A schematic model for the study of fragmentation of scissors M1 strength in deformed nuclei

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 596 ( 1996) 30-52

A schematic model for the study of fragmentation of scissors Ml strength in deformed nuclei K. Heyde

a, C. De Coster a, S. Romboutsa,

S.J. Freeman b

a Vakgroep Subatomaire en Stralingsfysica, Institute for Theoretical Physics and Institute for Nuclear Physics, Proeftuinstraat 86, B-9000 Gent, Belgium b Schuster Laboratory, University of Manchester, Manchester Ml3 9PL, UK Received

10 July 1995; revised 21 August

1995

Abstract We study a fragmentation model in order to describe the distribution of a collective, scissorslike l+ state into a background of deformed two- and four-quasiparticle excitations. The coupling matrix elements for coupling the collective state to the underlying microscopic structures are studied. The fragmentation results are compared to the available data. One-nucleon transfer strength distributions are discussed with reference to the fragmentation of two-quasiparticle configurations. We also point out possible origins for intermediate structure in the fragmentation mechanism.

1. Introduction The study of nuclear magnetism has been particularly revived through the detailed studies of the magnetic dipole mode of motion during the last years. After the discovery of concentration of Ml strength at low energies with a mainly orbital character (scissors Ml strength near E, N 3 MeV) [ I-31, firm evidence on spin-Ml strength in the energy region 6-9 MeV has been accumulated in the strongly deformed rare-earth nuclei [ 4-61. More recently, discussions on a high-energy K” = I+ component [ 7,8] of the quadrupole isovector mode of motion have started, again in the light of new experiments [ 91. In spite of the success of many microscopic studies ( (Q)RPA, shell-model, . . .) of such l+ states, difficulties remain with respect to the fragmentation of the Ml strength in the various regions: low-energy scissor orbital part (A), spin-flip component (B) and high-energy scissors region (C). Depending on the theoretical starting point when considering the average field (deformed mean field, Nilsson deformed oscillator, deformed Woods-Saxon), on the use of residual interactions (schematic separable multipole forces 037%9474/96/$15.00 @ 1996 Elsevier Science R.V. All rights reserved SSDIO375-9474(95)00384-3

K. Heyde et al. I Nuclear Physics A 596 (I 996) 30-52

31

in the spin and isospin channels, self-consistent Skyrme forces, . . .) and on the projection or separation methods to single out the spurious rotational motion of the nucleus as a whole, non-negligible differences in the distribution of Ml strength in the various energy regions (A,B,C) show up (see Refs. [ 2-6,9] for a detailed reference list). The attempts to describe the intricacies of concentration of Ml strength by diagonalizing in an appropriate basis the residual interaction are fine for the strongest I+ states. It is however very difficult to reproduce, in a stable way, the often rapidly changing patterns that show up in the Ml strength distribution going from one nucleus to another. In the present article we take the opposite approach and study the fragmentation and spreading of a collective scissors-like state (which is not an exact eigenstate of the nuclear Hamiltonian by its specific construction), albeit in a schematic model, which is embedded in a background of more complicated two-quasiparticle (2qp) and fourquasiparticle (4qp) configurations. A discussion of the coupling mechanism may shed light on the way a collective magnetic dipole state becomes fragmented and spreads over the underlying background. Whereas in the 3 MeV region (region A) coupling to the 2qp states will be very important, in the spin-flip region (region B) the very high 4qp density is the major cause of the further spreading of 2qp It strength in an almost continuous way. In the present article we shall mainly concentrate on the low-energy “scissors” region. In Section 2 we discuss the general coupling mechanism and construct a schematic fragmentation model for the scissors state as fragmented over a background of 2qp and 4qp states with, in Section 3, application to the rare-earth Gd nuclei. In that section we also discuss the fragmentation of two-quasiparticle (2qp) strength with reference to the general features of single-nucleon transfer experiments. In Section 4 we refer to former studies on the fragmentation and spreading in the spin-flip 6-9 MeV energy region and point towards possible origins of intermediate structure in the spreading of magnetic dipole strength.

2. Fragmentation

model

When the density of states within the I$ configuration space becomes too high for rigorous calculations (diagonalization, . .) to be feasible, one can apply a method that is generally used in nuclear reaction theory in order to describe fragmentation of collective strength over a background of microscopic states [IO,111. Whenever the average coupling strength (v) between the collective state and the background microscopic states becomes larger than the average distance between the discrete levels, D = 1/p, a Breit-Wigner damping of collective strength over the microscopic background occurs described by a width I’ = 2%-(P)’ p )

(1)

and the strength function (the probability of finding a simple collective state i in a unit energy interval of the spectrum) has the shape

K. Heyde et al. I Nuclear Physics A 596 (1996) 30-52

32

Ea

Fig. I. Schematic representation of the fragmentation process. In the upper part, the unperturbed spectrum is shown. The thick bar denotes the collective state and the thin bars the more complex configurations. In the middle part, the thick bars show the strength of the collective state admixed into each of the complex states, obtained by diagonalizing the full Hamiltonian in a space containing both the collective and the complex configurations. The lower part gives the strength distribution separately. Taken from Ref. [ IO].

P;(E) =

r E)2+ (;q2’

-!-

2n-(E;

The mechanism

-

of fragmentation

is illustrated

and schematically

shown in Fig. 1.

If, however, the level density becomes too low for the above condition

to be valid, even

when considering an almost constant coupling matrix element between the collective state i( E;) and the microscopic underlying structure, one has to diagonalize the matrix Eo A . . . A

I

A EI

I ..

A

-----A’

I En I

I

I

A’

[Ol

-

(3)

I &+I 101

A’

A’ . . .

..

