The neutron-proton degree of freedom in the IBA model

189c

NuclearPhysicsA421(1984)189~-2@k North-Holland,Amsterdam

DEGREE OF FREEDOM IN THE IBA MODEL

THE NEUTROSPROTON

A.E.L. DIEPERINK Kernfysisch Versneller Instituut, Rijksuniversiteit 9747 AA Groningen, the Netherlands

Groningen,

Properties of the neutron-proton interacting boson model (IBA-2) are reviewed. In particular we discuss new features that arise from the neutron-proton degree of freedom which is not present in the original formulation of the IBA. We suggest that this neutron-proton degree of freedom plays an important role in the understanding of triaxial shapes of certain nuclei and in the description of collective magnetic dipole properties. 1. INTRODUCTION

The original version of the interacting boson model (IBA-1)lS2 was introduced

by Arima and Iachello to describe

collective

properties

nuclei in terms of six bosons carrying angular momentm

The success of this model can for a large part be attributed that a simple hamiltonian, interactions collective

provides

containing

parametrization

phenomena

to give a microscopic

of a large variety of

observed in nuclei. Later on,

motivated

by attempts

elaborate

version was introduced 3,4 in which a distinction

neutron and proton degrees

of freedom

foundation

for the IBA model a more

of this extended

IBA-I model the general have not been explored discuss

is made between

(IBA-2).

Whereas it has become clear that for most properties predictions

to the observation

only a few one- and two-body boson

a phenomenological

vibrational-rotational

of even-even

Li;O (8) and Ii;2 (d).

of low-lying

states the

version are very similar to those of the simpler

consequences

of the neutron-proton

in great detail.

some of the new features

degree of freedom

It is the aim of these lectures

that arise in this approach

to

and to present

some examples where this degree of freedom may play a role. 1.1. Qualitative

derivation

To start with I will briefly introduction

of IBA-2 hamiltonian summarize

of the IBA-2 mode13s4,

are a rather direct and necessary

consequence

model picture of a typical rare-earth protons are distributed

the ideas that underly the

since the effects

over different

I will discuss

later on

of these basic ideas. A shell

nucleus is that in which the neutrons major shells

0375-9474/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(see fig. l), with the

and

A.E.L. Dieperink / The Neutron-Proton

19oc

Degree of Freedom

effective interaction between the like nucleons being dominated by a strong monopole and weaker quadrupole pairing interaction and the one between the neutrons and protons by a strong quadrupole-quadrupoleforce.

FIGURE 1. Schematic shell model picture of a rareearth nucleus with protons in the 50 < 2 < 82 shell and neutrons in the 82 < N <

126 shell.

This picture suggests that an even-even nucleus can be described as a system of correlated pairs of neutrons and protons outside closed shells. To keep the many:body problem manageable it

is assumed that only L=O (S) and L=2 (D) pairs

need to be considered, i.e. the basis states are expressed as N-n

1S ’ P

n

Nn-nn nn Dn [v,L,],LM>,

‘D p[~pLpl S P

where Lk represent the respective angular momenta and Np(Nn) the number of valence proton (neutron) pairs. (The validity of this approximationhas been discussed by various groups5-6). In addition to further simplify the problem the nucleon pairs are treated as bosons. Although this mapping on itself does not necessarily constitute an approximation since in principle the effect of neglecting the Pauli principle can be compensated for by including higher-order terms in the hamiltonian and allowing for a state dependence of the parameters,inpractical applications this certainly involves some approximations,which may be difficult to control. d } are In the boson space the six neutron and six proton bosons (sn,d J++P' W the building blocks of the symmetry group G = U (')(6) Qp U(n)(6). The basis N %,B n [L,], LM> span the irreducible representastates 1 s?-~P d>[Lp] s," tions [Np] o [N,] of G. The most general one- and two-body hamiltonian in this space can be expressed In terms of the 36+36 generators (s+s i i, d;si, s&,,, dk;iv} (i=p,n). In order to avoid having to deal with a large ntrnberof parameters it has become customary to consider in phenomenologicalcalculations a more schematic hamiltonian guided by microscopic considerations2

A.E.L. Dieperink / The Neutron-Proton

H=E

+ spd;.;dp

g.s.

