Manifestation of quadrupole collectivity in the magnetic dipole strength

Nuclear Physics A549 (1992) 10 "1-124 North.Holland

NUCLEAR PHYSICS A

Manifestation of quadrupole collectivity in the magnetic dipole strength

K. Heyde and C. De Coster I Institute for Theoretical Physics and Institute for Nuclear Physics, Proeftuinstraat 86,

B-9000 Ghent, Belgium

A. Richter 2 and H.-J. WSrtche Institut fiir Kernphysik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany

Received 17 January 1992 (Revised 21 February 1992)

Abstract: We discuss the systematics of E2 and MI transition strengths as obtained from experimental data in the rare-earth region. The similarity is studied in the light of the observed dependence of E2 and M l transition strength on quadrupole deformation. Especially the "saturation" effect for these electromagnetic observables is discussed, starting from the Nilsson deformed-shel~ model. The data are compared to the present calculations, using a QTDA approach. The particular E2-M 1 correlation is investigated within the IBM-2 using a sum-rule approach.

1. Introduction

The study of electric quadrupole (E2~ and magnetic dipole (M1) transition properties in transitional and strongly deformed nuclei has shown a number of unexpected properties. Besides a saturation of the B(E2; 0~ ~ 2~-) value 1) as well as of the summed Ml strength, i.e., ~ f B ( M I ; 0~--> 1i+), below Ex-~4 MeV when passing from vibrational and transitional nuclei into the region of strongly deformed nuclei, a strong correlation between the above two properties was shown to exist 2). This correlation is shown in fig. 1 where the data are plotted against the P-factor as introduced by Casten 3). The latter quantity, which is a measure of the average number of interactions of the valence neutrons and protons outside closed shells, has been proven to be a convenient quantity in order to unify systematics in quantities

like Ex(2~), R = Ex(4~)/Ex(2~),... over a wide mass region. This general relationship most probably points towards a deeper connecSon

between these two quantities, and the smooth behaviour in both ~f B(M1; 0~ ~ 1:) and B(E2; 0~---> 2~) on Z and N suggest a possible common origin of collective nature, such as nuclear deformation. Starting from the unperturbed two-quasiparticle

Research Assistant of the N.F.W.O. 2 Supported in part by the Bundesministerium fiir Forschung und Technologie under contract number

06DA184|.

0375-9474/92/$05.00 O 1992 - Elsevier Science Publishers B.V. All rights reserved

104 K. Heyde et al. / Quadrupole collectivity

300

,_ 200

100

. - , 4

~- 3

m 2

~ ~ 0 0 a I | 1 " t - -

o 1~Er o i~-1~Dy

A 1~A~-l~Sm

n vLz'~'~6-~Nd

I O i i

0 2 4 6 P=NpN=/(Np+No)

I I _

0 I I I | I . . . . . ! I I

._•._

0

o ~Er

o 16°-I~Dy . I~-16°Gd

c] 1~'146-Z~Nd

I _ I...

( b ) -

! | I I I _ J _ _

2 4 6 8 P=NpNJ(Np+Nn)

Fig. 1. {a} Plot of the B{E2;0~---,2~-) ~'educed transition probability in c;ca-cven rare-earth nuclei, indicated in the figure, versus P. (b) Plot o f ~ f B(M1; 0~'~ l f*) for the low-lying levels (E,<~4 MeV) for

the nuclei as in (a) versus P. Taken from ref. 2).

(2qp) Nilsson model, it has already been pointed out 4,5) that such a correlation exists between the low-lying dominantly orbital M 1 strength and nuclear quadrupole deformation. It has recently been confirmed experimentally by Ziegler et al. 6).

Particumarly striking is the sudden and steep increase of E2 and summed M1 strength, occurring in the transition from spherical vibrational-like towards strongly deformed nuclei, followed by a saturation of the strength for the region of well- deformed nuclei.

In sect. 2, we first address the E2 saturation properties and its relation to 5c~ormation. In sect. 3, we discuss the saturation phenomenon for M1 properties and in sect. 4 we study, in more detail, a correlation starting from a sum-rule approach both in the interacting proton-neutron boson model (IBM-2) and in the

K. Heyde et al. / Quadrupole collectivity 105

shell model 7). Thereby a firm connection with the quadrupole proton-neutron energy in the ground state can be established.

2. Saturation in E2 properties and relation to deformation

The problem of saturation of B(E2; 0 ( ~ 2~) values has been addressed earlier by Casten et ai. ~), and more recently also by Otsuka et ai. s), starting from the IBM-2. Here, the B(E2; 0( --, 2~) is predicted to increase with the total boson number N for the dynamical symmetries, according to the expressions ~)

u(5):

SU(3):

0(6):

B(E2; 0~ ~ 2i ~) = e~.,,5 N,

B(E2; 0 ~ 2 ~ ) = e~ , ,N(2N+3) ,

B(E2; 0~- ~ 2~)= e~.,,N(N + 4 ) , (2.1)

and hence no saturation in the boson number N is predicted. Using the OAf mapping ~"), a method in which the I BM parameters are determined

by mapping of a (truncated) shell-model space onto the IBM boson space, the Pauli effect is included via renormalization of the parameters. When equating the appropri- ate matrix elements of the fermion operators in the SD subspace to the corresponding matrix elements in the sd boson space, Pauli-principle effects are incorporated into the IBM operators, in terms of Pauli factors, depending on the degeneracies of the shell-model orbitals. These factors become unity in the limit of no Pauli effect, i.e., for few nucleons distributed over many single-particle orbitals. In general, the Pauli factors decrease with increasing nucleon number, reflecting the very mechanism of Pauli blocking of a shell-model orbital due to the presence of other nucleons. It is in particular clear (see fig. 2) that the Pauli blocking lowers the effective charge value appreciably when approaching the mid-shell, hence compensating for the increas~ of the total boson number N and leading to saturation of B(E2; 0~ ~ 2~).

