a factor of approximately 0.5-0.6, depending on the use of the CAL or COP mixture. A more detailed discussion of these results may be found in ref. 9). We have also studied the ground state decay of the 1~" state in 2°6pb, predicted by True at 2.17 MeV. Calculations without a tensor force suggest that this state is almost

218

K. HARADA. AND S. PI'I'rEL

purely f~ i p~'L, so that the decay should be/-forbidden. In our formalism the transition can proceed as a result of the ~21x(P, ~) F211(r) term in the coupling potential. The COP and CAL mixtures yield B(M1) values of 0.21 x 10- s s.p.u, and 0.33 x 10- s s.p.u, respectively, approximately l0 s times weaker than the allowed transitions in 2°6pb. This retardation is much greater than that calculated for the /-forbidden p~l ~ f~-i decay in 2°7pb. It arises as a result of a cancellation in the contributions of the p~l and f~l components in the ground state wave function to the mixing amplitudes. The inhibition we calculate for this decay is consistent with the upper limit on its strength set by Metzger 16). 4. Exchange effects

In the core-particle coupling scheme, the possibility of exchange contributions is either neglected or renormalized into the collective parameter. In this section we outline an approach which permits us to treat these effects within the same framework described in sect. 2. In the present calculation, we are interested in evaluating matrix elements of the type ( j , - 1, (Jl J2 t)x + ; jmJ VIj- 1;jm), wherejlj~ 1 represent either the i~ i7~1 neutron spin-orbit pair or the lq h?t I proton spin-orbit pair. For the proton particle-hole pair, the treatment of sect. 2 is appropriate since exchange effects do not contribute. For the neutron particle-hole pair, it can be shown that the correct expression for this matrix element (denoted by A) is h = (-)1+ ~x/3(--~ 1) Z (--)k(2k+l)W(j~J'-~; kl) k

x ( ( j i ~ ; klVIj'i~," k ) - - ( - - ) J ' + ~ - k ( j i ~ ; klVli~ j,"" k)).

(10)

In the core-particle formalism, only the direct term (shown in fig. 3a) is calculated.

Io) Direct

(b) Exchange

Fig. 3. Graphs which contribute to MI core polarization in z°TPb. Fig. 3a shows the direct term 'and fig. 3b the exchange term. Only the neutron particle-hole part is shown. The proton particle-hole part only involves a direct term, similar to that shown in fig. 3a.

2o6.207pb POLARIZATION

219

To evaluate the exchange term, we consider the following transformed representation for the core-particle state: Ij '-1, (i÷iTtl)l.; j ) = ( - ) J ' - J ~

~/3(2J+I)W(-~-~--jj';1J)liT~1, (i~ j , - 1 ) j ; j ) .

(11)

We now evaluate the direct term for the matrix element (iTt I , ( i y j ' - l ) j i j [ VIj-l;j), graphically represented by fig. 3b. Denoting it by B, we obtain the following expression: B = (__)j+j'+l

+Jx/(2J+l)(2j+l) E (--)k( 2 k + 1)W(J&~X~J',/cl) k

× ( j i ~ ; klVli~,j'; k).

(12)

Multiplying B by the remaining factors in (11) and carrying out the sum over J leads to exactly the desired form for the exchange term in (10). Thus a calculation of the direct term in the transformed representation is equivalent to a calculation of the exchange term in the original representation. The principal complication in calculating the exchange terms is that many values of J can contribute. For each J, two sets of L and S values are allowed. Since the ( i y j ' - l ) ~ state has negative parity, we can have either L = J, S = 0 and L = J, S = l (if J is odd) o r L - - J - 1 , S-- landL=J+l,S-1 (if J is even). The modifications of the results of sect. 3 arising from a consideration of exchange effects are summarized in table 6. In general the changes are small. In all cases the TABLE6 Calculated results including exchange effects

1. 2°TPb magnetic moments COP 0.61(0.61) CAL 0.62(0.61) 2. 2°~pb B(M1) values in s.p.u. COP CAL 3. 2°ePb g(2 +)

COP CAL

0.030(0.028) 0.026(0.025)

--1.61(--1.58) --1.58(--1.52)

0.42(0.41) 0.40(0.36)

1.18(1.16) 1.17(1.12)

0.46(0.45) 0.45(0.40)

--1.58(--1.54) --1.53(--1.48)

0.0023(0.0029) 0.0026(0.0048)

4. 2°epb B(MI; 1+ -* 0 ~ ) in s.p.u.

COP CAL

0.18(0.18) 0.18(0.16)

In parentheses are the results calculated with the direct term only. effect is to slightly cancel the influence of the direct term. Thus the neglect of exchange terms cannot be given as a reason for not obtaining sufficient retardation. A further point to note is that exchange effects are larger for the CAL mixture than for the COP mixture and in fact serve to bring the two sets of results into closer agreement with each other.