Here the

first

represent

the

is

the

boson

+ sn

d,+.“dn + KQ;*)

term denotes effect

of

the ground

the neutron

equivalent

of

.Q(i)+

state

i-”

the

structure

principle

parameters

effects.

interactions majorana

The next

between

the

the second

pairing

xi

and third

fourth

(1.2)

t

represent

bosons,

terms

the

force,

can be shown to absorb

terms

like

(1.1)

interactions,

the quadrupole-quadrupole

c2) = d+ + sfdi cI + xi(dp#) f QLF i&h=i where

Ap’Jpp-b&Vnn+ AM.

energy,

and proton

191c

Degree of Freedom

presumably

and the

last

most

of

weaker

term is

the Pauli

residual

the

socalled

interaction

(1.3)

It

is

introduced

characterized 1 Np-Nni > of introduce label

following

reason.

two-row representations (P+n) subgroup U (6)cdP+6)

the

the basis

states

symmetric

In addition

character

grounds,

Since

force,

C, [U(6) 1 f

namely

to

are

these

the

Is

states

basis

there

states

can also

it

is convenient

N-n)

states

with is

to

> (FxZjl Np-Nni ), with

related

a mixed

as yet

to

F=Fmax-$ @

to higher

and

are

in

IBA-1.

for

the

neutron-proton

little

evidence

the quadratic

has been

be

( fl =Np+Nn and n=O,l,..

correspondencewith those

predicted

Since

which

1 N>(Ef

Clearly

-2M +N (N+S),

shift

QDU(n) (6)

have a 1-l

multiplets

the

[N-n,n,O]

F-spin3e4,

N and F.

(F < Fmax).

the majorana

U(m)(6),

with

and therefore

excited

symmetry latter

the

a quantum number called

totally

of

for by the

introduced excitation

Cssfmir mainly

invariant on empirical

energies

(see

fig.

2).

r---------J I

5.1 t

---i--y _---_--** /

-

-..----4.

2.

FMnx-’

FIGURE 2.

______

I

i F

1't

Mbw2

Schematic -spin

energy

multiplets;

upward shift

spectrum

in

the arrow due to majorana

terms

indicates

of

Fthe

inter-

action.

-2’

!

-----2*

-I

-0’ F’FMM

It under amount

should lJ(*n) of

be noted (6)

F-spin

that

the

hamtltonian

traasfo~ations admixing

In the

and therefore low-lying

(1.1)

is

in general

may mix F-spin. states

not

invariant

In that

can be controlled

case

by a

the

192c

A. E. L. D~~~er~~~f The ~~tr~~-~oton

Degree of Freedom

variation of the strength h in (1.1) (see section 4). 1.2. Behaviour of the parameters. From empirical calculations with the IBA-2 model it appears that most collective properties over a large mass region can be parametrized qualitatively by the hamiltonian (1.1) in terms of four parameters: - &n,~ (which are found to vary rather little with boson number) and the &P quadrupole structure parametersx and xn, which appear to vary strongly with P Np and N,, respectively (in general between xi - -1.0 in the beginning of a major shell and xi - +l.O at the upper end). 'Ihistrend is confirmed qualitatively by several microscopic calculations7-lo.A not yet explained feature is that phenomenologicallyone seems to need x N 71, to describe to P spectra of the heavier Pt and OS isotopes11 whereas in a (naive) microscopic picture" one expects that both neutrons and protons are hole like and therefore have x , x, > 0. As to the value of A empirical fits to energy P spectra yield only an indication for a lower limit (x - 0.06 MeV) while up to now there exist no quantitative microscopic predictions. In section 4.2 it is suggested that information OR collective Ml transitionsmay be used to further constrain the value of h. 2. DYNAMIC S~TRIES

