PHYSICAL REVIEW C VOLUME 36, NUMBER 4 OCTOBER 1987

Deformed nucleonic bags and nuclear magnetic properties

Afsar AbbasInstitut fur Kernphysik, Technische Hochshule Darmstadt, 6IOO Darmstadt,

Federal Republic of Germany(Received 20 February 1987&

Good fits to the magnetic moments of He and 'H are obtained when the nucleonic bags con-sisting of three quarks each are assumed to be deformed. Expressions for the magnetic dipolestrengths residing in the low-energy and the h, sectors are obtained. If, as manifested in the Euro-pean Muon Collaboration effect, the bags expand in the nuclear medium, then the net deforma-tion of a bag must also decrease in nuclei.

Several nagging problems of the nonrelativistic quarkmodel are resolved when one takes the bag to be deformedwith large D-state admixtures. ' Deformed bags havebeen studied by various groups. A recent analysis ofthe E2/M 1 ratio in yN h(1232) favors deformation.

Here we wish to study the magnetic properties of nucleiwithin the context of the nonrelativistic quark model withdeformed nucleonic bags.

We define the nucleon and the isobar wave functions asfollows:

I N) [1 PD(N)) i 2I Ns)+ [PD(N)] '"

I ND),

Ia& - [1 —PD(a) ] '" I ~s &

—[pD (g)/2] «2IA/1) + [pD (g)/2] it2

I pgi& (2)

The subscripts S and D represent the spherical s-wave andthe deformed D-wave parts, respectively. The superscripts(1) and (2) refer to the symmetry (70,2+) and (56,2+),respectively, for h.

Let us assume that our nucleus consists of A such de-formed nucleonic bags. We define the nuclear groundstate by the Slater determinant of these nucleons:

A

A + I (qqq), &,Qt a 1

(3)

where I (qqq), ), a =1,2, . . .A are the nucleon wave func-tions (1). The antisymmetrizing operator A acts onlywithin the nucleonic space.

We will assume that specific nuclear medium effectswill modify the parameters PD and P~~ from their free nu-cleon values. Now we define the M1 operator as follows:

3

M 1 - g g g [cr„(i)+ l„(i)l g (i)p, (i)a 1 asi 1 p

(4)

Here p labels the spherical components and i the threequarks in the nucleon bag a. Q is the quark charge, cr andi represent the spin and the angular momentum parts, re-spectively, and p~ eh/2m~c, where mz is the mass of thequark. For M1 transitions we need an extra factor ofd3/4~.

Using operator (4) with the wave function (1) for a sin-

gle nucleon one obtains the following magnetic moments:

p (n ) = ——', [1 —PD (N ) ]p~,p(p) = [1 —PD(N)]p. (6)

so It(p)/p, (n) = —2, i.e., not modifying the already suc-cessful SU(6) prediction. '

For the trinucleon system, one expects that the magnet-ic moment of the ground state is obtained from that of theodd nucleon. So p( He) =p(n) and p( H) =p. (p). Thesingle parameter PD(N) is expected to be modified in thenuclear medium. Therefore,

p ( He) = ——,' [1 —PD ( He) ]p~,

p('H) = [1 —PD('H)] pq,

(7)

(8)

Bp('He) p('He) —p(n)bp('H) p('H) —p(p)

In our model it is

[PD(N) PD(3 [P (N) —P ('H)]

(10)

which, with the above values, is —1.08. This comparesvery well to the experimental value of —1.1. For the sakeof comparison, note that Karl, Miller, and Rafelski' ob-tain —

3 for the same ratio.Can we understand these values of PD(A)'? One of the

ways of explaining the European Muon Collaboration

where PD(A ) refers to the deformation of the odd nucleonfor a particular nucleus.

We now determine this PD(A) by fitting to the experi-ments. So,

p( He) 1 —PD( He) lt(3H) 1 —PD( H)p(n) 1 PD(N) p. (p) 1 —PD(N)

Taking the experimental values p( He) - —2. 12, p(n)= —1.91, p( H) 2.98, and p(p) =2.79 (all in units ofpN) for the left-hand side and using PD(N) = —,', whichgives good fit to various physical quantities, ' we obtainPD( He) 0.168 and PD( H) =0.199.

Another headache for the theoreticians is to explain themagnitude of the deviations expressed by the following ra-tio:

1663 1987 The American Physical Society

1664 AFSAR ABBAS

in nuclei. The total strength is

S~+ =g~(n ) g 8(a) ( 0& )

a 1

=Tr(8t8) + (Tr8+ ) (Tr8) —Tr(8t8) —(Tr8)

(i2)where 8(a) refers to the large bracket part in (4). Thetrace

Tr(8) =(0~ g 8(a)

~0) (i3)

is taken with respect to the A-dimensional nucleon sub-space. 0t0 means it is for the same nucleon while in 0t0the nucleons may be different. We are ignoring the shelleffects. The excitations to the 6 sector are obtained bythe standard procedure 12 This taken away from Sx+gives the strength sitting in the low-energy N sector. Weobtain [in units of pv2 ( —,

' tt)]:

effect is to increase the effective confinement size of thenucleons in the nuclear medium. " If we accept this pic-ture, then we can easily accommodate Pp(A ) & Pp(N).In a deformed bag there is a larger surface area in the flat n

region than at the edges of (say) the pumpkin. If the pullon the surface is uniform, greater expansion will takeplace perpendicular to the flat part and so the deformationwill decrease.

