Quadratic spin-orbit forces and the deuteron magnetic moment

Naclear Physics A242 (1975) 141- 148; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

QUADRATIC SPIN-ORBW AND THE DEUTERON DONALD

FORCES

MA&ETIC

MOMENT

W. L. SPRUNG

Physics Dept., McMaster University, Hamilton, Ontario, Canada L&S 4MI Received 13 January 1975 Abstract: The contribution to the deuteron magnetic moment due to L-dependent components of the two-nucleon force is discussed, and evaluated for several potential models. The Reid soft core potential is expressed as a sum of central, tensor, spin-orbit, L2 and Q12 forces, and it is argued that this choice of components has a number of advantages.

1. Introduction In 1957, Feshbach ‘) pointed out that if the nucleon-nucleon force involves the operator S 1L or L, z, this will induce a correction to the deuteron magnetic moment. Recently there has been a flurry of interest in this subject ‘-‘), concentrated mainly on the spin-orbit contribution. Scheerbaum “) has emphasized the importance of the quadratic spin-orbit force (L,,) contribution and has deduced the formulae for evaluating it. He was unable to evaluate this contribution for the Reid soft core potential ‘) because of an uncertainly in how to divide the Reid force up into its various components. In this note I show that this division can be made quite unambiguously, and point out the advantages of a specific choice for the form of the quadratic spin-orbit force operator. 2. General form of the nucleon-nucleon interaction In a classic paper, Okubo and Marshak “) discussed the most general form of the nucleon-nucleon force allowed by the principles of rotational, parity, and timereversal invariance, plus the fact that any expression quadratic in the spin operators ui may be linearised. The most general central force is then a function V(r’, p2, L’) multiplied by one of the scalars (1, u1 - d, d - z2, d * a2r1 - 7”). Since one can construct from these the singlet and triplet projection operators, it follows that in each state of total spin (S) and isospin (T) one can have a separate central force VI&r’, p2, L2). Alternatively one could deal with an exchange mixture of W, M, B, H forces. It is generally assumed for simplicity that V,, should depend only on Y’, as a first approximation - i.e. the force is local. In the singlet-even space (S = 0, T = 1) this is known to be inadequate ‘): the potential acting on the ID, state is only about two-thirds as deep as in the ‘So state “). This can be accounted for either 141

142

D. W. L. SPRUNG

by allowing explicit non-locality ‘), or a dependence on p* [ref. “)I, or by allowing a dependence on L2. The latter is more convenient for people fitting phenomenological forces. The simplest form is V,, = IC&(r2)+ V,‘,z? = VS,*(r?)+ VfI(r2)L2. The data on the ‘G4 state is, to date, inadequate to determine whether or not an additional V&+L4term is needed Besides these central forces, there are spin-orbit and tensor forces. The former is of the form (1, d - z”} times Vu(r2,p2, L2)L. S. Again, from 1 and z1 * z2 the projection operators on states of good T can be constructed, so the spin-orbit force has an independent shape in the triplet-odd (T = 1) and triplet-even (T = 0) states. Since only the P- and D-waves are well known, the form V&(r’) is adequate in each case. The allowed tensor operators are constructed as the scalar product of a tensor of rank two in spin space contracted with a tensor of rank two formed from r, or p, or L: S 12 [email protected]&.@*, Pi2 = 3d * p tr2 * p-d Q,,

* u2 p2,

= +{cr’ . La2 - L+a2 - La’ * L)-a’

(0 - u2 L2.

These are scalars, at most linear in either of the ei, give zero acting on a singlet state, and may be multiplied by {1, r1 - T’ >. Thus the radial shapes V(r2, p*, L*) will be different in the T = 0, T = 1 states. This particular classification of the tensor forces was found essential in perturbation theory calculations on the N-N force ‘). The usual, or static, tensor force is S12. Its presence is required by the deuteron quadrupole moment. If an additional tensor interaction is required, one may choose either P,, or Q,, . The latter is preferred by people fitting potentials. In the absence of good information about the F-wave or G-wave forces, the radial form can be taken to depend only on r. By combining Q,, with e1 - 02L2, one can construct the alternative forms used by other authors. These are, in my notation L,2 = ${el - La2 - L+a* * La1 * L},

(2)

H,,

(3)

= _t{(e’ x L) * (u” x L)+ (u” x L) * (u’ x L)] =u 1 * a?--L12,

used by Hamada-Johnston

’ “) and Y,, = L12 - (S * Ly = (S*L)*+(s.L)-Lz = 3[L12 i-s * L- L2],

(4)