IN IBA-2

2.1. Introduction The concept of dynamic symmetries has found useful applications in many areas of physics. In particular the discussion of dynamic symmetries that occur in the IBA-1 model have illuminated very much its connection with geometrical models13. Consider a general hamiltonian H with group structure G, i.e. H can be written in terms of all generators of G, Suppose that a particular H can be written in terms only of (Casimir) invariants Ci of a complete chain of subgroups G 3 Gl 2 G2...: H = Ca iCi. Such cases are referred14 to as dynamic symmetries. In addition to providing a complete labeling of the basis states they allow one to solve the eigenvalue problem for H in analytic form. To be specific, in the case of IBA-2 the symmetry group is the direct product of the neutron and proton U(6) groups: G = U ('j(6)@ H(n)(6). A simple investigationshows that in this case a large variety of dynamic symmetries can arise even If one imposes the physical condition of rotational invariance, i.e. considers only group chains that contain O(3) as a subgroup13,14* For example, in the computer program NPBOSl' that is used to numerically diagonalize the most general IBA-2 hamiltonian the "weak coupling" basis iS used:

A.E.L. Dieperink / The Neutron-~oton

G 3 uL(p+5)C9U(n+5) .l. N “‘N

p' n

*dp

3 ;(P’(5)@$n+5)

r

"d n

z

P

= O(p+*+3)

> $‘)(3)(9$“)(3)

L P

n

193c

Degree of Freedom

L*

(2.1)

::

Physically more interesting chains are of the type13,14 U(5) G2

U(tin+6)

3

ZJ O(5)

I) O(3)

O(6) 2 O(5) 3 i SU(3)

3

(2.2a)

O(3)

(2.2bf

O(3)

(2.h)

and chains in which the coupling occurs at the level of the first subgroups

G>

U(P)(S)% U(n)(5) 3 U(5) 3 O(5) 3 O(3)

(2.3a)

0(p)(6)QI,O(n)(6) 3 O(6) 3 O(5) 3 O(3)

(2.3b)

SU(P)(3)@ SU(r+3)= W(3) 3

O(3)

(2.3~)

O(3)

(2.3d)

i su(p)(3)@ E+)(3)

3 su*(3) 3

The majorana interaction M, being related to the quadratic invariant of ~(p+*)(6), is diagonal in the chains (2.2) but not in (2.3); on the other hand a necessary condition for the latter to be applicable is Vpp=V,,=Vnp,which is at variance with the common assumption that there is a strong Q.Q force present between the unlike bosons only. 2.2. Illustrativeexample: SU(3) limit In order to illustrate some consequences of the neutron-protondegree of freedom I consider a schematic ham~ltonian13y16

H = C K,$Q;~).Q:~) + f L:l).L:l+ .KLL(?L(~',

(2-4)

which is expressed in terms of SU(3) invariants:

0#J(i)(3)l

= 2Q(2) qQ(2) + 3L(l).L(') ii i i'

(2.5)

and hence diagonal in the scheme (2.3c,d). Here Li ('I-,lO (d;.;l$('), L(1), L(1) + L(1) and Q(t) is given in (2.1) with xi= 4@7. By completing n' p , . . the square, i.e. by introducing the invariant of the SUCptn1(3) group the energy spectrum of (2.4) can be expressed as

E(~~~~~~~n~~~L~

= 4 I: kii +,,>O.; i

+k$

+x$&i + 3ChjCi))

+ kpn(k2 + p* +hp + 3(x-@)) +KLL(Ltl). The

(2.6)

essential point6 of the following discussion are not changed if we put in

(2.4) ~p~;~n~= $icpn,and~L= - gyps hamiltonian

,

i.e.

if we restrict ourselves to the

A.E.L. Dieperink / The Neutron-Proton

194c

H

=

%,,(Q;‘) + d2)).(QL2)+ Qi2)).

In order to determine distinguish

The allowed

(N=Np+N,).

to (A#)

(i) xp'x,*

the appearance

(2X-4,2)8

discussed

in section

mechanism

for example

(hn,un)

be denoted (A#)

the representations

(A ,p) values

by SU*(3) to distinguish

= (2Np,2Nn)$(2Np-1,2N,-1)

The angular momentun triangular

structure

angular momentum

realizations

of the with the

A,, and p,) denoted

it from the previous

N,=4

of the leading

N,=3

irreps (2Np,2Nn)

reminiscent

su" (3)

of the rigid triaxial

I 1

(left) and a perturbed strong

is

(right)

SU*(3) spectrltm;

(weak) E2 transitions.

given

in

ref.

