PHYSICAL REVIE%' C VOLUME 33, NUMBER 1 JANUARY 1986

Interacting boson mode1 with surface delta interaction between nucleons:Structure and interaction of bosons

C. H. DruceDepartment ofPhysics, Unloerstfy ofArizona, TQeson, Arizona 8572l

S. A. MoszkowskiDepartment of Physics, Uniuersity of California, Los Angeles, California 90024

(Received 22 July 1985)

The surface delta interaction is used as an effective nucleon-nucleon interaction to investigate thestructure and interaction of the bosons in the interacting boson model. %'e have obtained analyticalexpressions for the coefficients of a multipole expansion of the neutron-boson-proton-boson interac-tion for the case of degenerate orbits. A connection is made between these coefficients and the pa-rameters of the interaction boson model Hamiltonian. A link between the latter parameters and thesingle boson energies is suggested.

I. INTRODUCTION

The interacting boson model' has been an importantdevelopment in nuclear structure physics. In both itsforms, IBM-1 and IBM-2, the model has had remarkablephenomenological success. Furthermore, tremendous pro-gress has been made in formulating the thoiretical basisof the model. This has allowed a connection to be madebetween the phenomenological model and the underlyingmicroscopic structure, which is the nuclear shell model.

Much of the theoretical work on the IBM has focusedon the calculation of the parameters of the IBM-2 Hamil-tonian from a microscopic basis. A reasonable procedurewhich is often used for doing such a calculation, is thefollowing:

(1) A representation is developed of the bosons in theunderlying fermionic, shell model space; i.e., s and d bo-sons are represented as S and D pairs in the fermionspace.

(2) An effective interaction between the S and D pairsin the fermion space is chosen.

(3) A mapping is developed between the fermion pic-ture and the corresponding picture in the boson space.

(4) The parameters of the IBM Hamiltonian are calcu-lated and/or the spectrum of the particular system isdetermined.

the s-d space and degrees of freedom outside the s-dspace. This coupling plays an important part in the re-normalization of the parameters of the IBM Hamiltonian.It is usual to use a quadrupole-quadrupole interaction buthigher multipoles in the interaction are needed and haveto be put in by hand. This leads to more parameters inthe theory. An attractive feature of the surface delta in-teraction (SDI) (Ref. 4) is that all multipoles are includedand are determined by a single parameter, the strength.

II. BOSON-BOSON INTERACTION

We will describe a calculation in which we followed aprocedure such as outlined in steps 1 through 4 above.We will restrict ourselves to rather schematic situations inwhich we consider two boson, i.e., four particle configura-tions, where the nucleons occupy degenerate orbits.

We shall construct the bosons by determining the spec-trum arising from placing two identical particles in one orseveral different degenerate j orbits. If these particles in-teract via an SDI, then for each value of angular momen-tum A, only a single state of each A is shifted in energy.All the other states have zero interaction energy. We con-sider this lowest energy state of each A to be a boson ofangular momentum A. Thus for A=0, 2,4, . . . , we con-struct s, d, g, . . . , bosons, respectively.

The energy of a A boson is given by:s

The IBM is a drastic truncation of the full shell-modelcalculation. However, it is possible to ameliorate theseverity of the truncation by refining the above procedureat each step. For example, in addition to s and d bosons,other bosons, such as g, i, k, etc., can be included. It isalso possible to include noncollective degrees of freedomas well. It is hoped that much of the effect of configura-tions that lie outside of the IBM can be taken into accountby renormalization of the parameters of the IBM Hamil-tonlan.

The effective interaction is an important choice to bemade because, not only does it determine the interactionsbetween the bosons, but it governs the coupling between

j' Aeh= gg t (2j+1)(2j +1) ) ( G, (1)J J

where the sum j,j' extends over the set of degenerate or-bits and G is the strength of the SDI.

For the set of degenerate orbits, we consider severalchoices: (1) a single j shell; (2) two degenerate orbits, jiand ji+1; (3) two degenerate orbits, j, and j, +2; (4) adegenerate major oscillator shell, i.e., j& ———,', —,', . . . , j.

%e now consider a situation in which we have one neu-tron boson and one proton boson, each constructed in themanner described above, interacting via an SDI in a par-

33 330 1986 The American Physical Society

33 INTERACTING BOSON MODEL WITH SURFACE DELTA. . . 33l

ticular space of degenerate orbits.The SDI between particles 1 and 2 is written as:

VsDi =4WG5(QN )v ),where 6 is the strength.