. E ll+l?l

where A denotes the coupling strength between the collective state EO and the II 2qp configurations and A’ the coupling strength between the collective state ( Eo) and the m 4qp configurations. We thereby neglect the 2qp-4qp coupling. The above Hamiltonian can be diagonalized more easily by rewriting the eigenvalue equation for the energies E( as

K. Heyde et al. I Nuclear

Physics A 596 (I 996) 30-52

3.7

(4) i-n+I.n+r which contains plitudes as

a root Ei between each successive

cl+,coll(~)

=

1 +

i If we consider

pair of unperturbed

the scissors 1+ component

c, +.COil (i) describing

energies. The am-

over the spectrum

i is obtained

(5)

c k”1?,, (&

=

an equally

$,?

spaced unperturbed

spectrum

with spacing

constant coupling matrix element A with A > D, then the Breit-Wigner

D and a single form is recovered

[ 121. We shall apply the model in the study of the fragmentation of a collective IBM-2 mixed-symmetry 1+ “scissors” state into the background of all Ii 2qp and 4qp Nilsson configurations. In such a model, according to the value of A, the

from Eq. (5)

condition Section

A > D is not fulfilled and the secular Eq. (4) needs to be solved (see also 2.3).

In a first step ( Section 2.1) , we describe a model where all M 1 strength is concentrated in a single collective state. In a second step (Section 2.2), starting from the Nilsson model and the appropriate equilibrium deformations, the two-quasiparticle (2qp) and four-quasiparticle (4qp) densities for K” = l+ states have been determined for eveneven Gd nuclei. In a third step (Section 2.3) we determine and diagonalize the coupling matrix between

both spaces and the fragmentation

process is studied.

2.1. The collective Ml state The concentration be studied in general,

of microscopic

Ml

strength

into a strongly

collective

state can

using a schematic model (Brown and Bolsterli [ 131) and indicates that the various 2qp (or lp-lh) configurations add up in a coherent way with

each individual configuration appearing with an amplitude of -$=, where II is the total number of microscopic unperturbed configurations (see also the appendix). In reality, collectivity

of low-lying

l+ states is weaker, and Ml strength

states. Since it is our suggestion distribution

that a realistic

using a fully microscopic

approach

description

is spread over several

of the detailed

is very difficult,

Ml strength

if not impossible,

in

particular for reproducing the smaller B(M1) values, we study the fragmentation of a strongly collective Ml state over a background of microscopic states. As a model to generate a collective Ml scissors state we have used the proton-neutron interacting boson model (IBM-2) [ 14,151 as applied to even-even Gd nuclei. We shall not discuss the IBM-2 calculation in detail (see Ref. [ 161) but just present the basic outcome that gives both the Ml strength and the energy E.,( 1 +). In Fig. 2, for the I48 6 A < 158 Gd nuclei we present the variation in excitation energy for the I:, 1: and 1: states and indicate the corresponding Ml strength. It is clearly seen that

K. Heyde ef al.lNuclear

34

Physics A 596 (1996) 30-52

AGd

6L

N

3-

t

r. 7

AGd

6L

N

;2I z l-

I

I

lL8

I

152

,

I

o

I

156 MASS

’

lL8

152

156

NUMBERIA)----c

Fig. 2. The energy spectra for the nuclei ‘4Xm’5XGdpresenting the mixed-symmetric (left-hand part). On the right-hand side we indicate the B( Ml; 0: + 1:) strength parameters describing the IBM-2 Hamiltonian have been taken from Scholten [ 171.

I:,

1: and 1: levels

( 1: is very small). The

almost all l+ strength is concentrated in a single 1T state; (closely reproducing the pure SU(3) limit result) this state shall be used as a starting point to spread strength into the background of 2qp and 4qp l+ states in these same nuclei. The above results are obtained using a IBM-2 Hamiltonian with parameters as discussed by Scholten in Ref. [17]. 2.2. The 2qp and 4qp I+ densities In order to study the fragmentation of the collective 1+ states, as generated in Section 2.1, over the microscopic background of 2qp and 4qp 1+ states we first determine the deformed equilibrium ground-state configuration for the ‘48-‘58Gd nuclei and generate the intrinsic single-particle orbitals ( fl) . We use a deformed modified Nilsson harmonic oscillator model [ 181 (with quadrupole ~2 and hexadecapole e4 deformation) with K, ,u parameters as determined through the prescription given in Ref. [ 181. From the singleparticle energies e( ?? 2, ~4) we calculate the total potential energy surfaces (TEE) using the Strutinsky renormalization method [ 191 with the liquid-drop component calculated using the Seeger-Howard mass formula [ 201. The resulting TPE surfaces are given in Fig. 3 with contour lines in the (~2, ~4) plane. The dashed line now connects the points of minimal energy for each E:! value (minimized in the hexadecapole degree of freedom ~4) [ 211. One observes that, with increasing mass, a stable quadrupole deformed shape sets in and stabilizes near to E:! ‘v 0.25 in the strongly deformed ‘56,‘5*Gdnuclei. Starting from the above equilibrium values, we generate the single-particle deformed intrinsic states, characterized by KY and construct the unperturbed K” 2qp and 4qp configurations and energies. These can be depicted as 29

:

I(f&fi2)K"),

(6)

K. Heyde et al. I Nuclear

Physics A 596 (I 996) 30-52

-

OUADRUPOLE

DEFORMATION

35

IE21

------

Fig. 3. Total potential energy surfaces for the even-even Gd nuclei ( 148 < A 6 158). Contour lines, with intervals of 0.5 MeV, are drawn in the (~2.~4) plane. The dashed line connects minimum total potential energy versus ~4 for every ~2 value. The vertical axis gives the ~4 coordinate, the horizontal axis is the ~2 coordinate.