17.

.. .

has the

1

decomposition

by

case, are

. . . tB (2Np-4,2Nn+2)@

M 2126

3. An unperturbed

a complete

a g-boson18.

of SU(p)(3) are combined

Q (2Np-2,2Nn-2)(8

structure,

the thick (thin) lines indicate

+

of the IBA

of these SIJ(~+~)(3) irreps, which group will

I

FIGURE

(conjugate)

by interchanging

there are

E (MeV)

band

for these states will be

by introducing

(Lp@p)

ones of SU(") (3) (obtained

is

in the IBA-1 formalism.

&'7.

In this case one deals with two different

. The allowed

to irreps not

A novel feature

4. Here,1 note that also other generalizations

(ii) S"(P)(3) @ 5iP) (3) withxp=Tn=

SU(i)(3) groups:

correspond

degree of freedom.

to try to locate such a predicted

excitation

model can lead to K=l bands,

conjugate

+ (hp'Pp)Q

...$ (2N-2.1) d (2N-4,2)tB...

Kx= l+ band not present

Since it would be of great interest a possible

(ii) xp=-x,,= *&'7.

which are underlined

in the neutron-proton

of a (2N-2,l)

experimentally,

&'J,and

to

&1'7

from the tensor decomposition

= (2N,O) 8

The representations

symmetric

(2.7)

values of (h ,p) it is necessary

S"(") (3) with xp'xn=

(A,u) values follow

leading

totally

the allowed

two situations:

(i) SU(P)(3) 8

(An&n)

Degree of Freedom

rotor:

A. E. L. Dieperink / The Neutron-Roton

rotational

Degree of Freedom

bands for each K= O,Z,...,min(ZNp,2Nn)

with L-K, K+l,

or L=O,2,4,... (K=O) (see fig. 3a). Note the occurrence states belonging In practice

to different

one expects

of higher

to be perturbed

of the majorana

to a splitting

leads to an ordering

of the states into bands, connected

and weaker

E2 transitions

n

by symmetry

force is shown

fig. 3b; in addition

Q(2)

O+

irreps of W*(3).

this symmetry

The effect of the inclusion

Qw+ P

. .. . (K f 0) excited

breaking

(l.l), e.g. of the form E # 0, lKil # j/J7 and A # 0.

terms in the hamiltonian

intraband

195c

of the angular momentum

(matrix elements

(schematically) degeneracies

in it

by strong interband

of the operator

)-

2.3. Remarks It is worth noting

that the hamiltonian

(2.7) with quadrupole

structure

constants x =x =0 can also be solved in closed form, by using the relation (d+s + s+-d) 92): (d+s + s+d)(2) = N(N+4) - P[O(6)1 - f $[0(5)1, where PIO(6)l is the pairing operator

of O(6), P[O(6)]=

C2[O(5)] = A ; 3(dti)(h).(dQ)(h). =, contains three limiting situations (i) gU(3) (xp'xn= *tJ7), xn=O).

A more complete

a fourth dynamical chain

(ii) W*(3)

analysis

symmetry

(d+.d+-

Iherefore

(xp=fln=

l+'7),

of the general

realized

K

if

s+s+).(“d.z

-

ss),

and

eq. (2.7) with general

Q(2)

and (iii) O(6) (x, =

H (eq.(l.l))

shows that there is

=0, E#O corresponding

to the U(5)

(2.2a). "W.a10. "(51 . "S

FIGURE 4.

Ia&1

i

Shape phase diagram

’

corners

su'la------- _-.-_ W,,, rn,Dl %?' '_

for IBA-2; the

of the tetrahedon

correspond

to

dynamic symmetries.

0 ,.""SbM mlnr0,W One may conclude

that in the U(p)(6)@

four soluble limiting W*(3)

transitional

space there are

cases (fig. 4). In section 4 we will show that the O(6) -

region could be of interest

of nuclei with triaxial

3. RELATION

U(")(6) parameter

or y-unstable

for the description

of a class

features.

