Consistency of nuclear Dirac phenomenology with meson-nucleon interactions

NucltarPhysics A3Z9 (1979) 354-364 Q North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

CONSISTENCY OF NUCLEAR DIRAC PHENOMENOLOGY WITH MESON-NUCLEON INTERACTIONSt J. V. NOBLEtt Department of Physics, Unitxrsity of Washington, Statut, Washington 98195 Received 29 December 1978 (Revised 23 April 1979) Abstract: Thephenomenological relativistic theory of nuclear single-particledynamics is reviewed and its salient characteristics discussed. The phenomenological (relativistic) potential strengths necessary to secure agreement with the usual low-energy properties of nuclei are determined . These phenomenological potentials are then related to the time-independent meson fields generated by nuclei, and it is shown by direct calculation that the potential strengths predicted on the basis of renotynalized meson-nucleon couplings (as measured in, say, NN scattering) are consistent with those deduced empirically. Thus thephenomenological theory is shownto be consistent with amore microscopic approach, in agreement with the work of previous authors. The role of the p-meson is then examined, andit is shown that the time-independent p-field leads to isovector terms in both the central and spin-orbit terms of the equivalent Schrrodinger potential. The signs and magnitudes of these terms agree with those determined from fits to the isobaric analogue (p, n) reactions, or to the systematics of single-particle energies .

1. Introduction Until fairly recently, the dynamics of nuclear single-particle motion has been described by the nonrelativistic Schrôdinger equation, in which the presence of the other nucleons manifests itself through an average one-body potential, and through the Pauli exclusion of certain orbitals. Duerr t), Miller 2), Lee and Wick 3), Walecka `) and others s.e) have sought to place the phenomenology of the shell model on a more fundamental basis by treating the single nucleon motion relativistically-in practice this has meant assuming the wave functions are solutions of the Dirac equation, withpotentials given by the average, static meson fields generated by the other nucleons. The quantum fluctuations of the meson fields about their mean values are considered as perturbations in this conceptual scheme . The Dirac phenomenology, while by no means the complete relativistic quantum field-theoretic treatment which is our ultimate goal, nevertheless offers significant advantages by contrast with the Schrôdinger phenomenology. The first, as several authors have pointed out''a't), is that the spin-orbit interaction emerges naturally as a Thomas precession . Secondly, a substantial fraction of the observed energy t Supported in part by the US Department of Energy and the US National Science Foundation. tt Permanent address: Department of Physics, University of Virginia, Charlottesville, VA 22901 . 354

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dependence of tht phenomenological (Schrôdinger) optical potential can be understood as a relativistic kinematic effect, at least at energies up to lA0 MeV. This has also been discussed previously l'Z's'~. From the point of view of medium-energy physics, however, the most important advantage of the Dirac wave functions is that they contain adefinite prescription for constructing the "small" Diraccomponents in terms of the "large" ones . Whenever we calculate matrix elements for weak or electromagnetic processes or for the emission or absorption of pions we require such a prescription, either to evaluate directly the field-theoretic operators, or to construct effective operators suitable for calculations with nonrelativistic, model wave functions . Several authors have recently applied the Dirac phenomenology to problems in electromagnetic interactions' °''1), weak interactions 12), and meson production is. "'"), and we may anticipate further developments along these lines. In order to develop confidence in the predictions of the Dirac model for mediumenergy processes, we must first check whether its predictions of static and low-energy nuclear properties agree with what is already known. In this spirit, I have in a previous paper calculated the dipole moment of the time-like component . of the nuclear axial current (analogous to the charge dipole moment), which is measured in the asymmetry of polarized nuclear ß-decay i2) . This moment is interesting because (except for exchange-current effects is)) it measures directly the average magnitude of the "small" Dirac wave function components which, as we shall see, can differ substantially in the presence of interactions from their "free" values . (No discrepancy between theory and experiment was found.) The present article is a demonstration that the potential strengths (central and spin-orbit) derived from the usual shell model and low-energy nucleon-nucleus scattering, are consistent with those obtained by nonrelativistic reduction of a Dirac equation, wherein the (Disc) potentials enter as the time-independent average meson fields generated by a closed-shell nucleus. To the extent that some of this work has been carried out previously by other authors Z), the present discussion is intended to be pedagogical; however, this paper does present a new result, namely the. derivation of the (real) Lane potential'6) from the properties of the p-nucleon coupling. In subsequent planned articles, the absorptive and dispersive parts of the optical potential will be examined from several different points of view .

