Physics Letters B 305 ( 1993 ) 322-326 North-Holland

PHYSICS LETTERS B

Centrifugal rotational effects on the magnetic dipole strength distribution

K . Heyde and C. De Coster 1

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

Received 8 March 1992; revised manuscript received 31 March 1993

We point out that the centrifugal term originating from the rotational energy concentrates magnetic dipole 1 + strength in a dramatic way into a scissor-like state near Ex ~ 3 MeV. The importance of this term and its neglect in earlier studies is discussed. Applications to the realistic situation of 164Dy are carried out where residual quadrupole and spin interactions are not able to redistribute the strong concentration of 1 + strength in an appreciable way.

The magnetic dipole excitations have been a sub- ject of intensive studies, in part icular after the dis- covery of a strong isovector orbital 1 + excitation in 156Gd in ( e , e ' ) scattering experiments performed at Darmstadt [1]. Various at tempts have been carried out to understand the observation of low-lying (Ex < 3 MeV) 1 + strength, in particular, in deformed nuclei [2]. Both collective model studies (l iquid drop two- fluid model interpretat ion [3-5] , Interacting Boson Model calculation [6-8 ] ) and, more recently, micro- scopic two quasi-particle (2qp) RPA [9] and TDA calculations) [10] were performed to understand the concentration of M 1 strength near Ex ~ 3 MeV in many nuclei.

In the microscopic calculations, the single-particle motion is taken as the motion of nucleons in a de- formed field, characterized by the projection quan- tum number 12 of the angular momentum on the sym- metry axis of the nucleus. Through the collective, ro- tational energy, however, the single-particle motion becomes coupled to the collective, rotat ional motion of the core and both the single-particle motion with the specific nucleon-nucleon residual interactions to- gether with the rotat ional energy effects need to be considered on the same footing. Unti l now, most mi- croscopic studies have discarded the latter term.

Aspirant of the NFWO. Present address: SCK/CEN, Boeretang 200, B-2400 Mol (Belgium).

It is the aim of the present letter to highlight the importance of the par t ic le-core coupling effects and to show, via detailed numerical studies, that the major consequence is a stronger concentration of 1 + strength than could be reached with a 1 + 2qp model space only.

The Hamil tonian describing the single-particle mo- tion in a deformed field, the rotat ional energy and residual two-body nucleon interaction terms reads

H = Hs.p. + / / ro t + Hres • (1)

Neglecting in a first phase the effects of Hres, the total energy of the (~1~"~2) 1 + 2qp states becomes

E2qp ( 1 + ) = Elqp (Q1) + Elqp (Q2)

+ ((g21Q2) 1+ //2 ~-~ R2[ (QI.Q2)I +) . (2)

For even-even nuclei, the collective angular momen- tum R can be rewritten as R = I - Jl - J2 and so, the collective energy term gives rise to terms of the following type:

R 2 = 1 2 - j 2 - j ~ _ 2 I . j l -2I .~ +2s~ "~ • (3)

Here, the first three terms give rise to diagonal en- ergy contributions, the next two terms do not induce any coupling effects in the space of (g2~t22)K s = 1 + states since no K s = 1 + to K s = 1 + coupling is present. Only the last term, called the centrifugal ef- fect, coupling the single-particle angular momenta for

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Volume 305, number 4 PHYSICS LETTERS B 20 May 1993

the two particles induces a specific coupling between the various 2qp 1 + components. This latter term can be rewritten as follows:

2jl " J2 = jr,+ h , - + j l , - j2,+ + 2jl,z h,z • (4)

Here, one clearly observes a coupling mechanism act- ing among the various (t21 02 ) 1 + 2qp configurations because of the jl,+j2,- + jl,-j2,+ term, conserving the K n = 1 + quantum number.

Shimano and Ikeda [ 1 1 ] have discussed the effect of this centrifugal term in a very restricted model space of all (121122)1 + configurations originating for an even-even nucleus where only the 1 h ll/2 (n) and 1 i13/2 (v) single-particle orbitals are active. Their ma-

jor outcome is that this term is able to induce very pe- culiar phase-relations amongst the 1 + configurations resulting from diagonalizing the centrifugal term in the (I2102 ) 1 + model space. These phase relations are exactly identical to the phase relations obtained by acting with the "scissor" operator ( a~ ,+ - flf~,+ ) on the ground-state 10). They worked out this schematic model for 156Gd, using the above lhl 1/2 (7~), 1 i13/2 (v) model space with a concentration of 1 + strength in a single state with a B(M1,0~- -~ l f ) value of 2.55

/t 2 at E x ( l f ) = 2.74 MeV [11]. It was not clear, however, how residual interactions Hres might effect or even fully fragment the above striking result.