with corresponding

unperturbed

energies

(8)

(9)

K. Heyde et al. I Nuclear Physics A 596 (I 996) 30-52

36

2

1

0

1

L

0

2

4

6

8

:

1

0 0

2

4

6

O_L”

8

-E

,,plMeVI

2

4

6

8

-

Fig. 4. The level density for the 2qp and 4qp 0 ’, I’, 2+ and 3’ configurations according to the prescription given in Eqs. (7)-(9), starting from the Nilsson model for l”Gd. The numbers are given on a logarithmic axis (exponent given) and the density is smoothed in bars with a 100 keV interval width. Note that the lower end ( IO” : 1 level) has been shifted for practical reasons in making up the drawings.

where E(0) are the one-quasiparticle energies (including pairing) derived from the Nilsson single-particle energies ~(a) (see Refs. [ 18,211 for the prescription used to determine the one-quasiparticle energies and the pairing gap). In Figs. 4,5 and 6 we give these corresponding

densities

(represented

as histograms

containing

the number of

unperturbed levels per 100 keV bin for the appropriate K” value) in ‘48Gd (N = 84), 152Gd (N = 88) and ‘58Gd (N = 94) for the K" = 0 i,I+,2+ and 3+ configurations. The upper curves denote the 4qp densities which, for the N = 88 and 94 nuclei, rapidly increase towards 102-10” levels per 100 keV bin when approaching an excitation energy of 6 < E,, < 8 MeV. It is possible to fit a degenerate Fermi-gas density distribution, as given in Bohr and Mottelson citebohr69, p(E)

= i exp (2&j

,

(10)

where a and 6 are fitted to the 4qp density distribution. This has been carried out in order to study the fragmentation of I+ spin-flip strength over a background of do still arise, but for the 4qp configuration [22,23 1. In 14sGd , quite large fluctuations

K. Heyde ef al. I Nuclear

-E

Physics A 596 (I 996) 30-52

37

unp (MeV) -

Fig. 5. As Fig. 4 but for ls2Gd.

heavier nuclei, the density rapidly (above 4 MeV) approaches increasing function (in agreement with results from analytical gas model

a very smooth exponential studies in a simple Fermi-

[ 121).

2.3. Microscopic

fragmentation

Having constructed ing microscopic strengths.

the necessary

ingredients

2qp and 4qp I+ densities),

A rather good estimate obtained as follows (see coherent superposition of smaller by a factor &I as approximate result (l+,coll~v/l+,2qp)

(the collective

I + state and the underly-

we evaluate the typical elementary

coupling

of the coupling matrix elements (I’, colllV I+, 2qp) can be also the appendix) [24]. Since the I+ collective state is a many 2qp states (n), the coupling matrix element will be compared to the collective matrix element, so we obtain the

N ~(l+,coll~V~l+,coll). J;t

(11)

K. Heyde ef al. /Nuclear Physics A 596 (1996) 30-52

38

4

3

2

1

L 0

2

4

6

2

4

6

0 ~

6

0

a

-

F.._ _ unp,

IMeW

2

4

6

6

2

4

6

6

-

Fig. 6. As Fig. 4 but for lsXGd.

Detailed calculations of the left- and right-hand side matrix elements of Eq. ( 1 1 ), by solving the 2qp problem in realistic cases of many non-degenerate j-shells, indicate that the equality, expressed by Eq. ( II), is a reasonably good approximation [ 25,261. Because V is the difference between the full and the unperturbed Hamiltonian of the system, the matrix element on the right-hand-side of Eq. (11) can be approximated by $ (Et+,co~i- ).E1+,2c,~ From the detailed knowledge of the 2qp energies (typical number n counting both proton and neutron 2qp configurations in the energy interval up to E, s 10 MeV is of the order of 200) in the even-even Gd nuclei one can determine El +,zqp and so, a typical energy shift of the average l+ 2qp energy E1+,lqp to the collective 1’ scissors state of the order 0.7-0.9 MeV results. This number is quite reasonable when comparing with realistic 2qp calculations in even-even nuclei. In e.g. the even-even N = 82 nuclei [ 271, when calculating the collective 2+ state near E, = 1.4 MeV, a shift of 0.8-l MeV is observed from the unperturbed 2qp states near E, Y 2.2-2.4 MeV. Similar results have been obtained in the Sn nuclei [28]. Moreover, the concentration of many 2qp states in the energy interval 2.2-2.4 MeV (2 24 with A the pairing gap) is an indirect indication for the small value of the typical (I’, 2qplVI I+, 2qp) matrix elements ( 10-100 keV) [ 271. These latter matrix

K. Heyde et al. I Nuclear

Physics A 596 (I 996) 30-52

____I+ V

Fig. 7. The coupling Eq. (13)). elements

are,

matrix element denoting

in general,

the 2qp to 4qp coupling

much reduced compared

in order to derive a value of A’ (see

to regular two-body

because of the presence of the pairing factors [ 291. Combining a typical

(1 +, co111Vi l+, 2qp) coupling

A=+ fi

I+,coll-~)

matrix elements

all the above quantities,

matrix element

= 50 keV

(12)

results. One may argue that the shift of 0.7-0.9

MeV is quite a bit smaller compared

to the

energy shifts ( l- giant dipole isovector resonance, 3- isoscalar vibration in doublyclosed shell nuclei) that can be of the order of 2-2.5 MeV [ 301. This may reflect the fact that the l+ scissors state is more like a Oful, orbital collective mode and thus is mainly affected by that part of the 2qp energy spectrum [ 5,6]. In considering coupling to the 4qp configurations, we use second-order

perturbation

theory and obtain as a result *~=(l~,colllVI1+.2qp)~(l+,2qPlV/I+,4qP) = -$1+,

co111VI If, toll) -5+,2qp

1 -

El_e ,

c%PlwlP) +

(13)

