Relativistic nucleon-nucleus optical model

Volume 84B, number 1

PHYSICS LETTERS

4 June 1979

RELATIVISTIC NUCLEON-NUCLEUS OPTICAL MODEL L.G. ARNOLD and B.C. CLARK Department of Physics, The Ohio State University, Columbus, 0H43210, USA Received 26 January 1979 Revised manuscript received 14 April 1979

A microscopic relativistic optical model is developed and compared with the empirical optical model at low energies.

We have been using a relativistic model of the nuclear potential as a basis for optical model analyses of medium energy elastic scattering data [ 1 - 3 ] . The essential feature of this model is a mixture of a Lorentz scalar potential and the time-like component of a four-vector potential as may be obtained by direct coupling of the nucleon field to scalar and vector meson fields [4]. We have shown that the reduction in structure of medium energy cross section and analyzing power data with increasing energy in the range from 500 MeV to 2 GeV follows naturally from kinematic effects associated with this potential mixture. In the present letter * 1, we develop a microscopic mixed potential model for low energies and compare its predictions with properties of the nonrelativistic optical model. We find a good correspondence with both the real central and spin-orbit parts of the non-relativistic potential. The wave function for the relativistic model is damped analogous to the Perey damping effect for a nonlocal or velocity dependent potential. The necessity of specifying the Lorentz transformation character of a potential is fundamental to a nuclear optical model based on a relativistic wave equation. It is a consideration which applies equally to macroscopic models and microscopic theory; in the latter, it is a derived property which originates in the meson theory of the two-nucleon interaction. The Lorentz transformation character of a potential plays no role in the non-relativistic optical model or mod,l Preliminary reports of this work were given in ref. [5]. 46

ifications thereof derived on the premise of "minimal relativity." The elegant and successful theoretical framework for this model [6,7] rests on the use of a phenomenological two-nucleon interaction. As a result, some essential input from meson theory may be lost at the outset in the microscopic nonrelativistic theory. In the present letter, we demonstrate the relevance of the Lorentz transformation character of the optical potential. The optical potential considered here consists of two parts: one part is a potential U s which transforms as a Lorentz scalar; the other part, a potential U0 which transforms as the time-like component of a Lorentz four-vector. These potentials and the Coulomb potential Vc are used in the Dirac equation, [E-

V c - U 0 - { 3 ( m c 2 + Us) ] ~b = - i h c a .

V~ .

(1)

Their use is motivated by conservation laws as applied to real, static potentials and by meson exchange considerations [8] of the two-nucleon interaction. The potentials U s and U 0 are the only ones which survive these constraints in the mean field approximation for scattering by a spin-zero, isospin-zero target. The potential U 0 may be identified with the field of the neutral vector w meson with a mass of 783 MeV, while U s may be identified with a neutral scalar field arising from two-pion exchange processes. The latter is often simulated by a neutral scalar meson with a mass of about 500 MeV. Components of a Lorentz tensor potential survive the conservation law constraints, but are omitted from eq. (1) since tensor coupling of the w meson is thought to be small. The p meson with a

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PHYSICS LETTERS

mass of 770 MeV would be added in extending these ideas to nuclei with isospin greater than zero. Such an extension would result in coupled Dirac equations analogous to the Lane equations from nonrelativistic theory and would include components of a Lorentz tensor potential due to the tensor coupling of the p meson. Mesons heavier than a nucleon have been omitted. We construct potentials to be used in eq. (1) along traditional lines of nonrelativistic theory [9]. In the mean field approximation, they take the standard folding form, Us(r) = f o s ( I r - ,'l)Fs(r') dr',

(2)

=fo0(Ir - r'[)~'0(r') d r ' ,

(3)

Vc(r) = (Ze2/A) f Ir - r'l-l~o(r ') dr',

(4)

Uo(r )

where u(r) denotes an effective interaction in nuclear matter with a finite range form factor andp'(r) is an effective density which incorporates the finite size of projectile and target nucleons. We make the approximation,ffs(r) = [ps/PO] nm~Y0(r), where [ps/PO] nm is the scalar to baryonic density ratio in nuclear matter. Finite size corrections are made by relating p'0(r) to the finite nucleus matter density PO(r) through double folding over Pb(rp)Pb(rt), where Pb(r) is a nucleon structure profile with 0.8 fm rms radius. An empirical formula given by Negele [10] is used for PO(r). Finite range corrections are made by writing o(r) = tf(r), where t is the volume integral of the effective inter-

