Hyperfine Interactions 43 (1988) 15-34 15

NUCLEAR MOMENTS: AN EFFECTIVE PROBE OF NUCLEAR STUCTURE

K. HEYDE

Institute for Nuclear Physics, Proeftuinstraat 86, B-9000 Gent, Belgium

Magnetic dipole and electric quadrupole moments are

discussed in nuclei near doubly-closed she'll nuclei (the

T1 nuclei) and in nuclei along series of single-closed

shell nuclei (plus of minus a few nucleons) (the In

odd-mass and odd-odd nuclei). We discuss the

"additivity" rules for nuclear moments. We also address

the EO moment: the liquid drop model and the shell-model

are discussed and compared to measurements of nuclear

radii in the Ca, Sn and Pb region. In the latter region,

the importance of intruder states across the Z=82 proton

closed shell is emphasized.

i. INTRODUCTION

The nuclear many-body system, consisting of A nucleons,

strongly interacting via two-body interactions is a system with

many facets, depending on the "energy" resolution one is looking

with. Looking to details which are of the magnitude of the nucleus

itself, surface deformations (static or dynamic) can be studied.

On a somewhat smaller scale, the typical nucleon degrees of freedom

show up. Still increasing the energy which is used to probe the

nucleus, subnucleon degrees of freedom do start to show their

explicit appearance. Finally, one can come in the high-energy

realm observing quark degrees of freedom. Together with these

general characteristics related to the nuclear wave function, the

electromagnetic operators and the electromagnetic coupling

strengths (electric charge ep,en; the gyromagnetic factors gs,gl)

depend in a similar way on the detail with which we are studying

the nucleus. Thus, it is natural that in most cases, the

electromagnetic coupling strengths will deviate strongly from the

free values. This process of obtaining effective charges and

�9 .l.C. Baltzer A.G., Scientific Publishing Company

16 K, Heyde, Nuclear momenls: an effective probe o f nuclear structure

gyromagnetic factors is a general consequence of the fact that we

describe the nucleus within a limited model space and thus at a

certain level of detail. Relating effective operators to the free

operators has been studied in many detail before. I refer to the

book of Brussaard and Glaudemans, ch.16 /i/ and the references

therein. It is important to recall that the effective charges and

gyromagnetic factors can vary very much depending on the

approximations made. I illustrate this process in figure i, where

a full space ( all A nucleons) and a small model space (few

particle-hole excitations) are related to each other via the

equality

gI ,gs Ilree) ep,e n (It ee)

T~ V I~ V

= .- =

/ / �9 "x ;'~.~ .~.;<7.7.;'

) �9 :,; ),/..)~//~ �9 ,.:;.,>, ;;,, ,.,>, ,,<//

~, , / / / /

F U L L S P A C E MODEL SPACE

gl 'gs [eflechve} ep ,en[ elfec hve)

Fig.l. Schematic

illustration of how

effective "charges"

are related to the

free "charges"

depending on the

particular choice of a

model space relative

to the full A-nucleon

valence space

<~H 1oeff" [~H> = <~[0freel~ > ( i )

where, knowing the full and model space wave functions ~,~M

respectively, effective charges and effective gyromagnetic ratios

are determined in an implicit way. Using the above prescription,

for medium-heavy and heavy nuclei, the gyromagnetic ratio gR for a

collective magnetic operator ~=gR.J , can be obtained by equating

<~(I,2,.,A)(J)I~ gl(i).iz,i+gs(i)Sz,il~(l,2,.,A)(J)> i

H ( j ) l j z [I . l~coll (j)> , (2) gR <~coll.

K. Heyde, Nuclear moments." an effective probe of nuclear structure 17

where ~(I,2,..,A) (J) denotes the full A-nucleon wave function and

~Mcoll.(J ) the corresponding collective model wave function. Since

in the left-hand side of eq. (2), the intrinsic spin contributions

almost cancel, a value of gR~Z/A ,coming from the orbital part

only, results as an effective collective model gyromagnetic factor.

