N U C L E A R PHYSICS A

ELSEVIER Nuclear Physics A 633 (1998) 662-680

Pairing-plus-quadrupole model and nuclear deformation: A look at the spin-orbit interaction*

Jutta Escher l, Chairul Bahri 2, Dirk Troltenier, J.R Draayer Department of" Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Received 8 December 1997; revised 19 December 1997; accepted 20 December 1997

Abstract

The pairing-plus-quadrupole model, realized in the framework of the Elliott SU(3) scheme, is used to study the combined effects of the quadrupole-quadrupole, pairing, and spin-orbit interactions on ground state shapes of nuclear systems. Relevant measures for nuclear deformation are reviewed. Representation mixing induced by the symmetry-breaking pairing and spin-orbit forces is shown to soften the deformation. The angular momentum dependence of the results is discussed. @ 1998 Elsevier Science B.V.

PACS." 21.60.Cs; 21.60.Ev; 21.60.Fw Keywords: Pairing; Quadrupole-quadrupole interaction; Spin-orbit force; Nuclear deformation; SU(3) model; Shell model

1. Introduct ion

The pairing-plus-quadrupole model (PQM), first introduced by Bohr and Mottel-

son [1] and Belyaev [2] , has been widely used to reproduce both few-particle non-

collective and many-particle collective features of nuclei [ 3-5] . In shell-model descrip-

tions, which are based by definition on the notion of particles moving in an average

nuclear potential, the pairing and quadrupole-quadrupole interactions are the most im-

portant short-range and long-range correlations, respectively. Pairing explains the energy

* Supported in part by the U.S. National Science Foundation. J Present address: Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel, e-mail:

jutta@algebron.fiz.huji.ac.il. 2 Present address: Department of Physics, University of Toronto, Toronto, Ontario M5S IA7, Canada, e-mail:

bahri@physics.utoronto.ca.

0375-9474/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(98) 00115-8

J. Escher et al./Nuclear Physics A 633 (1998) 662-680 663

gap of about 1 MeV ("pairing gap") which occurs between the j r = 0 + ground state and a set of nearly degenerate states (J~ = 2 +, 4 +, 6 + . . . . ) in the spectra of even-even nuclei that are just a few nucleons from being closed shell systems. The quadrupole- quadrupole interaction, on the other hand, dominates for near mid-shell nuclei, is known to induce deformation [6], and hence plays a major role in describing rotational spectra. Studies suggest that nuclei can be described rather well in terms of competition between the pairing and quadrupole modes [4].

The pairing interaction has long been known to play an important role in the physics of many-body systems. It has been used to model superconducting phenomena in various branches of physics, such as nuclear physics, condensed matter physics, and particle physics [ 1,7,8]. The work of Bardeen, Cooper, and Schrieffer provides undoubtedly the most prominent example for the practical consequences of this interaction [7], although pairing was first realized in the nuclear physics context [9,10]. Ever since Jensen and Mayer explained the spin of odd nuclei as resulting from the last unpaired nucleon, the pairing interaction has been considered an essential ingredient for the description of nuclear properties, and even the simplest models show that nucleons tend to pair off. The BCS approximation [7] for handling pairing correlations works well when applied to systems with a large number of particles; however, because it violates particle number conservation, a straightforward application to finite systems such as nuclei must be considered with some caution. Although enhancements, such as imposing particle number projection on BCS solutions, can be used to extend the applicability of the BCS approach, it cannot be expected to yield exact results for real nuclei. In contrast, the single j-shell seniority coupling scheme [11,12] and its multiple j-shell generalization [ 1 3 ] provide an exact solution of the pairing problem. In this case the seniority quantum number (generalized seniority for the multiple j-shell case) v, which counts the number of unpaired nucleons, governs the complexity of the problem. Unfortunately, pair breaking in nuclei, as generated for example by the spin- orbit interaction, is known to be very important. Since the complexity of the scheme grows combinatorially with v, this approach is problematic for all but near closed shell

systems. The quadrupole-quadrupole interaction has been a standard ingredient in models that

aim at reproducing rotational spectra and nuclear deformations. It emerges (apart from a constant) as a leading contribution in the multipole expansion of any nuclear long-range potential. It is essential for modeling collective properties of nuclei such as reduced quadrupole transition rates and quadrupole moments. One model which makes use of the dominance of the quadrupole-quadrupole interaction in deformed nuclei is the SU(3) shell model. The model, in its simplest form, was first introduced by Elliott [6] who recognized that the three components of the orbital angular momentum operator L = ~ i li and the five components of the (algebraic) quadrupole operator a~, = ~ i qT,,, =

~ i ( r2 Y2~z (ri) 4- b4p2y21, (lSi) ) / b2, which is obtained through symmetrization of the "collective" quadrupole operator, ac = ~-~i qf = ~ Y'~i r2 y2** ( r i ) / b2, close t/z

under commutation and generate the group SU(3). It turns out that SU(3) is the sym-

664 J. Escher et al./Nuclear Physics A 633 (1998) 662-680

metry group of the three-dimensional many-particle harmonic oscillator and has SO(3), which is generated by the three components of the angular momentum operator L, as a subgroup. As a consequence, eigenfunctions of the three-dimensional harmonic oscillator can be characterized by their transformation properties under special unitary (SU(3)) symmetry transformations. Furthermore, since the system is rotationally invariant, the irreducible representations (irreps) of the special orthogonal group (SO(3)) provide additional quantum numbers for a complete characterization of the harmonic oscillator eigenfunctions. Single-particle wave functions of a nucleon in the r/th harmonic oscilla- tor shell, with angular momentum l and projection m, for example, can be denoted by ](r/0)lm), where (r/0), l, and m refer to irreducible representations of the the groups SU(3), SO(3), and SO(2), respectively. This labeling scheme can be generalized to nuclear many-body wave functions [6]. The advantage of the Elliott scheme lies in the fact that the usual shell-model space can be organized into subspaces where all states of a given subspace transform in the same manner under the action of SU(3), and thus can be labeled by the same SU(3) quantum numbers. In a Hamiltonian system with a strong quadrupole-quadrupole interaction states of the subspace belonging to the so- called leading irrep lie lowest in energy and the large shell-model space can be truncated in a very natural way.