Z. Nadear Disc phenomenologyy The phenomenological Dirac equation for a single nudeon has the form (fi = c =1) { -ia " O+ßM+ßU(r)+V°(r)-a " V(r)+iG~I,,rßys~ " zr(r)

J . V. NOBLE

356

where a, ß, ys, and Q=ay S ~ ysa are the usual Dirac matrices ; ~r(r) is a fouroomponent Dirac spinor ; U(r) ~ Garnre (r).

(2)

is the potential arising from the (static) scalar meson field nucleons ; zr(r) is the corresponding static pion field, and

e(r)

of the remaining (3)

V"(r)~Gv,~~(r)+G ov,vrrz " p~(r),

is tie sum ofw-and p-meson fields. The term involving RTw is a condensatiôn of the tensor-coupled term of the vector mesons : expanded, it is +GpxN[ißa ' OP (r) '

T+ßa "

ox (P(r) " T)]}.

(4)

As it happens, we shall initially neglect the tensor coupling terms for three reasons: Firstly, if the nucleons generating the static meson fields are in a spherically symmetric configuration, the space components of ~~(r) and p"(r) vanish by symmetry . Secondly, the tensor coupling constant of the ~-meson is either small or zero l'). The p-meson tensor coupling is not small, but for N = Z nuclei, p° (r) vanishes. Thirdly, the tensor terms are surface-peaked, off-diagonal (between "small" and "large" Dirac components), and are multiplied by (2M)-1, so they end up quite small. Again, assuming the source nucleus is spherically symmetric and spin-saturated, the average pion field zr(r) vanishes. Thus we may replace eq . (1) by the phenomenological equation

3. The equivalent nonrelatlvistic potentiel Let us now specialize to a nucleus with T = 0, so that the static p-meson contribution vanishes identically. To recover the nonrelativistic shell-model potential, as usual is) we write l, 1

~(r)=r

so that (k =1 + a " I),

°~~ri(f)a(~)

-iQ . ~~u(~)b(r)'

NUCLEAR DIRAC PI-~NOMENOLOGY

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Solving eq. (7a) for b (r) and inserting the result in eq . (7b), and making the substitution we find the Schrödinger-like form V°"_Up X"+[1 Wz_ 1 1 W_ 4 r 2E+M+U~+~ WQ ~ l]X=O, +(E-V~z-(M+U)z-l(1 with

W(r) _ (yo'(r) - U'(r))/(E +M+ U(r) - V°(r)) .

(9) (10)

From eq. (9) we immediately identify the nonrelativistic central potential v~f(r) = U(r)

+M V°(r) + (2M) -1( Uz(r) - ~(r))

V°"(r) - U"(r) -(SM) -' Wz(r)+(2Mr)-1W(r)+(4M)-lE+M+U(r)-

(r)'

(11)

and the spin-orbit potential

Let us suppose that U(r) and V°(r) have the same shape, U(r)=i1(0)f(r),

v°(r)=V°f(r),

(13)

with, say, the nuclear shape function having the usual Woods-Saxon form Then it is easy to determine that in order to fit the standard empirical low energy (E ~ M) shell-model parameters i~ v~ ~ -53 MeV,

v,.o. ~ 17 MeV,

U(0) must be large and negative, and V°(0) must be large and positive. Their difference will be almost M in magnitude, but their sum will be rather small, and negative . As a consequence, the last three terms in eq . (11) will contribute only one or two MeV at moat to the volume integral of (11), and so we neglect them in our rough fit. There are two remarks one should append here: FuBtly, because the aforementioned three terms contain derivatives, their Fourier transforms will contain high Fourier components, and hence they will dominate the large-momentum behavior of the Fourier transform of X(r). Thus, even though we drop these