Here, using average 2qp and 4qp energies (the latter density for G-T, w and 7~1, combinations of 2qp states in total, up to E, 5 10 MeV, is typically of the order of I 04), a difference AE 21 2 MeV (twice the pairing gap, the energy needed to create an extra 2qp pair) occurs. For the average (2qpIVI4qp) matrix element, which looks like an L-matrix element (Fig. 7) we use the value of typical L-matrix elements as determined by Kuo and Brown in the fp region [ 3 I] and a value of z 100 keV results. Finally, combining all the above results, the A’ coupling matrix element is typically of the order of A’ E 2.5 keV. The above values of A and A’ are probably reasonable estimates with a 20% error bar on A. The above slight increase just spreads the strengths over a somewhat larger interval but does not change Figs. 8 and 9 in a qualitative way. The final distribution of the collective l+ state over the 2qp and 4qp background results from diagonalizing the coupling energy matrix (Eq. (3)). The number of 2qp

40

K. Heyde et al. I Nuclear

N

-

Physics A 596 (1996)

30-52

1

.-

-u -2

-3

0

4

2

6

8

I-

-

Ex (MeVl-

Fig. 8. (a-c) The strength of the collective Ii state admixed into the background of 2qp and 4qp 1’ states, averaged over an interval of 100keV, obtained after diagonalizing the matrix of F.q. (3). Results are given for the nuclei ‘“*‘so.‘52Gd. The black bars indicate the strength above the I % level. The energy of the collective I + state is taken as the result from the IBM-2 calculation (Section 2.1 and Fig. 2).

K. Heyde et al. I Nuclear

Physics

A 596 (1996)

30-52

41

-1

-2

-3

0

2

4

6

0

2

4

6

0

t

-1

N .-

-0 -2

-3

Fig. 9. (a-c)

Same as Fig. 8 but now for the even-even

nuclei ‘54~‘s6~‘sKGd.

states (up to E.,< 10 MeV) is typically of the order 200 whereas the number of all 4qp states amounts to IO4 (up to E., < 10MeV). In determining the c~+.~~rtamplitude, we have used (a) an algorithm to search for the roots in between successive

unperturbed

energies by using Eq. (4), (b) diagonalized

the

energy matrix of Eq. (3) using the Lanczos method, and, (c) diagonalized Eq. (3) using an energy truncation of 4.5 and 5 MeV for the 4qp spectrum. In all cases, no significant

K. Heyde et al. I Nuclear

42

variations

Physics A 596 (I 996) 30-52

I2 distribution in the final (cl +,COil

in 100 keV bins could be observed

as they

are given in Figs. 8 and 9. This is so because it is mainly the first step of fragmentation: the l+ collective

scissors state into the 2qp background

the overall fragmentation The final result,

with strength

A that determines

picture.

the collective

obtained for the l+ spreading

strength

J~i+,~~lll~, averaged

over

100 keV bins, is

as shown in Figs. 8 and 9 for the Gd isotopes as discussed

below. Here, the black bars indicate the strength above the 1% level.

3. Results

in the ‘w158Gd nuclei

In Figs. 8a-c and 9a-c we show the results for the nuclei i48~150~1s2Gd and 154,i56*158Gd, respectively.

A general,

first observation

is that the strength

lower mass group is spread over almost it is on average ing to observe

1 MeV, whereas

spread over an energy interval that the strongest

up to the 1% level for the for the heavier mass group

of almost

2 MeV. It is also interest-

bin in Figs. 8 and 9 represents

typically

of the order

of 30-35%

of the collective l+ strength. When summing all intensities, defined by I~l+,~~ll(i) I2 for the various Gd nuclei studied here ( 148 < A < 158), Ci(E,<4 MeV) always 93-95% of the total collective l+ strength is recovered. This is corroborating the fact that the experimentally

observed

Ml

strength,

(C B (Ml ) ), strongly correlates with collective variables clear charge radii, . . .). A number of studies evaluating plicitly

contain

up to E., 2 4 MeV (quadrupole deformation, nuthe summed Ml strength ex-

this feature and indicate that most orbital collective

in the above energy interval (O-4 MeV) [ 32-371. Even though the model used here is schematic, mer (Q)RPA

summed

and shell-model

strength

is present

and does not intend to replace for-

studies, it may give some insight

in the way collective

strength can become highly fragmented over the background of shell-model 2qp and 4qp configurations. It is the rather low level density in the 2.0 < E,v 6 3.5 MeV region that causes the specific fragmentation

process to occur, a region where the Breit-Wigner

equation for spreading (Eq. (2) ) would not be allowed. In addition to a conventional collective mode, it has been shown that relatively two-quasiparticle

(2qp)

configurations

can generate

large Ml matrix elements

pure

to indi-

vidual states by virtue of a large convection current associated with the motion of protons in high-e orbitals [ 38,391. Such configurations are also subject to fragmentation into the background of other quasiproton configurations [ 221. These effects become apparent in the gross features of single-nucleon transfer experiments which provide information of the single-particle strength distribution in the final nucleus, since a nucleon can be transferred

with minimal

excitation

of other degrees

of freedom.