Algebraic theories, such as the SU(3) model, offer the greatest simplifications when the interaction under consideration is symmetry-preserving in the selected group labeling scheme, that is, when the Hamiltonian does not mix states belonging to different irreps of the relevant group chain. One speaks of an exact symmetry when the Hamiltonian commutes with all the generators of the symmetry group and of a dynamical symmetry when the Hamiltonian is written in terms of and commutes with the Casimir operators of a chain of groups. In both cases basis states belonging to inequivalent irreps do not mix, and the Hamiltonian has a block structure in a basis characterized by the irreps of the relevant groups. In situations where a symmetry-breaking Hamiltonian is involved it is possible to decompose the offending terms into basic parts which exhibit specific transformation properties. Provided the appropriate group coupling coefficients and the matrix elements of some elementary tensor operators are available, matrix elements of operators which connect inequivalent irreps can be determined and the problem can then be (at least in principle) exactly solved.

For the example of the nuclear SU(3) model, with both the SU(3) D SO(3) and SU(3) D SU(2) ® U(1) subgroup chains, coupling coefficients as well as matrix ele- ments of SU(3) unit tensors are available: The relevant mathematical and computational framework for both SU(3) D SO(3) and SU(3) D SU(2) ®U(1) coupling coefficients has been developed about 25 years ago [ 14,15], and a new user-friendly code for eval- uating matrix elements of SU(3) unit tensors has been placed into public domain in 1994 [ 16,17]. Using these tools, the pairing-plus-quadrupole Hamiltonian was recently investigated in the framework of the SU(3) shell model for the case of N = 2, 4 . . . . . 18 identical particles in a degenerate oscillator shell [18]. The restricted Hilbert space allowed for a schematic study of the quadrupole-quadrupole and pairing forces in a

J. Escher et al./Nuclear Physics A 633 (1998) 662-680 665

truncation-free environment.3 The previous study [ 18] investigated the competing and complementary features of these important two-body correlations and their effects on the shape of nuclear systems. Ground state deformations induced by pairing were found to be triaxial but rather soft, whereas the quadrupole-quadrupole interaction was confirmed to favor prolate or oblate shapes which are sharply defined. These findings only seem to contradict the traditional association of pairing and sphericity since: a) In the past, pairing has been primarily employed for an explanation of experimental features, such as the pairing gap, which occur in the spectra of near-closed - and thus only slightly deformed - nuclear systems; and b) The rather soft/3 and y deformations induced by pairing imply that the quadrupole-quadrupole interaction, which dominates in mid-shell nuclei, has a significant influence on the shape of the system, driving it to the maximum Pauli-allowed prolate or oblate limit.

In addition, the spin-orbit interaction, which breaks SU(3) symmetry and which was not considered in the previous study, is known to play an important role in nuclear physics. Spin-orbit correlations were first introduced independently by Mayer [ 10] and by Haxel, Jensen, and Suess [9] in order to explain shell closures and magic numbers. In any given harmonic oscillator shell, the spin-orbit force mainly affects the largest-j orbital by lowering it energetically. In heavy nuclei, where the effect is so strong that magic numbers deviate from the major shell closures of the harmonic oscillator, the pseudo-spin concept can be applied [ 19-26]. In this approach the largest-j level in each major shell is removed from active consideration and pseudo-orbital and pseudo-spin angular momenta are assigned to the remaining single-particle states. The set of all pseudo spin-orbit levels associated with an oscillator shell form a complete pseudo- oscillator shell of one quantum less, and the symmetry-breaking spin-orbit force is a weak residual interaction which can be neglected in a first approximation. In light nuclei, however, the spin-orbit interaction has to be taken into account explicitly in order to achieve agreement with experimental data.

Since the spin-orbit force affects the low-energy structure of many light nuclei, it is worthwhile to investigate its effects on the shape of nuclear systems. It is the goal of this contribution to extend the previous study [ 18] to include a spin-orbit term in addi- tion to the previously considered quadrupole-quadrupole and pairing interactions and to examine the complementarity and competition of these terms in determining the shape of the systems under investigation. Relevant measures of deformation are discussed in Section 2. In Section 3 the SU(3) schell model and its geometrical interpretation are re- viewed, and the realization of the pairing-plus-quadrupole Hamiltonian is demonstrated. Results of the study can be found in Section 4, and the findings are summarized in Section 5.

3 Whereas the quadrupole-quadrupole interaction takes a very simple (diagonal) form in the SU(3) scheme, the expression for the pairing operator has a rather complex form: pairing - unlike the quadrupole-quadrupole interaction - breaks the SU(3) symmetry.

666 J. Escher et al./Nuclear Physics A 633 (1998) 662-680

2. Measures of deformation

Deformed nuclei are usually characterized by experimentally observed quadrupole moments which significantly exceed theoretical single-particle predictions. However, spectroscopic quadrupole moments provide only an incomplete measure of nuclear de- formation. One reason for this is the fact that the quadrupole moment of a given state depends on both the angular momentum of the state and its intrinsic structure. A nuclear state with total angular momentum 0 or 1/2, for example, has a vanishing quadrupole moment, but might very well possess an intrinsic deformation. Furthermore, a nucleus with equal probabilities of being prolate and oblate has a zero quadrupole moment. Hence, a vanishing quadrupole moment does not necessarily signify a spherical charge distribution. Neither does a large quadrupole moment for a particular state establish that the state has a well-defined intrinsic shape, since the quadrupole moment does not give any information about the rigidity or softness of nuclear deformation.