4 June 1'979

action in nuclear matter and f(r) is the form factor with rms radius determined by the mass of the exchanged meson. The construction is completed by evaluating the volume integrals, t s and t 0, and the density ratio, [Ps/PO] nrn, with a relativistic theory of nuclear matter. We use Walecka's mean field theory [11 ] for this purpose. A comparison of Walecka's theory with empirical information from other sources is given in table 1. The volume integrals of the effective scalar and vector meson exchange interactions in nuclear matter are obtained by solving the nuclear matter equations for equilibrium at specified values of the binding energy and Fermi wave number. These volume integrals have the following properties: for kf = 1.42 fm -1 , their magnitudes are in reasonable agreement with two-nucleon OBEP volume integrals; however, their ratio to/t s and the related ratio JO/Js, which represent the balance between scalar attraction and vector repulsion of nuclear forces in nuclear matter, are somewhat smaller than empirical OBEP values for the two-nucleon system. For kf = 1.31 f m - 1, there is agreement for the volume integral ratios, but the magnitudes of the effective vol. ume integrals in nuclear matter are systematicallylarger than OBEP values. The optical potential calculations given below tend to support the use of the smaller value of kf. In order to compare the relativistic optical model at low energies with the nonrelativistic model, the Dirac equation needs to be reduced to a Schr6dinger equation for the upper (large) components of ~b. The

Table 1 Comparison of relativistic mean field theory of nuclear matter with empirical OBEP analyses of two-nucleon data and optical model analysis of medium energy p-4He elastic scattering data. E b (MeV), kf(fm-1 )

11.00, 15.75, 11.00, 15.75, empirical

1.42 1.42 1.31 1.31

to

ts

(MeV fm 3)

(MeV fm 3)

1573 1706 2130 2321 1713 b) +- 226

-2115 -2326 -2751 -3039 - 2 1 8 5 b) +- 254

to/t s

jo/Js a)

-0.744 -0.733 -0.774 -0.763 -0.783 b) +-0.020

-0.797 -0.793 -0.824 -0.819 -0.814 c) +-0.006

[Ps/Po] nm

0.933 0.925 0.940 0.932 0.962 d) +0.032

a) Ratio of volume integrals of Uo and Us; Jo/Js = toPo/tsp s in nuclear matter. b) Average of three independent OBEP calculations from ref. [12]. Here, t o = (~c)3g2o/l~20 and t s = - r~c~a~21U2 ~,. , bst s. The quoted uncertainty is the rms deviation (13% for t o and t s, 3% for to/ts). c) From medium energy p-4He optical model analysis of ref. [2]. The quoted uncertainty reflects the statistical uncertainty. d) From the empirical results for to/t s b) and Jo/Js c).

47

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result o f this reduction is as follows , 2 . The large components ~ku are related to the solution ¢ o f the SchrSdinger equation, 2m

r d-r Us° ~ " L ( ~ = T ¢ ,

(5) via @u = (M*)1/2¢, where the "effective mass" M* is given by (6)

M* = 1 - (U 0 - Us)/2mc 2 .

This relation between flu and ¢ is analogous to the Perey damping effect relation [13] for the SchrSdinger equation with a nonlocal or velocity dependent potential. The damping function in the reduction o f the Dirac equation is a consequence o f a velocity dependence that is relativistic in origin. The central potential in eq. (5) is Ucent = M * ( U 0 + Us) + ( T - Ve)(1 - M * ) + ( ' h 2 / 8 m ) [ ( V l o g M * ) 2 - 2V21ogM *] .

(7)

The first two terms have the same form as the effective mass approximation for a uniform field, while the remaining gradient terms correspond to surface corrections due to the elimination of the ~ ~ u / b r term in the reduction. The dominant part of the central potential is M * ( U 0 + Us); the term T(1 - M * ) gives rise to a linear kinetic energy dependence, while V c × (1 - 3 4 * ) is the Coulomb correction for protons [14] ; the gradient terms are small and will not be discussed ,2 Kinematic corrections, T/2mc 2 and VcF2mc 2, are neglected here. To obtain the exact reduction, replace m by m(1 + T/2mc 2) in eqs. (5)-(8) and Uo by U0 + F c in eq. (6) only.

4 June 1979

further. The s p i n - o r b i t potential is obtained from the derivative o f

Uso = -2mc21ogM*

U0 - Us.