WITH GEOMETRY.

Over the last years it has become clear that there exists a close relation between

the algebraic

collective

models

approach

of IBA and the geometric

in which the radius of the nucleus

of the five quadrupole

variables

a

:B

formulation

is parametrized

= R(J(l+z uuYk

of in terms

P this relation has been studied with a variety

(6 A)). In the case of IBA of methods 13,20-24. In

particular

level a connection

it was shown that at the classical

can

196c

A.E.L. Dieperink / The Neutror~-baron

Degree

of Freedom

conveniently be established by taking the expectation value of the hamiltonian with respect to coherent states24. Although in principle the application of these techniques can be extended straightforwardlyto IBA-2 in practice the number of classical variables becomes so large that a general analysis becomes rather tedious. Here I will only summarize the results of a simplified analysisl3,25* It is convenient to introduce coherent states for the representation [Np]@ INnI of U(P)(6)@ W(6): lNpapNnan> = (Np!Nn!)-' (T

N s; +ap.d;) '

(~~~~~~o.d~)~~i

0>, (3.1)

where (x and a o represent five in general complex collective quadrupole P variables for protons and neutrons, respectively. If we restrict ourselves to ground state properties one may take the a's real, and the function E(ap~,) = < NpapNnan 1Iii NPapNnan>

(3.2)

can be interpreted as a potential energy surface. By first transforming to intrinsic deformation (S,), triaxiality (vi> parameters, and Euler angles (Qi) and next to the center-of-massframe one finds that E(a ,a ) is a function P * of seven variables, namely S, and yi for neutrons and protons and the 3 angles which describe the relative orientation of neutron and proton systems. Minimization of (3.2) with respect to these variables defines the equilibrium shape, which can be characterized by the values S

, $, o, y

, y, o, A&Jo.

In particular we consider applications to the hamf;Ioniai (2.;;"

'

(i) SU(P)(3) 0 SU(")(3) 3 SU(3) limit. In the ground state one finds S, o f Sn o = 2/J3,yp o = yn o = 0(x?>,i.e. the intrinsic neutron and proton dis;ributi&s have axis; symmelry and furthermore the relative angle between the two symmetry axis, x, has the equilibrium value x0 = 0, i.e. the matter distribution has also axial symmetry. A more detailed semi-classicalanalysis in which also the momenta are considered reveals various small amplitude vibrational modes around the equilibrium, such as $- and y-vibrations, and also isovector modes in which the neutrons and protons move out of phase. Here I wish to point out only one interesting case, namely the oscillation between the neutron and proton symmetry axis in terms of the angle x (see Fig. 5). One can show that this mode, which after quantization carries one unit of angular momentum along the symmetry axis, can be identified with the lh = 2N-2, p = l> Kx * I+ band mentioned in section 2. Clearly its excitation energy, which in the algebraic approach is determined by the strength of the majorana interaction, is also related symmetry energy of the geometric models. (ii) W(P)(3) * Z(")(3) 3 W*(3) limit.

to the neutron-proton

A.E.L. Dieperink / The Neutron-Proton

Degree of Freedom

197c

In this case one finds y

Geometric interpretationof the l?=l+ band in the SU(3) limit in which the neutron and proton symmetry axis carry out a small amplitude oscillation.

The ground state can thus be regarded as a prolate (proton) and an ablate (neutron} axially symmetric deformed rotor coupled in such a way that their (2) (2) interaction). It is overlap is maximized (due to the attractive Qn . Q P possible to characterize the resulting mass distribution by a triaxial shape with triaxiality parameter; by using the relation ta< = J2 9, 2/R, U, are the intrinsic matter quadrupole distrib&ion~: Qm,2 and Qm,O Qmo= < 22 - z? - 3 > and Q, 2 =v‘$ < 9 - 3 >. For the special case Np = Nn

where

the triaxiality parameter has ;he value f = 30".