DEUTERON

MAGNETIC

MOMENT

143

used in the Yale potential ll). These have particular advantages in the potential fitting process but are less convenient for the following analysis. 3. Analysis of the Reid soft core potential Reid fitted the radial form of this potential in each partial wave separately. For the coupled 3S,-3D, states, the potential is a symmetric 2 x 2 matrix VLLe.The coupling potential V,, = V,, = J8 V, defines the tensor force, V,(r)&. The S-wave channel force V,, = Vc(,(r)defines the central force. The 3D, channel Vz, = I@) - 2&(r)+ V, has a residual component V,. From the 3D, and 3D3 waves we can extract residual components V, and V,,

The simplest T = 0 space, V,(r) can be shapes V,(r)

V(3D2) = Vc(r)+2V,(r)+

V,,

V(3D3) = V&)-&(r)+

V,.

(5)

model in accord with the discussion of sect. 2 is that in the S = 1, V = Vc(r)+ V,,(r)L’ + V&)S - L+ VT(r)S12 + Vo(r)Q12. The functions assumed to depend only on r, for reasons mentioned above. The radial and VT(r) are already fixed by V,, and V,,. In this model V, = -3VL,+6VL,+21Vc, V, = - V,,+6V’,-21

V,,

(6)

V3 = 2VL,+6VL,+6Vo. It is easy to check that (i) the splittings induced by V,, and Vo are orthogonal to and Vcis each other and to V,,; (ii) the statistically weighted (2J+ 1) average of VLvLs zero. Thus the role of VLL,just as in the singlet-even case, is to make the average D-wave force weaker than in the 3S, channel. A ‘spin-orbit’ type splitting is allowed for by VLl’,and a tensor-like splitting is allowed for by V,. Indeed, the weights for are simply proportional to those given by Vo. The point is that s12, C-2, +2, -$, the strength of V,(r) is already fixed by the S-D wave coupling; if the D-waves show a stronger (or weaker) tensor-like splitting, it must be supplied by Ve(r). Also, just as in the singlet-even case, one can elect to use a p2 or an L2 dependence, here the choice between P,, and Q,, is made for convenience. The solution of eq. (6) gives 6OI’,, = -9V1--5V2+14V3, 9OV,, = 3v,+5v2+7v3,

(7)

18OVo = 3v,--5v2+2v3. For the Reid soft core potential, the residuals V, are sums of Yukawas, Vi(r) = ~c~eMnX/x with ranges n = 2,3,4,6 pion Compton wavelengths. The coefficients

D. W. L. SPRUNG

144

TABLE 1 Force constants for the Reid soft core potential 3

n=2

VI VZ V3 VLS VLL vQ

0

0

-325.588 -297.868 -21.6035 -34.3335 6.7234

871 0 -72.5833

48.3889 -24.1944

4 -2126.73 2484.26 4068.2 1061.23

383.539 -59.2505

6 8139.3 -6577.3 - 11012.0 -3242.25

-950.585 196.002

ci are given in table 1, along with the corresponding coefficients c,”for the potentials V,, V,, and V, which are the solutions of eq. (7). The result for V,, agrees with Scheerbaum 4), but he was unable to separate VLLfrom Vc, or VP from Vr, because he did not recognize that Vc and Vr are defined by the 3S, and S-D coupling potentials. These results are not entirely satisfactory. The spin-orbit force is generally associated with the p-meson, and we should expect it to be mainly repulsive and of range n = 4 or 6 pion masses. Because Reid did not anticipate the above analysis, he used n = 2, 3 terms in fitting the 3Dz state and the 3D3 state. These long-range terms get mixed up in eq. (7) and appear in V,,(r). Probably Reid could refit his 3D, state potential to escape these long-range components. Certainly the 3D3-3G3 waves allow considerable latitude in fitting a potential. Comparing the last two lines of table 1, we see that except for n = 3, there is a factor of about (-5) between the forces V,,(r) and V&r). The force could roughly be written as V(r)[Qrz--5L2] = -3P(r)[L’+&]. (8) Thus, it is nearly just the sum of an L2 force and a force of the Hamada-Johnston form. That the amplitudes are nearly equal is an interesting coincidence. In the potential fitted by de Tourreil and myself 12), we also found the L2force stronger than the V, force by a factor of (- 3). These are the only two attempts which have treated the ratio of the L2 to the Qiz force in an unbiased manner. 4. Correction to deuteron magnetic moment The deuteron magnetic moment correction induced by the L-dependent forces has been derived by Scheerbaum “). His results appear to be correct except for the factors of (2m/m,), m being the reduced mass of the N-P system and mp the proton mass. There should be only one such factor, coming from the relation rl = urn/m, relating the position of particle 1 (the proton) to the relative separation vector r = r1 -r2. The mp in the denominator is absorbed into eh/2m,c, the proton Bohr magneton. The m in the numerator is used to make 2m/h2 I$(,) = u,(r), the potential in units of fme2.