A more complete characterization of intrinsic nuclear states can be achieved with the help of rotationally invariant products of the electric quadrupole operator [27-29]. Electromagnetic multipole operators are spherical tensors and therefore angular momen- tum zero-coupled products of such operators can be constructed. Rotational invariance implies that the expectation values of these products are frame-independent which leads to significant simplifications. The electric quadrupole operator, for example, when eval- uated in a principal axis frame, is completely characterized by two parameters, Q and O, since its components can be written as: Q20 = Q cosO, Q2,+2 = ~zQsinO, and Qz,+1 = 0. Zero-coupled multiple products of the quadrupole operator are simple func- tions of the two parameters Q and O. Expectation values of these invariants can be determined experimentally from measured E2 reduced matrix elements through the use of intermediate state expansions. The method is completely model-independent but its accuracy is limited by the availability of experimental data. Model calculations yield exact values for the rotational invariants, which may then be compared to experimental results.

Given the expectation values of the various rotationally invariant products of the quadrupole operator, the centroids, variances, skewnesses, excesses, etc. associated with the (Q, O) distribution of the expectation value of the electric quadrupole moment can be determined directly for the nuclear state under consideration. Various methods can then be employed to relate the resulting distributions to (/3, y) shape distributions. Kumar [ 27 ], for example, uses the concept of an equivalent ellipsoid, the charge and moments of which equal those of the nucleus under investigation, to relate the model- independent nuclear moments to intrinsic shapes. Without assuming a specific nuclear model, axial symmetry, or smallness of deformation, he provides a prescription for obtaining measures of intrinsic nuclear deformation. The method is valid for spherical, deformed, and intermediate nuclei, and can be applied to any (even-even, odd-A, or odd-odd) nucleus, provided enough E2 matrix elements are available. It is also possible to employ model assumptions in order to relate the (Q, O) distributions to nuclear shapes. In the present work the nuclear quadrupole moments are calculated in the Eiliott

J. Escher et al./Nuclear Physics A 633 (1998) 662-680 667

SU(3) model, and the relation to nuclear deformations is established via the connection between the Elliott model and the Geometric Collective Model. Details of this method are outlined in Section 3.

3. SU(3) schell model picture

Realizing the PQM in the framework of the Elliott SU(3) scheme allows for a geometrical interpretation of the many-nucleon states via a relation between the invariants of the SU(3) group and those of the Geometric Collective Model. In the Elliott model, which is an algebraic theory based on the three-dimensional harmonic oscillator, basis states are labeled according to their properties under transformations of the unitary group U(g2), where/2 = ( r /+ 1 ) (77 + 2 ) /2 denotes the degeneracy of the r/th major oscillator shell under consideration, the special unitary group SU(3), and the angular momentum group SO (3) :

Iq>) = IN[ f ]a ( A/z)rL, S; JM) . (1)

Here N gives the number of particles in the r/th shell, [ f ] labels the irreducible representation (irrep) of U(J2), (A/z) refers to the irrep of SU(3), L and S are the orbital and spin angular momenta of the system, respectively, and J is the total angular momentum with projection M along the z-axis of the laboratory frame. The quantum numbers a and K are additional labels which are needed to distinguish between multiple occurrences of (,~/z) in a given [ f ] symmetry and multiple L values in a given (A/z) irrep, respectively.

A shell-model description of rotational nuclear motion can be given since the su(3) Lie algebra associated with the SU(3) group contracts to rot(3) -- [RS]so(3), the algebra associated with the rotational limit of the Geometric Collective Model. The su(3) algebra is generated by the three components of the orbital angular momentum operator L = ~-,i li and the five components of the Elliott (or algebraic) quadrupole operator Q~ = Y'~i ~ = ~ ~-]~i ( r2 Y2# ( ?i ) -~" b4 p2yz~z (]3i))/b 2, where the sum runs over all particles in the valence shell, /z = - 2 , - 1,0, 1,2, and the oscillator length is given by b = v~/mto; the generators of rot(3) are the components of L and those of the "collective" quadrupole operator, QC = ~ i q~, = ~ Y'~i rZY2,( Pi) / bz" Within a major oscillator shell the matrix elements of QC and Qa are identical, however QC couples states belonging to the r/th shell with those of the rfth shell with r f = r/4-2, whereas the matrix elements of Qa between states belonging to different shells vanish. It has been demonstrated [30,31 ] that the invariants tr [ (Q¢)2] and tr [ (QC)3], with tr [O] denoting the trace of the operator O, of ROT(3) = [R5]SO(3), the Lie group associated with rot (3), and those of SU(3), namely C2 (A/z) and C3 (a/z), the expectation values of the second and third order Casimir operators, can be linearly related to each other. This, in turn, results in a direct connection between the microscopic quantum numbers A and/x and the collective shape variables/3 and 3' [ 30,31 ]:

668 J. Escher et al./Nuclear Physics A 633 (1998) 662-680

0 ° Z

/ / / 1 / I / I / J / 7 s ~ 1 1 / / / / / 1 . . ' 7 / 6

k ~ / ' l l l l l l /-.// l l s ~ / 1 1 1 1 1 l . - 7 1 1 1 4 /~

' / '

/ / / /

0 2 4 6 8 1 0

Fig. 1. A traditional (,87) plot, where ,8 is the radius vector and y the azimuthal angle, demonstrates the relationship between the collective shape variables (~Sy) and the SU(3) irrep labels (A/z). The (#y) vary continuously (/3/> 0, 0 ~< "y ~ 60 ° ), while 3. and/1, take on positive integer values 0nly, as is indicated with the help of a grid, with each node corresponding to a (A/Z) pair.