In the main part of this contribution, I will concentrate on

regions of nuclei near doubly-closed shells or along single-closed

shell nuclei since in these regions, there is some hope of

obtaining rather good model wave functions that will allow a rather

stable determination of effective charges and effective

gyromagnetic ratios. In retrospect, we come in a situation where a

precise determination of nuclear moments can sometimes give unique

information on the the nuclear structure of such nuclei. Also,

besides the more standard magnetic dipole and electric quadrupole

moments, I will discuss some features restulting from E0 moments.

2. MAGNETIC DIPOLE AND ELECTRIC QUADRUPOLE MOMENTS NEAR CLOSED

SHELLS

It is in nuclei near doubly-closed shells that, within the

nuclear shell-model, a single, rather pure shell-model

configuration most often dominates the nuclear structure of

low-lying excited states. Because , even in such nuclei, a largely

restricted shell-model valence space is used, gs factors will

deviate largely from the free nucleon values due to

core-polarization and meson exchange effects/2/. Staying within a

set of isotopes or isotones near closed shells , one can expect

such corrections not to change eratically so that specific

variations of ~ can still give indications of the nuclear wave

functions and thus of the characteristics describing the nuclear

interactions. To illustrate this, we shall discuss

(i) the odd-mass T1 nuclei very close to N=126,

(ii)the odd-mass In nuclei, having Z=49 , as a function of neutron

number.

18 K. Heyde, Nuclear moments: an effective probe, o f nuclear structure

2.1. Odd-mass T1 nuclei

Recently, the magnetic moment of the 1/2 + level in 207TI ,

having a hole in the 208pb nucleus, was measured by Neugart et

al./3/ .In the light of the above discussion, a clear-cut variation

of N(I/2 +) in 207TI is observed, relative to the corresponding

moment ~(i/2 +) in the lighter 201-205TI nuclei. This reflects

relevant nuclear structure information on how the 1/2 + state varies

as a function of neutron number N /4/ (see also table i). Expecting

core-polarization and meson exchange corrections not to change

much, hinted by the very small variation in the inteval 201~A~205,

something else must be causing the variation in ~(i/2 +) when

approaching 207TI. The explanation can be found in

particle-vibration coupling which determines the collective admix-

Table i

The 2 + excitation energy in even-even Pb nuclei, the

related core-coupling amplitude ~ and the experimental and

theoretical ~(I/2+i) dipole moments in 201-207TI.

~exp. ~th. Ex(2+l )

207 <-0.135 1.876(5) 1.876 4.09

205 -0.339 1.6382134(7) 1.663 0.803

203 -0.368 1.622257(1) 1.629 0.899

201 1.61(2)

ture 12+| in the 1/2 + T1 wave functions. Using

particle-core coupling, this leads to a magnetic dipole moment for

a general hole state [j-l> with the inclusion of the 12+|

core-coupled component of

~(j-l) . ~(j-l) + ~2 <2+|174 (3)

in second-order perturbation theory. For j=3sl/2 and j'=2d3/2,

this has been studied in detail by Arima and Sagawa /5/ and one

gets

K. Heyde, Nuclear nToments: an effective probe of nuclear structure 19

~v (4) - " J1---o -i -

e3Sl/2 -(c2~3/2 + h~2(APb))

where ~u is the strength of the proton-neutron interaction

~wuQ~'Qu ,~nlj the proton single-hole energy and ~2(Apb) the 2+1

excitation energy in the nucleus Apb. Determining all these

quantities in the expression of eq.4, it is the large variation in

Ex(2+I ) in going from 208pb to 202-206pb that causes a large

variation in ~ and thus also in the value of ~(I/2+).

x - - - -

2~/2 2 ' 4-

-1 3 s 1/2

3 s'1~

J ; -1

3sl/2

Fig,2. Second-order core-

polarization corrections to the

magnetic dipole moment ~(i/2+i ) in

odd-mass TI nuclei

In evaluating the second-order core-polarization effect (see

fig.2), Arima and Sagawa obtain /5/ the matrix element

<2+| [~i 12+| (using

~exp. (2+)=0.14~N and ~exp.(3/2+l)=0.41~N ). Thereby, the following

values of ~ and ~(see table i) are obtained, clearly explaining the

importance of second-order core-polarization corrections.