220

K. HARADA AND S. PITTEL

The reason we have been able to explicitly treat these effects is because of our microscopic treatment of the core states. Frequently these states are treated in a collective scheme described macroscopically ~8). In these cases, exchange effects are usually neglected. The results of table 6 suggest that little error is introduced in this way. 5. Discussion

We describe in this paper a calculation of the effects of magnetic dipole core polarization in the Pb region. The particular aim of this study was to provide a consistent description of magnetic dipole moments and MI transition rates in 2°6Pb and 2°TPb. The model we use to describe these effects is a mixture of the conventional shell model and the core-particle coupling model. While we use microscopic shell-model wave functions to describe the states, the actual calculation follows more closely the weak-coupling scheme. Probably the main advantage of this approach over a pure shell-model description is one of clarity. Many of the qualitative features discussed in sect. 2 could not have been observed in a pure shell-model approach in which all matrix elements are related directly to two-body matrix elements. Furthermore, by coupling to physical states of the core, one has perhaps a more physical picture of what is happening. Nevertheless the final numerical results of the two approaches should be essentially the same. In general the reproduction of the experimental data is qualitatively very gratifying but quantitatively only fair. In all cases considered, the changes arising from core polarization are not large enough to account for the deviations from experiment. Thus it appears that some other effects are also contributing. One possible suggestion is that higher order corrections, analogous to those discussed by Mottelson for E2 transitions 19), might increase the effect of magnetic dipole core polarization. It is doubtful however that they would modify our results significantly. An alternative mechanism which could lead to modifications in magnetic dipole moments and transition strengths is 0- core polarization (e.g. from mixed-parity Hartree-Foek calculations 20)). In this approach the single-hole state I(/,j) - t ) is modified by a component involving a hole in the (I__.1, j) orbital coupled to a 0excitation of the core. Such admixtures lead to magnetic moments which lie between the Schmidt lines, and hence give effects in the proper direction. We have employed in these calculations a Gaussian interaction with both the CAL and COP mixtures of Gillet. Both give almost identical results, particularly when exchange effects are included. Furthermore from table 2 we saw that both mixtures focus the MI strength in 2°sPb in the "upper" 1+ state. In order to distinguish which mixture is more appropriate for describing nuclei in this region, one must find a property for which the two predict different results. One possible means is through an inelastic scattering experiment on 2°sPb. In such an experiment the CAL mixture would predict a strong excitation of the "upper" state, while the COP mixture would predict the "lower" state to be populated more strongly.

2°6'2°7pbPOLARIZATION

221

The authors would like to express their gratitude to Dr. F. R. Metzger for his continued interest in this study and for his many helpful comments. One of the authors (K.H.) wishes to thank the Bartol Research Foundation for its hospitality. References I) A. Arima and H. Horie, Prog. Theor. Phys. 12 (1954) 623; H. Noya, A. Arima and H. Horie, Suppl. Prog. Theor. Phys. 8 (1958) 33 2) R. J. Blin-Stoyle and M. A. Perks, Prec. Phys. Soc. 67A (1954) 885 3) A. M. Green and T. R. Sundius, preprint 4) v. (3illet, A. M. Green and E. A. Sanderson, Nucl. Phys. 88 (1966) 321 5) ~.. A. Broglia, A. Molinari and B. Sorensen, Nucl. Phys. A109 (1968) 353 6) A. Bohr and B. R. Mottclson, Mat. Fys. Medd. Dan. Vid. Selsk. 26 (1952) 14 7) G. R. Satchler, Nucl. Phys. AI00 (1967) 481 8) N. K. (31endenning, Prec. of the Int. School of Physics Enrico Fermi, Varenna, Italy, course 40, p. 332 (other references are listed therein) 9) K. Harada and S. Pittel, to be published in Phys. Lett. 10) e.g.A. Bohr and B. R. Mottelson, Nuclear structure, eel. 1 (W. A. Benjamin, New York, 1969) 11) W. W. True, Phys. Rev. 168 (1968) 1388 12) O. Hausser and D. Ward, Bull. Am. Phys. Soc. 15 (1970) 805 13) (3. Chilosi, R. A. Ricci, J. Touchard and A. H. Wapstra, Nucl. Phys. 53 (1964) 235 14) I. Hamamoto, Nucl. Phys. A135 (1969) 576 15) F. C. Zawislak and J. D. Bowman, Nucl. Phys. A146 (1970) 215 16) F. R. Metzger, Bull. Am. Phys. Soc. 15 (1970) 546 and preprint 17) P. Richard, N. Stein, C. D. Kavaloski and J. S. Lilley, Phys. Rev. 171 (1968) 1308 18) See e.g. (3. Brown, Unified theory of nuclear models (North-Holland, Amsterdam, 1965) pp. 121-125 19) B. R. Mottelson, Prec. of the Int. School of Physics Enrico Fermi, Varenna, Italy, course 15, p. 44 20) K. Bleuler, Prec. of the Int. School of Physics Enrico Fermi, Varenna, Italy, course 36, p. 464 (other references listed therein)