(tr [ (QC)2]) = 3k2~2 ~_~ C2(A#) = ~ [A 2 + A/£ +/LL 2 + 3(A + /Z) ] ,

(tr[(QC)3]) = 3k3/33cos3y,-÷C3(Atz) = ~(a-~z)(a+2~z+3)(2a+jz+3), (2)

where the constant k = ~ A ( r 2) with A being the number of nucleons in the nucleus and (r 2) the mean square radius of the system. The exact relation between (/3y) and (A/z) is given by:

k/3cosy = (2A + / z + 3 ) / 3 ,

k/3siny = ( / z + 1 ) / v ~ , (3)

which implies that each SU(3) irrep (A/z) corresponds to a unique geometrical shape (/3y). This correspondence can be illustrated with the help of a grid superposed on the well-known (/3y)-plane of the Geometric Collective Model as is shown in Fig. 1. 4

In order to obtain a measure for the "average" deformation of a state one needs to deduce values for the collective model /3 and y variables. Since /3 and y are simply averages of the microscopic observables, one can use Eqs. (3) to write

2 2

3 3 k /3v cos3yp = 4(C3)~, (4)

where {O)p ___ (qGJO[9"~} denotes the expectation value of the operator O in the vth eigenstate of the system. To extract information regarding the intrinsic deformation of the J~} configurations, the eigenstate I~} can be decomposed into SU(3) basis states,

I~'~) = Y~ c~il~i), (s) i

4 In the oscillator picture, core configurations couple to the SU(3) irrep (00), and the (A/z) irreps that can occur are determined by the number of valence protons or neutrons under consideration.

J. Escher et al./Nuclear Physics A 633 (1998) 662-680 669

where the states 14) are as defined in Eq. (1). Since C2 and C3 are diagonal in the SU(3) basis it is particularly simple to determine their expectation values in any calculated eigenstate and hence the shape of the nuclear system under investigation.

In addition to evaluating the average /3- and y-values for a nuclear state, it is also necessary to determine whether the deformation is very sharply defined or rather soft. To this end higher moments of the/3, and y . distributions of calculated eigenstates I$',,) of the relevant Hamiltonian were calculated. More specifically, by using standard error analysis methods, the width A/3~ can be shown to be given by

(AC2),, A/3~ = $/3. (C--~ T-2 ' (6)

while AT. can obtained from the expression

3 //3(ac2) (7) Acos3Tv=cos yvVlk2(C2),, +4 ) + ( (-C-~3)v .]

Here (AO)v =-- tr stands for the square root of the variance or second central moment, /x2, of the distribution of an operator O, where the nth central moment of a distribution

is defined by

= ( g ' . l ( O - (8)

The Hamiltonian of the PQM consists of three parts:

H = Hs + VQ + Ve, (9)

where H~ = ~ j ejhj includes the harmonic oscillator mean field, as well as all relevant single-particle effects, such as the spin-orbit and orbit-orbit terms. The ej denotes the single-particle energy of the angular-momentum-j orbital and hj is the number operator counting the fermions in that level, the sum being over all j levels of the shell(s) under consideration. VQ denotes the quadrupole-quadrupole interaction with strength X, VQ = -½xQ • Q, and Vp is the pairing interaction with strength G. Since the application is restricted to a single major oscillator shell, Q can be replaced by the Elliott quadrupole operator Q = QC = Qa, and - in the SU(3) basis (Eq. (1)) - the term VQ reduces to diagonal form, VQ = - ½ x [ 6 C 2 - 3L 2 ] with eigenvalues EQ(A,/z, L) = -½X[4{.~ 2 + a/z + / z 2 + 3.~ + 3/z} - 3L(L + 1) ].

In order to write down an expression for the SU(3) symmetry-breaking term Hs, a second quantization formulation is employed. Specifically, the number operator hj is written as

hj = ~-~ (~elhjl/3)a~a/s , (10) c~,3

where ~ and [3 stand for a set of quantum numbers labeling single-particle states (r//jm), and at~ and a,~ are fermion creation and annihilation operators, respectively, which obey the standard fermion anticommutation relations. Using

670 J. Escher et al./Nuclear Physics A 633 (1998) 662-680

if r/~ = r/a, 1,~ = l~, j,~ = j# = j, m,, = m E, and j = l,~ belongs to the r/~th major oscillator shell and ('~l~Jl~) --0 otherwise, the number operator becomes:

hj t = Z a(~o)1½Jma(r/O)1½Jm " ( 1 1 ) nl

Employing SO(3) and SU(3) recoupling techniques, one finds that the number operator can be written as [32]:

hj= ~ ( - 1 ) j+I/2+L(2j + 1) ~ ½ l L (~.~)KL

× ((r/0) l; (Or/) l[1(AA) KL) (1,1)I/(MOKLS;J=M--O (12) rF/ - -

where (4-[]-)p denotes a SU(3) D SO(3) reduced coupling coefficient, and (la)ll... is a 7~'ff vs

one-body SU(3) unit tensor, the left superscripts of which denote its particle (creation, annihilation) operator character. More specifically, the operator is defined as follows:

r/r/,(I'I)II(A#)KLS;JMv, = [a t(r/0) ~1 X ~I(O~,)½](A#)~cLS;JM . (13)

Analogously, one can express the pairing interaction in terms of fermion creation and annihilation operators:

Vp = - ¼ a Z - + - + a a ,,~u~ B /3' (14) at~

were ~ and /~ denote the time-reversed partners of the single-particle states a and 8, respectively, and G is the strength of the pairing interaction. EmploYing SO(3) and SU(3) recoupling techniques it can be shown that Vp may be expressed in terms of SU(3) irreducible tensor operators [ 18]:

G vp=-~- ~ ~~-'P~,{{(alU,)(#2a2)}po(,~o#o)} (AI/~I)(A2.v,2) T/~ t po(ao~o)

X (2,2)~[po(Ao#o)Ko=llo=O,so=O;k---O ,m' (,l~)s:0(m,l~)s2=o , (15)

where (2'2)11" is a two-body SU(3) unit tensor: fir/' v'...

( 2,2 ) ~[ po ( Ao tzo ) tco= l lo=O,so--O;k=O 7~r// ( hi fl, l ) Sl ----0(/~2 A2 ) S2-----0

~t ] (/"t2"~2) ] PO(AOI~O)KO=IIo=so----O;O = [[a~ x a t ] ('l'~') x [ar/, x ~r/,j Jo , (16)

and the coefficient P,m' {-" "} involves a sum over the product of three SU(3) D S0(3 ) reduced coupling coefficients: 5

5 Note that for a single major oscillator shell calculation the phase factor drops out since the single particle orbital angular momentum values are then either all even or all odd.