We can also expand the SDI in multipoles, i.e.,

VsDi ——G[1+5P2(cos8)v )v )+9P4(cos8)v )v )+ ] .

(3)

In a similar fashion, we can make a multipole expan-sion for the interaction between the neutron boson andproton boson, namely

V.-p = V.-b --p-b,FO+ 5F2F2(cos8„p)+9F4&g(cos8„p) + (4)

We do not specify here a relation between 8& )v and1 2

8„&, but require the equality of matrix elements of thetwo-neutron and two-proton system with those of one-neutron and one-proton boson. Thus we have mapped thefermion picture onto the boson representation. This isknown as the Otsuka, %rima, and Iachello (OAI) map-ping.

If the angular momentum of each boson is labeled byAz, where p =n or p, then the inatrix elements ofV„b „~b „can be evaluated in terms of the Fo, Fi,F4, etc. The matrix elements are given by:

Ap Ap K An An EC Ap A J(A„AP',J

i V„P iA„'AP', J)=(—) A„A@A„APg (2I(. +1) () 0 () () () ()

K0 0 0 0 0 0 A'„A,' K (5)

where A=(2A+1)' 'By using an SDI as the origin of the effective n-

boson —p-boson interaction, we can evaluate numericallyall the matrix elements on the left-hand side of Eq. (5).Diagonalization of the Hamiltonian matrix then yields thetwo-boson spectrum.

One can also invert Eq. (5) and evaluate the parameters,F», of the n-boson —p-boson interaction in terms of thetwo-boson matrix elements. Notice that all the radial in-formation is contained in the F» and thus the F» dependon the structure of the bosons and on the single particleconfigurations. We will use F», F», and F» to denotecoefficients of the Eth multipole that have different radi-al components. However, there is considerable simplifica-tion if the SDI is used. For example, we find that always:

0=4

where G is the strength of the SDI.The other free parameters are

F2 ——v'(1l5)(A„=O, Ap ——0;0i V„p i

A„'=2,Ap——2;0),

(7a)

F2 ——( —', )(A„=2,Ap ——2,0i V„p i

A„'=2,A@=2;0)

—( ~ )(A,=2,Ay=2;2~ V„p ~

A„'=2,A' =2;2)+F0,(7b)

and M is the Majorana operator given by:

Mwr $2(d«sn

des�«)

(d«sm des«)

+ g g (dt df' )(»)(d d )(»)

E=1,3(8c)

V„represents extra terms that are usually neglected inthe IBM Hamiltonian. We can write V„as:

V= g Y (dg )' '(d„d )'

M =0,2,4

(Sd)

We can link the parameters of V„~ given in Eq. (7) withthe usual IBM parameters given in Eq. (8). These expres-sions are given in Appendix A,

In an earlier publication, one of us (Moszkowski)showed that for certain matrix elements, the coefficientsof the n-boson —p-boson interaction are related to the sin-

gle boson energies. It can be shown that for matrix ele-ments were A„=A~=0 and A„'=A~ =K&0 [see Eq. (5)],the parameters of V„~ in Eq. (4) are given by:

To make contact with the IBM, we rewrite V„p interms of the usual IBM parameters, ' i.e.,

(8a)

Here Eo is a constant, Qz' is the quadrupole operator

given by:

(8b)

F = —v'(7zlo)(A„=O, A, =2;2~ V„, ~

A„'=2,A,'=2;2)ex

I'g ——46 (9)

= —&(7/10)(A, =2,Ap ——0;2~ V„p ~

A„'=2,Ap ——2;2),

(7c)

F4 ——( —,' )(A„=2,Ap ——2;0 i V„p i

A„'=2,Ap ——2;0)

+(—,' )(A„=2,Ap ——2;2

i V„p iA„'=2,Ap ——2;2)

—( —„)Fo .

where e~ and eo are the energies of J=E and J=0 bo-sons, respectively. The energies ez are given explicitly byEq. (3). A derivation of the result (9) based on a physicalinterpretation of the bosons is given in Appendix B.

Equation (9) also holds for matrix elements that haveAn 0~ Ap ++0~ An E~ Ap =0 as well as for matrixelements having A„=X&0 Ap 0 A 0 Ap K.

The parameters F» for matrix elements involving U =2

332 C. H. DRUCE AND S. A. MOSZKOWSKI 33

III. RESULTS

Table I shows a comparison of the ratios:

~x scand; X=2 4Fo eo

' (10)

which are equal when Eq. (9) holds. Numerical values aregiven for the four different configurations of single parti-

bosons only, i.e., no s bosons, are somewhat more difficultto obtain analytically but can still be obtained in closedform, at least for degenerate configurations with an SDI.These expressions are given in Ref. 5.

