PHYSICAL REVIE% C VOLUME 8, NUMBER 3 SEPTEMBER 1973

Magnetic Moment of the Deuteron and Nucleon-Nucleon Spin-Orbit Potentials

Shankar Mukherjee and Radhey Shyam~Physics Department, Banaras Hindu University, Varanasi, India

(Received 29 August 1972; revised manuscript received 30 April 1973)

The change in the magnetic moment of the deuteron arising from the presence of the spin-orbit potential in the nucleon-nucleon interaction has been estimated for the Yale, Hamada-Johnston, and Reid potentials. The present estimates are more close to experiment thanthose obtained previously from some old fashioned potentials.

The object of the present note is to supplementthe work of Feshbach' who previously estimatedthe change in the magnetic moment of the deuteronarising from the presence of a spin-orbit term inthe nucleon-nucleon interaction. He estimated theabove quantity for the spin-orbit term present inthe nucleon-nucleon interaction given by Signelland Marshak' and by Gammel and Thaler, ' andused a simple Hulthen wave function and only theS state of the deuteron. Estimates of Feshbach'swith the aforesaid potentials were quite high andcould not, therefore, be fitted to the experimentalvalue of the deuteron magnetic moment. But in thepast few years, several nucleon-nucleon potentialsobtained by fitting the scattering data at energiesbelow meson-production threshold have been pub-lished in the literature. The most successful andmost widely used are the ones suggested by theYale group headed by Breit, ' Hamada and Johnston(HZ), ' and Reid. ' Therefore, it has become neces-sary to redo the calculation with these realisticnucleon-nucleon models using corresponding pre-cise deuteron wave functions and including the Dstate of the deuteron.

The introduction of spin-orbit forces leads to amodification of the magnetic moment operator and,therefore, to a deviation of the calculated deuteronmagnetic moment from the familiar expression

(4)y=~ [u(x)+-,'vTS„s(x)]y',~4x

where N(x) and s)(x) are the radial wave functionsfor the S and Dstates, respectively; x= p.r; p,= 0.6999 fm ', the inverse Compton wavelengthcorresponding to the weighted average of the pionmasses; y„' is the triplet spin wave function; andS» is the usual tensor operator; we obtain

(~v) =)zt) v-'(&slav(x)ls)

—~&s(2v(x) (n}+~&D(Pv(x) (a}j .1

(5)

The following forms of V(x) are used in the pre-sent calculation:

Yale group, 'V(x) =Z„s„e '*/x"

where the constants a„are listed in Table I of Ref.

Hamada- Johnston, '

J, are the orbital, spin, and total angular momen-tum operators, respectively, in units of S. Asimilar expression has also been obtained by Eder. 'This contribution has to be added to the right-handside of E(I. (I). Inserting the deuteron wave func-tion

where p,„and p.~ are the magnetic moments of theneutron and proton, respectively, while p~ is theD-state probability. The experimental value of theleft-hand side of E(I. (I) is

As shown by Feshbach, the contribution to the deu-teron magnetic moment (Ep)z~ as arising from thespin-orbit term (L F)V(~) of the nucleon-nucleoninteraction is given by

(ap, )s(;=- {V(r)[(S7) —(S.JQr']),16kg

where x is the internucleon distance, and L, S, and

where the constants GI and b~ are given in Ref.5;Reid, '

4x

V(x) = V09.9Ix

An approximate analytic deuteron wave functioncorresponding to the Yale potential is given byKottler and Kowalski' and that for the Hamada-Johnston potential is given by McGee. ' The wavefunction for the Reid potential is available in thenumerical form. However, we have used here

1150 S. MUKHER JEE AND R. SH YAM

more precise deuteron wave functions obtained innumerical form by solving the Schrodinger equa-tion. We note that the correct deuteron bindingenergy and quadrupole moment are obtained foreach of these potentials. Accuracy of the wavefunction is further tested by evaluating the root-mean-square radius of the charge distributionof the deuteron, which for Yale, HJ, and Reidpotentials is 1.953, 1.96, and 1.943 fm, respec-tively. These values are in excellent agreementwith the experimental value of 1.96 fm obtainedfrom electron scattering measurements. Theradial function in each case is normalized suchthat

J [M(x)'+ w(x)'] dx = 1,0

'where ~0 is the hard-core radius which takes thevalues 0.35, 0.343, and 0.01 fm for Yale, HJ, andReid potentials, respectively. We note immediate-ly that, since all potentials are negative definite,(Ap)~ is negative. Since this is to be added to theright-hand side of Eq. (1), it is clear that p~ needsto be reduced in order to match the experimentalvalue [Eq. (2)]. Clearly, if (hg)~~ is too largein absolute value, it will become impossible to ob-tain Eq. (2).