J. Escher et al./Nuclear Physics A 633 (1998) 662-680 671

P ~ ' {{ ( a , ~ l ) ( m a 2 ) }p0 (a0~0) }

= ~-~(--1) l - l 'v / (21-1- 1)(2l ' + 1)((r/0)/; (r/O)tll(a~m) lO) IP

× ((r/'o)l'; (r/'O)l'll(a2~2) lO/((a~m) lo; (~2a2) 1oll(ao/~o) lO/~. (17)

The SU(3) coupling coefficients as well as reduced matrix elements of SU(3) unit tensors which occur in the the above expressions for hj and Vp can be calculated with the help of existing computer codes [ 15,17]; thus matrix elements of the symmetry- breaking spin-orbit and pairing interactions can be evaluated for any specific SU(3) application.

4. Results

It is well known that the quadrupole-quadrupole interaction plays an important role in driving the nuclear many-body system toward strongly deformed shapes, most of

which are experimentally determined to be prolate. In the language of the Geometric Collective Model, most deformed nuclei exhibit an energy minimum near the y = 0 ° axis in the (/3y)-plane. Pairing, on the other hand, has traditionally been (implicitly or explicitly) associated with sphericity, a notion that has been challenged by the re-

sults of some recent studies. It was found that seniority-zero states, which are fully paired eigenstates of the pairing operator, may very well possess an intrinsic quadrupole

moment [33,34] and, furthermore, a detailed investigation of the eigenstates of the pairing operator has demonstrated that pairing favors triaxially deformed many-particle

configurations [ 16,18]. To study the influence of the spin-orbit interaction on these results, the full Hamil-

tonian (Eq. (9)) was considered for the case of N = 6 identical particles in the r /= 3 shell. Since the spin-orbit force has the strongest effect on the largest-j orbital, the

energies E~/2, E3/2, and e5/2 of the levels PU2,P3/2, and f u z were fixed at 0 and e7/2 of the fT/2-1evel was varied from 0 to - h w .

The results are displayed in Fig. 2, which gives the (kfl, y ) values for a separation

de7~2 = 0, hw/10, hw/2, and hto of the f7/2 orbital from its (PU2,P3/2, f5/2) partners. The leftmost marks of each curve (indicated by a solid disk) represent a vanishing

quadrupole-quadrupole interaction (that is X = 0), and from left to right the parameter X increases in increments of 0.0005 hw, with the rightmost point corresponding to

X = 0.02hw.6 The lengths of the radial line segments which cross the bold curves at

(kfl , 9") values corresponding ( = 0.0005,0.0010 . . . . . 0.0200hto provide a measure of the widths of the associated/3 distributions: The line segments run from (k ( /3 -A/3) , y) to ( k ( f l + Af t ) ,3 ' ) . For the sake of clarity the spreads of the y distributions have not been indicated here, but will be discussed below.

6The pairing interaction strength G was fixed at 0.01 hto, a value which corresponds to about 0.1 MeV in the J)~-shell and leads to a pairing gap of 1 MeV ~ 0.1 hto.

672 J. Escher et aL /Nuclear Physics A 633 (1998) 662-680

E7/2 =~

0 2 4 6 8 10

~ A87/2 = ho)/10

0 2 4 6 8 10

~ AE7/2 = "ho)/2

0 2 4 6 8 10

/ A~7/2 = 11(0

0 2 4 6 8 10

Fig. 2. A plot of (Kfl, y) for calculated eigenstates for six particles in the fp-shell. The strength of the pairing interaction is held constant at G = 0.01ho~, while the quadmpole-quadrupole force increases in increments of 0.0005hw from the left (X = 0, indicated by a solid disk) to the right (X = 0.02hw) of each curve. Four graphs are shown, corresponding to different strengths of the spin-orbit interaction. Specifically, from top to bottom, the graphs correspond to Ae7/2 = 0, hw/10, ho)/2, and hw, respectively. A measure for the spread in the /~-distribution at each point (xp , y) is provided as well: The length of a radial line segment, which runs from (K(/~ - ,4/~), y) to (K(,8 + Aft), y) and crosses the bold curve at (xfl, ~'), indicates the width of the corresponding ,8-distribution.

The results show that for both vanishing and finite spin-orbit interaction strengths, the quadrupole-quadrupole interaction drives the system towards prolate shapes. However, while for a weak spin-orbit force (Ae7/2 = 0 ---+ h(o/10) the quadrupole term quickly dominates the behavior of the system, for Ae7/2 ~ hw/3 (not shown), hto/2 . . . . . hto one finds that the spin-orbit term counteracts this effect; the quadrupole-quadrupole interaction has to reach a certain minimum strength (Xo) in order to significantly affect the shape of the nucleus. However, once this strength Xo, which depends on Ae7/2

J. Escher et aL /Nuclear Physics A 633 (1998) 662-680 673

(X0 ~ 0.004hw for A~7/2 = hw/3, 0.006hoJ for Ae7/2 = hoJ/2, and 0.012hw for Ae7/2 = heo) is reached, the Q • Q term quickly drives the system towards its maximum (prolate) elongation. For a weak spin-orbit force this maximum is easily reached (see the top part of the figure), while a large ACT~2 requires a strong Q. Q term to force the system into its maximally prolate (or oblate) deformed state. It is also noteworthy to observe that increasing the spin-orbit interaction decreases the/3-deformation, while the expectation value for y is less affected. That is, the spin-orbit term has a strong effect on the elongation of a nucleus but it does not change its triaxiality significantly, while the primary effect of the quadrupole-quadrupole interaction is to drive a given system away from triaxiality, towards prolate (N < S2) or oblate (N > / 2 ) shapes.