DEUTERON

MAGNETIC

145

MOMENT

TABLE 2 Coefficients of the radial integrals for the deuteron magnetic moment correction,

cf. eq. (9)

Force OL L'

QlZ LIZ

R12 Y1Z

Contribution

TABLE3 to the deuteron magnetic moment from the L-dependent components phenomenological potentials (in units of efi/2m,c)

Force

a=LS

RSC HJ Yale TS 72 TRS 74

0.00214 0.00240 0.00662 0.00764 0.00949

of several

(APL LZ 0.00271

Q12

L12

HII2

-0.00783 -0.00975

0.00236 0.00315

-0.00573 -0.00722

Total

&,

pn

-0.00298 -0.00735 0.00662 0.00427 0.00542

3.60 2.79 5.39 4.96 5.17

6.47 6.95 6.96 5.45 5.80

Also shown are the actual percent D-state (pn) and the value &, consistent with no meson current contributions.

For the spin-orbit force one has, in units of eh/2m,c, with


(9)

U(Y), w(r) being the S-wave and D-wave radial functions for the deuteron “). For the forces quadratic in L, as remarked by Feshbach ‘), the (S . . . S) matrix elements will have zero coefficient. The coefficients for other forces are given in table 2. The zero correction for the Yale force in the bottom line was noted by Sheerbaum; it implies that the forces L,, and (S +IL)* give the same correction. It can be seen, from eqs. (2)-(4), that the relations between the various operators quadratic in L are reflected in table 2. The magnetic moment corrections for several phenomenological potentials are listed in table 3. The values for the HJ and Reid potentials do not quite agree with those of Scheerbaum ‘); perhaps this is because I recalculated the wave functions and have more points available. By interchanging the various wave functions, it was found that the values depend mostly on the potential and only by 5-10 % on the details of the deuteron wave function. In my review talk on the deuteron 3), I was guilty of the oversight of identifying V, for the Reid potential

146

D. W. L. SPRUNC

with - 3V,,(r), so the value 0.0066 nm. reported there for the LS contribution is in error. However, as explained above, the value found here is probably too small. The long-range (two- and three-pion masses) terms found in V,, are probably spurious. Being attractive they reduce the LS contribution too much. The two super soft core potential models, TS and TRS I’? 13) give rather similar results, even though the second model was fitted using an i? and an Lr2 force. All of the models agree in having the average D-wave force less attractive than the central force - ie. there is a repuhive r? component. All agree in having an attractive Q,, component, so the tensor splitting in the D-waves is on balance less than is implied by the tensor force alone. (The matrix elements of Q,, are opposite in sign to those of S1,; see the appendix.) The strength of the Lz force is generally somewhat greater than that of the L,, force, so the form H,, is not by itself adequate. 5. Discussion The deuteron magnetic moment may be written

wherep, is the percentage D-state, (dp)r is the sum of the L-dependent contributions and (LIP),,,is the contribution from meson currents and other effects. In the absence of these latter two contributions, one obtains the classic relation 0.0223 = 0.5361 pb giving 4.16 % D-state. A positive contribution from ApT increases the value of p,, which is consistent with the absence of meson current effects. This value is listed as j&,in the last column of table 3. In no case is this value actually equal to the percentage D-state calculated for that potential, although for the SSC potentials the discrepancy is smaller than for the others. For the Yale po~ntial gives pD = 6.96 %, for example. The Hamada Johnston potential is particularIy far out, because it combines a high PD = 7 % with a large negative (&)r. Every potential model therefore requires some contribution from meson currents to explain the deuteron magnetic moment; the largest in the case of HJ and the smallest for TS. From the spread of values in any column of table 3 it can be seen that (dp), is not too well determined, other than in sign and order of magnitude. In the case of the LS contribution the reason is quite clear. The value of I/Ls is zero acting on an S-state, so it is determined by the D-wave data. Due to the centrifugal barrier, the short-range part of VLs is left quite arbitrary. Yet in eq. (9) one requires the integra1 of V,with the S-wave deuteron wave function. This allows considerable variability in the result. Essentially the same effect applies to the other forces. The choice of L2 and Qlz as the L2 dependent operators has the advantage of a clear separation between the average weakening of the D-wave phase shifts (L’) and their tensor splitting (~2,~). All of the models include both types either explicitly or implicitly. The Hamada-Johnston force, through use of Hi2, arbitrarily gives them a fixed ratio of strength, which is not borne out by any of the other models.