The integrals in Eq. (5) have been evaluated nu-

merically, and values of each term in Eq. (5) bothfor old and new potentials are given in Table I.The value of (ap)~~ has been calculated using theabove integrals and is also listed for each poten-tial in Table I, along with the previous estimatesof Feshbach. We notice that the values of (np. )z~obtained previously by Feshbach using Signell-Marshak and Gammel-Thaler potentials, are solarge as to preclude the possibility of matching theexperimental value of the deuteron magnetic mo-ment, even if pD were reduced to zero. On theother hand, in the present calculation we see thatthe changes in the magnetic moment of the deu-teron arising from the spin-orbit term in Yale,HJ, and Reid potentials are much smaller than

the values obtained by Feshbach with Signell-Marshak and Gammel- Thaler potentials. Theoverestimation of (b,p. )~ by Feshbach could, there-fore, be partly attributed to the choice of the nu-

cleon-nucleon interaction, the neglect of the Dstate of the deuteron, and the simplicity of thedeuteron wave function; whereas the models forthe nucleon-nucleon interaction used in the pre-sent calculation are more refined, and so also arethe deuteron wave functions. Further, we see thatthe contribution of the SD interference and that ofthe DD term are quite large, making the values of(b p, )~ much smaller than those obtained previous-ly. However, it is recognized that the contribu-tion to the deuteron magnetic moment of the spin-orbit term present in the nucleon-nucleon inter-action is still quite significant. We have also es-timated the D-state probability pD after applyingthe correction (d, p, )z~ to the experimental value[Eq. (2)], and obtained the values 1.8, 3.3, and2.5$ for Yale, HJ, and Reid potentials, respec-tively. The above values of pa show undoubtedlyan improvement over the earlier calculation ofFeshbach, but they are much smaller than thevalues of ps (nearly 8-7%) present in the Yale,HJ, and Reid potentials. Though the suggestionof Feshbach has been straightforwardly carriedout here, appropriate considerations with respectto nonuniqueness of electromagnetic interactionsare yet absent in both the calculations. First, wehave assumed here that the center of charge of thedeuteron precisely follows the motion of the pro-ton which enables one to substitute (p —eA/c) for

p in the derivation of Eq. (3), which may not becorrect procedure in computing the interactionof a photon with a deuteron via exchange of ameson. " It is necessary, therefore, to find ex-change-current contributions to the deuteron mag-netic moment. " Adler" has made a quantitativeestimate of the meson-exchange contribution tothe static magnetic moment of the deuteron and hasshown that the mesonic exchange currents appearto be capable of explaining the discrepancy between

TABLE I. The contribution to the deuteron magnetic moment (Dp)&z arising from the spin-orbit term of the different types of nucleon-nucleon interaction models.

Interaction (s lx'v ix& ls& &s lx'v&x& lD& (D lx'v(x& ID&

(& P) gL,

(P~)

Signell-Mar shakGammel- Thale rYaleHamada-JohnstonB,eidLAD (P~ + Pp)]exp

Feshbach.

6.764.321.91650.51811.1750

0.92040.19720.3622

" Present work.

0.46070.07820.1109

-0 0560—0.0360 ~

—0.0123 b

-0.0035'—0.0080 ~

—0.0224

MAGNETIC MOMENT OF THE DEUTERON. . . 1151

the experimental and previous theoretical valuesof the deuteron magnetic moment. By exchangecurrent we mean those processes which do notinvolve the interaction of the y ray with the in-dividual dressed neutron or proton, but with thetwo-nucleon system.