(8)

The familiar Thomas form o f the s p i n - o r b i t potential is recovered from the approximation on the right. An interesting feature of the relativistic model at low energies is that the effective mass M* is independent o f energy for static U s and U 0 potentials *a. As a result, the s p i n - o r b i t potential is independent o f energy and the energy dependence o f the central potential is linear. These features o f the model are in qualitative agreement with properties o f the empirical optical potential at low energies. In addition, since Uso depends on M*, an intrinsic property o f the model is a direct connection between the s p i n - o r b i t and central potentials, particularly for the energy dependent and Coulomb correction terms of the central potential. This connection allows an unambiguous test of the model because the s p i n - o r b i t potential and the components o f the central potential are treated as independent in low-energy optical model analyses. Table 2 is a comparison of volume integrals for the effective potentials in eqs. (7), (8) calculated from the relativistic optical model with empirical volume integrals determined from analyses o f proton and neutron scattering on 40Ca at low energies. For U s and ,a A mixture of static Us and Uo potentials is compatible with Weisskopf's theorem [ 15 ] on the necessity of a nonstatic single particle potential for saturation of nuclear matter. In the relativistic model, the nonstatic effect required by Weisskopf's theorem comes from the potential mixture as determined by meson exchange considerations. Nonstatic corrections to Us and Uo from several sources can be anticipated within the framework of this model.

Table 2 Comparison of volume integrals for the effective potentials in eqs. (7), (8) from the relativistic optical model with empirical volume integrals for 4°Ca. E b (MeV), kf(fm-1 )

Ucent(T= 0) a) (MeV fm 3)

1 - M* (fma )

Vc(1 - M*) (MeV fm a )

Uso b) (GeV fm a )

15.75, 1.42 15.75, 1.31 empirical c)

- 387 -418 -485 +- 30

2.1 2.7 2.7 -+0.4

-16 -21 - 2 2 -+ 3

4.3 (3.9) 5.9 (5.2) 5.7 +- 1.3

a) The volume integral of a potential U is defined by J/A = A -1 f U(r) dr, where A is the mass number of the target. b) Bracketed values are from the approximate form, Uso ~ Uo - Us. c) From ref. [16] for protons and ref. [17] for neutrons. 48

Volume 84B, n u m b e r 1

PHYSICS LETTERS

U0 potentials determined from equilibrium at kf = 1.31 fm -1 , the agreement between calculated and empirical volume integrals is good; the agreement is better for kf = 1.31 fm -1 than for kf = 1.42 fm -1, indicating a preference for the smaller value. The connection predicted between Uso and the ( T - Vc) × ( 1 - M * ) terms of the central potential is in excellent accord with the independently determined empirical volume integrals. Note that the uncertainty in the empirical volume integral of Uso is large enough to obscure a distinction between the logM* radial form of the model and the approximate Thomas form. This comparison shows that the relativistic model with effective interactions determined from the mean field theory of nuclear matter [11 ] gives an adequate account of the integrated strengths of the 40Ca optical potential. A possible deficiency is that the calculated volume integral of Ucent is about 15% smaller than the empirical value. The M* (U 0 + Us) term in Ucent is sensitive to changes in the nuclear matter input due to the large cancellation of scalar and vector potential energies. A small change in the magnitudes o f J 0 and Js which preserves their ratio removes the deficiency without destroying the agreement between other calculated and empirical volume integrals. The geometry of the optical potential, as represented by the rms radii of its terms, provides further basis for comparison of the relativistic model with phenomenology. It has been known for some time that the rms radius of the central potential is systematically larger than the rms radius of Uso. The rms radii of the Coulomb correction and energy dependent terms have not been determined from optical model analyses, but are believed to be nearly equal and comparable to the central potential rms radius rather than the rms radius of Uso. These empirical results may be summarized as follows:

4 June 1979

40Ca). The reason that (Ro2pt) is sensitive to such a small geometrical effect can be seen from the approximate relation, 2 t) (Rop X

--

(R2o)

3(/~/kt0c)2

[(1 -Jo/Js)/(l

+Jo/Js)][(IlO/kts) 2 - 1] ,

(10)

which reveals an order of magnitude enhancement in ( R 2 p t ) - (R2so) f r o m t h e f a c t o r (1 -

Jo/Js)/(1 +Jo/Js);

this factor would be 1 for a pure potential, but is about 10 for the mixed potential model with Jo/Js = - 0.814. As shown in fig. 1, (R2pt)1/2. is extended significantly beyond the matter radius (R2) 1/2 by this enhancement of the mass difference; in contrast, (R2) 1/2 is hardly influenced by the mass difference. The mixed potential model yields good agreement between calculated and empirical rms radii for/~s in the neighborhood of 560 MeV. In the empirical optical potential, the geometry of the Coulomb correction and energy dependent terms is tied to the central potential geometry; the spinorbit geometry is considered anomalous [18]. The prediction of the relativistic model for the geometry of the Coulomb correction and energy dependent terms differs from this nonrelativistic picture. In the rela50

T 2

T

T

I/2

4.5

1

T

< %:~/==0.8 fm /% = 783 MeV

~

t~ 40

....

v

-- :..:...:....