It seems that He is a more loosely bound system thanH is. [B.E. = —7.77 MeV, r(rms) =1.88 fm for He,

compared to B.E. = —8.48 MeV, r(rms) =1.70 fm forH.] Since there is a greater space available for a bag to

expand in He than in H, it is therefore reasonable thatthe change in Pp (N) is larger in the former case.

It has been discussed in literature' that in general thevalues of the magnetic moments of nucleons inside a nu-cleus are larger than in the free space. Our discussion ofthe magnetic moment of A =3 system can be easily car-ried over to these cases.

Because it is strangeness conserving, the magnetic di-pole operator can excite only N and h. degrees of freedom

I

Ssrt+ =[3 + 9 Pp(N) —3 Pp(N) ](N+Z)+Pp(N)['9 ——', Pl)(N)]Z

+ 9 [1 PD(N) ] ~(/v —1)/2, 1+2 [1 PD (N) ] ~{z—1)/2, l

S~1 =—2+[1 Pp(N)] [1 ——Pp(/s) l + QPD(N)PD(I))I2 (N+Z),2 iz+ i

2

Sm&+ —S

(i4)

(is)

(16)

Here I= [0, 1,2, 3, . . . ]. The 6 function ensures that theseterms act only for the odd-N or odd-Z nuclei. FollowingRef. 2, we can express Pp(A) in terms of Pp(N),

Pp (/J. ) =2PD (N)/[I +PD (N) ], (i7)

to reduce the number of parameters to one. We thus ob-tain an expression for the magnetic dipole strength sittingin the low-energy sector in nuclei which depends upon oneparameter Pp (N).

For example, for Pp(N) =0.1 we obtain 23pN for Siand 38pN for Ca. These are higher than the present ex-perimental values' which are -7pN and -6pN, respec-tively, for the two nuclei. One has to go to Pp(N) -0.04for Si and Pp(N) -0.02 for Ca to obtain values whichare comparable to the experiments. From this, it appears

I

that the nucleonic deformation goes down with the massnumber.

In passing, let us mention that with the wave function(2) the magnetic moment of 5++ is

P(a++) =2 Pp(a)—in units of p~. Using Eq. (17) and Pp(N) = —,', we predictthe p(h++) to be 1.6pv. This value is within the limits ofthe analysis of Ref. 14; however, an updated analysis isneeded to clarify the point.

This work is supported by the Bundesministerium fiirForschung und Technologie, Federal Republic of Ger-many.

1S. L. Glashow, Physica A 96, 27 (1979).V. Vento, G. Baym, and A. D. Jackson, Phys. Lett. 102B, 97

(1981).J. L. Cortes and J. Sanchez Guillen, Phys. Rev. D 22, 1775

(1980).4R. Shanker, C. S. Warke, W. Greiner, and B. Miiller, J. Phys.

G 8, L33 (1982); R. L. Viollier, S. A. Chin, and A. K. Ker-man, Nucl. Phys. A407, 269 (1983); Z. Y. Ma and J. Wam-bach, Phys. Lett. 132B, 1 (1983); K. Hahn, R. Goldfam, audL. Wilets, Phys. Rev. D 27, 635 (1983).

sG. E. Brown, Prog. Part. Nucl. Phys. 8, 147 (1982).R. Davidson, N. C. Mukopadhyay, and R. Wittman, Phys. Rev.

Lett. 56, 804 (1986).7D. Flamm and F. Schoberl, Introduction to the Quark Model

of Elementary Particles (Gordon and Breach, New York,1982).

sH. R. Fiebig and J. Wambach, Nucl. Phys. A384, 371 (1982).9R. G. Sachs and J. S. Schwinger, Phys. Rev. 70, 41 (1946).' G. Karl, G. A. Miller, and J. Rafelski, Phys. Lett. 143B, 326

(1984).' F. E. Close, R. G. Roberts, and G. G. Ross, Phys. Lett. 129B,

346 (1983); R. L. Jaffe, Comments Nucl. Part. Phys. 13, 39(1984); P. Mathieu and P. J. S. Watson, Phys. Lett. 148B,473 (1984); L. S. Celenza, A. Rosenthal, and C. M. Shakin,Phys. Rev. Lett. 53, 892 (1984).

'2A. Bohr and B. R. Mottelson, Phys. Lett. 100B, 10 (1981).'3A. Richter, Phys. Scr. T5, 63 (1983).'4P. Pascual aud R. Tarrach, Nucl. Phys. B134, 133 (1978).