Inspired by the work of Shimano and Ikeda [ 11 ], we have re-examined, in detail, the various differ- ent terms of the Hamiltonian (eq. (1) ) which were already included in our previous calculations [10] (centrifugal Jl "J2 term, spin-spin and quadrupole- quadrupole residual interactions, the use of an en- larged model space (121£22) 1 + encompassing for pro- tons the N = 3, 4, 5 and for neutrons the N = 4, 5 and 6 oscillator shells). In that calculation, the impor- tance of the particular j t "J2 terms was not recognized although present, and so did not receive the necessary attention we like to emphasize in the present letter. We now clarify the crucial role played by the centrifugal term Ji "J2 in enforcing the scissor picture, even in the presence of effective proton-proton, neutron-neutron and proton-neutron effective interactions. This lat- ter aspect was not fully emphasized and clarified in our previous paper [ l 0,12 ]. The collective energy is evaluated transforming the strong coupling wave func- tions ](£Q1 ~2 )1 +) into a basis where the collective Jc and single-particle angular momenta (JlJ2)J are good

quantum numbers (weak-coupling basis). So, the ex- perimental core energies E(Jc) can be used and all effects : centrifugal effects (Jr " J2 ), Coriolis effects (Jl" I and J2 " I ) and recoil terms (fl, , /~)can be han- dled in an easy and effective way. (We loosely called in ref. [ 10] "Coriolis interaction" the sum of the ac- tual Coriolis and centrifugal interactions.) The contri- bution from the rotational, collective energy Hamil- tonian Hrot can be obtained as

(IM, K = 12, + 02 [ nro, I IM, K = ~2; + 12~ )

1 - 21 + 1 y ~ 2 E ( J c ) (2Jc + 1) (JK, JcOIIK) 2

J,Jc

x ~ A(jl t21,j2122;JK), (5) II ,Jl ~2,J2

where A (...) is a shorthand notation for an expres- sion containing occupation numbers of the various or-

~a and bitals and the Nilsson expansion coefficients ctj is given in detail in ref. [10].

The results for 164Dy where the M1 strength distri- bution and overlap with the scissor configuration is evaluated are shown in fig. 1. Both the unperturbed M 1 strength distributions (upper figures), the TDA results using various quadrupole pairing strengths [ 12] and the results also incorporating the centrifu- gal effects contained in the Hrot term are drawn. The same conclusions as reached by Shimano and Ikeda [11 ] remain but now for realistic situations. The concentration of 1 + strength is illustrated at best in the lower part of fig. 1 in the scissor overlap intensities and is very dramatic. At the same time, a shift to higher energies relative to the pure 2qpl + diagonalization results. So, we stress the fact that the inclusion of centrifugal effects enforces the effect already induced by the residual interaction. The M 1 strength is such that almost all proton and neutron 2qp 1 + --, 0 + transitions are brought into coherence (see table 1 ). An important point which is very clear already now is the major effect of the inclusion of the collective energy //rot, a term which may not be ne- glected if one takes the coupling of the single-particle motion to the deformed core into account.

A final interesting question is related to the geomet- ric interpretation of the centrifugal term in inducing correlations between the orbital motion of the quasi- particle configurations that constitute the 1 + state.

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Volume 305, number 4 PHYSICS LETTERS B 20 May 1993

4 I

UNPERTURBED 2OP K '= I ~ 3"

G2p = 0.00455

2" G2n = 0.00294

o I . J l

4

t 2 -

o~

1

0 2

1.0

~ " 1.0

~ 0 S

0.0 2

i Q1DA

C2p= 0.00455

G2n ~ 0.00294

J a , I . i 3

I (~DA + C~']'IREFUOAL

G2p~ 0.00455

i

(~IDA

G2p= 0.00455

O2n = 0.00294

I I , I i

i CrfDA ÷ ~ O A L

G2p= 0.00455

G2n = 0.002g~.

I

( 8 ) 4 I

UNPERTURBED 2QP K'= 1 * 3 "

02p= 0.00682

2 - G2n = 0.00441

1 -

0, I , :1:

(b) '

3

2

1

I o

Q'rDA

G2p= O.00682

G2n = 0.00441

I

h 4

(c) 4 QTDA ÷ C~'¢n~vuG,/g,.

3 G2p= 0.00682

G2n = 0.00441

2

1

o I 3

(d)~o

O.5

O.0

QTDA

G2p= 0.00682

G2n = 0.00441

I l t l I T , 3

( e ) , o

0 . 5

0 . 0 4

i OTDA + CENTRIFUGAl.,

G 2p = 0.00682

G2n = 0.00441

I, h

Excitation Energy (MeV)

Fig. 1. M 1 strength distribution as well as the overlap for the 1 + TDA wave functions with a scissor 1 + state. The figures indicated from upper to lower figure show (a) the unper- turbed M1 2qp strength (b) the results after a full diagonalization of the Hamilton exclud- ing Hrot and (c) including also the centrifu- gal term by including the rotational term//rot. In parts (d) and (e) the respective overlaps with the collective 1 + scissor state are given. The difference between left- and right-hand fig- ure is in the strength of the proton (neutron) quadrupole-pairing strength. Left side G2p = 0.00455 MeV, G2n = 0.00294 MeV (values deduced from a delta-force interaction). Right side G2p = 0.00682 MeV, G2n = 0.00441 MeV (values deduced including the finite range of more realistic interactions).