338

J . V. NOBLE

terms at this stage because they will have negligible effect on the gross properties of nuclei (binding energies, nuclear sizes, and moments), they should not be forgotten permanently. Second, the tacit assumption that the shapes of static nuclear scalar and vector meson fields are the same is not quite correct-in a self-consistent calculation i.z) we should find e (r) ac (g.s.~~t(r)ßi/r(r)~g.s.),

(15)

m°(r) ac (g~s .~~rt(r)~G(r)~g~s .)

(16)

whereas

In principle this difference will change the central potential slightly, and will add some structure to the spin-orbit potential, but in practice it leads to small effects at low energies, as we shall see. From eq. (11) and eq . (13) one obtains the relation U(0)+V°(0)+(2M)-1(UZ(0)-V°~(0)) = -53MeV,

(17)

and from eq. (12) and eq . (13) one finds (evaluating the factor (2M+ U(r)- V°(r)) - ' at r = R and taking r° =1 .1 fm) [V°(0)-U(0)]/[1+(4M)-1(U(0)-V°(0))]=934 MeV .

(18)

Solving these equations we obtain the phenomenological potential well depths U(0) _ -420 MeV,

V°(0) = 328 MeV.

(19)

4. ~~NIIc~oscopic" calculation of tèe ieoecalar potentiale We examine next the microscopic origin of the scalar and neutral vector fields. Applying the I,agrangian formalism to derive classical equations of motion for the static meson fields, we find J dsr~ exP ( rm'r-rF ) (g.s.~~rt(r')ß+G(r~)~g~s .), (20) U(r) ~ Gerrrrs (r) = 4,rr ) dar~ exP ( Ir~lI -rIl) (g~s .~~rt(r')+l~(r')IB~s.). zJ

Vd (r) ~ Gxxw°(r) (G4,r_

(21)

To obtain U(0) and V,°, (0), we integrate with respect to r, and use eq . (13), to get U(0)=-3 G"~ m~ ~ B~ 4vr (mer°) A Vo

(GrrN)2 ~. (0) = 3 4~r (~ro)

(22)

NUCLEAR DIRAC PHENOMENOLOGY

35 9

The factor B/A is defined by the relation B

A

J

d3r(g.s .~~rt(r)ß+~(r)~g~s .)

d3r(g.s.~~rt(r)~(l)Ig~s .) J and it is realtively easy to show that

,

1-(T)M-1 > B/A > 1-4(T)M-1,

(23)

(24)

where (Tj is the average nucleon kinetic energy, --24 MeV. The average value of the upper and lower bounds in (24) is then 0.94, which is what weuse in eq. (22). Now, before we can compare the predicted scalar and vector potentials with those extracted empirically, we must correct for the effects of antisymmetrization . Since it is only possible to exchange a meson between nucleons within a meson Compton wavelength of each other, the exclusion principle implies that an external (or valence) proton can interactwith all the neutrons in its vicinity, but with only half the nearby protons (those with opposite spin projection); and vice versa for external neutrons . Thus, for a nucleus with N = Z, the computed average meson fields should be reduced by a factor . â. Moreover, as we shall see below, the exclusion principle gives rise to an "induced" symmetry energy when N ~ Z, which has nothing to do with charge exchange . Taping the values of the coupling constants and masses from a recent tabulation of OBE fits to NN scattering'), we have Gz "~ =14.2, m~ _ 735 MeV, 4~r (25) (G"~ )? =12.9, m =783 MeV. 4Tr B/A from

and calculate (including the exclusion effect) U(0) _ -321 MeV,

V~m~ (0) = 273 MeV

(26)