0.08

~ 0 . 0 4

Z

c~

0.02

I I I I I ! I

Sn /

o.oo ~__1 ~L_ I 1 J _ 50 58 68

1

l

74 82

Neutron number

Fig. 2. The strength B(E2; 0~ ~ 2~-) divided by the neutron boson number for the Sn isotopes. The solid line corresponds to the results of OAI mapping, the dashed line denotes the limit of no Pauli effect. The

points are observed values ~). Taken from ref. x).

106 K. Herde et aL / Quadrupole coilectivi O,

Casten et aL !) have accounted for this saturation by introducing an effective boson number N~.t~, which has been derived from an estimate of the average quadrupole proton-neutron interaction energy in the 0 + ground state (quantity called Sp.), within the Niisson deformed-shell model as

s o . - 4 K X O2(f2 )Q (f2.) -" = v ~ v o . , (2.2) . ¢ ] ~ , 0 , ,

where K denotes the strength of the quadrupole interaction and Q2(f2p) gives the intrinsic quadrupole moment of the Nilsson orbital f2,, (p the charge label), and v~,, the respective occupation probability of that particular Nilsson orbital. Thi~ interaction energy saturates as a function of the number of valence protons and neutrons, as a consequence of the gradual filling of Nilsson levels with different projections f2 obeying the Pauli principle. Indeed, for prolate nuclei - as most deformed nuclei are ~2) _ the first few nucleons outside the closed shell start filling the orbitals with low f2 and thus large, negative intrinsic quadrupole moments and give rise to a large and negative Sp. contribution. As more nucleons are added, higher-f2 levels are filled with smaller and possibly positive intrinsic quadrupole moments, slowing down the increase in the proton-neutron interaction energy. This process is illustrated in fig. 3.

It was pointed out by Casten ~3) that the average binding energy of the proton- neutron interaction is, to a good approximation, proportional to the product of the number of valence protons and neutrons and, therefore, from this microscopic

I - I I /

/ 88

÷ 9 0

3 0 0 " - o 92 N p N n / . × 94

a 9 6

' • 98

,oo / . ,

; v 1 0 2 / _ A v A ~ A t

£'~ 2 0 0 " - ' 104 Z A ® v e ' -J,

° 7 ° i • 1 IO0~ + , f " . -j

0 I 0 0 2 0 0 3 0 0

Np N n

Fig. 3. Plot of the calculated values of I%oi, normalized to NrN., for the rare-earth region. The diagonal I%.I : N,N. is also shown. Both the initial linearity of ISo.l with NoN" and the subsequent saturation

are evident. Taken from ref. i).


estimate of the proton-neutron ground-state energy, effective valence nucleon numbers could be derived, thereby reproducing the B(E2; 0 ~ --> 2 ~) saturation rather well.

One can address the saturation problem in the E2 reduced transition probabilities from a different viewpoint too. In a collective approach, deformed nuclei are described as quadrupole deformed objects for which the nuclear surface is parameterized by expressing the radius vector as

R = Ro(l + aoo Y~o°' + fl y~2,), (2.3)

with a deformation parameter fl, which is related to the intrinsic quadrupole moment to first order via the expression ~4)

3 ZR2o ~ (2.4) - 543-

The value of Ro is determined by the condition that the volume enclosed by the spheroid defined in eq. (2.3) is equal to the volume of a sphere with radius Ro. For transitions within a rotational band, in particular for the 0~ ~ 2 ~ E2 transition, one derives the model-dependent expression ~4)

5 B(E2; 0~-~ 2 ~ ) = ~ e2Q 2 . (2.5)

167r

One often uses these relations to deduce a value for the equilibrium quadrupole deformation 13 from the experimental B(E2; 0~---> 2~) values (see, e.g., ref. ~s)).

In fig. 4, the equilibrium ground-state quadrupole deformation is shown for the rare-earth nuclei*. It becomes clear that a steep rise in deformation is followed by a saturation at about 8 =0.25 for A---160-180. One can explain this behaviour, starting from the rigorous calculation of the potential-energy surfaces using the Sm~finsky method ~9.2o). The latter starts from the liquid-drop model (LDM) used to describe the stability of the nuclear ground state against deformation, by calculat- ing the potential energy surface. Since the LDM describes average properties of the nucleus, it will be independent of the shell structure to a large extent. Specific shell corrections have to be added, corrections that take into account the varying density of single-particle levels in the vicinity of the Fermi level (energy) and this as a function of deformation and for different proton (Z) and neutron (N) number. This correction gives rise to oscillations in the LDM potential energy surface and eventually leads to an equilibrium shape that deviates from the LDM prediction.