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Volume 305, number 4 PHYSICS LETTERS B 20 May 1993

Table 1 M1 strength of the strongest excitation in 164Dy at 3.04 MeV from a TDA calculation with residual interactions, including monopole-, quadrupole pairing and the centrifugal term originating from the collective rotational energy.

(~e j ) I2 :t (g ' j ' )t2 '~' INnz A),r IN' n'zA')r,, M1 contr.

v ( 1 h9/2 ) 3 /2- ( 2f7/2 ) 5/2 - 1521 ) + 1523) - 0.005 (1h9/2)3/2- (lh9/2)5/2- [521)+ 1521)+ 0.288 (3P3/2) 1/2- (lh9/2)3/2- 1521)_ [521)+ -0.055 (li13/2)5/2 + ( lit3/2)7/2 + 1642) + 1633) + 0.189 ( lil3/2)7/2 + (li13/2)9/2 + 1633) + 1624) + 0.116 ( li13/2)3/2 + (lit3/2)5/2 + 1651) + [642) + 0.029

rr (2d3/2) I/2 + (2d5/2) 3/2 + 1411)_ 1411)+ 0.283 ( 1 g7/2 ) 5/2 + ( 1 g7/2) 7/2 + 1413) _ 1404)- 0.054 ( 1 hl 1/2 ) 5 /2- ( l ht 1/2 ) 7 /2- 1532) + [523) + 0.408 (2d5/2)3/2 + (2d5/2)5/2 + 1411)+ [402)+ 0.252 ( 1 hl t/2 ) 7 /2- ( l ht 1/2 )9/2- 1523) + 1514) + 0.229 (lh11/2)3/2- (lh11/2)5/2- 1541)+ 1532)+ 0.039

, 3- axis

Fig. 2. Geometrical construction indicating the two pos- sible 2qp (I21122 ) configurations with K n = 1 + that be- come coupled through the centrifugal term of eq. (4) i.e. (12 + 1, -12 ) 1 + and (I2, - Q + 1 ). The orientation of the sin- gle-particle associated angular momenta are also indicated. The orbital reorientation induced through the coupling is indicated by the double arrows. The collective, rotational motion is presented as a rotation around the 1-axis.

I f we again concentrate on the more simple form of the centrifugal coupling as depicted in eq. (4), we re- mark that the orbitals (£21122) 1 + - (12 + 1 , -12 ) be- come coupled to the (12 , - t2 + 1 ) configuration and vice versa (fig. 2). This wobbling effect or orbital re- or ientat ion induced by the jl ,+ j2,- -I- J l , - J2,+ term with the whole system influenced by the rotat ional mot ion around the l-axis can be seen as a possible

microscopic quasi-particle approach to the collective macroscopic two-fluid model description. The precise relationship between these deformed microscopic and macroscopic pictures needs to be still better under- stood.

In conclusion, we have pointed out that the in- clusion of centrifugal coupling effects originating in the collective rotational motion to which the single- particle mot ion becomes coupled is instrumental in inducing a larger concentration of 1 + strength near Ex -~ 3 MeV as was anticipated before. This result was obtained for rare-earth nuclei treating on equal footing the 2qp residual interactions (quadrupole and spin terms), quadrupole and monopole pair ing and these centrifugal effects. The present calculation cor- roborates the results, obtained earlier by Shimano and Ikeda [ 11 ] who used a schematic calculation within a two-level lh11/2 (Tr) li13/2 (v) model space, only. We have been lead by the critical analysis given therein in order to suggestan alternative geometrical interpreta- t ion for the 1 + mode as an orbital reorientation wob- bling effect induced by the above centrifugal effect. A number of questions on how the present results con- nect to the earlier two-fluid rotor oscillatory models need to addressed.

The authors are grateful to A. Richter for bringing a preprint of Shimano and Ikeda to their at tention and for constant st imulating discussions and collabo- rative work on magnetic dipole strength in atomic nu-

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Volume 305, number 4 PHYSICS LETTERS B 20 May 1993

clei. They are indebted to the NFWO and IIKW for financial support. Part of this study was carried out with a NATO Research grant CRG 92/0011.

References

[1] D. Bohle et al., Phys. Lett. B 137 (1984) 27. [ 2 ] C. De Coster, Ph.D. Thesis ( University of Gent, 1991 ),

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[6] F. Iachello, Phys. Rev. Lett 53 (1984) 1427. [ 7 ] F. Iachello and A. Arima, The interacting boson model,

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[9] A. Faessler, R. Nojanov and F.G. Scholtz, Nucl. Phys. A 515 (1990) 237, and references therein.

[ 10 ] C. De Coster and K. Heyde, Nucl. Phys. A 529 ( 1991 ) 507; A 524 (1991) 441, and references therein.

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[12 ] C. De Coster and K. Heyde, Nucl. Phys. A 550 (1992) 563 (E).

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