In order for k/3 and y to yield a meaningful measure for the deformation of a nuclear system, their expectation values as well as the widths of their distributions need to be known. For the/3 deformation parameter these widths have already been indicated by the lengths of the line segments in Fig. 2. Fig. 3 illustrates how for different values of AeT/2 the widths of both the/3 and y distributions change as a function of the strength X of the quadrupole-quadrupole interaction. For Ae7/2 = 0 (vanishing spin-orbit force) and ALE7/2 = hw/ lO (weak spin-orbit force) and small x-values A/3//3 ~ 20%--25% and zly ~ 15 °, that is, the y-distribution has a considerable width, allowing for a wide spread in y. The softness of the deformation implies that even though pairing favors triaxial nuclear shapes, the quadrupole-quadrupole interaction is able to easily drive the system away from the triaxial pairing minimum towards prolate or oblate shapes. As Q • Q gets stronger and begins to dominate the Hamiltonian, the shape of the nuclear system becomes sharper, in particular in the k/3-variable. As shown in Fig. 3, with increasing spin-orbit interaction strength ( d e 7 / 2 = hw/2 -~ hoJ) the distributions in both k/3 and 3' become more spread out. The quadrupole-quadrupole interaction still pushes the system towards larger kfl and smaller y, but it is no longer able to clearly define the shape of the nucleus.

This behavior can be understood if one considers the symmetry-breaking effects of the spin-orbit force, as is done in Fig. 4. Shown is the eigenstate decomposition for the above system of N = 6 particles in the fp-shell, with the parameters in the Hamiltonian (9) chosen to be G = 0.01ha~, X = 0.005hw and 0.015hto, and de7~2 is varied from 0 to heo (case I: ACT~2 = 0; case II: Ae7/2 = hw/10; case III: de7~2 = hto/2; case IV: ACT~2 = hto). For a vanishing spin-orbit interaction (case I) the ground state of the system is a pure S = 0 state, with the (A/x) = (12,0) component being the dominant one in the expansion. The small amount of mixing between different SU(3) irreps with the same spin S is due to the pairing interaction, which is a spin scalar but breaks the SU(3) symmetry. Already for a small spin-orbit force (Ae7/2 ~ hw/lO, see case II) the symmetry-breaking effects of this interaction become clearly visible, and for a strong spin-orbit force (Z{e7/2 ---+ hO), see cases III and IV) the SU(3) structure of the system is completely destroyed. The symmetry-preserving influence of the quadrupole-quadrupole interaction, which is a SU(3) scalar, can be observed by comparing Fig. 4a with Fig. 4c. It is clear that there is much less mixing of different SU(3) irreps in the latter figure which illustrates the effects of increasing the strength

674 J. Escher et al./Nuclear Physics A 633 (1998) 662-680

8

6

4

2

~ o

8

6

4

2

0

8

6

4

2

0

Deformation parameter I~

A £ 7 / 2 = 0

A e 7 / 2 = ~ m / 1 0

50 °

40"

30 °

20"

]0 ~

0 o

-10"

5O °

40 °

30 ~

20 °

10 o

0 o

.10 °

T r i a x i a l i t y p a r a m e t e r 7

• i . . . . i . . . . I . . . . , . . . . u •

" A E T t 2 = 0 " . .%

' ~ %..

"... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

' : . AE7/2 = taCO/1 0

" . . .

~ ' h ~ h "'"" ..............

" " ,

",,.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ 2

. . , . . . . , . . . . i . . . . a . . . . a .

0.00 0.50 1.00 1.50 2.00

Z [ lO-2hm]

7 50 °

40 °

30 °

20 °

10"

00

-I(Y'

50 °

40 °

30"

20 °

10 °

0"

_10 o

. . . . . . . . . . . . . . . . . . . . . . . . • ...... AE7/2 = ' h t O / 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..---.... A £ 7 / 2 = ' h o )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " . . . \ ~* ~ * * ~ , ,

\ L ....... . . . . . . . . . . . . . . . . . .

-20" . , . . . . , . . . . , . . . . i . . . . , . 0.(10 0.50 1.00 1.50 2.00

Z [ 10-2hm]

Fig. 3. Expectation values for Kfl and % along with their uncertainty ranges (xfl ± zlrfl,), + A3,) for six particles in the fp-shell, as a function of the quadrupole-quadrupole interaction strength X. Four plots are shown, corresponding to different strengths of the spin-orbit interaction.

of the quadrupole-quadrupole interaction (from 2" = 0.005hto to X = 0.015h~o): For deT/2 = 0 (case I) the ground state is a nearly pure (99.7%) (a/z) = (12 ,0 ) state, and even for Ae7/2 = hto (case IV) the SU(3) symmetry structure is still visible: The ground state is dominated by the leading SU(3) irreps (A/z) = (12 ,0) (64.4%) and (A/z) = (10, 1) (21%), corresponding to S = 0 and S = 1, respectively.

Increasing the pairing interaction strength (for constant 2" = 0.005hta) causes mixing between different SU(3) irreps within each U(10) irrep [ f ] ; inequivalent spatial U(10) irreps, and thus different spins, are not connected by the pairing interaction since the

a)

c)

Intensity

e)

(

q

Ineeus£ty

J. Escher et al./Nuclear Physics A 633 (1998) 662-680

]c I I I I l l IV Cae.e

ao- ea

m 2o-

S~

I I I I I I IV I Case

675

S-2

5 = I

b S-O

I I I 111 IV z Cate

Fig. 4. Eigenstate decomposition for a system of N = 6 particles in the fp-shell. Shown are the contributions tcli[ 2 (summed over the multiplicity indices ~ and x) from different basis states kbi) to the ground state of the system for various values of G, X, and ZleT/2. More specifically, the parameters are G = 0.01hto, X = 0.005hto in a and b, G = 0.01h~o, 2( = 0.015hto in c and d, G = 0.02hto, X = 0.005hto in e and f, and de7~ 2 varies from 0 to 1 h~o (case I: Ae7/2 = 0, case II: A~7/2 = hto/10, case III: A~7/2 = h to /2 , and case IV: zl~7/2 = haO. The basis states kbi) are organized along the "basis state" axis according to their spin S and SU(3) content (A/z). The left section (parts a,c,e) illustrates the competition between the symmetry-enhancing quadrupole-quadrupole interaction and the symmetry-breaking spin-orbit and pairing forces, and the right section (parts b,d,f) distinguishes between the symmetry-breaking effects of the latter two interactions.