4. APPLICATIONS. From practical applications of the IBA-2 model it appeared that the predictions for properties of low-lying collective states are very similar to those for the IBA-I model. Naturally the question arises what are the explicit effects to be expected from the neutron-protondegree of freedom. I will discuss two examples, namely a possible interpretationof triaxial features observed in certain nuclei in terms of a perturbed SU"(3) limit, and the description of collective magnetic dipole properties. 4.1. Interpretationof triaxial properties. Although it appears that several nuclei have properties that can be interpreted in terms of the O(6) limit of IBA19 (the y-unstable limit) a more detailed analysis shows that in some cases also rigid triaxial features (Su*(3)> are present. Properties that appear to be sensitive to the nature of they- potential are the odd-even staggering of the y-band energies and certain B(E2) values connecting states in they- and g-bands. Schematic studies of the SUX(3) - O(6) transitional region in IBPr2 indicate that these quantities indeed depend strongly on the magnitude of x(x,- 3,) and the strength of the Q(2)*Q(2> interaction between the like bosons. From the point of view of the shell model underlying IBA-2 one expects that an SU*(3) situation occurs whenever the neutrons are ;;ticle-like and the proton hole-like (or vice P 106 versa); good examples are B~Ba7s (NP = 3, Nn = 7) and ,,sP&s (Np= 5, Nn = 5). Recently detailed infy;)ation on E2 matrix elements has been obtained from Coulomb excitation26 for ht,Rxs. A previous XBA-2 calculationz7for this

A.E.L. Dieperink / The Neutron-Proton

198c

Degree of Freedom

(2). Q(n2)+ Qs"'. Q(,2)interaction - - xn without the V QQ' Qn p yielded a spectrum and EZ-propertiestoo much y- unstable in character. It was nucleus with I

found2' that inclusion of VQQ led to an appreciable improvementfor both the clustering of the energies in they-band (which is intermediatebetween the O(6) and SU*(3) limits) and the diagonal Q-moments (see figs. 6,7). ! ;-.=o

'04Ru 3.0E(MeV1

8---.~, I .-’ 7--..,.-. .-. ‘1,-..

-.

a -. '._, I' I, 6-.._,:--

IO-

4___

s---.,,-’ .-. ~__.,

_

_..

_...

of ref. 27; B: revised IBA-2

a-...

‘I-

z.o -

_1

calculation28which includes \>

(4) @3--

OS--

G-

VQQ, C: rigid triaxial

~z::._._ -...z %-a$_ 02Km'.__. -.'. .-

_

2-3;“._-____._ tw

O2-

*

FIGURE 6. Energy spectrum of 104Ru; A: IBA-2 calculation

‘T

rotor26.

c+--

A

EXE

6

2_..__-___-._00

OT..

-. .-...T EXf? 6

am

a12

No %I

gjO.06.

w m

004.

2

4

I

6

6

2

4

6 I

6

v

I

FIGURE 7. Comparison of experimentalz6and calculated B2 properties in lo4Ru as a function of spin I; AR: denotes the asymmetric rigfd rotor result; the IBA-2 results are from ref. 27,and the IBA-2* results from ref. 28.

Another example of ay-soft n;&ieus where the agreement with experfment can be improved by including VQQ is FBI%. From fig. 8 where quadrupole moments in the g- and y-bands, which were deduced from recent Coulomb excitation measurementa2',are presented, it appears that neither the extreme y-soft nor the rigid trlaxial rotor model agree with experiment. (As mentioned above the empirical values I - T, for the Pt isotopes11 appear anomalous from the P simple shell model point of view. This may be ascribed to strong renormalizationeffects due to excitations of nrotons across the ti82 shell

A.E.L. Dieperink / The Neutron-Proton

Degree

ofFreedom

i99c

into the h9/2 orbit as has been discussed by Wood3*). aos?+ / I ! : / / I XVI FIGURE 8. Comparison of experimental" and calculated quadrupole moments in g- and y-band in lg4Pt; the IBA-2 result is from ref. 11, the IBA-2* calculation