° In this work, we used for the calculation the parameterization e2, following Nilsson ~6). Alternatively, one can use/3 or B, depending on the context or to facilitate comparison with experimental data [e.g., in ref. 6) B is used]. Many pararneterizations of deformation exist, which can all be related to each other. An overview is given in refs. ~4.~7). In particular, for the parameters used here,

- " ' "

~ 5 45 /32+ 675 /34 3 v . . . . .

i08 K. Heyde et al. / Quadrupole colleclivity

T . . . . .

L ]

o o6÷ i i

L

¢'M

002!

I i

000i 140

I j / / / 150 150

Mass number A

o

170 180

Fig. 4. Ground-state equilibrium quadrupole deformation as taken from Mfl ler and Nix ,s) as a function of mass number for the Nd (t-q}, Sm (/',), Gd (,t,}, Dy (O}, Er {O) and Yb (~7) isotopes.

6o-~

6 , (

o

3

~D

-o .4 -o.2 o.o 0.2 o.~ 0.6 0.8

Fig. 5. Nilsson diagram valid for neutrons. The single-particle orbitals are labelled by spherical quantum numbers (lj) at spherical shape or by means of asymptotic quantum numbers [Nn:A[I] at deformed

shapes. Taken from ref. 2=).


i ' ! I '

62~1TI 9z

\ " , , , \

0 \,,,\ \ ' ~ /

,,J I i I , ~ ,

-04 -0.2 0 2 0,/,

E 2

Fig. 6. Potential energy curves, for a sequence of Sm isotopes, from the calculations of ref. 23). The energy has been minimized with respect to e 4 and is shown as a function of e 2. Taken from ref. 22).

As an example, when inspecting the Nilsson level scheme (see fig. 5), one observes

that, as one moves away from the magic numbers starting from the spherical shape,

regions of low level density occur for deformation e2 ¢ 0. Hence deformed shapes

will become favoured, and as more nucleons are added, the shell correction becomes

rather stable. As a consequence a saturation of the deformation is observed near a

value of e2"0 .26 , or equivalently 8 =0.245, fl =0.30. This deformation value

o,4

0,2

0 t"q

- 0.2

- 0 . 4 . . . . . . . . .

50 60 70 80 go

N E U T R O N NUMBER, N

Fig 7. Shelkenergy diagram in the (e2, N) plane. Areas corresponding to negative shell energy are shaded, and the contour separation is 1 MeV. Taken from ref. 2~1.

110 g. Heyde et ai. / Quadrupole collectivity

O .Oq

0.04

LO

0.02

0.00

v 0 - - "~ ~ 0 ~

A ~g1" o

_ Z / "

/ tx m I m I n I 0 2 t. 6 8

P= Np Nn/[Np+N n)

Fig. 8. Ground - s t a t e equi l ib r ium q u a d r u p o l e d e f o r m a t i o n as t aken f rom ref. t8) as a func t ion o f P for the Nd (I-q), Sm (A) , G d (,A-), Dy (O) , Er (,~) and Yb (V) isotopes.

decreases subsequently when approaching the next magic closed shell. The saturation effect is illustrated in fig. 6 for the series of Sm isotopes 22). In fig. 7 the shell energy is shown as a function of deformation and neutron number (for Z =66). One can clearly see that, going towards mid-shell at N = 64, the minimum in the shell energy moves towards deformation e2 = 0.3, then decreasing again going towards a spherical shape for N = 82.

So, it becomes clear that saturation in the B(E2; 0 ~ - 2~) strength is intimately connected to the properties of nuclear deformation, saturating for the rare-earth nuclei mainly around a value of e2"-0.26 (6"-0.245, /3=0.30). To illustrate this even more clearly, we have plotted nuclear deformation, parameterized by 62, versus the P-factor, and obtain a very striking similar plot as for the E2 experimenta~ data (see fig. 8).

3. Systematics of M| properties and saturation

In a similar way as in the case of E2 properties (see sect. 2), the saturation found in the systematics of the experimental ~ r B ( M I ; 07--, 1~) in the rare-earth region cannot be understood and reproduced within the IBM-2. There, a linear dependence of the B(MI ; 0~ --, 1 ~ ) on the P-factor is predicted 9), i.e.,

u(5): B(M1; 0~---, 1 +,,.~.) = 0 ,

3 )2 ~ 8 N=N,, SU(3): B(Ml ;0? -~ l+m. .~ . )=~(g ,~ -g~ 2 N - 1 '

3 )2 3 N,,N,,. (3.1) 0(6): B(M 1; 0~- ~ 1 +m'"') = 4"--~ (g" - g" N +----~


Again, we look for a possible origin of saturation in a more microscopic description. It was discussed by De Coster and Heyde 4.5) that the low-lying (orbital) M1 strength depends on nuclear deformation via the pairing factor, appearing in the expression for a 2qp 1 + --> 0qp 0 + M 1 transition, i.e.,

3 B ( M 1 ; 0 + .--) 1 +) =~--~ (u,v2- + g:+lE=>l =. (3.2)

The latter expression indeed cuts out many 2qp 1 + -> 0qp 0 + transitions due to major quenching in the pairing factor and selects only a small interval of states around the Fermi level. These 2qp excitations - mostly of (l, j--> l, j ) type - constitute an "active valence region" of orbitals. Even for these selected orbitals, nuclear deformation plays a major role in tuning the quenching, introduced by the pairing factor, as is illustrated schematically in fig. 9. Indeed, as nuclear deformation increases, the Nilssoti orbitals belonging to a given (l, j) shell, spread out in energy, and therefore their occupation probabilities become quite different as compared to the situation of vanishing deformation. Hence, the pairing factor will be much larger for large deformation and will not be changing much (saturates) for these deformations.

t

E2--~

\

\

\ - - 7 ~ " - - " ~ -

V2-. - ,~. ~ V2 . -~ . . - ~ V 2 . - . ~ V2..~,.