L--l

L=J

J=L+l

J=L

J=

-L(LSl)

LZ

I-L(LI1)

tL+ls

LlZ

Only Siz has a non-diagonal element.

s=o

S=l

>.,qy;

TABLE 4

-2L(LS

L

2L(LSl)-1

-_(L+l)

HI2

1)

0

L(2L-

1)

3-4L(L-t1)

tL+f)t2L+3)

012

-Lw-1)

0

--L(LS

0

YIZ

1)

0

L

-1

-_(LSl)

L*S

Partial wave matrix elements of operators discussed in text, in LS’J representation

0

-2L/(2L+-3)

“t2

-2(L+1)/(2L-I)

S 12

3 ij

5 0

E

i4

r;l

u

148

D. W. L. SPRUNG

Continued research support from the National Research Council of Canada under operating grant A-3198 is gratefully acknowledged. I am grateful to Dr. R. R. Scheerbaum for communi~ting to me the 3D3- 3G3 states part of the Reid soft core potential, and for a preprint of his paper. Appendix Having derived the relations in table 2 properly following Feshbach’s arguments, we can see that simple considera~ons suffice to relate these relations to the result given in eq. (9) for the spin-orbit force. Consider the operator (S - L)‘. Since we look for terms linear in the magnetic field, only one at a time of the L is replaced by (r x A). Thus the coefficient of the (D . . . D) integral should be -6( - 3) 2 = 1, with a factor of two for two equal terms. For the (S . . . D} integral the coefficient is (-&/+)(3) = &/2. No factor of two occurs since, if S * L is on the same side as the 1s) state, it wilf give zero. The L2 force can be worked out easily, as the effective magnetic moment operator is proportional to L. To see that&, is equivalent to (Se L)” we write the Yale operator as Y12 = (S *L)“-$(a’

*icy-*(a”

*L)Z.

It is plausible that, acting on a triplet state symmetric in the spins, the last two terms should be equal to each other and should just cancel the first term, for S = $(ul +a’). Knowing the results for L,, and L? is sufficient to complete the table. These heuristic arguments may be helpful in rationalizing the results of table 2. The matrix elements of the operators discussed in sect, 2 are given in tabIe 4. It is seen that the matrix elements of Q, 2 are simply - 2{2L - 1)(2L + 3) times those of the static tensor operator S,,. Of course, SIz also couples the L = J_+ 1 states together while Q12 does not. References 1) H. Feshbach, Phys. Rev. 107 (1957) 1626 2) I. C. Bergstrom, Phys. Rev. C9 (1974) 2435 3) D. W. L. Sprung, Proc. Int. Conf. on the few body problem in nuclear and particle physics, Lava1 University, 1974 4) R. R. Scheerbaum, Phys. Rev. Cl1 (1975) 255 5) R. V. Reid, Jr., Ann. of Phys. 50 (1968) 411 6) S. Okubo and R. E. Marshak, Ann. of Phys. 4 (1958) 166 7) D. A. Giltinan and R. M. Thafer, Phys. Rev. 131 (1963) SOS; C. Lee, J. L. Gammei and R. M. Thaler, Nuclear forces and the few nucleon problem, vol. 1, ed. C. T. C. Griffith and E. A. Power (Pergamon, London, 1960) p. 43 8) A. M. Green, Nucl. Phys. 33 (1962) 218 9) D. W. L. Sprung, LNS technical report no. MIT-2098-201 (1965) IO) T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 11) K. E. Lassila, M. H. Hull, Jr., H. M. Ruppel, F. A. MacDonald and G. Breit, Phys. Rev. 126 (1962) 881 12) R. de Tourreil and D. W. L. Sprung, Nuci. Phys. A201 (1973) 193 13) R. de Tourreil, B. Rouben and D. W. L. Sprung, Nucl. Phys., in press