0

includes Vsa, and AR denotes the rigid asymmetric rotor. JO_...L-.IiI 29 49 B %

1 21 41 a7

One may speculate on the physical origin of the VQa interaction. Although this interaction between the like particles may already be present at the level of fermions in most microscopic IBA-2 approaches it is assumed that such a seniority breaking interaction is negligible. However, it could also be regarded as an induced renormalizationeffect coming from the truncation of the model space. For example, it can be shown3' that elimination of the Izi4g-boson in fig. 9 of both neutron and proton bosons are present leads within the IBA-2 model space in good approximation to an effective Q(2).($2) interaction between the proton bosons with a strength roughly proportional to the nunber of neutron bosons. FIGURE 9. Cl.d Rxample of a diagram which gives rise to P "

P

-0,v

an effective three-bodv interaction in

C&d

the sd space.

"

4.2. Magnetic dipole properties in IRA-2. In the simplest form of the collective model magnetic dipole moments in even-even nuclei are determined by the value of the g-factor, gR = f, and Ml transitions are forbidden. Nevertheless there exists a considerable amount of information that suggests that (i) there exist collective Ml transitions in even-even nuclei, and (ii) that there are appreciable deviations from the value gR = $ for magnetic moments of 2: states. Some attempts have been made to describe these effects in IBA-132*33 by assuming an Ml operator of the form T(Ml,u) = glL;%

. g2[QC2)A L(')];') + g3[ndAL(l)](l) cI

(4.1)

It is of interest to investigate whether this parametrizationwhich has been shown [32]

to describe quite well the spin dependence and relative magnitudes

2ooc

A.E.L. Dieperink / The Neutron-Proton l?egree of Freedom

of E2/Ml mixing ratios in rare-earth nuclei can be understood from a more fundamental point of view. In this respect it is worth noting that in the past Greiner had suggested34 that the origin of deviations from gR = Z/A as well as collective Ml transitions lies in a difference between the neutron and proton deformations. Whereas that idea, which effectively amounts to mixing with a hypothetical I?= l+band, leads to a spin dependence of Ml matrix elements very similar as those predicted by eq. (4.1) it does not explain the measured gR values very well. It is clear that the same idea, namely the exploitation of the neutronproton degree of freedom for the description of Ml properties, can be formulated in terms of the IBA-2 approach in a more general way. The most general one-body magnetic dipole operator in IBA-2 can be expressed as

(4.2) where gp and g, are the boson g-factors in units pN= &. 4.2.1. gR-factors of 2;'states. Recently we have analysed35*36gR-factors of 2: states in even-even nuclei in terms of IBA-2. It is instructive to note that a simple analytic formula for gR can be derived in case the states possess maximum F-spin. By rewriting (4.2) as +(l) -+(I)=1 E (gpNp+ gnNn) I(')+ (gp-gn) i (~~$l)-~~t~l)), gPLP + gnLn (4.3) and using the fact that the matrix elements of the second term at the r.h.s. of (4.3) vanish for totally symmetric states one finds

gR

= ; < L,M=Ll g,L;l; + gnL:l; 1L,%L > = (gpNp~nNn)/N. , I

(4.4)

In practice if eigenfunctionsof (1.1) are used one finds that the deviations from the estimate (4.4) are smaller than a few percent. It has been found35*36 that the general trend of the gR factors in the rareearth nuclei can be described quite well by (4.4) with rather constant values of the effective boson g-factors, gp(Np)- 1.0 f 0.2 pN and g,(N,)m-0.1 f 0.2 pN (see figs. 10,ll): in the lower half of the shell where neutrons and protons are particles gR is a decreasing function of Nn, whereas in the upper half where the bosons are built from holes gR increases with neutron number. (It has been suggested37*38that the deviations observed for the lighter gm and Nd isotopes might be related to the effects of the shell closure for 2=64 and neutron number N
A.E.L. Dieperink / The Neutron-Proton

Degree of Freedom

201c

FIGURE 10. Comparison of the experimental gR factors of 2: states in the 50<2<82 region with the result of the IBA-2 parametrization (4.2) (solid line). Also shown are the results of the cranking model"

(dashed curve), and

Kumar-Baranger3g(dashed-dotted curve).

i

FIGURE 11. The empirical values of the boson g-factors gp and gn [eq. (4.2)f (dotted curve) compared with the result of microscopic calculations36 (full and dotted lines).