Though we have discussed the example of 207TI in some detail,

similar variations in ~ should be observable /4/ for other

single-particle states near closed shells, where a similar

variation in the 2+i excitation energy is observed and where

single-particle configurations are present that allow for

particle-core coupled admixtures.

20 K. He),de, Nuclear moments." an effective probe o f nuclear structure

2.2. Single-closed shell nuclei (• 1 nucleon): application to In

nuclei

Studying lodger series of isotopes or isotones near to

single-closed shell nuclei is a good test for the single-particle

(or single-hole) character of the observed configurations since, as

discussed above, core-polarization and meson exchange corrections

do not present strong nor rapid variations in most cases. Since

for the variation with nucleon number (neutron number for series of

isotopes) the core characteristics for single-closed shell nuclei

almost remain constant up to the doubly-closed shell nuclei,

core-coupling admixtures show almost no variation in contrast to

the variation when going towards the doubly-closed shells (see

sect. 2.1)/6/.

As an example, we present the 9/2 + magnetic dipole moments,

measured by Eberz et al. /7/ for the odd-mass In nuclei, states

which are well described by the Ig9/2 -I proton hole configuration

(see fig.3). The very pronounced constancy of this magnetic dipole

6

E

:Z.

2

--___l

i

orb~ta { . -

J collect,ve

~ p,n

/ I I

Aln 49

0 '1 ' 1'20 I , , 10Z. 108 108 110 112 1 4 116 1~8 122 12/. 125 128

Fig.3. The experimental /~(9/2 + 1 ) moments (taken from Eberz et

al. /7/) compared to the particle-core coupling calculations /6/.

The orbital, spin and collective contributions are given

separately.

-i hole component and moment expresses the constancy of the ig9/2

the core admixture, which has a 0.4-0.5 amplitude

(;2+| )/6/. The slight up-sloping when approaching

the N=50 core is most probably reflecting the decreasing

K. Heyde, Nuclear moments." an effective probe qf nuclear structure 21

particle-core amplitude. The calculations were carried out using a

particle-core coupling model as described by Heyde et al./6/ and

using a value of gs elf" = 0.7 gs free"

In the odd-mass In nuclei, nuclear quadrupole moments have also

been measured by Eberz et al. /7/ and are compared with the

calculated values , again using particle-core coupling of ref./6/,

using a proton effective charge ep=l.5e, in figure 4. Here too, a

1.0

t 05

s

, , I i ; i ~ i i i -- i

c o l l e c | i v e

single- porlicle

lOt. 1,06 108 110 112 114 116 118 120 122 124 126 128 - - A - - - i , , . . -

Fig.4. The experimental Q(9/2 + 1 ) moments (taken from Eberz et

al. /7/) compared to the particle-core coupling calculations /6/.

The collective and single-particle are given separately

rather constant behaviour is observed reflecting the major lg9/2 -I

component. For the quadrupole moment, however, in lowest-order

perturbation theory, the collective admixtures give an important

contribution ( see fig.4 where the separate single-hole and

collective contributions are given) in contrast to the case of

magnetic dipole moments. In lowest-order , particle-core coupling

eQeff'(j) = Qs'p'(j)I ep + i0 a/,/4~.Ze.~2/hw 2 } (5)

induces a term /8/ where ep is the proton single-hole effective

charge, a describes the particle-core coupling strength and B 2 is

related to the B(E2;2+I~0+I) transition probability in a harmonic

approximation/9/. Approaching the closed shells at N=50 and N=82,

the B(E2) values start dropping, h~ 2 increases and the

particle-core coupling strength a decreases too, so the total

correction factor is a decreasing function from the mid-shell

(N=66) nuclei on and varies approximately as ~22/~2 (see figure

5).