These values agree with the empirical ones (19) about as well as can be expected, considering the uncertainties of Gdnr, the e-mass, the contributions of the 2~r and 3~r continua ; the possible effects of nonlinear sell-couplings of the meson fields ; and the uncertainties in the empirical potentials arising from our decision to evaluate the expression (2M + U(r) - V°(r))-' at the nuclear surface. Thus we may conclude that to this point the Dirac phenomenology is consistent with a more microscopic picture of the origin of nuclear forces . An important prediction of the preceding treatment, as noted in the introduction, is that the equivalent Schrôdinger potential should be energy dependent, with v~ _ -53 + 0.35 T MeV,

(27)

360

J. V . NOBLE v,.o =17 - 0.01

T MeV,

(28)

where T is the kinetic energy (of a nucleon scattering from a nucleus) . Only the energy variation of the central potential depth, vn is important in low energy and intermediate energy applications, obviously. In fact, the energy dependence of eq . (27) is slightly more rapid than that determined from optical-model fits to nucleonnucleus scattering below, say, T =100 MeV [ref. 19)]. We do not regard this as a serious discrepancy, since the observed rapid increase with energy of the imaginary part of the optical potential can be expected to lead to a negative, decreasing dispersive contribution to the real part, via the dispersion relation Re [Vac~(T) - ve(T)]= ~ P

J°

T'(T T') ~ [V°°`(TI)]~

(29)

The fact that much of the observed energy dependence of the real part of the Schrödinger optical potential can be accounted for as a relativistic kinematic effect has, of course, been remarked upon previously by several authors s" 9.ii) . One may note that such energy dependence is a consequence of classical relativistic mechanics: A point particle acted upon by forces derived from time-independent scalar and vector potentials will obey the equâtion of motion

as is well known z°"Zi)

5. Microscopic calculation of the Lane potentiel Now we turn to the possibility that the isospin of the "target" nucleus is unequal to zero, so the time-independent part of the p-meson field of eq . (3) no longer vanishes identically. Following the reasoning which led to eq . (21) we have -r~ (Givx)Z r 3 ~ exp ( - mnI r ) T(r Vp (r) ~ GoivxT ' P ° (r) = d Ir - r,l ), (31) 4,rr T ' J r where T(r) is an operator in the target space, T(r) _ ~t(r)~/r(r).

(32)

It turns out that the p-tensor term in eq . (4) does not contribute significantly to the isovector potential, despite the rather large value of Go,,t r/G~x found empirically l'). On the other hand, the quadratic p-meson term in eq . (11), as well as the p-ar cross term, must be included. By virtue of some straightforward isospin algebra we isolate the isovector part of the quadratic p-meson term : (33) [T ~ T(r)T ~ T(r')]T-1 = -2T ~ T(r)S(r-r').

NUCLEAR DIRAC PHENOMENOLOGY

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Thus the net isovector contribution to vGf(r) is

T

E i~ T (r) v~ f(r)= ~ V°v(r)-(2M)-'fVP(r)® V;(r)]z"i A

d3r

exp ( -2mP~r-r' ) T'(r') . ~r-r'~

(34)

Taking the volume integral once again, we find v (~1) = 3(Gornv)z mP ~ ~M-t(E- Vo (0 )+ ma (Crnrr)2 4~r ~ 2M 4~r ~ (mvro)

(35)

Using the average of several values l') for (GPV,,,n,)Z/4~r (0.71 ~ 0.06), we find v~1 ~ = 18t2+0.02T MeV.

(36)

22) using the A recent experimental study of the Lane mode1 16) by Patterson et al. (p, n) reaction at energies from 25 to 45 MeV yields values for v~'~ in substantial agreement with eq. (36~the average of their eleven fits (under various constraints) is v~ =17 + 0.08 T MeV, whereas their "best" fit gives v~ =17.7 MeV. To some extent this excellent agreement must be considered fortuitous because the imaginary Zz) is large, and would part of the Lane potential obtained by Patterson et al. v~'~ . [However, in the low-energy therefore be expected to contribute dispersively to 19"2z); regime the absorptive parts of optical potentials are strongly surface-peaked the corresponding dispersive parts would not be expected to contribute significantly to the volume integralof v~l~f(r).] Thetheoretical uncertaintyfrom this source will be of the same order as that associated with Gva,, ~ t2 MeV. Before turning to the isovector spin-orbit interaction, we examine briefly the question of the symmetry term in the semi-empirical mass formula, about 28(Nss) Z)Z/A MeV. As is well known this term derives partly from the exclusion effect on kinetic energies, giving