In Fig. 10 we show

ejectile momentum spectra from a series of (t,(u) proton pick-up reactions performed on stable odd-2 rare-earth targets at a beam energy of 35.5 MeV at the Daresbury Laboratory. The experimental aspects are discussed in detail elsewhere [ 401. The global features are similar to those obtained from ((Y, t) proton-stripping reactions on the same targets [ 411. In each case shown, the final nucleus is even-even and configurations

K. Heyde et al. I Nuclear

,.O #OO.

s.0

P.0

43

Physics A 596 (I 996) 30-52

Excitation Enorgy (MeV) 1.0 8.0

1.0

4.0

2:o

ZOO- 1%1o(t,a)‘~Dy

16sHo(t,a)1WDy

rooA=+L

1

lSO-

n

8.0

6.0

4.0

3.0

50

400

100 200

100 0

Fig. IO. Alpha-particle

momentum spectra obtained for the (1, a)

targets at an incident energy of 35.5 MeV O-4 MeV

(left part) and 2.5-6.5

MeV

reaction on three stable, odd-l: rare-earth

and a laboratory angle of 18”, for the excitation energy regions

(right part). There is no common normalization between the spectra.

The abrupt loss of counts at the extreme left of the left-hand set of spectra, and at both ends of the right-hand set, corresponds to the end of the active length of the focal-plane detector.

populated

are 2qp in nature since the reaction

proceeds to a good approximation

by a

single-step process. The general features of the spectra shown (see Fig. 10) are: region (I) up to an energy of E., z 24 where collective excitations dominate the spectrum, region (II) above 24 up to E., M 3 MeV where a sudden increase of transition strength in well-separated peaks shows up and region (III) above 3 MeV where the detailed peak structure changes in a rather smoothly

varying

background.

Peaks corresponding to transitions populating the ground-state rotational band in each nucleus are evident in the spectra shown in Fig. 10 at low excitation energies. These bands are populated by transfer involving the orbital which is occupied by the odd proton in the target nucleus. The pair gap, between the ground state and the first twoquasiproton excitation, can be seen and is also illustrated, in 16’Er, in more detail in Fig. 11. Within this gap low-lying collective excitations may be populated via twoquasiproton admixtures in their wave functions, if these are accessible to the reaction

44

K. Heyde el al. I Nuclear

Physics A 596 (1996) 30-52

-I-

0

Fig.

I I. Excited states and band structure in “‘Er up to r 2 MeV. The data are taken from NDS [ 431.

in question. Above an energy of approximately 211, corresponding to around 1.7 MeV in excitation, there is a sudden appearance of many peaks corresponding to transitions populating

intrinsic

2qp excitations

and rotational

bands

built upon them

(Figs.

10

and 11) [ 421. Between 3 and 3.5 MeV, the precise point depending on the particular final nucleus, the nature of the spectra changes quite dramatically. The strong transfer peaks are replaced suddenly by a more continuously varying strength, where peaks, if any, are small (< 5%) in comparison with transitions at lower energies. This loss of peak structure, and the fragmentation responsible for it, are an interesting phenomenon. In general, residual interactions will induce the mixing of 2qp states and the transfer strength

associated

with a particular

several’ final states. In the limit of strong fragmentation,

2qp configuration

will become

fragmented

over

the transfer reaction spectra would consist of small

peaks corresponding to many weak transitions, where a single peak corresponds to only a small fraction of the cross section associated with a particular 2qp configuration. If the level density is such that individual transitions could not be resolved experimentally, the resulting spectrum would tend to be smooth and featureless as in the case at high excitation energy in Fig. 10. Moreover, the fragmentation of 2qp strength observed in single-nucleon transfer spectra to even-even rare-earth nuclei is enhanced by mixing with four-quasiparticle (4qp) configurations which become energetically possible at approximately 44. From Figs. 4-6 one indeed observes a sudden and dramatic increase in the density of states close in energy to 44. It is this mixing with 4qp configurations, caused by seniority violating interactions, which is responsible for the change in the character of

K. Heyde et al.

I Nuclear

Physics A 596 (1996) 30-52

E

2 qP

I

+ lnteractlon

I

incoherent

2qP strength

Fig.

12. Schematic illustration of the fragmentation

of the 2qp strength, due to configuration

mixing and

coupling into the more complex background of 4qp configurations as a possible explanation for the loss of structure in the transfer data above z 3 MeV.

the spectra near to 3 MeV (see Fig. 12). Strong mixing

and high fragmentation

occur

suddenly simply due to the abrupt and significant increase in the density of states at or near to 44. The above model used to calculate the level densities using an unperturbed picture should be modified by the inclusion of (i) the 4qp analogue of the Gallagher splitting [ 441 and (ii) the pair-blocking

effect [ 451, which can be large when four orbitals

are blocked from the pairing interaction. These effects will shift the energy of 4qp configurations rendering the value of 4A only an approximate energy when such configurations become important. The onset of severe fragmentation and depends on the nucleus under consideration. The global

features

of the transfer

spectra are therefore

is therefore ill defined

suggestive

of considerable

fragmentation of 2qp configurations. Any state above an energy of approximately 44 contains, in general, considerable admixtures of 4qp admixtures in its wave function. It therefore appears that large Ml strength associated with a high-l orbital will be fragmented

over many states in an analogous

way to the collective

strength previously

discussed. Both mechanisms responsible for the generation of large Ml strength, highl protons or a collective enhancement, are severely fragmented and the observed Ml strength associated with any one state is likely to be smaller than expected.

K. Heyde et al. I Nuclear Physics A 596 (I 996) 30-52

46

1+

C

Fig. 13. Schematic representation of a fragmentation of doorway I+

process of the

I+

collective state through a number

states containing hexadecapole and quadrupole coupled configurations, causing intermediate

structure before fragmenting further into the group of microscopic 2qp configurations (from right to left).