22 K. Heycle, Nuclear moments." an effective probe o f nuclear structure

The presence of a possible sub-shell closure at N=64 is manifest

from the drop in the experimental quadrupole moment around this

particular value of N. The actual particle-core coupling

calculations take into account the ig9~2, 2Pi~2 , 2P~2 , if~ 2 hole

orbitals and octupole phonons besides the quadrupole excitations.

This more complete basis can result in slight deviations in the

behaviour of Q from the above simple relation, presented in fig.5.

Deducing a deformation parameter ~2 from the measured

quadrupole moment is a very model dependent procedure, at least

when using models for deformed nuclei in the Z=50 region where

quadrupole vibrational excitations are more likely to occur at low

energy. Therefore, we do not extract at present such quantities,

unless clear-cut evidence for stable, quadrupole deformation is

present.

(a) (b)

1 ---,,,

Fig.5. Zero-order (a) and lowest-order collective

contributions (b) to the single-particle

quadrupole moment. The dependence on mass number

for Qa aud Qa+b is also drawn in a schematic way.

In the odd-mass In nuclei, low-lying 1/2- and 3/2- levels

occur that contain the 2Pi/2 -I and 2P3/2 -I single-hole component to

a large extent. Here, core-coupling is even more important and

large deviations from the pure single-hole moments can be expected.

We give the 1/2-1 and 3/2-1 wave functions in ll7In /6/ as an

illustration of this importance of core-coupling e.g.

ll/2"l>-0.8612Pl/2-1>+O.3512+| + ....

13/2"l>-0.7412P3/2"l>-o.4212+|174 + ....

(6)

K. Heyde, Nuclear moments: an e//eclive probe of nuclear s{ruclure 2 3

Experimentally ,one-nucleon transfer spectroscopic factors towards

these levels in a llSsn(d,3He) ll7In reaction, corroborate this

conclusion /I0/ since SI/2-=1.49(5), $3/2 -= 2.28(8) with 1.45 and

2.18 as corresponding theoretical values /6/. Experimental values

for ~(i/2-) /7/ show,' at present, a very unexpected behaviour

leading to values outside of the Schmidt lines. A first

observation is that core-polarization seems unimportant for Pl/2

orbitals /ii/, an observation that is in line with the

particle-core coupling calculations. Also core components for the

appropriate wave functions do not make up for a large variation in

~(i/2-) (see ref./6/ ).

On the 3/2- level,little is known although some time ago, a

controverse showed up from a measurement of ~(3/2-I) in ll7In with

a value of 0.102(58) ~N /12/ ,completely at variance with the

calculated ~(3/2-i) value /6/ ( see figure 6). Out of a

measurement of the 6(3/2-i~i/2-i) mixing ratio in i17,119In, rather

I I I I I I

I1"/I n gR = Z~/'//

r

. . . . ~ ~ ~ g ~ = ~ _ _ _

,' ~ ~ I ' '

Fig.6. Variation of the core-coupling magne-

tic dipole moments p(3/2" I) and p(i/2- I) in

ll7In as a function of gs" The values for

both gR-0 and gR-Z/A are given. The

experimental value of i~(i/2- I) is also given

0 I

- - 9 S

good agreement with a value of gseff'=0.7 gs free results. This

would imply a magnetic moment #(3/2-1)=2.5 ~N" Bodenstedt et al.

/13/ later on traced back the problem to an incorrect value of the

lifetime of this 3/2-1 level, quoted as TI/2=192(16) ps. in

literature and which now becomes TI/2(3/2-I)~ 10ps., resulting in a

magnetic dipole moment N(3/2-I)~0.84 ~N which is in line with the

theoretical value.