0.3 sF(N-Z)2/AA.10(N-Z)Z/A MeV, partly from the exclusion effect on potential energies already mentioned in sect . 4 above, giving

-v~°~(N-Z)Z/6A~9(N-Z)2/A MeV,

and finally, from true isovector meson exchange, which gives v~l~(N-Z)Z/2A~9(N-Z)2/A MeV.

J. V . NOBLE

362

Clearly, the sum of the three contributions gives a good account of the empirical symmetry term . The corresponding single-neutron well depth is -53+36(N-Z)/A MeV, in reasonable agreement with the well depth extracted from a fit to the systematics of single-neutron spectra za) . 6. Microscopic cslcolation of the p-meson conMbation to the spin-orbit potential The phenomenological isovector spin~rbit interaction is the one effect to which the p-tensor coupling term of eq . (4) contributes significantly. In fact, the tensor coupling dominates the isovector Thomas precession term arising from the vector part of the phenomenological pNN coupling. Once again carrying out the nonrelativistic reduction of the Dirac equation, this time including the term -`G°nrrv

~LP °( r) ~ T]' 2M ßa ~

K =

GoNx/GPtvx

(37)

in the potential, we find the isovector spin~rbit interaction given by V'1°~ =4Tr

(G°'N)Z~K(1-

V°(0)/2M)-Z+(2+(U(0)- V°(0))/2M) -'] x (MZm~ô)-1

(38)

where the extra factor (1- V/M)-Z arises by analogy with interaction corrections to 2 the electromagnetic (isovector) anomalous magnetic moment of the nucleon' ) . The results are c~> - 3.3 MeV (VMD)

(39)

where VMD refers to K = 3.7, and EXP refers to l') K = 6.6 . Owing to the effect of the exclusion principle, we also expect an induced non-charge-exchange symmetry term in the spin-orbit interaction of the form -T3L(Z-N)/3A]V;°ô, ~ -5.7T3(Z-N)/A MeV.

(40)

This has the opposite sign from, but approximately the same magnitude as, the neutral part of the isovector spin orbit interaction, so we expect considerable cancellation . In other words, the spin-orbit interactions for neutrons and protons should be nearly identical, even in nuclei with a large neutron excess . On the other hand, a substantial isovector spin-orbit interaction should be (and apparently is 2s)) observable in charge-exchange reactions. Although some authors ~) have apparently assumed a symmetry correction to the spin-orbit potential with the same sign and magnitude, relative to v;°ô., as obtains for the symmetry correction to the central potential, there does not seem to be any hard evidence that such a term