4. Intermediate

structure

The model described above spreads the collective If state (generated IBM) directly within the background of 2qp and 4qp If configurations.

within

the

It might occur that, occasionally, in between the first step of fragmentation into the 2qp density, some intermediate coupling to a recognizable degree of freedom could show up, leading to some non-statistical intermediate structure [ lo,1 11. The hexadecapole degree of freedom could be such an intermediate step since the g-boson, used in the IBM, when coupled with a number of d-bosons, can generate a number of extra If states whereas the pure sd IBM only results into a single strong I+ state in the 2.5-4 MeV energy region [46-481. So, the sdg-boson states may act as natural “doorway” states because the mixed-symmetry states, by their very nature as a proton-neutron scissors configuration, could contain a non-vanishing dynamic structure [ 1.51 partly composed of hexadecapole configurations. The process is schematically shown in Fig. 13 where the fragmentation of the strongly collective l+ state into the 2qp l+ background is modulated by the presence of the 1+ sdg boson configurations. In making this argument more quantitative, we can indeed study the mixing of the mixed-symmetry IBM-2 l+ state with the lowest ld2g; 1’) configurations (which is symmetric in neutrons and protons, so we can study the latter states in IBM-l ) [ 151. In Fig. 14, we present the lowest 1+ states in ‘56Gd, ‘58Gd and ‘68Er resulting from the

K. Heyde et al. I Nuclear Physics A 596 (1996) 30-52

-

-

-

=-=-=

47

= __-. -

--3-

-

-

‘=Gd

2-

16*Er

‘=Gd

1+

0I

Fig. 14. Results of IBM-l freedom (s&

calculations for If

boson calculation).

states in ‘s6.‘SXGd and lmEr including the g&son

The parameters have been taken from Ref. [47]

dashed line gives the position of the experimentally scattering. The data are taken from Ref.

observed

[ I ] ( ‘56.‘5XGd)

sdg IBM. For ‘56,158Gd, the Hamiltonian where they have been determined lying states. In 16*Er the parameters

I+

level, most strongly excited in Ml

and from Refs.

and its parameters

by fitting energies are determined

degree of

(for the Gd nuclei). The electron

[ 4-6,49 ] ( laEr). are taken from Ref. [ 471,

and decay properties

of the low-

by fitting the K” = 3+ hexadecapole

[ 50,511. Beca;se of the truncation of the model space to one g-boson excitation an d using a g-boson energy eg FV 1.5 MeV this (to states In) and I(~d)~-‘g)) calculation is reliable only up to about 4 MeV. This calculation now gives an estimate band

of the number of additional If states, which is about 5 per MeV, in the region 2.5 < E., 6 4 MeV. The above density, taken with an average mixing matrix element coupling the mixed-symmetry 1+ state to the sdg IBM configurations ( (H,i,) = 100 keV [ I.5 1) results in a spreading width for the intermediate coupling process of I’1 = 2r(H,i,)*p( So, the qualitative

I’)

E 300 keV.

process of Ml fragmentation

(14) then becomes

(if sdg IBM states are

present) that of an IBM-2 mixed-symmetry l+ state, coupling to the l+ “doorway” states, resulting from coupling g- and d-bosons, and thus giving rise to intermediate

48

K. Heyde et al. I Nuclear Physics A 596 (1996) 30-52

structure

over an interval with Tl N 300 keV before further fragmenting

1’ background

into the 2qp

(see Fig. 13).

5. Conclusion In strongly deformed

rare-earth

nuclei, many attempts have been carried out in order

to describe both the global and detailed experimentally strength

distributions.

fragmentation,

Despite

and microscopic

both collective

model

observed

magnetic

descriptions,

QRPA, QTDA calculations,

dipole (Ml )

showing

a lack of

it remains difficult to obtain

a good insight in the basic processes at work in producing the observed fragmentation pattern in the low-energy region and also the spin-flip Ml strength near 6-8 MeV. In the present a schematic

article

description

we present

and discuss

of the fragmentation

a fragmentation

of a single,

model

collective

that aims at

Ml

state over a

background of many microscopic configurations: in increasing complexity we consider 2qp and 4qp states. The collective state here, is obtained from the Interacting Boson Model (IBM-2).

The microscopic

2qp and 4qp configurations

are constructed

as the nqp

excitations in a deformed Nilsson harmonic oscillator potential including the necessary pairing correlations. We subsequently study the coupling of the collective 1+ state to the 2qp and further on to the 4qp states. In order to obtain the fragmentation not use standard

perturbation

theory because,

pattern we do

at the lower energy side (2.5-3.5

MeV),

of 2qp I+ states is not too large compared to the coupling strength to allow the use of it. Instead, we diagonalize the full, albeit simplified, coupling matrix the level density

in order to obtain an indication the microscopic states.

of the distribution

of the collective

If

component

over

Results are presented for the even-even Gd nuclei ( 148 < A < 158). As a major outcome we find that the strongest peak (averaged over a 100 keV bin) never contains more than 30-35% 4 MeV, 93-95%

of the collective

component

of the total collective

Ml

but adding all Ml strength

strength

is obtained.

up to E, 2

In other words, the

present results concerning fragmentation and summed Ml strength are conform with the conclusions reached in a totally different way using sum-rule methods. We stress that the present calculations, that on one side make use of statistical arguments but also incorporate the underlying microscopic 2qp and 4qp distribution of 1 + levels, aim at obtaining some rather general results on fragmentation. It does not come in competition with all former detailed (Q)RPA and shell-model studies concentrating on studies of the 1+ strength distributions in near spherical and strongly deformed nuclei. The alternative mechanism for generation of Ml strength, the motion of protons in high-l quasiparticle states is also subject to fragmentation as evidenced by the general features of proton-transfer reactions. In these experiments, rather clear evidence for an enhanced dissipation of single-particle (or 2qp) strength over the background above 33.5 MeV shows up. This could indeed be due to the fast increase of 4qp configurations at these energies and may proceed through the 2qp to 4qp coupling. Finally, we propose a possible origin of some intermediate structure in the spreading