35-

R~>I/2

~

2

- - < R m >

~/2-

--

(R2pt) - (R 2) ~ 3 fm 2, (R 2 ) "~ (R 2 ) "~ (Ro2pt) •

(9)

The experimental uncertainties in (R2pt) and (R 2) are about 1 fm 2. A systematic difference between (R2pt) and (R 2) is attained in the relativistic model as a consequence of the mass difference between the scalar and vector mesons. This mass difference leads to a very small difference in the rms radii of U0 and Us (~0.05 fm for

30

~ 300

400

500

600

1 700

L 800

900

/zs (MeV) Fig. 1. Dependence of optical potential rms radii on the mass of the scalar meson. (R2m)1/2 is the r m s radius of the 4°Ca matter distribution. (R~) 1/2 is the rms radius of U o,
49

Volume 84B, number 1

PHYSICS LETTERS

tivistic model, the geometry of these terms is tied to the geometry of Uso through M*. The result, (11) is a natural consequence of this connection. The rms radius of the central potential is larger in the relativistic model because of the scalar-vector meson mass difference. Thus, there is a rather clear cut difference between the relativistic and nonrelativistic optical models , 4 . To date, there have been no experimental tests of the geometry of the Coulomb correction and energy dependent terms. In this letter, we have summarized features of a relativistic optical model at low energies. We conclude that the Lorentz transformation characteristics of nuclear interactions, as determined from elementary meson exchange considerations, are relevant to a first principles understanding of the optical potential. We thank R.L. Mercer, B. Mulligan, J.V. Noble and R.G. Seyler for helpful discussions. ,4 This difference between relativistic and nonrelativistic models may have some bearing on the Nolen-Schfffer anomaly [ 19].

References [1] R.L. Mercer, L.G. Arnold and B.C. Clark, Phys. Lett. 73B (1978) 9; Bull. Am. Phys. Soc. 22 (1977) 616. [2] L.G. Arnold, B.C. Clark and R.L. Mercer, Phys. Rev. C19 (1979) 917.

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4 June 1979

[3] B.C. Clark, L.G. Arnold and R.L. Mercer, Bull. Am. Phys. Soc. 23 (1978) 571; 23 (1978) 964. [4] H.-P. Duerr, Phys. Rev. 103 (1956) 469. [5] L.G. Arnold, B.C. Clark and R.L. Mercer, Bull. Am. Phys. Soc. 22 (1977) 1029; 23 (1978) 571; 23 (1978) 931. [6] J.-P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev. C16 (1977) 80, and references therein. [7] F.A. Brieva and J.R. Rook, Nucl. Phys. A297 (1978) 206, and references therein. [8] G.E. Brown and A.D. Jackson, The Nucleon-nucleon interaction (North-Holland, Amsterdam, 1976), Ch. 10, Appendix B. [9] P.E. Hodgson, Nuclear reactions and nuclear structure (Oxford U.P., London, 1971) Ch. 6; see also F. Petrovich, in: Microscopic optical potentials, ed. H. yon Geramb (Springer, New York, 1979) Vol. 89, p. 155. [10] J.W. Negele, Phys. Rev. C1 (1970) 1260. [11] J.D. Walecka, Ann. Phys. (NY) 83 (1974) 491. [12] A. Gersten, R. Thompson and A.E.S. Green, Phys. Rev. D3 (1971) 2076; G. Schierholz, Nucl. Phys. B40 (1972) 335; K. Erkelenz, K. Holinde and R. Machleidt, Phys. Lett. 49B (1974) 209. [13] F.G. Percy, in: Direct interactions and nuclear reaction mechanisms, ed. E. Clementel and C. Villi (Gordon and Breach, New York, 1963) p. 125. [14] G.R. Satchler, in: Isospin in nuclear physics, ed. D.H. Wilkinson (North-Holland, Amsterdam, 1969) p. 389. [15] V.F. Weisskopf, Nucl. Phys. 3 (1957) 423; Rev. Mod. Phys. 29 (1957) 174. [16] W.T.H. van Oers, Phys. Rev. C3 (1971) 1550; W.T.H. van Oers et al., Phys. Rev. C10 (1974) 307. [17] J. Rapaport et al., Nucl. Phys. A286 (1977) 232. [18] D.W.L. Sprung and P.C. Bhargava, Phys. Rev. 156 (1967) 1185. [19] J.A. Nolen Jr. and J.P. Schiffer, Ann. Rev. Nucl. Sci. 19 (1969) 471.