24 K. Heyde, Nuclear t.oments." apt effective probe o f nuclear structure

So, magnetic dipole moments as calculated for rather simple

configurations near closed shells can even be helpful to correct

for measurements in a way rather independent of details of the

calculation.

Similar studies can be carried out for other series of

single-closed shell (• 1 nucleon) nuclei e.g. the Sb nuclei, the

N=81 and N=83 nuclei,..

3. ADDITIVITY RULES IN ODD-ODD NUCLEI

Magnetic dipole - a n d electric quadrupole moments have been

measured in many odd-odd nuclei near closed shells (odd-odd In,

odd-odd Sb , odd-odd TI,.. nuclei). For such nuclei, starting

again from a rather simple configuration for both the odd-proton

and the odd-neutron nuclei ,the known moments and using simple

angular momentum recoupling techniques /14/, rather general

"additivity" rules can be derived and have been used in determining

the composed moments.

If we call the eigenstate in the odd-proton nucleus IJp> with

~(Jp), Q(Jp),.. the corresponding moments and IJn> tl%e eigenstate

for the odd-neutron nucleus with ~(Jn),Q(Jn),.. the corresponding

moments, under the assumption of weak coupling in obtaining the

eigenstate IJ>=IJp69Jn;J> in the odd-odd nucleus, one obtains the

expressions

t ' ( J ) = 2 [ Jp Jn + p Jn - J ( J + l ) (7 )

Q(J ) = - 0 . ( 2 J + l ) x

-3 n o J n - Jp 0 Jp

(8)

K. Heyde, Nuclear moments." an effective probe o f nuclear structutv 25

These methods have been tested with a quite remarkable success in

the odd-odd In mass region by Eberz et al. /7/for many magnetic

dipole- and electric quadrupole moments ( see also refs. /15L23/

for a detailed discussion on the possible configurations).In such

an approach, rather complicated proton and neutron states are

combined under the assumption that these states are not modified

very much when coupling to form the final state in the odd-odd

nucleus. In odd-odd nuclei near closed shells, when rather few

configurations with the same Jp and Jn value are present in the

odd- proton and odd-neutron nuclei respectively, a rather good

applicability of the additivity rules is expected. For the

odd-neutron case, where N (the neutron number) varies over a rather

large interval 51~N~79 , a less unambigous situation results. If

have the odd-proton nucleus eigenstates Ijp(i)> and the we

odd-neutron eigenstates IJn(3)> ~ , where we can have different

(Jp,i) values and (Jn,j) values in a single nucleus, weak-coupling

can be strongly violated i.e. one obtains wave functions /24/

Ij~176 - Ijp(1)| + Za(Jp (i),Jn (j) ;J) Ijp(i)| ;J> (9)

In this case, extra components from configuration mixing in the

final nucleus result These terms give rise to extra

"polarization" terms with respect to the original zero-order term.

A good , qualitative estimate of configuration mixing is obtained

by studying the number of final states J in the odd-odd nucleus.

If there is only a single J state over an interval of = 1 MeV,

weak-coupling most probably will be a good approximation. If, on

the other hand, many J levels result at a small energy separation,

chances for large configuration mixing are more likely to occur.