K. Heyde et al.

of l+ collective invoking

I Nuclear

39

Physics A 596 (I 996) 30-52

strength

into the microscopic

the hexadecapole

degree of freedom

background

of 2qp and 4qp states by

as an “intermediate”

step. Coupling

the

original collective I+ state into configurations made up of hexadecapole and quadrupole (sdg configurations) excitations may cause non-statistical fragmentation over the 2qp4qp background

with the “hexadecapole”

Even though the model a better insight is obtained of scissors

1+ strength.

fragmentation

states acting as doorway

and results presented when approaching

We aim at applying

states.

are at best qualitative,

the important

problem

we think that

of fragmentation

the present ideas also to the study of the

process of scissors strength in odd-A deformed

nuclei.

Acknowledgements The authors Laboratory

are grateful

to collaborators

on the experimental

here and which was supported

programme,

at Manchester

University

some of the results

by the UK Science

and Daresbury

of which are used

and Engineering

Council.

One of

us (S.F.) wishes to acknowledge receipt of a SERC studentship. Three of the authors (C.D.C., K.H. and S.R.) are most grateful to the NEW0 for financial support during this study. They are very much indebted to A. Richter for constant discussions and a detailed exchange of ideas and remarks on an earlier version of the manuscript in order to better understand the delicate process of fragmentation of l+ scissors strength. They like to thank B. Mottelson, D. Brink, R. Leonardi from the ECT*, Trento where the final elements

of this article where put together

Isospin Excitations in Nuclear Structure” important discussions.

Appendix. A simple particle-hole

during

the workshop

and, more in particular,

model for proton-neutron

B. Mottelson

“Spin

and

for some

mixed-symmetry

states

A schematic model involving proton and neutron excitations can be used to construct a mixed-symmetry state, for example, using a proton and neutron particle-hole ( 1p- 1h) basis. If we choose a short-range a-interaction obtain the two-body matrix elements (jTj,‘IVlj~j~-‘)

(j,j~‘lVl_i,j~‘)

=

for the residual two-body

interaction,

we

D - E,

=D,

(A.11

where D and E denote the direct (D) and exchange (E) matrix element, respectively (see Fig. A.1). Starting from the S-interaction IDI = IEl and therefore for the proton lp-I h case. we obtain a vanishing matrix element whereas for the proton lp-1 h to neutron 1p1h coupling matrix element, the direct term remains. If we now consider an ideal case in which the proton 1p-l h and neutron lp-I h have the same unperturbed energy some interesting results show up from studying the secular equation. If all energies are

50

K. Heyde et al. I Nuclear Physics A 596 (1996) 30-52

;st

.v v j;’

Jn

----

j;

-1

:l

jn

-1 JK .I

J

.s-1

.I

JR

JII

Jll

J,

---_

.A -1

JV

Jv

Fig. A. I. The direct and the exchange matrix elements for proton (neutron) for the matrix element between a proton

I p- I h

I p- I h states

and the direct term

and a neutron I p- I h state.

degenerate, an energy matrix of the form of Eq. (A.2) results with a block diagonal structure in the z--n-, Y--V systems and a constant coupling matrix [D] in the z---v part, u

37.

-G

[Ol

..

I

I

CD1

-q I

[Ol

I IDI

(A.2)

e

I

[Ol

‘..

I [Ol

-e

Besides a large number of eigenstates remaining at the unperturbed energy q, one single state arises where all proton configurations are in phase and all neutron configurations are in phase, but with the two blocks having structure reads as IcoW = Cain jTr

I AK’) f C ajJ.ivj,‘) 1

opposite

phase. So, the wave function

(A.3)

i”

with (A-4) where n is the total (proton + neutron) number of configurations contributing to the collective state. For a model with all ,!$ = 2 MeV and a constant coupling matrix element of D = -0.2 MeV, using 10 states, a wave function (see Eq. (A.3) ) results with amplitudes

K. Heyde et al. I Nuclear Physics A 596 (I 996) 30-52

P*N

Fig. A.2. Outcome of a diagonalization for the unperturbed system of 5 proton I p- I h and 5 neutron I p- I h configurations, all at the same unperturbed energy of E, = 2 MeV, and using a coupling matrix element with strength D = -0.2 MeV.

aj,=aj,-

-_

hzO.316

at an energy EC+'= 1.O MeV, EC-'= 3.0MeV, so a shift of 1 MeV has been induced with respect to the unperturbed situation as illustrated in Fig. A.2. Even though this model is a highly schematic model there is a suggestion of a relationship between a microscopic proton and neutron structure and the creation of collective

in- and out-of-phase

nuclear two-body trum is clearly describing

interaction. too stringent,

mixed-symmetry

motion

using a reasonable

Even though the assumption some of the general proton-neutron

assumption

concerning

of a degenerate

characteristics

states in a realistic

the

energy spec-

could remain

when

situation.

References [II D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Lo Iudice, F. Palumbo and 0. Scholten, Phys. Lett. B 137 (1984)

27.

[21 P von Brentano, A. Zilges, R.-D. Her&erg,

U. Kneissl, J. Margraf and H.H. Pitz, Nucl. Phys. A 577 (1994) l9lc. 131 U. Kneissl, J. Margraf, H.H. Pitz, P von Brentano, R.-D. Herzbcrg and A. Zilges, Prog. Part. Nucl. Phys. 34 ( 1995) 285. [41 A. Richter, Nucl. Phys. A 522 ( 1991) 139~. I51 A. Richter, in: The Building Blocks of Nuclear Structure, ed. A. Covello (World Scientific, Singapore, 1993) p. 33.5. [61 A. Richter, in: Proc. Int. Conf. on Nuclear Shapes and Nuclear Structure at Low Excitation Energies, eds. M. Vergnes, D. Goutte, PH. Heenen and J. Sauvage (Editions Front&es) p. 253. [71 R. Nojarov, A. Faessler and M. Dingfelder, Phys. Rev. C ( 1994) submitted. [81 N. Lo Iudice and A. Richter, Phys. Lett. B 228 ( 1989) 291. 191 A. Richter, Progr. Part. Nucl. Phys. 34 ( 1994) 261.