Fig. 9. Schematic illustration of the spreading of Nilsson deformod single-panicle energies e(.Q) as a function of the position of the Fermi level according to th~ ~:quihbrium deformation. The situations '~1,/t4 correspond to almost spherical shapes whereas '~2, ,~3 are obtained for strongly deformed shapes (insert). The occupation probability distribution v2(~) for the four different situations ;~!, ,~2, ,~3, ,~4 are shown in the lower part with the MI pairing reduction factor as given by eq. (3.2) that almost disappears

at ~1, ,~4 and reaches a much larger value at ;t 2 and ,~3-

112 K. t teyde et al. / Quadrupole collectivity

In fig. 10a, we present the summed M1 orbital strength below E x - 4 M e V as obtained from detailed QTDA calculations 24,25). The results from the above schematic description are well reproduced in these more realistic calculations and indicate saturation for a number of nuclei near the value of P = 7-8 and with a summed orbital M1 strength near 2 .5g~. In fig. 10b then, we summed all M1 strength up to E, - 4 MeV (to comply with the data points obtained and also expressing the fact that most M1 strength in this particular energy region is mainly of orbital character*). The theoretical QTDA values though, exhibit a much slower rise with

(al

c,,~z ~ 3

~2 o m &,q

I

0 v~o

o/--.

0 2 /-, 6 8 P = Np Nn/[Np'~Nn )

~z

r-

~2 rn

I' 0

Ibl

~ ' o

! I i I 0 2 4 6 8

P = Np Nn/{Np+Nn } ~ "

Fig. 10. (a) Summed orbital MI strength and (b) summed MI strength below E~--=4 MeV as obtained in QTDA calculations versus P, for the Nd ([--1), Sm (A), Gd ( * ) , Dy (O), Er (O) and Yb (V) isotopes.

The solid line is drawn to guide the eye.

* The difference between MI and orbital strength may seem quite large in some cases, but it should be noted that the limit E, -=- 4 MeV is set artificially, and hence some low-lying spin-flip contributions could come in.

K. Heyde et al. / Quadrupole collectivity 1 13

P than the data in fig. 1 indicate. This might be partially due to the use of the Nilsson model as the underlying basis for a description of 1 + states and the corresponding B(M| ) strength, even in the region of transitional nuclei in between vibrational and strongly deformed nuclei.

We like to stress here that the deformation dependence of the M 1 strength results indirectly from pairing effects, in contrast with the behaviour of the B(E2; 0~--, 2 : ) value which is proportional to the square of the intrinsic quadrupole moment and hence is a direct manifestation of the ground-state quadrupole deformation of the

nucleus. As mentioned in the Introduction, recent (y, y') experiments on the Sm isotopes 6)

suggest a quadratic dependence of the summed low-lying M1 strength on nuclear deformation. Starting from the Nilsson 2qp picture, we plot the dependence of the occupation probabilities on nuclear deformation for a representative example of ( I , j ~ l, j ) transitions (see fig. 11) and observe an almost linear variation of c 2 on

E2 (or ,5) for relatively small deformations. Hence, we can write approximately

a (Ul v 2 - u2v,) 2-- `52(1 + - + • • • ) , (3.3)

5̀

where a stands for a numerical value resulting from the precise linear relationship. Furthermore, for transitions between a group of Nilsson single-particle states (excluding pairing, which has been treated separately) of (I,j--, i , j ) type, using the

10 15/"5 m, I'T,

_ . . . . . . . . . . . . . . . . - ~ ~

~ 7/21-o

O 0 , I I I t [/-.13 5 / 2 1 I

000 005 010 015 020 025

e210) ~-

Fig. 11. Occupation probability 0 2 versus equilibrium quadrupole deformation for proton |qp states in ~54Sm, for (0) shells close to the Fermi level. The states are labelled with the spherical quantum numbers nO in the spherical limit, and with the asymptotic Nilsson quantum numbers [Nn.Af2] at large

deformation.

1 14 K. Heyde et al. / Quadrupole coilectivi(v

single-particle energies as obtained in the asymptotic limit, one finds 26)

B(MI)~--SA 4/3 . (3.4)

Combining eqs. (3.3) and (3.4), a quadratic dependence on nuclear deformation (to leading order in the deformation parameter) results. In fig. 12 we show our QTDA results for the Sm isotopes in comparison with the data. The linear behaviour with 8 2 is indeed reproduced, and quantitatively the theoretical summed orbital M1 strength is in line with the experimental data.