In using the additivity method, it is of the utmost importance

to use the odd-mass moments as close as possible to the

"unperturbed" odd-mass nuclei that are used to carry out the

coupling in obtaining the final odd-odd nucleus. In the odd-odd In

nuclei, more in particular for the (ig9/2-1(n) lhll/2(u))8-

configurations, some problems occur when comparing the measured

moments and the "additivity" moments (see fig.7). If one considers

a pure ig9/2 -I proton-hole configuration and a lhll/2 neutron

26 K. Heyde, Nuclear monTents: an effective probe of nuclear structure

one-quasi particle configuration ( and linear filling of the lhll/2

orbital with n valence neutrons), then the dipole, respectively

quadrupole moment would vary like /14/

i) p(8")= a p(ig9/2- + b.~(lhll/2) , ( i o )

Q(8-)= a'.Q(ig9/2 -I) + b'.(12-2n)/10. Q(lhll/2) ( n )

i s i J I i I

10. io0 t E t. 3

I I I

112 11/. 116 118

i i I I I I i

/ r ~

o o o ..o o o

I i [ I 120 122 12/. 126

Fig.7. Variation of ~(8"i) and

Q(8- 1 ) in the odd-odd In nuclei

(112 s A s 126). The experimental

values (1) are taken from Eberz et

al. /7/, the additivity moments (D)

(eqs. 7 and 8) are obtained using

the discussion as given in ref. /7/.

which means, a constant value for #(8-) and a linear increase in

Q(8-) with n, the number of neutrons filling the lhll/2 orbital. In

the more specific case of more orbitals filling at the same time,

some modifications to this simple dependence on particle number can

be expected( starting of filling the 3Sl/2,2d3/2 orbitals before

N=76 and early filling of the lhll/2 orbital before N=64). This

will result due to the pair correlations and the resulting pair

distribution of neutrons over the five neutron single-particle

states /25/.

K. Heyde, Nuclear nloments." an effective probe o f nuclear structure 27

4. E0 MOMENTS: NUCLEAR RADII NEAR CLOSED SHELLS.

The monopole moment gives information on the nuclear radius ,

its variation along series of isotopes and isotones and on the

nuclear structure information contained therein. The E0 moment is

therefore a very direct and clear-cut probe of sudden onset of

deformation in nuclei with many valence neutrons and protons /4/.

Keeping to single-closed shell nuclei, e.g. the Ca,Sn,Pb nuclei or

series of isotones, a very specific behaviour of the nuclear radius

can be expected on the basis of the nucleus considered as a

spherical object, able to undergo vibrational excitations /26/.

Expanding the nuclear radius-vector as

R-Ro (I+ iZ;giYlo (R)) ,

a value of the E0 moment

(12)

<r2> - <r2>o(l + 5/4~. Z <~A2>) , (13)

where <r2>o=3/5.ro 2 A 2/3 and Z<#A2> describes the deformation

ability of the nucleus.The above values give rise to an isotopic

shift, A<r2> that can be expressed as

A <r2> - 7. -25 r2 A-I/3 + 7' ~ r20 A2/3 A(ZA ~2) (14)

with n=~'= 0.5 in the spherical droplet model /27/. The value

Z<~ 2> ,which near spherical Z=20,50,82 nuclei is mainly of

quadrupole origin ( but can contain other multipoles too /28/)

results in a parabolic-like behaviour, being zero or vanishing at

the closed shells ( see fig.8 for the Sn nuclei /29,30/).

In fig.8, the dynamic deformation deduced from B(E2) values in the

even-even Sn nuclei are smaller than the analogous quantity derived

from nuclear radii. This deviation could be attributed to the

following two effects :

28 K. Ho,de, Nuclear montents." (tn effective probe o f nucle(n" structure

025

I 0 20

oo. 0 15

g

o~o

I 0 05

0 O0

c

o

o

_I I

50 55

�9 = I(8% ''~ fr0mB(EO) u

(~17>"1 trom B(E21 a

i I I L I 50 65 70 75 B0

- - n e u t r o n n u m b e r N =

Fig. 8. Deformation l)~n:ame ter 132 , obtained from the

experimental <r2> values in the even-even Sn nuclei (~,[3)

(taken from ref, /29/). Comparison is made with the /32

values, obtained from B(E2;2I i ~ 0+1 ) values using a

harmonic oscillator assumption (.). The figure is taken

from Eberz et al. /29/.