52

K. Heyde et al. I Nuclear

Physics A 596 (1996) 30-52

[IO] C. Mahaux, PF. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Reports [I I ] J.P. Jeukenne and C. Mahaux, Nucl. Phys. A I36 ( 1969) 49.

[ 121 [ 131 [ 141 [ 15) [ 161 [ 171 [ 181 [ 191 [20] [2 I ] [ 221 [ 231 [24] [25] [26] [27] [28] [29]

[ 301 [3 I ] [ 321 [33] [34] [35] [36] [37] [38] [39] [40]

[ 41 ] [42] [ 431 [44] [ 451

A. Bohr and B. Mottelson, Nuclear Structure, vol. I (Benjamin, GE. Brown and M. Bolsterli, Phys. Rev. Lett. 3 ( 1959) 472.

I20 ( 1985) I.

New York, 1969).

E Iachello, Phys. Rev. Lett. 53 ( 1984) 1427. 0. Scholten et al., Nucl. Phys. A 438 ( 1985) 41. Interacting Bosons in Nuclear Physics, ed. F. lachello (Plenum,New York, 1979). 0. Scholten, Ph.D. Thesis, University of Groningen ( l980), unpublished. G. Nilsson, CF. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C. Gustafson, I.L. Lamm, P Moller and B. Nilsson, Nucl. Phys. A I31 ( 1969) I. M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44 ( 1972) 320. PA. Seeger and W.M. Howard, Nucl. Phys. A 238 ( 1975) 491. K. Heyde, J. Jolie, P. Van Backer, J. Moreau and M. Waroquier, Phys. Rev. C 29 ( 1984) 1428. C. De Coster, Ph.D. Thesis, University of Gent ( l992), unpublished. C. De Coster, K. Heyde and A. Richter, Nucl. Phys. A 542 ( 1992) 375. T. Koeling and F. Iachello, Nucl. Phys. A 295 ( 1978) 45. S. Yoshida, Phys. Rev. 123 (I961 ) 2122. S. Yoshida, Nucl. Phys. 38 ( 1962) 380. M. Waroquier and K. Heyde, Nucl. Phys. A 164 (1971) 113. M. Waroquier, J. Ryckebusch, J. Moreau, K. Heyde, N. Blasi, S.Y. van der Werf and G. Wenes, Phys. Reports I48 ( 1987) 249. T.T.S. Kuo, E.U. Baranger and M. Baranger, Nucl. Phys. 79 ( 1966) 513. K. Heyde, The Nuclear Shell Model, Study Edition (Springer Verlag, Berlin, 1994). T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 ( 1966) 40; A I I4 ( 1968) 241. W. Ziegler, C. Rangacharyulu, A. Richter and C. Spieler, Phys. Rev. Lett. 65 ( 1990) 2515. J.N. Ginocchio, Phys. Lett. B 265 ( 1991) 6. K. Heyde, C. De Coster, A. Richter and H.J. Wortche, Nucl. Phys. A 549 ( 1992) 103. K. Heyde, C. De Coster, D. Ooms and A. Richter, Phys. Lett. B 312 ( 1993) 267. N. Lo Iudice and A. Richter, Phys. Lett. B 304 ( 1993) 193. P, von Neumann-Cosel, J.N. Ginocchio and A. Richter, preprint (1995). I. Hamamoto and S. Aberg, Phys. I&t. B I45 ( 1984) 163. S.J. Freeman, R. Chapman, J.L. Durell, M.A.C. Hotchkis, F. Khazaie, J.C. Lisle, J.N. MO, A.M. Bruce, PV. Drumm, R.A. Cunningham, D.D. Warner and J.D. Garrett, Nucl. Phys. A 554 ( 1993) 333. S.J. Freeman, R. Chapman, J.L. Durell, M.A.C. Hotchkis, F. Khazaie, J.C. Lisle, J.N. MO, A.M. Bruce, PV. Drumm, R.A. Cunningham and D.D. Warner, Nucl. Phys. A 552 ( 1993) IO. S.J. Freeman, Ph.D. Thesis, University of Manchester, 1990, unpublished. B. Elbck and P. Tjom, Advances in Nuclear Physics, vol. 3 (Plenum, New York, 1969). Nuclear Data Sheets 53 ( 1988) 231. C.J. Gallagher, Phys. Rev. 126 ( 1962) 1525. I? Ring and P. Schuck, The Nuclear Many-Body Problem (Springer Verlag, Berlin, 1980).

[ 461 P. Van Isacker, K. Heyde, M. Waroquier and G. Wenes, Phys. L&t. B 104 ( 1981) 5. [47] [48]

I? Van Isacker, K. Heyde, M. Waroquier and G. Wenes, Nucl. Phys. A 380 (1982) 383. K. Heyde, P Van Isacker, M. Waroquier, G. Wenes, Y. Gigase and J. Stachel, Nucl. Phys. A 398 ( 1983) 235. [ 491 A. Richter, private communication. [ 501 D.D. Warner, R.F. Casten and W.E. Davidson, Phys. Rev. C 24 ( I981 ) I7 13. (5 I ] I? Van Isacker, unpublished.