Similar results have been obtained on a microscopic basis by Hamamoto and Magnusson _,7) using a QRPA approach for the series of Sm isotopes, although too much strength at lc, w energies is predicted, especially for '-~4Sm. Starting from the results of B~s and Broglia 26), the authors calculate the influence of the pairing factor [following a method proposed by Bohr and Mottelson for the calculation of the moment of inertia 2s)] and reach the same conclusions.

Another interesting approach is given by Garrido et al. 29), using the harmonic- o~cillator model and a microscopic formulation for the scissors state. They predict scissors-like excitations around Ex=2-3 MeV for which the strength is largely fragmented and proportional to 8 (respectively 82 when including pairing), besides a higher-lying scissors excitation around Ex = 20 MeV for which the strength is proportional to 8 2. Microscopic results obtained in the projected Hartree-Fock- Bogoliubov model (PHFB) for the Nd and Sm isotopes, are in good agreement with experiment and reproduce the 8 2 dependence of the low-lying M1 strength.

z.O

3O

U4

1 0

O0

/ 1/./. ,1/~8 - 15/.

_ Sm / ~ /

/ j 2 / /o

I I I 002 OOZ. 006 008

5 ~ ~.

Fig. 12. Comparison between the calculated summed M 1 strength using the QTDA and the experimentally summed M I strength 6) as a function of the square of nuclear deformation 8 2. We present the total summed MI strength (o and solid line) and the summed orbital MI strength (© and dashed line). The

solid and dashed lines are fits to the theoretical values.

K. Heyde et ai. / Quadrupole collectivity 115

As far as collective descriptions are concerned, in the TRM of Lo ludice and Palumbo 30.3~), protons and neutrons are assumed to form rigid bodies, performing out-of-phase rotational oscillations. In this semi-classical treatment, the magnetic dipole transition probability is depending on the restoring-force constant and the moment of inertia, i.e. B(M1)l'ocx/-C~. The former being proportional to 8 2, and for a rigid moment of inertia, the B(M1) becomes proportional to &

However, in recent papers 32,33) it was shown that much improvement in the agreement with the data is found when accounting for pairing correlations in the calculation of the moment of inertia, following a method described in ref..,8). This then leads to a quadratic dependence of the moment of inertia on nuclear deformation, and hence B(Ml)l'cx:8 2. It again points to the fact that pairing is crucial in order to describe the systematics of magnetic dipole strength throughout a mass region covering a transition from spherical, vibrational-like towards deformed nuclei.

Rohozifiski and Greiner 34) describe the magnetic dipole excitation mode within a collective model, assuming different proton and neutron deformations, resulting in a relative motion, which consists in neutron skin vibrations. They too predict a quadratic dependence of magnetic dipole transition probability on nuclear deformation. It is thereby realized that when using a collective geometrical model, the collective surface motion can be associated mainly with the valence nucleons. The "valence property" is a natural consequence of the interplay of pairing and deformation effects as was discussed in sect. 3.

Also in the papers by Lo ludice et al. 32.33), the TRM is alternatively extended by assuming an inert core, where only the mass external t~ this core is allowed to rotate. Subsequently a proportionality B(M1)I' oc 8 3/2 was predicted. For the B(M1) value, the prediction is close to the original TRM prediction, when pairing correlations are included, which is in support of the above interpretation of collective surface motion. However, it should be mentioned that this approach is less successful in predicting the form factor. Therefrom it is concluded that the success of the 1BM, compared to the semi-classical approach, is not only to be attributed to the inert core, which is present in the geometrical limit. The IBM, in constructing the boson space, has indeed monopole and quadrupole correlations built in.

Using the latter algebraic approach in an extended version, the sdg IBM-2 model, Mizusaki et al. 35) can reproduce the behavior of the M 1 excitation from the ground state 0~ for the series of Sm isotopes. They conclude that the drastic change of the M I excitation strength can be obtained very naturally as a consequence of the increase of boson number. This in turn refers to the integrated proton-neutron interaction strength 3) and hence to nuclear deformation. Within the IBM-2 as well, Ginocchio 36) calculates the non-energy-weighted sum r~,~e and indicates the proportionality to the expectation value of the d-boson number in the ground state. In terms of the IBM/3, the mean value ~ is proportional to/32/(1 +/32) [refs. 37,38)].

From the above discussion, starting both from a microscopic and collective approach, it becomes clear that the dependence of the low-lying orbital M 1 strength

116 K. Ho,de et aL / Ouadrupole collectivio,

on nuclear deformation, mainly results from pairing effects. The latter effects induce quenching, tuned via the quadrupole deformation, and thereby preferentially select "valence" orbits, pointing to the M I excitations as a "surface phenomenon".

Both from experimental and theoretical findings, an (approximate) proportionality relation ~f B(M1; 0~--> lf+)oc 3-" is deduced, which is one way of interpreting the correlation with the electric quadrupole transition probability B(E2;0~-->2~). Especially this common saturation property can be understood. In the next section, we deduce an analytic expression for the correlation of E2 and M1 strength, originating from the proton-neutron interaction energy. We derive this by evaluating the M1 sum rule w/thin the IBM-2.