- the 2~ ~ O~ E 2 transition in the even-even Sn nuclei has been

analysed using a pure harmonic quadrupole vibrator model. The 2~

level is clearly not so collective since in microscopic

calculations, it can be described mainly as a linear combination

of two-quasi particle excitations. Moreover, E2 strength

pertaining to the quadrupole vibrations can remain in some of the

higher-lying 2 + levels,

- from eq. (14), higher multipole deformation effects could be

present in determining the nuclear radius (A=3,...). When

analyzing the total deformation effect as caused by the

quadrupole deformation only, differences with other methods for

extracting ~2 values can result.

Empirical fits to the nuclear radius have been carried out ,

along series of proton single-closed shell nuclei(plus or minus a

few nucleons) , mainly using a quadratic expression in the number

of valence nucleons /4/, maximizing at mid-shell (taking the Sn

region with N=66 as a reference value) like

K. Heyde, Nuclear momenls: an effeclive probe o f mlclear sll'uclla'e 29

<r2> - a(N-66) + b(N-66) 2 + 1 / 2 . c ( 1 - ( - 1 ) N) (15)

with a=0.0643(3)fm2,b=-l.13(8) .I0 -3 fm 2 and c=-0.020(I) fm 2 for the

Sn nuclei /29/ (see fig.9 where results for the Cd /31/and In

/7,15/ nuclei are shown, too). Referring to the N=50 and N=82

closed shells, the above expression can be rewritten as

<r2> = a(N-50) + b (N-50)(N-82) + I12.c(I-(-i) N) + d (16)

It can be shown , for proton-closed shell nuclei with neutrons

filling a single j-shell and described by a (j)nj n configuration

that the proton-neutron residual interaction , using a multipole

expansion, "polarizes" the proton core into a new state /32/ .

~ o

^

V

spher,cul drop[el \ . -

N=66 , " / i . ~4 Jl 4 ' ' ~

I w P d

Sn , # ' r J ~ i)t

�9 ~ 4" , <1 /,,' i . ~ 4" C d /..'" F r"

/ 4 , j / , / j "

.." f4,'

5'o is 6'o A ;o 7's 8'o - - nculron number N

Fig. 9. The nuclear radii

<r2> for the even-even Cd

/31/, even- even Sn /29/ and

odd-mass In /7/ nuclei. The

spherical droplet behaviour is

given as well as parabolic

fits including odd-even

effects, using eq. 16. The

figure is taken from Otten

/4 / .

[O> = [Jc = 0, Jn = J ; J> + Z~ JT'Jc=J0'Jn=l' ;J> (17)

30 K. Heyde, Nuclear moments: an effective probe of nuclear structure

Using standard shell-model techniques /14/, it is mainly the

proton-neutron quadrupole component that causes core polarization

and thus an increase in the nuclear radius The radius

corresponding with this new wave function becomes

<~1~ =~1~> - <o1~ =~1o> 1 1

+ z =7,o,j<olx r217,Jc-O,Jn-J ;J> 7,Jc-O,J'-J i

Z , a . r ' J - J ^ J - J ' ; J [ ~ r217 ' J - J ^ J - J ' ; J > ' c u' n 7,7,,j0,j,aT,J0,J ,J0,J '<7' c u' n 1 (18)

This expression (18) can be brought into the form

<r2>n-<OlZr.210>= <0 t ri210> ~n i i ~ + + ~i/2.n(n-l) + ~[I/2.n} (19)

with [i/2.n] defined as the largest integer not exceeding n/2.

When using the value of E=I<jlIT(J0) I Ij>I2/2j+I,~=-4E/(2j-I) and ~=

2E(2j+I)/(2j-I) which holds -for J0>0 and even, expression 19

finally becomes

<r2>n - <OIEr210>+2~/(2j-l) (n(2j+l-n) -I/2.(l-(-l)n)(j+ 1/2)) (2o) i i

which is very similar to the empirical expressions used before.