NUCLEAR DIRAC PHENOMENOLOGY

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improves the fit to the systematics of single-neutron spectra. Thus, the result of the preceding calculation, in which it was found that neutron and proton spin~rbit interactions should be equal and independent of neutron excess, must stand as a prediction. It is perhaps worth emphasizing that, with the conventions used here for the definition of the strength of the spin-orbit potential, v,.o., the value of the isovector spin-orbit strength predicted by eq . (38) (using the empirical p-nucleon tensor coupling constant) is about a factor of two smaller than that found experimentally by Gosset etal. ~), but in very good agreement with what would have been predicted by folding the empirical two-body interaction into an appropriate nuclear matterdensity function ss). This is no surprise, of course, since the empirical meson-nucleon coupling constants and masses (in the case of the p) are determined by fitting the same on-shell NN scattering data which are used to derive the empirical potentials . However, it isworth noting that in the absence of the p-nucleon tensor coupling, the isovector spin-0rbit strength would have been a mere 0.24 MeV [with Gosset's convention zs) about 0.13 MeV], in very serious disagreement with experiment. That is, the experiment of Gosset etal. may be interpreted as strong direct evidence for the tensor rho-nucleon coupling. The factor of two disagreement alluded to above should not be taken too seriously, since the X z/lrl minimum found in this experiment (and used to determine v;lô. ) is rather flat . [The difference in convention alluded to above is that between the spin-orbit potential of ref.'9) and that of ref. zs)-the former is 2.73 times larger than the latter .] 7. Condasio~ The exercises described in this paper were motivated by a desire to check to what extent the Dirac phenomenology agrees with our knowledge of the general properties of nuclei, as well as with our rather detailed knowledge of the phenomenological meson-nucleon couplings. The fact that no inconsistency has yet been found gives us confidence in the further application of this model to nuclear phenomena at intermediate energies, where a relativistic theory is essential. I am grateful to G. A. Miller and L. G. Arnold for helpful conversations, to J. S. Blair for pointing out ref. zs), and to the Department of Physics of the University of Washington for its hospitality. Reterences 1) H. P. Duerr, Phye . Rev. 103 (1956) 469 2) L. D. Miller, Ann. of Phys. 91(1975) 40 ; L. D. Miller and A. E. S. Green, Phys. Rev. CS (1972) 241

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3) T. D. Lee and G. C. Wick, Phys . Rev. 1D9 (1974) 2291 ; T. D. Lee, Rev. Mod. Phys. 47 (1975) 267 4) J. D. Walecka, Ann. of Phys. 83 (1974) 491 5) R L Mercer, L. G.Arnold and B. C. Clark, Phys . Lett. 73B (1978) 9; Bull . Am. Phys. Soc. 23 (1978) 931 ;964 6) J. Bogota and J. Rafelaki, Phys . Lett . 71B (1977) 22 7) D. R. Inglis, Phys . Rev. SO (1936) 783 8) W. H. Furry, Phya. Rev. 50 (1936) 784 9) R. Humphriea, Nucl . Phys . A182 (1972) 580 10) J. L. Friar, in Mesons in nuclei, ed. M. Rho and D. L. Wilkinson (North-Holland, Amsterdam,1979) 11) J. V. Noble, Phys. Rev. C17 (1978) 2151 12) J. V. Noble, Axial and magnetic tests for nuclear dirac wave functions, Phys. Rev. C, in press 13) L. D. Miller and H. J. Weber, Phys. Lett. 64B (1976) 279; Phys. Rev. C17 (1978) 219 14) J. L. Friar, Phys . Rev. C15 (1977) 1783 15) K. Kubodera, J. Delorme and M. Rho, Phys . Rev. Lett . 40 (1978) 755 16) A. M. Lane, Phys . Rev. Lett. 8 (1962) 171 ; Nucl . Phys . 3S (1962) 676 17) M. M. Nagels, J. J. DeSwart, H. Nielsen, G. C. Oadea, J. L. Peterson, B. Tromborg, G. Gustafson, A. C. Irving, C. Jarlskog, W. Pfiel, H. Pilkuhn, F. Steiner and L. Tauacher, Nucl. Phys. B109 (1976) 1 18) M. L. Goldberger and K. M. Watson, Collision theory (Wiley, New York, 1964) p. 420ff 19) See, e.g., A. Bohr and B. R. Mottelson, Nuclear structure, vol. l (Benjamin, NewYork,1969) p. 237 20) G. Nôrdstrom, Ann . der Phys . 42 (1913) 540 21) R P. Feynman, Lectures on gravitation (California Institute of Technology, 1962, unpublished) 22) D. M. Petterson, R. R. Doering and R. Galonsky, Nucl. Phys. A1b3 (1976) 261 23) A. de-Shalit and H. Feshbach, Theoretical nuclear physics (Wiley, New York, 1974) p. 127 24) Ref.' P. 239 25) J. Gosset, B. Mayer and J. L. Escudié, Phys . C14 (1976) 878