4. Correlation between E2 and MI strength: An |BM-2 sum-rule approach

One can express the linear energy-weighted M1 sum rule in terms of a double commutator of the M I operator with the nuclear hamiltonian as

Z B ( M 1 . + = _ A , ¢) ~x/3(0~][[H, 7"(M1)], (4.1) . f

This sum rule can be evaluated for a general shell-model hamiltonian 7,39). The right-hand side of eq. (4.1) then results in a one-body term proportional to the expectation value of the spin-orbit interaction in the ground state, and a two-body term resulting from the residual interaction.

Starting from the l BM-2 hamiltonian describing proton-neutron mixed-symmetry modes of motion ~.4,,.4~) containing the one-body boson energies ca,, ca,, and a

quadrupole-quadrupole interaction two-body force, the energy-weighted M1 sum rule of eq. (4.1) was evaluated in ref. 42). In this sect. 4, we present the main results as well as the major steps in deriving the double commutator within the framework of the IBM-2 [ref. 4_~)].

More precisely, we start from the hamiltonian A A A A / ~

Hl = e,Gn,t~. + ed,na, + K=,.Q= " Q,, + ~,,, (4.2) A

with M~., the Majorana operator and for the boson number and quadrupole operator the following expressions

n 4 = ( d d),, , (4.3)

Q, , : ( s+~l+d+s) ' ,~ '+X, , (d d),, , (4.4)

one can evaluate the double commutator, using the IBM-2 Ml operator, 1 _ _

The one-body term e,lB,~ results in a vanishing contribution to the right-hand side of eq'aaticn (4.1). On the other hand, the quadrupo!e-interaction term gives rise te

K. Heyde et al. / Quadrupole collectivity

the following expression for the do,Jble commutator, i.e., A • A

Q,,, ¢(MI)], ¢(MI)1 ̀ °'

-3--p-0 { g~,~,, [ -~x/3 ( d + s + s + d ) ~ ' 4'rr

-2v~ ~+ 2 2 X"(d+a)7' " (~"

+ 2g=g,,K~,,~v/~ 0.,," 0 .} ,

+(=coy)

117

(4.6)

o r

= _ 3 v ~ , A (g= - g,,)-K=,,Q= • (~,,. (4.7)

Starting from the more general F-spin invariant IBM-2 hamiltonian 9.40,41)

(4.8)

w i t h /~d~ -" /~d,,, the evaluation of the double commutator is somewhat more involved but results into the particularly simple expression

[[K0" Q, T(M1)], 7"(M1)] '°' - 3 x/~ (g~-g,,)-KQ~. (~,,, 17"

(4.9)

where 0 = 0~ + 0,,. So, in both cases, we obtain for the energy-weighted I~] 1 sum rule

~B(M1;O~->lt)EJlr)=2--~ [ K J=' (g=-g,.)2(O~lO=.Q,,lO,)+m=,., (4.10) . f"

for the hamiltonians H: and H2 along the upper and lower part, respectively and where rn=,, is the contribution to the energy-weighted sum rule of the Majorana term.

From the expression (4.10) one observes a strong correlation between the M1 energy-weighted sum rule and the strength of the deformation driving quadrupole proton-neutron force. The expectation value in the 0 + ground state of the latter force is a measure for the quadrupole deformation energy which can also be related to the corresponding binding energy in a quadrupole-deformed mean field such as the Nilsson model 43).

We will now evaluate the ground-state expectation value appearing on the right- hand side of expression (4.10). Inserting a complete set of intermediate 2 + states, called 2t+), the expectation value becomes also

E <2;110 10¢><2;110,, I0;>, (4.11) t

118 K. Heyde et al. / Quadrupole collectivity

If we make the tacit assumption that the states describing the 0~ and 2~ have good F-spin quantum number F = Fm,~ = ½(N~ + N,), the reduced matrix ele~;nts can be related to the one for the full Q~ + ¢), operator 4o), since

<FlIQ, Ill:> N, (4.12) <FII + Q.IIF>- N~ + N."

We finally obtain the result that*

Y~B(M1;O-~-->I~)Ex(lf)=cY~B(E2;O-~->2f)+m,,,, (4.13) f f

where the electric quadrupole operator is defined by the following expression:

~'(E2) = ee,(O,, + Or) ,

with

= - - - - - ~ e . b , (4.14) een T~ " (g=-g")2(N=+ Nv)-J

in order to make the right-hand side of eq. (4.13) equal to the left-hand-side M1 energy-weighted sum rule. Because of the dimension mismatch between both sides of eq. (4.13) a correction factor c = ~ MeV/(e. b) 2 needs to be introduced. In the expression (4.14) K=,,, K and g~r, ~, are then the dimensionless values of the corresponding quadrupole strength and boson gyromagnetic factors. For a more general, independently chosen effective electric charge on the right-hand side of eq. (4.13), when evaluating the B(E2; 0~- -> 2}-) values within the framework of the IBM-2, we obtain a proportiona|ity between the two sides of the expression. The numerical factor 9/2~r in eq. (4.14) stems from the particular choice of the T(MI) operator as given in eq. (4.5), which was first used by lachello 44) and has been consistently used since then in the IBM-2 calculations. Using this choice, boson gyromagnetic factors g~ -~ ISZN and g~--- 01-~g have been derived by Sambataro et al. 4s) using the OAI ,,napping !o) in relating fermion and boson M 1 matrixelements. Using a different numerical factor instead of x/3/4~" will result, through the OAI mapping, in different boson gycomagnetic factors since only the product ~ g~= gtR,'(p = 7r, v) is determined. This will lead, though, to the same numerical value of the effective value ee, as defined in eq. (4.14).