6 7 6 J. Escher et al./Nuclear Physics A 633 (1998) 662-680

0.8

0.6

0 0.4

, I , ,

1

@)6 o J=O

I I I I

I I I l i I l i ~ l l u l l i

" l " ~ . L

X "%, ' ,

I

I , i , , | i , , ,

O A ~ 7 n = 0

. . . . O--- - A£7/2 = f a o Y l 0

-- A- -AeT/ 2 = faoY2

- - a - - - AE7/2 = "I'10~

0.2 ".

~ - ~ . . ^ _ A ^ ~ A"k--~-~ .,.-~ ~ L!"-I-

l , , , , I , , , , l , , , , l , , , , I , , , ,

-0.50 0.00 0.50 1.00 1.50 2.00 2.50

Z [ 10-2"~10~ ]

Fig. 5. Overlap of the calculated ground state for 6 particles in the fp-shell with the seniority v = 0 state 1(f7/2) 6 v = 0). Four graphs are shown, corresponding to zle7/2 = O, h~ / lO, h~o/2, and ho). The strength of the pairing interaction was fixed at G = 0.01ho).

pairing operator is a scalar in spin space. This is illustrated in the right portion of

Fig. 4 (b, d, and f): The basis states are ordered as in the left section of Fig. 4, but in addition the spin assignments are indicated. Each spin corresponds to a particular

U(10) irrep: S = 0 ~ [ f ] = [222], S = 1 ~ [ f ] = [2211], S = 2 ~ [ f ] = [21111], and S = 3 ~-~ [ f ] = [111111]. One finds that increasing G from 0.01haJ (Fig. 4b) to

0.02hw (Fig. 4f) , while keeping the strength of the quadrupole-quadrupole interaction at

a constant 0.005hw, enhances mixing within each U(10) irrep [ f ] , whereas an increase

in the strength of the spin-orbit interaction results in mixing S :~ 0 components into the

ground state wave function: For de7/2 = 0 (case I) the ground state is a pure S = 0 state,

for Ae7/2 = hto/lO (case II) S = 1 (and very small S = 2) admixtures appear, and for

/re7/2 >~ h w / 2 (cases III and IV) one also find S = 2 and S = 3 contributions. And again,

a comparison of Fig. 4b with Fig. 4d shows that the quadrupole-quadrupole interaction

counteracts this trend: Admixtures of S/> 1 are suppressed for a stronger quadrupole- quadrupole force, although the ground state exhibits some S = 1 contributions (see cases

III and IV). In light of the fact that both pairing and the spin-orbit interaction break the SU(3)

symmetry, it is reasonable to consider alternative truncation approaches, such as the se- niority coupling scheme. In this context it is interesting to study whether the ground state of the simple nuclear system under consideration here has any significant resemblance to a seniority-zero state. This issue is addressed in Fig. 5. Shown is the overlap of the ex-

act ground state of the Hamiltonian system discussed above with a state I(f7/2) 6 v = OI

which describes six identical nucleons in the single fT/2-shell, coupled to seniority v = 0. One finds that for a strong spin-orbit splitting (ACT/2 >~ h~o/2, see top two curves in

J. Escher et aL /Nuclear Physics A 633 (1998) 662-680 677

the figure) and a weak quadrupole-quadrupole interaction, there is significant overlap; the ground state of the system does indeed look very much like a seniority-zero state. Increasing the strength of the quadrupole-quadrupole interaction decreases the overlap, more significantly for a small spin-orbit splitting than for a large one. (This result is in agreement with the findings shown in Fig. 3, which illustrate that the quadrupole-

quadrupole interaction has to reach a certain minimum strength, the size of which depends on the spin-orbit force, in order to noticeably affect the shape of the nuclear system.) Fig. 5 also underscores the importance of the spin-orbit interaction. For a very weak quadrupole-quadrupole interaction and a degenerate fp-shell (Ae7/2 = 0) there is

about 20% overlap between the exact ground state and the seniority-zero state, whereas

a small amount of spin-orbit splitting Ae7/2 = hw/10 increases the overlap to over 90%. The effect of the spin-orbit interaction, in particular for a small quadrupole-quadrupole

interaction, is a decrease in the mixing of the largest single-j orbital (here f7/2) with the other single-j orbitals in the shell (here (fs/z,P3/2,Pl/z)) , thus a decrease in the /3-deformation and the collectivity of the system, and an increase in the overlap with the associated seniority-zero state [34].

The Q • Q term, on the other hand, is found to counteract the symmetry-breaking effects of both the two-body pairing and the one-body spin-orbit forces. Hence the

results confirm that for a nuclear system where the quadrupole-quadrupole interaction plays a dominant role (as signified experimentally by rotational spectra), the SU(3)

basis becomes very valuable since it lends itself to a straightforward truncation. For

systems with a strong spin-orbit force it has been shown that the pseudo-SU(3) scheme yields good solutions to the nuclear many-body problem [21,35,36]. In both cases, however, if there are strong pairing correlations present, mixing between different irreps becomes very important and thus complicates the search for a suitable truncation scheme.

In Fig. 6 the angular momentum dependence of the above findings is investigated for four particles in the fp-shell. The curves in the top portion of the plot show the results for J = 0 and Ae7/2 = 0, ho . ) /10 , and hw. The system is found to behave

similarly to the six-particle system described above: An increase in the strength of the quadrupole-quadrupole interaction leads to larger values of k/3 while y decreases.