4.2.2. Magnetic dipole transitions. The second term in eq. (4.3) describes Ml transitions between collective states. In fact if the values of gp and g, are taken from the analysis of

A.E. L. Dieperink / The Neutron-Proton

202c

magnetic

moments

distinguish between ground

no new parameters

state and a l+ excited

collective

perturbation

It is obvious

/(E,-Es).

between

symmetric

the denominator

in strength

interactions.

Preliminary

investigations

compared

to experiment.

states

depends

by various

(in and on the

(a): on A,

strongly

F-spin symmetry

# en, K # x, and P P of the various nn, np and pp two-body

differences

IBA-2 parameters

element

in (l.l), such as E

also possible

the conventional

matrix

low-lying

with mixed

of the latter depends

is determined

terms that could be present

the O+

between

of components

(s) and mixed symmetry

In the IBA-2 approach

the value of the numerator

strong transitions

(ii) weaker transitions

that the strength

to

for example between

theory) on the ratio of an off-diagonal

energy difference

breaking

(i) possibly

F-spin symmetry, state,

It is convenient

states that take place through admixtures

n-p symmetry.

whereas

are involved.

two types of Ml transitions:

states with different

Degree of Freedom

in the Sm isotopes

the calculated

Ml strength

indicate

that with

is too large

This can easily be improved by increasing

the value of

A from 0.06 MeV (as used in many IBA-2 fits) to A _ 0.20 MeV. With the latter value the excitation becomes

energy E, of the lowest

l+ state in the SU(3) region

E, y 3 MeV.

It is a highly

interesting

mixed neutron-proton

question

symmetry

whether

character

l+ states which do not occur in the one-fluid predicted

in all limiting

is only possible interest

to consider

by IBA-2, such as collective

models.

cases of IBA-2 excitation

if the ground

Whereas such states are

with the Ml operator

state contains d-bosons.

some characteristics

of IBA-2 in more detail.

one can locate states with a

predicted

Whereas

Therefore

(4.2)

it is of

of the K=l+ band in the SU(3) region

its excitation

energy

in IBA-2 depends very

much on the value of the strength A of the majorana

interaction

give a simple estimate

To this end I consider

intrinsic

for the transition

states corresponding

strength.

it is easy to the

to the (h,u) = (2N,O) and (A,u) = (2N-2,l)

representations. of SU(3):

1g.s > = (N~!N,!) -3

(b;,,jNp (b”,, o)Nnl o>,

and

IK=~> = h

d+ p,l bp,O- (Np/N)+d+,,l bn,O)

where

bk,O = &

-I-

operator

((N,/N)+

(4.2) in the intrinsic < K=l

Therefore

(k=p,n).

(s; + 12 <,o)

The matrix

(4.5)

I g-s>, elements

(4.6) of the Ml

frame are given by

1 T1(l)1 K=O > = (2NpNn/N)+(3/4,)t(gp-gn)

one finds in the adiabatic

limit

[u,].

(4.7)

A.E. L. Dieperink / The Neutron-Proton

B(M1, O++l+) = 21
Using the values gp and g, as obtained B(M1) strength

(NP = 7, N, = 5) amounts

to approximately

(gp-gn)z

moments

for a typical rare-earth &N2.

in inelastic

This prediction

electron

(4.8)

[n,].

nucleus

one 156 64Gd

has stimulated

scattering.

Preliminary

provide evidence

results41 (B(M1) -

v

203~

from the fit to magnetic

finds that the predicted

a recent search for Ml strength

Degree of Freedom

1.5&i) at Er-

for relatively strong Ml transitions 156,158Gd 3.10 MeV in . Also the study of the measured

form factor as a function

of the momentum

transition

rather than a spin-flip

has an orbital

the geometric

picture

of this model

transfer

suggests character

that this in agreement

with

(see fig. 5).

ACKNOWLEDGEMENT. The author

is indebted

Pure Research

to the Netherlands

(Z.W.O.) for providing

Organisation

financial

for the Advancement

of

support.

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KVI