In applications to rare-earth nuclei, the product K=,N=N,/(N:~+ N,,) 2 is very smoothly varying and stays near to a value of -0.02 for the region of nuclei 146-152Nd, 148-154Sm, ~-~°-~SgGd. Indeed, the value K=,, increases for N and/or Z approaching

the closed ~hell 46) and the pcoduct ( N=/ N)( PC,/ N ) decreases for N or Z approaching the closed shell, compensating quite well to an almost constant value.

Using the parameter values obtained from fits to the nuclear properties in the Nd, Sm and Gd isotopes 46), one can determine the value of ee,. The results are presented in table 1 and illustrate that the effective electric charge defined by eq.

* See Note added in proof.


TABLE 1

The boson effective charge een, derived from eq. (4.14), using the r ,~ parameters from Scholten 46). The g-factors are chosen as g~ = l/.t N

and g,, = 0/zN [refs. 4o.41)]

N

Z 86 88 90 92

Nd 0.148 0.130 0.111 0.126 Sm 0.127 0.113 0.114 0.120 Gd 0.106 0.103 0.115 0.119

(4.14), surprisingly gives values eef~ that are of the same order as ee~ values used in independent, numerical IBM-2 studies in the rare-earth region 46). One finds indeed typical values of the effective charge used in the standard IBM-2 calculations for the rare-earth mass region 46,47). Numerical calculations with the parameters from ref. 46) taking the Majorana contribution into account have been carried out for the Sm nuclei. The implications on the M I energy-weighted sum rule as well as a discussion of the relative importance of the quadrupole-quadrupole and Majorana components 48) will be discussed in ref. 50).

We now incorporate a number of approximations to make the relation in eq.

(4.13) more useful. (i) In transitional and, in particular, in strongly deformed nuclei, most of the

summed E2 strength resides in the first 2 + state. So, within a good approximation, the sum of the right-hand 3ide in equation (4.13) can be restricted to the 2~" level only, a state for which F = Frnax holds to a very good approximation 4o).

(ii) In the energy-weighted sum rule, since we discuss the IBM-2 and as such, collective orbital scissors-like states are considered, most MI strength is coming from a very restricted interval near Ex"-3 MeV [refs. 6"2)]. So, an average energy Ex(1 ÷, coll.) can be taken out of the sum on the left-hand side of eq. (4.13)*. We note that, when plotting the energy-weighted as well as the nan-energy-weighted M 1 sum rule, using the experimental B(M 1) values and the corresponding excitation energies versus the experimental B(E2; 0~---> 2~) values, one obtains approximately a straight line in both cases (see fig. 13). The slopes, however, exhibit a ratio

of about 3. These results support the above approximation. Using the above assumptions, we finally obtain an approximate relation connecting

the summed M1 strength and the B(E2; 0~--> 2~') reduced transition probability as

Y~ B ( M l ; 0~- ~ l.~)Ex(1 +, coll.)= cB(E2; 0~- ~ 2~-). f

(4.15)

This relation (4.15) is born out by experimental B(M1) and B(E2) values, as is clear from fig. 13 and comes about from the precise structure of the hamiltonian

* See Note added in proof.

120 K. Heyde et al. / Ouadrupole collectivity

~ Z

Z 0

t ~

?

0

1.-4 ~

p.,.

i

. .

÷ ~ ÷ . . . . B ( E 2 . 0 ~ 2 ~ ) ', ';"J.u ) . . . . . .

-!

" i

/ /

/

/ / ./

/ / J

q ' l )

(b;

~'* 2 ~ ) ( W u ) - - - - - ~ -

Fig. 13. (a) Energy-weighted sum of MI strength below E~ = 4 MeV and (b) summed M! strength below E ~ = 4 MeV versus B(E2; 0~---, 2~-). The solid lines are fits to the data points. The data point for tMDy

which shows a peculiar behaviour of this nucleus (see fig. !), is excluded from the figures.

(using a quadrupole-quadrupole force) and the form of the IBM-2 M1 operator. In fig. 14, the M1 and E2 values obtained in the numerical calculations for the Nd, Sm and Gd isotopes are plotted against each other, and a straight line is fitted to

r the calculated points, from which slope we obtain an effective charge e , : t r = 0.16 e. b*. Besides this collective model approach, similar results have also been ohtair~ed

within the shell model by Zamick and Zheng 7). In this microscopic approach the strength of the quadrupole-quadrupole interaction - known as a deformation driving fo-~'e - is seen as an indication of collectivity. The summed orbital M1 strength is shown to saturate for large values of the quadrupole-quadrupole strength. Further- more, a relation is found between the linear energy-weighted sum rule for low-lying orbital M I strength and the expectation value of the neutron-proton quadrupole- quadrupole interaction in the ground state, which is very similar to eq. (4.10), and

" See Note added in proof.


r ~ Z

: : t

> 5

÷ ,,.-.