This method , Used by Zamick /33/and Talmi and Thieberger before

/34/, has been applied to the Ca,Pb nuclei and a number of nuclei

in the Z=50 region ( see fig.9)even though a single j-shell is only

well founded in the Ca region. In most cases, fits using the more

general expression of eq.19 with ~,~, and ~ taken as three

independent parameters, give a good description of the overall

behaviour of <r2> and of the odd-even staggering .

We like to point out that the shell-model approach, giving an

explicit form of the core polarizibility (containing at the same

time odd-even mass effects ,see eq.19) and the geometric

polarizibility ( see eq.13) are of a very analogous form although

expressed within a different model description.

K. Heyde, Nuclear moments: an effective probe of nuclear structure 31

I would like to conclude this section that sometimes, even

near to the closed shell regions, important deviations from the

smooth behaviour of E0 moments can result and thus signal a

dramatic change of the nuclear radius: the Au nuclei (with Z=79

close to the Z=82 Pb nuclei) form a nice example of such changes

where at N=I06 (A=195) a sudden increase of <r2> relative to the

N=I08 (A=197) value shows up /35/. That it is not a minor effect

is illustrated in figure i0. It signals particle-hole excitations

across the Z=82 closed shell no longer via a dynamic (polarization)

effect but now remaining as a permanent deformation of the final

state. This results in much larger effects on <r2> than just

present in the odd-even staggering. Such excitations have been

called intruder states /9/ and correspond to a 4h-lp configuration,

rather than a 3h configuration ( Pt + 1 particle compared to Hg + 1

hole) in the Au nuclei. Very recent measurements would indicate a

high value also in A=193 (N=I04) for the Au nuclei /38/ indicating

0.0--

10/. 106 108 I~0 112 11/. 1~5 118 120 1�89 12/. 1�89 - - N - -

t ~- _ 0 . 0 ~. E

L .

V

Fig.10. The nuclear radii for

odd-mass and odd-odd Au nuclei,

taken from Wallmeroth et al.

/35/. We also give the radii for

the even-even Pb nuclei (taken

from Anselment et al. /36/ and

Dinger et al. /37/), the

spherical droplet model and a

parabolic fit, using the

parameters given by Talmi /32/ in

the same figure.

that near mid-shell, the intruder configuration remains the ground

state in the odd-mass Au nuclei for at least some units.

The Pb nuclei themselves should show, if these arguments of

p-h excitations across the Z=82 shell are correct, low-lying 2p-2h

0 + excited states near the neutron mid-shell configuration with

32 K. Heyde, Nuclear inolnenls." all effective probe (~f nuclear slrllellll'e

eventually effects on the nuclear radius. Such low-lying 0 + states

have indeed been observed in the neutron deficient Pb nuclei by the

Leuven group /39/ and thus corroborate the idea of low-lying

paticle-hole excited 0 + intruder states. Such excitations

correspond to shapes that have much larger deformation compared

with the regular, spherical ground-state shape constituted by the

Z=82 closed core and a number of neutron valence holes in the N=126

core. If we inspect the Pb <r2> values, (see fig.10) indeed, in the

region where the 0 + intruder state is coming low, (Ex(0+2) ~ 1.0

MeV) , deviations from the spherical droplet behaviour and a

parabolic fit, carried out independelty to the Pb nuclei with

I16~N~126 , towards larger radii shows up. Calculations for making

these observations more quantitative are in progress.

ACKNOWLEDGEMENTS

The author is grateful to J.L.Wood, R.A.Meyer and M.Huyse for

constant education on nuclear structure, also on moments and in

particular on the importance of the study of intruder states in

nuclei. He is indebted to H.Huber for discussions on nuclear

magnetic dipole and electric quadrupole moments and to H.-J.Kluge

for discussions on E0 moments in the Au region. This work was

supported by a research grant RG-NATO 86/0452, the III<W and NFWO.

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