The latter effect is not as prominent as it is for the N = 6 system, which is due to

the fact that the maximally prolate deformed Pauli-allowed irrep is (,~/z) = (8 ,2) , whereas for six particles it is (A/z) = (12,0) . In both cases, the introduction of

a spin-orbit force (A~7/2 = 0 ---+ h a ) ) results in reduced expectation values for k/3 while y is not significantly affected. The curves in the middle and bottom portions of Fig. 6 show for the N = 4 example that the situation for J = 2 is very similar to the case of J = 0, and the results for J = 4 differ slightly. The general trend for J = 4 agrees with the findings for J = 0 and J = 2, although the asymptotic behav- ior of the curves seems to indicate a new development. For Ae7/2 = hw/lO and ho) and ,t' ---+ oc, the curves approach the point (kfl, y) = (7.08,8.18°), while the limit point for Ae7/~ = 0 lies at (kfl, y) = (7.21, 13.9°). In the limit X --~ c~ three ba- sis states of the N = 4 system become degenerate, namely ]N[f]ce(,t/z)KL, S; J) = 14122] I (8 ,2 )1 4,0;4) , 1412212(8,2) 1 4,0;4), and 141211] l ( 9 , 0 ) 1 3, 1;4). Since nei-

678 J. Escher et al./Nuclear Physics A 633 (1998) 662-680

0

• I " I " I I I I

0 2 4 6 8

• AETt 2 = 0

- - * " - AE7f 2 = ' l ~ O / 1 0

. . . . e - - - - AE7/2 = 1~1~

0 2

I I I • I I

4 6 8

0 2

I • I I I I

4 6 8

Fig. 6. A plot of (Kfl, y) for calculated eigenstates for four particles in the fp-shell with angular momentum J = 0 (top), J = 2 (middle), and J = 4 (bottom). The strength of the pairing interaction was held constant at G = 0.01hto. In the figure the quadrupole-quadmpole interaction strength increases in increments of 0.001hto from the left (X = 0) to the right (X = 0.02F~o)). Each part of the figure shows three curves for both fl and y, corresponding to different strengths of the spin-orbit interaction (Ae7/2 = 0, hto/10, and hto).

ther the quadrupole-quadrupole interact ion nor pair ing mixes different U ( 3 ) symmetr ies

( I f ] i r reps) , the lowest eigenstate for Ae7/2 = 0 and X ~ oo is a pure (A/ t ) = (8, 2)

state, which is associated with (kf l , y ) = (7.21, 13.9°) . The sp in -orb i t force, on the

other hand, mixes different spins ( and thus [ f ] i rreps) , hence for Ae7/2 = hto/lO and

hto an increase in X leads to a combina t ion of the above basis states which corresponds

to (kfl , y ) = (7.08, 8 .18°) . Thus, aside from peculiar effects which can be ascribed to

the mix ing of degenerate states, the trends observed for the J = 0 case survive higher

angular momenta . Fur thermore, if one considers the widths of the k ~ and y distr ibutions

for N = 4 (no t shown here) , it turns out that for J = 0, 2, and 4 the spread in both the

kfl and y variables displays a behavior similar to that discussed for the N = 6, J = 0

case.

J. Escher et al./Nuclear Physics A 633 (1998) 662-680 679

5. Summary

The pairing-plus-quadrupole model has been investigated in the framework of the SU(3) shell model for the case of N = 4 and 6 identical particles in a single ma- jor harmonic-oscillator shell. The restricted Hilbert space allowed for a schematic study of the competing quadrupole-quadrupole, pairing, and spin-orbit interactions in a truncation-free environment. The study focused primarily on nuclear deformation. Mea- sures for nuclear shapes were reviewed and the calculation of their expectation values as well as of the widths of the corresponding distributions in the SU(3) shell model was outlined.

While the quadrupole-quadrupole interaction preserves and enhances the SU(3) sym- metry of nuclear systems, both pairing and the spin-orbit force break SU(3). Since pairing is a scalar in the spin space, it only mixes different (,t/z) irreps of SU(3) but not different [ f ] irreps of U(J2). A description of those triaxially deformed many- particle configurations which pairing favors requires a large number of SU(3) basis states; hence the k/3 and T deformations induced by pairing are rather soft, with d ~ / ~

20%-25% and T on the order of 10 °. It is the softness of this deformation that allows the quadrupole-quadrupole interaction to easily drive a given nuclear system away from a triaxial pairing minimum towards prolate or oblate shapes. The quadrupole term, on the other hand, is diagonal in the SU(3) scheme, and therefore, as Q • Q begins to dominate the Hamiltonian, it not only drives the system towards larger k/3 and smaller (larger) T for shells which are less (more) than half filled, but it also sharpens the shape of the nucleus. Only for mid-shell nuclei, when the Exclusion Principle forces the most deformed SU(3) irrep to be triaxial, the two forces actually favor the same (triaxial) distribution [ 37 ].

The spin-orbit interaction breaks both the SU(3) symmetry and (by mixing different spins) the U(3) symmetry of nuclear systems. A weak spin-orbit force pushes the system toward smaller kfl-values while not affecting the T-deformation significantly, whereas a strong spin-orbit term succeeds in softening the k/3 and T deformations to the point where a shape can no longer be clearly defined. These trends were shown to be fairly angular momentum independent. Although the symmetry-breaking effects of the spin-orbit interaction are strong, they do not preclude the use of the SU(3) scheme, since for systems with a dominant spin-orbit force the pseudo-spin concept can be applied which then leads to a pseudo-realization of SU(3). The advantage of this realization is that the symmetry-breaking spin-orbit interaction in the new representation is weak enough to yield good pseudo-SU(3) quantum numbers. The challenge that remains is to find an analogous concept that will allow for a proper treatment of the pairing interaction in heavy nuclei.

680 J. Escher et aL /Nuclear Physics A 633 (1998) 662-680

Acknowledgements

The au tho r s g ra te fu l ly a c k n o w l e d g e suppor t f r om the Ins t i tu te o f Nuc l ea r Phys i c s in

Seat t le , w h e r e par t o f this r e sea rch was comple ted . One o f the au thors (J. E . ) wou ld

also l ike to t h a n k A m i r a m Lev ia t an and the Racah Ins t i tu te o f Phys ics , J e rusa lem, for

suppor t d u r i n g the f inal wr i t e -up o f this paper.

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