A 0 A

M

I I I i j

50 100 150 200 250

B I E 2 } ~ / e 2 ( a r b un i ts ) -,,-

Fig. 14. Energy-weighted magnetic dipole strength to the first mixed-symmetry I + excitation B(M 1; 0~ --, l +~.) versus electric quadrupole strength to the first symmetric 2 + excitation B(E2; 0~-,'. 2~), as obtained in numerical IBM-2 calculations, using the parameters of re['. 46), for the Nd (I-l), Sm (A) and Gd (~-)

isotopes. The solid line represents a fit of the relation (4.15) to the calculated points.

hence indirectly linked to the linear non-energy-weighted sum for the E2 strength. Therefore, the above results are of rather general value.

There remains a problem though, with the large B(M1) value measured in 164Dy (see fig. 1). The smooth variation in the summed M 1 strength cannot accommodate such a large and sudden change in a single nucleus. Since the sum-rule equality, expressed in eq. (4.13), mainly accounts for the orbital part of the MI sum rule at lower energies (the derivation of eq. (4.13) was carried out within the IBM-2*), the deviation might hint towards important M 1 spin-flip contributions 4,,).

5. Conclusion

(i) Concluding, we have studied the saturation of B(E2;0;--,2~) and ~t B ( M I ; 0 ~ I ~ ) (with Ex(I~)~<4MeV) reduced transition probabilities when moving from quadrupole vibrational towards strongly deformed rare-earth nuclei. This is related to the saturation in the equilibrium quadrupole deformation in nuclei as verified starting from the Nilsson deformed single-particle model. The above saturation, in retrospect, via the pairing factor in eq. (3.2) implies a saturation for the summed B(MI) strength and this in an indirect way. The B(E2;0; ~2~) saturation, because of its quadrupole deformation dependence, is then a direct manifestation of the above saturation in equilibrium ground-state deformation.

* Also within the shell model, Zamick and Zheng v) point out that for a spin-independent central two-body interaction, such as the quadrupole-quadrupole force, only the o:bital strength contributes to the sum rule.

122 I(. Heyde et al. / Quadrupole collectivity

(ii) We have also studied the correlation between E2 and MI electromagnetic properties in rare-earth nuclei. It was shown in sects. 2 and 3 that the saturation properties in both the B(E2; 0~--> 2~-) and the summed M1 strength ~r B(M1; 0~ 1~) are intimately connected to the saturation property of nuclear quadrupole deformation at the equilibrium shape. This property follows fi'om the microscopic shell structure of the nucleus. For the E2 properties the connection is a direct one whereas for the M1 properties there exists an implicit relation through the pairing factor that determines the particular M1 strength.

We have studied the M1-E2 correlation analytically, starting from the proton- neutron interacting boson model and have obtained a general expression relating the energy-weighted M1 sum rule to the summed E2 strength in a given nucleus. One can approximate this relation for orbital M1 strength, located in the energy region around Ex ~-3 MeV, into an expression showing a proportionality between the summed M1 strength below Ex=4 MeV and the B(E2; 0~-~ 2~-) value. In this particular energy region, spin-flip excitations that mainly contribute to the M1 sum rule at higher excitation energies (5 <~ Ex << 9 MeV) are expected to contribute only in a minor way. Such a relation is indeed born out by the experimental data on MI and E2 strength in rare-earth nuclei.

Following the quadratic dependence of the B(E2; 0~--2~) reduced transition probability on nuclear deformation, the same dependence is implied for the summed orbital M 1 strength below E, = 4 MeV. This is indeed found experimentally for the series of Sm isotopes and supported by microscopic QTDA, QRPA and PHFB calculations as well as semi-classical descriptions in terms of the scissors mode or neutron skin vibrations.

The authors are grateful to R.F. Casten for interesting discussions on the scaling properties and to the referee for raising a number of interesting points. They like to thank O. Scholten for collaborating on various aspects of sum rules in the IBM-2. One of the authors (K.H.) is grateful to the Joint Institute for Heavy-lop~ Research at Oak Ridge and UNISOR for support in the final stages of this work.

Note added in proof

After finishing the present article, we found a possibility to take the Majorana force into account in an elegant way, under the constraint of good F-spin. We can write eq. (4.13) as

Z B(M1; Oi + ~ | ; ) [ E , ( I ~ ) - A N ] = E B(E2; 0 ~ 2 ~ ) . t .f

Here, A denotes the strength of the Majorana force and N the total boson number. Also, the right-hand side of eq. (15) of ref. 42) has to be multiplied by a factor of 10.

Even though the Majorana-force contribution to the energy-weighted sum rule can become non-negligible, the functional relationship between the linear energy-


weighted sum rule and the E2 strength due to the quadrupole-quadrupole proton- neutron force is not modified in any important way. We only have to make use of a renormalized (increased) effective electric charge. A full treatment of sum rules in the interacting boson model is in progress 50).

Regarding approximation (ii) to eq. (4.13), one should in fact extract Ex(1 + , co l l ) -AN. However, this can be corrected for by using a renormalized effective charge e~tr = ee,[ Ex/(Ex - AN)] '/2 The renormalization factor [ E J ( E x - A N ) ] ~/2 is for all nuclei of the order of 1.45, giving an effective charge eeyy~0.11 e. b.

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Manifestation of quadrupole collectivity in the magnetic dipole strength