To summarize, we have established a connection be-tween the parameters Ex of V„b „~b, „ in Eq. (4) andthe parameters of the IBM Hamiltonian in Eq. (8).Furthermore, we have a numerical procedure for calculat-ing the Fx [see Eq (7.)]. However, the interesting featureis that, for certain matrix elements, the Ex have a simple,analytical form, namely that given by Eq. (9). The goalnow is to see if the exact result given by Eq. (9) for thecase where s bosons are involved can be exploited to ob-tain simple results for the case where only d bosons areinvolved. If this is possible, we will have established asimple, analytical procedure for determining the parame-ters of the IBM Hamiltonian in terms of the strength ofthe SDI.

cle orbitals that we have considered. Table II gives theanalytical expressions for these ratios.

These results indicate that Eq. (9) holds approximatelyfor the case of a single j shell and for the case of a degen-erate major oscillator shell, i.e., j;=—,', —,', . . . , j. Theagreement is slightly worse for the case of two degenerate

j shells ji and ji+2. The case of two degenerate j shellshaving ji and ji+1 apparently causes the greatest viola-tion of Eq. (9). In the limit that j~ ao, we see that theseratios are equal. This limit, namely that the A's of the bo-sons are small compared to the maximum single particle j,can also be interpreted as a semiclassical limit.

The approximate agreetnent of the ratios in Eq. (10)means that, at least for certain cases, we have a way of es-

timating the values of the parameters of the IBM Hamil-tonian by knowing the values of the single boson energies.

The task ahead is to generalize these simple estimates tothe more reahstic case of nondegenerate single particle or-bits. Furthermore, it would be interesting to know if theapproximate connection between the parameters of theIBM Hamiltonian and the single boson energies has anygreater significance.

ACKNOWLEDGMENTS

One of us (C.H.D.) would like to acknowledge usefuldiscussions with Bruce Barrett. This work was supportedin part by NSF Grants Phy-84-05172, Phy-82-08439, andPhy-84-20619.

TABLE I. Coefficients of a multipole expansion of the boson-boson interaction for different configu-rations.

F'4e2/ep

Fl4e4/ep

32

00.800000

(i) Single degenerate j shell52

0.32650.91430.18370.3809

0.58050.95240.32650.4675

92

0.71990.96970.40500.5035

1 32~2

(ii) Taro degenerate j shells j and j+13 5272

5 72~2

7 92&2

F24e2/ep

F44e4/ep

1.96001.600000

0.18361.14290.32661.1429

0.53781.06670 AAAA

0.7273

0.71261.03900.49180.6474

(iii) Two degenerate j shells j and j+21 5 3 72&2 272

5 92 l 2

F24e2/ep

Fl4e4/ep

1.74111.88570.13540.2857

2.07972.27300.49540.7350

2.01112.37800.62991.4831

(iv) Degenerate oscillator shell j;= 2, 2, . . . , j3 1 5 1 7

2&2 t,o2 2—to—2 2

1 9—to—2 2

F24e2/ep

Fl4e4/ep

1.96001.600000

2.46932.28570.67480.9524

2.77772.66671.36111.5758

2.98352.90911.83901.9953

33 INTERACTING BOSON MODEL %'ITH SURFACE DELTA. . . 333

TABLE II. Boson-boson multipole coefficients (analytic expressions).

to j

eo

1 (2j —1)(2j +3}4 (2j)(2j+2)

1 (2j —1)(2j+1}4 (2j —2}(2j+2)

(2j —1)(2j +2)

eq

eo

1 (2j —3) (2j+5)4 (2j)'(2j +2)'

9 {2j—3)(2j —1)(2j +3){2j+5)64 (2j —2)(2j)(2j +2)(2j +4}

9 (2j —3) (2j+5)(2j)'(2j +2)'

1 (2j —4) (2j+4)4 (2j —2) (2j+2)

9 {2j—3)(2j —1)(2j+1)(2j+3)64 (2j —4}(2j—2)(2j +2)(2j +4}

9 {2j 3 )2(2j + 5 }~

64 (2j —2) (2j+2)

{4j—1){4j+4)'

1 (2j —3)(2j —1)(16j+35)8 (2j)(2j +2)(2j +4)

APPENDIX A: IBM PARAMETERSIN TERMS OF MULTIPOLE COEFFICIENTS

ments apply only in spherical nuclei and not in the de-formed region.