Adler" calculated the pm mesonic-exchange-cur-rent correction to the magnetic form factor forelectron-deuteron scattering. For zero momentumtransfer, this correction [b,G„~(0)] was then as-sociated with a mesonic contribution to the deu-teron's static magnetic moment. He obtainedn,G„~ (0) in terms of a one-dimensional weightedintegral over the S and D radial wave functions ofthe deuteron. The following numerical estimatesof AG„~ (0) were obtained by him using the deuter-on's wave functions corresponding to the Yale andH J potentials:

AGs~(0) = [+(0.94 to 1.88) x10 '] p, „, Yale;

AG„, (0) =- [*(0.90 to 1.80) x10 '] iJ,„,Hamada- Johnston.

(10)

It is to be noted that these results involve an arbi-trary sign and a factor of about 2 uncertainty dueto the one- or two-pole fit to the nucleon isovectorform factor. Further, the deuteron wave functionscorresponding to the Yale, HJ, and Reid potentials,when compared, are found to be quite similar.Therefore, it is expected that the value of the me-sonic contribution to the deuteron's magnetic mo-ment for the Reid potential will not show morethan the 5% model dependence observed by Adlerfor the Yale and Hamada-Johnston potentials.

In the above calculation Adler overestimatedAG„~(0) because of the uncertainties in the valueof the coupling constant (g~,&'/4v) that existed at

that time. He used the value (g~~'/4v) =0.18 inrationalized units. The analysis of the experi-mental data on pion photoproduction by Donnachieand Shaw" later indicated that the value of thecoupling constant (g ~'/4w) =0.15+0.05. A morerecent analysis of the data on pion photoproductionshows that the value of the coupling constant usedby Adler is about 20% larger. " [The value ofg „& is to be multiplied by the proper Clebsch-Gordan coefficient in the calculation AG&~ (0).]Nevertheless, the contribution to the deuteron'smagnetic moment coming from the meson exchangecurrent is more than that coming from the presenceof the spin-orbit force in the nucleon-nucleon inter-action. However, it is clear that the requirementthat the magnetic moment of the deuteron be cor-rectly predicted will be useful in determining theform and the meaning of the spin-orbit potential.

Lastly, we would like to emphasize the followingpoints. (1) The mesonic-exchange effect discussedhere constitutes one particularly simple type ofrelativistic effect. Additional corrections to thecurrent operator can also be expected to play aninteresting role. (2) The spin-orbit force is oneof the many possible velocity-dependent potentials.(3) The finite size of the proton has been neglectedin the calculation. Therefore, a more definiteknowledge about the correction to the deuteronmagnetic moment could be obtained after the aboveeffects are incorporated in the calculation.

ACKNOWLEDGMENTS

One of the authors (SM) is grateful to ProfessorGregory Breit for many useful discussions on thework of the Yale group and to Professor B. M.Udgaonkar for his kind hospitality at the TataInstitute of Fundamental Research, India.

*Research fellow, Council for Scientific IndustrialResearch, India.

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(1957).3J. L. Gammel and R. M. Thaler, Phys. Rev. 107, 290

(1957).4K. E. Lassila, M. H. Hull, Jr., H. M. Ruppel, F. A.

McDonald, and G. Breit, Phys. Rev. 126, 881 (1962).~T. Hamada and I. D. Johnston, Nucl. Phys. 34, 382

(1962).R. V. Reid, Jr. , Ann. Phys. (N.Y.) 50, 411 (1968).

~G. Eder, Nuclea~ I'orces (Massachusetts Institute ofTechnology Press, Cambridge, Mass. , 1968), p. 48.H. Kottler and K. L. Kowalski, Nucl. Phys. 53, 334(1964).

9I. J. McGee, Phys. Rev. 151, 772 (1966).~ L. Heller, in The Taboo-Body I orces in Nuclei, edited

by S. M. Austin and G. M. Crawley (Plenum, New York,1971), p. 79.

~~R. Bryan, private communication. Authors are in-debted to Professor Ronald Bryan for his useful com-ments.

~2R. J.Adler, Phys. Rev. 141, 1499 (1966); 169, 1192(1968).A. Donnachie and G. Shaw, Ann. Phys. (N.Y.) 37, 333(1966).J. Lefrancois, in Proceedings of the 1971 InternationalSymPosium on Electron and Photon Interactions at High

Energies, edited by ¹ B. Mistry (Laboratory of Nu-clear Studies, Cornell University, Ithaca, New York,

1972), p. 52.