Ho=I'0

a =F2,Ca=0

X=X =X„=—v'(10/7)F2'/Fg,

g, = —( —,' )F' + (

—", )F' + ( —,' )F2&',

gi ——( ~~ )F2+( ~~ )F4 —( 7 )F2X

F() ——(—", )F'i+( ', )Fg FiX— —

I'2 ———( ,", )F'i+( —„)F—4+(—„)Fi&,Y~=( —,", )Fg+( ,', )F4 —( —, )FiX—

(A 1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

Other authors have been coming to the conclusion that

g2 ——0, based on fits to data. Of course, all these argu-

The parameters of the IBM Hamiltonian in Eq. (8) canbe expressed in terms of the parameters Fir given in Eq.(7). These are displayed below.

APPENDIX B: BOSON-BOSON INTERACTIONFOR NUCLEONS IN DEGENERATE ORBITS

WITH SDI

This is a derivation of the expression

Fx =4«x/eo (B1)

based on an analysis of the nucleon pair wave functions,and in particular of the correlations induced by the sur-face delta interaction (SDI). For the sake of mathematicalsimplicity, we neglect spin in our considerations; includ-ing spin does not change the results.

We consider now two particles in a collection of degen-erate orbits interacting via an SDI with strength G. Foreach value of angular momentum L, only a single state ofeach L is shifted in energy. We consider this lowest ener-

gy state of each L to be a boson of angular momentum L.The energy of this shifted state is

1 1' LeL ——gg(21+1)(21'+1) G . (B2)

I 1'

The corresponding wave function is

1 1' L t' I.]I]L]]r(1,2)=(M/v Q) yy/1' 0 0 0 yy, I &] (8]i]t]])&] (8i,y2) i (B3)

where l=(21+1)', Q=g(21+1) is the total pair degeneracy, and i is a normalization constant. The labels 1 and 2refer to the two particles, respectively.

For L =0 or s bosons, in particular, we have the following:

(1,2) =(M/v Q) g g( —) Y'] (8],P])I'] (8z, ])]]i)l m

=(~/4mv Q) g (21 +1)P](cos8]z)I

using the addition theorem for spherical harmonics.Appropriately normalized, this becomes

]Ijoo(1,2) =(1/v Q) g (21+ 1)P](cos8]z) .

For an S-pair, the two particle density is

334 C. H. DRUCE AND S. A. MOSZKO%'SKI 33

p(1,2) = ~%~(1,2)~

'=(1/Q) g g (21 + 1)(2!'+1)Pi(cosSi&)Pi (cos8,2)

E E'

1 /' L=(1/Q)ggg(2L+l)(21+1)(2l'+1) 0 0 PL, (cos8&2)

L E E'

Notice that setting L =0 in (B2) gives The intrinsic two particle density is simply the product:

eo ——QG pi(1,2) =pi(1)pi(2) . (89)

and the comparison of (82) and (B6) yields:

p(1,2) = g (eq /eo)(2L + 1)PL, (cos8i2) .

Thus the two particle wave function for an S pair is sim-

ply related to the SDI energies for all the pairs.The two nucleons forming an S-state pair have conju-

gate wave functions, and thus the same single particledensities. I.et us define the "intrinsic" single particle den-sity by:

pi(1)= y (ei, /eo)' (2L +1)PL,[cos(8i —8,)],L

where 8i can be regarded as the angular coordinate of bo-son I.

The actual two particle density may be obtained byaveraging over all possible directions Si of the boson, i.e.,

p(1,2)= J p(i1, 2)d Qi/4n . (810)

Next, we use the intrinsic densities to calculate the in-teraction between bosons. Let I and II denote, respective-ly, the angular coordinates of the two bosons. Nucleons 1

and 2 are associated with boson I, while 3 and 4 are asso-ciated with boson II. Now 1 and 2 have the same densi-ties, and so do 3 and 4. Thus the boson-boson interactionis four times the interaction energy between any pair, aslong as each boson has one nucleon represented; for exam-ple, nucleons 1 and 3. For an SDI, nucleons 1 and 3 onlyinteract if they are at the same angle.

Thus we can write:

V~,.~.=46Jpi(1)pii(1)d Ql/4~

=46 g g (eL, /eo)' (eq /eo)' (2L+1)(2L'+1)Pq[cos(8& —8,)]L L'

XPg [cos(8i —Sn)]dQi/4ir

=46 g (eq /eo)(2L + 1)PL, [cos(8i—Sii)]

If we now expand Vb „b „as a multipole expansion: we verify the result that was to be proven:

Vga ——g F»(2E+ 1)P»[cos(Si—Sn)]K

(B12) F» ——46e» /eo .

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